Solid State Chemistry

Solid State Chemistry

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Solid State Chemistry

by R.C. Ropp Warren New Jersey USA

2OO3

ELSEVIER AMSTERDAM - BOSTON- HEIDELBERG- LONDON- NEW YORK- OXFORD PA R IS - S A N D I E G O - S A N F R A N C I S C O - S I N G A P O R E - S Y D N E Y - T O K Y O

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

9 2003 Elsevier Science B.V. All rights reserved.

This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier's Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-maih [email protected]. You may also complete your request on-line via the Elsevier Science homepage (http://www.elsevier.com), by selecting 'Customer Support' and then 'Obtaining Permissions'. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 207 631 5555; fax: (+44) 207 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier Science is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier's Science & Technology Rights Department, at the phone, fax and email addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. First edition 2003 Library of Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for.

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Preface

Most of the

books

concerning

"Solid

State

Chemistry"

that

I have

e xamin ed have been w r i t t e n for the specialist or for the advanced s t u d e n t . I have c o m p o s e d this book with the novice in mind. That is, those w h o have little b a c k g r o u n d but w i s h to begin to learn about the solid state and w h a t it entails can start at the be gi nn i ng and build u p o n their knowledge

and

experience.

This

includes

biotechnology, p h a r m a c e u t i c a l a n d p r o t e o m i c s

those

working

own

in

the

fields. At this point in

time, they have s e q u e n c e d the h u m a n g e n o m e and are trying to define t h e s t r u c t u r e of proteins. It is m y opinion t h a t the material in this book will be helpful to t h e m as well. With this in mind, I have p r e s e n t e d the information in this book in a f o r m t h at can be easily u n d e r s t o o d . I t h i n k that it is quite i m p o r t a n t t hat any s t u d e n t of the body of knowledge

t h a t we call "science"

needs to be

cognizant of the history a nd effort t hat has been m a d e by those w h o p r e c e d e d us. It was Newton who said: "If I have seen further, it is because I have stood on the s h o u l d e r s of giants". Thus, I have tried to give a s h o r t history of each particular s e g m e n t of solid state theory and technology. I have enjoyed c o m p o s i n g the material in this book and t r u s t t h a t t h e reader can use it as a self-study to build u p o n his (her)

store of usable

knowledge. I t h a n k m y wife, Francisca Margarita, profusely for her s u p p o r t during t h e time t h a t it has t a k e n to finish this c o m p o s i t i o n . R.C. R o p p March 2 0 0 3

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vii

Table off contents Preface Table of Contents CHAPTER 1 - The Phase Chemistry of Solids 1 . 1 . - Phase Changes of Solids, Liquids and Gases 1.2.- Differences Between the Three States of Matter a. The Gaseous State b. The Liquid State c. The Solid State 1.3.- The Close-Packed Solid 1.4. Phase Relations Between Individual Solids References Cited Problems for Chapter 1

PaBe i iii

2 9 12 15 17 23 27 28

CHAPTER 2- Determining the Structure of Solids 2.1.- Scientific Basis for Determining the Structure of Solids 2.2.- Solid State Structure Conventions and Protocols 2.3.- How to Determine the Structure of Compounds 2.4.- Symmetry Distribution of Crystals 2.5.- Phase Relationships Among Two or More Solids SUGGESTED READING Problems for Chapter 2

31 31 45 55 61 64 69 69

CHAPTER 3- Defects in Solids 3.1.- The Defect Solid a. The Point Defect in Homogeneous Solids b. The Point Defect in Heterogeneous Solids c. The Line Defect d. The Volume Defect 3.2.- Mathematics and Equations of the Point Defect a. The Plane Net 3.3.- Non-Stoichiometric Solids 3.4.- Defect Equation Symbolism

71 73 74 72 82 85 88 88 95 98

viii

3.5.- S o m e Applications For Defect C h e m i s t r y a. P h o s p h o r s

99 100

3.6.- Defect Equilibria a n d Their E n e r g y 3.7. Defect Equilibria in Various Types of C o m p o u n d s a. S t o i c h i o m e t r i c Binary C o m p o u n d s of MXs b. Defect C o n c e n t r a t i o n s in MXs C o m p o u n d s

101 103 104 107

3.8.- The Effects of Purity (And I m p u r i t i e s )

110

Suggested Reading

112

P r o b l e m s for C h a p t e r 3

113

A p p e n d i x 1.

C o n c e n t r a t i o n s of Defects in Non-Ionized a n d

Non-Stoichiometric Compounds A. Defects in N o n - S t o i c h i o m e t r i c MXs + d C o m p o u n d s A p p e n d i x I I. Analysis of a Real Crystal Using the Thermodynamic Method A. The AgBr Crystal with a Divalent Impurity, Cd 2+ B. Defect Disorder in AgBr- A T h e r m o d y n a m i c A p p r o a c h AoDendix III. Statistical M e c h a n i c s a n d the Point Defect CHAIrI'ER 4- M e c h a n i s m s a n d Reactions in the Solid S t a t e

115 115 118 118 120 124 129

4. I.- P h a s e C h a n g e s 4.2.- The Role of P h a s e B o u n d a r i e s in Solid State R e a c t i o n s

130 132

4.3.- Reaction Rate P r o c e s s e s in Solids 4.4.- Defining H e t e r o g e n e o u s Nucleation P r o c e s s e s

138 140

4.5.- The T a r n i s h i n g R e a c t i o n

146

4.6.- Fick's Laws of Diffusion 4.7.- Diffusion M e c h a n i s m s 4.8. - Analysis of Diffusion R e a c t i o n s

148 151 156

4.9. - Diffusion in Silicates 4. I0.- Diffusion M e c h a n i s m s W h e n the Cation C h a n g e s V a l e n c e

161 171

P r o b l e m s for C h a p t e r 4 A p p e n d i x I- Math Associated with Nucleation Models of 4 . 4 . 1 . A p p e n d i x II- Math Associated w i t h Incipient Nuclei G r o w t h A p p e n d i x III- H o m o g e n o u s Nucleation P r o c e s s e s A p p e n d i x IV- Diffusion E q u a t i o n s Relating to F u n d a m e n t a l Vibrations of the Lattice

175 177 182 184 188

CHAPTER 5- Particles and Particle Technology 5.1.- Sequences in Particle Growth 5.2.- Sintering, Sintering Processes and Grain Growth 5.3.- Particle Size 5.4.5.5.5.6.5.7.-

Particle Distributions Particle Distributions and the Binomial T h e o r e m Measuring Particle Distributions Analysis of Particle Distribution P a r a m e t e r s A. The Histogram B. Frequency Plots C. Cumulative F r e q u e n c y D. Log Normal Probability M e t h o d 5.8.- Types of Log Normal Particle Distributions A. Unlimited Particle Distributions B. Limited Particle Distributions C. Particle Distributions with Discontinuous Limits D. Multiple Particle Distributions Case I: Fluorescent Lamp Phosphor Particles Case II: T u n g s t e n Metal Povcder 5.9.- A Typical PSD Calculation 5.10.- Methods of Measuring Particle Distributions A. The Microscope- Visual Counting of Particles

B. Sedimentation Methods C. Electrical Resistivity-The Coulter Counter D. Other Methods of Measuring Particle Size Permeability Gas adsorption Particle size by laser r e f r a c t o m e t r y Suggested Reading Problems for Chapter 5 Chapter 6. - Growth of Crystals 6. I.- Methods of Growth of Crystals 6.2.- Furnace Construction 6.3.- Steps in Growing a Single Crystal

191 192 193 203 207 209 213 217 218 218 219 220 222 223 223 224 225 226 228 229 232 233 237 241 245 245 245 247 248 249 251 252 253 258

260 270 274 275 278 282

Czochralski Growth of Single Crystals 6.5.- The Bridgeman-Stockbarger Method for Crystal Growth 6.6.- Zone Melting as a Means for Forming Single Crystals 6.7.- Zone Refining 6.8.- The Impurity Leveling Factor 6.9.- The Verneufl Method of Crystal Growth 6. I0.- Molten Flux Growth of Crystals 6.11.- Hydrothermal Growth 6.12.- Vapor Methods Used for Single Crystal Growth 6.13.- Edge Defined Crystal Growth 6.14.- Melting and S t o i c h i o m e t r y

285 288 292 294 296

6.15.- Actual Imperfections in Crystals 6.16.- Electronic Properties of Crystals A. Conductivity in Ionic Compounds 6.17.- Silicon Single Crystals and Integrated Circuits A. Silicon

299 302 303 308 308

.4.

-

B. Silicon as a S e m i - C o n d u c t o r C. Semiconductor Devices D. Integrated Circuits E. Manufacture of Integrated Circuits F. Steps in the Manufacture of Integrated Circuits G. Film Deposition H. Impurity Doping

310 311 313 315 318 328 328

I. Lithographic P a t t e r n i n g J. E t c h i n g K. A Final Look at the IC Manufacturing Process L. Crystal Growth and Crystal Defects Affecting IC's 6.18.- Future Methods for Manufacture of Integrated Circuits 6.19.- Pushing the Limits of Semi-Conductor Technology 6.20.- The Solar Cell A. Crystalline, PolycrystaUine and Amorphous Solar Cells B. Thin Film Solar Cells 6.21.- Piezo-electric materials and Their Uses A. Applications Sonar

329 330 330 334 337 338 343 345 350 351 352

xi

Medical U l t r a s o u n d Micro-positioning and M i c r o - m o t o r s

353

Piezoelectric T r a n s f o r m e r s

353 353

Active Noise a n d Vibration D a m p i n g

353

SUGGESTED READING

353

References on Silicon Devices

354

Problems for C h a p t e r 6

354

Chapter 7- M e a s u r e m e n t of Solid State P h e n o m e n a

357

7.1.- Methods of M e a s u r e m e n t of Solid State Reactions A. Differential T h e r m a l Analysis (DTA)

357 358

B. Differential S c a n n i n g Calorimetry 7.2.- Utilization of DTA and DSC

374 376

A. Applications of DTA

376

B. Uses of DSC 7.3.- T h e r m o g r a v i m e t r y

380 381

7.4.- D e t e r m i n a t i o n of Rate Processes in Solid State Reactions

389

A. Types of Solid State R eact i ons B. The Freeman-Carroll Method applied to DTA Data

389 392

C. The Freeman-Carroll Method Applied to TGA Data 7.5.- Dilatometry 7.6.- T h e r m o m e t r y

393 394 401

7.7.- Application of Dilatometry to Plastic Materials

403

7.8.- Optical M e a s u r e m e n t s of Solids A. Defining Light

405 405

B. M e a s u r e m e n t of Color

409

C. The Nature of Light

410

D.- Absorbance, Reflectivity and T r a n s m i t t a n c e

411

E. M e a s u r e m e n t of Color I. The S t a n d a r d Observer II. The Nature of C h r o m a

415 417 417

III. Intensity and S c a t t e r i n g IV. Color Processes and Color Matching S y s t e m s

418 420

F. The S t a n d a r d Observer and The First Color-Comparator 4 2 1 I. T r i s t i m u l u s Coefficients

426

xii

II. Chromaticity Coordinate Diagrams 7.9.- Color Spaces A. The Munsell Color T r e e B. Color Matching and MacAdam Space R e c o m m e n d e d Reading Problems for Chapter 7

42 43 43 43 43 44

Chapter 1

ii

....

The P h a s e C h e m i s t r y of Solids To u n d e r s t a n d the solid state, we m u s t first u n d e r s t a n d how m a t t e r exists. T h a t is, m a t t e r w a s originally defined as "anything t h a t occupies s p a c e a n d h a s weight". This definition w a s m a d e in the e i g h t e e n t h c e n t u r y w h e n little w a s k n o w n a b o u t "matter" . T h u s ,

it w a s defined in t e r m s of its general

s h a p e a n d physical c h a r a c t e r i s t i c s . We will delve into the "ouflding-blocks" of m a t t e r a n d how solids originate. It is a s s u m e d t h a t you have some initial knowledge c o n c e r n i n g p h y s i c s a n d c h e m i s t r y as well as s o m e ability in m a t h e m a t i c s . The m o r e difficult m a t h e m a t i c a l t r e a t m e n t s are reserved to the a p p e n d i c e s of certain c h a p t e r s . Originally, in the 18th century, w h e n the "Scientific Revolution" w a s j u s t getting started, science w a s referred to as "Natural Philosophy". R e s e a r c h e r s were i n t e r e s t e d in n a t u r a l science, including w h a t we call today: Physics, Chemistry, Biology a n d Biochemistry. All of t h e s e scientific disciplines deal with

molecules

having various

physical

aspects

and

properties

(even

proteins). The earliest w o r k dealt with air, w a t e r a n d solids since t h e s e were the easiest to m a n i p u l a t e in t h o s e times. Note t h a t we r e g a r d all of their scientific a p p a r a t u s to be c r u d e by o u r s t a n d a r d s .

Yet, even the m o s t

s o p h i s t i c a t e d i n s t r u m e n t s (by their criteria) h a d to be built b y h a n d . Even so, early w o r k e r s k n e w t h a t m a t t e r existed in three forms, i.e.- solids, liquids a n d gases. W a t e r w a s s t u d i e d extensively b e c a u s e it w a s e a s y to freeze a n d boil. As we will show, m a n y of o u r basic m e a s u r e m e n t s

a n d scientific

s t a n d a r d s r e s u l t e d from s u c h studies. Reversible t r a n s f o r m a t i o n s b e t w e e n the t h r e e forms (phases) of m a t t e r , are called "Changes of State". The science related to t h e s e p h a s e c h a n g e s lies in the r e a l m of "Physical Chemistry" while their a c t u a l "chemistry", or "reactiveproperties", lies either in the r e a l m of "Inorganic" or "Organic" chemistry. Inorganic

materials

include

both

m a t e r i a l s have a c a r b o n b a c k b o n e .

elements

and

compounds.

Organic

Nearly all inorganics are solids at

a m b i e n t t e m p e r a t u r e . At elevated t e m p e r a t u r e s , t h e y t r a n s f o r m or melt to form a liquid a n d t h e n a gas. Most of t h e s e p h a s e c h a n g e s are reversible. A

[]

few are g a s e s at r o o m t e m p e r a t u r e , b u t liquids are rare. In contrast, organic compounds

can

be

either

solid,

liquid,

or gaseous.

Although

organic

c o m p o u n d s do u n d e r g o some C h a n g e s - o f - S t a t e easily a n d reversibly, their t h e r m a l stability is a problem. If one tries to i n d u c e a c h a n g e of s t a t e in an organic c o m p o u n d , one u s u a l l y finds t h a t liquid organic c o m p o u n d s will freeze a n d / o r sublime. Notwithstanding, m o s t organic solids d e c o m p o s e if the g a s e o u s state is sought. This is due to their relatively low t h e r m a l stability, i.e.- "they burn". Organic h y d r o c a r b o n c o m p o u n d s u s u a l l y r e a c t with

oxygen

and

convert to c a r b o n

t e m p e r a t u r e s of 200 ~

to 600 ~

stable at r a n g e s of 1500 ~ interested

in

inorganics

and water

at r a t h e r

low

while m a n y inorganic c o m p o u n d s are

to 2 0 0 0 ~ because

dioxide

of

a n d even higher. We will be m o r e their

greater

thermal

stability.

Additionally, inorganic c o m p o u n d s have physical properties t h a t c a n n o t be d u p l i c a t e d in the organic d o m a i n of chemistry, a n d vice-versa. 1.1.- PHASE CHANGES OF SOLIDS, LIQUIDS AND GASES The b e s t w a y to u n d e r s t a n d p h a s e c h a n g e s is to e x a m i n e those observed with water, i.e.- t h o s e involving the molecule, H20. We are, of course, familiar with the three s t a t e s of water, n a m e l y ice, w a t e r a n d s t e a m , since these forms are e n c o u n t e r e d daily. The a c t u a l difference b e t w e e n t h e m is a m a t t e r of energy. To illustrate this fact, let u s calculate how m a n y calories are required to c h a n g e ice into w a t e r into s t e a m . This p r o b l e m is typical of those c o n c e r n i n g Changes-of-State. In solving s u c h a problem, we will also define certain c o n c e p t s a n d c o n s t a n t s relating to its solution. For example, we k n o w t h a t w a t e r (a liquid) will c h a n g e to ice (a solid) ff its internal temperature

falls below a certain t e m p e r a t u r e .

Likewise, ff its

i n t e r n a l t e m p e r a t u r e rises above a certain point, w a t e r c h a n g e s to s t e a m (a gas). B e c a u s e w a t e r is so a b u n d a n t on the Earth, it w a s u s e d in the p a s t to define C h a n g e s of S t a t e a n d even to define T e m l ~ r a t u r e

Scales. However,

the concept of "heat" is also involved, a n d we need to also define the perception of h e a t as it is u s e d in this context. Note t h a t defining h e a t implies t h a t we have a reproducible w a y to m e a s u r e t e m p e r a t u r e . A great deal of w o r k w a s r e q u i r e d in the p a s t to r e a c h t h a t stage. First, you have to e s t a b l i s h t h a t certain liquids e x p a n d w h e n heated. T h e n you m u s t e s t a b l i s h

how m u c h t h e y expand. You c a n do this b y malting a glass capillary {a glass t u b e with a u n i f o r m small-bore hole in it} a n d t h e n m e a s u r i n g how far the liquid m o v e s from a given t e m p e r a t u r e point to a n o t h e r . The easiest w a y to do this is to establish the freezing point of w a t e r a n d its boiling point on y o u r "thermo" meter. Lastly, you t h e n d e t e r m i n e if y o u r c h o s e n liquid e x p a n d s in a linear m a n n e r b e t w e e n t h e s e two t h e r m a l points (For a history, see p. 401}. There are two h e a t factors involved in a n y p h a s e change. T h e y are: "heat capacity", i.e.- Cp or C v , a n d "heat of transformation", u s u a l l y d e n o t e d as H. The former factor is c o n c e r n e d with i n t e r n a l t e m p e r a t u r e

c h a n g e within

a n y given m a t e r i a l (solid, liquid or gas) w h e r e a s t h e latter is involved in c h a n g e s of s t a t e of t h a t material. The a c t u a l n a m e s we u s e to describe H d e p e n d u p o n the direction in w h i c h the p h a s e c h a n g e occurs, vis: 1.1.1.-

C h a n g e s of S t a t e in W a t e r TEMPERATURE CHANGE OF STA.TE ice to w a t e r

HLHEAT OF: Fusion

~ 0

OF 32

w a t e r to ice

Solidification

0

w a t e r to s t e a m

Vaporization

100

212

32

s t e a m to w a t e r

Condensation

100

212

The terms, heat, h e a t c a p a c i t y a n d h e a t of t r a n s f o r m a t i o n were originally defined in t e r m s of water. The earliest concept w a s t h a t of "heat", b u t it w a s soon f o u n d t h a t every m a t e r i a l h a d its own "heat capacity". T h a t is, the u n i t of "heat" (or i n t e r n a l energy) itself w a s originally defined as the a m o u n t of energy r e q u i r e d to raise the t e m p e r a t u r e of 1.00 g r a m of w a t e r b y 1.00 degree Celsius, the u n i t of h e a t energy being set equal to one {1.0) calorie. Heat c a p a c i t y itself w a s originally defined as the a m o u n t of h e a t r e q u i r e d to raise the t e m p e r a t u r e

of one {1) cubic c e n t i m e t e r (cc.) of w a t e r (whose

density w a s later defined as 0.9999 @ 3.98 ~ by one (1.00) degree. However, after tbJs concept h a d b e e n defined, it w a s f o u n d t h a t the s a m e a m o u n t of h e a t d i d n o t raise the i n t e r n a l t e m p e r a t u r e of other m a t e r i a l s to the

same

degree.

This led to the

concept

of h e a t

capacity.

Heat

of

t r a n s f o r m a t i o n w a s discovered s o m e w h a t later. How t h e s e c o n c e p t s arose c a n be illustrated as follows.

S u p p o s e you decide to solve the p r o b l e m of defining the c o n c e p t s of h e a t c a p a c i t y a n d h e a t of t r a n s f o r m a t i o n . The former is hhe a m o u n t of energy required to c h a n g e the t e m p e r a t u r e of a m a t e r i a l while the latter relates to how m u c h energy is n e e d e d to c h a n g e from one p h a s e to another. Defining t h e s e c o n c e p t s is a n intellectual r e s e a r c h exercise in itself. In doing this work, it is clear t h a t you m u s t be able to define s o m e sort of m e a s u r e of energy, or heat, itself (even t h o u g h you have not yet defined a u s a b l e scale of heat). Since w a t e r is plentiful (it covers a b o u t 75% of the E a r t h ' s surface), you decide to u s e w a t e r as a s t a n d a r d . You begin by distilling w a t e r to obtain a p u r e p h a s e . You will find t h a t this is difficult since even the a p p a r a t u s you u s e will c o n t a m i n a t e the distilled w a t e r ff you are not careful. After observing its behavior

as

its t e m p e r a t u r e

is changed,

you

decide

to

set u p

a

t e m p e r a t u r e scale in t e r m s of water. In this work, you define the meltW_g point of ice as 0 . 0 0 0 ~

a n d the boiling point of w a t e r as 100.000 ~

(C is

"Celcius", n a m e d after the r e s e a r c h e r who did the original w o r k in 1734). You do this b e c a u s e t h e s e are easily observable points in a n a r b i t r a r y scale of t e m p e r a t u r e as related to h e a t (and more i m p o r t a n t l y are reproducible a n d reversible points). As we said, the m o s t i m p o r t a n t p a r t of this w o r k is to obtain a s a m p l e of p u r e water. M u c h effort w a s e x p e n d e d in the p a s t in a c c o m p l i s h i n g this goal. T a k i n g y o u r p u r e p h a s e of water, you t h e n m e a s u r e its d e n s i t y as a function of t e m p e r a t u r e . Note t h a t you have already defmed a "specific volume" for u s e in y o u r work, ie- cubic c e n t i m e t e r = cc. You d e t e r m i n e t h a t the point of m a x i m u m density of w a t e r o c c u r s at 3 . 9 8 0 ~ d e c r e a s e s b o t h as you a p p r o a c h 0 . 0 0 ~

a n d t h a t the d e n s i t y of w a t e r

a n d its boiling point of 100.00 ~

You t h e n assign the a r b i t r a r y value of 1.000 g r a m / 1 . 0 0 0 ml. of w a t e r (or 0 . 9 9 9 8 g r a m / c c , as later more precise m e a s u r e m e n t s have revealed) as the d e n s i t y of w a t e r at its t e m p e r a t u r e of m a x i m u m density. By t h e n m e a s u r i n g the a m o u n t

of h e a t r e q u i r e d to raise the t e m p e r a t u r e

of 1.000 cubic

c e n t i m e t e r of w a t e r b y 100th of y o u r a r b i t r a r y t e m p e r a t u r e range, i.e. 1.000 ~

by

you have also defined y o u r s t a n d a r d u n i t of h e a t (and called it

1.000 calorie). You have now defined "heat" itself as the a m o u n t of energy r e q u i r e d to raise 1 cc. of w a t e r by one degree as one calorie. However, the t e m p e r a t u r e at

w h i c h you m e a s u r e d y o u r s t a n d a r d calorie w a s not specified. For this reason, you choose the t e m p e r a t u r e r a n g e of 14.5 ~ to 15.5 ~ b e c a u s e you have d e t e r m i n e d t h a t this r a n g e is one w h e r e the d e n s i t y of w a t e r r e m a i n s relatively uniform. T h u s , the s t a n d a r d calorie w a s originally defined as the a m o u n t of h e a t r e q u i r e d to raise the t e m p e r a t u r e of o n e g r a m of w a t e r from 14.5 ~ to 15.5 ~

at a c o n s t a n t p r e s s u r e of one (1) a t m o s p h e r e . Note t h a t in

y o u r r e s e a r c h to define heat, h e a t c a p a c i t y a n d h e a t of t r a n s f o r m a t i o n , you have discovered several a n o m a l i e s of w a t e r t h a t h a d not b e e n k n o w n before y o u r w o r k w a s accomplished. Next, you now m e a s u r e o t h e r m a t e r i a l s to deterrrdne how m u c h energy is required to raise their t e m p e r a t u r e by one degree a n d find t h a t the a m o u n t of energy ,caries from m a t e r i a l to material. In this way, you e s t a b l i s h the notion of h e a t capacity. However, you also note t h a t e a c h m a t e r i a l h a s its own i n t e r n a l energy (heat capacity) at a given t e m p e r a t u r e in relation to t h a t of water. The next step t h a t you t a k e is to s h o w t h a t the t e m p e r a t u r e of ice does n o t c h a n g e as it m e l t s (or t h a t the t e m p e r a t u r e of w a t e r does not c h a n g e as it freezes). This t a k e s several very careful m e a s u r e m e n t s to e s t a b l i s h this fact. It is this concept t h a t e s t a b l i s h e d t h a t there are two k i n d s of "heat" involved in Changes-of-State, n a m e l y t h a t of h e a t of t r a n f o r m a t i o n a n d h e a t capacity. T h u s , the i n t e r n a l t e m p e r a t u r e of a m a t e r i a l c h a n g e s in a linear m a n n e r as energy is a d d e d until the point of t r a n s f o r m a t i o n (phase change) occurs. Then, the i n t e r n a l t e m p e r a t u r e does n o t c h a n g e u n t i l the p h a s e c h a n g e is complete. Reiterating, h e a t of t r a n s f o r m a t i o n , or H, is involved in c h a n g e of form of the m a t e r i a l while h e a t c a p a c i t y relates to its i n t e r n a l c h a n g e in t e m p e r a t u r e as it a p p r o a c h e s a n o t h e r point of change. Both of t h e s e c o n s t a n t s are b a s e d u p o n the s t a n d a r d of energy, or heat, of one (1.00) calorie. Heat c a p a c i t y is also k n o w n as t h e r m a l capacity. We t h u s label h e a t c a p a c i t y as Cp, m e a n i n g the t h e r m a l c a p a c i t y at c o n s t a n t pressure.

Sometimes, it is also called specific heat, m e a n i n g the ratio of

t h e r m a l c a p a c i t y of a n y given m a t e r i a l to t h a t of water, defined as 1.000.

This is the a m o u n t of h e a t in calories t h a t it t a k e s to raise a n y given material 1.00 ~ Originally, C v , the t h e r m a l capacity at c o n s t a n t v o l u m e w a s also used, b u t its u s e is rare n o w a d a y s . The r e a s o n for this is t h a t m o s t m a t e r i a l s e x p a n d over a given t e m p e r a t u r e range, a n d this complicates a n y a c c u r a t e m e a s u r e m e n t (since you have to c o m p e n s a t e for this c h a n g e in volume). We have now d e m o n s t r a t e d t h a t there are two types of "heat" involved in the physical c h a n g e s of a n y given material, one involved in c h a n g e of internal t e m p e r a t u r e of the material a n d the other d u r i n g its t r a n f o r m a t i o n . We specify its internal energy as the "heat" of a material, u s i n g HS,L,G. where S, L, or G refer to either solid, liquid or gas. Note t h a t this internal energy differs from the h e a t of t r a n s f o r m a t i o n , It. (This concept is p e r h a p s one of the m o r e difficult o n e s for one to grasp. However, the internal energy is t h a t energy the s u b s t a n c e h a s b e t w e e n HS,L,G a n d Cp is: I.I.2.-

at a given temperature).

{ H = Cp. T}S,L,G, or

Thus,

the relation

ikI-Is,L,G = Cp(S,L,G) ATs,L,G

where the a c t u a l state of m a t t e r is either S, L, or G. (Note t h a t Cp will differ d e p e n d i n g u p o n the state of m a t t e r involved). In contrast, t h e i n t e r n a l t e m p e r a t u r e the material undergoes

of a material does not change

as

a c h a n g e o f s t a t e . (Thus, its internal energy does

not c h a n g e at t h a t point). Therefore, for a c h a n g e of state between solid a n d liquid, we w o u l d have: 1.1.3-

{ I ~ - HI.} = Cp (T(s) - T(L) ) or tdtI = Cp AT

Note t h a t AIt here is a h e a t of t r a n s f o r m a t i o n involved in a c h a n g e o f s t a t e w h e r e a s zkHs,L,G refers to c h a n g e in ~ t e r ~ d h e a t for a given state of matter. In the case of water, w e have defined its heat capacity, whereas its

heats of transformation were measured. These c o n s t a n t s r e p r e s e n t the relative a m o u n t s of energy n e e d e d to c a u s e c h a n g e s in form of the molecule, H20, from one p h a s e to a n o t h e r state. Their values are given in the following values as s h o w n in 1.1.4. on the next page:

1.1.4.Ice:

0.5 c a l / g m / ~ 80 c a l . / g m . (S ~ L)

Water:

1.00 c a l / g m / ~ 5 4 0 c a l . / g i n . {L ~ G)

Steam:

0.5 c a l . / g i n / ~

We c a n n o w c a l c u l a t e the a m o u n t of energy r e q u i r e d to raise o n e g r a m o f i c e at - 10 ~

to form o n e g r m n o f s t e a m at 110 ~

This is s h o w n in the

following: 1.1.5.- CALORIES REOUIRED TO CHANGE 1 G r a m OF ICE TO STEAM FORM Ice

AT -10 to 0

Ice

0

Water

0 to 100

Water

0

Ca 0.5

C ~ 5.0 cal.

...... 1.0

100 cal.

. . . . . .

to S t e a m

&H_K

ADDED

---

5.0 cal.

80 cal.(fusion)

85.0 cal.

---

100.0 cal.

540 cal.

5 4 0 . 0 cal.

(vaporization) Total =

730.0 cal.

In all cases, AItx is specified in t e r m s of the type of c h a n g e of s t a t e occurring, while Cp AT is the c h m a g e i n i n t e r n a l h e a t w h i c h o c c u r s as the t e m p e r a t u r e rises (or falls). At a given c h a n g e of state, all of t h e energy goes to the c h a n g e of s t a t e a n d t h e t e m p e r a t u r e does not c h a n g e until t h e t r a n s f o r m a t i o n is complete. In t h e n e x t section, we shall see w h e r e the energy goes. The s a m e t r a n s f o r m a t i o n s t a k e place r e g a r d l e s s of the n a t u r e of t h e a t o m s composing

the

compound,

unless,

as

we

said

above,

the

compound

d e c o m p o s e s to s o m e o t h e r form. Let u s now e x a m i n e t h e t h e r m a l c h a n g e s we have d e s c r i b e d above in t e r m s of a g r a p h s h o w i n g t h e c h a n g e in a c t u a l t e m p e r a t u r e . This is i l l u s t r a t e d in the follo,~mg d i a g r a m , given on the n e x t page as 1.1.6. Note t h a t we have plotted the calories a d d e d on a s h o r t e n e d scale so as to

get the total calories a d d e d to the s y s t e m on one page. The s a m e n u m b e r s given in 1.1.5. are u s e d in 1.1.6. b u t it is easier to see t h a t the calories involved in C h a n g e - o f - S t a t e p r e d o m i n a t e over t h o s e w h i c h m e r e l y c h a n g e the i n t e r n a l energy, or t h e t e m p e r a t u r e , of the m a t e r i a l (here, ice or steam). The next c o n c e p t w h i c h we w i s h to e x a m i n e is t h a t of the differences b e t w e e n the three s t a t e s of m a t t e r , gases, liquids a n d solids. In t h i s case, we will fred very significant differences in their e n e r g y content, n a m e l y t h a t t h e g a s e o u s form is t h e m o s t energetic while the solid h a s t h e l e a s t energy.

1,2.- DIFFERENCES B ~ E N

THE THREE STATES OF MATTER

In order to define s u c h differences, we m u s t first s h o w how t h e s e s t a t e s differ physically from one another. We will s t a r t with gases, t h e n liquids a n d finally solids. As we shall see, the m a j o r difference b e t w e e n these s t a t e s is a m a t t e r of energy, the solid having the least energy of all. G a s e o u s molecules are free to roam, w h e r e a s m o l e c u l e s in the liquid s t a t e are b o u n d together a n d molecules in the solid state are b o u n d a n d ordered into a tight-knit structure. a. The G a s e o u s State The g a s e o u s s t a t e h a s b e e n defined as "a state of m a t t e r in w h i c h the s u b s t a n c e e x p a n d s readily to fill a n y c o n t a i n i n g vessel" (1). W h a t this m e a n s is t h a t a n y collection of molecules in the g a s e o u s state is free to move in all directions a n d t h a t the g a s e o u s molecules will fill a n y c o n t a i n e r in w h i c h t h e y are confined. For the H20 molecule in a g a s e o u s state (which h a s 3 a t o m s per molecule), there will be 9 degrees of freedom (since there are 3 d i m e n s i o n s in w h i c h it c a n move). These c a n be divided into 3-translational, 3-vibrational a n d 3-rotational degrees of freedom. These axe s h o w n in the following diagram, given as 1.2.1. on the next page. Note t h a t the drawings are exaggerated from the a c t u a l condition to illustrate the c h a n g e s in vibrational a n d rotational states. T h e s e molecules are free to move in a n y of three directons, a n d c a n rotate a n d vibrate in v a r i o u s m o d e s in a n y of the three directions as shown. Their energy s t a t e s are quantized, b u t well s e p a r a t e d in energy as s h o w n in the following diagram, also given on the next page as 1.2.2. Note t h a t the drawings are exaggerated from the actual condition to illustrate the c h a n g e s in vibrational a n d rotational states. These molecules are free to move in a n y of three directons, a n d c a n rotate a n d vibrate in v a r i o u s m o d e s in a n y of the three directions as shown. In this diagram, we have s h o w n 3 s e p a r a t e vibrational s t a t e s with rotational s t a t e s s u p e r i m p o s e d u p o n them. For o u r g a s e o u s w a t e r molecule w h i c h h a s three a t o m s per molecule, (1 oxygen a n d two h y d r o g e n atoms), the 3 vibrational degrees of freedom will have (2J+ 1 = 7) rotational s t a t e s s u p e r i m p o s e d u p o n

10

t h e m (J is the n u m b e r of a t o m s in the molecule, having q u a n t i z e d vibrational states).

If the

molecule

happened

to be

NH3,

then

the

expected

number

of

vibrational s t a t e s would be nine. If we m e a s u r e the a b s o r p t i o n s p e c t r a of a n y molecule, M X 2 , in the infra-red region of the s p e c t r u m , we will obtain

11

r e s u l t s similar to t h o s e s h o w n in the d i a g r a m given above, the exact region of the s p e c t r u m d e p e n d i n g u p o n the type of molecule p r e s e n t (i.e.- m o l e c u l a r vceight). It t h u s s h o u l d be clear t h a t e a c h g a s e o u s molecule is free to move in a n y of the three d i m e n s i o n s until it collides with either the walls of the container, or with s o m e o t h e r molecule. The average d i s t a n c e t h a t e a c h molecule m o v e s before collision is called the "mean free path". The m e a n free p a t h will be a function of b o t h the t e m p e r a t u r e a n d the p r e s s u r e of the gas. This concept arose from the Kinetic T h e o r y of G a s e s w h i c h in t u r n arose from Avagadro's Hypothesis. In 1811, Avogadro p o s t u l a t e d t h a t equal v o l u m e s of g a s e s c o n t a i n equal n u m b e r s of molecules (at a given t e m p e r a t u r e

a n d pressure). Following this, J o u l e

explained in 1843 t h a t the p r e s s u r e of a gas is c a u s e d b y the i n t e n s e m o t i o n of the molecules w h i c h b o m b a r d the walls of the container. The exact p r e s s u r e w a s p r o p o s e d to be proportional to the speed a n d m o m e n t u m of these molecules. Both Avogadro's a n d J o u l e ' s theories were d i s p u t e d over a n u m b e r of y e a r s on several g r o u n d s until 1906 w h e n J e a n Perrin directly observed, a n d Einstein explained, the Brownian m o v e m e n t in gases. These arguments,

coupled with direct scientific observation, finally served to

e s t a b l i s h t h e s e theories as Laws. However, it w a s Maxwell in

1848 who s h o w e d t h a t molecules have a

distribution of velocities a n d t h a t they do not travel in a direct line. One experimental m e t h o d u s e d to s h o w this w a s t h a t a m m o n i a molecules are not detected in the time expected, as derived from their calculated velocity, b u t arrive m u c h later. This arises t "ore the fact t h a t the a m m o n i a molecules

interdiffuse a m o n g the air moi~.cules b y i n t e r m o l e c u l a r collisions. The m o l e c u l a r velocity calculated for N:i3 molecules from the w o r k done b y J o u l e in 1843 w a s 5.0 x102 m e t e r s / s e c , at r o o m t e m p e r a t u r e . This implied t h a t the odor of a m m o n i a o u g h t to be detected in 4 millisec at a d i s t a n c e of 2.0 m e t e r s from the source. Since Maxwell observed t h a t it took several m i n u t e s , it w a s fully obvious t h a t the m o l e c u l e s did not travel in a direct p a t h . Analysis b y C l a u s i u s in 1849 s h o w e d t h a t the mmmonia molecules travel only s o m e 0.001

cm. b e t w e e n collisions with air molecules at a t m o s p h e r i c

p r e s s u r e , in time intervals of a b o u t 10 -10 sec. between collisions. This m e a n t

12

t h a t t h e y d e s c r i b e a long a n d intricate p a t h in the p r o c e s s of a c q u i r i n g a d i s p l a c e m e n t of several m e t e r s . The m a t h e m a t i c s involved are intricate a n d we will not p r e s e n t t h e m here. However, t h e y a r e available for t h o s e who w i s h to s t u d y t h e m (2). Nevertheless, t h e fact t h a t t h e r e is a n average d i s t a n c e t h a t m o l e c u l e s travel b e t w e e n collisions h a s given u s the c o n c e p t of the Mean Free P a t h (which is the average d i s t a n c e b e t w e e n collisions). A b y - p r o d u c t of the w o r k b y Perrin a n d Einstein w a s the first reliable e v a l u a t i o n of Avogadro's n u m b e r , the n u m b e r of m o l e c u l e s in a mole. The b e s t c u r r e n t value is believed to be: 6.02204531 x 1023 m o l e c u l e s per k i l o g r a m mole. It is well to note, at this point, t h a t all of t h e s e o b s e r v a t i o n s are the r e s u l t of m a n y h o u r s of w o r k by prior investigators from the p a s t . T h u s , we have their experience a n d i n t u i t i o n to d r a w u p o n for a n y w o r k t h a t we m a y c a r r y forward to t h e benefit of m a n k i n d . b. The Liquid S t a t e In the liquid state, the m o l e c u l e s are still free to move in t h r e e d i m e n s i o n s b u t still have to be confined in a c o n t a i n e r in the s a m e m a n n e r as the g a s e o u s s t a t e if we expect to be able to m e a s u r e t h e m . However, there are i m p o r t a n t differences. Since the m o l e c u l e s in the liquid s t a t e have h a d energy

removed

from

them

in

order

to

get

them

to

condense,

the

t r a n s l a t i o n a l degrees of freedom are found to be restricted. This is d u e to the fact t h a t t h e m o l e c u l e s are m u c h closer together a n d c a n i n t e r a c t with one a n o t h e r . It is this i n t e r a c t i o n t h a t gives the liquid s t a t e its u n i q u e properties. T h u s , the m o l e c u l e s of a liquid are not free to flow in a n y of the three directions, b u t are b o u n d by i n t e r m o l e c u l a r forces. T h e s e forces d e p e n d u p o n the electronic s t r u c t u r e of the molecule. In the c a s e of water, w h i c h h a s two electrons on the oxygen a t o m w h i c h do n o t p a r t i c i p a t e in the b o n d i n g s t r u c t u r e , the m o l e c u l e h a s a n electronic m o m e n t , i.e.- is a "dipole". This r e s u l t s in a p r o p e r t y w h i c h we call fluid viscosity since the m o m e n t of e a c h m o l e c u l e i n t e r a c t s with all of its n e a r e s t neighbors. Yet, the s a m e vibrational a n d r o t a t i o n a l s t a t e s are still p r e s e n t b u t in a different form. T h a t is, t h e y are m u t a t e d forms of the s a m e energy levels t h a t we found in the g a s e o u s state. This is i l l u s t r a t e d in t h e following diagram:

13

Note t h a t in the above d i a g r a m there axe still vibrational s t a t e s b u t t h a t the rotational s t a t e s are "smeared" one into the other. There is little t r a n s l a t i o n a l m o t i o n for the w a t e r molecules within the interior of the liquid u n l e s s t h e y escape from the liquid p h a s e . If t h e y do so, we call this "evaporation" (This m a y be c o n t r a s t e d to the escape of molecules from a solid w h i c h we call "sublimation"). :Most liquids do have a def'med v a p o r p r e s s u r e w h i c h m e a n s t h a t molecules c a n a n d do escape from the surface of the liquid to form a gas. This is a n o t h e r r e a s o n t h a t the properties of a liquid vm3r from those of the g a s e o u s state. Hence, we still have the vibrational a n d rotational degrees of freedom left in the liquid, b u t not t h o s e of the t r a n s l a t i o n a l mode. A r e p r e s e n t a t i o n of 'water molecules in the liquid state is p r e s e n t e d in the

following diagram,

s h o w n as 1.2.4. on the next page. (Ignore the o p e n s p a c e s since this is merely a simulation). As s h o w n in this diagram, the w a t e r molecules flu the c o n t a i n e r a n d also have a free surface from w h i c h they c a n escape. T h u s , we conclude t h a t the molecules of a liquid are free to slide p a s t one a n o t h e r b u t the overall a s s e m b l a g e of molecules does n o t have a definitive form, except t h a t of the c o n t a i n e r u s e d to hold it. For this reason, a liquid h a s b e e n defined as "a s u b s t a n c e or state of m a t t e r w h i c h h a s the capacity to flow u n d e r extremely small s h e a r s t e s s e s to conform to the s h a p e of a n y

14

confining vessel, b u t is relatively incompressible and lacks the capacity to expand without limit" (I). 1.2.4.-

Therefore, as we change the state of matter, the translational degrees of freedom in liquids become severely restricted in relation to those of the gaseous state. And, the vibrational a n d rotational degrees of freedom appear to be s o m e w h a t restricted, even t h o u g h m a n y of the liquid vibrational and rotational states have been found to be quite similar to those of the gaseous state.

15

c. The Solid S t a t e If we n o w r e m o v e m o r e

ener~r

from t h e

liquid,

it finally r e a c h e s

a

t e m p e r a t u r e w h e r e it "freezes", t h a t is - it converts to a solid. W h a t h a p p e n s , in a m o l e c u l a r sense, is t h a t the m o l e c u l e s b e c o m e o r d e r e d . A n o t h e r w a y to s a y this is t h a t t h e y form a lattice-like framework. A r e p r e s e n t a t i o n of the solid s t a t e is s h o w n in ~ e foUowing diagram:

In this case, the m o l e c u l e s are f o u n d to have eu-rmnged t h e m s e l v e s in orderly rows. A l t h o u g h we c a n see only t h e top layer, t h e r e are several layers below with the exact s a m e a r r a n g e m e n t (This is not the exact a r r a n g e m e n t f o u n d in "ice" b u t is a stylized r e p r e s e n t a t i o n of the solid s t a t e of water). It s h o u l d

thus

be

clear t h a t

as we c h a n g e

the

state

of m a t t e r ,

the

t r a n s l a t i o n a l degrees of f r e e d o m p r e s e n t in g a s e s b e c o m e r e s t r i c t e d in liquids ~md d i s a p p e a r

in solids.

For g a s e o u s

molecules,

both

~ibrational

and

r o t a t i o n a l degrees of freedorn are p r e s e n t while t h o s e of the liquid s t a t e are modified to the point w h e r e only ~ibrational s t a t e s c a n be said to truly-free s t a t e s . The s a m e c a n n o t be said for m o l e c u l e s in the solid state. In t h e solid,

16

only the vibrational s t a t e s remain, b u t t h e y do not resemble t h o s e of the g a s e o u s a n d liquid states. This is illustrated as follows:

Note t h a t t h e s e vibrational s t a t e s in the solid are not recognizable in t e r m s of those of the g a s e o u s or liquid states. And, the rotational s t a t e s a p p e a r to be completely absent. It h a s b e e n d e t e r m i n e d t h a t solids have quite different vibrational s t a t e s w h i c h are called "phonon modes". These vibrational s t a t e s are

quantized

vibrational m o d e s within the solid s t r u c t u r e w h e r e i n the

a t o m s all vibrate

together

in a specific pattern. T h a t is, the vibrations have

clearly defined energy m o d e s in the solid. The n u m b e r of p h o n o n m o d e s are limited a n d have b e e n described as " p h o n o n b r a n c h e s " w h e r e two types are present, "optical " a n d "acoustical". (These n a m e s arose due to the original m e t h o d s u s e d to s t u d y t h e m in solids). For the solid state, there will be a specific n u m b e r of p h o n o n b r a n c h e s f o u n d in the vibrational s p e c t r u m of a n y given solid, which d e p e n d s u p o n the n u m b e r of a t o m s c o m p o s i n g the solid. The n u m b e r of b r a n c h e s f o u n d in the p h o n o n m o d e s can be f o u n d from the following equations, given in 1.2.7. on the next page. C o n t r a s t this s i t u a t i o n with those of b o t h the liquid state a n d the g a s e o u s state. (What do we m e a n b y "quantized p h o n o n m o d e s in the solid?- the vibrations have specific a m o u n t s of energy a n d these m o d e s a p p e a r only as r e s o n a n t vibrations- i.e.- the molecules or a t o m s vibrate together only- at certain frequencies,

d e p e n d i n g u p o n the m a s s

or the

17

molecules (atoms or ions) a n d the chemical b o n d s holding the s t r u c t u r e together). 1.2.7.- N u m b e r of B r a n c h e s of P h o n o n Dispersi0.n: Equations:

Optical = 3 y -3

Acoustical = y a t o m s / m o l e c u l e

P h o n o n States:

For w a t e r with 3 a t o m s per mole c.ule: Acoustical

=3

Optical

= 6 (i.e.- [3x31 -3)

The m a j o r difference, then, between the 3 p h a s e s we have d i s c u s s e d is t h a t the solid c o n s i s t s of a n a s s e m b l a g e of

close-packed

molecules

w h i c h we have s h o w n to have a r i s e n - w h e n we r e m o v e d e n o u g h energy from the molecules so as to c a u s e t h e m to c o n d e n s e a n d to form the solid state. Let u s n o w examine the properties of a t o m s or molecules w h e n t h e y are crowded together to form a "close-packed" solid. 1.3.- THE CLOSE PACKED SOLID 'We have already said t h a t the solid differs from the o t h e r s t a t e s of m a t t e r in t h a t a long range ordering of a t o m s or molecules h a s appeared. To achieve long r a n g e order in a n y solid, one m u s t s t a c k a t o m s or molecules in a s y m m e t r i c a l w a y t o c o m p l e t e l y fill all of the space available. This is not a trivial m a t t e r since solids require t h a t all of the a t o m s be a r r a n g e d in a synm]etical p a t t e r n hi t h r e e d i m e n s i o n s . T h u s , if we could actually see these a t o m s in a solid, we w o u l d find t h a t t h e y are c o m p o s e d of specific "building blocks", w h i c h we shall call "propagation models" or "Units". (Actually, it is now possible to directly observe the p a c k i n g of a t o m s in solids, b u t t h a t is a n o t h e r story, t h a t of the atomic-force microscope). S u c h m o d e l s m u s t be entirely synm]etrical in t h r e e - d i m e n s i o n a l space so t h a t we c a n a r r a n g e t h e m properly to form a 3 - d i m e n s i o n a l solid. B e c a u s e of this limitation, we find t h a t only certain types of p r o p a g a t i o n m o d e l s will work. And, in doing so, we c a n gain f u r t h e r insight into the properties of a solid. To u n d e r s t a n d tt'ds, consider the following discussion.

18

Of t h e t h r e e - d i m e n s i o n a l m o d e l s available to u s , only c e r t a i n s h a p e s c a n b e u s e d to f o r m a s y m m e t r i c a l solid. T h e s e are e v e n - n u m b e r e d s e t s of a t o m s , a r r a n g e d in e i t h e r of t h e following forms1.3. I.- Atomic F o r m s S u i t a b l e for A s s e m b l i n g Long D i s t a n c e A r r a n g e m e n t s T e t r a g o n a l = 4 a t o m s per Unit H e x a g o n a l = 6 a t o m s p e r Unit Cubic

= 8 a t o m s p e r Unit

T h e s e specific s h a p e s are s h o w n in t h e following d i a g r a m :

Note t h a t t h e r e are 4, 6 or 8 a t o m s p e r Unit, b u t o d d - n u m b e r e d u n i t s of 5, 7, or 9 a t o m s p e r Unit are n o t u s e d . If s u c h U n i t s are tried, o n e finds t h a t t h e y c a n n o t be fit t o g e t h e r in a t h r e e - d i m e n s i o n a l p a t t e r n w h i c h h a s long r a n g e o r d e r a n d s y m m e t r y (If y o u do n o t believe this, t r y it yourself. You will Fund t h a t a five-atom Unit, w h i c h is t h e h e x a g o n a l Unit given above m i n u s one atom,

cannot

be

stacked

together

without

losing

part

of t h e

three-

d i m e n s i o n a l space. In o t h e r w o r d s , t h e r e will b e "holes" in t h e s t r u c t u r e ) . You m i g h t w o n d e r w h y we did n o t specify e i t h e r 1, 2 or 3 a t o m s p e r Unit. T h e r e a s o n lies in t h e fact t h a t 1 a t o m , or 2 a t o m s are n o t t h r e e - d i m e n s i o n a l

19

b u t are t w o - d i m e n s i o n a l (a fact t h a t you c a n a s c e r t a i n b y glueing s o m e pingpong balls together to m a k e individual m o d e l s - t h e s e c o m p o n e n t s c a n be u s e d to form t h e three Units given in

1.3.2.). Note t h a t we have now

e s t a b l i s h e d t h a t only specific s h a p e s c a n be u s e d to a s s e m b l e a solid. The next q u e s t i o n t h a t n e e d s to be a n s w e r e d is how the solid w o u l d a p p e a r if we could see the a t o m s directly. There

are,

in

heterogeneous.

general,

two

kinds

of

solids,

homogeneous

and

The former is c o m p o s e d from a t o m s t h a t are all the s a m e

a n d the latter from a t o m s not the s a m e . If the a t o m s are all of one kind, i.e.one of t h e e l e m e n t s , t h e p r o b l e m is s t r a i g h t forward. S e t s of eight a t o m s , e a c h set a r r a n g e d as a cube, will g e n e r a t e a cubic s t r u c t u r e . Two sets of three a t o m s , e a c h set of t h r e e a r r a n g e d in a triangle, will p r o p a g a t e a hexagonal pattern with three dimensional symmetry. E l e m e n t a l solids h a v i n g a t e t r a g o n a l s t r u c t u r e are very few a n d it is e a s y to a s c e r t a i n t h a t m o s t of the e l e m e n t s form s t r u c t u r e s t h a t are either cubic or hexagonal, b u t rarely tetragonal. The r e a s o n for this is t h a t t e t r a g o n a l u n i t s are m o r e conducive for the c a s e w h e r e not all of t h e a t o m s are the s a m e , i.e.the h e t e r o g e n e o u s case. One e x a m p l e of this is t h e c a s e w h e r e we have 1 p h o s p h o r o u s atom, c o m b i n e d w i t h 4 oxygen a t o m s to form the p h o s p h a t e Unit, i.e.- P04. A n o t h e r case m i g h t be w h e r e we c o m b i n e 1 c a r b o n a t o m w i t h 3 oxygen a t o m s to form a c a r b o n a t e Unit. In t h e first case, we have a t e t r a g o n a l Unit, w i t h the p h o s p h o r o u s a t o m at t h e c e n t e r of the 4 a t o m s c o m p o s i n g a tet_rahedron. In the s e c o n d case, the c a r b o n a t o m aligned w i t h 3 oxygen a t o m s w h i c h lie in t h e form of a t e t r a h e d r o n . In the first case, the t e t r a h e d r o n arises b e c a u s e of s p a t i a l preferences, w h e r e a s in the s e c o n d case, we k n o w t h a t t h e c a r b o n a t o m prefers to form t e t r a h e d r a l b o n d s . T h u s , it s h o u l d be clear t h a t the specific s t r u c t u r e f o u n d in a solid arises either b e c a u s e of s p a t i a l c o n s i d e r a t i o n s , or b e c a u s e of b o n d i n g p r e f e r e n c e s of certain atoms comprising the structure. In forming a solid from a t o m s or molecules, p a r t of t h e p r o b l e m lies in t h e fact t h a t we m u s t h a n d l e a very large n u m b e r of a t o m s or molecules. For example, 6.023 x 1023 (60.23 septiUion) a t o m s c o m p r i s e one mole a n d we m u s t s t a c k e a c h of t h e s e in a s y m m e t r i c a l m a n n e r to form the c l o s e - p a c k e d

20

solid. (As a m a t t e r of c o m p a r i s o n ,

calcium carbonate, whose molecular

weight is a b o u t 100 g r a m s / m o l e , c a n be held in b o t h of y o u r c u p p e d h a n d s . This a m o u n t c o n t a i n s the 6.0 septillion (1028) molecules). There is also a n o t h e r i m p o r t a n t factor. T h a t is, in building a solid s t r u c t u r e one finds t h a t s o l i d s t r u c t u r e s are b a s e d o n t h e l a r g e s t a t o m p r e s e n t , as well as how it s t a c k s together (its valence) in space filling-form. For m o s t inorganic c o m p o u n d s ,

this is the o x y g e n

atom,

e.g.- oxides,

silicates,

p h o s p h a t e s , sulfates, borates, t u n g s t a t e s , v a n a d a t e s , etc. The few exceptions involve chalcogenides, halides, hydrides, etc., b u t even in t h o s e c o m p o u n d s , the s t r u c t u r e is b a s e d u p o n aggregation of the largest atom, e.g..- the sulfur a t o m in ZnS. Zinc sulfide exhibits two s t r u c t u r e s ,

sphalerite-

a cubic

a r r a n g e m e n t of the sulfide atoms, a n d wurtzite- a h e x a g o n a l a r r a n g e m e n t of the

sulfide

atoms.

Divalent

zinc

atoms

have

essentially

the

same

coordination in b o t h s t r u c t u r e s . The following d i a g r a m s h o w s how s u c h a solid w o u l d look on a n atomistic scale:

W h a t we have s h o w n is the surface of several rows of a t o m s c o m p o s i n g the

21

solid s t r u c t u r e . We see the o u t e r m o s t layer of the s t r u c t u r e w h i c h is likely to be the oxygen a t o m s c o m p o s i n g the c o m p o u n d . Let u s n o w consider p r o p a g a t i o n u n i t s in their space-filling aspects. As we defined t h e m above, they are solid state building blocks t h a t we c a n s t a c k in a symmeh--ical form to infinity. Thus, 4 - a t o m s will form a 3 - d i m e n s i o n a l t e t r a h e d r o n (half a c u b e is oniv l - 2-dkmensional) w h i c h is a valid p r o p a g a t i o n unit. This m e a n s t h a t we c a n take tetrahech-ons a n d fit t h e m together 3dimensionally to form a s y m m e t r i c a l s t r u c t u r e w h i c h extends to infini~-. However, the s a m e is not true for 5-atoms, which foiTns a four-sided pyramid. This s h a p e c a n n o t be completely fitted together in a s y m m e t r i c a l a n d space-filling mmnner, b e c a u s e w h e n we s t a c k these p}Tamids containing 5 atoms, we find t h a t their t r a n s l a t i o n a l properties preclude formation of a s}Tnmetrical s t r u c t u r e b e c a u s e there is lost space between the p y r a m i d s which r e s u l t s in holes in hhe long-range s t r u c t u r e . However, if one m o r e a t o m is a d d e d to the pyramid, we t h e n have a n o c t a h e d r o l i w h i c h is space-filling with t r a n s l a t i o n a l properties. This r e s u l t s in a hexagonal s t r u c t u r e . Going further, c o m b i n a t i o n s of seven a t o m s are asymmetrical, b u t eight a t o m s form a cube w h i c h c a n be p r o p a g a t e d to infu-'dty to form a cubic s t r u c t u r e . Note t h a t b y t u r n i n g the top layer of four a t o m s b y 45 ~ (see 1.3.1.), we have a hexagonal u n i t w h i c h is related to the hexagonal u n i t c o m p o s e d of 6 atoms, two triangles atop of each other. By t~g ping-pong balls and" gluing t h e m together to form the p r o p a g a t i o n u n i t s s h o w n above, one c a n get a b e t t e r perspective between cubic a n d hexagonal close-packing. Although we c a n c o n t i n u e with more a t o m s per p r o p a g a t i o n unit, it is easy to s h o w t h a t all of those are related to the four basic p r o p a g a t i o n u n i t s found in the solid state, to wit: 1.3.4.-

Propagation Units Usually F o u n d ~ So!i'ds Tet_rahedron

(4)

Octahedron

(6)

Hexagon

(6 or 8)

Cube

(8)

22

We t h u s conclude t h a t s t r u c t u r e s of solids are based, in general, u p o n t h e s e four (4) b a s i c p r o p a g a t i o n units, w h i c h c a n be s t a c k e d in a syrranetrical a n d space-filling form to n e a r infinity. The synm~etry will be t h a t of the largest a t o m in the s t r u c t u r e , u s u a l l y oxygen in inorganic solids. Variation of s t r u c t u r e d e p e n d s u p o n w h e t h e r the o t h e r a t o m s forming the s t r u c t u r e are larger or smaller t h a n the basic p r o p a g a t i o n u n i t s c o m p o s i n g the s t r u c t u r e . In m a n y c a s e s t h e y are smaller a n d will fit into the i n t e r s t i c e of the p r o p a g a t i o n unit, illustrated b y the PO4 - t e t r a h e d r o n m e n t i o n e d above. In this case, the 1~+ a t o m is small e n o u g h to fit into the center (interstice) of the t e t r a h e d r o n formed b y the four oxygen atoms. If we c o m b i n e it with Er 3+ (which is slightly smaller t h a n PO4), we obtain a tetragonal s t r u c t u r e , a sort of elongated c u b e of high s y m m e t r y . But if we combine it with La 3+ , w h i c h is larger t h a n PO4 3- , a monoclinic s t r u c t u r e with low s y m m e t r y results. There still is one a s p e c t of p h a s e c h e m i s t r y t h a t we have not yet a d d r e s s e d . T h a t is the case w h e r e m o r e t h a n

one solid p h a s e

exists. The basic

properties of a solid include two factors, n a m e l y composition a n d s t r u c t u r e . We will a d d r e s s s t r u c t u r e s of solids in the next chapter. The composition of solids is one w h e r e the individual c o n s t i t u e n t s will vary ff the solid is h e t e r o g e n o u s . T h a t is, the two types of inorganic solids v a r y according to w h e t h e r t h e y are h o m o g e n e o u s or h e t e r o g e n e o u s . This is s h o w n in the following: 1.3.4.- Properties of Solids Variation In H o m o g e n e o u s Solids:

Elemental

S t r u c t u r e only

H e t e r o g e n e o u s Solids:

Compounds

S t r u c t u r e a n d Composition

W h a t this m e a n s is t h a t elements (e.g.- metals) c a n have m o r e t h a n one s t r u c t u r e . For example, Fe exists in 4 s t r u c t u r e s , i.e.- Fe is t e t r a m o r p h o u s . a-Fe is the one stable at r o o m t e m p e r a t u r e . But, it t r a n s f o r m s to ~-Fe, t h e n y-Fe a n d finally ~ -Fe as the t e m p e r a t u r e is raised. These c h a n g e s are all reorganizations

in

packing

d e n s i t y before

the

melting

temperature

is

reached. In c o n s t r a s t , h e t e r o g e n e o u s solids u s u a l l y exist in one s t r u c u r a l

23

modification (a few c h a n g e s t r u c t u r e as t e m p e r a t u r e is increased) b u t t h e y exist in m a n y compositions, d e p e n d i n g u p o n how t h e y were formed. For example, a large n u m b e r of c a l c i u m silicates are known, including: 1.3.5.- Known C a l c i u m Silicates Ca2SiO 3

Ca 3 Si20v

Ca4Si2Ov(OH)2

CaSi205

Ca2SiO4

Ca9Si6021(OH)

CaSiO 3

Ca4(ShO 17)(OH)~.

This is only a partial list (I c o u n t e d 58 k n o w n c o m p o u n d s

of varying

composition). E a c h h a s its own composition a n d sWacture. In order to differentiate a m o n g s u c h complicated s y s t e m s , i.e.- o x y g e n a t e d c o m p o u n d s of c a l c i u m a n d silicon, we r e s o r t to w h a t is called a "phase-diagram". A p h a s e d i a g r a m s h o w s t h o s e c o m p o u n d s w h i c h are formed w h e n varying m o l a r ratios of CaO a n d SiO 2 are r e a c t e d together. 1.-4. P h a s e Relations Between Individual Solids To differentiate a n d to be able to d e t e r m i n e the differences b e t w e e n the p h a s e s t h a t m a y arise w h e n two c o m p o u n d s are p r e s e n t (or are m a d e to r e a c t together), we u s e w h a t are t e r m e d " p h a s e - d i a g r a m s " to illustrate the n a t u r e of the i n t e r a c t i o n s b e t w e e n two solid p h a s e compositions. Consider the following. S u p p o s e we have two solids, "A" a n d "B". It does not m a t t e r w h a t the exact composition of e a c h m a y be. A will have a specifc melting point (ff it is stable a n d does not d e c o m p o s e at the M.P.) a n d likewise for the c o m p o u n d , B. We f u r t h e r s u p p o s e t h a t the M.P. of B is h i g h e r t h a n t h a t of A. F u r t h e r m o r e , we s u p p o s e t h a t A a n d B form a solid solution at all v a r i a t i o n s of composition. W h a t this m e a n s is t h a t from 100% A- 0% B to 50% A-50% B to 0% A-100% B, the two solids dissolve in one a n o t h e r to form a completely h o m o g e n o u s single p h a s e . If we plot the M.P. of the system, we find t h a t t h r e e different c u r v e s could result, as s h o w n in t h e following diagram, given as 1.4.1. on the n e x t page. In this case, the m e l t i n g point of the ideal solid solution s h o u l d i n c r e a s e linearly as the ration of B / A increases. However, it u s u a l l y does not.

24

Either a negative deviation or a positive deviation is regularly observed. In any p h a s e diagram, composition is plotted against temperature. In this way, we can see how the interactions between p h a s e s change as the t e m p e r a t u r e changes a n d the behavior as each solid p h a s e then melts. Either two-phase or three p h a s e sys t em s can be illustrated. This is shown in the following:

Here, the two-phase diagram is simplified to show a hypothetical p h a s e involving "A" and "B" c o m p o u n d s which form a solid solution from 100% A to 100% B. The solid solution is labelled as "r The melting t e m p e r a t u r e of A is higher t h a n t h a t of B. Therefore, the melting t e m p e r a t u r e of a drops as the composition becomes richer in B. At specific t e m p e r a t u r e s on the diagram (see 1. & 2.), a two-phase s y s t e m appears, t h a t of a liquid plus t h a t of a. Finally, as the t e m p e r a t u r e rises, the melt is h o m o g e n o u s a n d the solid, a , h a s melted. In the t h r e e - p h a s e system, only the relationship between A, B

25

a n d C c a n be i l l u s t r a t e d on a t w o - d i m e n s i o n a l drawing. A t h r e e - d i m e n s i o n a l d i a g r a m w o u l d be r e q u i r e d to s h o w the effect of t e m p e r a t u r e as well. The p h a s e d i a g r a m s we have s h o w n are b a s e d u p o n the fact t h a t A a n d B form solid m u t u a l l y soluble solid s t a t e solutions. If t h e y do not, i.e.- t h e y are not m u t u a l l y soluble in the solid state, t h e n the p h a s e d i a g r a m b e c o m e s m o r e complicated. As a n example, consider the foUowing, w h i c h is the c a s e of limited solid solubility b e t w e e n A a n d B. (N.B.- s t u d y the following d i a g r a m s carefully)"

Here, we have the c a s e w h e r e A & B form two slightly different c o m p o n d s , a a n d ~. The composition of a is AxBy while t h a t of ~ is AuBv , w h e r e the s u b s c r i p t s indicate the ratio of A to B (these n u m b e r s m a y be whole n u m b e r s or t h e y m a y be fractional n u m b e r s ) . At low B c o n c e n t r a t i o n s , a exists as a solid (the left side of the diagram). As the t e m p e r a t u r e increases, a m e l t is o b t a i n e d a n d a r e m a i n s a s a solid in the melt (L) {This is indicated by the details given j u s t below the s t r a i g h t line in the diagram}. As the t e m p e r a t u r e is increased, t h e n a m e l t s to form a u n i f o r m liquid. In the middle c o n c e n t r a t i o n s , r a n d g exist as two s e p a r a t e p h a s e s . At 75% A-25% B, ~ m e l t s to form a liquid p l u s solid a w h e r e a s j u s t the opposite o c c u r s at 25% A a n d 75% B. W h e n A e q u a l s B, t h e n b o t h a a n d ~ m e l t together to form the liquid melt, i.e.- the "eutectic point".

26

In two similar c a s e s b u t differing cases, i.e.- a s y s t e m with a melting point m i n i m u m a n d a n o t h e r with a different type of limited solid solubility, the behavior differs as s h o w n in the following diagram:

In the case on the left, a composition exists w h i c h in w h i c h a m i n i m u m melting point exists. Again, A a n d B form a w h i c h partially melts to form a + L. However, two forms of a also exist, a compositional a r e a k n o w n as a "miscibility gap", i.e.- " a l a n d a2". But at higher t e m p e r a t u r e s , b o t h of t h e s e melt into the single p h a s e , a. Finally, we obtain the melt p l u s a. Note t h a t at a b o u t 80% a i - 2 0 % a ~ , a melting point m i n i m u m is seen w h e r e it

melts

directly i n s t e a d of forming the t w o - p h a s e system, a + L. In the case of limited solid solubility, the p h a s e b e h a v i o r b e c o m e s m o r e complicated. Here, b o t h a a n d ~ form (where the a c t u a l composition of a is AxBy while t h a t of ~ is A u B v , as given before. Note t h a t the values of x a n d y change, b u t t h a t we still have the a phase. The s a m e holds for u a n d v of the phase). At low A c o n c e n t r a t i o n s , a exists alone while ~ exists alone at higher B c o n c e n t r a t i o n s . A region exists w h e r e the two p h a s e system,

+

exists. We notew t h a t as the t e m p e r a t u r e is raised, a two p h a s e s y s t e m is also seen consisting of the melt liquid p h u s a solid p h a s e . The p h a s e behavior s h o w n on the right side of the d i a g r a m arises b e c a u s e the two p h a s e s , A a n d B, have limited solubility in each other.

27

Now, let u s consider the case w h e r e three (3) s e p a r a t e p h a s e s a p p e a r in the p h a s e diagram, s h o w n as follows:

In this case, we have three (3) s e p a r a t e p h a s e s t h a t a p p e a r in the p h a s e diagram. T h e s e p h a s e s are a , ~, a n d o , w h o s e c o m p o s i t i o n s are: ct = AxBy, = AuBv a n d ~ - AcBd, respectively (the v a l u e s of x, y, u, v, c, a n d d all differ from e a c h other so t h a t AxBy is a specific c o m p o u n d as are the others). By s t u d y i n g this p h a s e d i a g r a m carefully, you c a n see how t h e individual p h a s e s relate to e a c h other. As you c a n see, a p h a s e d i a g r a m c a n b e c o m e quite complicated. However, in m o s t c a s e s involving real c o m p o u n d s , the p h a s e d i a g r a m s are u s u a l l y simple. Those involving c o m p o u n d s like silicates c a n be complex, b u t those involving alloys of m e t a l s s h o w simple behavior like limited solubility. R E F E R E N C E S CITED I. "Dictionary of Scientific a n d Technical Terms" - D.N. Lapedes- Editor in Chief, McGraw-Hill, New York (1978)

28

(2) "Enclyclopedia of Physics- 2 n d Ed." - Edited b y R.G. Lerner & G.L. Trigg, VCH Publishers, NY (1990). PROBLEMS FOR CHAPTER 1 i. Given I 0 . 0 g r a m s of ice at - 65 ~

calculate the n u m b e r of calories

required to c h a n g e it to s t e a m at 240 ~ 2. Do the s a m e for 24.0 g r a m s of ice at -11 ~ converted to boiling water. 3. Look u p the following h e a t capacities: Br2 CO2 PCI3 CH4 NH3 C2H4 NH3 CO2

- gases - liquids - solids

4. List aU of the metallic e l e m e n t s a n d their crystal s t r u c t u r e s . 5. List the n u m b e r of expected vibrational a n d rotational m o d e s for the following g a s e o u s molecules: Br2

co2 PCI3

cI-14 C2H4 C2H6 6. Draw the following p l a n e s for the cubic lattice (see C h a p t e r 2 for help): {101} {222} {101} {301}.

29

7. Given the foUowing phase diagram, label the individual phases present, assu.~ that three phases are present. Use a , 13, and o as symbols for the three phases:

This Page Intentionally Left Blank

31

Chapter 2 D e t e r m i n i n g t h e S t r u c t u r e of Solids This c h a p t e r will p r e s e n t m o r e a d v a n c e d topics t h a n t h o s e of the first c h a p t e r in t e r m s of d e t e r m i n i n g the s t r u c t u r e of solids. C o n s e q u e n t l y , you will gain s o m e

knowledge

of h o w

one

goes

about

determining

the

s t r u c t u r e of a solid, even if you n e v e r have to do it. 2 . 1 - S C I E N T I F I C DETERMINATION OF THE STRUCTURE OF SOLIDS In this section,

we will p r e s e n t

the

basis developed

to explain

the

s t r u c t u r e of solids. T h a t is, t h e c o n c e p t s t h a t w e r e p e r f e c t e d in o r d e r to a c c u r a t e l y d e s c r i b e h o w a t o m s or ions fit t o g e t h e r to form a solid p h a s e . This w o r k was a c c o m p l i s h e d by m a n y p r i o r w o r k e r s w h o e s t a b l i s h e d t h e r a t i o n a l e u s e d to define the s t r u c t u r e of a s y m m e t r i c a l solid. As you will recall, we said t h a t the basic difference b e t w e e n a gas, liquid a n d t h a t of a solid lay in the o r d e r l i n e s s of the solid, c o m p a r e d to t h e o t h e r p h a s e s of the s a m e m a t e r i a l .

We

have

already

indicated

that

solids

can

have

several

forms

or

s y m m e t r i e s . To e l u c i d a t e the s t r u c t u r e of solids in m o r e detail, at l e a s t t h r e e p o s t u l a t e s apply: First, the f o r m a t i o n of a solid r e s u l t s from a s y m m e t r i c a l "stacking" of a t o m s to n e a r infinity from a t o m s or m o l e c u l e s with s p a c i n g s is m u c h s m a l l e r t h a n t h o s e f o u n d in the liquid or g a s e o u s state. S e c o n d l y , if we w i s h to gain an i n s i g h t into h o w t h e s e a t o m s are a r r a n g e d in the solid, we n e e d to d e t e r m i n e w h a t k i n d of p a t t e r n t h e y form w h i l e in close p r o x i m i t y . T h i r d l y , we c a n t h e n relate o u r p a t t e r n

to o t h e r

structures

and thus

define the s y m m e t r y of solids in g e n e r a l . One w a y to a p p r o a c h a solution of the last two p o s t u l a t e s is to define t h e

32

s t r u c t u r e of any given solid in t e r m s of its l a t t i c e p o i n t s , W h a t this m e a n s is t h a t by s u b s t i t u t i n g a p o i n t for e a c h atom(ion) c o m p o s i n g the s t r u c t u r e , we find t h a t t h e s e p o i n t s c o n s t i t u t e a latticework, i.e.- t h r e e - d i m e n s i o n a l solid, having c e r t a i n s y m m e t r i e s (Examples of the s y m m e t r i e s to w h i c h we refer are given in 1.3.2. of C h a p t e r 1). A lattice is not a s t r u c t u r e per se. A lattice is d e f i n e d as a s e t of t h r e e d i m e n s i o n a l p o i n t s , having a c e r t a i n s y m m e t r y . T h e s e p o i n t s may, or m a y

not, be totally o c c u p i e d by the a t o m s c o m p o s i n g the s t r u c t u r e . C o n s i d e r a cubic s t r u c t u r e s u c h as t h a t given in the following diagram:

Here, we have a set of p o i n t s o c c u p i e d by a t o m s (ions) a r r a n g e d in a simple

three-dimensional

cubic

pattern.

The

lattice

directions

are

defined, b y c o n v e n t i o n , as x , y & z. Note t h a t t h e r e are eight (8) c u b e s in our example. In o r d e r to f u r t h e r define o u r cubic p a t t e r n , we n e e d to analyze b o t h t h e s m a l l e s t Unit a n d h o w it fits into the lattice. We call the s m a l l e s t Unit a "unit-cell" a n d find t h a t t h e u n i t - c e l l is the s m a l l e s t c u b e in the d i a g r a m .

33

We also n e e d to define h o w large t h e u n i t - c e l l is in t e r m s of b o t h t h e l e n g t h of its s i d e s a n d its v o l u m e . We do so b y d e f i n i n g

the

unit-cell

d i r e c t i o n s in t e r m s of its "lattice u n i t - v e c t o r s " . T h a t is, we define it in t e r m s of t h e x, y, & z d i r e c t i o n s

of t h e u n i t cell w i t h specific v e c t o r s

h a v i n g d i r e c t i o n s c o r r e s p o n d i n g to:

X~X

Z-~ ~

w i t h t h e l e n g t h of e a c h u n i t - v e c t o r b e i n g e q u a l to 1.0. (Our n o t a t i o n for a v e c t o r h e n c e f o r t h is a l e t t e r w h i c h is "outlined", e.g.- ~ is d e f i n e d h e r e as t h e u n i t - c e l l v e c t o r in t h e "x" d i r e c t i o n ) .

As you p r o b a b l y r e m e m b e r ,

a

"vector" is specified as a line w h i c h h a s b o t h d i r e c t i o n a n d d u r a t i o n f r o m a given p o i n t . We c a n n o w define a set of v e c t o r s called " t r a n s l a t i o n " v e c t o r s , i.e.- a , ~, a n d ~ in t e r m s of t h e following: 2.1.2.-

a. = a x ~

+ a y :y + a z g

b=bxz

+t~]] +bz~

= Cx :~

+ Cy ]

+ Cz

Here, t h e a , b, a n d ~ v e c t o r s are n o w d e f i n e d in t e r m s of t h e a, b, a n d c lattice c o n s t a n t s in e a c h of t h e x, y, a n d z d i r e c t i o n s of 2.1.1., w h e r e a, b, a n d c are lattice c o n s t a n t s . T h e y are real n u m b e r s ~ n g s t r o m (A) d i s t a n c e u n i t s , i.e.- 10- 8 c m .

corresponding

to

Note t h a t we n o w have: a as a v e c t o r in t e r m s of x, y a n d z v e c t o r s as a f u n c t i o n of t h e lattice d i s t a n c e s in t h e t h r e e d i r e c t i o n s ,

ax,

ay

& az.

Here, a = ax, ay & az, i.e.- a is c o m p o s e d of c o m p o n e n t s in e a c h of t h e t h r e e d i r e c t i o n s . T h e s a m e h o l d s for b, a n d ~ v e c t o r s .

34

In g e n e r a l , we u s e o n l y t h e lattice c o n s t a n t s to define t h e solid s t r u c t u r e ( u n l e s s we

are a t t e m p t i n g

to d e t e r m i n e

its synm~etry). We c a n

then

define a s t r u c t u r e factor k n o w n as t h e t r a n s l a t i o n vector. It is a e l e m e n t r e l a t e d to t h e u n i t cell a n d d e f i n e s t h e b a s i c u n i t of t h e s t r u c t u r e . We will call it ~ . It is d e f i n e d a c c o r d i n g to t h e following e q u a t i o n : 2.1.3.-

=

n l & + n2 5

w h e r e n l , n2 , a n d n3

+ n3

are i n t e r c e p t s of t h e u n i t - v e c t o r s , ~ , y , a , o n

t h e x, y, & z - d i r e c t i o n s in t h e lattice, r e s p e c t i v e l y . T h e u n i t cell vohxme is t h e n d e f i n e d as: 2.1.4.-

V

Vector

=

{ ~-

]

x a }

(This is a " d o t - c r o s s " v e c t o r p r o d u c t ) .

n o t a t i o n is b e i n g u s e d h e r e b e c a u s e t h i s is t h e

e a s i e s t w a y to

define t h e unit-cell. T h e r e a s o n for u s i n g b o t h u n i t lattice v e c t o r s a n d translation vectors parameters

lies in t h e

in t e r m s

fact t h a t we c a n n o w

of a, b, a n d c (which

specify

unit-cell

are t h e i n t e r c e p t s

of t h e

t r a n s l a t i o n v e c t o r s o n t h e lattice). T h e s e ceil p a r a m e t e r s are v e r y useful since t h e y specify t h e actual l e n g t h a n d size of t h e u n i t cell, usually in A., as we shall see. A l t h o u g h t h e cubic s t r u c t u r e looks like t h e s i m p l e s t o n e of t h o s e p o s s i b l e , it actually is t h e m o s t c o m p l i c a t e d in t e r m s of s y m m e t r y . W h a t t h i s m e a n s is t h a t we c a n "spin" t h e lattice by h o l d i n g

it at a c e r t a i n

point

and

r o t a t i n g it a r o u n d t h i s axis w h i l e still m a i n t a i n i n g t h e s a m e a r r a n g e m e n t of a t o m s in space. (Take a c u b e a n d do t h i s for yourself). T h e c u b i c l a t t i c e gives

rise

to

a great

synm~etrical l a t t i c e s

many (As we

symmetry will

see,

elements if t h e

in

lattice

to

less

is e l o n g a t e d

contrast

and

d i s t o r t e d f r o m cubic, s o m e of t h e s e synm~etry e l e m e n t s do n o t arise). In 1 8 9 5 , R 6 n t g e n e x p e r i m e n t a l l y d i s c o v e r e d "x-rays" a n d p r o d u c e d

the

first p i c t u r e of t h e b o n e s of t h e h u m a n h a n d . This w a s followed b y w o r k by von Laue in 1912 w h o s h o w e d t h a t solid c r y s t a l s c o u l d act as d i f f r a c t i o n gratings depended

to f o r m

symmetrical

patterns

of "dots"

whose

arrangement

u p o n h o w t h e a t o m s w e r e a r r a n g e d in t h e solid. It w a s s o o n

35

realized t h a t the a t o m s f o r m e d

"planes" w i t h i n the solid. In 1913,

Sir

William H e n r y Bragg a n d his son, William L a w r e n c e Bragg, analyzed t h e m a n n e r in w h i c h s u c h x-rays w e r e reflected

by p l a n e s of a t o m s in t h e

solid. T h e y s h o w e d t h a t t h e s e reflections w o u l d be m o s t i n t e n s e at c e r t a i n angles, a n d t h a t the values w o u l d d e p e n d u p o n the d i s t a n c e b e t w e e n t h e p l a n e s of a t o m s in the crystal a n d u p o n the w a v e l e n g t h of t h e x-ray. T h i s r e s u l t e d in t h e Bragg equation:

2.1.5.-

n~. = 2d sin 0

w h e r e d is t h e d i s t a n c e in a n g s t r o m s (A = 10 -8 cm) b e t w e e n p l a n e s a n d 0 is the angle in d e g r e e s of the reflection. In Bragg's x - r a y diffraction equation, i.e.- the a n g l e , 0 , is actually t h e angle b e t w e e n a given p l a n e of a t o m s in the s t r u c t u r e a n d the l m t h of the x - r a y b e a m . The unit, "d", is defined as the d i s t a n c e b e t w e e n p l a n e s of the lattice a n d k is t h e wavelength

of the

radiation.

It

was

Georges

Friedel

who

in

1913

d e t e r m i n e d t h a t the i n t e n s i t i e s of reflections of x - r a y s from t h e s e p l a n e s in the solid could be u s e d to d e t e r m i n e the s y m m e t r y of the solid. T h u s , by convention, we usually define planes, not points, in t h e lattice. We have i n t r o d u c e d the t e r m "symmetry" in the solid. Let u s n o w e x a m i n e exactly w h a t is m e a n t by t h a t t e r m . To illustrate this c o n c e p t ,

examine

the s y m m e t r y e l e m e n t s of o u r cube, given in the follovcing d i a g r a m , given on the n e x t page as 2 . 1 . 6 . In this case, we c a n s h o w t h a t the p l a n e s of the cubic lattice are d e f i n e d by m o v i n g various d i s t a n c e s in t h e lattice from t h e origin. W h a t is m e a n t by this t e r m i n o l o g y is t h a t as we move from the (0,0,0) origin j u s t 1 . 0 0 unit-cell d i s t a n c e in the x" d i r e c t i o n to the (1,0,0) point in the cubic lattice, we have defined t h e {100} p l a n e (Note t h a t ,are are u s i n g t h e n l i n t e r c e p t of the unit-cell). In a like m a n n e r , if we move in t h e "y" d i r e c t i o n a d i s t a n c e of j u s t 1 unitcell, we have defined the {010} plane. Using the the "z" d i r e c t i o n s gets us a set of o n e - d i m e n s i o n a l p l a n e s (See line A in 2 . 1 . 6 . ) .

36

By m o v i n g I unit-cell d i s t a n c e in the b o t h the x- a n d y - d i r e c t i o n s , t h e {110) h a s b e e n defined, etc. (Line B a b o v e - n o t e t h a t we have n o t i l l u s t r a t e d the {101} plane). Moving 1 unit-cell in all t h r e e d i r e c t i o n s t h e n gives u s t h e {111} plane. In a l i k e m a n n e r , we c a n o b t a i n the {200}, {020}

37

a n d {002} planes. A set of p l a n e s s u c h as the {003} a n d {004} series a r e not as c o m m o n , b u t do exist in s o m e solids. Also s h o w n are the r o t a t i o n a l axes of the cubic lattice, s h o w n as the diad, t r i a d a n d t e t r a d axes. "What this m e a n s is ff ,are spin t h e c u b e on its face, we u s e the t e t r a d axis to d e t e r m i n e t h a t it t a k e s a total of 4 t u r n s to b r i n g a n y specific c o r n e r b a c k to its original position. Likewise, the t r i a d axis u s e s t h e c o r n e r of the c u b e where

the lattice is m o v e d

a total of t h r e e

times,

a n d t h e diad axis

e m p l o y s the diagonal a c r o s s t h e cube to p e r f o r m the synm~ehur o p e r a t i o n . T h u s , t h e p l a n e s of the lattice are f o u n d to be i m p o r t a n t

and can be

defined by m o v i n g along one or m o r e of t h e lattice d i r e c t i o n s of the unitcell to define t h e m . Also i m p o r t a n t are the s y m m e t r y o p e r a t i o n s t h a t c a n be p e r f o r m e d w i t h i n the unit-cell, as we have i l l u s t r a t e d in the p r e c e d i n g diagram. T h e s e give rise to a total of 14 different lattices as we will s h o w below. But first, let us confine o u r d i s c u s s i o n to j u s t the simple cubic lattice. It t u r n s out t h a t t h e m e t h o d u s e d to decribe the p l a n e s given above for the cubic lattice c a n also be u s e d to define the p l a n e s of all of the k n o w n lattices, by u s e of t h e so-called "Miller Indices", w h i c h are r e p r e s e n t e d by: 2.1.7.-

{h,k,l}

T h e w a y t h a t Miller I n d i c e s a r o s e c a n be u n d e r s t o o d by c o n s i d e r i n g t h e h i s t o r y of crystal s t r u c t u r e work, a c c o m p l i s h e d by m a n y i n v e s t i g a t o r s . In 1921, E~wald d e v e l o p e d a m e t h o d of c a l c u l a t i n g the s u m s of diffraction i n t e n s i t i e s from different p l a n e s in the lattice by c o n s i d e r i n g called t h e

"Reciprocal

Lattice".

The

reciprocal

lattice

w h a t is

is o b t a i n e d

by

d r a w i n g p e r p e n d i c u l a r s to e a c h p l a n e in the lattice, so t h a t the axes of t h e r e c i p r o c a l lattice are p e r p e n d i c u l a r to t h o s e of the c r y s t a l lattice. This has t h e r e s u l t t h a t t h e p l a n e s of the r e c i p r o c a l lattice are at right angles (90 ~) to the real p l a n e s in the unit-cell. For o u r cubic lattice, t h e r e c i p r o c a l lattice -would look as s h o w n in t h e following d i a g r a m , given as 2.1.8 on t h e n e x t page.

38

In this diagram, a series of h e x a g o n - s h a p e d p l a n e s are s h o w n w h i c h are orthogonal, or 90 degrees, to each of the c o r n e r s of the cubic cell. E a c h plane c o n n e c t s to a n o t h e r plane (here s h o w n as a rectangle) on e a c h face of the unit-ceU. T h u s , the faces of the lattice unit-cell a n d t h o s e of t h e r e c i p r o c a l unit-cell c a n be s e e n to lie on the s a m e plane while t h o s e at the c o r n e r s lie at right angles to the c o r n e r s . The n o t i o n of a r e c i p r o c a l lattice arose from Ewald who u s e d a s p h e r e to represent

how the x-rays i n t e r a c t with any given lattice plane in t h r e e

d i m e n s i o n a l space. He e m p l o y e d w h a t is now called the Ewald Sphere to s h o w how reciprocal space could be utilized to r e p r e s e n t diffractions of xrays by lattice planes. Ewald originally r e w r o t e the Bragg e q u a t i o n as: 2.1.9.-

sin 0

=

n k/ 2d {hkl}

=

1 / d {hkl} 2/X

Using this equation, Ewald applied it to the case of the diffraction s p h e r e w h i c h we s h o w in the following d i a g r a m as 2.1.10.

on the n e x t page.

S t u d y this d i a g r a m carefully. In this case, the x-ray b e a m e n t e r s

the

s p h e r e e n t e r s from the left a n d e n c o u n t e r s a lattice plane, L. It is t h e n diffracted by the angle 20 to the p o i n t on the s p h e r e , P, w h e r e it

39

is r e g i s t e r e d as a diffaction p o i n t on the r e c i p r o c a l lattice. The d i s t a n c e b e t w e e n p l a n e s in the r e c i p r o c a l lattice is given as 1/dhkl w h i c h is r e a d i l y o b t a i n e d from the diagram. It is for t h e s e r e a s o n s , we c a n u s e the Miller Indices to indicate p l a n e s in the real lattice.

The

reciprocal

lattice

is useful in

defining

some

of the

electronic

p r o p e r t i e s of solids. T h a t is, w h e n we have a s e m i - c o n d u c t o r (or even a c o n d u c t o r like a metal), we find t h a t the e l e c t r o n s are confined in a b a n d , defined by the reciprocal

lattice. This h a s i m p o r t a n t

effects u p o n t h e

c o n d u c t i v i t y of any solid a n d is k n o w n as the "oand theory" of solids. It t u r n s out t h a t the r e c i p r o c a l lattice is also the site of the Brfllouin zones, i.e.- the "allowed" electron e n e r g y b a n d s in the solid. How this o r i g i n a t e s is e x p l a i n e d as follows. The free electron r e s i d e s in a q u a n t i z e d e n e r g y well, defined by k (in w a v e - n u m b e r s ) . This r e s u l t c a n be derived from the S c h r o e d i n g e r waveequation. However, in the p r e s e n c e of a periodic a r r a y of e l e c t r o m a g n e t i c p o t e n t i a l s arising f r o m the a t o m s c o n f i n e d in a crystalline lattice,

the

energies of the e l e c t r o n s f r o m all of the a t o m s are severely limited in orbit a n d are r e s t r i c t e d

to specific allowed e n e r g y b a n d s . T h i s p o t e n t i a l

o r i g i n a t e s from a t t r a c t i o n a n d r e p u l s i o n of the e l e c t r o n c l o u d s from t h e periodic array of a t o m s in the s t r u c t u r e . S o l u t i o n s to this p r o b l e m w e r e

40

made

b y B l o c h in

1930

who

showed

they

had

the

form

(for

a one-

as m o d i f i e d

by the

dimensional lattice). 2.1.11.-

=

e ikx u ( x ) - o n e d i m e n s i o n a l

~ k (a) =

where

e ika u k (X) - t h r e e d i m e n s i o n a l

k is t h e w a v e n u m b e r

of the

allowed

band

l a t t i c e , a m a y b e x, y o r z, a n d uk (x) is a p e r i o d i c a l f u n c t i o n w i t h t h e s a m e p e r i o d i c i t y as t h e p o t e n t i a l . O n e r e p r e s e n t a t i o n

is s h o w n in t h e f o l l o w i n g :

41

We have s h o w n the least c o m p l i c a t e d

one w h i c h

t u r n s o u t to be t h e

simple cubic lattice. S u c h b a n d s are called "BriUuoin" zones a n d , have said,

are the

allowed

energy

bands

of e l e c t r o n s

in

as w e

any given

crystalline latttice. A n u m b e r of m e t a l s a n d simple c o m p o u n d s have b e e n s t u d i e d a n d t h e i r Brilluoin s t r u c t u r e

determined.

However,

when

one

gives a r e p r e s e n t a t i o n of t h e e n e r g y b a n d s in a solid, a "band-model" is usually p r e s e n t e d . T h e following d i a g r a m s h o w s t h r e e b a n d m o d e l s : 2.1.13.- E n e r g y B a n d M o d e l s

In the solid, e l e c t r o n s r e s i d e in the valence b a n d b u t c a n be excited i n t o the c o n d u c t i o n b a n d by a b s o r p t i o n of energy. The e n e r g y gap of various solids d e p e n d s u p o n the n a t u r e of the a t o m s c o m p r i s i n g the solid. S e m i c o n d u c t o r s have a r a t h e r n a r r o w e n e r g y gap (forbidden

zone) vchereas

t h a t of i n s u l a t o r s is wide (metals have little or no gap). Note t h a t e n e r g y levels of the a t o m s "A" are s h o w n in the valence b a n d . T h e s e will vary d e p e n d i n g u p o n the n a t u r e a t o m s p r e s e n t . We will n o t delve f u r t h e r into this a s p e c t h e r e

since it is the s u b j e c t of m o r e

electronic a n d optical m a t e r i a l s .

advanced studies

of

42

R e t u r n i n g to t h e u n i t - c e l l , we c a n also utilize t h e v e c t o r m e t h o d to d e r i v e t h e origin of t h e Miller I n d i c e s . T h e g e n e r a l e q u a t i o n for a p l a n e in t h e l a t t i c e is: 2.1.14.-

~ x,y,z o ~

= d

w h e r e ~ x,y,z is t h e l a t t i c e u n i t v e c t o r in a n y of t h e t h r e e d i m e n s i o n s , ~ is t h e g e n e r a l v e c t o r of t h e l a t t i c e f r o m t h e origin a n d d is, of c o u r s e , t h e distance

between

planes

of

the

lattice

from

the

origin.

A better

p e r s p e c t i v e is s h o w n in t h e following d i a g r a m :

If ~ x,y,z is n o r m a l to t h e real l a t t i c e p l a n e (as in t h e r e c i p r o c a l Lattice}, we c a n define t h e x, y a n d z d i r e c t i o n s of t h e u n i t cell in t e r m s of t h e u n i t cell v e c t o r s a n d t h e c o r r e s p o n d i n g

intercepts

of t h e r e c i p r o c a l

lattice,

i.e.- h, k, a n d 1: 2.1.16.-

x=~/h

y=]

/k

z=a

/1

T h e n , t h e l a t t i c e u n i t v e c t o r will be: 2.1.17.-

~ h,k,1 = 1/ {h 2 + k 2 + 12} 1/2 {11 ~, + k :~ + 1 ~}

w h e r e we h a v e c o n v e r t e d t h e x, y a n d z d i r e c t i o n s into i n t e r c e p t s of t h e reciprocal

l a t t i c e cell. As we s h a l l see,

this equation

is also useful in

d e r i v i n g e q u a t i o n s d e s c r i b i n g s y n m l e t r y o p e r a t i o n s a n d p l a n e s p a c i n g s of t h e real l a t t i c e .

43

To s h o w t h a t t h e above d i s c u s s i o n is t r u e , c o n s i d e r t h e following. Let us set u p a p l a n e of a c u b i c lattice as s h o w n in t h e following d i a g r a m :

In t h i s case, we have s h o w n a {310} p l a n e of a c u b i c lattice. Note t h a t w e m o v e 3 s t e p s of "a" in t h e x - d i r e c t i o n

a n d o n e in t h e "y" d i r e c t i o n

to

define t h i s plane. S i n c e a p l a n e is t w o - d i m e n s i o n a l , t h i s r e p r e s e n t a t i o n is a c c u r a t e . T h e e q u a t i o n of a n y o n e of t h e e l e m e n t s of t h i s p l a n e is t h u s : 2.1.19.-

a = 3x + y

a n d if we h a d b or c (both of w h i c h e q u a l "a" in a c u b i c lattice), we c o u l d t h e n u s e a , b, & c to r e p r e s e n t t h e l e n g t h s of t h e s i d e s of t h e u n i t - c e l l w h i c h in t u r n define t h e p l a n e s w i t h i n t h e u n i t ceil. For t h e cubic l a t t i c e , t h e n we have: 2.1.20.-

Note

d = 1/(h 2 + k 2 + 1 2)1/2 . [ h ~ + h : y + h ~ ]

that just three

2.1.6.).

points

have been

u s e d to define

our plane

T h e original d e f i n i t i o n w a s given by W h e w e l l in

G r a s s m a n in 1829, b u t w a s p o p u l a r i z e d b y Miller in 1839.

1825

(see

a n d by

Since t h r e e

p o i n t s of a lattice c a n be u s e d to define a plane, it is obvious t h a t s u c h

44

p l a n e s c a n be d e f i n e d b y u s i n g o n l y t h r e e p o i n t s in s p a c e , i.e.- a , b & c, to c h a r a c t e r i z e t h e s e t of p l a n e s w i t h i n t h e l a t t i c e . Miller I n d i c e s are t h e r e c i p r o c a l s of t h e i n t e r c e p t s , a , b & c, of t h e c h o s e n p l a n e on t h e x , y , z - d i r e c t i o n s in t h e l a t t i c e (and r e l a t e to t h e n l, n2 a n d n3 i n t e r c e p t s w h i c h we u s e d in o u r d e f i n i t i o n of t h e unit-cell). How t h e y a r e u s e d is s h o w n in t h e following table: 2.1.21.-

P o i n t s on t h e L a t t i c e a n d C o r r e s p o n d i n g Miller I n d i c e s a,

b,

c

MILLER I N D I C E S

(1, 0, O)

{100}

( 1 / 2 , 0, 0 )

{200}

( o , ~/2, o)

{o2o}

(0, O, 1 / 2 )

{002}

Here, we h a v e d e f i n e d t h e s u m t o t a l of t h e i n t e r c e p t s in t e r m s of t h e u n i t cell d i s t a n c e b e i n g s e t e q u a l to 1.00, i.e.2.1.22.-

a/(1/h)

+b/(1/k)+c/(1/k)

= 1

Thus, these intercepts

are given in t e r m s of t h e a c t u a l u n i t - c e l l l e n g t h

f o u n d for t h e specific

structure,

and not the lattice

itself. T h e

Miller

I n d i c e s are t h u s t h e i n d i c e s of a stack of planes w i t h i n t h e lattice. P l a n e s are i m p o r t a n t in solids b e c a u s e , as we will see, t h e y are u s e d to l o c a t e a t o m p o s i t i o n s w i t h i n t h e lattice s t r u c t u r e . The

Final f a c t o r to c o n s i d e r

in o u r s t u d y of h o w to

define

the

s t r u c t u r e of solids is t h a t of t h e a n g l e b e t w e e n t h e x, y, a n d z d i r e c t i o n s in t h e lattice. directions

In o u r e x a m p l e

so far, all of t h e a n g l e s w e r e

9 0 ~ in all

(also called "orthogonal"). If t h e a n g l e s are n o t 9 0 ~ t h e n w e

h a v e a d d i t i o n a l l a t t i c e s to define. For a given u n i t - c e l l d e f i n e d b y t h e u n i t cell l e n g t h s , a, b & c, t h e c o r r e s p o n d i n g a n g l e s have b e e n d e f i n e d as: 2 . 1 . 2 3 . - N o t a t i o n U s e d for Del'ming Angles in t h e Solid L a t t i c e a=2~/n ~=2n/n 7=2~/n~

I

2

45

w h e r e ~ is t h e angle in t h e x - d i r e c t i o n , ~ is in t h e '~ d i r e c t i o n , etc. T h e i n t e r c e p t s are t h o s e of t h e lattice a n d n o t of t h e u n i t - c e l l .

Note t h a t c~ , ~,

Y = 9 0 o for t h e cubic lattice as given above as 2 . 1 . 1 .

This c o m p l e t e s o u r d i s c u s s i o n of t h e b a s i s a n d f a c t o r s d e v e l o p e d b y p a s t i n v e s t i g a t o r s to d e s c r i b e a n d c o n c e p t u l i z e t h e s t r u c t u r e of solids. You will n o t e t h a t we have n o t yet fully d e s c r i b e d t h e s y m m e t r y factor of solids. The r e a s o n for t h i s is t h a t we u s e s y m m e t r y f a c t o r s to c h a r a c t e r i z e s o l i d structure

without

resorting

to

the

theoretical

basis

of

structure

d e t e r m i n a t i o n . T h a t is, we h a v e a s t a n d a r d m e t h o d for c a t e g o r i z i n g s o l i d s t r u c t u r e s . We say t h a t salt, NaC1, is cubic. T h a t is, t h e Na § ion a n d t h e CI ion are a l t e r n a t e l y a r r a n g e d in a c l o s e - p a c k e d c u b i c s t r u c t u r e . T h e n e x t section now investigates these structure protocols. 2 . 2 . - Solid S t a t e S t r u c t u r e C o n v e n t i o n s a n d P r o t o c o l s We h a v e a l r e a d y d i s u s s e d s t r u c t u r e f a c t o r s a n d s y n m l e t r y as t h e y r e l a t e to t h e p r o b l e m of d e f i n i n g a c u b i c u n i t - c e l l a n d find t h a t still a n o t h e r f a c t o r exists if o n e is to c o m p l e t e l y define c r y s t a l s t r u c t u r e of solids.

This turns

o u t to be t h a t of t h e i n d i v i d u a l a r r a n g e m e n t of a t o m s w i t h i n t h e u n i t - c e l l . This t h e n gives u s a total of t h r e e (3) f a c t o r s are n e e d e d to define a g i v e n lattice. T h e s e c a n be s t i p u l a t e d as follows: 2 . 2 . 1 . - F a c t o r s N e e d e d to Fully Define a Crystal L a t t i c e

I

-

unit-cell a x e s , intercepts and angles

II - r o t a t i o n a l s y m m e t r y III - localized s p a c e g r o u p s y m m e t r y In t h e first F a c t o r , if t h e cell l e n g t h s are n o t equal, i.e. a ,

b

,

c, w e

no l o n g e r have a c u b i c lattice, b u t s o m e o t h e r type. A n d if t h e a n g l e s a r e no l o n g e r 9 0 ~ i.e.- ~ , b , g , t h e n w e have a c h a n g e in t h e lattice t y p e as well.

Additionally, we c a n have t h e s i t u a t i o n w h e r e r = ~ ,

and~,~,y,

or alternatively, a , ~

= Y, a = [3 ,

Y

anda,

y, ~,

~ = y

~,y.The

n u m b e r of c o m b i n a t i o n s t h a t w e c a n m a k e f r o m t h e s e 3 - l e n g t h s a n d 3angles is s e v e n (7) a n d t h e s e define t h e 7 u n i q u e lattice s t r u c t u r e s w h i c h

46

are called BRAVAIS LATFICES after the o r i g i n a t o r of the c o n c e p t . T h e s e have b e e n given specific n a m e s : 2.2.2.-

ORTHORHOMBIC

CUBIC HEXAGONAL

MONOCLINIC

T~GONAL

TRICLINIC

TRIGONAL We can t h e r e f o r e a r r a n g e the seven (7) s y s t e m s into a h i e r a r c h y b a s e d on s y m m e t r y , as s h o w n in the following d i a g r a m 2.2.3.-

IHierarchy of Crystal systems I

IHexag~

1 c~bic

j

!

!

IT~'g ~

I Orthorhomb, ci I M~176

I

I

ITriclinic I

I

Each System is a Special Case of Those Above It The lattice with the h i g h e s t s y m m e t r y is listed at the top, with the least s y m m e t r i c a l lattice b e i n g s h o w n at the b o t t o m .

E a c h of t h e s e

seven

lattices m a y have sublattices, the total being 14. The

following diagram, given on the next page as 2.2.4.,

shows

the

s y m m e t r y of t h e s e 14 Bravais space lattices in t e r m s of the seven crystal s y s t e m s given in 2.2.3. The specific r e s t r i c t i o n s are listed in Table 2-1 on the following page

along with

the

relation

between

axes a n d angles

47

48

a s s o c i a t e d w i t h e a c h s t r u c t u r e . C o n t r a s t b o t h t h e d a t a in T a b l e 2-1 a n d t h e d i a g r a m s in 2 . 2 . 4 .

in o r d e r

to get a c l e a r n o t i o n of t h e s y m m e t r y

f a c t o r s involved in t h e 14 Bravais l a t t i c e s .

TABLE 2- I T H E F O U R T E E N (14) BRAVAIS LATYICES IN T H R E E D I M E N S I O N S . N U M B E R OF

RESTRICTIONS

.LATTICES SYSTEM

ON UNIT C E I L

IN SYSTEM

Cubic

LATI'ICE SYMBOLS AXES AND ANGLES

P (primitive)

a=b=c =13

I(body-centered)

= "?' = 9 0 ~

F(face - c e n t e r e d ) Hexagonal

a=b~c (~ = ~ = ~ ((~ = ~ = 9 0 ~ (y = 1 2 0 o)

Tetragonal

a=

b r c =1~

Trigonal

R

=y=90

~

a=b=c (~ = ~ = 9 0 ~ 9 0 o > ~/ < 120 o

Orthorhombic

P

a~b,c

C

a = ~=~, = 9 0

~

I F Monoclinic Triclinic

P

as

C

a=

b~c 7 =90~

a,b#c

T h e f a c t o r s given in b o t h 2.2.4. a n d T a b l e 2-1 a r i s e d u e to t h e u n i t - c e l l axes, i n t e r c e p t s a n d

a n g l e s involved for a given c r y s t a l l a t t i c e s t r u c t u r e .

Also given a r e t h e l a t t i c e s y m b o l s w h i c h a r e g e n e r a l l y u s e d . T h e a x e s a n d a n g l e s given for e a c h s y s t e m a r e t h e r e s t r i c t i o n s on t h e u n i t cell to m a k e

49

it c o n f o r m to the p a r t i c u l a r type of lattice. T h e y c o m p l e t e l y define t h a t s t r u c t u r e except for the s y m m e t r y . These

14 Bravais Lattices are u n i q u e in t h e m s e l v e s .

If we a r r a n g e t h e

crystal s y s t e m s in t e r m s of s y m m e t r y , the c u b e h a s the h i g h e s t s y r m n e t r y a n d the triclinic lattice, the lowest s y m m e t r y , as we s h o w e d above. T h e s a m e h i e r a r c h y is m a i n t a i n e d in 2.2.4. as in Table 2-1. The s y m b o l s u s e d by c o n v e n t i o n in 2.2.4. to d e n o t e the type of lattice p r e s e n t are P

=

Primitive

I = I n v e r s i o n (or b o d y - c e n t e r e d ) F = Face-centered What tl~As m e a n s is t h a t the primitive lattice is c o m p o s e d of p o i n t s at t h e c o r n e r s of t h e lattice, w h e r e a s the i n v e r s i o n lattice h a s an additional p o i n t at the c e n t e r

of the

lattice,

i.e.-

"body-centered".

Face-centered

has

p o i n t s in the m i d d l e of e a c h face of the lattice in addition to t h o s e at t h e c o r n e r s of the lattice. If we n o w apply r o t a t i o ~ f l s y m m e t r y (Factor II given in 2.2.1) to the 14 Bravais lattices, we obtain the 32 P o i n t - G r o u p s w h i c h have the factor of s y m m e t r y i m p o s e d u p o n the 14 Bravais lattices. The s y m m e t r y e l e m e n t s t h a t have b e e n u s e d are: 2.2.5.-

Rotation axes Plane s y m m e t r y

< horizontal

vertical

Inversion s y m m e t r y ( m i r r o r ) We have already i l l u s t r a t e d r o t a t i o n axes in 2.1.6. Plane s y m m e t r y involves s y m m e t r y s u c h as t h a t of the h e x a g o n a l faces given above in 2.2.5. We will now e x a m i n e i n v e r s i o n or m i r r o r s y m m e t r y . One type m i r r o r s y m m e t r y is s h o w n in the following diagram, given as 2.2.6. on the n e x t page. If o n e h a s a s t r u c t u r e w h e r e t h e a t o m s are a r r a n g e d on the {001} p l a n e b u t are i n v e r t e d as s h o w n in the diagram, it is easy to see t h a t a m i r r o r p l a n e exists b e t w e e n t h e m . However, the a r r a n g e m e n t as s h o w n is

50

2.2.6.-

llnversionMirror Symmetry I

inverted. It is this k i n d of r o t a t i o n a n d

inversion t h a t will be s e e n to be

vital in fully d e s c r i b i n g the s t r u c t u r e of solids. We have already d e s c r i b e d

crystal s t r u c t u r e s in t e r m s

of p r o p a g a t i o n

units, i.e.- t r a n s l a t i o n , in w h i c h the crystal is c o m p o s e d of a small n u m b e r of individual a t o m s a r r a n g e d in a u n i t w h i c h is p r o l i f e r a t e d to form t h e solid. E a c h a s s e m b l y of atoms, i.e.- p r o p a g a t i o n units, is r e l a t e d to any o t h e r by one of four s i m p l e o p e r a t i o n s . T h e s e are: 2.2.7.- The Four Symmetry_ O p e r a t i o n s T r a n s l a t i o n , i.e.- from one plane to a n o t h e r Rotation about an axis of the crystal Reflection a c r o s s a p l a n e Inversion t h r o u g h a p o i n t T h e s e are the four m a i n o p e r a t i o n s r e q u i r e d to define the s y m m e t r y of a crystal s t r u c t u r e . The m o s t i m p o r t a n t is t h a t of t r a n s l a t i o n since each of the o t h e r p r o c e d u r e s ,

called s y m m e t r y o p e r a t i o n s , m u s t be c o n s i s t a n t

with the t r a n s l a t i o n o p e r a t i o n in the crystal s t r u c t u r e . T h u s , the r o t a t i o n o p e r a t i o n m u s t be t h r o u g h an angle of 2n / n, w h e r e n = 1, 2, 3, 4 or 6.

51

Note t h a t we have a l r e a d y s h o w n t h a t the cubic lattice c a n be r o t a t e d only in this m a n n e r , e x c e p t i n g six. We find t h a t t h e n u m b e r 6 c o m e s from t h e h e x a g o n a l lattice itself w h e r e the a r r a n g e m e n t of a t o m s is obvious (see 2.2.4.). E a c h of the o p e r a t i o n s involving crystal a s s e m b l y u n i t s c r e a t e s a special g r o u p : T r a n s l a t i o n a l symanetry o p e r a t i o n s g e n e r a t e the 14 Bravais lattices. The rotational o p e r a t i o n s g e n e r a t e a total of 32 Point G r o u p s d e r i v e d f r o m t h e s e s y m m e t r y o p e r a t i o n s on the 14 Bravais lattices. The reflections and inversions s y m m e t r y o p e r a t i o n s g e n e r a t e

a total of

231 groups, called Space Groups w h i c h i n c l u d e the 32 Point Groups. In

Point

Groups,

one

point

of the

lattice

remains

invarient

under

s y m m e t r y o p e r a t i o n s , i.e.- t h e r e is no t r a n s l a t i o n involved. S p a c e G r o u p s are s o - n a m e d b e c a u s e in e a c h g r o u p all t h r e e - d i m e n s i o n a l s p a c e r e m a i n s i n v a r i e n t u n d e r o p e r a t i o n s of the group. T h a t is, t h e y c o n t a i n t r a n s l a t i o n c o m p o n e n t s as well as the t h r e e s y m m e t r y o p e r a t i o n s . "We will not d w e l l u p o n t h e 231 S p a c e G r o u p s since t h e s e relate to d e t e r m i n i n g the e x a c t s t r u c t u r e of the solid. However, we will s h o w h o w the 32 Point G r o u p s relate to crystal s t r u c t u r e of solids. In 1890, Schoenflies f o r m u l a t e d a g r o u p of symbols d e s c r i b i n g the various r o t a t i o n s possible for Point Groups. T h e s e w e r e r e p l a c e d in 1936 by t h e International group

S y m b o l s or H e r m a n n - M a u g u i n

s y s t e m could

not

be

extended

to

Symbols, describe

since Space

the

point

Groups.

A

c o m p a r i s o n of the Point G r o u p Symbols are given in the following Table, given on the

next

page

as Table

2.2.

The

arrangement

shown

for

Schoenflies symbols in 2.2.8 on t h e n e x t page is inverted. It is this k i n d of r o t a t i o n a n d inversion t h a t will be s e e n to be vital in i~dlly d e s c r i b i n g t h e s t r u c t u r e of solids. S u b s c r i p t s in Table 2-2 are u s e d to i n d i c a t e the d e g r e e of the r o t a t i o n a l axes p r e s e n t , i.e.- C3 -- a three(3) fold r o t a t i o n a l axis. Horizontal s y n m l e t r y is i n d i c a t e d by "h" while vertical s y m m e t r y = v.

52

Table 2.2. - C o m p a r i s o n of Schoenflies a n d I n t e r n a t i o n a l S y m b o l s

crystaJ S y s t e m Name Cubic

Geometry a=b=c

r

Hexagonal

Tetragonal

Trigonal

Point Group S y s t e m

=7=90~

Bravais i

International

T

23

P

0 Th

432

I

m3

F

43m

Oh

m3m

a=13*c

C6

6

D6

622

(~=13 = 9 0 ~ (7 = 120 o)

C2h

6/m

C6v

6ram

C3h D3h

6 6m2

D6h

6/m

C4

4

b, c a = ~ = y = 9 0 ~ D4

i

or62m

422

C4h C4v

4/m

$4

D2d

4 4m2

D4h

4/mmm

4mm or 4 2 m

C3

a=b=c

D3 a = 13 = 9 0 ~ 9 0 o > y < 1 2 0 ~ C3v

32 3m

&

C3i

3 & 3 21m

D3d Orthorhombic

a~b~c a=

Monoclinic

as

~ =

b*c

a=~ Triclinic

7

=90~

a~b~c

,a,*

~'7

=

D2 9 0 ~ C2v

mm2

C

D2h

mmm

F

C2

2

P

7 C3

in

C

222

C2h

2/m

C1

1

q

,

,,

.

.

.

.

.

1

i

Lattice

Schoenflies

Td

a=

,,

P

53

In the Schoenflies notation, the e l e m e n t s defined a s 2.2.8.-

C = r o t a t i o n axis only D = d i h e d r a l (rotation p l u s d i h e d r a l r o t a t i o n axes) I = inversion s y m m e t r y T --- t e t r a h e d r a l s y m m e t r y 0-

octahedral symmetry

In t h e H e r m a n n - M a u g u i n Symbols, the s a m e r o t a t i o n a l axes are i n d i c a t e d , plus any inversion s y r m n e t r y t h a t m a y be p r e s e n t . T h e n u m b e r s i n d i c a t e the n u m b e r of r o t a t i o n s p r e s e n t , present

a n d the

inversion

m shows that a mirror

symmetry

is i n d i c a t e d

s y m m e t r y is

by a b a r over t h e

,_,-.

n u m b e r , i.e.- 5 . At this point, you m a y find t h a t the s u b j e c t of syrranetry in a c r y s t a l s t r u c t u r e to be confusing. However, by s t u d y i n g the t e r m i n o l o g y carefully in Table 2-2, one c a n begin to sort o u t t h e various l a t t i c e s t r u c t u r e s a n d the s y m b o l s u s e d to d e l i n e a t e t h e m . All of the crystal s y s t e m s c a n be described

by u s e of e i t h e r

Schoenflies

or H e r m a n n - M a u g u i n

symbols,

c o u p l e d w i t h t h e u s e of the p r o p e r g e o m e t r i c a l symbols. In 1965, specific d i a g r a m s w e r e developed by W e i n r e i c h to illustrate t h e 32 p o i n t - g r o u p

symmetries.

Appropriate

Schoenflies symbols w e r e

also

given. T h e s e d r a w i n g s are given on the following page as 2.2.10. If t h e y are e x a m i n e d differences

closely,

it b e c o m e s

in s y m m e t r y o p e r a t i o n s

easier

to a s s e s s

involved

and compare

for e a c h

the

type of l a t t i c e

s t r u c t u r e . As an example, c o n s i d e r the following. In the C l h

point group, t h e r e is a one-fold rotation axis p l u s h o r i z o n t a l

syTmnetry (horizontal mirror). Note also the difference b e t w e e n C l h a n d C2h. It is easy to see t h a t t h e s e point g r o u p s are r e l a t e d to one a n o t h e r , b u t t h a t t h e y specify quite different r o t a t i o n a l s y m m e t r i e s . In c o n t r a s t , C2 has a 2-fold r o t a t i o n axis b u t no h o r i z o n t a l s y m m e t r y (but C2h

has horizontal

s y m m e t r y ) . C o n t r a s t the C2v point g r o u p w i t h vertical s y m m e t r y to the C2

54

55

a n d C2h point g r o u p s . The D2h point g r o u p h a s a two-fold r o t a t i o n axis, a horizontal m i r r o r , a n d 2 diad r o t a t i o n axes. C o m p a r e also the Cv a n d D3v p o i n t g r o u p s a n d the C3h a n d D3h g r o u p s . Now e x a m i n e the s y m m e t r y e l e m e n t s for the cubic lattice. It is easy to s e e t h a t the n u m b e r

of r o t a t i o n

synmletry elements

is quite high. This is t h e

elements,

plus horizontal

and vertical

r e a s o n w h y the

Cubic

S t r u c t u r e is p l a c e d at the top of 2.2.3. Even t h o u g h the lattice p o i n t s of 2 . 2 . 1 . are deceptively e l e m e n t s are n o t

simple

for the

cubic

structure,

the

symmetry

T h e r e is one o t h e r factor c o n t r i b u t i n g to the overall s y m m e t r i e s of t h e lattice s t r u c t u r e . This factor involves t h e local s y m m e t r y of the a t o m i c g r o u p s w h i c h actually form the s t r u c t u r e . E x a m p l e s are the "solid-state building blocks" given above, e.g.- the t e t r a h e d o n - the group, PO43" a n d the o c t a h e d r o n - like group, NbO6-. It is easy to see t h a t if a s t r u c t u r e is composed

of s u c h building blocks,

t h e y will i m p o s e

s y m m e t r y on the lattice, in addition present. The

result

is t h a t

to the o t h e r

F a c t o r III of 2 . 2 . 6 .

given

a local s t r u c t u r a l

symmetries

already

above i m p o s e s

further

s y m m e t r y r e s t r i c t i o n s on the 32 point g r o u p s a n d we obtain a total of 2 3 1 space groups. We do not i n t e n d to delve f u r t h e r into this a s p e c t of l a t t i c e c o n t r i b u t i o n s to crystal s t r u c t u r e of solids, a n d t h e factors w h i c h c a u s e t h e m to vary in form. It is sufficient to k n o w t h a t t h e y exist. Having covered the e s s e n t i a l p a r t s of lattice s t r u c t u r e , we will e l u c i d a t e how o n e goes about d e t e r m i n i n g the s t r u c t u r e for a given solid. 2-3: HOW TO DETERMINE T H E STRUCTURE OF COMPOUNDS We w a n t to review how one goes about actually d e t e r m i n i n g t h e s t r u c t u r e of a given solid. T h e r e are two factors we n e e d to c o n s i d e r : 1) w h a t are the s t e p s r e q u i r e d in actually d e t e r m i n i n g a s t r u c t u r e ? 2) w h a t k i n d of i n f o r m a t i o n is o b t a i n e d ?

56

Since the c r y s t a l is a p e r i o d i c

a r r a y of a t o m s in w h i c h the i n t e r a t o m i c

d i s t a n c e s a n d i n t e r p l a n a r s p a c i n g s are of t h e s a m e o r d e r of m a g n i t u d e as t h e w a v e l e n g t h s of t h e readily available x-rays, e.g.- t h e K a r a d i a t i o n s , w e can use a target elements.

This

of t h e gives

x-ray tube composed

rise

to

charactersitic

of o n e

of t h e

monochromatic

heavier radiation

d e f i n e d by: 2.3.1.-

Mo = 0 . 7 1 1 fi,- Cu = 1 . 5 4 1 8 ft,- a n d Cr = 2 . 2 9 1 ft,.

A crystal therefore

acts as a t h r e e - d i m e n s i o n a l

diffraction

g r a t i n g for

t h e s e x-rays, a n d t h r e e e q u a t i o n s (the Laue e q u a t i o n s ) m u s t be satisfied if t h e r e is to be c o n s t r u c t i v e i n t e r f e r e n c e of t h e s e m o n o c h r o m a t i c x-rays. T h e Laue e q u a t i o n s to be e m p l o y e d are: 2.3.2.-

a(a-cto ) = h k

b (~-fio) = k k

c (7- ~/o ) = 1 k

w h e r e a, b a n d c are the r e p e a t d i s t a n c e s of t h e lattice, a a n d a o (etc.), are

the

direction

cosines

for

the

diffracted

and

incident

beams

r e s p e c t i v e l y , a n d h, k, a n d 1 are i n t e g e r s defining t h e o r d e r of t h e p a r t i c u l a r diffracted b e a m . k is, obviously, t h e w a v e l e n g t h b e i n g d i f f r a c t e d . It w a s W. L. Bragg w h o s h o w e d t h a t the above e q u a t i o n s are e q u i v a l e n t to t h e c o n d i t i o n for reflection of t h e x - r a y s b y the p l a n e w i t h i n d i c e s h k l , namely: 2.3.3.-

n k

= 2 d sin Ohkl

H e r e , 0 is the angle b e t w e e n plane,

hkl. Also, d is the

the i n c i d e n t

perpendicular

p l a n e s . If we w i s h to d e t e r m i n e

(or reflected) distance

b e a m for t h e

between

successive

t h e exact s t r u c t u r e of a n y given solid, w e

n e e d to follow the following s t e p s : I. Obtain a suitable x - r a y material. 2. D e t e r m i n e

pattern

t h e exact i n t e n s i t i e s

u s i n g a single

crystal of t h e

of the lines or p o i n t s

p a t t e r n by i n t e g r a t i o n of t h e diffracted e n e r g y .

in t h e

57

3. Calculate the values of d, the d i s t a n c e b e t w e e n a d j a c e n t p l a n e s in the crystal lattice by u s i n g the Bragg E q u a t i o n . 4 . D e t e r m i n e t h e s p a c i n g s of the lines in the p a t t e r n , a n d from t h e s e the d i m e n s i o n s of t h e unit-cell. 5. Scale o u r c a l u l a t e d i n t e n s i t i e s to o u r o b s e r v e d i n t e n s i t i e s by: E Ic = E Io a n d calculate a reliability factor, called R, from: R = E (Io - Ic ) /Z Io 6. D e t e r m i n e t h e s t r u c t u r e factor, F ( h k l ) 2 for the p a t t e r n . 7. Refine the s t r u c t u r e factor until it a p p r o a c h e s zero. Nowadays, m o s t of t h e s e s t e p s are done

using a Computer.

e n e r g y diffracted by {hkl} p l a n e s is p r o p o r t i o n a l

The

total

to the s q u a r e of t h e

s t r u c t u r e factor, viz2.3.4.-

E (hkl)

= F (hkl) 2

The s t r u c t u r e factor itself is e x p r e s s e d as the s u m of e n e r g y diffracted, over one unit-cell, of the individual s c a t t e r i n g factors, fl, for a t o m s l o c a t e d at x, y a n d z. Having done this, we c a n t h e n identify t h e exact locations of the

a t o m s (ions) w i t h i n

t h e unit-cell,

its p o i n t - g r o u p

symmetry,

and

crystal s y s t e m . This t h e n c o m p l e t e s o u r p i c t u r e of t h e s t r u c t u r e of t h e material. However, one is usually m o r e i n t e r e s t e d in identifying the n a t u r e of t h e c o m p o u n d by x - r a y analysis a n d does not usually w i s h to identify t h e s t r u c t u r e in detail. To do this, we p r o c e e d as follows. First of all, we s t a r t w i t h La~O3 and/~d203. After fh-ing a 1:11 m i x t u r e of the two, we u s e an xray diffraction i n s t r u m e n t to o b t a i n a series of diffraction lines as d e p i c t e d in t h e following diagram, given as 2.3.5. On the n e x t page. Because of the physical g e o m e t r y of the x - r a y diffraction g o n i o m e t e r ( a n g l e - m e a s u r i n g device), one o b t a i n s values of 20 directly. T h e i n t e n s i t i e s are r e a d from t h e diffraction c h a r t , scaled to the m o s t i n t e n s e

58

2.3.5.-

IA Typical X-ray Diffraction Pattern Obtained From a Diffra~,.tometer]

0

I0

20

30

40 50 :28 Degrees

60

70

80

90 100

line. If we do not k n o w w h a t the n a t u r e of the m a t e r i a l is, t h e n we c a n use the o b t a i n e d p a t t e r n to d e t e r m i n e its p r o b a b l e c o m p o s i t i o n . U s i n g the s e t of diffraction lines, we pick the t h r e e m o s t i n t e n s e l i n e s in t h e p a t t e r n . By r e f e r r i n g

to the

"POWDER DIFFRACTION FILE", p u b l i s h e d

AMERICAN SOCIETY FOR TESTING AND MATERIALS, 1916 Philadelphia,

Penna.

19103,

we

can

look

up

the

most

by t h e

Race St., probable

c o m p o s i t i o n . Usually, we will k n o w how the m a t e r i a l w a s m a d e a n d t h e c o m p o n e n t s u s e d to m a k e it. If not, we c a n analyze for c o n s t i t u e n t s . In this case, we w o u l d find La a n d AI, a n d w o u l d s u r m i z e t h a t we have an oxidic c o m p o u n d , since we m a d e the c o m p o u n d by firing the two o x i d e s together. T h e r e are two c o m p o u n d s possible, n a m e l y - LaA103 a n d ~ 1 1 0 1 8 . (How do we know?.- Consult the v o l u m e s p u b l i s h e d by the A m e r i c a n C e r a m i c Society e n t i t l e d "Phase D i a g r a m s for Ceramicists" for La203 a n d A1203 w h i c h s h o w t h a t only two c o m p o u n d s form w h e n t h e s e two oxides are reacted). In the Diffraction File, the 3 mc~'t i n t ~ m ~ l i n e s of ~ O 3

are d = 2 . 6 6 A

(I00), 3 . 8 0 A (80) a n d 2.19 A (80). w e fred t h a t this set of lines does n o t fit o u r p a t t e r n . We do Find t h a t the p a t t e r n given for L a ~ I I O I 8 is i d e n t i c a l to the

x-ray p a t t e r n

we

obtained.

Thus,

we

have

characterized

our

material. Then, the {h,k,l} values c a n be calculated from special f o r m u l a s developed for this p u r p o s e . T h e s e are given in the following Table 2-3. T h e s e e q u a t i o n s allow us to calculate {h,k,l} values, once we k n o w the d

59

values a n d s t r u c t u r e ,

as c a l c u l a t e d f r o m t h e 20 v a l u e s f o u n d f r o m

the

diffractometer, using the Bragg equation. T a b l e 2- 3 p l a n e S p a c i n g s for v a r i o u s L a t t i c e G e o m e t r i e s

CUBIC

TETRAGONAL

I/d 2 = h 2 + k 2 + 12 / a 2 I / d 2 = h 2 + k 2 / a 2 + 12 / c 2 tIE~_4,GONAL ' ORTI-IORIIOMBIC I / d 2 = 4 / 3 [(h 2 + k 2 + 1 2 ) / a 2] + 1 2 / c 2 I / d 2 = (h 2 / a 2) + ( k 2 / b 2) +(12/c 2) RIIOMBOIiEDRAL iltllll

iiiii

1 / d 2 = ( h 2 + k 2 + 1 2 l s i n 2 a + 2 ( h k + k l + h l ) ( c o s 2 ~ c o s a ) / c x 2 (1- 3 cos 2 a + 2 cos 3 a) MONOCLINIC ............... ,,,

i.,,1

1 / d 2 = { 1 / s i n 2 p}{ h 2 / a 2 + (k 2 s i n 2~) / [ b 2 + 1 2 / c 2 - 2 h l c o s ~ / a c ] TRICLINIC i / d 2 = I / V 2 { S I I h 2 + $22 k 2 + $33 12 + 2 S l 2 h k + 2 $ 2 3 k i + 2 S l 3 h l i

w h e r e : V = v o l u m e of u n i t cell, $22 = a 2 c 2 sin 2 ~,

$33

$23 = a 2 b c (cos ~ c o s y

S ll =

I|l

b 2 c 2 s i n 2(~ , S 2 2 = a 2 c 2

s i n 2 ~,

= a 2 b 2 sin2y, S12 = abc 2 (cos (~ cos ~ - cos y ), -cos

~)and

S13

= ab 2 c ( c o s y c o s ~

For t h e h e x a g o n a l c o m p o s i t i o n given above, i.e.- t h e c o m p o u n d

- cos~)

LaA]llO18

,

t h e {hkl} v a l u e s w e r e o b t a i n e d b y t r y i n g c e r t a i n v a l u e s in t h e h e x a g o n a l f o r m u l a a n d s e e i n g if t h e r e s u l t s give valid n u m b e r s , c o n s i s t a n t w i t h t h e n u m b e r s u s e d . T h a t is, a trial a n d e r r o r m e t h o d w a s u s e d to o b t a i n t h e c o r r e c t r e s u l t s . For e x a m p l e , o n e ,]could s t a r t w i t h {100} a n d d e t e r m i n e w h a t value of d c o n f o r m s to t h i s p l a n e . T h e n , {200} arid {300} w o u l d b e u s e d , etc. If we do this, we o b t a i n t h e v a l u e s s h o w n in T a b l e 2 - 4 as: TABLE 2-4 C o n v e r s i o n of 2 0 V a l u e s of t h e Diffraction 2O I / Io d {hld} 2 0 8.02 16 11.02 002 36.18 16.1 4.81 004 39.39 18.'86 32 4.71 001 40.94 20.12 28 4.41 012 42.79 22.07 10 4.03 45.01 013 24.23 21 3.67 001 53.36 24.55 11 3.63 014 58.57 32.19 44 2.78 I I0 60.07 32.50 15 2.76 008 67.35 33.23 15 2.70 112 71.61 34.02 100 2.64 017 95.42 .

.

.

.

P a t t e r n to {hkl} V a l u e s I / Io d {hkl} 74 2.48 I 14 2.29 023 27 2.20 0010 29 2. i'1 025 66 026 2.61 46 15 1.72 029 36 1.58 127 1.54 60 0211 1.39 48 220 1.32 0214 18 1.04 2214 14 .

.

60

This allows us to d e t e r m i n e the u n i t cell l e n g t h s for o u r c o m p o u n d as: Hexagonal: Additionally, we

ao = 5 . 5 6 . / k -

c a n list the

bo = 2 2 . 0 4 A

x-ray p a r a m e t e r s

and convert

them

to

s t r u c t u r a l factors as shown: T h e s e values are averaged over all of t h e reflections u s e d for calculation. Note t h a t this p a t t e r n h a s several p l a n e s w h e r e the "d" value is m o r e t h a n ten. Let u s c o n s i d e r one o t h e r example. S u p p o s e we o b t a i n e d the following set of 20 v a l u e s a n d intensities for a c o m p o u n d . These are given in the following Table 2-5: TABLE 2- 5

I/Io 96 100 56 50 14 18 14

DIFFRACTION LINES AND INTENSITIES OBTAINED 2 0 in degrees I/Io 2 0 in degrees 29.09 I0 90.55 48.40 17 97.80 48.40 19 118.26 57.48 11 121.00 60.28 7 144.34 78.38 9 148.49 80.78

In looking over the Table, we Fmd t h a t the first t h r e e

values are t h e

s t r o n g e s t diffraction lines. After calculating "d" values a n d looking u p t h e set of s t r o n g lines w h i c h c o r r e s p o n d to o u r set, we find t h a t the p r o b a b l e c o m p o u n d is CdO2, or c a d m i u m peroxide. This c o m p o u n d t u r n s o u t to be cubic in s t r u c t u r e , with ao

= 5.313 A .

W h e n we calculate the {h,k,1}

values of the diffracting planes, the s t r o n g e s t line is f o u n d to be {200}. We c a n t h e n

make

the

determination

that

since

Cd 2+ is a s t r o n g l y

diffracting a t o m (it h a s high atomic weight, w h i c h is one way of s t a t i n g t h a t it h a s m a n y e l e c t r o n

shells, i . e . - l s 2 2s 2 2 p 6 3s 2 3 p 6 3 d l0 4 s 2 4p6

4dlO), the s t r u c t u r e is p r o b a b l y f a c e - c e n t e r e d

cubic. Indeed,

this t u r n s

o u t to be the case. In the u n i t cell, Cd a t o m s are in the special p o s i t i o n s of:

{0,0,0},

{1/2,1/2,1/2};

0,1/2.1/2};

{1/2,1/2,0}.

There

are

four

m o l e c u l e s per u n i t cell. We could c o n t i n u e f u r t h e r so as to calculate intensities, I c , u s i n g atomic s c a t t e r i n g factors already p r e s e n t in p r i o r

61

l i t e r a t u r e . We w o u l d scale o u r c a l u l a t e d i n t e n s i t i e s to o u r o b s e r v e d i n t e n s i t i e s by X Ic = X I o . We t h e n calculate a reliability factor, called R, from R = X (Io - Ic ) /Y Io .

A low value i n d i c a t e s t h a t o u r selection of

lattice p a r a m e t e r w a s correct. K not, we c h o o s e a slightly different value a n d apply it. The details of t h e p r o c e d u r e for d e t e r m i n i n g exact s t r u c t u r e a n d atomic p o s i t i o n s in the lattice are well k n o w n , b u t are b e y o n d t h e scope of this C h a p t e r . 2.4, - SYMMETRY DISTRIBUTION OF CRYSTALS I n o r g a n i c m a t e r i a l s do n o t often p o s s e s s large unit-cells b e c a u s e t h e i r c h e m i c a l f o r m u l a e are m u c h s i m p l e r t h a n organic m o l e c u l e s .

Unit-cell

v o l u m e s for organic c r y s t a l s c a n be as large as 1 0 , 0 0 0 ,~3 a n d p r o t e i n s t e n d to be at least 10 to 100 t i m e s larger t h a n that. Most i n o r g a n i c cell v o l u m e s are in the r a n g e of 1 0 - 1 0 0 0 A 3 b u t SiC c a n have cell v o l u m e s up to 1500 A 3. C e r t a i n v i r u s e s (which are close to p r o t e i n s in c o m p o s i t i o n ) c a n have cell dimensions

u p to 3 6 0 0 A 3 a n d cell v o l u m e s of 46 billion ,Aft. P r o t e i n s

(which are actually p o l y p e p t i d e c h a i n s c o n s i s t i n g of a m i n o acids l i n k e d by p e p t i d e b o n d s ) c a n have m o l e c u l a r w e i g h t s in the millions. T h e h u m a n protein, h e m o g l o b u l i n , side-chains

called

is c o m p o s e d

"alpha"

and

two

of four p o l y p e p t i d e s , differing

side

two i d e n t i c a l

chains

of

similar

c o m p o s i t i o n called "beta". E a c h a l p h a c h a i n h a s 141 a m i n o acids a n d e a c h b e t a c h a i n h a s 146 a m i n o acids, giving a total m o l e c u l a r w e i g h t of about 64,500.

It is one of t h e s i m p l e r biological m o l e c u l e s . W h e n c r y s t a l l i n e

m a t e r i a l s are s u r v e y e d as to tile type of s y m m e t r y t h e y exhibit in the solid state, we find t h a t only t h o s e c o m p o u n d s w i t h s i m p l e c o m p o s i t i o n s have high sylmnekry, i.e.- t h e y crystallize in the cubic form. T h o s e t h a t do not, i n c l u d i n g m o s t organics, t e n d to form c r y s t a l s of low s y m m e t r y as s h o w n in the following table, given on t h e n e x t page. The p e r c e n t a g e s are b a s e d u p o n 5 5 7 2 inorganic, 3 2 1 7 organic a n d 2 2 4 p r o t e i n c o m p o u n d s . T w o - t h i r d s (66%) of the i n o r g a n i c c o m p o u n d s have s y m m e t r i e s h i g h e r t h a n o r t h o r h o m b i c w h e r e a s 85% of t h e o r g a n i c s do not.

62

Table 2-6 Distribution of Crystalline Materials Among the Seven Crystal Systems* System Inorganic Organic Proteins Triclinic 2% 6% 2% Monoclinic 14 49 35 O r t h o r h o m b i c 18 30 43 Trigonal 12 4 6 Tetragonal 14 6 6 Hexagonal 11 2 2 Cubic 30 4 5 * This data was taken from "Crystal Data"- Am. Crystallographic Assoc. Monograph - W. Nowacki- Ed. (1967) For inorganics, the leading space groups are: Fm3m, Pnma, P 6 3 / m m c , Fd3m, P 21/c, Pm3m, C2/c and R3m. Since complicated formulae tend to produce low symmetry, m a n y of the organic crystals appear in the lowest symmetry catagories like monclinic or triclinic. About 80% of inorganic and 60% of organic c o m p o u n d s t r u c t u r e s are centric, that is- they have a center of symmetry. Most protein s t r u c t u r e s are not centric since nearly all living systems, including humans, have a "handedness". The h a n d e d n e s s arises from the replication process in which helices (e.g.- proteins, RNA and DNA) wind and unwind, t h e r e b y t r a n s m i t t i n g genetic code. Helices possess a screw-axis s y m m e t r y so that most crystals of biological origin belong to space groups having s c r e w axes. Among the proteins, the most populous space groups are: P 21 21 21 (34%), P2 i,(23%), C2 (11%), and P 21 21 2 (7%). In biological molecules, inversion symmetry and mirror planes are virtually non-existant. Screw axes are also common among crystals of the simpler organic compounds. But, m a n y of these have mirror and inversion s y m m e t r y as well. The most c o m m o n space groups for organic c o m p o u n d s are: P 21/c (26%), P 21 21 21

(13%), P 21 (8%), and C2/c (7%).

Most of the differences between inorganic, organic and protein crystals lies in the fact that ionic forces (found mostly in inorganic s t r u c t u r e s ) depend upon interatomic distances but not on angle. In contrast, covalent bonds (such as those which p r e d o m i n a t e in organic and p r o t e i n

63

compounds) depend

upon

angle a n d

therefore

the crystal(s) is usually

acentric, i.e.- w i t h o u t a c e n t e r of syrranetry. The h u m a n b o d y is c o m p o s e d

of e u k a r y o t i c cells. S u c h cells have an

a s s o r t m e n t of i n t r a c e l l u l a r m e m b r a n e - b o u n d

organeUes w h i c h carry o u t

the f u n c t i o n s of t h a t p a r t i c u l a r cell. E x a m p l e s rrdght be liver cells vs: b r a i n cells. Every cell in the h u m a n b o d y c o n t a i n s c h r o m o s o m e s . A m o n g the m a n y o r g a n i s m s t h a t have s e p a r a t e sexes, t h e r e are two basic types of c h r o m o s o m e s : a u t o s o m e s a n d sex c h r o m o s o m e s . A u t o s o m e s control t h e i n h e r i t a n c e of all the c h a r a c t e r i s t i c s a n d f u n c t i o n s of the cell except t h e s e x - l i n k e d ones, w h i c h are c o n t r o l l e d by the sex c h r o m o s o m e s . H u m a n s have 22 pairs of a u t o s o m e s a n d one pair of sex c h r o m o s o m e s . living cell to p e r s i s t

from one g e n e r a t i o n

to a n o t h e r ,

For any

it m u s t

store

i n f o r m a t i o n a n d u s e it to p a s s to the next g e n e r a t i o n of cells. AU cells have

solved

this

problem

by

coding

information

in

certain

large

molecules, i.e.- deoxyribonuclecic acid= DNA) a n d ribonucleic acid= RNA. The c o m p o s i t i o n s of DNA a n d RNA have b e e n s h o w n to c o n s i s t solely of the 4 nucleic acids, Adenine (A), T h y m i n e (T), G u a n i n e (G) a n d Cytosine (C). The overall code of the c h r o m o s o m e s is called hhe " h u m a n g e n o m e " . In 1953, Crick a n d W a t s o n solved the s t r u c t u r e of DNA. With the help of Rosalin Franklin, a co-worker, (G+C a n d A+T) f o r m e d

they s h o w e d t h a t p a i r s of nucleic acids

a spiral m o l e c u l e

of high m o l e c u l a r weight in

w h i c h a double s t r a n d was t w i s t e d into a helix. In 1990, w o r k w a s s t a r t e d to c h a r a c t e r i z e the h u m a n g e n o m e w h i c h h a d b e e n s h o w n to c o n s i s t of about 3 billion b a s e pairs. The final r e s u l t was a n n o u n c e d in the year 2000. All of the c h r o m o s o m e s have b e e n c h a r a c t e r i z e d . The h u m a n genorne h a s b e e n s h o w n to c o n t a i n s o m e 3 0 , 0 0 0 g e n e s (which are s e c t i o n s of t h e c h r o m o s o m e w h i c h code for specific proteins). E a c h cell p r o d u c e s the type of p r o t e i n s n e e d e d for it to function. The function of mRNA is to t r a n s f e r i n f o r m a t i o n from t h e DNA, so as is to fix the Emits of the p r o t e i n n e e d e d . The vast m a j o r i t y of the p r o t e i n s f o u n d in living o r g a n i s m s are c o m p o s e d of only- 20 d i f f e r e n t k i n d s of a m i n o acids, r e p e a t e d m a n y t i m e s a n d s t r u n g t o g e t h e r in a p a r t i c u l a r order.

E a c h type of p r o t e i n h a s its o w n u n i q u e s e q u e n c e of

a m i n o acids. This s e q u e n c e ,

k n o w n as its p r i m a r y s t r u c t u r e ,

actually

64

determines

the s h a p e a n d f u n c t i o n of the protein.

The s e c o n d a r y a n d

t e r t i a r y s t r u c t u r e refers to the looping a n d folding of the p r o t e i n c h a i n b a c k u p o n itself.

This

overall

form

determines

the

actual

chemical

function of the individual protein. S o m e p r o t e i n s are e n z y m e s as well.

Although t h e r e are b u t 3 0 , 0 0 0 different g e n e s in the h u m a n body, it has b e e n s h o w n t h a t as m a n y as 2 0 0 , 0 0 0 p r o t e i n s are p r o d u c e d for u s e in t h e individual h u m a n cells. T h u s , it is clear t h a t the d a t a given in T a b l e 2 - 6 m a y be r e g a r d e d as p r e l i m i n a r y . Time will s h o w w h e t h e r the o b s e r v a t i o n s m a d e in Table 2-6 r e g a r d i n g p r o t e i n s t r u c t u r e are c o r r e c t .

2.5.- P h a s e R e l a t i o n s h i p s A m o n g Two .or More Solids

Up to this point, we have c o n s i d e r e d only one solid at a time. H o w e v e r , w h e n two (2) or m o r e solids are p r e s e n t , t h e y can form quite c o m p l i c a t e d s y s t e m s w h i c h d e p e n d u p o n the n a t u r e of e a c h of the solids involved. T o differentiate

a n d to be able to d e t e r m i n e

the differences

between

the

p h a s e s t h a t m a y arise w h e n two c o m p o u n d s are p r e s e n t (or are m a d e to react together), we u s e w h a t are t e r m e d " p h a s e - d i a g r a m s " to illustrate t h e n a t u r e of the i n t e r a c t i o n s b e t w e e n two solid p h a s e c o m p o s i t i o n s . You will note t h a t s o m e of this m a t e r i a l w a s p r e s e n t e d earlier in C h a p t e r 1. It is presented

here

again to f u r t h e r

emphasize

the

importance

of p h a s e

diagrams.

C o n s i d e r the following. S u p p o s e we have two solids, "A" a n d "B". It d o e s not m a t t e r w h a t the exact c o m p o s i t i o n of e a c h m a y be. A will have a specifc m e l t i n g p o i n t (ff it is stable a n d does not d e c o m p o s e at the M.P.) a n d likewise for the c o m p o u n d , B. We f u r t h e r s u p p o s e t h a t the M.P. of B is h i g h e r t h a n t h a t of A. F u r t h e r m o r e , we s u p p o s e t h a t A a n d B form a solid s o l u t i o n at all v a r i a t i o n s of c o m p o s i t i o n . What this m e a n s is t h a t f r o m 100% A-0% B to 50% A-50% B to 0% A-100% B, the two solids dissolve in one a n o t h e r to form a c o m p l e t e l y h o m o g e n o u s single p h a s e . If we p l o t the M.P. of the s y s t e m , we find t h a t t h r e e different c u r v e s could result, as s h o w n in the following diagram, given as 2.5.1. on the next page.

65

In this case, the melting point of the ideal solid solution shoul d i n c r e a s e linearly as the ration of B/A increases. However, it usually does not. E i t h e r a negative deviation or a positive deviation is regularly observed.

In any p h a s e diagram, c o m p o s i t i o n is plotted against t e m p e r a t u r e . In this way, we cart see how the i n t e r a c t i o n s b e t w e e n p h a s e s change as t h e t e m p e r a t u r e changes and the behavior as each solid p h a s e t h e n m e l t s . Either two - p h as e or three p h a s e s y s t e m s can be illustrated. This is s h o w n in the following:

66

Here, the t w o - p h a s e d i a g r a m is simplified to s h o w a h y p o t h e t i c a l p h a s e involving "A" a n d "B" c o m p o u n d s w h i c h form a solid solution from 100% A to 100% B. The solid solution is labelled as "~". The m e l t i n g t e m p e r a t u r e of A is h i g h e r t h a n t h a t of B. T h e r e f o r e , t h e m e l t i n g t e m p e r a t u r e of r d r o p s as the c o m p o s i t i o n b e c o m e s r i c h e r in B. At specific t e m p e r a t u r e s on the d i a g r a m (see 1. & 2.), a t w o - p h a s e s y s t e m a p p e a r s , t h a t of a liquid plus t h a t of r Finally, as the t e m p e r a t u r e r i s e s , the m e l t is h o m o g e n o u s a n d the solid, r

h a s m e l t e d . In the t h r e e - p h a s e

s y s t e m , only the r e l a t i o n s h i p b e t w e e n A, B a n d C c a n be i l l u s t r a t e d on a t w o - d i m e n s i o n a l drawing. A t h r e e - d i m e n s i o n a l d i a g r a m w o u l d be r e q u i r e d to s h o w the effect of t e m p e r a t u r e as well. The p h a s e d i a g r a m s we have s h o w n are b a s e d u p o n the fact t h a t A a n d B form solid m u t u a l l y soluble solid s t a t e solutions. If t h e y do not, i.e.- t h e y are not m u t u a l l y soluble in the

solid

state,

then

the

phase

diagram

b e c o m e s m o r e c o m p l i c a t e d . As an example, c o n s i d e r the following, w h i c h is the c a s e of limited solid solubility b e t w e e n A a n d B:

Here, we have the c a s e w h e r e A & B form two slightly d i f f e r e n t c o m p o n d s , a a n d I~. T h e c o m p o s i t i o n of a is AxBy while t h a t of ~ is AuBv , w h e r e the s u b s c r i p t s i n d i c a t e the ratio of A to B (these n u m b e r s m a y be whole n u m b e r s or t h e y m a y be fractional n u m b e r s ) . At low B

67

c o n c e n t r a t i o n s , a e x i s t s as a solid (the left side of t h e d i a g r a m ) . As t h e t e m p e r a t u r e i n c r e a s e s , a m e l t is o b t a i n e d a n d a r e m a i n s as a solid in t h e m e l t (L) {This is i n d i c a t e d by t h e details given j u s t below t h e s t r a i g h t l i n e in t h e diagram}. As t h e t e m p e r a t u r e is i n c r e a s e d , t h e n ~ m e l t s to form a u n i f o r m liquid.

In t h e

middle

concentrations,

a

and

~ exist

as t w o

s e p a r a t e p h a s e s . At 75% A-25% B, ~ m e l t s to form a liquid p l u s solid c~ w h e r e a s j u s t t h e o p p o s i t e o c c u r s at 25% A a n d 75% B. W h e n A e q u a l s B, t h e n b o t h a a n d ~ m e l t t o g e t h e r to form t h e liquid m e l t , i.e.- t h e " e u t e c t i c point". In two s i m i l a r c a s e s b u t differing c a s e s , i.e.- a s y s t e m w i t h

a

m e l t i n g p o i n t m i n i m u m a n d a n o t h e r w i t h a different type of l i m i t e d solid solubility, the b e h a v i o r differs as s h o w n in t h e following d i a g r a m :

In t h e c a s e on t h e left, a c o m p o s i t i o n exists w h i c h in w h i c h a m i n i m u m m e l t i n g p o i n t exists. Again, A a n d B form a w h i c h partially m e l t s to f o r m + L. However, two forms of a also exist, a c o m p o s i t i o n a l a r e a k n o w n as a "miscibility gap", i.e.- " a l a n d a2". But at h i g h e r t e m p e r a t u r e s ,

b o t h of

t h e s e m e l d into t h e single p h a s e , a. Finally, we o b t a i n t h e m e l t + a. N o t e t h a t at about 80% A-20% B, a m e l t i n g p o i n t m i n i m u m

is s e e n w h e r e

m e l t s d i r e c t l y i n s t e a d of f o r m i n g t h e t w o - p h a s e s y s t e m , a + L.

68

In t h e c a s e of l i m i t e d solid solubility, t h e p h a s e b e h a v i o r b e c o m e s m o r e c o m p l i c a t e d . Here, b o t h r a n d ~ f o r m ( w h e r e t h e a c t u a l c o m p o s i t i o n of is AxBy w h i l e t h a t of ~ is AuBv , as given before. Note t h a t t h e v a l u e s of x a n d y c h a n g e , b u t t h a t we still h a v e t h e a p h a s e . T h e s a m e h o l d s for u a n d v of t h e ~ p h a s e ) . At low A c o n c e n t r a t i o n s ,

r e x i s t s alone w h i l e ~ e x i s t s

alone at h i g h e r B c o n c e n t r a t i o n s .

exists where

system,

+

A region

exists. We n o t e t h a t as t h e t e m p e r a t u r e

t h e two p h a s e

is r a i s e d , a two p h a s e

s y s t e m is also s e e n c o n s i s t i n g of t h e m e l t liquid p h u s a solid p h a s e . T h e p h a s e - b e h a v i o r s h o w n o n t h e r i g h t side of t h e d i a g r a m a r i s e s b e c a u s e t h e two p h a s e s , A a n d B, h a v e l i m i t e d solubility in e a c h o t h e r . Now, let u s c o n s i d e r t h e c a s e w h e r e t h r e e (3) s e p a r a t e p h a s e s a p p e a r in t h e p h a s e d i a g r a m . In t h i s c a s e , w e h a v e t h r e e (3) s e p a r a t e p h a s e s t h a t a p p e a r in t h e p h a s e d i a g r a m . T h e s e p h a s e s are r ~, and o, whose c o m p o s i t i o n s are: r = AxBy, ~ = AuBv a n d ~ = AcBd , r e s p e c t i v e l y ( t h e values of x, y, u, v, c, a n d d all differ f r o m e a c h o t h e r so t h a t AxBy specific c o m p o u n d as a r e t h e o t h e r s ) . T h i s s h o w n as follows:

is a

69

By studing this phase diagram carefully, you can see how the individual phases relate to each other. As you can see, a phase diagram can b e c o m e quite complicated. However, in most cases involving real compounds, t h e phase diagrams are usually simple. Those involving c o m p o u n d s like silicates can be complex, but those involving alloys of metals show simple behavior like limited solubility. Now, let us r e t u r n to a further discussion of the properties of solids. Summarizing to this point, we have shown that only certain propagation units can be stacked to infinity to form c l o s e - p a c k e d solids. We have also shown how the units fit together to form specific solids with specific symmetries. T h e structure of solids has also been reviewed in some detail, including differences in crystal s y m m e t r y of inorganic and organic compounds. WTlat we find is that the solid contains stacking defects. T h a t is, we find that we cannot form the solid without encountering some sort of defect as we stack molecules to near infinity to form our "perfect" solid. Obviously, this arises because of the application of the 2nd Law of T h e r m o d y n a m i c s which involves entropy. The next chapter analyzes t h e defect solid in some detail. SUGGESTED READING 1. "Solids-Elementary Theory for Advanced Students"- G. Weinreich, J. Wiley & Sons, Inc., New York (1965). 2. "Solid State Physics" - A.J. Dekker, Prentice-Hall, Cliffs, NJ (1958). (See Chaps. 1,2 &3 in particular).

Inc., Englewood

3. "Stereographic Projections of the Colored Crystallographic Point Groups", J.A. McMillan, Am. J. Phys., 35, 1049 (1967). 4. "Imperfections in Crystal", J.M. Honig, J. Chem. Ed., 34, 224 (1957). 5. "The Physical Chemistry of Solids", R.J. Borg and G.J. Dienes, Academic Press, NY (1992). 6. "Solid State Chemistry", Prentice-Hall ,N. B. Hannay, Englewood Cliffs, NJ (1967)

70

P r o b l e m s for C h a p t e r 2.. I. R e d r a w the Figure given in 2. I. 15. a n d 2. i. 18 so as d e t e r m i n e the e q u a t i o n for the {420} plane. 2. Define the synm~etry e l e m e n t s for:

C2h

C6v D3h D4 & D2d

3. Given the following "d-spacings" for the diffraction lines of a given cubic powder, calculate the {h,k,l} v a l u e s of the p l a n e spacings. Intensity

2 0 in Degrees

96

29.09

I00

33.70

56

48.4

50

57.48

18

78.38

19

118.26

4. Iron (Fe) h a s the b o d y - c e n t e r e d s t r u c t u r e at 298 ~

a n d a d e n s i t y of

7.86 g m with a lattice s p a c i n g of ao = 2.876 A. Calculate Avogaddro's n u m b e r from these data.

71

Chapter 3

II

Defects in Solids In the first c h a p t e r , we defined the n a t u r e of a solid in t e r m s

of its

building b l o c k s p l u s its s t r u c t u r e a n d s y m m e t r y . In the s e c o n d c h a p t e r , we defined how s t r u c t u r e s of solids are d e t e r m i n e d .

In this c h a p t e r , w e

will e x a m i n e how the solid actually o c c u r s in Nature. C o n s i d e r t h a t a solid is m a d e up of a t o m s or ions t h a t are h e l d t o g e t h e r

by c o v a l e n t / i o n i c

forces. It is axiomatic t h a t a t o m s c a n n o t be pried t o g e t h e r a n d forced to form a periodic s t r u c t u r e w i t h o u t m i s t a k e s b e i n g m a d e . The 2 n d Law of T h e r m o d y n a m i c s d e m a n d s this. S u c h m i s t a k e s seriously affect the overall p r o p e r t i e s of the solid. T h u s , defects in the lattice are p r o b a b l y the m o s t i m p o r t a n t a s p e c t of the solid s t a t e since it is i m p o s s i b l e to avoid d e f e c t s at t h e atomistic level. Two factors are involved: 1) T h e e n t r o p y effect 2) The p r e s e n c e of i m p u r i t i e s in any given solid. W h a t we m e a n by this is t h a t the c h e m i c a l p r o p e r t i e s of the solid are not d e t e r m i n e d solely by its s t r u c t u r e b u t also by the n a t u r e of its c h e m i c a l c o m p o s i t i o n a n d in p a r t i c u l a r b y any d e f e c t s t h a t m a y be p r e s e n t . We will find t h a t b o t h physical a n d c h e m i c a l

properties

of a solid are largely

d e t e r m i n e d by the type a n d n a t u r e of defects p r e s e n t . inorganic

It is axiomatic in

c h e m i s t r y that:

'~rhe p e r f e c t solid d o e s n o t e x i s t in Nature and its reactive p r o p e r t i e s are d e t e r m i n e d , to a great extent, by the d e f e c t s p r e s e n t in its structure". We have a l r e a d y s t a t e d t h a t s o m e defects are r e l a t e d to the entropy of the

solid, a n d t h a t a perfect solid w o u l d violate the s e c o n d law of t h e r m o d y n a m i c s . T h e 2 n d law s t a t e s t h a t zero entropy is only possible at absolute zero t e m p e r a t u r e . However, m o s t solids exist at t e m p e r a t u r e s far

are defect-solids. The defects are usually "point defects", w h i c h are a t o m i s t i c above absolute zero. T h u s , m o s t of the solids t h a t we

encounter

72

defects in the lattice s u c h as a "vacancy" w h e r e the lattice s t r u c t u r e l a c k s an a t o m (ion). E n t r o p y effects are generally a s s o c i a t e d with point defects in the lattice. However,

e n t r o p y c a n form "stacking defects"

due e i t h e r

to "slipped"

p l a n e s in the solid or to t h r e e - d i m e n s i o n a l "faults" t h a t o c c u r r e d in t h e s t a c k i n g p r o c e s s . . How is this possible? You m i g h t c o n s i d e r that, for any given solid, as the t e m p e r a t u r e

rises from absolute zero, m o s t of t h e

e n e r g y g a i n e d goes into v i l m ~ o m ~ l e n e r g y (recall w h a t we said in the first c h a p t e r about the differences b e t w e e n gases, liquids a n d solids in t e r m s of the t h r e e

degrees

of f r e e d o m of a molecule or a s s e m b l y of atoms).

I n c r e a s i n g the e n e r g y in a solid does not r e o r d e r the lattice a n d d e c r e a s e the i n h e r e n t point defects. (See A p p e n d i x III at the e n d of this c h a p t e r ) . For example, an a t o m of a gas h a s t h r e e d e g r e e s of f r e e d o m (the t h r e e spatial c o o r d i n a t e s of the atom) a n d will, therefore, have an average total energy

of 3 / 2 k T .

For an a t o m in a solid,

vibratory m o t i o n

involves

p o t e n t i a l e n e r g y as well as r e s t r i c t e d kinetic energy, a n d b o t h m o d e s will contribute a term

1 / 2 k T , r e s u l t i n g in an average total e n e r g y of 3 k T .

T h u s , it is the e n t r o p y of m i x i n g t h a t forces the c r e a t i o n of a c e r t a i n n u m b e r of v a c a n t lattice p o s i t i o n s above 0.0 ~ natural

result

of t h e r m o d y n a m i c

equilibrium

Hence, v a c a n c i e s are t h e and

not

the

result

of

a c c i d e n t a l g r o w t h or s a m p l e p r e p a r a t i o n . The s e c o n d significant factor arises from the fact t h a t no solid is e v e r 100% pure. It is this lack of p u r i t y w h i c h gives rise to lattice d e f e c t s . C o n s i d e r the fact t h a t

those

inorganic

compounds

which

h i g h e s t p u r i t y k n o w n to date are about 9 9 . 9 9 9 9 9 9 9 9 %

are of t h e

pure. T h a t is, t h e

p u r i t y is about 1 / 1 0 p a r t per billion, i.e.- 1 x 10 -10. But, the c o m p o u n d still c o n t a i n s about 1014 a t o m s per mole, i.e.- per 6 . 0 2 x 1023 a t o m s , i.e.in t e r m s of a t o m s actually p r e s e n t in a m o l e : Impurity = Solid

=

100,000,000,000,000 atoms

= 1.7 x 1 0 lO

602,000,000,000, 000,000,000,000 atoms

Additionally, ff t h e s e i m p u r i t y a t o m s are n o t of the s a m e valence as t h o s e

73

of the h o s t c o m p o u n d , t h e n v a c a n c i e s a n d o t h e r lattice defects will b e present. We will be c o n s i d e r i n g p r i m a r i l y inorganic solids b u t m u s t k e e p in m i n d t h a t t h e s a m e p r i n c i p l e s also apply to organic solids. T h e r e f o r e , we i n t e n d to e x a m i n e the n a t u r e of point defects in t e r m s of t h e i r t h e r m o d y n a m i c s , equilibria a n d the e n e r g y r e q u i r e d for t h e i r formation. It will be s e e n t h a t point

defects

follow the

same

physical c h e m i s t r y

laws t h a t

apply to

inorganic c o m p o u n d s a n d physical p r o p e r t i e s in g e n e r a l . Of necessity, we c a n n o t be exhaustive, a n d t h e r e are m a n y t r e a t i s e s w h i c h deal solely with the t h e r m o d y n a m i c s of the point d e f e c t . 3.1- THE D E F E C T SOLID We have s h o w n t h a t by s t a c k i n g a t o m s or p r o p a g a t i o n u n i t s t o g e t h e r ,

a

solid w i t h specific

a

s y m m e t r y results. If we have done this properly,

perfect solid s h o u l d r e s u l t w i t h no holes or defects in it. Yet, the 2 n d law of t h e r m o d y n a m i c s

demands

that

a certain

number

of point

defects

(vacancies) a p p e a r in the lattice. It is i m p o s s i b l e to o b t a i n a solid w i t h o u t s o m e sort of defects. A p e r f e c t solid w o u l d violate this law. The 2 n d law s t a t e s t h a t z e r o e n t r o p y is only possible at absolute zero t e m p e r a t u r e . Since m o s t solids exist at t e m p e r a t u r e s far from absolute zero, t h o s e t h a t we e n c o u n t e r a r e defect-solids.

It is n a t u r a l to a s k w h a t the n a t u r e of

t h e s e defects m i g h t be. C o n s i d e r the surface of a solid. In the interior, we see a c e r t a i n s y m m e t r y which

depends

u p o n the

surface from t h e interior,

structure

of t h e solid. As we a p p r o a c h

the

the s y n m l e t r y b e g i n s to c h a n g e . At the v e r y

surface, t h e s u r f a c e a t o m s see only half t h e s y m m e t r y t h a t t h e i n t e r i o r a t o m s do (and half of the b o n d i n g as well). R e a c t i o n s b e t w e e n solids t a k e place at the surface. T h u s , the surface of a solid r e p r e s e n t s

a defect in

itself since it is not like the i n t e r i o r of t h e solid. In a t h r e e - d i m e n s i o n a l three

major

types

of

solid, we c a n p o s t u l a t e t h a t t h e r e defects,

having

either

one-,

tvr

o u g h t to b e or

three-

74

dimensions. Indeed,

this is exactly t h e c a s e f o u n d for defects in solids.

Specific n a m e s have b e e n given to e a c h of t h e s e t h r e e t y p e s of d e f e c t s . Thus

a

one-dimensional

dimensional

defect

defect

is

a "line" or "edge"

called

a

defect

"point"

defect,

a

two-

and a three-dimensional

defect is called a "plane" or "volume" d e f e c t . Point

defects

are c h a n g e s

at a t o m i s t i c levels,

while

line

and volume

defects are c h a n g e s in s t a c k i n g of p l a n e s or g r o u p s of a t o m s ( m o l e c u l e s ) in t h e s t r u c t u r e . Note t h a t t h e a r r a n g e m e n t ( s t r u c t u r e ) of t h e individual a t o m s (ions) are not affected, only t h e m e t h o d

in w h i c h t h e s t r u c t u r e

u n i t s are a s s e m b l e d . Let us n o w e x a m i n e e a c h of t h e s e t h r e e

t y p e s of

defects in m o r e detail, s t a r t i n g w i t h t h e o n e - d i m e n s i o n a l lattice d e f e c t a n d t h e n w i t h t h e m u l t i - d i m e n s i o n a l defects. We will f'md t h a t s p e c i f i c t y p e s have b e e n f o u n d to be a s s o c i a t e d w i t h e a c h type of d i m e n s i o n a l defect w h i c h have specific effects u p o n t h e stability of t h e solid s t r u c t u r e . a. The Point Defect in H o m o g e n e o u s Solids Point

defects

determine

were

mentioned

in a prior

chapter.

We n o w

need

to

how t h e y affect t h e s t r u c t u r e a n d c h e m i c a l r e a c t i v i t y of t h e

solid state. We will begin by identifying t h e various defects w h i c h c a n arise in solids a n d later will s h o w h o w t h e y c a n be m a n i p u l a t e d to o b t a i n d e s i r a b l e p r o p e r t i e s n o t found in n a t u r a l l y f o r m e d solids. Since we have a l r e a d y defined solids as e i t h e r h o m o g e n e o u s a n d h e t e r o g e n e o u s , let us look first at the h o m o g e n e o u s type of solid. We will first r e s t r i c t d i s c u s s i o n to solids w h i c h

are s t o i c h i o m e t r i c ,

our

a n d later will e x a m i n e

solids w h i c h c a n be classified as " n o n - s t o i c h i o m e t r i c " , or having an e x c e s s of one or a n o t h e r of one of t h e b u i l d i n g b l o c k s of t h e solid. T h e s e o c c u r in s e m i - c o n d u c t o r s as well as o t h e r t y p e s of e l e c t r o n i c a l l y or optically active solids. Suppose

you

were

given

the

problem

of identifying

defects

in

a

h o m o g e n e o u s solid. Since all of t h e a t o m s in this type of solid are t h e s a m e , t h e p r o b l e m is s o m e w h a t simplified over t h a t of t h e h e t e r o g e n e o u s solid (that is- a solid c o n t a i n i n g m o r e t h a n one type of a t o m or ion). After

75

s o m e i n t r o s p e c t i o n , you could s p e c u l a t e t h a t the h o m o g e n e o u s solid c o u l d have the foUowing t y p e s of p o i n t defects-

3.1.1.- T y p e s of Point Defects E x p e c t e d in a H o m o g e n e o u s Solid

~-

Vacancies

Substitutional Impurities

Self-interstitial

Interstitial I m p u r i t i e s

On the left, in 3. I . i . , are the two types of p o i n t defects w h i c h involve t h e lattice itself, while the o t h e r s involve i m p u r i t y a t o m s . I n d e e d , t h e r e

do

not s e e m to any m o r e t h a n t h e s e four, a n d indubitably, no o t h e r s have b e e n observed. Note t h a t we are limiting o u r defect fan~ly to point d e f e c t s in t h e lattice a n d are i g n o r i n g line a n d v o l u m e defects

of the lattice.

T h e s e four p o i n t defects, given above, are i l l u s t r a t e d in the following diagram, given as 3.1.2. on the n e x t page. Note t h a t w h a t we m e a n by a n "interstial" is an a t o m t h a t c a n fit into t h e s p a c e s b e t w e e n the m a i n a t o m s in the crystalline array. In this case, w e have s h o w n a h e x a g o n a l lattice a n d have labelled e a c h type of p o i n t defect. Note t h a t we have s h o w n a v a c a n c y in o u r h e x a g o n a l lattice, as well as a foreign interstitial a t o m w h i c h is small e n o u g h to fit into the i n t e r s t i c e between

the

atoms

of the

structure.

Also s h o w n

are

two

types

s u b s t i t u t i o n a l a t o m s , one larger a n d the o t h e r s m a l l e r t h a n the

of

atoms

c o m p o s i n g the p r i n c i p a l h e x a g o n a l lattice. In b o t h cases, the h e x a g o n a l p a c k i n g is d i s r u p t e d due to a "non-fit" of t h e s e a t o m s in the s t r u c t u r e . Additionally, we have i l l u s t r a t e d a n o t h e r type of defect

t h a t c a n arise

w i t h i n t h e h o m o g e n e o u s lattice of 3.1.2. (in addition to the v a c a n c y a n d s u b s t i t u t i o n a l i m p u r i t i e s t h a t are b o u n d to arise). This is called the "selfinterstitial". defect.

Note t h a t it h a s a decisive effect on the

Since

the

atoms

are

all the

same

size,

the

structure

at t h e

self-interstitial

i n t r o d u c e s a l i n e - d e f e c t in the overall s t r u c t u r e . It s h o u l d be evident t h a t t h e line-defect i n t r o d u c e s a difference in p a c k i n g o r d e r since the c l o s e p a c k i n g a t t h e a r r o w s h a s c h a n g e d to cubic a n d t h e n r e v e r t s to h e x a g o n a l in b o t h lower a n d u p p e r rows of a t o m s .

76

It m a y be t h a t this type of defect is a m a j o r c a u s e of the line or edge t y p e of defects t h a t a p p e a r in m o s t h o m o g e n e o u s solids. In c o n t r a s t , the o t h e r defects p r o d u c e only a d i s r u p t i o n in the localized p a c k i n g o r d e r of t h e h e x a g o n a l lattice, i.e.- the defect does not e x t e n d t h r o u g h o u t the lattice, b u t only close to the specific defect. Now, s u p p o s e t h a t we have a solid solution of two (2) e l e m e n t a l

solids.

Would the point defects be the same, or not? An easy way to visualize s u c h point defects is s h o w n in the following diagram, given as 3.1.3. on t h e next page. It is well to note h e r e t h a t h o m o g e n e o u s lattices usually involve m e t a l s or solid s o l u t i o n s of m e t a l s (alloys) in c o n t r a s t to h e t e r o g e n e o u s lattices w h i c h involve c o m p o u n d s s u c h as ZnS.

77

Here, we u s e a h e x a g o n a l l y - p a c k e d r e p r e s e n t a t i o n of a t o m s to d e p i c t t h e close-packed

solid.

In

this

case,

we

have

shown

h o m o g e n e o u s solids. T h a t is, o n e solid is c o m p o s e d

both

types

of

of t h e s a m e s i z e d

a t o m s w h i l e t h e o t h e r is c o m p o s e d of two d i f f e r e n t sized a t o m s . On t h e r i g h t a r e t h e t y p e s of p o i n t d e f e c t s t h a t c o u l d o c c u r for t h e s a m e sized a t o m s in t h e lattice. T h a t is, given a n a r r a y of a t o m s in a t h r e e d i m e n s i o n a l lattice, only t h e s e

two t y p e s of l a t t i c e p o i n t d e f e c t s

could

o c c u r w h e r e t h e size of t h e a t o m s are t h e s a m e . T h e t e r m " v a c a n c y " is self-explanatory but "self-interstitial" means into a s p a c e b e t w e e n

that one atom has slipped

t h e r o w s of a t o m s (ions).

In a l a t t i c e w h e r e

the

a t o m s a r e all of t h e s a m e size, s u c h b e h a v i o r is e n e r g e t i c a l l y v e r y difficult u n l e s s a s e v e r e d i s r u p t i o n of t h e l a t t i c e o c c u r s (usually a " l i n e - d e f e c t " r e s u l t s . T h i s b e h a v i o r is q u i t e c o m m o n solids.

In

a like

manner,

if t h e

in c e r t a i n t y p e s of h o m o g e n e o u s

metal-atom

were

to

have

become

m i s p l a c e d in t h e l a t t i c e a n d w e r e to h a v e o c c u p i e d o n e of t h e i n t e r s t i t i a l

78

positions, as s h o w n in the different sized a t o m solid, t h e n the lattice is d i s r u p t e d by its p r e s e n c e at the interstitial position. This type of d e f e c t has b e e n o b s e r v e d . S u m m a r i z i n g , t h r e e types of p o i n t defects are evident in a h o m o g e n e o u s lattice. In addition to the Vacancy, two types of s u b s t i t u t i o n a l defects can also be

delineated.

Both are

direct

s u b s t i t u t i o n s in the

"lattice",

or

a r r a n g e m e n t of the atoms. One is a smaller atom, while the o t h e r is l a r g e r than

the

atoms

comprising

the

lattice.

In

both

cases,

the

lattice

a r r a n g e m e n t affects the h e x a g o n a l o r d e r i n g of the lattice a t o m s a r o u n d it. The lattice is s e e n to be affected for m a n y lattice d i s t a n c e s . It is for this r e a s o n t h a t c o m p o u n d s c o n t a i n i n g i m p u r i t i e s s o m e t i m e s have quite different c h e m i c a l reactivities t h a n the p u r e s t ones. T h a t also h a s an effect u p o n the c h e m i c a l reactivity of the solid. However, the i n t e r s t i t i a l i m p u r i t y does not affect the lattice o r d e r i n g at all. Now, let us look at a n o t h e r type of defect in the solid. Let u s c o n s i d e r the h e t e r o g e n e o u s lattice b. The Point Defect in H e t e r o g e n e o u s Solids The situation c o n c e r n i n g

defects

in h e t e r o g e n e o u s

inorganic

solids is

similar to t h a t given above, except for one very i m p o r t a n t factor, t h a t of c h a r g e on the atoms. Covalent inorganic solids are a rarity while ionicity or partial ionicity s e e m s to be the n o r m . T h u s , h e t e r o g e n e o u s solids are usually c o m p o s e d of c h a r g e d moieties, half of w h i c h are positive (cations) a n d half negative (anions). In general, the total c h a r g e of the c a t i o n s will equal t h a t of the a n i o n s (Even in the case of s e m i - c o n d u c t o r s , w h e r e t h e total of the c h a r g e s is not zero, the excess c h a r g e (n- or p- type) is s p r e a d over the whole lattice so t h a t no single atom, or g r o u p of atoms, ever h a s a c h a r g e different from its neighbors).

In a given s t r u c t u r e ,

usually s u r r o u n d e d by anions, a n d vice-versa.

c a t i o n s are

Because of this, we can

r e g a r d the lattice as b e i n g c o m p o s e d of a cation s u b . l a t t i c e a n d an Rnlon mfl~lattice. ( R e m e m b e r w h a t w a s s t a t e d in C h a p t e r 1 c o n c e r n i n g the fact t h a t m o s t s t r u c t u r e s are o x y g e n - d o m i n a t e d ) . What we m e a n by a "sublattice" is i l l u s t r a t e d in the following diagram:

79

In this case, we have s h o w n b o t h the

cation

and

anion "sub-lattices

separately, a n d t h e n the c o m b i n a t i o n . It s h o u l d be clear t h a t all positive c h a r g e s in the cation sub-lattice will be b a l a n c e d by a like n u m b e r of negative c h a r g e s in the a n i o n sub-lattice, even if excess c h a r g e exists in one or the o t h e r of the sub-lattices. However, if an a t o m is m i s s i n g , t h e overall lattice r e a d j u s t s to c o m p e n s a t e for this loss of charge. If t h e r e is a different

atom

compensation

present,

having

a

differing

charge,

the

charge-

m e c h a n i s m again m a n i f e s t s itself. T h u s , a cation with an

extra c h a r g e n e e d s to be c o m p e n s a t e d by a like anion, or by a n e a r e s t n e i g h b o r cation w i t h a lesser charge. An e x a m p l e of this type of c h a r g e c o m p e n s a t i o n m e c h a n i s m for a divalent cation sub-lattice w o u l d be t h e following defect e q u a t i o n : : 3.1.5.-

2 C a 2+ ~

Li +

+

Sb3+

w h e r e the Sb 3+ a n d Li+ are s i t u a t e d on n e a r e s t n e i g h b o r cation sub-lattice sites, in the divalent Ca 2+ sub-lattice. Note t h a t a total c h a r g e of 4+ e x i s t s on b o t h sides of the above e q u a t i o n .

Thus, the charge compensation mechanism represents the single most important m e c h ~ n i ~ n which operates within the defect ionic solid.

80

We find t h a t the n u m b e r a n d t y p e s of defects, w h i c h c a n a p p e a r in t h e h e t e r o g e n e o u s solid, are limited b e c a u s e of two factors: 1) The c h a r g e - c o m p e n s a t i o n factor 2) The p r e s e n c e of two s u b - l a t t i c e s in the ionic solid. These

factors r e s t r i c t

the n u m b e r of p o i n t

defect

t y p e s we

need

to

c o n s i d e r in ionic h e t e r o g e n e o u s lattices (having b o t h c a t i o n s a n d anions p r e s e n t ) . For ionic solids, the following t y p e s of defects have b e e n f o u n d to existS c h o t t k y defects (absence of b o t h cation a n d anion) Cation or a n i o n v a c a n c i e s F r e n k e l defects (Cation vacancy p l u s s a m e cation as interstitial) Interstitial i m p u r i t y a t o m s (both cation a n d anion) S u b s t i t u t i o n a l i m p u r i t y a t o m s ( b o t h cation a n d anion) T h e s e defects are i l l u s t r a t e d in the following diagram, given as 3.1.6. on the next page. Note that, in general, a n i o n s are larger in size t h a n cations due to t h e extra e l e c t r o n s p r e s e n t in the former. A h e x a g o n a l lattice is s h o w n in 3.1.6. w i t h b o t h F r e n k e l a n d S c h o t t k y defects, as well as s u b s t i t u t i o n a l defects. T h u s , if a cation is m i s s i n g (cation vacancy) in the cation sublattice, a like anion will be m i s s i n g in the a n i o n sub-lattice. This is k n o w n as a S c h o t t k y defect (after the first investigator to note its e x i s t e n c e ) . In the case of the F r e n k e l defect, the

"square" r e p r e s e n t s

cation w a s s u p p o s e d to reside

lattice

in the

before

where

it m o v e d

the

to its

interstitial position in the cation sub-lattice. Additionally, "Anti-Frenkel" defects c a n exist in the anion sub-lattice. The s u b s t i t u t i o n a l defects are s h o w n as the s a m e size as the cation or anion it displaced. Note t h a t if t h e y were not, the lattice s t r u c t u r e w o u l d be d i s r u p t e d from r e g u l a r i t y at the p o i n t s of i n s e r t i o n of the foreign ion.

81

All of t h e s e point defects are i n t r i n s i c arise due to t h e p r e s e n c e

to t h e h e t e r o g e n e o u s

solid, a n d

of b o t h cation a n d a n i o n sub-lattices. T h e

factors r e s p o n s i b l e for t h e i r f o r m a t i o n are e n t r o p y effects ( s t a c k i n g faults) a n d i m p u r i t y effects. At t h e p r e s e n t

time, the h i g h e s t - p u r i t y m a t e r i a l s

available still c o n t a i n about 0.1 p a r t p e r billion of various i m p u r i t i e s , y e t are 9 9 . 9 9 9 9 9 9 9

% pure.

i m p u r i t y a t o m s p e r mole.

Such

a solid will still c o n t a i n

about

So it is safe to say t h a t all solids

1014

contain

i m p u r i t y a t o m s , a n d t h a t it is unlikely t h a t we shall ever be able to obtain a solid w h i c h is c o m p l e t e l y pure and d o e s n o t contAln d e f e c t s .

82

c. The Line Defect In a t h r e e - d i m e n s i o n a l lattice, we have o b s e r v e d p l a n e s of a t o m s (or ions) c o m p o s i n g the lattice. Up to now, we have a s s u m e d t h a t t h e s e p l a n e s m a i n t a i n a c e r t a i n relation to one a n o t h e r . T h a t is, we have s h o w n t h a t t h e r e are a set of p l a n e s as defined by the {hkl} values, w h i c h in t u r n d e p e n d s u p o n the type of Bravais lattice t h a t is p r e s e n t . However, we find t h a t it is possible for t h e s e rows of atoms to "slip" from their e q u i l i b r i u m positions. This gives rise to a n o t h e r type of lattice defect defects".

In the following diagram, we p r e s e n t

called "line

a h e x a g o n a l lattice

w h i c h a line defect is p r e s e n t :

Here, rows of a t o m s c o m p r i s i n g a h e x a g o n a l lattice are shown. The first

in

83

four layers are c o m p o s e d However,

of c l o s e - p a c k e d

at t h e p o i n t s h o w n w h e r e

h e x a g o n a l layers of a t o m s .

the line-defect

axes c r o s s in t h e

d i a g r a m , a row of a t o m s is rrdssing. This h a s c h a n g e d t h e p a c k i n g of t h e two a t o m layers c l o s e s t to t h e m i s s i n g row of a t o m s to t h a t of cubic closep a c k i n g . It is t h i s l a c k of c o n t i n u i t y w h i c h c a u s e s t h e Hne defect b e c a u s e one layer h a s s l i p p e d from its e q u i l i b r i u m position. While it rrdght n o t s e e m t h a t this s i t u a t i o n is serious, it c a u s e s a s t r a i n to a p p e a r in t h e lattice s t r u c t u r e in w h i c h a c o m p r e s s i o n is p r e s e n t on one side of t h e l i n e defect a n d a s t r a i n on t h e o t h e r . A r e p r e s e n t a t i o n of this is s h o w n in t h e following d i a g r a m :

Note t h a t a row of a t o m s is m i s s i n g . This h a s t h e effect of i n t r o d u c i n g a tension

on

the

lattice

since

the

underneath

rows

c o m p e n s a t e for t h e lack of t h e one row. T h e r e f o r e , the upper

of a t o m s

have to

a t e n s i o n is p l a c e d on

rows of a t o m s b e c a u s e of t h e m a n d a t e d

change

in l a t t i c e

s p a c i n g s in t h e lower rows, w h e r e a c o m p r e s s i v e force is p r e s e n t .

84

There

have b e e n several c a s e s w h e r e

observe

line

imperfections

it h a s b e e n possible to d i r e c t l y

by suitable

e x a m i n a t i o n of the surface in reflected

preparation

and

microscopic

light. One e x a m p l e is the MgO

crystal. MgO is a cubic crystal a n d it is possible to e t c h it along t h e { 100} direction

(this is the d i r e c t i o n along the x-axis). W h a t is o b s e r v e d is a

series of surface lines of specific length. The r e a s o n t h a t t h e s e d e f e c t lines s h o w u p is t h a t t h e y are m o r e d i r e c t i o n of the edge dislocation.

easily e t c h e d

by acid along t h e

Note t h a t in its s i m p l e s t form, an e d g e

dislocation is an o m i s s i o n of a line of a t o m s c o m p o s i n g the lattice. T h e a r e a w i t h i n the lattice a r o u n d the line defect is u n d e r b o t h c o m p r e s s i o n a n d t e n s i o n due to the difference in a t o m - d e n s i t y as one p a s s e s t h r o u g h it in a d i r e c t i o n p e r p e n d i c u l a r to the line defect. A n o t h e r view of the s a m e type of defect is s h o w n in the following d i a g r a m :

Here, the axis of slip is s h o w n in the h e x a g o n a l lattice as being on, or n e a r the surface of the a r r a y of a t o m s . In the cubic lattice, a slip p l a n e is d e p i c t e d w h e r e a line of a t o m s is m i s s i n g a n d the lattice h a s m o v e d to a c c o m m o d a t e this type of lattice d e f e c t . It s h o u l d be clear t h a t the p r e s e n c e of line defects in a crystal l a t t i c e leads to a d i s r u p t i o n of the c o n t i n u i t y of the lattice j u s t as the p r e s e n c e of point

defects

affects the p a c k i n g of a given lattice.

The

line

defect,

85

however, is a physical c h a n g e in t h e lattice w h e r e a s t h e p o i n t defect can be classified as a chernical change. T h a t is, the p o i n t c h a n g e in the electronic

defect

causes a

s t r u c t u r a l a r r a n g e m a n t of the a t o m s while t h e

line defect h a s no effect on the electronic p r o p e r t i e s . d. The Volume Defect The

in

two

d i m e n s i o n s . Let u s s u p p o s e t h a t a line defect h a s a p p e a r e d while

volume

defect

is s o m e w h a t

more

difficult

to

visualize

the

crystal s t r u c t u r e w a s forming. This w o u l d be a s i t u a t i o n similar to t h a t already s h o w n in 3.1.3. w h e r e a line defect w a s s h o w n . The c o m p r e s s i o n t e n s i o n a r e a of the defect h a s a definitive effect u p o n the growing crystal a n d c a u s e s it to d e f o r m a r o u n d the line defect. This is s h o w n in t h e following d i a g r a m :

In one case, we have s h o w n a b l o c k r e p r e s e n t a t i o n of the g r o w t h p a t t e r n of the crystal a n d in the o t h e r an a t o m i s t i c r e p r e s e n t a t i o n . What h a p p e n s is t h a t a series of "steps" a p p e a r a r o u n d t h e line defect a n d the crystal b e g i n s to a s s u m e a spiral g r o w t h p a t t e r n . T h a t is, the p o i n t s s h o w n as "a", b e i n g at least one a t o m u n i t h i g h e r in the crystal, begin to grow over t h e a t o m u n i t s at "o". This r e s u l t s in a g r o w t h s i t u a t i o n similar to t h a t s h o w n

86

in the following d i a g r a m . Here, the spiral p a t t e r n of g r o w t h is e v i d e n t , a n d p r o c e e d s from the line defect dislocation. Both a top view a n d a side view are given. This type of crystal g r o w t h arises from the p o i n t of t h e line defect w h i c h h a s i n t r o d u c e d the c o m p r e s s i o n - t e n s i o n factor into t h e growing lattice.

This type of v o l u m e defect in the crystal is k n o w n as a "screw dislocation", so-called b e c a u s e of its t o p o g r a p h y . Note t h a t the spiral dislocation of t h e growing lattice d e p o s i t s a r o u n d the line defect at fight angles to the line defect. In the following diagram, given as 3.1.12.

on the n e x t page, a n o t h e r

r e p r e s e n t a t i o n is shown, detailing how the dislocation line (line d e f e c t ) becomes a screw-dislocation. Note t h a t "b" in this d i a g r a m is the s a m e as t h a t in 3.1.8. B e c a u s e e d g e a n d v o l u m e defects

propagate

throughout

the

lattice,

t h e y affect t h e

physical p r o p e r t i e s of the solid, w h e r e a s it is the point defects t h a t affect the c h e m i c a l

properties

of the

solid. T h e s e

latter p r o p e r t i e s

electrical a n d resistive, optical a n d reactivity p r o p e r t i e s we c a n n o w classify d e f e c t s in solids

as:

EXTRINSIC - Edge & Volume (physical) INTRINSIC - Point ( c h e m i c a l )

include

of solids. T h u s ,

87 84

It is the i n t r i n s i c defects t h a t have t h e m o s t i n t e r e s t for us, since t h e y affect

the

havelittle

chemical effect.

properties

Extrinsic

of the

defects

are

solid

while

the

proper

extrinsic study

defects

for

those

i n t e r e s t e d in the m e c h a n i c s of solids, p a r t i c u l a r l y m e t a l s . A b r o a d variety of c o m m e r c i a l p r o d u c t s are b a s e d u p o n c o n t r o l l e d p o i n t defects.

These

color-television,

include

transistors,

integrated

circuits,

photosensors,

f l u o r e s c e n t l a m p s , j u s t to n a m e a few. None of t h e s e

w o u l d be possible

without

control

of point

defects.

As a n

example,

c o n s i d e r single crystal silicon. A boule is g r o w n about 2 0 0 to 3 0 0 ram. in d i a m e t e r (8 to 12 inches). T h e s e weigh about 2 0 0 Ibs before t h e y are sliced into fiat discs. The native defects in silicon g r o w n from the m e l t are v a c a n c i e s a n d interstials, i m p u r i t i e s a n d oxygen e n t e r i n g the c r y s t a l from the g r o w t h c o n d i c t i o n s

(from the SiO2 crucible u s e d to hold t h e

liquid m a s s a n d from t h e a h m o s p h e r e [here, the i n e r t gas u s e d to b l a n k e t the

growth

chamber

does

not

eliminate

all of t h e

surface-adsorbed

g a s e o u s molecules]). Si v a c a n c i e s a n d i n t e r s t i t u a l s t e n d to a g g l o m e r a t e to form "surface "pits" w h i c h i n t e r f e r e w i t h t h e f o r m a t i o n of the i n t e g r a t e d circuits on the surface of t h e silicon disc as t h e y are b e i n g m a n u f a c t u r e d .

88

J u s t recently, it w a s a n n o u n c e d t h a t crystal g r o w t h c o n d i t i o n s could be modified so t h a t v a c a n c i e s could be n e a r l y e l i m i n a t e d in the final c r y s t a l lattice. This i n c l u d e d oxygen a t o m s w h i c h were s u b s t i t u t e d as 0 - - i n t h e lattice a n d c a u s e s t a c k i n g faults (Note t h a t p a r t of t h e oxygen is usually lost as volatile SiO). This m i g h t n o t s e e m i m p o r t a n t until one realizes t h a t c o n d u c t i v e line w i d t h s on the silicon surface have b e e n n a r r o w i n g o v e r the y e a r s from 0 . 3 5 ~ n in 1995 to 0.13 ~ in 2002. Since the average atomic size is of the o r d e r of a n g s t r o m s , i.e.- 10 s c m or 0.01 ~rn, it predicted

t h a t line w i d t h s of i n t e g r a t e d circuits will r e a c h 0.07 ~rrt in

2 0 1 0 or t h a t of the individual a t o m s . Additionally, iron as Fe "~, one of t h e m o s t i n s i d i o u s of i m p u r i t i e s

in single

crystal silicon

(Si~),

has been

c o n t r o l l e d to < 1 0 i~ a t o m s per c m "~ Since silicon c o n t a i n s r o u g h l y 5 x 10 ~ a t o m s per c m a, this c o r r e s p o n d s to a p u r i t y of 2 x 1 0 i~ or 2 p a r t s p e r million-million (2 p a r t s per quadrillion). T h u s , control of intrinsic d e f e c t s is one of the m o s t i m p o r t a n t of the factors c o n t r i b u t i n g to the p r o p e r operation

of i n t e g r a t e d

circuits

devices

(computers,

and

all d e v i c e s

c o n t a i n i n g electronic circuits). 3.2.- Math.ematics a n d E q u a t i o n s of the Point Defect A c o n s i d e r a b l e b o d y of scientific w o r k h a s b e e n a c c o m p l i s h e d in the p a s t to define

and characterize

point

defects.

One

major

reason

is

that

s o m e t i m e s , the e n e r g y of a p o i n t defect c a n be calculated. In o t h e r s , t h e c h a r g e - c o m p e n s a t i o n w i t h i n the solid b e c o m e s a p p a r e n t . In m a n y cases, if one deliberately a d d s an i m p u r i t y to a c o m p o u n d to modify its p h y s i c a l p r o p e r t i e s , the c h a r g e - c o m p e n s a t i o n , intrinsic to the defect formed, can be p r e d i c t e d .

We are now r e a d y to d e s c r i b e

t h e s e defects in t e r m s of

their e n e r g y a n d to p r e s e n t e q u a t i o n s d e s c r i b i n g their equilibria. One way to do this is to u s e a "Plane-Net".

This is simply a t w o - d i m e n s i o n a l

r e p r e s e n t a t i o n w h i c h u s e s s y m b o l s to replace the s p h e r i c a l i m a g e s t h a t we u s e d above to r e p r e s e n t the a t o m s (ions) in the s t r u c t u r e . 8, THE PLANE N E T To begin, let us call the cation "M" a n d the anion "X". Although t h e y will be c h a r g e d in the h e t e r o g e n e o u s lattice, let u s ignore c h a r g e

for t h e

89

m o m e n t (or we c a n imagine t h a t e a c h h a s a single c h a r g e as in the NaCI lattice, i.e.- M + a n d X-). If we a r r a n g e b o t h the M cation a n d the X anion in a cubic array, we w o u l d have the foUowing r e p r e s e n t a t i o n of a s t r u c t u r e . This, we call a " p l a n e net". 3.2.1.-

[ A RepresentatiOn of a Plane N e t ]

ITypeso.f,,Defects]Mx M X M X

MX MXMXMX

ix M X M X M XM• X M X MXMXMX

.....

XMXMXM MXMXMX XMXMXM MXMXMX XMXMXM M MXMX MXMXMX XMXMXM XMXMXMXM MXMXMX MXMXMX XMX MXMX1VI XMXMXM XMX MXMXMXMXMXM [X-]IVIX M X M X M X M X M X M XMX MXMXMXMXMXM

XMXM MXMXMXMX [~~--~X M F]M X M X M MXMXMXMX

x

~

xMx M

It is also easy to see t h a t we c a n s t a c k a series of t h e s e "NETS" three-dimensional

solid. We c a n also s u p p o s e

t h a t the

to form a

same

type of

defects will arise in o u r Plane Net as in e i t h e r the h o m o g e n e o u s o r h e t e r o g e n e o u s solid a n d so p r o c e e d to label s u c h defects as M i , m e a n i n g an i n t ~ t i t i a l

cation. In the s a m e way, we label a c a t i o n v a c a n c y as VM,

a n d Xi is the i n t e r s t i t i a l anion, a n d Vx is t h e s y m b o l for the a n i o n vacancy. Also s h o w n are t h e symbols, Ms a n d Ks, w h i c h d e n o t e t h e s u r f a c e sites of the cubic s t r u c t u r e of the Plane Net. If we are dealing w i t h

an ionic

s t r u c t u r e , we w o u l d also e x p e c t to have defects w h i c h are c h a r g e d s i n c e the i n c o r p o r a t i o n of e i t h e r c a t i o n s or a n i o n s ha-ring differing c h a r g e s as

substitutional ions into either of the sub-lattices is to be expected (See 3.1.3. given above). Additionally, b o t h surface sites a n d i n t e r n a l ions can have c h a r g e s differing f r o m the m a j o r i t y of the c a t i o n s or anions.

90

T h e s e , t h e n , are t h e s e t of p o s s i b l e d e f e c t s for t h e P l a n e Net, following s u m m a r i z e s t h e t y p e s of i n t r i n s i c d e f e c t s e x p e c t e d . we h a v e u s e d t h e labelling: V = v a c a n c y ;

and the Note t h a t

i = i n t e r s t i t i a l ; M -- c a t i o n site;

X = a n i o n site a n d s = s u r f a c e site. We h a v e a l r e a d y s t a t e d t h a t s u r f a c e s i t e s are special.

Hence,

t h e y are i n c l u d e d

in o u r l i s t i n g of i n t r i n s i c

defects. 3.2.2-

CHARGED PARTICLES

VACANCIES

e- ,

V M , VX, V + M , V - x

SURFACE S I T E S

INTERSTITIAL SITI~S Mi,

Xi , M + i ,

X'i

/1;+

Ms, Xs, M+s, X's

Two i t e m s of i n t e r e s t c a n be n o t e d I) We c a n h a v e p u r e v a c a n c i e s in t h e l a t t i c e 2} T h e s e v a c a n c i e s c a n b e c h a r g e d . T h e s a m e c a n be s a i d for i n t e r s t i t i a l s i t e s a n d s u r f a c e s i t e s as well. W h a t t h i s m e a n s is t h a t t h e p o i n t d e f e c t c a n a c q u i r e a c h a r g e . For e x a m p l e , let u s c o n s i d e r a c a t i o n v a c a n c y in ZnS. T h e v a c a n c y c a n b e r e p r e s e n t e d a s : V z n 2 + . However, w i t h a V z n 2 + p r e s e n t , a l a c k of c h a r g e of 2+ e x i s t s in t h e lattice. T h i s deficit of c h a r g e m u s t b e c o m p e n s a t e d in t h e l a t t i c e b y a like a m o u n t of n e g a t i v e c h a r g e s u c h as a Vs2- . However, w e h a v e a l r e a d y s t a t e d t h a t t h e Vzn2+ p o i n t d e f e c t c a n also a c q u i r e a c h a r g e s u c h as V + z n 2 + . In t h i s c a s e we h a v e a total deficit Obviously, t h e p r e s e n c e

in c h a r g e

of 3+.

of t h e a n i o n v a c a n c y , Vs2- , is n o t sufficient to

m a i n t a i n c h a r g e c o m p e n s a t i o n in t h e lattice. T h e r e f o r e

some other mode

of c h a r g e c o m p e n s a t i o n is r e q u i r e d . O n e p o s s i b i l i t y is, of c o u r s e , t h e c h a r g e d a n i o n v a c a n c y , V-s2_ . T h e r e a r e o t h e r s as well. T h e c h a r g i n g of s u r f a c e s i t e s a n d i n t e r s t i t i a l s is n o t as c l e a r as t h a t of v a c a n c i e s , a n d w e will n o t dwell o n t h e m f u r t h e r at t h i s p o i n t . One m i g h t t h i n k t h a t p e r h a p s V M a n d V+• o u g h t to b e i n c l u d e d in o u r list of

vacancies.

However,

a

negatively-charged

cation

vacancy

alone,

p a r t i c u l a r l y w h e n it is s u r r o u n d e d b y n e g a t i v e a n i o n s , w o u l d n o t b e v e r y

91

stable. N e i t h e r

should a positively-charged

anion v a c a n c y be any m o r e

stable. E i t h e r a r r a n g e m e n t w o u l d r e q u i r e high e n e r g y stabilization to exist. T h e r e f o r e , we do not i n c l u d e t h e m in o u r listing. However, a VM could c a p t u r e a positive c h a r g e to b e c o m e V+M a n d likewise for the Vx which

then becomes

a V-x.

Both of t h e s e

o u g h t to be stable w h e n

s u r r o u n d e d by the o p p o s i t e l y c h a r g e d sites of the o t h e r sub-lattice ( a n d i n d e e d t h e y are as w e have i n d i c a t e d above). One s h o u l d note t h a t w h e n a c h a r g e d - i n t e r s t i t i a l is p r e s e n t in one sublattice, the o t h e r sub-lattice will c o n t a i n e i t h e r a l i k e - c h a r g e d i n t e r s t i t i a l or a l i k e - c h a r g e d s u b s t i t u t i o n a l ion w h i c h exactly b a l a n c e s t h e total c h a r g e p r e s e n t in the lattice. S u c h e q u a t i o n s m i g h t i n c l u d e one or b o t h of t h e following e q u a t i o n s : 3.2.3.-

[ M i 3+]Mi + ~ [Xi 3"]Xi" [ Mi 3+]Mi + ~

X3-x"

The first e q u a t i o n d e s c r i b e s the e q u i l i b r i u m b e t w e e n two i n t e r s t i t i a l s i t e s with e x c e s s charge, one on e a c h sub-lattice, while the o t h e r shows

the

equilibria b e t w e e n

the

interstitial

cation

equation

a n d t h a t of t h e

s u b s t i t u t i o n a l anion, b o t h h a v i n g a like e x c e s s charge. Obviously, various c h a r g e - c o m p e n s a t i n g c o m b i n a t i o n s of all of the 6 basic lattice defects a r e possible, i.e.- the VM ,Vx ,Mi, X4, Ms , a n d Xs d e f e c t s , as well as b e t w e e n the s u b s t i t u t i o n a l c a t i o n s or a n i o n s h a v i n g a c h a r g e differing from t h o s e c o m p r i s i n g the lattice. The n u m b e r of c o m b i n a t i o n s is thus: C = 6!/(6-2)! , or C = 3 0 Additionally, we c a n have at l e a s t four (4) t y p e s of c h a r g e d v a c a n c i e s a n d i n t e r s t i t i a l sites as given in 3.2.2. (Charged surface sites are n o t c o m m o n b u t are i n c l u d e d h e r e for the s a k e of c o m p l e t e n e s s ) . T h i s gives rise to 12 m o r e c h a r g e defect e q u a t i o n s or a total of 42 defect e q u a t i o n s t h a t we c a n write!

92

The

positive

electronic

hole

requires

further

equivalent of the electron,

explanation, e-,

n+

is the

positive

in solids a n d t h e y a n n i h i l a t e

each o t h e r u p o n r e a c t i o n : 3.2.4-

e-

+

n+

=

energy

The following d i a g r a m s h o w s h o w the positive hole c a n exist in the solid.

Note t h a t we have labelled all t h r e e p l a n e s a n d have r e m o v e d Plane "2" so as to s h o w the p r e s e n c e of 70Gf131 in the lattice. This type of c h a r g e s u b s t i t u t i o n a l defect is u s e d extensively in the S e m i c o n d u c t o r I n d u s t r y to

93

p r o d u c e i n t e g r a t e d circuits by deliberate "doping" of at om s w hi ch have a valency differing from t h a t of the major s t r u c t u r a l element, i.e. Si as Si 4§ A second kind of electronic defect involves the electron. Let us s u p p o s e t h a t the second plane of the cubic lattice has a vac~m~ i nst ead of a substitutional i m p u r i t y of differing valency. This m a k e s it possible for t h e lattice to cap tur e and localize an e x t r a n e o u s electron at the vacancy site. This is s h o wn in the following diagram. The c a p t u r e d electron t h e n endows the solid s t r u c t u r e with special optical p r o p e r t i e s since it can absorb p h o t o n energy. The s t r u c t u r e t h u s b e c o m e s optically active. T h a t is, it absorbs light w i t hi n a well-defined b a n d and is called a "color-center" since it i m p a r t s a specific color to the crystal.

The alkali halides are noted for their p r o p e n s i t y to f o r m color-centers. It has been found t h a t the p e a k of the b a n d c h a n g e s as the size of the cation in the alkali halides increases. There a p p e a r s to be an inverse rel at i on b etween the size of the cation (actually, the polarizability of the cation) a nd the p e a k energy of the absorption band. These are the two types of electronic d e f ect s t h a t are found in crystals, n a m e l y positive "holes" and negative "electrons", and their p r e s e n c e in the s t r u c t u r e is related to t h e fact t h a t the lattice t e n d s to b e c o m e u p o n the type of defect p r e s e n t .

charge-compensated,

depending

94

R e t u r n i n g to t h e s u b j e c t of lattice defect f o r m a t i o n , we

can now proceed

to w r i t e a s e r i e s of defect r e a c t i o n s for t h e d e f e c t s w h i c h we f o u n d for o u r

plane net" 3.2.7.-

SCHO~

:

MX ~ M l . 5

Xl_5

FRENKEL :

MX --~ M1-5 X

ANTI-FRENKEL:

MX ~

F - CENTER:

MX ~ M X l - 5

MXI-5

+ 5Vm

+ 5Mi

+

+ 5Vx

+ 5Vx 5Vm

+

5Xi

+ {Vx/e-} +

5/2

x2~

In t h i s case, we u s e 5 as a s m a l l f r a c t i o n since t h e actual n u m b e r of d e f e c t s is s m a l l in r e l a t i o n to t h e overall n u m b e r of ions actually p r e s e n t . For t h e F-Center, the brackets enclose

the

complex

consisting

of a n e l e c t r o n

c a p t u r e d at a n a n i o n vacancy. Note t h a t t h e s e e q u a t i o n s e n c o m p a s s all of the

mechanisms

t h a t we

have p o s t u l a t e d

for e a c h

of t h e

individual

r e a c t i o n s . T h a t is, we s h o w t h e p r e s e n c e of v a c a n c i e s in t h e S c h o t t k y c a s e a n d i n t e r s t i t i a l c a t i o n s for t h e F r e n k e l c a s e involving e i t h e r t h e c a t i o n o r anion. T h e latter, involving a n i n t e r s t i t i a l a a i o n is called, by c o n v e n t i o n , t h e "Anti-Frenkel" case. The defect r e a c t i o n involving t h e "F-Center" is also given. Actually, t h e f o r m a t i o n of a n F - c e n t e r

is m o r e

complicated

t h a n this.

A l t h o u g h F - c e n t e r s c a n be f o r m e d by several m e t h o d s , t h e b e s t w a y to do so is b y e x p o s i n g a s o d i u m c h l o r i d e c r y s t a l to s o d i u m m e t a l v a p o r s . In t h a t case, t h e following defect r e a c t i o n s have b e e n o b s e r v e d to t a k e p l a c e : 3 . 2 . 8 . - Defect

R e a c t i o n s for t h e NaCI Crystal Na o ,g_... -7 Na + + e Na + + { N a +,C1-} # 2 { N a + I C 1 - , VCI]}

VC1

+

e- ~

[ Vc1 / e-]

These equations illustrate how the crystal responds

to t h e p r e s e n c e

of

s o d i u m vapor, i.e.- e x c e s s Na + , by f o r m i n g a n i o n v a c a n c i e s , to f o r m t h e Fcenter. Up to t h i s point, we h a v e only c o n s i d e r e d s t o i c h i o m e t r i c lattices. Let us

95

now c o n s i d e r n o n - s t o i c h i o m e t r i c lattices. W h a t we m e a n by this is that, in n o n - s t o i c h i o m e t r i c crystals, t h e h e t e r o g e n e o u s lattice c a n c o n t a i n an excess of either the cation or the anion in c o n t r a s t to the h o m o g e n e o u s lattice w h i c h m a y only c o n t a i n an excess of charge. W h a t h a p p e n s t h e n is t h a t the h e t e r o g e n e o u s lattice r e s p o n d s by c r e a t i o n of even m o r e l a t t i c e defects. 3-3 : NON-STOICHIOMETRIC SOLIDS In a n o n - s t o i c h i o m e t r i c crystal, the lattice m a y have e i t h e r excess c h a r g e or excess cations a n d / o r a n i o n s s i t u a t e d in the lattice. C o n s i d e r the s e m i conductor,

Ge. It is a h o m o g e n e o u s

solid a n d is e x p e c t e d

to c o n t a i n

excess charge. The defect r e a c t i o n s a s s o c i a t e d w i t h the f o r m a t i o n of ptype a n d n - t y p e lattices are: 3.3.1-

n-tylm:

Ge + 5As

-% [ Ge/Ass]

+

Be-

l~type:

Ge+SGa

-% [Ge/Ga5 ]

+

8~+

w h e r e 8 is a small excess over s t o i c h i o m e t r y . The excess c h a r g e s s h o w n are s p r e a d over the entire lattice, as s t a t e d before. In the h e t e r o g e n e o u s

solid, a different

mechmnism concerning

charge

d o m i n a t e s . If t h e r e are a s s o c i a t e d vacancies, a different type of e l e c t r o n i c defect, called the "M-center", prevails. In this case, a m e c h m n i s m s i m i l a r to t h a t already given for F - c e n t e r s o p e r a t e s , except t h a t two (2) e l e c t r o n s o c c u p y n e i g h b o r i n g sites in the crystal. The defect e q u a t i o n for f o r m a t i o n of the M - c e n t e r is: 3.3.2.-

M-center:

KC1 ~ KCII_5

+

5/2 C12 ~ + 5/4 [ V-c1 I V'Cl ]

The following diagram, given as 3.3.3. on the n e x t page, i l l u s t r a t e s t h e "M-Center": This

type of e l e c t r o n i c

defect,

found

only in

heterogeneous

solids,

c o n s i s t s of two associated, n e g a t i v e l y - c h a r g e d a n i o n vacancies, one o n e a c h p l a n e as s h o w n .

96

T h e y are usually j o i n e d along the {I I0} p l a n e of t h e lattice of the facecentered

salt crystal, a l t h o u g h we have not s h o w n t h e m this w a y ( T h e

{I00} p l a n e is i l l u s t r a t e d in the diagram). Note t h a t e a c h v a c a n c y has c a p t u r e d an e l e c t r o n in r e s p o n s e to the c h a r g e - c o m p e n s a t i o n m e c h a n i s m w h i c h is operative for all defect r e a c t i o n s . In this case, it is the anion w h i c h is affected w h e r e a s in the "F-center", it w a s t h e cation w h i c h was affected (see e q u a t i o n 3.2.8. given above). T h e s e

associated, negatively-

c h a r g e d , v a c a n c i e s have quite different a b s o r p t i o n p r o p e r t i e s t h a n t h a t of the F - c e n t e r . Additionally, the h e t e r o g e n e o u s solid will c o n t a i n defects in its lattice, as n o t e d above. This involves h e t e r o g e n e o u s solids c o n t a i n i n g i m p u r i t i e s a n d we u s e the s a m e type of n o t a t i o n given above. For the case of an AgCI c r y s t a l c o n t a i n i n g the Cd 2+ cation as an i m p u r i t y , we have: 3.3.4-

2 Ag+

~

[ Cd2+Ag+ ' VAg+]

w h e r e t h e Cd 2+ cation o c c u p i e s one Ag + site a n d the v a c a n c y t h e other, as n e a r e s t n e i g h b o r s . T h i s is an e x a m p l e of a h e t e r o t y p r

impurity system

since different c a t i o n s are involved. Although the v a c a n c y o c c u p i e s o n e site, it is n o t c h a r g e d since t h e total c h a r g e differential lies on the Cd2+Ag site. Note t h a t the symbol t h a t we have u s e d d e n o t e s

a doubly-charged

97

c a d m i u m cation on a m o n o v a l e n t silver cation site in the lattice. T h i s i l l u s t r a t e s the fact t h a t t h e n o r m a l lattice of an i n s u l a t o r c o m p o u n d t e n d s to c h a r g e c o m p e n s a t e itself fully, b u t excess c h a r g e does not a p p e a r . T h e same

is n o t

true

with

semi-conducting

compounds,

which

m a y be

h e t e r o g e n e o u s or h o m o g e n e o u s in n a t u r e . A n o t h e r h e t e r o t y p e e x a m p l e is CdCI2 c o n t a i n i n g Sb3+. Here, we c a n w r i t e at least t h r e e different e q u a t i o n s involving defect equilibria: 3.3.5.- Defect R e a c t i o n s Involving the CdCI2:Sb 3+ S y s t e m 2 Cd 2+

--7~

Sb 3+ +

2 Cd2+

--Tz'-

Sb3+

2 Cd2+

-7~

Sb3+

n+

+

+

VCd

V+Cd

+

Li+

In the last equation, we have the i n s t a n c e w h e r e c h a r g e - c o m p e n s a t i o n has o c c u r r e d due to i n c l u s i o n of a m o n o v a l e n t cation. A vacancy does not f o r m in this case. All of t h e s e e q u a t i o n s are c a s e s of i m p u r i t y s u b s t i t u t i o n s in the n o r m a l lattice.. A n o t h e r g e n r e is the so-called h o m o t y p e i m p u r i t y s y s t e m . One e x a m p l e is the

s u b s t a n c e , n i c k e l o u s oxide, w h i c h

is a p a l e - g r e e n

insulator w h e n

p r e p a r e d in an i n e r t a t m o s p h e r e . If it is r e h e a t e d in air, or if a m i x t u r e of NiO a n d Li + is r e h e a t e d in an i n e r t a t m o s p h e r e , the NiO b e c o m e s a b l a c k semi-conductor.

This

is a classical

example

of the

effect

of d e f e c t

r e a c t i o n s u p o n the i n t r i n s i c p r o p e r t i e s of a solid. The defect r e a c t i o n s involved are3.3.6.-

2 N i 2+ ~

[ N i 3+ / V N i ]

2 N i 2+ ~

Ni 3+ +

+

n+

Li +

This b e h a v i o r is typical for t r a n s i t i o n m e t a l s w h i c h easily u n d e r g o c h a n g e s in valency in the solid state.

T h a t is, h o m o t y p e i m p u r i t y s y s t e m s usually

involve c a t i o n s w h i c h can u n d e r g o valence c h a n g e s easily, i.e.- t r a n s i t i o n m e t a l cations. In the NiO case, we have gone f r o m an i n s u l a t o r to a s e m i c o n d u c t o r j u s t b y i n t r o d u c i n g lattice defects t h r o u g h a t h e r m a l effect.

98

Having i n t r o d u c e d

these

examples,

let u s n o w e x a m i n e

a method

of

s y m b o l i s m u s e f u l in c h a r a c t e r i z i n g defect r e a c t i o n s in solids. 3 . 4 . - D E F E C T EQUATION SYMBOLISM Whether

you realize

s y m b o l i s m for d e f e c t s

it or

not,

we

and defect

have already

developed

our

own

r e a c t i o n s b a s e d o n t h e P l a n e Net.

It

m i g h t b e well to c o m p a r e o u r s y s t e m to t h o s e of o t h e r a u t h o r s , w h o h a v e also c o n s i d e r e d wrote

the

represent

t h e s a m e p r o b l e m in t h e p a s t . It w a s R e e s (1930)

first m o n o g r a p h

on d e f e c t s

in solids.

Rees

t h e c a t i o n v a c a n c y , as did Libowitz (1974).

used

who

a b o x to

This has certain

a d v a n t a g e s s i n c e w e c a n w r i t e e q u a t i o n 3 . 3 . 5 . as s h o w n in t h e following: 3.4. I-

S y m b o l i s i z m U s e d for Defect E q u a t i o n s

2Cd=+

7-7

+C d

2 Cd 2+

-Tz'--

~-~

2 Cd 2+

z_ --7

~

I

+

+

Ca

+

Cd

Ni

+

Cd

i Cd

+

~+

Likewise, t h e o t h e r e q u a t i o n s of 3.3.6. w o u l d b e c o m e : 3.4.2-

2 Ni 2+

z._ -7

Ni3+ +

2 Ni2+

z._

Ni3+

[

[N i + n+ §

+

[~-] N i

A l t h o u g h t h e r e s u l t s a r e e q u a l as far as utility is c o n c e r n e d , c o n t i n u e to u s e o u r s y m b o l i s m , for r e a s o n s w h i c h will b e c o m e

we

shall

clearer

l a t e r on. T h e following T a b l e is a c o m p a r i s o n of defect s y m b o l i s m , as u s e d b y p r i o r A u t h o r s . Note t h a t o u r s y m b o l i s m m o s t r e s e m b l e s t h a t of Kroeger, b u t n o t in all a s p e c t s . T h e s e p r i o r a u t h o r s also c o n s i d e r e d o t h e r i n t r i n s i c d e f e c t s t h a t we h a v e n o t t o u c h e d , "anti-structure"

occupation.

namely interstices

and the so-called

99

Table 3- i C o m p a r i s o n of P o i n t Defect N o t a t i o n b y V a r i o u s A u t h o r s Rees{ 19301 Kroeffer [ 1 9 5 4 ) Libowitz ( 1 9 7 4 ) v Cation Site Vacancy:

DM

Anion Site Vacancy:

~x

VM

[~M

Vx

Cation I n t e r s t i t i a l

AM

Mi , M+i

Mi

Anion Interstitial

AX

Xi,

Xi

X- i

Negative Free C h a r g e

e

e-

e-

Positive Free C h a r g e

p

h+

h+

Interstices

~V.

Unoccupied Interstitial

---

V.1

Anti-structure Occupation

---

aV.

I

1

A

M M , X X, M x , X M

T h e l a t t e r d e a l s w i t h a n i m p u r i t y _Rnlon o n a c a t i o n site c o u p l e d w i t h an i m p u r i t y c a t i o n o n a n a n i o n site, b o t h w i t h t h e p r o p e r c h a r g e . We have mentioned

interstices

b u t n o t in detail. T h e y

appear

as a f u n c t i o n of

s t r u c t u r e (Refer b a c k to C h a p t e r 1 - D i a g r a m 1.3.2.). T h e r e

is o n e in a

t e t r a h e d r o n , four in a b o d y - c e n t e r e d c u b e , a n d six in a s i m p l e cube. T h u s , r in {c~Vi} is 1, 4 or 6, r e s p e c t i v e l y . We shall n e e d t h i s s y m b o l later, as well as V i , t h e u n o c c u p i e d i n t e r s t i t i a l . Before

we

proceed

to

analyze

defect

reactions

by a m a t h e m a t i c a l

a p p r o a c h , let u s c o n s i d e r a n a p p l i c a t i o n s of solid s t a t e c h e m i s t r y . In t h i s e x a m p l e , t h e effect of a defect o n t h e p r o p e r t i e s of t h e solid is d e s c r i b e d . 3-5 - SOME APPLICATIONS FOR D E F E C T C H E M I S T R Y In o r d e r to u n d e r s t a n d s o m e of t h e a s p e c t s of t h e s e e x a m p l e s , we n e e d to d i s c u s s a c e r t a i n a m o u n t of b a c k g r o u n d for e a c h a r e a of t h e solid s t a t e being considered.

I00

a. P h o s p h o r s : P h o s p h o r s are i n o r g a n i c m a t e r i a l s w h i c h c o n v e r t i n c i d e n t r a d i a n t e n e r g y to visible light w i t h i n a device. T h e device c h o s e n c a n be a c a t h o d e - r a y tube, i.e.- a television tube, or a f l u o r e s c e n t lamp. A p h o s p h o r c o n s i s t s of a m a t r i x modified by a n additive c h o s e n so t h a t it b e c o m e s optically active w i t h i n the matrix, or c o m p o u n d . This is an e x a m p l e of a s u b s t i t u t i o n a l i m p u r i t y in a lattice w h e r e i n

t h e additive, usually called an "activator",

i n t r o d u c e s a lattice defect t h a t is optically active. However,

the a d d e d

i m p u r i t y still follows all of the r u l e s found for defects in a lattice, as s h o w n by the following e x a m p l e . In the prior l i t e r a t u r e , it w a s found (Kinney-

1955)

t h a t Ca2P207 (i.e.-

c a l c i u m p y r o p h o s p h a t e ) could be activated by Sb 3+ to form a p h o s p h o r : Ca2P207: Sb.02. This f o r m a l i s m actually i n d i c a t e s the c o m p o s i t i o n : [(Cao.99, Sbo.ol)2 P2 07)]. where

a solid-state

solution

of c a l c i u m

pyrophosphate

and

antimony

p y r o p h o s p h a t e is i n d i c a t e d . T h e b r i g h t n e s s r e s p o n s e of this p h o s p h o r w a s m o d e r a t e w h e n excited by ultraviolet r a d i a t i o n b u t w a s i m p r o v e d four t i m e s by t h e addition of Li+. T h e o p t i m u m a m o u n t of Li+ p r o v e d to be t h a t exactly equal to the a m o u n t of Sb 3+ p r e s e n t in the p h o s p h o r , i.e.2 Ca 2+ ~-~ Sb 3+ + Li+. The

explanation

lies

in

f o r m a t i o n of the p h o s p h o r

the

defect

itself. The

reactions defect

controlling

reactions

the

occurring

w e r e found to be the s u b s t i t u t i o n of a trivalent cation on a divalent site a n d t h e defects r e a c t i o n s t h e r e b y associated. This is s h o w n in the following table w h i c h c o m p a r e s t h e s e two m e t h o d s of p r e p a r i n g s u c h m a t e r i a l s . In this case, the i n c r e a s e in b r i g h t n e s s w a s f o u n d to be r e l a t e d to the a m o u n t of activator actually b e i n g i n c o r p o r a t e d into the lattice. proportional

It to

is well k n o w n t h a t p h o s p h o r b r i g h t n e s s is t h e n u m b e r s of Sb 3+ ions (activators) actually

i n c o r p o r a t e d into cation sites of t h e p y r o p h o s p h a t e s t r u c t u r e .

101

Table 3 - 2 Defect R e a c t i o n s W h i c h Occur in the P h o s p h o r : (Ca.99, Sb.ol )2 P2 07). PHOSPHOR 2 Ca 2+ z.._ 2 Ca 2+

Sb3+Ca or 2 Ca 2+ ~ ~Sba+ca

+ VCa + ~+ Sb3+Ca + V+Ca +

Li+ca

BRIGHTNESS 25 % 100 %

P h o s p h o r s are p r e p a r e d by h e a t i n g the i n g r e d i e n t s at high t e m p e r a t u r e (> 1000 ~

to o b t a i n a c o m p o u n d having high crystallinity. The s i n t e r i n g

p r o c e s s d e c r e a s e s e n t r o p y a n d is c o u n t e r p r o d u c t i v e to the f o r m a t i o n of v a c a n c i e s in the p y r o p h o s p h a t e lattice. In the a b s e n c e of Li + , vacancyf o r m a t i o n w a s r e t a r d e d w i t h the r e s u l t t h a t the a m o u n t of Sb 3+ w h i c h could be i n c o r p o r a t e d into activator sites actually d e c r e a s e d . In c o n t r a s t , four t i m e s as m a n y activator ions were i n c o r p o r a t e d into the lattice w h e n the c h a r g e - c o m p e n s a t i n g Li + ions w e r e p r e s e n t on n e a r e s t n e i g h b o r sites. Since it w a s n e c e s s a r y to have a v a c a n c y w h i c h could form the c h a r g e c o m p e n s a t e d site, i.e.- V+Ca , r e q u i r e d for the i n c o r p o r a t i o n of t h e impurity, Sb3+ca2+, t h e s e lattice sites did not form, due to t h e s i n t e r i n g p r o c e s s m a n d a t o r y for f o r m i n g the p h o s p h o r . Note t h a t we have w r i t t e n two defect r e a c t i o n s for the case of vacancy f o r m a t i o n in Table 3-2. P y r o p h o s p h a t e is an i n s u l a t o r a n d the f o r m a t i o n of a p o s i t i v e l y - c h a r g e d vacancy is m u c h less likely t h a n the v a c a n c y p l u s a free positive charge. This b r i n g s us to a rule f o u n d in defect c h e m i s t r y t h a t s e e m s to be universal, namely: ALTHOUGH MORE THAN ONE DEFECT REACTION MAY BE APPLICABLE TO A GIVEN SITUATION, PREVAILING

ONLY ONE IS USUALLY FAVORED BY T H E

THERMODYNAMIC

AND

ELECTRON-COMPENSATION

CONDITIONS. 3 , 6 , - DEFECT EQUILIBRIA AND THEIR ENERGY J u s t as c h e m i c a l r e a c t i o n s c a n be d e s c r i b e d a n d c a l c u l a t e d in t e r m s of t h e r m o d y n a m i c c o n s t a n t s a n d c h e m i c a l equilibria, so c a n we also d e s c r i b e

102

defect f o r m a t i o n in t e r m s of equilibria. U s i n g t h e Law of Mass ,action, viz3.6.1-

Law of Mass Action: bB + cC

~ dD + e E

9K =

adD fieE / abB aCc

w h e r e t h e e q u i l i b r i u m c o n s t a n t , K, is w r i t t e n in t e r m s

of t h e a c t i v i t i e s

a n d coefficients of t h e e q u a t i o n given o n t h e left side of 3.6.1. We c a n also w r i t e a n e q u a t i o n for a n y given defect

in t h e lattice. For t h e i n t r i n s i c

defects in a n MX lattice, we w o u l d have: 3.6.2.-

F r e n k e l Defects {for t h e lVlX c r y s t a l )

Mx 3.6.3.-

-7

Mi

z_.

+

VM

9 KF

=

aM i av M / aM

S c h o t t k y Defects (for t h e MX c r y s t a l ] MX

~

V M + VX

. KSh

= aVM aVM / aMx

Note t h a t we have also specified t h e s e e q u i l i b r i u m c o n s t a n t s in t e r m s of t h e activity of t h e ~ i a t e d

defects.

We c a n also w r i t e t h e r m o d y n a m i c

e q u a t i o n s for t h e s e d e f e c t s : 3.6.4.-

Chemical Thermodynamics

AG = AH - T A S 3.6.5.-

=-RTlnK

K=

exp

A S / R - exp - A H / R T

Defect T h e r m o d y n a m i c s Kd = exp

ASd / R - exp - A H d / R T

w h e r e d refers to t h e specific defect. Note t h a t we m u s t k n o w e i t h e r AHd or ASd in o r d e r to c a l c u l a t e t h e e q u i l i b r i u m c o n s t a n t . Alternately, if we k n o w t h e e q u i l i b r i u m c o n s t a n t (or c a n e s t i m a t e it), we n e e d to k n o w only o n e of t h e o t h e r s to c a l c u l a t e t h e e n e r g y associated

with

the

defect.

The

following

summarizes

k n o w l e d g e we h a v e d e v e l o p e d above for t h e S c h o t t k y a n d F r e n k e l

the

103

defects: I. T h e r e

is an Activation E n e r g y for defect

formation.

In m a n y

cases, this e n e r g y is low e n o u g h t h a t defect f o r m a t i o n o c c u r s at, o r slightly above, r o o m t e m p e r a t u r e . 2. Defects m a y be d e s c r i b e d in t e r m s of t h e r m o d y n a m i c c o n s t a n t s a n d equilibria. The

presence

vibrational f r e q u e n c i e s

of defects

changes

both

the

local

in the vicinity of the defect a n d the local

lattice configuration a r o u n d the d e f e c t . One q u e s t i o n we m a y logically a s k is h o w are we to k n o w w h a t t y p e s of defects will a p p e a r in a given solid? T h e a n s w e r to this q u e s t i o n is as follows:

IT HAS BEEN FOUND: "There are two associated effects on a given solid w h i c h have o p p o s i t e effects on s t o i c h i o m e t r y . Usually, one involves the cation site and the other the anion site. Because of the d i f f e r e n c e s in d e f e c t - f o r m a t i o n - e n e r g i e s , t h e c o n c e n t r a t i o n of o t h e r d e f e c t s is ummlly negligible". T h u s , if F r e n k e l Defects p r e d o m i n a t e in a g i v e n solid, o t h e r defects are usually not p r e s e n t .

Likewise, for the S c h o t t k y

applies for ~ i a t e d types

of

Defect.

Note

that this

defects. If t h e s e are not p r e s e n t , t h e r e will still be 2

defects

present,

each

having

an

opposite

effect

upon

stoichiometry.

3 . 7 - DEFECT EQUIIJBRIA IN VARIOUS TYPES OF COMPOUNDS Up

to now,

hypothetical concepts

to

we

have b e e n

example more

of the

concerned solid

complicated

with

state. ones

c o m p o u n d s . We will u s e the p r e c e p t s

the

We will such

as

MX c o m p o u n d now

extend

binary

and

a l r e a d y developed

as a these

ternary

for the s i m p l e

MX c o m p o u n d . For t h e s a k e of simplicity, we r e s t r i c t ourselves to b i n a r y c o m p o u n d s , t h a t is- one cation a n d one anion.

An e x a m p l e of a t e r n a r y

104

c o m p o u n d is AIg(S, w h e r e A a n d B are different cations, a n d S is a s m a l l whole n u m b e r . Our e x a m p l e of a b i n a r y c o m p o u n d will be:

MXs In t h i s case, we m a y have: MX 2 , MX 3 or

even MX4. We c a n d i s t i n g u i s h

b e t w e e n four s t a t e s for t h e s e h y p o t h e t i c a l c o m p o u n d s , i n c l u d i n g s p e c i f i c c o m b i n a t i o n s of t h e s e s t a t e s , to wit: s t o i c h i o m e t r i c vs: n o n - s t o i c h i o m e t r i c non-ionized

vs: i o n i z e d

This gives us four c o m b i n a t i o n s :

1) Non-ionized S t o i c h i o m e t r i c 2) Ionized S t o i c h i o m e t r i c 3) N o n - i o n i z e d N o n - s t o i c h i o m e t r i c 4) Ionized N o n - s t o i c h i o m e t r i c

We will

deal

stoichiometric

with case

the and

first 2)

two

cases,

ionization

of

namely defects

1) in

defects

in

the

stoichiometric

c o m p o u n d s . Since t h e c a s e of n o n - s t o i c h i o m e t r i c c o m p o u n d s c a n be v e r y c o m p l i c a t e d , we will p r e s e n t t r e a t m e n t of t h e last two in t h e A p p e n d i x at t h e e n d of this C h a p t e r . a. .STOICHIOMETRIC BINARY ..COMPOUNDS OF MXs In t h e real world of defect c h e m i s t r y , we find t h a t in a d d i t i o n to t h e s i m p l e defects, o t h e r t y p e s of defects c a n also a p p e a r , d e p e n d i n g

upon

t h e type of c r y s t a l we are d e a l i n g with. T h e s e m a y be s u m m a r i z e d as s h o w n in t h e following, given as 3.7.1. on t h e n e x t page. A c c o r d i n g to o u r n o m e n c l a t u r e , as u s e d in t h e table, V M is a v a c a n c y at an M c a t i o n site, etc. The first five p a i r s of defects given above have b e e n o b s e r v e d e x p e r i m e n t a l l y in solids, w h e r e a s t h e last four have not. T h i s a n s w e r s t h e q u e s t i o n p o s e d above, n a m e l y t h a t defects in solids o c c u r in pairs.

105

3.7.1-

Defects in the MXs C o m p o u n d PAIRS OF DEFECTS ...... i'i'

ii ii

i

i

i

Schottky

VM

+ VX

Frenkel

VM

+ Mi

Anti-Frenkel Anti-Structure

Xi XM

+ Vx + MX

Vacancy- ~ m t i S t r u c t u r e

VM

+

-

i

i

i'

i

i

ii

MX i

i

i

Mi

Interstitial

+

A n ~ - S t r u c t u r e I n t e r s t i t i a l MX + Anti-Structure Interstitial A n t i - S t r u c t u r e Vacanc~r

Xi

Xi i

XM + XM +

Mi Vx

We have now i n t r o d u c e d into our n o m e n c l a t u r e

a distinction b e t w e e n

s t r u c t u r e and a n t i - s t r u c t u r e defects. What this m e a n s is t h a t s t a c k i n g faults can s o m e t i m e s result in the formation of XM and M• defects. T h e s e have been recognized as high-energy defects since they exist as cations (anions) s u r r o u n d e d by a complete positive (negative) charge. In fact, it is difficult to see j u s t how they could exist. However, there have been s o m e cases s tu d ied w he r e they do exist, and so we have included t h e m in our listing of defects although we originally st at ed t h a t s u c h defects are n o t likely to be very prevalent. However, note t hat these defects generally exist as a vacancy plus the a n t i - s t r u c t u r e defect, not as associated antis t r u c t u r e defects together, solid.

as we showed above for the h o m o g e n e o u s

Table 3-3, given on the next page, s u m m a r i z e s the various pairs of defect s possible for binary c o m p o u n d s .

Equilibria

are

given

along with

the

a p p r o p r iate equilibrium c ons t ant . Note t h a t these equations are r a t h e r simple and can be u s e d to specify the equilibrium c o n s t a n t s for t h e s e defects p r e s e n t in t h e lattice. These types of defects have been o b s e r v e d and s tu d ied in the c o m p o u n d s given u n d e r "Example" in t h i s Table. T h e s e are

the

major

types

of defects

to be

expected

in

most

c o m p o u n d s , wh e r e the n u m b e r of sites in the lattice is c o n s t a n t .

i norgani c

106

TABLE 3-3 POSSIBLE DEFECTS IN THE MXs COMPOUND (For a C o n s t a n t N u m b e r of Sites) Type of Defect

Defect Pair E x a m ' ) l e Defect E q u a t i o n

Schottky Frenkel

VM + Vx

,

VM + Mi

ZnO

,

Equilibrium Constant

0 --~ VM + S Vx +(~Vi

KSh = VM Vx S V i a

MM + o~Vi-~ VM + Mi K F - V M Mi / MM Vi a

,

Anti-Frenkel

Xi + Vx

Lfla2 Xx + [~ Vi ~ Xi + Vx

KAF = Xi VX / XX Vi~

,

Anti-Struct. If t h e r e

XM+MX

AuZn MM + Xx ~ Mx + XM KAS = M x X M / M M Xx

is an excess of sites p r e s e n t

in the lattice,

s o m e w h a t different (and m u c h m o r e complicated).

the

s i t u a t i o n is

In at least one case,

one type h a s b e e n o b s e r v e d a n d so we c a n n o t c o m p l e t e l y ignore t h e m , even t h o u g h t h e y are fairly r a r e . Table 3-4 gives a c o m p l e t e listing of t h e s e types of defects a n d the n a m e s given to t h e m . TABLE 3 - 4 (Excess N u m b e r of Sites) Name of

Defect

Defect

Pair Type Vac.-Struc. VM + MX

Defect E q u a t i o n 5MM ~ 6Mx +(1-6) VM

Equilibrium Constant KVS =

MSXVMI+Svi a / M5+ (x IV i

Struct.Vac.

Vx + XM

6Xx~XM ( 1 +6)Vx +ctVi

KSV =XM VX 1+5 Vi a/XSx

Interstitial

Mi + Xi

6(MM+SXx) --~Mi+S Xi

Ki = M i X i S V i l + 5 + a [MM 5 XxS5 1

- ( 1 +6+a)Vi

Interstitial

Mx+Xi

MX KIS = MX Xi 1+SVi(1+5+a) / MM Xx 1+5 +( 1 +6)Xi( 1 +8+a)Vi

Mi+XM

(I+6)MM

Struct. StructInterstial These

/

five defects

MM + ( I + 6 ) X x

+

~

SXx

(1+6)Mi + 5XM

KSI = MiXM a / MM 1+5 XXS

are b a s e d u p o n an excess in the n u m b e r of sites

available. This excess we call "8". Note t h a t we are not s p e a k i n g of t h e ratio of c a t i o n s to anions, i.e.- s t o i c h i o m e t r y , b u t of the total n u m b e r of sites. To see how this is possible, c o n s i d e r the V a c a n c y - S t r u c t u r e type.

107

We have a VM (a cation vacancy) a s s o c i a t e d w i t h an M x , an M a t o m on an anion site.The total n u m b e r of a t o m s r e m a i n s c o n s t a n t , b u t t h e r e is an e x c e s s of cations, notably M x . F o r t u n a t e l y , we do not have to deal w i t h t h e s e e q u a t i o n s very m u c h b u t i n c l u d e t h e m for the s a k e of c o m p l e t e n e s s . Note t h a t we have u s e d a h y p o t h e t i c a l c o m p o u n d to r e p r e s e n t

all of t h e

possible c o m p o u n d s t h a t we m i g h t e n c o u n t e r . In o r d e r to relate t h e s e e q u a t i o n to the real world, t h e following table lists the e q u a t i o n s given in Table 3-3 in t e r m s of a CaC12 crystal. TABLE 3- 5 POSSIBLE DEFECTS IN THE CaC12 COMPOUND (For a C o n s t a n t N u m b e r of Sites) Type of Schottky

VCa+ VC1

0 --~ VCa + 2Vc1 +(1Vi

Equilibrium Constant KSh = VCa V2C1 V i a

Frenkel

Vea+ Cai

MM + (~Vi ~ VM + Mi

KF =VCa Cai / Caca Vi a

ClCl + ~ Vi ~

KAF = Cli VC1 / C1c1Vi~

Defect

Anti-Frenkel Anti-Struct.

Defect Pair

Cli + VC1

Defect E q u a t i o n

Cli + VC1

C1Ca+Cac1 C~2a + C1C1--~ C1Ca+Cac1

KAS = C1Ca-~ac1

~a+

C1c1

We have u s e d CaC12 b e c a u s e it is a familiar m a t e r i a l a n d the e q u a t i o n s w h i c h r e s u l t are r a t h e r simple. Note t h a t in m o s t cases, the e q u i l i b r i u m c o n s t a n t is given in t e r m s of the r a t i o of defects to the n u m b e r of ions in t h e i r correct position in t h e lattice. Let us n o w c o n s i d e r the c o n c e n t r a t i o n of defects t h a t m i g h t a p p e a r in t h e MXs crystal. b . . D E F E C T CONCENTRATIONS IN ~

COMPOUNDS

To see h o w t h e s e e q u a t i o n s m i g h t be used, c o n s i d e r t h e following. First, s u p p o s e we w a n t to be able to d e t e r m i n e the n u m b e r of intrinsic d e f e c t s in a given solid. Since p a i r s of defects p r e d o m i n a t e in a given solid, t h e n u m b e r of e a c h type of i n t r i n s i c defect, Ni (M) or Ni (X), will equal e a c h

108

other, For S c h o t t k y defects in the MXs crystal, we have: 3.7.2-

Ni(VM)

= Ni(S Vx)

This m a k e s o u r m a t h e m a t i c s s i m p l e r since we can rewrite the S c h o t t k y e q u a t i o n of Table 3-3 as: 3.7.3-

0 -~ Ni (VM) + S Ni (VM) + a Vi

Here, we have e x p r e s s e d the c o n c e n t r a t i o n as the ratio of defects to t h e n u m b e r of M- a t o m sites (this h a s c e r t a i n a d v a n t a g e s as we will see). We can t h a n rewrite the defect equilibria e q u a t i o n s of Table 3-3

a n d 3-4 in

t e r m s of n u m b e r s of i a t r i m ~ r defect c o n c e n t r a t i o n s , s h o w n as follows: S o m e of t h e s e e q u a t i o n s are c o m p l i c a t e d a n d we n e e d to e x a m i n e t h e m in m o r e detail so as to d e t e r m i n e h o w t h e y are to be used. 3.7.4.-

EQUILIBRIUM CONSTANTS a n d INTRINSIC DEFECTS

SCHO~:

kSh = Ni (S N i ) 2

= S s NiS+l

FRENKEL:

kF

ANTI-FRENKEL:

kAF = Ni 2 / (S- Ni)(r

ANTI-STRUCT:

kAS = Ni 2 / ( 1 - N i ) ( S - N i ) S

VAC.-STRUCT:

kvs

= S s (S + 1 )S+l. Ni2S+l / (S-Ni -S Ni )S

STRUCT-VAC:

ksv

= (S+ 1)S+l NiS+2 / (S - N i - S Ni)

INTERSTIAL:

kI

= S S Ni S+I / (a - N i - SNi) a+S+I

= Ni 2 / (1- Ni) (r

Ni)

,.

Ni)

E q u a t i o n 3.6.10. given above s h o w s t h a t intrinsic defect c o n c e n t r a t i o n s will i n c r e a s e with i n c r e a s i n g t e m p e r a t u r e a n d t h a t t h e y will be low for high E n t h a l p y of defect formation. This arises b e c a u s e the e n t r o p y effect is

a positive

exponential.

exponential Consider

the

while

the

enthalpy

effect

following

examples

of

is

various

a

negative types

c o m p o u n d s a n d the n a t u r a l defects w h i c h m a y o c c u r ( d e p e n d i n g how the c o m p o u n d s w e r e originally f o r m e d ) :

of

upon

109

TiO

is c u b i c w i t h t h e NaCI s t r u c t u r e . A s a m p l e w a s a n n e a l e d at 1 3 0 0 ~

D e n s i t y a n d X-ray m e a s u r e m e n t s r e v e a l e d t h a t t h e i n t r i n s i c d e f e c t s w e r e S c h o t t l ~ in n a t u r e (VTi + VO) a n d t h a t t h e i r c o n c e n t r a t i o n w a s 0 . 1 4 0 . I n t h i s c a s e , S = 1 so t h a t : 3.7.5.-

KSh

= 0.0196 = 2 x I 0 -2

T h i s c r y s t a l is q u i t e d e f e c t i v e s i n c e i o u t of 7 T i - a t o m - s i t e s ( 0 . 1 4 -I) is a v a c a n c y , a n d l i k e w i s e for t h e o x y g e n - a t o m - s i t e s . A n o t h e r e x a m p l e isFrom intrinsic

thermodynamic

defects

were

measurements,

Anti-Frenkel

in

it w a s

nature,

i.e.-

found

that

the

(Hi

VH).

An

+

e q u i l i b r i u m c o n s t a n t w a s c a l c u l a t e d as: 3.7.6.-

kAF

--

3 . 0 x 10 -4

at a t e m p e r a t u r e of 6 0 0 ~

T h i s c o m p o u n d h a s t h e c u b i c fluorite s t r u c t u r e w i t h o n e o c t a h e d r a l i n t e r s t i c e p e r Ce a t o m . T h e r e f o r e , a = 1, a n d S = 2 for C e l l 2 . We c a n therefore write3.7.7.-

kAF

=

Ni 2 / (2- Ni )(1- Ni)

Ni

=

2.4 x 10 -2

= 3 . 0 x I 0 -4

or

(600 ~

T h i s m e a n s t h a t I o u t of 4 2 h y d r i d e a t o m s is i n t e r s t i t i a l , a n d I o u t of 8 4 h y d r i d e - a t o m - s i t e s is v a c a n t . Let u s n o w r e v i e w w h a t w e h a v e c o v e r e d c o n c e r n i n g s t o i c h i o m e t r i c b i n a r y compoundsI. We h a v e s h o w n t h a t d e f e c t s o c c u r in p a i r s . T h e r e a s o n for t h i s lies i n t h e c h a r g e - c o m p e n s a t i o n p r i n c i p l e w h i c h o c c u r s in all ionic solids.

110

2.

Of

the

nine

defect-pairs

possible,

only

5

have

actually

been

e x p e r i m e n t a l l y o b s e r v e d in solids. T h e s e areSchottky Frenkel Anti-Frenkel Anti-Structure Vacancy-Structure. 3. We have given d e f e c t - e q u a t i o n s for all n i n e types of defects, a n d t h e E q u i l i b r i u m C o n s t a n t s t h e r e b y associated. However, calculation of t h e s e equilibria would r e q u i r e values in t e r m s of e n e r g y at e a c h site, values w h i c h are difficult to d e t e r m i n e . A b e t t e r m e t h o d is to c o n v e r t t h e s e EC e q u a t i o n s to t h o s e involving n u m b e r s of each type of intrinsic defect, as a ratio to an intrinsic cation or anion. This w o u l d allow u s to calculate t h e actual n u m b e r of i n t r i n s i c defects p r e s e n t in the crystal, at a s p e c i f i e d temperature.

Let us n o w s u m m a r i z e w h a t we have c o v e r e d : 1. We have s h o w n t h a t defect e q u a t i o n s a n d equilibria c a n be w r i t t e n for t h e MXs c o m p o u n d , b o t h for s t o i c h i o m e t r i c a n d n o n s t o i c h i o m e t r i c cases. 2. In addition, we have s h o w n t h a t f u r t h e r defect f o r m a t i o n c a n be i n d u c e d by e x t e r n a l r e a c t i n g species, a n d t h a t t h e s e act to f o r m specific types of defects, d e p e n d i n g u p o n the c h e m i c a l n a t u r e of t h e crystal lattice. 3. We have also s h o w n t h a t the i n t r i n s i c defects c a n b e c o m e ionized to form s p e c i e s not found in n a t u r e . 3-8,-

THE E F F E .CI'S OF PURITY [AND. I M P U R I T I E S )

Our s t u d y h a s led us to the point w h e r e we c a n realize t h a t t h e p r i m a r y effect of i m p u r i t i e s in a solid is the f o r m a t i o n of defects, p a r t i c u l a r l y t h e F r e n k e l a n d S c h o t t k y types of a s s o c i a t e d defects. T h u s , the p r i m a r y effect

III

obtained in a high purity solid is the m i n i m i z a t i o n of defects. I m p u r i t i e s , particularly those of differing valencies t h a n those of the lattice, cause c h ar g ed vacancies a n d / o r interstitial sites. "We can also increase t h e reactivity of a solid to a certain extent by m a k i n g it m o r e of a defect crystal by the addition of selected impurities. It is not so a p p a r e n t as to w h a t h a p p e n s to a solid as we cont i nue to purify it. To u n d e r s t a n d this, w e need to examine the various grades of purity as we normally e n c o u n t e r them. Although we have e m p h a s i z e d inorganic c o m p o u n d s t h u s far (and will co n tin u e to do so), the s a m e principles apply to organic crystals as well. COMMERCIAL GRADE is usually about 95% purity (to o r i e n t ourselves, w h a t we m e a n is t h a t 95% of the m at eri al is t h a t specified, w i t h 5% being different (unwanted-?) material. Laboratory or "ACS-REAGENT GRADE" averages about 99.8% in purity. The GRADES listed are n a m e d for the usage to w h i c h they are applied, and are usually m i n i m u m purities r e q u i r e d for the particular application. Fiber optic materials are c u r r e n t l y p r e p a r e d by chemical vapor d e p o s i t i o n t e c h n i q u e s b e c a u s e any h a n d l i n g of m a t e r i al s i n t r o d u c e s i m p u r i t i e s . 3.8. I.- GRADES OF PURITY FOR COMMON CHEMICALS_ I m p u r i t y Atoms p e r GRADE

%

Commercial

95

50,000

Laboratory

99.8

2000

Luminescent

.

Mole of C o m p o u n d

Dgm

....

99.99 .

.

100 .

. .

1.2 x 1021 6 x 1019 .

Semi-conductor

99.999

Crystal G r o w t h

99.9999

1

6 x 1018 6 x 1017

Fiber-Optics

99.999999

0. 01

6 x 1015

,,,

10

3.0 x 10 22

F u r t h e r m o r e , this is the only way found to date to p r e p a r e the r e q u i r e d materials at this level of purity. The frontiers of purity a c h i e v e m e n t p r e s e n t l y lie at the fraction of p a r t s per

quadrillion

level.

However,

because

of E n v i r o n m e n t a l

Demands,

analytical m e t h o d o l o g y p r e s e n t l y available far exceeds this. We can n o w

112

analyze m e t a l s a n d anions at the femto level (parts p e r quadrillion = 10 -15) if we w i s h to do so.

Nevertheless, it is becoming apparent that as h/gh purity inorganic solids are being obtained, we observe that their physical properties may be somewhat different than those usually accepted for the same compound of lower Imrity. The h i g h e r - p u r i t y c o m p o u n d m a y u n d e r g o solid state r e a c t i o n s s o m e w h a t different t h a n t h o s e c o n s i d e r e d "normal" for the c o m p o u n d . If we r e f l e c t b u t a m o m e n t , we realize t h a t this is w h a t we m i g h t e x p e c t to o c c u r as w e obtain c o m p o u n d s (crystals) c o n t a i n i n g far fewer intrinsic d e f e c t s . It is p r o b a b l y true t h a t m a n y of the d e s c r i p t i o n s of physical a n d solid state r e a c t i o n m e c h a n i s m s n o w existing in the l i t e r a t u r e are only partially correct.

It would a p p e a r

that part

of the

frontier

of k n o w l e d g e

C h e m i s t r y of The Solid State will lie in m e a s u r e m e n t

for

of physical a n d

c h e m i c a l p r o p e r t i e s of inorganic c o m p o u n d s as a function of purity. S.u~ested Reading 1. A.C. D a m a s k a n d G.J. Dienes, Point Defects in Metals, G o r d o n Breach, New York ( 1 9 7 2 ) . 2. G.G. Libowitz, "Defect Equilibria in Solids", Treatice

&

on Solid State

Chem.- (N.B. Hannay- Ed.), I, 3 3 5 - 3 8 5 , ( 1 9 7 3 ) . 3. F.A. Kroeger, The Chemistry of Imperfect Crystals, N o r t h - H o l l a n d , Amsterdam (1964). 4. F.A. Kroeger & H.J. Vink in Solid State Physics, Advances in Research and Applications (F. Seitz & D. TurnbuU-Eds.), pp. 3 0 7 - 4 3 5 ( 1 9 5 6 ) . 5. J.S. A n d e r s o n

in Problems

of Non-Stoichiometry (A. Rabenau-Ed.),

p p . 1 - 7 6 , N. Holland, A m s t e r d a m ( 1 9 7 0 ) . 6. W. Van Gool, Principles of Defect chemistry of Crystalline Solids, Academic Press, New York ( 1 9 6 4 ) . 7. G. Brouwer, "A General A s y m m e t r i c Solution of Reaction E q u a t i o n s C o m m o n in Solid State Chemistry", Philips Res. Rept., 9 , 3 6 6 - 3 7 6 (1954)

113

Pairs in Ionic Crystals", Phys. Rev., 1 1 2 , 5 4 - 5 5

8. A. B. Lidiard, " r (1958). 9. J.S. A n d e r s o n ,

"l'he C o n d i t i o n s of E q u i l i b r i u m of N o n s t o i c h i o m e t r i c

C h e m i c a l C o m p o u n d s , Proc. Roy. Soc. (London), 1 1 8 5 , 6 9 - 8 9 ( 1 9 4 6 ) . 10. N.N. G r e e n w o o d , Ionic Crystals, Lattice Defects & Non-Stoichtometry, Butterworths, London (1968). PROBLEMS FOR CHAFTER 3 I. Identify the t y p e s of p o i n t defects likely to a p p e a r in a h o m o g e n e o u s solid.

Contrast

those

to t h e

typical defects

which

may appear

in

a

h e t e r o g e n e o u s solid. 2. Given the s t r o n t i u m

chloride

crystal, write

the

defect

reaction(s)

e x p e c t e d ff l i t h i u m chloride is p r e s e n t as an i m p u r i t y . Do likewise for t h e a n t i m o n y chloride i m p u r i t y . Also, write t h e defect r e a c t i o n s e x p e c t e d if b o t h i m p u r i t i e s are p r e s e n t in e q u a l q u a n t i t i e s .

3. D r a w o n e or m o r e "plane-nets" for the "P" cation c o m b i n e d w i t h a "U" anion. I n d i c a t e all of the possible

defects

t h a t c a n a p p e a r . Write t h e

symbol of e a c h as you p r o c e e d . I n c l u d e p a i r s of defects as n e e d e d .

4. Write e q u a t i o n s for as m a n y of the t h i r t y (30) defects r e a c t i o n s of y o u r "PU" plmne-net as you can. Do not forget t h e d e f e c t - p a i r s . 5. For a divalent lattice c o n t a i n i n g b o t h Ca 2+- a n d S = , write all 30 of t h e e q u i l i b r i u m e q u a t i o n s if b o t h As3+ a n d CI- are s u b s t i t u t i o n a l i m p u r i t i e s in the lattice. ( I n c l u d e all of t h e possible v a c a n c i e s a n d i n t e r s t i t i a l sites). Be s u r e to i n c l u d e t h e site w h e r e e a c h ion is s i t u a t e d . 6. Calculate the activation e n e r g y at r o o m t e m p e r a t u r e for a cubic c r y s t a l c o n t a i n i n g 3 x 10 -3 v a c a n c i e s .

114

7. Rewrite all of the e q u a t i o n s in Table 3-2 in t e r m s of the c o m p o u n d , CaCI2, a n d c o m p a r e t h e m to t h o s e given in Table 3-3. What effect d o e s n o n - s t o i c h i o m e t r y have u p o n the n u m b e r s a n d types of defects t h a t m a y be p r e v a l e n t ? Do the s a m e for the c o m p o u n d AsCI3. 8. Write a series of e q u a t i o n s for the F r e n k e l defect, similar to t h o s e given for the S c h o t t l ~ defect, i.e.- E q u a t i o n s 3.7.2 to 3.7.3. 9. Using a "MX" plane net, illustrate the defects s h o w n in 3.7.1. for t h e h y p o t h e t i c a l c o m p o u n d , MX. Do the s a m e for the c o m p o u n d , M X 2 , u s i n g the configuration: X-M-X. I0. Given: kSh = 3 x I 0 3

for CaCI2, calculate the n u m b e r of i n t r i n s i c

defects p r e s e n t in this crystal. If CaCI2 is f a c e - c e n t e r e d

cubic, u s e t h e

s a m e e q u i l i b r i u m c o n s t a n t to calculate the intrinsic Frenkel, A n t i - F r e n k e l a n d I n t e r s t i a l defects e x p e c t e d in this crystal. 11. The native defects in ZnO are: V=zn, Z n i , VO, Zn+i, VO+ , & V z n . a. Write the defect r e a c t i o n s leading to their f o r m a t i o n . b. Write the defect r e a c t i o n s for ZnO t h a t h a s b e e n s u b j e c t e d to a r e d u c i n g a t m o s p h e r , i.e.- a r e a c t i o n with H2. Predict the i n t r i n s i c defects w h i c h will be p r e s e n t . 12. Draw a h e t e r o g e n e o u s lattice, u s i n g circles a t o m p o s i t i o n s in a simple

cubic lattice.

a n d s q u a r e s to i n d i c a t e

Indicate

both

Schottky

and

F r e n k e l defects, p l u s the simple lattice defects. Hint- u s e b o t h cation a n d anion sub-lattices. 13. The following v a c a n c y p a r a m e t e r s have b e e n m e a s u r e d for a solid as: AH v = 2 2 0 0 cal / mol ASv = 2.0 k Calculate the vacancy c o n c e n t r a t i o n at 160 ~ A p p e n d i x III)

a n d at 298 ~

(Hint: see

115

14. The following vacancy p a r a m e t e r s have b e e n m e a s u r e d for gold as: AH v = 2 1 . 6 2 kcal / mol ASv= 2.0 K Calculate the v a c a n c y c o n c e n t r a t i o n at the m e l t i n g p o i n t of 1063 ~ a n d at 298 ~

Appendix I. C o n c e n t r a t i o n s of Defects in Non-Ionized N o n - S t o i c h i o m e t r i c C o m p o u n d s This

presentation

is p r e s e n t e d

for t h o s e w h o w i s h

to

examine

the

m a t h e m a t i c s of b o t h n o n - s t o i c h i o m e t r i c intrinsic- defect c o m p o u n d s a n d the ionization of defects in b o t h s t o i c h i o m e t r i c

and nonstoichiometric

c o m p o u n d s as r e p r e s e n t e d by:

MXs• A. DEFECTS IN NON-STOICHIOMETRIC MXs + 8 COMPOUNDS We now p r o c e e d as we did for the s t o i c h i o m e t r i c case, n a m e l y to d e v e l o p defect- c o n c e n t r a t i o n e q u a t i o n s for the n o n , s t o i c h i o m e t r i c case. C o n s i d e r the effect of A n t i - F r e n k e l defect p r o d u c t i o n . F r o m Table 2-1, we get KAF, with its a s s o c i a t e d equation,

kAF

9 In Table 2-2, we u s e Kxi

for X-

interstitial sites. C o m b i n i n g these, we get:

App 1.1.-

KAF = KVx " Kxi

When both Vx

and

Xi

=

Ni 2 / (S-Ni} (r

coexist

in

the

s t o i c h i o m e t r y (from 3.7.10.) b e c o m e s : Appl.2.-

5

= [Xi] -

[Vx]

Using the e q u i l i b r i u m c o n s t a n t of 3.7.9., i.e.-

Ni )

lattice,

the

deviation

from

116

App 1.3.-

Kv x =

Px2 I/2 [Vx] /Xx

=

pX 2 1/2 [Vx] / S - [Vxl

a n d t h e a p p r o p r i a t e o n e f r o m Table 3-2 {i.e.- Kxi ), we get {assume for s i m p l i c i t y t h a t S = ~ = 1) 9 App 1.4.-

5

= ct Px21/2 K x i / p x 2 Kxi + 1

-

S K v x / Kv x + Px21/2

We c a n r e a r r a n g e t e r m s in App 1.4. to obtain: App 1.5.-

Kxi {I- 5} p x 2- 6( K v x K x i + 1) Px2 I/2 -Kvx 2 {1- 6 }

U s i n g App I . I . a n d Ni << App 1.6.-

=

0

I, we n o w obtain:

Ni 2 (1- 5}px 2- 6 Kv x Px21/2-

Kvx 2 ( I - 6) = 0

Solving for p x 2 yields 9

App 1.7.p x 2 = KVx 2 ( 82+ 2 N i ( l -

Since

at s t o i c h i o m e t r i c

62 ) + 6 1 6 2 + 4 N i ( I - 5 2 ) ] I 1 2 1 2 N i 4 (I-6) 2

composition,

5 must

e q u a l zero,

this

rather

f o r m i d a b l e e q u a t i o n r e d u c e s to:

App 1.8.where

Px2 o

Px2~ = Kv x 2 ! Ni2 is t h e p r e s s u r e of X2

gas in e q u i l i b r i u m w i t h t h e MXs

c r y s t a l at t h e s t o i c h i o m e t r i c c o m p o s i t i o n . T h i s gives u s t h e o p p o r t u n i t y to divide App 1.7. by App 1.8. to obtain:

App 1.9.- Px2 /Px2 ~

= 82 + 2 Ni (I- 62) + 6162 + 4 Ni { I - 6 2 } 1 2 N i 2 { i - 5 ) 2

We c a n t h e r e f o r e c a l c u l a t e 5 in t e r m s of t h e ratio of Px2 to Px2 o a n d Ni. T h i s r e s u l t is s h o w n in t h e following d i a g r a m ;

117

H e r e i n is s h o w n h o w 8 c h a n g e s f r o m n e g a t i v e to p o s i t i v e at t h e h i g h e r p r e s s u r e r a t i o s. For a h y p o t h e t i c a l MXs c o m p o u n d (S = 1), w h i c h c o n t a i n s A n t i - F r e n k e l d e f e c t s , to o b t a i n a 0 . 7 % d e v i a t i o n f r o m t h e s t o i c h i o m e t r i c c o m p o s i t i o n r e q u i r e s a p r e s s u r e i n c r e a s e of s o m e 5 0 0 0

fold, w h e n

the

o r i g i n a l i n t r i n s i c d e f e c t c o n c e n t r a t i o n is 10 -4 . H o w e v e r , if it is 10 -2 , o n l y a two-fold i n c r e a s e in p r e s s u r e is n e e d e d

to c a u s e t h e s a m e effect on a

d e v i a t i o n of 0 . 7 % , :i.e.- 6 = + 0 . 0 0 7 in MXs+8.

This means that a

hlghly d e f e c t i v e

t h e f o r m a t i o n of m a n y m o r e defects

s t r u c t u r e is m u c h m o r e p r o n e t o c a u s e

defects

than one contslniug

only a few

118

A l t h o u g h we will n o t t r e a t t h e o t h e r t y p e s of p a i r s of defects, it is well to n o t e t h a t s i m i l a r e q u a t i o n s c a n also b e d e r i v e d

for t h e o t h e r i n t r i n s i c

defects. W h a t we h a v e really s h o w n is t h a t e x t e r n a l r e a c t a n t s c a n c a u s e f u r t h e r c h a n g e s in t h e n o n - s t o i c h i o m e t r y of t h e solid. Let u s n o w c o n s i d e r ionized defects.

It s h o u l d be c l e a r t h a t a n e x t e r n a l g a s e o u s f a c t o r h a s a

m a j o r effect u p o n d e f e c t f o r m a t i o n . T h e e q u a t i o n s given above are v e r y c o m p l i c a t e d a n d r e p r e s e n t m o r e closely w h a t actually h a p p e n s in t h e r e a l w o r l d of defect f o r m a t i o n in c r y s t a l s .

APPENDIX I I ANALYSIS OF A REAL CRYSTAL USING A THERMODYNAMIC M E T H O D In t h e n o n - s t o i c h i o m e t r i c the m a t h e m a t i c s

case where

i o n i z a t i o n of d e f e c t s is t h e n o r m ,

b e c o m e too c o m p l i c a t e d so t h a t t h e e q u a t i o n s are n o t

solvable. However, we c a n u s e a t h e r m o d y n a m i c r e s u l t s we w a n t . We will p r e s e n t u s e in p h o t o g r a p h i c

film

method

to o b t a i n t h e

h e r e t h e c a s e of silver b r o m i d e w h o s e

highlights

the

u s e of d e f e c t

chemistry

for

practical purposes.

I. T H E At!Br CRYSTAL WITH A DIVALENT IMPURITY, Cd 2+ v

C o n s i d e r t h e crystal, AgBr. B o t h c a t i o n a n d a n i o n are m o n o v a l e n t , i.e.- Ag + a n d B r - . T h e a d d i t i o n of a d i v a l e n t c a t i o n s u c h as Cd 2+ s h o u l d i n t r o d u c e vacancies, VAg,

into

mechanism.

maintain

To

the

c r y s t a l , b e c a u s e of t h e electro-neutrality,

we

charge-compensation prefer

to

define

the

s y s t e m as: App. 2 - 1 . -

(1-6) Ag + Br- : 6 Cd 2+ S =

F o r t u n a t e l y , AgBr is e a s y to g r o w as a single crystal, u s i n g S t o c k b a r g e r Techniques.

Possession

and

facilitates o u r m e a s u r e m e n t

measurement

of d e f e c t s .

T h e d e f e c t s we e x p e c t to find are:

of a single

crystal

greatly

119

App. 2-2.-

WAg , Agi , e- , n+, CdAg , a n d

[CdAg, VAg]

The last defect is one involving two n e a r e s t n e i g h b o r cation l a t t i c e - s i t e s The following Table p r e s e n t s t h e defect r e a c t i o n s g o v e r n i n g this case.

TABLE 2-App D E F E C ~ REACTIONS IN THE AgBr CRYSTAL CONTAINING Cd 2+ a.

0

b. c.

AgA4~ + ( x V i AgAg + (~ Vi

d.

e_ VAg

z._

-7 ~.--

-7

Ag i

+ 71;+ + Agi

VAg

+ Agi +

--7

Agi + + eBrBr + VBr

e.

1 / 2 Br2

-7

f.

CdA_g ~'-7

CdAg+ + e-

g.

CdAg + VAg

L._

[CdAg, VAg]

h.

CdAg+ + WAg-

-7

lea N , v~l

i. For C o n s t a n t Cd CdT = CdAg + CdAg+ + [CdAg , WAg] = K j. For E l e c t r o - n e u t r a l i t y e- +VAg- = ~+ + A g i +

+ CdAg+

E x p e r i m e n t a l l y , we find t h a t if we fix the Cd 2+ c o n t e n t at s o m e c o n v e n i e n t level, it is n e c e s s a r y to a n n e a l the AgBr c r y s t a l s at a fixed t e m p e r a t u r e for t i m e s long e n o u g h to achieve c o m p l e t e equilibrium. If t h e t e m p e r a t u r e is c h a n g e d , t h e n b o t h type a n d relative n u m b e r s of defects m a y also c h a n g e . S i n g l y - c h a r g e d defects p r e d o m i n a t e , i.e.-

App. 2-3.-

2VAg

=

Agi +

+

CdA~

T h e s e in t u r n m a y form the c o m p l e x : App. 2-5.-

CdAg+ +

App. 2-6.-

VAg ~

WAg-~

WAg

[CdAg

+ a+

, VAgl

120

B. DEFECT DISORDER IN A~Br- A THERMODYNAMIC Ap.PROA.CH. v

To i l l u s t r a t e yet o n e a p p r o a c h to analysis of defect f o r m a t i o n , c o n s i d e r t h e i n f l u e n c e of Br2 gas u p o n defect f o r m a t i o n in AgBr. T h e

free e n e r g y of

f o r m a t i o n , AG, is r e l a t e d to t h e r e a c t i o n :

App. 2 - 8 . -

Ago + 112 Br2 {g) ~

AgBr{g)

{AG~

}

T h i s c a n be r e w r i t t e n as: App. 2 - 9 . -

aAg p l / 2

Br 2

=

exp AG ~

/ RT

It m a k e s n o difference as to w h i c h of t h e activities w e use. If w e n o w fix PBr 2 at s o m e low value, we Fred t h a t t h e ~ / e crystal, as i n f l u e n c e d b y t h e e x S e m a / f a v t a r ,

App. 2- I 0 .

Agi + , WAg , Bri-

d e f e c t s in o u r AgBr

PBr2, will be-

, VBr + , e

and

~+

B e c a u s e of t h e h i g h e l e c t r o s t a t i c e n e r g y r e q u i r e d to m a i n t a i n t h e m in an ionic c r y s t a l s u c h as AgBr, w e c a n safely i g n o r e t h e following lac~aible defects:

App. 2 - 1 1 .

AgBr + , AgBr ++ ,

BrAg

, BrAg = .

If we h a v e t h e r m a l d i s o r d e r at r o o m t e m p e r a t u r e

(I do n o t k n o w of any

c r y s t a l for w h i c h t h i s is n o t t h e case), t h e n we c a n e x p e c t t h e f o l l o w i n g defect r e a c t i o n r e l a t i o n s :

App. 2 - 1 2 . -

Agi +

=

WAg"

Bri

=

Agi +

VAg-

=

VBr +

Bri -

=

VBr +

121

At e q u i l i b r i u m , t h e following e q u a t i o n s arise, as s h o w n in A p p . 2 - 1 3 . App. 2 - 1 3 . -

P-gAg

+ 1 / 2 Br2 {g) ~

K6 =

WAg-

AgBr{s) + VAg

+ n+

~+ / p l / 2 B r 2

T h i s gives u s a t o t a l of eight (8) c o n c e n t r a t i o n s to calculate. T h e y involve: App. 2 - 1 4 . -

Ag~,

BrBr , A g i ,

Bri

, WAg- , VBr + , e- ,

n+

O u r p r o c e d u r e is to set u p a site b a l a n c e in t e r m s of lattice m o l e c u l e s , i.e.- AgBr: App. 2 - 1 5 . -

AgAg , + WAg- + e-

= 1

(e-

BrBr

= 1

(~+

+ VBr + + ~+

~-

Ag-Ag) Br+Br )

S i n c e Br2 {gas} is t h e d r i v i n g force for defect f o r m a t i o n , w e n e e d also to c o n s i d e r d e v i a t i o n f r o m s t o i c h i o m e t r y , 6 . T h u s , w e also set a Ag1=8 Br balance: App. 2 - 1 6 . -

Aga.g + A g i +

+e-

BrBr + Bri-

=

+~+

1+6

=

1

To m a i n t a i n e l e c t r o - n e u t r a l i t y , t h e following e q u a t i o n is a p p l i c a a b l e : App. 2 - 1 7

.-

Agi +

+ VBr +

+ n+

=

Bri-

+ WAg

+ e-

We also set u p t h e following e q u a t i o n s : App. 2 - 1 8 . -

Ag~

+

r

~

Agi +

BrBr

+

r

~

Bri-

+~+

~

e-

0

+ VAg+

VBr +

Ke

= Agi+VAg-/ Vi a

Ky = B r i - V B r + / V i a K~ =

e-

~+

Note t h a t we h a v e d i s t i n g u i s h e d b e t w e e n t h r e e (3) s i t u a t i o n s , to wit: a. E l e c t r o - n e u t r a l i t y b. T h e r m a l D i s o r d e r

122

c. N o n - s t o i c h i o m e t r y (excess c a t i o n ) These

are t h e eight e q u a t i o n s

(App. 2-12.

to App.

2-18.)

required

to

c a l c u l a t e t h e defect c o n c e n t r a t i o n s a r i s i n g f r o m t h e effects of t h e e x t e r n a l factor, PBr2-

From

measurements

migration

of

of

charged

conductivities,

species),

transfer

lattice

numbers

constants

and

(electro-

experimental

d e n s i t i e s , it h a s b e e n s h o w n t h a t F r e n k e l defects p r e d o m i n a t e

(Lidiard -

1957). T h i s m e a n s that:

App. 2 - 1 9 . -

Agi + , WAg-

F u r t h e r m o r e , VAg-

>>

Bri-

,

VBr +

_~ Agi + so t h a t ill t e r m s of o u r e q u i l i b r i u m c o n s t a n t s

we get: App. 2-20.-

FOR F R E N K E L D E F E C T S :

K~ >> K~

a n d K~

T h u s , we n e e d only to c o n s i d e r t h e above two (2) defects,

>> K8

n a m e l y - VAg-

a n d Agi + , since t h e y are t h e m a j o r c o n t r i b u t o r s to n o n - s t o i c h i o m e t r y . c a l c u l a t i n g POBr 2 as before ( w h e n 6 = 0 ,

see 2 . 7 . 2 2 & 2.7.23.),

By

we c a n

e x p r e s s o u r overall defect e q u a t i o n as:

App. 2-21 .-

pl/2Br2 / (POBr2) 1/2 = {6 / 2e +[ ( 1+ 8/2e )211/ :2 } {6/218 + [ 1+(8/2[~)211 / 2 }

B e c a u s e of t h e c o n d i t i o n s given in App. 2-14., t h e first half of t h e e q u a t i o n can be set equal to one. equih'brium

App. 2 - 2 2 . -

constants

e ~

NotethRtwe

are using

8 ,

~

,

and

y

as the

. i.e. -

Ke 1/2

~

In t h e r e m a i n i n g p a r t of t h e e q u a t i o n ,

-~

K~ 1/2

y ~

K71/2

6 >> ~ . By t a k i n g l o g a r i t h m s , w e

123

c a n t h e n obtainApp. 2 - 2 3 . -

1/21n

/po

PBr 2

=

Br2

ln5

- In [3

T h i s r e s u l t t h e n l e a d s us to a plot of the effect of p a r t i a l p r e s s u r e of Br2 on t h e deviation from s t o i c h i o m e t r y , 5 , for the AgBr crystal. (This w o r k is d u e to G r e e n w o o d - 1 9 6 8 ) . For ~ = 0 , t h e r e is a point of inflection w h e r e the slope of the line is defined by the e q u i l i b r i u m c o n s t a n t , i.e.- ~

=

K~Y2 . The larger t h i s

value, the flatter is the curve. All r e l a t i o n s r e g a r d i n g the defects c a n n o w be derived. The m a j o r defects t u r n o u t to be-

App. 2-24.-

VAg-

e-

F__

w

Agi +

---

K5 ( 1 / K e ) 1/2

pl/2

~

(Ks) 1/2 (1/Ks) K~ ( 1 / P B r 2 ) 1/2

Br 2

T h i s s h o w s t h a t b o t h n+ arid e- are rrdnority defects d e p e n d e n t on PBr2. The following gives the s t a n d a r d E n t h a l p y a n d E n t r o p y of t h e s e d e f e c t r e a c t i o n s , a c c o r d i n g to Kroeger ( 1965)-

App. 2 - 2 5 . AgAg ~

D E F E C T REACTION

AS

Agi + + VAg ( F r e n k e l )

Ag/~ + BrBr --7

VAg"

+

AH

Cal. / mol / ~ 25.6

VBr +

Kcal. / mol. 29.3

- 13.3

36

25

78

4.9

25.4

,

0~

e- + n+

1/2Br2

+ AgAg ~ AgBr + "gAg

It is a p p a r e n t

.....

that the Frenkel

p r o c e s s are t h e p r e d o m i n a t i n g

+ n+

process

coupled

with

the

electronic

m e c h a n i s m s in f o r m i n g defects in AgBr

t h r o u g h the a g e n c y of e x t e r n a l r e a c t i o n w i t h Br2 gas.

124

A Final c o m m e n t : we c a n use t h e s e t h e r m o d y n a m i c values to calculate t h e e q u i l i b r i u m c o n s t a n t s a c c o r d i n g to: App. 2 - 2 7 . -

Ki =

exp-

AGi o / R T

a n d c a n also obtain the activity of the silver a t o m in AgBr from App. 2-9. By u s i n g e q u a t i o n App. 2-9., we c a n s h o w : App. 2 - 2 8 .

For aAg

= I , @ T = 277 ~

For PBr2 = I atm.

@ 277 ~

9 6 = + 1012 9 6 =- I0 7

w h e r e the p l u s or m i n u s indicate an excess or deficit of the silver a t o m in AgBr. This r e s u l t is due to W a g n e r ( 1 9 5 9 ) . Apt)endix I ! I Statistical M e c h a n i c s a n d the Point Defect The m a t h e m a t i c a l m e t h o d , statistical m e c h a n i c s , w a s developed by J a m e s Clerk Maxwell of S c o t l a n d a n d Ludwig B o l t z m a n n of G e r m a n y to p r e d i c t a n d explain

the

measurable properties

of m a c r o s c o p i c

systems,

i.e.-

solids, on the basis of the p r o p e r t i e s a n d b e h a v i o u r of the m i c r o s c o p i c constituents

of t h o s e

systems,

i.e.-

atoms.

Statistical m e c h a n i c s ,

for

example, i n t e r p r e t s t h e r m a l e n e r g y as the e n e r g y of atomic p a r t i c l e s in disordered energy

states and temperature

is s h a r e d

among

such

as a q u a n t i t a t i v e m e a s u r e

particles.

Statistical

mechanics

of h o w draws

heavily on the laws of probability, so t h a t it does not c o n c e n t r a t e on t h e b e h a v i o u r of every individual particle in a m a c r o s c o p i c s u b s t a n c e b u t o n the average b e h a v i o u r of a large n u m b e r of particles of the s a m e k i n d . One law of statistical m e c h a n i c s

s t a t e s that,

in a s y s t e m in t h e r m a l

equilibrium, on the average, an equal a m o u n t of e n e r g y will be a s s o c i a t e d with e a c h i n d e p e n d e n t

e n e r g y state. This law s t a t e s specifically t h a t a

s y s t e m of particles in e q u i l i b r i u m at absolute t e m p e r a t u r e T will have an average e n e r g y of 1 / 2 kT a s s o c i a t e d with e a c h degree of f r e e d o m ( w h e r e k

is the

Boltzmann

constant).

In

addition,

any degree

of f r e e d o m

c o n t r i b u t i n g p o t e n t i a l e n e r g y will have a n o t h e r 1 / 2 k T a s s o c i a t e d w i t h it.

125

For a s y s t e m of n d e g r e e s of f r e e d o m , of w h i c h t h a v e a s s o c i a t e d p o t e n t i a l e n e r g i e s , t h e total a v e r a g e e n e r g y of t h e s y s t e m is 1 / 2 ( n + t ) k T . For a n a t o m in a solid, v i b r a t o r y m o t i o n involves p o t e n t i a l e n e r g y as w e l l as k i n e t i c e n e r g y , a n d b o t h m o d e s vail c o n t r i b u t e a t e r m 1 / 2 k T , r e s u l t i n g in a n a v e r a g e total e n e r g y of 3kT. T h u s , it is t h e e n t r o p y of m i x i n g t h a t forces t h e c r e a t i o n of a c e r t a i n n u m b e r of v a c a n t l a t t i c e p o s i t i o n s above 0.0

~

Hence,

vacancies

are

the

natural

resultof

thermodynamic

e q u i l i b r i u m a n d n o t t h e r e s u l t of a c c i d e n t a l g r o w t h or s a m p l e p r e p a r a t i o n . In o r d e r to u s e t h i s m e t h o d , o n e s t a r t s b y d e f i n i n g t h e n u m b e r of s t a t e s t h a t m a y b e p r e s e n t . Let NL b e t h e t o t a l n u m b e r of l a t t i c e s i t e s a n d Nv b e t h e n u m b e r of v a c a n c i e s . T h e n u m b e r of w a y s t h a t Nv v a c a n c y - s i t e s c a n b e a r r a n g e d o n NL s i t e s will b e a s i m p l e c o m b i n a t i o n , d e f i n e d b y a m i x i n g entropy: App3.1.-

g~ =

NL! Nv f {N~ - N v) !

T h e m i x i n g e n t r o p y is d e f i n e d by: App3.2.-

ASM = k In g~

In o r d e r to solve c o m b i n a t o r i a l e q u a t i o n s like A p p 3 .1,, a m e t h o d

called

S t i r l i n g ' s a p p r o x i m a t i o n for l a r g e n u m b e r s is u s e d . T h i s givesApp3.3.where

N L

AS M _= k [ N L In N L - Nv In N v - (N L - Nv) ln()] is t h e n u m b e r of l a t t i c e s i t e s a n d N v is t h e n u m b e r of v a c a n c i e s .

It is well to n o t e h e r e t h a t App3.4.-

N L > >

Nv SO t h a t t h e e n t r o p y of m i x i n g will be-

AS M= IN v In N L - N v In N v ] or

AS M = - k N v l n ( N v / N L) = - R X v l n X ~

w h e r e Xv is t h e q u o t i e n t of

Nv/N

L .

T h i s e x p r e s s i o n is o n l y valid for n o n -

126

interacting

d e f e c t s . A n d , it is t h e s a m e a s t h e

entropy

of m i y d n g of a n

i d e a l s o l u t i o n , vis: App3.5.-

AS M= -R [X^ In X. + Xv In X v l

S i n c e X . is a n y a t o m in t h e solid, X A -= i, t h e e n t r o p y of m i x i n g b e c o m e s : App3.6.-

AS M= - R X v In X v

T h e c o m p l e t e H e l m h o l t z f r e e e n e r g y of t h e s y s t e m is t h e n w r i t t e n : App3.7.-

AFv = - Nv [AEv- TASW] + RT [ [ N , - NvI In N_a._:_NNv + N v In __Nv_l [ N L N L N~I~KLI

w h e r e AS v

is t h e c h a n g e in v i b r a t i o n a l e n t r o p y a r i s i n g f r o m a c h a n g e i n

v i b r a t i o n a l e n e r g y a r o u n d t h e v a c a n r l a t t i c e s i t e . If w e m i n i m i z e

the free

e n e r g y w i t h r e s p e c t to t h e v a c a n c y c o n c e n t r a t i o n , w e g e t : App3.8.-

0Fv

=0=AE

v-TAS~v+RTIn

Nv N L -'N v

H e r e , AEv a n d ASv a s t h e s t a n d a r d i n t e r n a l e n e r g y a n d s t a n d a r d e n t r o p y o f t h e d e f e c t s , r e s p e c t i v e l y . T h i s g i v e s u s a final r e s u l t of: App3.9.-

In

Nv_ = In X v = - AE v - T&.Sv NL- Nv

As w e

have already stated,

RT NL >>

Nv so

that

the

atomic

fraction

of

v a c a n c i e s isApp3.10.-

X v = e x p (-AF v /RT) ,~

OF:

Xv

=e

exp {-aGv / R T )

_ AE/RT

S i n c e , AGv = hFv + PAV (at c o n s t a n t p r e s s u r e ) , w e c a n t a k e t h e d e r i v i t i v e of A p p 3 . 1 0 . to o b t a i n :

127

App3.11-

0(AGv / RT} = d In AGv OT-1 dT 1

AH v R

w h i c h is a f o r m of t h e G i b b s - H e l m h o l z e q u a t i o n . Note t h a t we c a n u s e t h e s a m e s t a t i s t i c a l m e c h a n i c a l a p p r o a c h to c a l c u l a t e S c h o t t s k y p a i r s , F r e n k e l p a i r s , d i v a n c i e s (which are a s s o c i a t e d v a c a n c i e s ) , i m p u r i t y - v a c a n c y c o m p l e x e s , a n d line d i s l o c a t i o n - p o i n t d e f e c t c o m p l e x e s .

This Page Intentionally Left Blank

129

M e c h a n i s m s a n d R e a c t i o n s in t h e Solid S t a t e We have i n v e s t i g a t e d the s t r u c t u r e of solids in the s e c o n d c h a p t e r a n d t h e n a t u r e of point defects of t h e solid in t h e t h i r d c h a p t e r . We are n o w r e a d y to d e s c r i b e how solids react. T h i s will i n c l u d e t h e m e c h a n i s m s involved w h e n solids form b y r e a c t i o n from c o n s t i t u e n t c o m p o u n d s . We will also d e s c r i b e s o m e m e t h o d s of m e a s u r e m e n t a n d h o w one d e t e r m i n e s e x t e n t a n d r a t e of the solid s t a t e r e a c t i o n actually t a k i n g place. We will also s h o w how t h e p r e s e n c e

a n d / o r f o r m a t i o n of p o i n t defects affect reactivity in

solid s t a t e r e a c t i o n s . T h e y do so, b u t n o t in the m a n n e r t h a t you m i g h t s u s p e c t . We will also s h o w h o w solid s t a t e r e a c t i o n s p r o g r e s s , p a r t i c u l a r l y those

involving silicates w h e r e

several d i f f e r e n t

phases

appear

as a

function of b o t h time a n d relative ratios of r e a c t i n g c o m p o n e n t s . Solids are g e n e r a l l y c o n s i d e r e d

c h e m i c a l l y i n e r t at r o o m t e m p e r a t u r e s

a n d the m o s t c o m m o n - p l a c e e v i d e n c e is often overlooked. T h a t is, solids do not a p p e a r to be reactive until t h e y are h e a t e d . However, t h e a t o m s o r ions c o m p r i s i n g

solids are u n d e r

constant vibratory motion

with

the

lattice a n d c a n "diffuse" from site to site. If v a c a n c i e s are p r e s e n t , t h e y are c o n t i n u a l l y b e i n g "filled" a n d "emptied" even at r o o m t e m p e r a t u r e . T h o s e solids b a s e d u p o n iron (Fe) u n d e r g o c o n t i n u o u s oxidation to form a layer of "rust". T h u s , solids are n o t c o m p l e t e l y stable a n d are u n d e r c o n t i n u o u s c h a n g e over t i m e . It s h o u l d be clear by n o w t h a t inorganic solids {which c o n s i s t of a t o m s b o u n d t o g e t h e r by b o t h covalent a n d ionic forces} do n o t r e a c t by e i t h e r c h a n g i n g the b o n d i n g w i t h i n a m o l e c u l a r s t r u c t u r e or by r e a c t i n g o n e - o n one in a mobile p h a s e s u c h as a liquid, as do organic c o m p o u n d s . Solids c a n only r e a c t at t h e haterfaee of a n o t h e r solid, or in the c a s e of a liquidsolid r e a c t i o n , r e a c t w i t h the liquid m o l e c u l e at the solid i n t e r f a c e . For a solid w h i c h exists as a powder,

a useful m e n t a l c o n c e p t is t h a t of

p i n g - p o n g bails of several colors. Here, we i m a g i n e t h a t r e a c t i o n r e s u l t s w h e n the layers b e c o m e mixed. F u r t h e r m o r e , r e a c t i o n c a n only t a k e p l a c e

130

b etween neighboring balls of differing color, provided they are n e a r e s t neighbors. Thus, it should be clear t hat if we pour balls of the same color into a j a r and t h e n p o u r a n o t h e r layer of c o n t r a s t i n g color on top, r e a c t i o n can only take place at the interface of the layers, between bails of differing colors. One color can only "diffuse" into the other layer by replacing a ball of the o t h e r color. Our first step will be to delineate k n o w n solid state reaction m e c h a n i s m s . M e c h a n i s m s Relating to Solid State Reactions Phase c h a n g e s Formation of p h a s e b o u n d a r i e s Rate process c h a n g e s in solids Nucleation Diffusion p r o c e s s e s Diffusion-controlled solid state reactions. We can further separate t h e s e reaction m e c h a n i s m s into two types, t h o s e involving a single p h a s e (homogeneous) and those involving two different p h a s e s (heterogeneous). We will examine each of these in turn, s t a r t i n g with p h a s e changes. By p h a s e changes, we m e a n t r a n s f o r m a t i o n s of f o r m or s t r u c t u r e within the solid state. 4,1, - PHASE CHANGES We can distinguish seven (7) types of p h a s e t r a n s f o r m a t i o n s . For t h e h o m o g e n e o u s solid, these include the types of changes given in 4.1.1. on the next page. These seven t r a ns f or m at i ons, w h i c h involve j u s t o n e composition, are familiar to most. We t o u c h e d u p o n t h e m briefly in t h e first c h a p t e r w h e n we c o n s i d e r e d the t r a n s f o r m a t i o n s involving w at er. Now consider p h a s e c h a n g e s involving two solids and the types of r e a c t i o n m e c h a n i s m s w hi c h m a y occur. These are called h e t e r o g e n e o u s r e a c t i o n s and are p r e s e n t e d in Table 4-1, also given on the next page.

131

4.1.1.-

PHASE TRANSFORMATIONS (SINGLE PHASE) DESIGNATION

CHANGE Gas to Liquid

Condensation

G~L

Gas to Solid

Condensation

G~S

Liquid to Gas

Evaporation

L~ G

Liquid to Solid

Solidification

L~ S

Solid to Gas

Sublimation

S ~ G

Solid to Liquid

Melting

S~L

Solid to Solid

Polymorphic Transformation

S~S

TABLE 4 - I TYPES OF SOLID STATE H E T E R O G E N E O U S REACTIONS Final Product E x a m p l e of Reaction. Type of R e a c t i o n 3 AuCI ~ AuCI3 + 2 Au 1. D e c o m p o s i t i o n CaCO3 ~ CaO + C02 A~B+C N H4CI ~ N H3 ~ + HCI 2 Agl + Hgl2 ~ Ag2 HgI4

2. S y n t h e s i s A+B~C

MgO + A1203 ~ Mg•J2 04 MgO + S i 0 2

~ MgSi03

S + S

C u + AgCI ~

CuCI+ Ag

A+B~C+D

S+G

BaC03 + T i 0 2 ~

4. C o n s e c u t i v e A~ B~C

S + S

i) La203 + 11 A1203 ~

3. S u b s t i t u t i o n

Note

2 ~03

+ 10A1203

ii) 2 l_aAl03 + 10 A1203 ~ ~ 1 1 0 1 8

In b o t h 4.1.1. a n d Table 4-1, S , respectively.

BaTi03 + 0 0 2

that

we

have

L & G refer to solid, liquid a n d gas, also

classified

these

heterogeneous

m e c h a n i s m s in t e r m s of t h e s a m e PHYSICAL CHANGES given above for h o m o g e n e o u s t r a n s f o r m a t i o n s . For t h e m o s t part, t h e initial m a t e r i a l will be a solid while t h e n a t u r e of t h e final p r o d u c t will vary a c c o r d i n g to t h e type of m a t e r i a l u n d e r g o i n g solid s t a t e r e a c t i o n . Review t h e s e e x a m p l e s carefully. T h e y are typical of t h o s e e n c o u n t e r e d in solid s t a t e c h e m i s t r y . T h u s , "A" r e a c t s to f o r m "B +C" ( D e c o m p o s i t i o n ) or

132

"A + B" r e a c t s to form "C" (Synthesis"). etc. T h e s e r e a c t i o n s cover m o s t of t h o s e n o r m a l l y f o u n d in solid s t a t e c h e m i s t r y .

Once you have m a s t e r e d

t h e s e t y p e s of r e a c t i o n s , you will be able to identify m o s t solid s t a t e reactions

in t e r m s

of the r e a c t a n t s

a n d p r o d u c t s . This is i m p o r t a n t ,

especially w h e n you m a y be trying to form an new c o m p o s i t i o n n o t well k n o w n in solid state science. Addionally, you m a y w i s h to form a specific c o m p o s i t i o n by a new

method

to see

ff it h a s s u p e r i o r

physical o r

c h e m i c a l p r o p e r t i e s over t h a t s a m e m a t e r i a l f o r m e d by a different solid state m e c h a n i s m or reaction. In m a n y cases, this h a s b e e n f o u n d to be t r u e a n d this factor h a s b e e n r e s p o n s i b l e for several scientific a d v a n c e s in the solid-state. T h a t is- ff you c a n fund a different m e t h o d for m a k i n g a material, its p r o p e r t i e s m a y prove to be s u p e r i o r to t h a t already k n o w n . 4 . 2 . - T H E ROLE OF PH/~_SE BOUNDARIES IN SOLID STATE REACTIONS C o n s i d e r two (2) solids, A a n d B. It s h o u l d be obvious t h a t t h e y will r e a c t only w h e n in close proximity, a n d usually w h e n t h e y are i n t i m a t e l y m i x e d a n d h e a t e d (if t h e y are powders). What this m e a n s is i l l u s t r a t e d by t h e following d i a g r a m :

In the case w h e r e two p a r t i c l e s are involved, t h o s e t h a t are n o t in c l o s e p r o x i m i t y will not react, w h e r e a s t h o s e t h a t are close will u n d e r g o solid state r e a c t i o n with ease. This is due to the fact t h a t c a t i o n s a n d / o r a n i o n s from

one

structure

must

be

transported,

or

interchange

by

some

m e c h a n i s m , to the o t h e r s t r u c t u r e in o r d e r to form a c o m p l e t e l y n e w c o m p o u n d . T h u s , t h e degree

of d i s p e r s i o n a n d m i x i n g of one r e a c t i n g

solid w i t h a n o t h e r is i m p o r t a n t to the overall m e c h a n i s m of solid s t a t e

133

reaction. This c a n n o t be o v e r e m p h a s i z e d since lack of dispersion is the p r i m a r y cause for p r o d u c t i o n of u n w a n t e d p h a s e s (or lack of p r o d u c t i o n t h a t is, w h e n the solid state reaction

does not proceed

according to

e xp ectio n ) . As an example, c o n s i d e r the following. S u p p o s e we have a crucible halffilled with a powder. We now fill the crucible with a n o t h e r powder of different c o m p o s i t i o n and t h e n heat the Failed crucible. Any solid state reaction which does occur can only do so at the b o u n d a r y of the two layers of powders. If the reaction is: A + B ~ AB, t h e n we find t hat the r e a c t i o n product, which is also a solid, forms as a p h a s e b o u a d a l ~ b e t w e e n the two layers. The same condition exists in a solid state reaction bet w een two crystalline particles having differing compositions. That is, they can only react at the interface of each particle. This is illustrated in the following diagram, which is a model of how the c o m p o n e n t s react t h r o u g h a p h a s e boundary:

In this case, we have given b o t h the star'ring conditions and those of t h e i n t e r m e d i a t e stage of solid state reaction. It should be clear t h a t A r e a c t s with B, and vice versa. Thus, a p h a s e b o u n d a r y is formed at the interface of the b u lk of each particle, i.e.- between A and AB, and bet w een B and AB. The p h a s e boundary, AB, t h e n grows o u t w a r d as shovm above. Once the p h a s e b o u n d a r y is established, t h e n each reacting specie m u s t diffuse t h r o u g h the p h a s e {AB} to reach its opposite p h a s e b o u n d a r y in order to react. That i s - * A " m u s t diffuse t h r o u g h "AB" to the p h a s e b o u n d a r y

134

between

"AB" a n d "B" before it can r e a c t with "B" to form "AB" a n d

f u r t h e r i n c r e a s e the d i m e n s i o n s of "AB" at the e x p e n s e of b o t h "A" a n d "B'.

It

is

quite

important

that

you m a s t e r

this

concept

which

is

f u n d a m e n t a l t o w a r d s u n d e r s t a n d i n g b o t h p h a s e b o u n d a r i e s of solid s t a t e r e a c t i o n a n d diffusion c o n d i t i o n s t h a t prevail. However,

this is not h o w it o c c u r s in Nature. We have p r e s e n t e d t h e

above c o n c e p t

because

it

occur.

is

easier

Thus,

the

to

understand

overall

solid

than state

the

actual

conditions

which

reaction

is

dependent

u p o n the rate of diffusion of the two (2) species. T h e s e two

r a t e s may, or m a y not, be the same. The r e a s o n t h a t A a n d / o r B do n o t react in the middle, i.e.- the p h a s e {AB}, is t h a t AB h a s a c e r t a i n o r d e r e d s t r u c t u r e w h i c h p r o b a b l y differs from either A or B. But t h e r e is a m o r e i m p o r t a n t r e a s o n w h i c h is not easily i l l u s t r a t e d in any d i a g r a m . The actuality is t h a t any given a t o m of A d o e s n o t m o v e t h r o u g h AB to t h e p h a s e b o u n d a r y of A a n d AB. W h a t really h a p p e n s is t h a t any given a t o m displaces o n e of t h e A a t o m s in {AB}. The d i s p l a c e d A-atom t h e n c a u s e s a n o t h e r d i s p l a c e m e n t by a "hopping" m o t i o n . The s t a t e d d i s p l a c e m e n t t h e n travels to the right until the o t h e r edge of the p h a s e b o u n d a r y is e n c o u n t e r e d . There, r e a c t i o n of the final, d i s p l a c e d A - a t o m o c c u r s with a B - a t o m to form AB. S i m u l t a n e o u s l y , d i s p l a c e d B - a t o m s are diffusing to t h e left by the s a m e "hopping" m o t i o n . T h u s , the walls of {AB} move in a t h r e e - d i m e n s i o n a l direction at the e x p e n s e of b o t h of the atomic v o l u m e s of A a n d B. A d e p i c t i o n

of this m e c h a n i s m

is given in the following

diagram, given as 4.2.3. on the next page. Here,

we

have s h o w n

an "A" ion

(shown

as a b l a c k s p h e r e )

which

a p p r o a c h e s the surface of "AB" a n d displaces an "A" ion in this solid p h a s e . A series of "hops", i.e.- from "1" to "7", t h e n o c c u r s in the AB-phase w i t h the final "A" ion e n d i n g w i t h i n the B - p h a s e w h e r e a d i s p l a c e m e n t in t h e n o r m a l l y cubic "B" p h a s e occurs. At the s a m e time, the d i s p l a c e d "B" ion is diffusing in the o p p o s i t e d i r e c t i o n by a series of "hops", i.e.- from "a" to "e" to the interface of "AB" with "A". Note t h a t the "A" p h a s e is s h o w n as a h e x a g o n a l p h a s e while "B" is a cubic p h a s e (as is the "AB" phase). It s h o u l d be clear t h a t the rate of diffusion of "A" will differ from t h a t of "B".

135

Note also t h a t a c h a n g e in s t r u c t u r e h a s o c c u r r e d as the AB a r r a n g e m e n t grows. Let u s t a k e a n o t h e r example, u s i n g the s a m e type of model, in w h i c h A is a gas a n d B is a solid. A d i a g r a m i l l u s t r a t i n g this m e c h a n i s m is given as follows:

Here, only

one

i n t e r f a c e forms as the r e a c t i o n t a k e s place initially at t h e

surface of "B". Once t h e AB-phase h a s formed, g a s e o u s A a t o m s m u s t t h e n diffuse t h r o u g h to the p h a s e b o u n d a r y in o r d e r for the r e a c t i o n to o c c u r . T h e wall of {AB} t h u s m o v e s in two (2) d i m e n s i o n s i n s t e a d of t h e t h r e e , in c o n t r a s t to the c a s e of 4 . 2 . 2 . C o n s i d e r a n o t h e r e x a m p l e involving difflasion a n d reactivity, s u c h as t h e r e a c t i o n b e t w e e n p a r t i c l e s of:

136

4.2.5.-

BaO + SiO2 =~, BaSiO3

In t h i s case, t h e s t r u c t u r e of BaO c o n s i s t s of d i s c r e t e a t o m s of 139_2+ a n d 0 = , a r r a n g e d in a c u b i c s t r u c t u r e . S i 0 2 , o n t h e o t h e r h a n d , c o n s i s t s of S i 0 4 tetrahedra

bound

at

the

apices

through

mutual

oxygen

atoms,

and

a r r a n g e d in a t h r e e - d i m e n s i o n a l n e t w o r k . T h i s is s h o w n in t h e f o l l o w i n g diagram:

We Find t h a t t h i s solid s t a t e r e a c t i o n is v e r y slow, even at 8 0 0 ~

, and

o c c u r s at t h e i n t e r f a c e of t h e two t y p e s of p a r t i c l e s . T h e r e a c t i o n is s l o w b e c a u s e it is d i f f u s i o n - l i m i t e d . W h a t is h a p p e n i n g is t h a t s i n c e t h e silican e t w o r k is t h r e e - d i m e n s i o n a l l y b o u n d , t h e ozfly re.action t h a t o c c u r s is c a u s e d b y t h e diffusion of Ba2+ a t o m s w i t h i n t h e n e t w o r k , as s h o w n in t h e following:

137

Note

that

diffusion o c c u r s

only in o n e

d i r e c t i o n b e c a u s e the

silica-

t e t r a h e d r a are n o t free to m o v e . What is actually h a p p e n i n g is t h a t t h e three-dimensional

network

of t e t r a h e d r a

is being r c a r r t m g e d to f o r m

a n o t h e r s t r u c t u r e . This i l l u s t r a t e s the fact t h a t the actual s t r u c t u r e a n d composition

of t h e

two

reacting

species

are

the

major

factor

in

d e t e r m i n i n g the n a t u r e a n d s p e e d of the solid state r e a c t i o n . On the o t h e r h a n d , if we u s e BaC03 as a r e a c t a n t , vis: 4.2.8.-

BaC~3 + Si02 ~ BaSiO3 + CI~)2 II

we find t h a t the r e a c t i o n is very fast. This h a s n o t h i n g to do w i t h t h e g a s e o u s CID2 p r o d u c e d , b u t w i t h the form of the BaO particles. In t h i s case, the initial r e a c t i o n is: 4.2.9.-

BaC03 ~

BaO + O32

The BaO is p r o d u c e d in the form of very small p a r t i c l e s of n e a r l y a t o m i c p r o p o r t i o n s w h i c h r e a c t i m m e d i a t e l y to form the tdlicatr rate of r e a c t i o n is p r o p o r t i o n a l to the n u m b e r of a u r

Actually, t h e

produced per unit

volume. A n u c l e u s is a p o i n t w h e r e a t o m s or ions have r e a c t e d a n d b e g u n the f o r m a t i o n of the p r o d u c t s t r u c t u r e . In the c a s e of the BaO r e a c t i o n , the n u m b e r of nuclei f o r m e d per u n i t of time is small a n d f o r m a t i o n of the s t r u c t u r e is difl~asion limited. In the case of Ba(X)3 d e c o m p o s i t i o n , t h e a t o m i c - p r o p o r t i o n e d BaO r e a c t s n e a r l y as fast as it is f o r m e d so t h a t t h e n u m b e r of nuclei p e r u n i t v o l u m e is e n o r m o u s l y i n c r e a s e d .

It is t h u s

a p p a r e n t t h a t if we w i s h to i n c r e a s e solid state r e a c t i o n rates, one way to do so is to u s e a d e c o m p o s i t i o n r e a c t i o n to s u p p l y the r e a c t i n g s p e c i e s . we will f u r t h e r a d d r e s s this type of r e a c t i o n later on in o u r d i s c u s s i o n . Note

also t h a t we

have j u s t i n t r o d u c e d

the

concepts

n u c l e a t i o n in o u r s t u d y of solid state r e a c t i o n p r o c e s s e s .

of nuclei

and

Our next s t e p

will be to e x a m i n e s o m e of the m a t h e m a t i c s u s e d to define r a t e p r o c e s s e s in solid state r e a c t i o n s . We will not delve into the p r e c i s e e q u a t i o n s h e r e b u t p r e s e n t t h e m in A p p e n d i c e s at the e n d of this c h a p t e r . But first, w e n e e d to e x a m i n e r e a c t i o n rate e q u a t i o n s as a d a p t e d for t h e solid state.

138

4.3.-

REACTION RATE P R O C E S S E S IN SOLIDS

As s t a t e d

above, we c a n classify solid s t a t e r e a c t i o n s

as b e i n g

either

h o m o g e n e o u s or h e t e r o g e n e o u s . T h e f o r m e r involves r e a c t i o n s by a s i n g l e c o m p o u n d w h e r e a s t h e l a t t e r involves r e a c t i o n s b e t w e e n

two d i f f e r e n t

c o m p o u n d s . As s t a t e d above in Table 4-1, t h e r e are four (4) t y p e s of s o l i d state reactions. 4.3.1.-

T w e s of Solid S t a t e R e a c t i o n s I. D e c o m p o s i t i o n :

A~B+C

2. S y n t h e s i s :

A+B~C

4. S u b s t i t u t i o n a l :

A+B~C+D

4. C o n s e c u t i v e :

A~B~C

Rate p r o c e s s e s for t h e solid s t a t e are g e n e r a l l y d e f i n e d in t e r m s of a r a t e , r, a n d a v o l u m e , V, usually a m o l a r v o l u m e . T h u s , we have: 4.3.2.-

Homogeneous:

r = d n i / d t - 1/Vt

Heterogeneous:

r = d V t / d t 9 1/Vf

w h e r e ni is initial m o l e s , Vt is v o l u m e at time, t, a n d Vf is final v o l u m e . The

r e a s o n t h a t t h e s e e q u a t i o n s differ is t h a t a r e a c t i o n

between

two

s o l i d s m a y p r o p a g a t e a fim~ product h a v i n g a l a r g e r or s m a l l e r v o l u m e t h a n t h a t of t h e original r e a c t i n g species, w h i l e t h a t of a s i n g l e compound r e m a i n s ~ l b l e . If w e define t h e f r a c t i o n d e c o m p o s e d at a n y t i m e as: 4.3.4.-

x - Vt / Vf

We c a n e x p r e s s t h e s e v o l u m e s in t e r m s of x so as to get: 4.3.5.-

d x / d t = k l (T) 9f(x)

w h e r e k l is t h e r a t e c o n s t a n t for t h e e q u a t i o n in 4 . 3 . 4 . a n d f(x) r e l a t e s to

139

the definition given in 4.3.2. This e q u a t i o n , i.e.-4.3.5., t h e n h a s the s a m e general form as t h o s e u s e d in the s t u d y of k i n e t i c s of c h e m i c a l r e a c t i o n s . F r o m t h e Kinetic Theory, w h e n a g a s e o u s s y s t e m goes from an initial s t a t e to a fmal state, t h e r e a c t i n g species m u s t c o m e close e n o u g h to react. At the m o m e n t of "joining", we have the "activated complex". By u s i n g t h i s concept, we can o b t a i n s o m e g e n e r a l e q u a t i o n s useful to us. The c o n c e p t of the "activated complex" is i l l u s t r a t e d in t h e following diagram:

It is easy to see t h a t the free e n e r g y of the activated complex,

G*, is

h i g h e r t h a n t h a t of t h e initial state. K* is the e q u i l i b r i u m c o n s t a n t of t h e activated complex, a n d k is the B o l t z m a n n c o n s t a n t . Then, we c a n w r i t e e q u a t i o n s d e s c r i b i n g b o t h AG a n d the activated state as: 4.3.7.-

AG = R T I n K * k l = k T / h exp.(-AG*/RT)

where

= k T / h exp.(AS*/R) e x p . ( - A H * / R T )

the s t a r (*) refers to the activated species.

F r o m the Clausius-

Clapyeron equation, we have: 4.3.8.-

Eo - R T 2 d(In kl) / dt = AH* + R T

w h e r e Eo is an i n t e r n a l energy. By defining a f r e q u e n c y factor as:

140

4.3.9.-

Z

= k T / h exp ( - E o * I R T )

We c a n s i m p l y e x p r e s s t h i s r e s u l t s i m i l a r to t h a t of 4 . 3 . 7 . to obtain: 4.3.10.-

kl

= Z exp. ( - E o * / R T )

w h i c h is t h e A R R H E N I U S EQUATION for t h e r a t e c o n s t a n t , k l. We c a n

general

t r a n s f o r m t h e above e q u a t i o n s into

e q u a t i o n s as well,

and use

t h e m to define t h e v a r i o u s t y p e s of r e a c t i o n s given above, n a m e l y : 4.3.11.-

RATE EQUATIONS FOR SOLID STATE R E A C T I O N S Simple nth.0rder kI nA ~ B

.Rate E q u a t i o n -dXA / d t = k l xn A

Parallel R e a c t i o n s kl k2 { A ~ B} a n d { A

Rate Equation - d x A / d t = k l X A + k 2 XA

~C}

Consecutive kl k2 A ~B

Rate Equation -dxA/dt = klXA

~C

- k2xB

T h r e e t y p e s of r a t e e q u a t i o n s a r e s h o w n h e r e . T h e s e r a t e e q u a t i o n s c a n b e used

for

quite

measurement

complicated

approach

is

reactions,

needed.

How

but we

a

specific do

this

method is

critical

or to

d e t e r m i n i n g a c c u r a t e e s t i m a t i o n of t h e p r o g r e s s of a solid s t a t e r e a c t i o n . We will d i s c u s s s u i t a b l e m e t h o d s

in a n o t h e r c h a p t e r . We n o w r e t u r n to

t h e s u b j e c t of n u c l e a t i o n so t h a t we c a n a p p l y t h e r a t e e q u a t i o n s g i v e n above to specific c a s e s . First, we e x a m i n e h e t e r o g e n e o u s p r o c e s s e s . 4 . 4 . - DEFINING H E T E R O G E N E O I I S NUCLEATION P R O C E S S E S If we are i n t e r e s t e d in t h e n u c l e a t i o n of a p a r t i c l e p r i o r to c o m p l e t i n g t h e solid s t a t e r e a c t i o n , we n e e d to d i s t i n g u i s h b e t w e e n s u r f a c e a n d v o l u m e n u c l e a t i o n of t h e p a r t i c l e , s i n c e t h e s e a r e t h e m a j o r m e t h o d s of w h i c h w e c a n p e r c e i v e . S e v e r a l c a s e s a r e s h o w n in t h e following d i a g r a m .

141

In this diagram, we classify t h e various t y p e s of h e t e r o g e n e o u s solid s t a t e r e a c t i o n s t h a t c a n occur. Note t h a t b o t h t r a n s f o r m a t i o n to solid a n d fluid p r o d u c t s are diagramn-led. Also s h o w n are v a r i a t i o n s w h e r e surface a n d / o r v o l u m e n u c l e a t i o n is involved. W h e n solids r e a c t a n d n u c l e a t i o n is the n o r m , t h e y do so either by surface

142

or v o l u m e n u c l e a t i o n . Model B involves surface nucleation w h e r e i n surface b e c o m e s

n u c l e a t e d in one spot, i.e.- two r e a c t i n g

particles

the in

close proximity. This is s h o w n in 4.4.2. Here, two t o u c h i n g p a r t i c l e s have i n i t i a t e d g r o w t h of a n o t h e r p h a s e by n u c l e a t i n g at t h e point w h e r e e a c h particle t o u c h e s the o t h e r . 4.4.2.- Model B w h e r e R e a c t i o n Only T a k e s Place at the Point W h e r e Two Particles T o u c h

As s h o w n

in 4.4.1.,

two

stages

usually o c c u r

in

surface

nucleation,

n u c l e a t i o n a n d t h e n nuclei growth. If surface n u c l e a t i o n is fast (Model C) it is likely d u e to r e a c t i o n of a gas with the solid particle. The r e a c t i o n of a liquid is the o t h e r possibility, i.e.4.4.3.-

AS + BG ~ (AB)s AS + BL ~ (AB)s

If it slow, t h e n n u c l e a t i o n is likely to be due solely to proximity. Model D is an e x a m p l e of volume nucleation w h e r e d e c o m p o s i t i o n

of a solid is

involved w h e r e a s Model E is t h a t involving gas or liquid n u c l e a t i o n of t h e solid. Note t h a t if n u c l e a t i o n does not occur, the solid r e a c t s u n i f o r m l y t h r o u g h o u t its whole v o l u m e (Model F). However, this m o d e is r a r e a n d the n u c l e a t i o n stages are m o r e likely to occur. We will not dwell u p o n h o w t h e s e n u c l e a t i o n m o d e l s w e r e derived a n d will only p r e s e n t the r e s u l t s here.

One is r e f e r r e d

to A p p e n d i x

I

wherein

one

m a t h e m a t i c s u s e d to obtain t h e n e t - r e s u l t . The factors t h a t we m u s t c o n s i d e r i n c l u d e the following: I) The f o r m a t i o n of an i n c i p i e n t embryo, E 2) N u m b e r s of e m b r y o s w h i c h f o r m 3) The g r o w t h of the e m b r y o into a n u c l e u s , N

can

study the

143

4) G r o w t h of n u c l e i 5) Grow~h a n d t r a n s f o r m a t i o n of nuclei into t h e final p r o d u c t of t h e solid s t a t e r e a c t i o n by f o r m a t i o n of p h a s e b o u n d a r i e s . W h a t we m e a n by an e m b r y o is s h o w n in the following d i a g r a m w h e r e i n the d i m e n s i o n s are d i s t o r t e d from the actual case in o r d e r to i l l u s t r a t e our point.

In this case, we m u s t differentiate

between

the

walls of the

rapidly

forming e m b r y o a n d t h o s e of t h e i n c i p i e n t n u c l e u s . T h i s d r a w i n g s h o w s the f o r m a t i o n of an e m b r y o E , w h i c h t h e n grows into a n u c l e u s , N, w i t h i n the crystallite volume, A (none are d r a w n to scale). T h e c h a n g e in f r e e e n e r g y in going from B to A win be a function of b o t h v o l u m e a n d surface a r e a (shape). We m u s t d i s c r i m i n a t e b e t w e e n t h e interfacial surface e n e r g y between

B a n d A, a n d t h e i n t e r n a l e n e r g y of C. T h e n u c l e u s , N, t h e n

grows by e x p a n d i n g its walls to form a n e w s t r u c t u r e ( c o m p o u n d ) .

We

note t h a t the g r o w t h of nuclei is a s s o c i a t e d with the m o t i o n of a p h a s e b o u n d a r y , as d e s c r i b e d above. We will n o t give the m a t h e m a t i c s involved h e r e b u t r e s e r v e t h e m for A p p e n d i x II at t h e e n d of this c h a p t e r . However, we n e e d to point o u t t h a t the r a t e of n u c l e a t i o n , I, is defined as:

144

4.4.5.-

I - N u m b e r of n u c l e i / U n i t t i m e / U n i t v o l u m e

a n d x d t is t h e g r o w t h

r a t e of t h e nuclei, as in 4 . 4 . 4 . T h e

final r e s u l t

involves: 4.4.6.which

rcrit is

required

the

= - 2 Y / AGv equation

G~bs-Thompson

defining

the

critical

radius

for a n e m b r y o to form. We also c a n r e l a t e t h e s e e q u a t i o n s to

n u m b e r s of n u c l e i b e i n g f o r m e d a n d t h e i r g r o w t h . T h i s is i l l u s t r a t e d as follows in 4 . 4 . 7 . , given o n t h e n e x t p a g e . In t h e s i m p l e case, B g r o w s at t h e e x p e n s e of A. However, in t h e s e c o n d c a s e involving p a r a l l e l r e a c t i o n s , t w o e m h r y c m f o r m f r o m A. I n t h e t h i r d case, at l e a s t two s t e p s are involved in t h e

two c o n s e c u t i v e

reactions

w h e r e "A" t r a n s f o r m s to "B" w h i c h t r a n s f o r m s to "C". In T a b l e 4-2, we controlled

show both phase-boundary controlled

a n d diffusion-

n u c l e a t i o n r e a c t i o n s as a f u n c t i o n of b o t h c o n s t a n t a n d z e r o

rate of nucleation.

TABLE 4 - 2 NUCLEATION AS A FUNCTION OF T H E SHAP_E FACqY)R, 0 DIFFUSION CONTROLLED [ ~ {t) = D I t ]

PHASE BOUNDARY CON'I~OLLED [ ~ {t) = k2 t r ] 0

r

Constant Rate spheres

E

r

E

4

3 E2 + E1

2.5

3 / 2 ED + E1

3

2 E2 + E1

2.0

E2 + ED

2

E2 + E l

4

3 E2

1.5

3

2 E2

1.0

of N u c l e a t i o n ,,

plates needles i

Zero

Rate

ii

of s p h e r e s

1.5

1.5 ED + E1 i

1.5 ED

Nucleation plates needles i

2 jl

i

E2 iii

i

0.5

ED 0.5 ED

i

145

R e f e r r i n g to 4.4.4. a n d Table 4-2, we see t h a t the s h a p e factor, 0, h a s a m a j o r effect u p o n r a n d t h e e x p o n e n t of time, activation energies. E1 is the activation e n e r g y w h e r e a s E2 is t h a t for n u c l e i g r o w t h . ED

t, as well as on t h e for nuclei f o r m a t i o n ,

is t h e activation e n e r g y for

d i t I u ~ o n g r o w t h . Note t h a t w e have n o w directly a s s o c i a t e d diffusion as a m e c h a n i s m in nuclei f o r m a t i o n a n d t h a t we h a v e a l r e a d y given e x a m p l e s

146

above c o n c e r n i n g solid state r e a c t i o n m e c h a n i s m s c o n c e r n i n g

diffusion

(see 4.2.2 for e x a m p l e ) . Actually, t h e r e

are t h r e e

in

(3) m a j o r classifications for nuclei g r o w t h

h e t e r o g e n e o u s solid state reactions. Restating, t h e s e are: 4.4.8.-

Phase-boundary Controlled Random Growth Diffusion C o n t r o l l e d

We have already dealt with two of these. Section 2 dealt with f o r m a t i o n of a p h a s e b o u n d a r y while we have j u s t c o m p l e t e d

Section 4 c o n c e r n i n g

nuclei g r o w t h as r e l a t e d to a p h a s e b o u n d a r y . We will c o n s i d e r diffusion m e c h a n i s m s in nuclei a n d d i f f u s i o n - c o n t r o l l e d solid state r e a c t i o n s at a later p a r t of this c h a p t e r . T h e s e are areas of solid state c h e m i s t r y w h e r e c o n s i d e r a b l e e x p e r i m e n t a l w o r k h a s already b e e n a c c o m p l i s h e d , p a r t i c u l a r l y in the a r e a of t a r n i s h i n g b e c a u s e of t h e obvious c o m m e r c i a l i n t e r e s t in s u c h r e a c t i o n s . We are n o w r e a d y to c o n s i d e r s o m e solid state r e a c t i o n s t h a t relate m o r e directly to the real world.

These

include the t a r n i s h i n g r e a c t i o n

and

Fick's Laws of Diffusion. Both of t h e s e scientific areas have b e e n r i g o r o u s l y studied

because

of

their

importance

in

revealing

how

diffusion

m e c h a n i s m s are r e l a t e d to everyday solid state r e a c t i o n s w h i c h o c c u r on a daily basis. 4 . 5 , THE TARNISHING REACTION In 1902, W a g n e r p u b l i s h e d an analysis, b a s e d on diffusion r e a c t i o n s , of t h e oxidation of the surface of a metal. His i n t e r p r e t a t i o n

has remained

a

classic in solid state diffusion analysis. The surface of a m e t a l c o n s i s t s of m e t a l a t o m s b o u n d to the i n n e r s t r u c t u r e by a series of h y b r i d - b o n d s . If oxygen gas is p r e s e n t (air), the m e t a l can form an oxide coating: 4.5.1.-

M + 1/202

~

MO

147

In s o m e cases, the o x i d e - c o a t i n g p r o t e c t s the surface from f u r t h e r o x i d e buildup. One e x a m p l e is t h a t of a l u m i n u m w h e r e an oxide coating a p p e a r s a l m o s t i n s t a n t a n e o u s l y once the p r i s t i n e

surface is e x p o s e d to air. Yet,

t h e r e are m a n y c a s e s w h e r e the oxide layer c o n t i n u e s to b u i l d u p until t h e m e t a l is totally c o n s u m e d (One e x a m p l e is t h a t of iron a n d "rust"). How is this possible?

Wagner hypothesized

that both

metal

and

oxide

ions

d i f f u s e d Lhrough the m e t a l oxide layer so as to build u p the layer t h i c k n e s s f r o m b o t h s i d e s . The following d i a g r a m is one r e p r e s e n t a t i o n of s u c h a

mechanism:

In this case, the partial p r e s s u r e , p, of oxygen gas is a limiting factor in the overall r e a c t i o n to form the oxide film, MO, w h o s e t h i c k n e s s , x, is shown. It s h o u l d be a p p a r e n t , from the above diagram, t h a t t h e m o r e x i n c r e a s e s , the slower will be the c h a n g e in the x - d i m e n s i o n (since it t a k e s longer for the ions to diffuse), i.e.4.5.3.-

dx/dt

~ 1/x

or: d x / d t = K - 1 / x

The i n t e g r a t e d form is: x 2 = Kt + C , or: x = D AFt, w h e r e D is t h e Diffusion Coefficient. DiI~on.

Equation

4.5.3.

is called

the

Pal~lic

Law of

If the g r o w t h of a p h a s e can be fitted to this equation, t h e n it is

likely t h a t the p r i m a r y r e a c t i o n m e c h a n i s m involves simple diffusion.

148

Diffusion m e c h a n i s m s involve the following defect reactions: 4.5.4. -

I n t e r c h a n g e of A t o m s Vacancy Hopping Interstitial M o v e m e n t Dissociative E x c h a n g e

We have already d i s c u s s e d m o s t of t h e s e m e c h a n i s m s . In the first, Mx a n d XM i n t e r c h a n g e m a y be involved, w h e r e a s the last involves a m e c h a n i s m similar to t h a t given in 4.2.2. Note t h a t the s e c o n d type will only a p p e a r only if the individual ions involved are

small e n o u g h to fit into

the

i n t e r s t i c e s of the lattice. 4 . 6 . - FICK'S LAWS OF DIFFUSION About 1942, Fick f o r m u l a t e d laws w h i c h d e s c r i b e d diffusion p r o c e s s e s in solids, similar to t h a t already p r e s e n t e d for the t a r n i s h i n g reaction.

To do

this, Fick h y p o t h e s i z e d t h a t a t o m s (ions) w o u l d "hop" f r o m site to site in the m a n n e r we have already d e s c r i b e d in 4.2.2. a n d 4 . 2 . 4 . C o n s i d e r a lattice having a t h r e e - d i m e n s i o n a l set of planes. If we have t h i s t h r e e - d i m e n s i o n a l lattice, it is easy to see t h a t t h e r e are six (6) ways for an a t o m (ion) in the x - p l a n e to move to the y-plane. T h i s is s h o w n in t h e following illustration, given as 4.6.1. on the next page. By defining nA as the n u m b e r of A a t o m s p r e s e n t per u n i t volume, z as t h e time of stay at any given site, 1/~ as the f r e q u e n c y of j u m p s a n d J x ~ y

as

the n u m b e r of j u m p s per u n i t time, Fick w a s able to derive two laws r e g a r d i n g diffusion in solids. To do this, he also defined: I - N u m b e r of n u c l e i / U n i t t i m e / U n i t volume. Using his definitions, we c a n derive a series of e q u a t i o n s w h i c h r e s u l t s in Fick's First Law of D i i ~ s i o n .

149

T h e d i a g r a m given above d e f i n e s t h e p a r a m e t e r s

n e c e s s a r y to define t h e

e q u a t i o n s . How t h i s is d o n e is s h o w n in T a b l e 4-4, given o n t h e n e x t p a g e , w h e r e t h e i n d i v i d u a l s t e p s n e c e s s a r y are s e t f o r t h . T h e r e s u l t is: 4.6.2.-

FICK'S F I R S T LAW:

J = - D dc/dx

w h e r e J is t h e as t h e total n u m b e r diffusion coefficient

for

the

process.

of j u m p s If s u c h

that occur, exists,

a n d D is t h e

then

dc/dx

,

a

c o n s t a n t , a n d we m u s t t h e n s u b t r a c t two d i f f e r e n t j u m p r a t e s , i.e.4.6.3.-

J1 -J2

=-AxdJ/dx=-~x{dc/dt)

If t h e l a s t t e r m is t a k e n o n l y

in r e s p e c t

to t h e x d i m e n s i o n ,

then we

arrive at t h e s e c o n d law, viz4 . 6 . 4 . - FICK'S SECOND LAW: [0c/0tlx = 0 {D 0c/0x} or ( 0 c / 0 t } x = D ( 0 2 c / 0 x 2)

150

Table 4-4 H o w F i c k ' s F i r s t L a w of D i f f u s i o n Is D e r i v e d Ste: 1.

Here

we

terms

have defined

of

the

E q u a t i o n s l e a d i n g to F i c k ' s L a w s in Jx~y = 1 / 6 - 1 / T ( n A R . a ) possible

Jx

number

--

of

y

" j u m p s " , i.e.- 1 o u t of 6 w a y s , T - t h e time

that

numbers

it

takes

to

do

of a t o m s p r e s e n t

cross-sectional

so,

the

and

the

a r e a of t h e l a t t i c e i n

question. .

We then define the concentration

o f nAty)

= nA + a d n A / d x

A - a t o m s in t h e y - p l a n e to b e : .

.

C o m b i n i n g t h e s e t w o e q u a t i o n s g i v e s J x ~ y = {nA + a dnA I d x ) ( a R l 6T) us: If

we

assume

that

the

J

opposite

=

Jx~

y

-

Jy~

x

d i r e c t i o n of h o p p i n g is also valid, i.e.- J y__, x , t h e n w e h a v e : .

U s i n g J as t h e t o t a l n u m b e r

of j u m p s

J = aR/6~(nAnA+adnA/dx-aR/6)

t h a t o c c u r , w e c o m e to: .

We now define a diffusion coefficient "D"

.

=-a

D-

2 R / 6~ 9 d n A / d X

a2/6T

as:

If w e h a v e t h e i n t e r s t i t i a l 4.6.3.)., which

then is

we

must

the number

c a s e ( s e e D ------ a a 2 / "I: ( i n t e r s t i t i a l ) (z,

include

of i n t e r s t i c e s

in t h e l a t t i c e : I

.

9.

+ 10.

J

T h i s g i v e s us:

=-DRdnA

/dx

If w e n o w d e f i n e c o n c e n t r a t i o n in dc/dx = R dnA/dX t e r m s of n A , i.e.- c ---- RnA , so t h a t : then

we get FICK~

F i r s t Law:.

J

= - D dc/dx

................... i ......................................................................................................................... ; ........................................................................................................

(Note t h a t p a r t i a l d i f f e r e n t i a l s a r e u s e d in t h e 2 n d L a w (see 4 . 6 . 4 . above) s i n c e t h e y r e l a t e o n l y to t h e "x" d i m e n s i o n -

given

similar equations may

151

be derived for t h e "y" a n d "z" d i m e n s i o n s ) . Both of t h e s e Laws are u s e d e x t e n s i v e l y in solid s t a t e c h e m i s t r y . We c a n also s h o w t h a t t h e diffusion coefficient, D, u s e d in t h e s e Laws is t e m p e r a t u r e d e p e n d e n t , a n d c a n fit it to an A r r h e n i u s e q u a t i o n : 4.6.5.-

D = Do exp - E / k T

S o m e t i m e s , a m o r e useful t e r m is: 4.6.6.-

ED = - k l d(InD) / d ( 1 / T )

w h e r e ED is t h e diffusion e n e r g y involved in t h e p r o c e s s . However, t h e s e e q u a t i o n s are only u s e d w h e n t h e Fick E q u a t i o n s fail to fit t h e diffusion d a t a o b t a i n e d . It is possible to also s h o w t h a t t h e diffusion p r e s e n t in any c o m p o u n d is d i r e c t l y r e l a t e d to t h e f u n d a m e n t a l (zero p h o n o n line) of t h e lattice. The derivation is given in A p p e n d i x III. at t h e e n d of t h i s c h a p t e r . 4 . 7 . - DIFFUSION M E C H A N I S M S To this point, vce have e x a m i n e d diffusion g r o w t h in t e r m s of n u c l e a t i o n a n d e m b r y o f o r m a t i o n . Let u s n o w explore t h e actual s p e c i e s w h i c h diffuse in t h e l a t t i c e . WE WILL FIND THAT .aLL DIFFUSION MOTION OCCURS BY D E F E C T MOVEMENT

IN THE LATTICE .

C o n s i d e r t h e following d i a g r a m , given as 4.7.1. on t h e n e x t page, s h o w i n g two t y p e s of self-diffusion. mechanisms,

Self diffusion

vacancy and interstitial.

can occur

by at least

Both are "hopping"

two

motions,

as

is t h a t one site is e x c h a n g e d

for

d e s c r i b e d above. W h a t we see is in b o t h m e c h a n i s m s

a n o t h e r b y t h e defect u n d e r g o i n g a "jump" in t h e lattice n e t w o r k .

152

4.7.1.-

IMECHANISMS OF SELF-DIFFUSION

[INTERSTITIAL]

[VACANCY J

XXXX X X X X X X-if] X XXXX

XXXX XXXX

XX~X XX_ ~ x~xXX XX

XXXX

X X X X

X X X X

B E FO RE

AFTE R

xI-IX x

B E FO RE

I

X

AFTE R

Actually, we m u s t a c c o u n t for all t y p e s of defects,

X

including

charged

s p e c i e s (see 2.2.1.). To do this, let u s r e c o n s i d e r t h e t a r n i s h i n g r e a c t i o n (Section 4 , 5 , given above), u s i n g t h e g e n e r a l e q u a t i o n : 4.7.2.-

M + 1 / 2 X 2 =,, MX

S u c h a r e a c t i o n w o u l d o c c u r if we e x p o s e d

a m e t a l s u r f a c e to e i t h e r

oxygen or chlorine. A MX film w o u l d b u i l d u p on t h e m e t a l s u r f a c e a n d g r o w t h of a trim w o u l d o c c u r by diffusion. In t h e initial d e s c r i p t i o n , w e i g n o r e d v a c a n c y a n d i n t e r s t i t i a l diffusion a n d p r e s e n t e d only t h e c h a r g e d p a r t i c l e s , M 2+ a n d O = as t h e diffusing s p e c i e s (see s e c t i o n 4.5.). In actuality, t h e m e t a l diffuses as t h e i n t e r s t i t i a l , Mi 2+, a n d t h e a n i o n as Oi = . One w o u l d e x p e c t

that r

a m o u n t s of e a c h s p e c i e

w o u l d diffuse in

opposite directions, thus preserving electroneutrality. While t h i s m a y be t r u e for t h e r e a c t i o n in 4.7.2., i.e. - Mi 2+ r

Xi =, w h a t of

t h e c a s e for BaSiO3 w h e r e diffusion w a s l i m i t e d to one d i r e c t i o n ? It is n o t r e a s o n a b l e to a s s u m e t h a t t h e solid w o u l d build u p a c h a r g e as t h e Mi 2+ ions are diffusing (and t h e SiO4 = ions are not) a n d we m u s t s e a r c h for compensating species elsewhere. o c c u r s by diffusion of c h a r g e d ~ d r

It t u r n s o u t t h a t c h a r g e c o m p e n s a t i o n in t h e l a t t i c e .

To i l l u s t r a t e this, let us u s e t h e r e a c t i o n diagram:

in 4.7.2.

a n d t h e following

153

In this

diagram,

we

have s h o w n

both

charged

ions involved

in t h e

m i g r a t i o n a l d i f f u s i o n p r o c e s s as well as c h a r g e d v a c a n c i e s w h i c h also a d d to

the

overall

diffusion

process.

We m u s t

have

electroneutrality

in

diffusion, i.e.- p a i r s of d e f e c t s , a n d u s i n g the above example, we c a n w r i t e

nine e q u a t i o n s , of w h i c h the following is j u s t o n e case: 4.7.4.-

Mi + ~

Xi-

Here, two i n t e r s t i t i a l s are in equilibrium. We c a n also write

equations

involving c h a r g e only, or v a c a n c y plus c h a r g e d species. However, we have not c o n s i d e r e d r a t e s of diffusion in o u r model. In the BaSi04 case, t h e Ba2+will be very fast while the silicate ion will diffuse very slow (if at all). B e c a u s e of t h e vast differences in t h e t y p e s of diffusing species, t h e r e is no r e a s o n to e x p e c t all of t h e m to diffuse at t h e s a m e rate, p a r t i c u l a r l y w h e n we c o m p a r e e l e c t r o n s a n d vacancies. Actually, this a s p e c t of solid s t a t e r e a c t i o n h a s b e e n s t u d i e d in g r e a t detail a n d t h e K i r c h e n d a l l Effect deals w i t h this a s p e c t . To u n d e r s t a n d this effect, c o n s i d e r the following.

S u p p o s e we have a

s i t u a t i o n w h e r e A r e a c t s w i t h B to form a solid solution, AB. Let u s f u r t h e r s u p p o s e t h a t the diffusion r e a c t i o n is: 4.7.5.-

Ai + V A

~=> Bi + V B

T h u s , the v a c a n c i e s m i g r a t e along w i t h t h e i n t e r s t i t i a l a t o m s {ions)

154

in b o t h directions. In o u r case, we will ignore c h a r g e d species to simplify o u r explanation. Let us also s u p p o s e t h a t the v a c a n c i e s m i g r a t e

much

faster t h a n the interstitials a n d that: 4.7.6.-

DVA >> DVB

so that:

d c v A / d t >> dCvB/dt

Given t h e s e r e s t r a i n t s , we obtain a r a t h e r q u e e r result. This is i l l u s t r a t e d in the following diagram:

What we find is t h a t the difference in diffusion m e c h a n i s m s gives rise to the c r e a t i o n of n e w aite~ a c r o s s the diffusion zone a n d actually c a u s e s a d e f o r m a t i o n in the solid b e c a u s e the VA defects pile u p a n d finally b e c o m e a n n i h i l a t e d b e c a u s e of clustering. Acttml h o l e s a p p e a r in the solid due to vacancy diffusion. Note t h a t in this case, we are not f o r m i n g a newcompound

through

solid state

reaction,

b u t are

forming

a solid

solution of A a n d B. The K i r c h e n d a l l Effect h a s b e e n o b s e r v e d m a n y t i m e s , b u t o c c u r s m o s t often in r e a c t i o n s b e t w e e n m e t a l s w h i c h form alloys. T h e r e are t h r e e

(4) types of d i f f u s i o n - c o n t r o l l e d r e a c t i o n s possible for

h e t e r o g e n e o u s solid state r e a c t i o n s , viz4.7.8.-

TYPES OF DIFFUSION REACTIONS Simple Diffusion:

~l = k l

(t) 1/2

P h a s e - B o u n d a r y Controlled

~=k2

t

Material T r a n s p o r t

155

For s i m p l e

diffusion-controlled

reactions,

we

can

show

the

following

holds: 4.7.9.where

x 2 = 2 D Co t V M + (a) I/2 Co

= concentration

of c o n s t i t u e n t s at i n t e r f a c e ;

V = v o l u m e of

p r o d u c t AB p e r m o l e of r e a c t a n t s ; a = s u r f a c e layer t h i c k n e s s at i n t e r f a c e w h e n t = 0. It is also well to n o t e t h a t t h e final v o l u m e of t h e p r o d u c t , AB, m a y n o t be t h e s a m e as t h a t of t h e r e a c t a n t s vis.-

For p h a s e - b o u n d a r y c o n t r o l l e d r e a c t i o n s , t h e s i t u a t i o n differs s o m e w h a t . Diffusion of s p e c i e s is fast b u t t h e r e a c t i o n is slow so t h a t t h e diffusing s p e c i e s pile up. T h a t is, t h e r e a c t i o n to r e a r r a n g e t h e s t r u c t u r e is slow i n relation

to t h e

arrival of t h e

diffusing i o n s or a t o m s . T h u s ,

a phase-

b o u n d a r y (difference in s t r u c t u r e ) focus exists w h i c h c o n t r o l s t h e overall r a t e of solid s t a t e r e a c t i o n . T h i s r a t e m a y be d e s c r i b e d by: 4.7.10.-

d x / d t = k l ( s t / V o ) - ( I - ( l - x ) n ) = k i t /to

w h e r e st = i n s t a n t a n e o u s s u r f a c e area; Vo = original v o l u m e of p a r t i c l e s r e a c t i n g ; ro = original r a d i u s of p a r t i c l e s ; a n d n = 1, 1 !2, 1 / 3 for a 1d i m e n s i o n a l , 2 - d i m e n s i o n a l , or 3 - d i m e n s i o n a l r e a c t i o n . Material t r a n s p o r t f r e q u e n t l y involves a n e x t e r n a l g a s e o u s p h a s e , a n d a g e n e r a l f o r m u l a h a s n o t evolved. T h e above e q u a t i o n s , i . e . - 4 . 7 . 9 , diffusion m e c h a n i s m s ,

a n d 4 . 7 . 1 0 . s u m m a r i z e two of t h e t h r e e

o n e of w h i c h usually p r e d o m i n a t e s

in a n y g i v e n

c a s e {see 4.7.8.). But t h e s e e q u a t i o n s c o n t a i n q u a n t i t i e s t h a t are h a r d to m e a s u r e , or even e s t i m a t e . A m u c h b e t t e r w a y is to follow t h e m e t h o d

of H a n c o c k a n d S h a r p ( ~

1948). If o n e c a n m e a s u r e x, t h e a m o u n t f o r m e d in time, t, t h e n t h e

156

following e q u a t i o n applies: -Iogfin[l-xl]

4.7.1 I- H a n c o c k & S h a r p EquationType of diffus.ion

r a n g e of m

s i m p l e diffusion

0.57 - 0 . 6 2

nuclei g r o w t h

1.00-

phase boundary

1.25

=minT

+Ink

1.15 - 4.00

Here, we fund it n e c e s s a r y to be able to m e a s u r e t h e p r o g r e s s of a solid state reaction.

If we c a n do so, t h e n we c a n d e t e r m i n e

diffusion involved. I f - log (In(I-x)) value for t h e slope, m,

the

type

of

is p l o t t e d a g a i n s t In t, one o b t a i n s a

of t h e line w h i c h allows classification of t h e m o s t

likely diffusion p r o c e s s . Of c o u r s e , one m u s t be s u r e t h a t t h e solid s t a t e r e a c t i o n is p r i m a r i l y diffusion-limited.

Otherwise,

t h e analysis does n o t

hold. 4 . 8 , - ANALYSIS OF DIFFUSION REACTIONS Let u s n o w t u r n to diffusion in t h e g e n e r a l case, w i t h o u t w o r r y i n g about t h e e x a c t m e c h a n i s m or t h e r a t e s of diffusion of t h e various species. As an e x a m p l e to i l l u s t r a t e h o w we w o u l d analyze a d i f f u s i o n - l i m i t e d solid s t a t e r e a c t i o n , we u s e t h e g e n e r a l e q u a t i o n d e s c r i b i n g f o r m a t i o n of a c o m p o u n d w i t h s p i n e l (cubic) s t r u c t u r e a n d s t o i c h i o m e t r y : 4.8.1.-

AO + B203 ~ AB204

w h e r e A a n d B are two different m e t a l l i c ions a n d O is t h e oxygen a t o m (as t h e oxide). T h e r e are two (2) different c a s e s t h a t we c a n d i s t i n g u i s h , b o t h of w h i c h r e p r e s e n t p o s s i b l e diffusion m e c h a n i s m s in spinel. T h e s e involve e i t h e r a g a s e o u s or an ion diffusion m e c h a n i s m . Let us c o n s i d e r

the

ion-diffusion

mechanism

first.

As s h o w n

in t h e

following d i a g r a m , given as 4.8.2. on t h e n e x t page, b o t h A a n d B diffuse

157

t h r o u g h AB204 at the

s a m e rate. We c a n s h o w the p r o b a b l e diffusion

c o n d i t i o n s as specific solid s t a t e r e a c t i o n s , -r

Notice t h a t the partial r e a c t i o n s given in 4.8.3. are b a l a n c e d b o t h as to m a t e r i a l a n d charge. T h e s e are the r e a c t i o n s w h i c h o c c u r at the i n t e r f a c e (or p h a s e b o u n d a r y ) b e t w e e n reacting components,

the diffusing ions

a n d the

b u l k of t h e

as we have a l r e a d y i l l u s t r a t e d above. T h e r e

are at

least two o t h e r possible m e c h a n i s m s , as s h o w n in t h e following d i a g r a m :

158

In o n e case, t h e diffusion of A 2+ is m u c h f a s t e r t h a n B 3+ a n d in t h e o t h e r t h e o p p o s i t e is t r u e . Note t h a t c h a r g e - c o m p e n s a t i o n of m i g r a t i n g s p e c i e s is m a i n t a i n e d in all c a s e s . We c a n also i l l u s t r a t e a g a s e o u s t r a n s p o r t r e a c t i o n m e c h a n i s m in t h e s a m e m a n n e r , as s h o w n in t h e following d i a g r a m :

The r e a c t i o n s for t h i s c a s e (CASE III) are: 4.8.6.-

O= ~

1/202

+ 2e-

A 2+ + 2 e- + i / 2 0 2 + B203

~ AB204

In t h i s m e c h a n i s m w h e r e DA 2+ >> DB3+, t r a n s p o r t of e x t e r n a l o x y g e n gas is involved in t h e overall solid s t a t e r e a c t i o n , a c c o m p a n i e d b y e l e c t r o n i c c h a r g e diffusion. There

are

still

two

more

mechanisms

to

consider,

that

where

the

diffusion r a t e of B3+ is m u c h g r e a t e r t h a n t h a t of A2+: DB3+ >> DA2+ DA2+ >> DB3+ a n d t h e o p p o s i t e case, as s h o w n . In b o t h of t h e s e cases, t r a n s p o r t of 0 2 is

159

j u s t o p p o s i t e t h a t s h o w n in 4 . 8 . 6 . , a n d t h e r e a c t i o n s a r e : 4 . 8 . 7 . - CASE IV:

30 =

~

3/202

AO + 3 / 2 0 2 or:

20 =

~

02

+6e-

+ 6 e- + 2 B3+ +4e-

2 A 2+ + 0 2 + 4 e- + 2 B203 ~

This mechanism

AB204

2 AB204

is s h o w n i n t h e f o l l o w i n g :

We h a v e n o w p r e s e n t e d

all o f t h e 1 ~ ~ I r

diffusion reactions

in s p i n e l

s y n t h e s i s . T h e s e a r e s u m m a r i z e d a s follows: 4 . 8 . 9 . - Possible Diffusion M e c h a n i s m s in S p i n e l Ion D i f f u s i o n R e a c t i o n s : 3 A 2+ + 3 B203 ~ 3 AB204 + 2 B 3+ 2 B 3+ + 4 AO

~

AB204 + 3 A 2+

Conceivable Cases 1. DA2+ = DB3+ 2. DA 2+ >> DB 3+ (A 2+ + O = - for c h a r g e c o m p e n s a t i o n } 3.

DB4+ >> DA 2+ (B 3+ + O = - for c h a r g e c o m p e n s a t i o n }

160

4 . 8 . 9 . - Possible Diffusion M e c h a n i s m s in Spinel { c o n t i n u e d ) Gaseous T r a n s p o r t R e a c t i o n s : O= ~ 1 / 2 0 2 + 2 e A 2+ + 2 e ~ + 1 1 2 0 2 + B 2 0 3

~ AB204

Possible Cases 1. DB2+ >> DA 3+ ( t r a n s p o r t of c h a r g e only) 2. DA 3+ >> DB2+ ( t r a n s p o r t of c h a r g e only) It s h o u l d be clear, possibilities

by e x a m i n i n g

4.8.9.

carefully, t h a t

exist for the diffusion of r e a c t i n g

species

a number through

of

spinel

d u r i n g the solid state r e a c t i o n u s e d to form it. Which of t h e s e is m o r e likely will d e p e n d u p o n the exact n a t u r e of "A" as well as t h a t of "B". It w o u l d be too m u c h to expect t h a t the s a m e diffusing m e c h a n i s m w o u l d exist for all types of spinels (yes, t h e r e are m a n y t y p e s of spinels, s i n c e "spinel" is a p a r t i c u l a r c o m p o s i t i o n , i.e.- AB204, having a cubic s t r u c t u r e . What we n e e d to do is to

d e t e r m i n e w h i c h of t h e s e c a s e s is the m o s t

relevant. To do this, we n e e d to g e n e r a t e s o m e e x p e r i m e n t a l data. T h u s , we m e a s u r e f o r m a t i o n rate in air, p u r e oxygen gas a n d t h e n in an i n e r t gas. If the r a t e s do not differ significantly, t h e n we c a n rule out gaseous transport mechanisms. including

electrical

There

conductivity,

are o t h e r

transference

t e s t s we

numbers

can

and

apply,

thermal

e x p a n s i o n . Although t h e s e s u b j e c t s have b e e n i n v e s t i g a t e d in detail, w e shall not p r e s e n t t h e m h e r e . It s h o u l d be clear t h a t a n u m b e r of m e c h a n i s m s exist. Which of t h e s e d o m i n a t e s will, as we s t a t e d before, will d e p e n d u p o n the n a t u r e of b o t h "A"

and

"B".

However,

it

has

been

observed

throughout

many

investigations t h a t one m e c h a n i s m s e e m s to d o m i n a t e . We will not delve through these works. It suffices to say that: "For the most part, in oxygen-domlnnted

hosts, dlf~-~on by small cations prevails and that d l f f ~ o n of chargecompensated pairs predominates".

161

It is for this r e a s o n t h a t we write t h e solid s t a t e diffusion r e a c t i o n for t h e m a g n e s i u m a l u m i n a t e spinel as follows: 4.8.10.-

3 M g 2+ + 4 A1203 =,, 3 MgAI204 + 2 AI 3+ 2 AI 3+ + 4 MgO 4MgO

~

+ 4 AJ203

MgA1204

+ 3 M g 2+

~ 4 MgAI204

Note t h a t t h e ions given in 4 . 8 . 1 0 . c a n c e l o u t in t h e equations, a n d t h a t only the

overall solid

state reaction

remains.

However,

it

has been

d e t e r m i n e d t h a t diffusion is limited by: 4 . 8 . 1 1 . - Diffusion M e c h a n i s m : lh~x~kel Pair:. AIAI + ooVi ~

VAI + Ali 3+

In this case, the F r e n k e l p a i r diffusion, i.e.- VAI + Ali 3+, p r e d o m i n a t e s a n d is f ~ t ~ r t h a n a n y o t h e r possible m e c h a n i s m . 4.9.-

DIFFUSION IN SILICATES

We have p r e s e n t e d spinel s y n t h e s i s b e c a u s e t h e s e s y s t e m s w e r e s t u d i e d first a n d c a n be u n d e r s t o o d in a s i m p l e m a m i e r . Various silicate s y s t e m s have also b e e n s t u d i e d a n d it h a s b e e n d e t e r m i n e d

t h a t t h e y are m o r e

r e p r e s e n t a t i v e of t h e g e n e r a l c a s e involving solid s t a t e s y n t h e s i s r e a c t i o n s t h a n spinel. Let u s e x a m i n e t h e following s i m p l e silicate r e a c t i o n : 4.9.1.-

2 CaO + SiO2 ~

Ca2SiO4

Calcium oxide, in p r o p e r p r o p o r t i o n , r e a c t s w i t h silica to form c a l c i u m orthosflicate. In t e r m s of t h e spinel case, we w o u l d e x p e c t to see t h e following diffusing s p e c i e s : 4.9.2.-

Ca2+ , S i 4 + , O

,e- ,p+

Using o u r model, we w o u l d illust a t e t h e diffusion r e a c t i o n s w i t h t h e

162

following diffusion c o n d i t i o n s : 4.9.3.- M e c h a n i s m s of Diffusion a n d R e a c t i o n B e t w e e n CaO a n d Si02

In t h i s case, we have two c o n c o m i t a n t m a t e r i a l s , CaO a n d Si02, r e a c t i n g t o g e t h e r to form t h e c o m p o u n d , c a l c i u m o r t h o s i l i c a t e , w h i c h exists as a p h a s e b o u n d a r y b e t w e e n t h e five diffusing species. We c a n h y p o t h e s i z e at least t h r e e c a s e s involving diffusion. In t h e first case, b o t h Ca 2+ a n d 0 = diffuse t o g e t h e r in t h e s a m e d i r e c t i o n . In t h e s e c o n d case, Ca 2+ a n d Si 4+ diffuse in o p p o s i t e d i r e c t i o n s while t h e t h i r d c a s e involves 02 t r a n s p o r t along w i t h t h e diffusion of Ca 2+ w h i c h t h e n r e a c t s w i t h t h e Si02. T h e s e t h r e e m e c h a n i s m s m i g h t s e e m to be valid b u t f u r t h e r i n v e s t i g a t i o n reveals t h a t actual diffusion is c o n t r o l l e d

by t h e n a t u r e of t h e l a t t i c e

s t r u c t u r e w h i c h involves t h e silicate t e t r a h e d r o n .

Additionally, we find

163

t h a t the actual diffusion r e a c t i o n w h i c h o c c u r s is quite different t h a n t h a t of the spinel case. This is s h o w n in the following d i a g r a m : 4.9.5.-

T h u s , the r a t e of diffusion h a s b e e n a s c e r t a i n e d to be: RATES OF DIFFUSION FOR SILICATE-BASED COMPOUNDS Rate of species diffusion: Ca 2+ ~- 0 = >> Si 4+ This is c a u s e d by the fact t h a t the Si 4+ is tied u p in the f o r m of silica tetrahedra

where

some

of the

oxygen

atoms

are

shared

within

the

s t r u c t u r e a n d are not free to move. A r e p r e s e n t a t i o n of this is given as follows: 4.9.6.Diffusion Reaction 2 Ca 2+ 20

These

tetrahedra

=

2Ca2++

are tied t o g e t h e r

20

at the

= + S i O 2 --r Ca2 SiO 4

corners

so t h a t

a silicate

"backbone" f o r m s the s t r u c t u r e . The m e t a l c a t i o n s form "bridges" b e t w e e n b a c k b o n e - l a y e r s a n d are m u c h m o r e free to move. However, it is well to note t h a t a small a m o u n t of silicate d o e s move, b u t the exact n a t u r e of t h e diffusing specie c a n n o t be quantitatively defined (It m a y d e p e n d u p o n t h e n a t u r e of the c o m p o u n d s b e i n g formed. Most probably, the diffusing specie is actually SiOn b u t the c h a r g e of e a c h actual specie m a y vary). In

164

o r d e r to avoid confusion, we will c o n t i n u e to use Si 4+ to indicate silicate diffusion.

T h u s , w e find that t h e predominating diffusion m e c h a n i s m for ion di~ion in C a 2 ~ 0 4 s y n t h e s i s is that of Ca 2+ and t h e o x i d e ion, 0 •. We now c o m e to the m o s t i m p o r t a n t p o i n t of this Section. Up to now, w e have a s s u m e d t h a t the p h a s e w i t h i n the p h a s e b o u n d a r y , as given in t h e d i a g r a m s , is iaviolable a n d not s u b j e c t to change. T h a t is, once t h e Ca2Si04 h a s formed, no f u r t h e r r e a c t i o n can occur. While this m a y be t r u e for s o m e solids, it certainly is n o t t r u e for Ca2Si04, a n d i n d e e d for m o s t of the o t h e r s y s t e m s t h a t we m i g h t study. The way we d e t e r m i n e

ff f u r t h e r r e a c t i o n is possible is to c o n s u l t t h e

p h a s e d i a g r a m of the s y s t e m . If a c o m p o u n d is stable, t h e n it h a s a finite probability of forming d u r i n g t h e solid state reaction. Let u s n o w s u p p o s e t h a t f u r t h e r r e a c t i o n d o e s take p l a c e at the p h a s e b o u n d a r y of CaO a n d Ca2Si04, a n d also at Ca2Si04 a n d Si02. We will call t h e s e n e w p h a s e s "X" a n d "Y". This gives u s the s i t u a t i o n s h o w n in the following:

Here, we have r e a c t e d o u r diffusing species, Ca 2+ a n d O = , w i t h the initial p r o d u c t of o u r reaction, Ca2SiO4 in one direction, a n d Si 4+ a n d O = in t h e o t h e r to derive two entirely new r e a c t i o n species. It w o u l d t h u s a p p e a r t h a t "X" s h o u l d be Ca2Si05 a n d "Y" s h o u l d a p p e a r as Ca3Si207. A f u r t h e r look at the p h a s e d i a g r a m s h o w s t h a t the former s t o i c h i o m e t r y d o e s n o t exist, b u t the latter does. Because t h e

diffusing species,

Ca 2+ a n d O=,

165

c a n n o t r e a c t a c c o r d i n g the above-given reaction, they continue to d 1 ~ until they reach the vicinity of the pyrosllicate, Ca3Si207. There, t h e r e a c t i o n at this p h a s e - b o u n d a r y is: 4.9.8.-

Ca2+ + O = +

Ca3Si207 ~

2 C~2Si04

But t h e r e is n o t h i n g to stop f u r t h e r diffusion of t h e s e species a n d t h e y c o n t i n u e to diffuse to the vicinity of the SiO2 p h a s e b o u n d a r y , w h e r e t h e r e a c t i o n is: 4.9.9.-

Ca 2+ + O= + SiO2 ~

CaSiO3

T h u s , the finR! possible r e a c t i o n is t h a t vchich f o r m s the m e t a s i l i c a t e s t o i c h i o m e t r y . In the o p p o s i t e direction, we also have the diffusing species, Si 4+ a n d O=. T h e s e r e a c t w i t h the n e a r e s t p h a s e to give CaSiO4, as s h o w n in the following d i a g r a m :

Note t h a t we have t w o dlfr~,~on r e a c t i o n s one p r o c e e d i n g

w h i c h f o r m the m e t a s i l i c a t e ,

from the cationic species a n d the o t h e r f r o m anionic

species. T h i s h a s an a m a z i n g effect on the overall reaction, as s h o w n in the following diagram, given as 4.9.11. on the n e x t page. Note carefully the s e q u e n c e of i n t e r m e d i a t e r e a c t i o n s t h a t have o c c u r r e d , as d e p i c t e d involves

in this diagram. We see t h a t the first solid s t a t e r e a c t i o n

formation

of

the

orthosilicate.

But

this

is

very

quickly

t r a n s f o r m e d into m e t a - a n d pyro-silicates. However, it is the m e t a s i l i c a t e which

predominates

as the solid state

reaction

reaches

its m a t u r i t y .

Finally, it p r e d o m i n a t e s over all o t h e r forms of silicate p r e s e n t .

166

But, we still have small a m o u n t s of t h e o t h e r s p r e s e n t . Let us now r e c a l l the prevailing diffusion c o n d i t i o n s , as given in 4.9.4., i.e.4.9.12.-

Ca2+ =_ 0 = >> Si4+

Because of this diffusion m e c h a n i s m , we observe the above s e q u e n c e

of

diffusion r e a c t i o n s w h i c h o c c u r with time, as given in 4.9.12. It is actually possible to observe t h e s e r e a c t i o n s by the use of a polarizing m i c r o s c o p e hot stage w h e r e i n

the r e a c t i o n s are c a u s e d to o c c u r by h e a t i n g w h i l e

o b s e r v i n g the final s t r u c t u r e s f o r m e d via the x-ray diffraction p a t t e r n s . The above d i a g r a m is s o m e w h a t c u m b e r s o m e a n d it is m u c h easier to illustrate t h e s e r e a c t i o n s by a graph.

This is s h o w n in the following

diagram, given as 4 . 9 . 1 3 . on the n e x t page. In the s e q u e n c e of diffusion r e a c t i o n s , we note t h a t Ca2SiO4 is f o r m e d imanediately, followed by Ca4Si207. Both ImgiIt to d i ~ p l ~ a r w h e n CaSiO3

167

b e g i n s to form. N e a r t h e end, CaSi03 b e c o m e s t h e m a j o r p h a s e p r e s e n t , b u t we n e v e r r e a c h t h e p o i n t w h e r e j u s t ONE COMPOUND r e m a i n s . We always o b t a i n a m i x t u r e . T h i s p o i n t c a n n o t be o v e r s t r e s s e d a n d is o n e of t h e m o s t i m p o r t a n t p o i n t s to b e m a d e in t h i s c h a p t e r . T h e overall r e a c t i o n m e c h a n i s m s

are d i f f u s i o n - c o n t r o l l e d ,

and the total

solid s t a t e r e a c t i o n c a n be s u m m a r i z e d as follows: 4.9.14.-

8 CaO + 4 S i 0 2

~

4 Ca2SiO4

Ca2Si04 + S i 0 2

~

2 CaSiO3

3 Ca2SiO4 + SiO2

~ 2 Ca3Si207

2 Ca3~t207 + 2 SiO2 ~ 8CaO + 8SIO2 Although

these

reactions

~

6 CaSiO3 8CaSiO3

are w r i t t e n

to

show

the

formation

of t h e

m e t a s i l i c a t e , w e a l r e a d y k n o w t h a t n o n e of t h e s e r e a c t i o n s ever c o m e s to completion.

There

are c o m p e t i n g

side-reactions. We s t a r t w i t h a 2:1

168

s t o i c h i o m e t r y of CaO a n d SiO2, b u t e n d u p w i t h a s t o i c h i o m e t r y w h i c h is m o s t l y 1:1, t h a t of t h e m e t a s i l i c a t e . A n o t h e r w a y to look at t h i s p h e n o m e n o n

is t h a t t h e diffusion c o n d i t i o n s

favor t h e f o r m a t i o n of t h e m e ~ e ,

a t e , a n d t h a t t h e "excess Ca" is t a k e n

u p in t h e f o r m a t i o n of c o m p o u n d s

having a "calcium-rich" stoichiometry

(in c o m p a r i s o n to t h a t of t h e m e t a s i l i c a t e ) . However, if we s t a r t w i t h a s t o i c h i o m e t r y of 1:1 CaO to SIO2, we d i s c o v e r t h a t t h e s a m e c o m p o u n d s f o r m as before, a n d a m i x t u r e of c o m p o u n d s is still o b t a i a e d . T h e o n l y d i f f e r e n c e is t h a t t h e r e are s m a l l e r a m o u n t s of Ca2SiO4 a n d Ca3Si207 present! The only

c o n c l u s i o n t h a t we c a n d r a w is t h a t d i f f u s i o n - c o n t r o U e d

solid

s t a t e r e a c t i o n s t e n d to p r o d u c e m i x t u r e s of c o m p o u n d s , t h e r e l a t i v e r a t i o of w h i c h

is r e l a t e d

to

their

thermodynamic

s t a b i l i t y a t t h e re.action

t e m p e r a t u r e . Obviously t h e n , if w e c h a n g e t h e t e m p e r a t u r e of r e a c t i o n , w e would

expect

to

see

somewhat

different

mixtures

of

compounds

produced. Let u s n o w look briefly at a n o t h e r s i m i l a r s y s t e m w h e r e we will s t a r t w i t h a s t o i c h i o m e t r y of : 1.00 BaO to 1.00 S i 0 2 (in m o l e s ) . T h i s m i x t u r e w o u l d b e e x p e c t e d to r e a c t as: 4.9.15.-

BaO + SiO2 ~

w i t h t h e diffusion c o n d i t i o n :

BaSi03

Ba 2+ , 0 = >> Si 4+ . We c a n s h o w t h e s o l i d

s t a t e r e a c t i o n b e h a v i o r again b y a c h a r t , given as 4 . 9 . 1 6 . on t h e n e x t p a g e . O n e m i g h t t h i n k t h a t s i n c e Ba 2+ is a m u c h l a r g e r ion t h a n Ca 2+ , it w o u l d diffuse slower. S u c h is n o t t h e c a s e as c a n b e s e e n b y c o m p a r i n g t h e xaxis of 4 . 9 . 1 4 . w i t h t h a t of 4 . 9 . 1 6 . in t e r m s of r e a c t i o n t i m e .

169

It is r e m a r k a b l e t h a t in t h i s c a s e , w e s t a r t e d w i t h a I : i s t o i c h i o m e t r y a n d e n d e d u p w i t h a c o m p o u n d t h a t h a s a 2:1 s t o i c h i o m e t r y ! It s h o u l d b e c l e a r b y c o m p a r i n g

t h e e x a m p l e s for c a l c i u m silicate a n d

b a r i u m silicate t h a t o n e c a n n o t p r e d i c t h o w t h e d i f f u s i o n - c o n t r o l l e d s o l i d s t a t e r e a c t i o n s will p r o c e e d

since they are p r e d i c a t e d

upon the relative

t h e r m o d y n a m i c s t a b i l i t y of t h e c o m p o u n d s f o r m e d in e a c h s e p a r a t e p h a s e . T h e s e r i e s of d i f f u s i o n - c o n t r o l l e d r e a c t i o n s are for t h e c a s e of t h e s o l i d s t a t e r e a c t i o n b e t w e e n BaO + S i 0 2 as given in 4 . 9 . 1 6 . above. T h e s e s o l i d s t a t e r e a c t i o n s c a n b e s u m m a r i z e d as: 4.9.17.-

(a) 2 13a2+ + 2 0 =

+ SiO2

~

Ba2SiO4

(b) Ba2SiO4 + Si 4+ + 2 0 =

~

BaSiO3

B e c a u s e t h e r a t e of R e a c t i o n (a) is so m u c h f a s t e r t h a n t h a t of (b), we e n d u p w i t h Ba2SiO4 a s t h e m a j o r p r o d u c t . Brat, w e always e n d up w i t h a m i x t u r e of b a r i u m silicate s t o i c h i o m e t r i e s .

170

THIS

BRINGS US TO A MAJOR AXIOM FOR C H A V r E R - 4

REGARDING

SOLID STATE REACTIONS, VIS"DIFFUSION-CONTROLLED REACTIONS BETWEEN REFRACTORY O X I D E S ALWAYS R F ~ U L T S IN M I X T U R E S O F C O M P O U N D S

WHAT

THIS

MEANS

IS

THAT

WE

CANNOT

"

PREPARE

PURE

C O M P O U N D S BY REACTING OXIDES T O G E T H E R ! By now, you are p r o b a b l y w o n d e r i n g

h o w to avoid s u c h a m e s s . T h e r e

o u g h t to b e a w a y to e v a d e t h i s s i t u a t i o n , a n d i n d e e d t h e r e is. We c a n f i n d at l e a s t four (4) m e t h o d s to do so. T h e y i n c l u d e : 4.9.18.-

1. U s i n g a g a s e o u s r e a c t a n t 2. U s i n g a flux 3. P r o m o t i n g a s u p e r - r e a c t i v e c o m p o n e n t 4. U s i n g a p r e c i p i t a t e d p r o d u c t to act as t h e r e a c t i o n b a s e

Note t h a t all of t h e s e m e t h o d s a t t e m p t to b y p a s s t h e d e p e n d e n c e

of t h e

solid s t a t e r e a c t i o n u p o n diffusion. But, u s i n g a g a s e o u s r e a c t a n t m a y n o t b e p r a c t i c a l in all c a s e s . And, s o m e t i m e s it is h a r d to f'md a flux w h i c h d o e s n o t i n t e r f e r e w i t h t h e r e a c t i o n . A flux is d e f i n e d as follows: 4.9.19.-

DEFINITION OF A FLUX

"A m a t e r i a l w h i c h m e l t s lower t h a n t h e solid s t a t e r e a c t i o n temperature, allows

d i s s o l v e s o n e or m o r e

material

entering

transport

to

the

of t h e c o m p o n e n t s reaction

into t h e solid s t a t e r e a c t i o n .

zone,

and

without

Preferably,

the

end-

p r o d u c t s h o u l d b e i n s o l u b l e in t h e flux". We c a n also e m p l o y a s u p e r - r e a c t i v e p r o d u c t so as to o b t a i n 1 0 0 % of t h e desired

product.

follows.

In

this

E x a m p l e s are given in 4 . 9 . 2 0 . case,

we

are

varying

the

on the

ratios

of

next

p a g e as

reacting

p r o p o r t i o n s in a d e l i b e r a t e m a n n e r to p r o d u c e s e p a r a t e c o m p o u n d s .

molar

171

4.9.20.2 BaCO 3

+ SiO2

~

Ba2SiO4 ( I 0 0 % yield)

BaCO3

+ SiO2

~

BaSiO3

3 BaCO3

+ SiO2

~

Ba3SiO5 ( I 0 0 % yield)

(100% yield)

T h e m e c h a n i s m involved is as we have a l r e a d y d e s c r i b e d in 4.2.8., i.e.-

The

very

fine

particles

of

nearly

atomic

i m m e d i a t e l y . B e c a u s e t h e BaO p r o d u c t area, t h e solid s t a t e r e a c t i o n is

not

proportions

react

almost

h a s an e x t r e m e l y large s u r f a c e

diffusion limited. In t h e last m e t h o d

(see 4.9.18.), we m i g h t u s e a p r e c i p i t a t e d p r o d u c t to u s e as t h e basis to form t h e d e s i r e d c o m p o u n d . We m i g h t p r e c i p i t a t e : 4.9.22.-

Ba 2+ + HPO4 =

~

Bat-IP04

w h i c h is t h e n r e a c t e d to form t h e o r t h o p h o s p h a t e as t h e d e s i r e d p r o d u c t , e.g. 4.9.23.If

2 BaHPO4 + BaO

~

Ba3(P04)2 + H 2 0

b a r i u m o r t h o p h o s p h a t e is f o r m e d in t h i s m a n n e r , we c a n g u a r a n t e e

t h a t t h e final solid s t a t e r e a c t i o n p r o d u c t will c o n s i s t of 100% of t h e desired product. 4 . 1 0 . - DIFFUSION MECHANISMS WHEN A . . _ . . CATION CHANGES VALENCE Before we leave t h e s u b j e c t of diffusion, let u s e x a m i n e t h e c a s e w h e r e a c h a n g e of v a l e n c e s t a t e of t h e c a t i o n o c c u r s . T h i s m e c h a n i s m

is q u i t e

172

c o m m o n for t h o s e solid s t a t e r e a c t i o n s involving t h e t r a n s i t i o n m e t a l s of the

Periodic

Table.

In s t u d i e s

of solid

state

reactions,

this

situation

c o n t i n u e s to be t h e s u b j e c t of m u c h i n v e s t i g a t i o n b e c a u s e of t h e u n u s u a l s i t u a t i o n s w h i c h are e n c o u n t e r e d . C o n s i d e r t h e r e a c t i o n : 4.10.1.-

NiO + A1203 =r NiA1204

T h i s w o u l d give u s t h e following ditKmion d i a g r a m :

In t h i s case, we have 4 p o s s i b l e diffusion m e c h a n i s m s all p r o d u c i n g NiA1204. We c a n n o t say w h i c h of t h e diffusion p r o c e s s e s is f a s t e r at t h i s point, or w h i c h m e c h a n i s m prevails. F r o m w h a t we have s t a t e d p r e v i o u s l y , it is likely to be t h a t of A13+ a n d Ni2+ diffusing in o p p o s i t e d i r e c t i o n s . But, s u p p o s e s o m e of t h e N i 2+ oxidizes to N i3+ , vis4.10.3.-

Ni 2+ = Ni 3+ + e-

T h i s is a c a s e w h e r e a t r i v a l e n t c a t i o n is p r e s e n t in a d i v a l e n t lattice ( s e e

173

C h a p t e r 3).The shown4.10.4.-

following defect

2 Ni2+Ni ~i

=

+e-

2 Ni2+ N i

= + e-

=

r e a c t i o n s w o u l d t h e n be operative as

Ni3+Ni

+ VNi + P+

VNiNi3+Ni

+ VNi

+ P+

This w o u l d give us the foUowing diffusion diagram:

Note t h a t N i 2+ does n o t m i g r a t e b u t gives rise to a trivalent species plus a vacancy. This

situation

gives

rise

to

a complicated

set

of diffusion

conditions: a. The c h a r g e d vacancy is one of the m i g r a t i n g s p e c i e s . b. The c h a r g e d vacancy c o m b i n e s with N i3+ to form

a defect

c o m p o u n d , NiV-N iA1204. c. The divalent ion, Ni 2+, is also p r o d u c e d in the diffusion r e a c t i o n s .

174

d. S i n c e DNi 3+ >> Dv N i ' Ni 3+ diffuses o p p o s i t e l y to AI 3+ in t h e above s e t of a c t u a l diffusion c o m p o u n d arises, i,e,- N i2/~204.

reactions,

as s h o w n

another

Note t h a t t h e c h a r g e d ~ac~axmy diffuses as o n e of t h e r e a c t i n g s p e c i e s f o r m t h e defect c o m p o u n d . T h i s s i t u a t i o n is q u i t e c o m m o n s t a t e c h e m i s t r y of c o m p o u n d s c o n t a i n i n g m u l t i v a l e n t t r i v a l e n t N i 3+ also g i v e s rise to a n e w c o m p o u n d , N i A I 0 4 .

new

to

in t h e solid cations. The Yet t h e s a m e

r e a c t i o n s given in 4 . 1 0 . 2 . are also OPERATIVE. Thus, the overall reaction a c t u a l l y tak/mg p l a c e is a c o m b i n a t i o n of 4 . 1 0 . 2 . a n d 4 . 1 0 . 5 . LET US NOW SUMMARIZE T H E AREAS W E HAVE COVERED SO FAR: a. T y p e s of Solid S t a t e R e a c t i o n s b. F o r m a t i o n of P h a s e B o u n d a r i e s b e t w e e n R e a c t i n g S o l i d s c. N u c l e a t i o n R a t e P r o c e s s e s d. T h e Critical R a d i u s a n d T r a n s f o r m a t i o n f r o m E m b r y o to N u c l e u s . e. Diffusion Laws a n d M e c h a n i s m s f. Analysis of D i f f u s i o n - L i m i t e d Solid S t a t e R e a c t i o n s T h e s e s u b j e c t s i n c l u d e t h e m a j o r i t y of t h e c a s e s we will e n c o u n t e r in o u r s t u d y of solid s t a t e r e a c t i o n s a n d t h e i r c o m p l e x i t y . Let u s n o w s u m m a r i z e t h e i n f o r m a t i o n we h a v e c o v e r e d : 1) We c a n define t h e e x a c t n a t u r e of a solid s t a t e r e a c t i o n . 2) Before a n y solid s t a t e p r o c e s s c a n o c c u r , n u c l e i m u s t f o r m so as to allow t h e r e a c t i o n to go to c o m p l e t i o n . 3)

In

some

cases,

an

embryo

must

form

before

the

nucleus

f o r m a t i o n is c o m p l e t e . 4)

Nucleation

nucleation)

can

occur

within

a n y given

or be i n d u c e d b y a n y o u t s i d e

i n c l u d i n g t h e p r e s e n c e of a foreign solid.

solid

factor

(homogeneous

(heterogeneous),

175

5) In a diffusion-limited reaction, m o v e m e n t of species occurs by a "hopping" motion. 6) If a solid state reaction is diffusion-lirrdted, if is unlikely that we can obtain 100% of any product, and will always obtain a mixture of compounds whose relative ratio will depend t h e r m o d y n a m i c stability at the firing t e m p e r a t u r e .

upon

their

7) Several different types of species, including various solid state defects, diffuse and form a phase boundary of reaction, which may further react to form specific compositions. Although we have covered m e c h a n i s m s relating to solid state reactions, the formation and growth of nuclei and the rate of their growth in both heterogeneous and homogeneous solids, and the diffusion p r o c e s s e s thereby associated, there exist still other processes after the particles have formed. These include sequences in particle growth, once t h e particles have formed. Such sequences include: I m p i n g e m e n t of nuclei Ostwald Ripening (coarsening) Sintering Formation of Grain Boundaries We will reserve a discussion of these subjects to the appendices of this chapter. Any s t u d e n t who has interest in these subjects amy brouse in t h e descriptions given. PROBLEMS FOR CI-L~J~I'ER 4 I. Write equations for all of the p h a s e s changes given in 4.1.1. Use A for heat added or subtracted, as n e e d e d . 2. Write equations for the four (4) primary types of heterogenous solid state reactions usually e n c o u n t e r e d . Do not use those already given in Table 4-1.

176

4. Given t h a t Gd203 r e a c t s with A1203 to form GdAIO3, d r a w a d i a g r a m s h o w i n g the r e a c t i o n

conditions,

the p h a s e b o u n d a r y f o r m e d

and the

diffusion c o n d i t i o n s likely to prevail in the solid state r e a c t i o n . 4. If iron m e t a l w o u l d oxidize to form a h o m o g e n o u s oxide layer w i t h o u t flaking off, d r a w a d i a g r a m s h o w i n g the r e a c t i o n conditions, the p h a s e b o u n d a r y formed a n d the diffusion c o n d i t i o n s likely to prevail in the solid state r e a c t i o n . 5. Given t h a t the d e c o m p o s i t i o n of SrCO 3 is a simple n th o r d e r r e a c t i o n a n d t h a t the rate c o n s t a n t is 4 x 10 -4 m o l / s e c , at 865 ~ a m o u n t of SrO f o r m e d

calculate t h e

in 10.0 m i n u t e s from the original

1.0 mol of

powder, if the r e a c t i o n is a f i r s t - o r d e r r e a c t i o n . E x p r e s s your r e s u l t in %. If the r e a c t i o n were s e c o n d - o r d e r

(which it is

not), w h a t w o u l d be the % r e a c t i o n ? 6. Write solid state r e a c t i o n s for the s y n t h e s i s of: a. b. c. d.

M g2SiO4 Ba0.95Pb0.05 Si205 (Sr, Mg)4(PO4) 2 MgWO4

e. CalOCI2(PO4) 6 f. Cd(BO2) 2 g. Homilite- Ca2(Fe,Mg,Mn)B2Si2OI0 h. N a t r o s i l i t e i. ZnS (specify c o n d i t i o n s ) j. S t e r n b e r g i t e - AgFe2S4 (specify c o n d i t i o n s ) k. SrHPO4 (state c o n d i t i o n s of s y n t h e s i s ) 7. Classify the types of solid state r e a c t i o n s c o m p o u n d s in the above p r o b l e m .

you u s e d

to

form

the

8. For t h e spinel, Hercyanite, d r a w a d i a g r a m illustrating the p r o b a b l e ion diffusion p r o c e s s e s , give the diffusion c o n d i t i o n s a n d t h e diffusion

177

r e a c t i o n s if the diffusion coefficents for t h e cation a n d anion are about equal. 9. For t h e spinel, H e r c y a n i t e , d r a w a d i a g r a m i l l u s t r a t i n g t h e p r o b a b l e ion diffusion

processes,

give

the

diffusion

conditions

and

the

diffusion

r e a c t i o n s if the diffusion coefficents for t h e cation a n d a n i o n are: a. DA2+ >> DB4+ b. DB3+ >> DA2+ I0. For the spinel, H e r c y a n i t e , a s s u m e t h a t t h e c a t i o n c h a n g e s in v a l e n c e state.

Then,

processes,

draw

give the

a diagram diffusion

illustrating conditions

the and

probable the

ion

diffusion

diffusion reactions,

i n c l u d i n g t h e effect of i n d u c e d c r y s t a l lattice d e f e c t s . Av~ndix

I

M a t h e m a t i c s Associated w i t h N u c l e a t i o n Models of 4.4. I. In o r d e r to s h o w h o w a t h e o r y for the n u m b e r of nuclei f o r m e d p e r u n i t time c a n be built up, we c o n s i d e r t h e following. First, we a s s u m e t h a t a n u c l e u s will be s p h e r i c a l (to m i n i m i z e surface energy). T h e v o l u m e will be: Ap.l.l.-

V = 4 / 3 7r r3

We u s e r d t as a p a r t i c l e r a d i u s involving time, t, to define a g r o w t h r a t e a n d also define N as the n u m b e r of nuclei f o r m e d in t = y. T h e n t h e volume, V(t), f o r m e d as a f u n c t i o n of t i m e will be: Ap. 1.2.-

V(t)

= f {4n 13 0 f { r d t l x * ( d N l dt)t = y} d y

This is t h e g e n e r a l form in t h e c a s e w h e r e we have s p h e r i c a l nuclei. If w e do n o t have s p h e r i c a l nuclei, t h e n the e q u a t i o n m u s t be modified to: Ap. 1.3.-

V(t)

= f { 0 f [ r d t l ~-- o ( d N / dt)t = y ldy

178

It is easily s e e n t h a t 0 is a g e o m e t r i c a l

or shape factor in A p . l . 2 .

and

Ap. 1.4. a n d we define k as a n e x p o n e n t for t h e r a d i u s of t h e nuclei, x d t (it is 4 in A p . l . 2 .

(for t h e s p h e r i c a l case) a n d 2 in A p . l . 4 . for t h e g e n e r a l

case. In e i t h e r case, we have b o t h t h e v o l u m e of t h e n u c l e u s a n d t h e n u m b e r of nuclei f o r m e d as variables. Let u s n o w r e c o n s i d e r o u r n u c l e a t i o n m o d e l s of 4.4.1., specifically M o d e l s B, D a n d E. T h e s e involving r a n d o m

are e x a m p l e s of p h a s e - b o u n d a r y c o n t r o l l e d nucleation.

We n o w

assume

an exponential

growth embryo

f o r m a t i o n law (see 4.4.7), w i t h isotopic g r o w t h of nuclei in t h r e e d i m e n s i o n s a n d k2 as t h e r a t e c o n s t a n t . By suitable m a n i p u l a t i o n of 4 . 4 . 6 . , we c a n get (by Taylor S e r i e s e x p a n s i o n ) : Ap. 1.4.- RANDOM NUCLEATION -- Vt/Voo = 87r N o k 2 3 / V o k l 3 exp. [(-klt) - ( - k l t ) 2 / 2 ! + ( - k i t ) 3 / 3 ! ] w h e r e we have only given t h e first t h r e e t e r m s of t h e series. If No is large, t h e n k I will be small. T h i s gives us:

Ap.l.5.-

= {n N o k 2 3 k l / 3 V o } t 3

w h e r e ~ is t h e f l ~ c t i o n r e a c t e d . T h i s is o n e of the c o n c e p t s given in 4 . 4 . 1 . If k l is large, t h e n t h e first t e r m p r e d o m i n a t e s , a n d we get:

Ap.l.6.-

--- { 4 n N o k 2 4 k l / 3 V o } d~ / d t =

t3 ,and,

{4n N o k 2 3 k 1 ! 3 V o }

t2

The effective interfacial a r e a for o u r n u c l e i w o u l d be (1- ~), so that:

Ap.l.7.But since: extended obtain:

= 1 - exp. {4n No k23 k l / 4 Vo} t 3 d~

- (I - ~)

d~ex

(where

d~ex

is d e f i n e d

nuclei f o r m a t i o n ) , we c a n r e a r r a n g e A p . l . 6 .

as t h e r a t e of

and integrate

to

179

Ap.l.8.-

f d[~ex = f d [ ~ / ( 1 - [ ~ )

= -In(l-~)

We n o t e t h a t the above e q u a t i o n s enable u s to calculate ~, the fraction of nuclei f o r m e d , in t e r m s of a r a t e c o n s t a n t a n d the time, t. Note again t h a t k l is the r a t e c o n s t a n t for a u c l e i f o r m a t i o a w h e r e a s k2 is the r a t e c o n s t a n t for nuclei g r o w t h . Although the above is c o m p l i c a t e d ,

it does aptly illustrate the various

m e c h a n i s m s involved w h e n a t o m s (ions) m i g r a t e by diffusion a n d s t a r t to form a n e w s t r u c t u r e by f o r m a t i o n of i n c i p i e n t e m b r y o s , t h e n nuclei a n d finally t h e g r o w t h of p h a s e b o u n d a ~ e s . T h e r e is a n o t h e r w a y to a p p r o a c h the s a m e p r o b l e m . We c a n define a v o l u m e of d o m a i n s (nuclei) r e a d y to grow at time, y. If we use: k2 3 (t- y), this m e a n s t h a t k l is large. Following t h e s a m e m e t h o d as given above, w e arrive at:

Ap.l.9.-

[~ex = 4 / 3 n k 2

3f(t

-y)3 Idy

w h e r e I = r a t e of n u c l e a t i o n for a u n i t v o l u m e so t h a t Idy is set equal to t h e total n u m b e r of nuclei p r e s e n t . All o t h e r t e r m s have b e e n p r e v i o u s l y defined. T h u s , ff k l is truly a c o n s t a n t , t h e n : Ap.l.10.-

( 1 - [~) - n /3 k2 3 k l t 3

But if nuclei are a l r e a d y p r e s e n t , t h e n : Ap.l.ll.-

-In(l-~)

=4/3nNo

k2 3 t 3 = k ' t r = Z e x p . (-E/RT) t r

w h e r e r is an e x p o n e n t w h i c h d e p e n d s u p o n the s h a p e factor, 0, of t h e r e a c t i o n (as s h o w n above) a n d k' is a r a t e c o n s t a n t i n c o r p o r a t i n g all of t h e r e l e v a n t c o n s t a n t s t o g e t h e r ( i n c a s e we do n o t have a s p h e r i c a l n u c l e u s ) . We c a n s h o w h o w various m e c h a n i s m s a n d s h a p e factors affect the t i m e factor a n d e n e r g i e s involved in nuclei f o r m a t i o n a n d g r o w t h .

180

We c a n d i s t i n g u i s h b e t w e e n the i n t e r f a c e s t r u c t u r e of t h e n u c l e u s , N, a n d the c r y s t a l b e i n g t r a n s f o r m e d ,

/k T h i s c a n be s e e n

in the

following

diagram:

Here, we s h o w t h r e e differences in the interface b e t w e e n the n u c l e u s , N, a n d the original crystal, /~ We find t h a t in the first case, the l a t t i c e s m a t c h fairly closely a n d are c o h e r e n t . In the s e c o n d case, t h e r e is s o m e c o r r e s p o n d e n c e b e t w e e n the lattices. But the i n c o h e r e n t c a s e s h o w s little m a t c h i n g of the two lattices. It is t h e r e f o r e n o t s u r p r i s i n g t h a t the interface sh--ucture h a s a large effect

181

on the interfacial energy, b u t also u p o n t h e t h e r m o d y n a m A c s of c h a n g e w h i c h i n f l u e n c e s t h e r a t e of f o r m a t i o n of e m b r y o s , E, w h i c h t h e n f o r m nuclei. T h e i n t e r n a l e n e r g y (AGmax) of t h e e m b r y o is found in t h e s a m e way: Ap.l.13.-

W-AGmax

= i/3(4n

rcrit ~') = 1 6 / 3 { n

73/AGv}

w h e r e Gv is the difference of free e n e r g y b e t w e e n t h e n u c l e u s w i t h i n t h e crystal; N is the n u m b e r of nuclei; k is t h e nuclei volume; [~ is a s h a p e factor; o is t h e interfacial t e n s i o n or surface energy; a n d V is t h e i n t e r f a c e e x c h a n g e energy. Using t h e r m o d y n a m i c s , we c a n also define: Ap.l.14.-

AGv = RT In P/Pequil " i / V c r i t ,

then:

AGv = HV { {T - T e q u i l ) / T e q u i l } w h e r e equil, refers to the e q u i l i b r i u m state, Vcrit is t h e critical v o l u m e of t h e embryo, a n d b o t h Gv a n d HV still refer to t h e difference b e t w e e n t h e ~.ates of B a n d C. Accordingly, it is easy to arrive at: Ap. I. 15.-

I = Z exp.(- Eo / RT) exp.(- W / RT)

w h e r e Eo is the e m b r y o m o t i o n energy. If No is t h e n u m b e r of original e m b r y o nuclei p r e s e n t , t h e n : Ap.l.16.-

I = d N n l d t = k l (No - N) a

w h e r e k l is a r a t e c o n s t a n t a n d 6 is equal to t h e n u m b e r of s t e p s involved in t h e i n t e r f a c e c h a n g e of str~Jcture. T h u s , we have: Ap.l.17.-

Nn = No [ 1 - exp. (-k! t)]

or:

I -- k l N o

exp. ( - k i t )

T h e s e e q u a t i o n s allow u s to calculate b o t h n u m b e r s of e m b r y o s b e i n g f o r m e d and t h e e n e r g y involved in t h e i r f o r m a t i o n . T h u s , t h e i n t e r f a c e s t r u c t u r e affects t h e r a t e of n u c l e i f o r m e d and the r a t e of t r a n s f o r m a t i o n

182

of A to N (which t h e n grows to form the "B"-phase). We can now c o r r e l a t e Ap. 1.12. a n d 5 of Ap. 1.16. with the c a s e s given in Ap. 1.12. Note t h a t w e have a s s u m e d t h a t N 2 / 3 / N ~ 1 in the calculation.

ADvenflix I! M a t h e m a t i c s A s s o c i a t e d with Incipient Nuclei G r o w t h C o n s i d e r the t r a n s f o r m a t i o n to form a n u c l e u s in t e r m s of the rate i n s t e a d of a volume. We define: Ap.2.1.- Rate of nucleation{I) ----the n u m b e r of nuclei f o r m e d p e r u n i t t i m e per u n i t volume. If the n u c l e a t i o n is r a n d o m , we will have the h o m o g e n e o u s case; if it is specific

(with

heterogeneous

foreign

or

phase

boundary

walls),

we

have

the

case, vis-

Ap.2.2.

The above d r a w i n g illustrates the f o r m a t i o n of an n u c l e u s N w i t h i n t h e crystallite, /k (not d r a w n to scale in the above diagram). The c h a n g e in free e n e r g y in going from A to N will be a function of b o t h v o l u m e a n d surface a r e a (shape). We m u s t d i s c r i m i n a t e b e t w e e n the interfacial surface e n e r g y b e t w e e n A a n d N, a n d the i n t e r n a l e n e r g y of N. T h u s , the c h a n g e in free e n e r g y will be:

Ap.2.3.-

AG =Nn(GV

+ or} ~ + 13N2/3 y

183

w h e r e Gv is t h e d i f f e r e n c e of free e n e r g y of N w i t h i n A; N n is t h e n u m b e r of nuclei; k is t h e n u c l e i v o l u m e ; ~ is a s h a p e factor; o is t h e i n t e r f a c i a l t e n s i o n or s u r f a c e energy; a n d u is t h e i n t e r f a c e

exclmuge energy. T h e

e q u a t i o n Ap. 1.3., given above is u s e d h e r e , i.e.-

Ap.2.4.-

V{t)

= f { 0 [ f x d t 12 ( d N / d t ) t = y } d y

T h i s is t h e g e n e r a l f o r m in c a s e we do n o t h a v e s p h e r i c a l nuclei. If we do:

Ap.2.5.-

V(t)

= y{ 0 f{

We will a s s u m e a s p h e r i c a l

4n/3 [j'xdt

13(dN/dt)t=y}dy

s h a p e factor a n d u s e t h e l a t t e r of t h e t w o

e q u a t i o n s . S i n c e t h e first p a r t of Ap.2.4. is t h e v o l u m e f a c t o r w h i l e t h e l a s t p a r t is t h e s u r f a c e factor, we c a n do t h i s w i t h s o m e j u s t i f i c a t i o n . For a s p h e r i c a l n u c l e u s , i.e.- A p . 2 . 5 . , we g e t :

Ap.2.6.-

AG

=Ni 4/3nr

3(GV

+ ~} + 4 n r 2 N 2/3 Y

We n o w d i f f e r e n t i a t e Ap. 2.6. so as to o b t a i n zero s u r f a c e e n e r g y , i.e.-

Ap.2.7.-

d(A G ) / d r = 0 = 4n Ni r2 (Gv + o)

- 8 n r N 2/3 Y

If we h a v e zero surface e n e r g y , t h e n d ( ~ G ) / d r = 0 a n d o = 0. T h u s w e have: Ap.2.8.-

4n r 2 ni Gv + 8 n r N 2/3 Y= 0

orwhich

is

r crit = - 2 Y / A G v the

Gibbs-Thompson

equation

defining

the

critical

radius

r e q u i r e d for a n e m b r y o to form. Note t h a t w e h a v e a s s u m e d t h a t N 2/3 /N 1 in t h e c a l c u l a t i o n . T h e i n t e r n a l e n e r g y { A Gmax = W) is f o u n d in t h e s a m e way:

Ap.2.9.-

W = 1 / 3 (4n rcrit Y) -- 1 6 / 3 { n ~' 3 /AGv}

184

w h e r e we have u s e d similar c r i t e r i a t o simplify t h e r e s u l t i n g e q u a t i o n . Since from t h e r m o d y n a m i c s : Ap.2.10.-

AGv = RT In P/Pequil " 1/Vcrit, t h e n : AGv = HV { {T -Tequil /Tequfl }

w h e r e equfl, refers to the e q u i l i b r i u m state, Vcrit is the critical volume, a n d b o t h Gv a n d Hv

still refer to the difference b e t w e e n t h e ~

of A

a n d B. Using e q u a t i o n s a l r e a d y developed, it is easy to arrive at: Ap.2.11.-

I = Z exp (- Eo / RT) exp ( - W / R T )

w h e r e Eo is the e m b r y o m o t i o n energy. If No is t h e n u m b e r of original e m b r y o nuclei p r e s e n t , t h e n : Ap.2.12.-

I = d N / d t = k l (No - N) ~

w h e r e k I is a r a t e c o n s t a n t a n d 8 is equal to the n u m b e r of s t e p s involved in the i n t e r f a c e c h a n g e of s t r u c t u r e . T h u s : Ap. 2 . 1 3 -

N --No [ 1 - e x p ( - k i t ) ] or: I --- k l N o

exp ( - k i t )

We n o t e t h a t the g r o w t h of nuclei is a s s o c i a t e d with the m o t i o n of a p h a s e boundary. This c o m p l e t e s o u r analysis of t h e g r o w t h of e m b r y o s . A a ~ n f l l x III

H O M O G E N E O U S NUCLEATION P R O C E S S E S In a

m a n n e r similar to t h a t u s e d for h e t e r o g e n e o u s

define diffusion a n d nuclei g r o w t h for a already

stated

that

homogeneous

r e a c t i o n s , we c a n

homogeneous

nucleation

can

be

solid. We have contrasted

to

h e t e r o g e n e o u s n u c l e a t i o n in t h a t the f o r m e r is r a n d o m w i t h i n a ~ e c o m p o u n d while t h e l a t t e r involves m o r e t h a n one p h a s e or c o m p o u n d .

185

H O m o g e n e o u s n u c l e a t i o n m i g h t apply to p r e c i p i t a t i o n p r o c e s s e s w h e r e homogeneous

nucleation

must

occur

~onta~eousIy

before

any

p r e c i p i t a t i o n p r o c e s s c a n occur. Since p r e c i p i t a t i o n of c o m p o u n d s f r o m solution is a c o m m o n i n d u s t r i a l p r o c e s s , we i n c l u d e it h e r e . To begin, we k n o w t h a t a c h a n g e in free e n e r g y m u s t o c c u r s i m u l t a n e o u s l y as a c h a n g e in p h a s e (nucleation) occurs. This will be r e l a t e d to the total volume, V, a n d t h e total surface a r e a of b o t h nuclei a n d particles, E, visAp.3.1.-

A G = Vag

+ Eo

w h e r e VAg is the tree e n e r g y p e r u n i t v o l u m e a n d o is the surface e n e r g y of e a c h nuclei. If we have s p h e r i c a l nuclei, t h e n :

Ap.3.2.-

AG=4/3nr

3 Ag + 4 n r 2 o

If t h e total free energy, AG, does not c h a n g e w i t h nuclei radius, i.e.d(AG) I dr

is defined as zero, t h e n :

Ap.3.3.where

r crit r*crit

= - 2 o / Ag

is the

critical

r a d i u s for n u c l e u s f o r m a t i o n

similarity of this e q u a t i o n to t h e

Gibbs-Thompson

d e s c r i b e s t h a t of the h e t e r o g e n e o u s solids).

radius

critical

of

nuclei

(Note

of Ap. 1.10. growth

the

which

between

In t h e following table, given on the n e x t page, we s u m m a r i z e t h e m e t h o d s u s e d to o b t a i n the n e c e s s a r y e q u a t i o n s d e s c r i b i n g nuclei g r o w t h a n d t h e k i n e t i c s of g r o w t h . In the last equation, I is still defined as: I = N u m b e r of nuclei p e r Unit time p e r Unit volume. We find t h a t t h e quantity, N S* fe, is a p p r o x i m a t e l y e q u a l to 1046 per second.

p e r cc.

186

Table 4 - 3 Derivation of F o r m u l a s D e s c r i b i n g H o m o g e n e o u s Nuclei F o r m a t i o n 1. Using the e q u a t i o n s in Ap.3.1. & Ap.3.2., i.e.- AG = Vag + Xo and ~G=4/3nr

AG*=

16n o 3 / 3 ( A g ) 2

(where AG* = c h a n g e in free e n e r g y at the critical r a d i u s , r*,

3 Ag + 4 n r 2 o

for n u c l e u s formation).

we c o m b i n e t h e m to obtain: 16 n o 3 To 2 / 3

2, We t h e n define: Ag - Ah(T-To)/To = tub(AT/To) to

AG*=

obtain:

individual nuclei)

3. The n u m b e r of nuclei c a n be

N* = N e x p .

(/kh) 2 A T 2

(ah is the e n t h a l p y of the AG*/kT

d e s c r i b e d by a B o l t z m a n n d i s t r i b u t i o n , i.e.4. By defining dN/ dt as a

d N l d t = S*fe exp. AG*/ k T

f r e q u e n c y {t is time), we get:

(G*= free e n e r g y of t h e n u c l e u s , S * = critical s u r f a c e interface fe = I n t ~ e i a l e n e r g y

5. C o m b i n i n g t h e s e two e q u a t i o n s finally gives us-

I -- d N * / d t = N S * fe exp.{- A G * / k T

We c a n solve this graphically, as follows-

+ AGD / k T )

187

In t h i s c a s e , we plot T vs: r * so as to d e t e r m i n e t h e value of T* a n d T vs: I = dN/dt,

i.e.-

the

change

in

numbers

of n u c l e i

with

time

so as to

d e t e r m i n e t h e t e m p e r a t u r e of m a x i m u m f o r m a t i o n of n u c l e i . It c a n be s e e n in A t h a t t h e value of r * is r e m a r k a b l y c o n s t a n t over a w i d e r a n g e of t e m p e r a t u r e s ,

b u t t h a t it s t a r t s to a p p r o a c h

infinity at s o m e

c r i t i c a l t e m p e r a t u r e , T*. In c o n t r a s t , t h e n u m b e r of n u c l e i p r o d u c e d in B is m a x i m u m at s o m e p a r t i c u l a r t e m p e r a t u r e . T h i s s h o w s u s t h a t a l t h o u g h the critical radius does not c h a n g e temperature,

the

numbers

s i g n i f i c a n t l y over

of h o m o g e n e o u s

nuclei

a wide

range

produced

is

of

very

dependent upon temperature. Let u s n o w e x a m i n e t r a n s f o r m a t i o n k i n e t i c s for t h e g e n e r a l c a s e , t h a t is, the

homogeneous

c a s e for n u c l e i g r o w t h

a fte r t h e y h a v e f o r m e d .

We

assume: a. T h e n u c l e a t i o n is r a n d o m . b. T h e n u c l e a t i o n a n d g r o w t h r a t e a r e i n d e p e n d e n t

of e a c h o t h e r .

c. T h e s p h e r e s g r o w u n t i l i m p i n g e m e n t . T h e g e n e r a l e q u a t i o n s w e will u s e are:

Ap.3.5.where

d x / d t = k l ( 1 -x) n

a n d : x = 1- exp. (- k l t )t

all t e r m s are t h e s a m e as a l r e a d y given in S e c t i o n 3 . 3 . We solve

t h e s e e q u a t i o n s b y a T a y l o r s e r i e s , i.e.-

Ap.3.6.-

-In(l-x)

= 1 +kt/2!

+kt2/3I

+kt 3/4!

If w e n o w d e f i n e T as a n i n s t a n t a n e o u s m o m e n t of t i m e , w e c a n g e t : Ap.3.7.-

( d x / dt)t = -c

T h i s c a n b e a r r a n g e d to:

= K n e -ki t

188

Ap.3.8.-

( dx/dt)t=l;

= nklt nl(1-

x)

If x is small, t h e n -

Ap.3.9.-

dx/dt = nklt nl

By d e f i n i n g s p h e r e s w h i c h are n u c l e a t e d at time, ~, t h e t i m e of g r o w t h b e c o m e s : (t - ~). If we n o w define Xext as t h e e . x ~ h m i e a m o u n t p e r n u c l e u s before g r o w t h s t a r t s (note t h a t it is n o t a r a d i u s ; in fact, it is t h e s a m e x given in Ap.3.5.), t h e n w e have: Ap.3.10.-

Xext = f

4 / 3 u U 3 {t - ~)I d~

w h e r e U - - v o l u m e of t h e nuclei. If we t h e n f u r t h e r define xf as t h e final a m o u n t of v o l u m e per sphere at i m p i n g e m e n t w h e r e g r o w t h s t o p s , i.e.- xf = 1- Xext, t h e n we get t h e J o h n s o n - M e h l e q u a t i o n for s l o w nu 9 Ap.3.11.-

xf = 1- exp. {( - 1 1 3 n I U

3 } t 4}

If t h e n u c l e a t i o n is fast, t h e A v r a m i e q u a t i o n is o p e r a t i v e : Ap.3.12.-

xf = 1- exp. {{ - 1 ! 3 n Nv U 3 } t 3}

w h e r e Nv is d e f i n e d as t h e n u m b e r of n u c l e i p e r u n i t v o l u m e . It is t h e s e two e q u a t i o n s w h i c h h a v e b e e n u s e d e x t e n s i v e l y to d e s c r i b e h o m o g e n e o u s n u c l e a t i o n , p a r t i c u l a r l y in p r e c i p i t a t i o n p r o c e s s e s . 9

A o o e n d i x IV _

_

Diffusion E q u a t i o n s R e l a t e d to F u n d a m e n t a l V i b r a t i o n s of t h e L a t t i c e R e f e r r i n g to t h e Fick E q u a t i o n s of diffusion, let u s n o w r e a x a m i n e

I/T ,

t h e j u m p f r e q u e n c y , so as to r e l a t e diffusion p r o c e s s e s to lattice v i b r a t i o n p r o c e s s e s . The reciprocal of t h e t i m e of s t a y is t h e j u m p f r e q u e n c y a n d is r e l a t e d to t h e dilh~sion coefficient by: AP.4.1.-

D = a 2 16"r, = a a 2 1 T

= a~a 2

189

where w = I/•

~ a n d I: is t h e j u m p f r e q u e n c y (The o t h e r f a c t o r s r e m a i n

t h e s a m e as a l r e a d y defined), w c a n b e r e l a t e d to t h e free e n e r g y of t h e l a t t i c e via a B o l t z m a n n d i s t r i b u t i o n , i.e.AP.4.2.-

- Wo a exp (-AGo / rt) = ~o r exp (-AS / R - AH / RT)

b y a s e r i e s of m a t h e m a t i c a l m a n i p u l a t i o n s . It t u r n s o u t t h a t Wo is the

atomic vflrational frequency or t h e zero phonon m o d e of t h e lattice, a n d that: AP.4.3.-

Wo = 112n ~ r ( K / m )

w h e r e K is t h e elastic c o n s t a n t of t h e l a t t i c e a n d m is t h e reduced m a s s of the

atoms composing

diffusion

the

lattice.

Note

t h a t we

have just related

the

coefficient, D, to t h e z e r o - p h o n o n m o d e of t h e lattice. T h i s is

r e a s o n a b l e s i n c e t h e j u m p f r e q u e n c y o u g h t to b e a f u n c t i o n of t h e p h o n o n v i b r a t i o n a l c o n s t a n t as w ell.

This Page Intentionally Left Blank

191

Chapter 5 Particles a n d Particle T e c h n o l o g y Solids ap p ear in one of two forms, either

as crystals or powders.

The

difference is one of size, since m a n y of the p o w d e r s we use are in reality very fine crystals. This, of course, d e p e n d s u p o n the m a n n e r in which the solid is prepared. Nevertheless, m o s t solids t h a t we e n c o u n t e r in the real world are in the form of powders. That is, they are in the form of discrete

small

particles of varying size. Each particle has its own u n i q u e d i a m e t e r and size. Additionally, their physical p r o p o r t i o n s can vary in s h a p e from s p h e r e s to needles.

For a given powder,

all grains will be the same shape, but t h e

particle s h a p e a n d size can be altered by the m e t h o d u s e d to create t h e m in the first place. Methods of particle formation include: Precipitation Solid state r e a c t i o n Condensation In this Chapter, we i n t e n d to investigate m e c h a n i s m s of particle growth. We wiU, in the next chapter, a d d r e s s the m e t h o d s u s e d to form (grow) single crystals an d how they have been utilized. Certain p r o p e r t i e s of particles, including m e a s u r e m e n t of size, will be d i s c u s s e d later in this chapter. But first, we need to define exactly w h a t we m e a n by a particle. In the previous chapter, we covered m e c h a n i s m s relating to solid state reactions, formation of nuclei a nd the rate of their growth in b o t h h e t e r o g e n e o u s a nd h o m o g e n e o u s solids. Still other p r o c e s s e s exist after particles have formed, including s e q u e n c e s in particle growth. We have already m e n t i o n e d p r e c i p i t a t i o a as one m e t h o d for obtaining d i s c r e t e particles. As indicated in the previous chapter, nucleation in p r e c i p i t a t i o n p r o c e s s e s can be h o m o g e n e o u s (no outside influences) or h e t e r o g e n e o u s (by specific outside coercion). In precipitation p r o c e s s e s to form a di st ri but i on of particle sizes, it is probably a c o m b i n a t i o n of b o t h m e c h a n i s m s .

192

5 . 1 . - SEQUENCES IN PARTICLE GROWTH We have already s h o w n t h a t embryo formation leads t o nucleation, and t h a t this n u cleatio n preceeds any solid state reaction or change of state. In a like m a n n e r , nuclei m u s t form in order for any precipitation process to p r o c e e d . Once formed, these nuclei t h e n grow until i m p i n g e m e n t of the g r o w i n g particles occurs. I m p i n g e m e n t implies t h a t all of the n u t r i e n t supplying t h e particle growth has been u s e d up. T hi s m e c h a n i s m applies to b o t h solid state reaction a n d precipitation p r o c e s s e s to form p r o d u c t particles. Then a process k n o w n as O ~ w a l d R i l ~ n l n g takes over. This process occurs even at ro o m t e m p e r a t u r e . The s e q u e n c e s in particle growth are: 5.1.1.-

Embryo f o r m a t i o n Nucleation Nuclei G r o w t h Impingement Ostwald Ripening (coarsening) Sintering For m at i on of Grain Boundaries

We have already covered the first three in some detail.

Impingement involves

the point where the growing particles actually t o u c h each other and have u s e d up all of the n u t r i e n t which h a d originally c a u s e d t h e m to s t a r t growing. Ostwald r i p e n i n g usually occurs bet w e e n particles, following i m p i n g e m e n t , wherein

larger particles grow at the expense

of the smaller ones.

One

example of this would be if one h a d a precipitate already formed in solution. In m a n y cases, the smaller particles redissolve and r e p r e c i p i t a t e on t h e larger ones, causing t h e m to grow larger. Interfacial tension b e t w e e n t h e particles is the driving force, a n d it is the surface area t h a t b e c o m e s minimized. Thus, larger particles, having a total lower surface area, i n c r e a s e at the expense of n u m e r o u s smaller particles which have a relatively high surface area.

193

Ostwald ripening differs from nuclei growth in t h a t the relative size and n u m b e r s of particles change, w h e r e a s in nuclei growth, the n u m b e r s of particles growing from nuclei do not change. Sintering, on the o t h e r hand, is an entirely different process, and usually occurs w h e n external heat is applied to the particles. 5.2. - SINTERING, SINTERING PROCESSES AND GRAIN GROWTH Sintering of particles occurs w h e n one h e a t s a s y s t e m of particles to an elevated t e m p e r a t u r e .

It is c a u s e d by an i nt eract i on of particle

surfaces

w h e r e b y the surfaces fuse t o g e t h e r and form a solid m a s s . It is related to a solid state reaction in t hat sintering is governed by diffusion processes, but no solid ~cat~ r e a c t i o n , or change of com p osi t i on or state, takes place. T h e best way to illustrate this p h e n o m e n o n is to use p o r e growth as an example. When a s y s t e m of particles, i.e.- a powder, is h e a t e d to high t e m p e r a t u r e , the particles do not u n d e r g o solid state reaction, u n l e s s there is m ore t h a n one c o m p o s i t i o n pr e s ent . Instead, the particles t h a t are t o u c h i n g each o t h e r will sinter, or fuse t o g e t h e r , to form o n e larger particle. S u c h a m e c h a n i s m is illustrated in the following diagram, given as 5.2.1. on the next page. As can be seen, voids arise w h e n the particles fuse together. The void space will d e p e n d u p o n the original s h a p e of the particles. In 5.2.1., we have shown spherical particles w hi c h p r o d u c e only a few voids. Additionally, an overall change in the total volume of the particles, a n d t hat of the fusion product, occurs. Mostly, the c ha nge is negative, but in a few cases, it is positive. Experimentally, this c h a n g e is a very difficult p r o b l e m to m e a s u r e . What has b e e n done is to form a long thin bar or rod by p u t t i n g the p o w d e r into a long thin mold a n d p r e s s i n g it in a hydraulic p r e s s at m a n y tons p e r square inch. One can t h e n

measure

the

change

of volume i n d u c e d

by

sintering, by m e a s u r i n g a change in length as related to the overall length of the rod.

194

It h a s b e e n found t h a t t h e s i n t e r i n g of m a n y m a t e r i a l s c a n be r e l a t e d to a p o w e r law, a n d t h a t t h e ~arlxflmge, AL, c a n be d e s c r i b e d 5.2.2.-

AL / L o

by:

= kt m

w h e r e k is a c o n s t a n t , Lo is t h e original l e n g t h , t is t h e t i m e of s i n t e r i n g , a n d t h e e x p o n e n t , m , is d e p e n d e n t u p o n the m a t e r i a l b e i n g i n v e s t i g a t e d . It h a s also b e e n f u r t h e r d e m o n s t r a t e d t h a t a c o n s i d e r a b l e difference e x i s t s b e t w e e n t h e s i n t e r i n g of Rne a n d c o a r s e p a r t i c l e s , as s h o w n in t h e following d i a g r a m , given as 5.2.3. on t h e n e x t page. In t h i s case, t h e "fine" p a r t i c l e s a n d t h e "coarse" p a r t i c l e s w e r e s e p a r a t e d so t h a t t h e difference in size b e t w e e n individual p a r t i c l e s w a s m i n i m i z e d . T h a t is, m o s t of t h e individual p a r t i c l e s in e a c h fraction w e r e

almost the same

size. Both t h e fine a n d c o a r s e p a r t i c l e s have a s i n t e r i n g slope of 1 / 2 b u t it is t h e c o a r s e p a r t i c l e s w h i c h s i n t e r to form a solid having a d e n s i t y c l o s e s t to

195

theoretical density. This is an excellent exaumple of the effect of pore volume, or void formation, and its effect u p o n the final density of a solid formed by powder c o m p a c t i o n and sintering techniques. Quite obviously, the fine particles give rise to m a n y m or e voids t h a n the coarser particles so t h a t t h e attained density of the final s i nt er ed solid is m u c h less t h a n for the solid p r e p a r e d using coarser particles. It is also clear t hat ff one wishes to obtain a s i n t e r e d p r o d u c t with a density close to the theoretical density, one needs to start with a particle size distribution having particles of varied d i a m e t e r s so t h a t void volume is m i n i m i z e d . The next illustrate aggregate s h o wn in

subject we will discuss is t hat of grain growth. The si m pl est way to this factor is t h r o u g h the s i nt eri ng behavior of aggregates. An is defined a large particle, c o m p o s e d of m a n y small particles, as the following diagram, given as 5.2.4. on the next page.

In this diagram, two steps are implicit. It is t h e aggregates, c o m p o s e d of very fine specks, w h i c h sinter to form larger grains (particles).

196

But, since m a n y particles are growing at the same time, growth occurs until i m p i n g e m e n t . The formation of b o u n d a r i e s b e t w e e n the particles (grains) growing t h u s results. It is these grains which form the final sintered whole. Note that the crystallographic orientation of each grain differs from that of its neighbors, as illustrated in the above diagram. When a system of very fine particles is formed, the interfacial tension is high due to the very high surface area present. Agglomerates (weak surface e n e r g y interchange) will form, or a 4 j [ ~ ~ a t e s (strong surface energy i n t e r c h a n g e ) can result, especially ff the Ostwald ripening m e c h a n i s m is slow, or is inhibited. I m m e d i a t e removal of a precipitate from its "mother liquor" is one example w h e r e the likelihood of aggregate formation is e n h a n c e d . S i n t e r i n g t h e n p r o d u c e s both pore growth a n d grain growth. This m e c h a n i s m also applies to powder compaction p r o c e s s e s w h e r e aggregates m a y be p r e s e n t in the powder. Sintering t h e n leads to grain growth as well. It should be clear, then, that the precipitation process n e e d s to be controlled carefully in order to p r o d u c e a material c o m p o s e d of particles of a desired configuration. Both Ostwald riping a n d sintering can be utilized to obtain a particle of desired size, d i m e n s i o n s and particle habit. Industrial technologists have t a k e n advantage of these particle forming a n d altering

197

m e c h a n i s m s . One example of this type of particle growth is described as follows. In fluorescent lamps, a layer of p h o s p h o r s is applied to the inside of a glass tube by m e a n s of a s u s p e n s i o n of particles, i.e.- the h a l o p h o s p h a t e p h o s p h o r having a composition of: (MsF, CI(PO4)s: Sb ~+ Mn 2+ (plus m i n o r a m o u n t s of other p h o s p h o r s to achieve certain "colors"). It has b e e n d e t e r m i n e d that the lamp b r i g h t n e s s and duration of light o u t p u t (maintenance) is highly d e p e n d a n t u p o n how well the the internal surface of the glass is covered by the particles. By m a i n t a i n i n g precipitation conditions so t h a t small t h i n s q u a r e s of CaHP04 result from the precipitating solution, a m a x i m u m coverage of the glass is achieved. Precipitation occurs by adding a solution of (NH4)2HP04 to a solution of CaCI~. The resulting precipitate is consists of very fine particles a n d is not very crystalline. As a m a t t e r of fact, the p a r t i c l e s were usually ill-defined a n d b o r d e r e d u p o n a m o r p h o u s . The solution t e m p e r a t u r e is t h e n raised so t h a t Ostwald ripening can occur. As the larger crystaUites begin to grow, the smaller ill-defined crystallites, having a m u c h larger surface area, dissolve a n d r e p r e c i p i t a t e u p o n the larger ones. T h e resulting single-crystal squares, i.e.- El, have an average size of about 25~m. The solid state reaction to form the h a l o p h o s p h a t e p h o s p h o r is: 5.2.5.-

6 CaHPO 4 + 3 CaCY)~ + ~ 2

~

2 C ~ F (P04) a

w h e r e we have not shown the Sb20~ a n d MnCOs a d d e d as "activators". T h e h a l o p h o s p h a t e t h u s p r o d u c e d follows the crystal habit of the m a j o r ingredient, in this case that of CaHP04 habit. When applied, the thin s q u a r e s lie fiat a n d overlap on the glass surface. During sintering at about 1200 ~ the CaHP04 does not disintegrate b u t u n d e r g o e s an internal r e a r r a n g e m e n t to form (M~207 while m a i n t a i n i n g the s a m e crystal habit. The CaCX)3 disintegrates into small particles (like BaCOn) while CaF2 exhibits a sublimation p r e s s u r e at the firing t e m p e r a t u r e . The resulting solid state reaction to form the h a l o p h o s p h a t e p r o d u c t thus d e p e n d s u p o n the crystal habit of the major ingredient, CaHPO4.

198

The same type of m e c h a n i s m s apply to other materials. Even where a m e t a l is melted an d t h e n cast, nucleation leads to formation of m a n y fine p a r t i c l e s in the sub-solidus (partially solidified) state. This leads to grain growth in t h e solid metal, thereby lowering its strength. Sometimes, special additives are added to the melt to slow nucleation during cooling, t hereby increasing t h e s t r e n g t h of the metal p r o d u c t . Another example is our old friend, BaC03. If we fire this solid c o m p o u n d in air at a very high t e m p e r a t u r e for a long time, we get several changes. First, it d e c o m p o s e s to very free particles of BaO. These fine particles have a large surface area, and with c o n t i n u e d ruing sinters to form larger particles. Eventually, the particles get big enough, a n d the porosity decreases to t h e point wh er e grain b o u n d a r i e s begin to form between particles. The grains sinter together to form a large particle with m a n y grain boundaries. It should be again be e m p h a s i z e d t h a t each grain in the large particle is essentially a small single crystal with its lattice oriented in a slightly different d i r e c t i o n from th at of it neighbors (see 5.2.4.). Let us now consider the t h e r m o d y n a m i c s of sintering. There are two types of sintering which are d i s t i n g u i s h e d by the change in volume which occurs. These are: 5.2.6.-

NO SHRINKAGE :

dV/dt = 0

WITH SHRINKAGE :

d V / d t = f(V)

As we have already said, the c ha nge in volume, from initial state to final state, can be positive or negative, but is usually negative. The driving force is a d e c r e a s e in Gibbs free energy, AG. It is related to both the interfacial t e n s i o n (surface energy), V, a n d the surface energy of the particles (which is r e l a t e d to their size), viz5.2.7.-

d G = y dA

199

C o n s i d e r a m o r e f a m i l i a r e x a m p l e , t h a t of a d r o p l e t s i t t i n g u p o n t h e s u r f a c e of a l i q u i d . T h e d r o p l e t h a s a r a d i u s , r, a n d t h e r e a r e n - m o l e s of liquid w i t h i n it w i t h a m o l a l v o l u m e , V. To f o r m t h e d r o p l e t r e q u i r e s a n a m o u n t , nV, of t h e liquid, w h e r e V is t h e f r a c t i o n a l m o l a r v o l u m e of t h e d r o p l e t . T h i s gives us: 5.2.8.-

nV = 4 / 3 n r 3

a n d t h e c h a n g e in free e n e r g y to f o r m t h e d r o p l e t is: 5.2.9.-

dG = A G d n

= y dA

If w e n o w d i f f e r e n t i a t e t h e s e e q u a t i o n s , i.e.5.2.10.-

V o l u m e of Droplet: n = 4n r 3 / 3 V

so t h a t : (In = 4n /V r 2 d r

and: S u r f a c e A r e a of Droplet:

A = 4n r 2 so t h a t :

dA = 8n r d r

we c a n p u t all of t h e e q u a t i o n s t o g e t h e r so as to yield t h e Kelvin e q u a t i o n for c h a n g e in free e n e r g y as a f u n c t i o n of t h e r a d i u s of t h e s p h e r i c a l p a r t i c l e : 5.2.11.-

AG = 2 y

V/r

O n e c a n do t h e s a m e for a c u b i c p a r t i c l e , in fact for a n y s h a p e factor. S i n c e t h e c h e m i c a l p o t e n t i a l , p , is r e l a t e d to AG, w e u s e t h e following e q u a t i o n s : 5.2.12.-

-Po 2y

=AG=RTlnp/po

V /r

or: p = p o

=

RTInp/po exp (2 y V / r R T )

If w e n o w define ~p = p - Po , t h e n :

200

5.2.13.-

P/Po

=

1 + Ap/po

M a t h e m a t i c a l l y , In (I + Ap / Po ) -= AP / Po (within 5% ff ~ p / Po

>_ 0 . I ) . T h u s

we get t h e a p p r o x i m a t e e q u a t i o n : 5.2.14.-

Ap/po

=

2yV

/r-I/RT

It is t h i s e q u a t i o n w h i c h h a s b e e n u s e d m o r e t h a n any o t h e r t o

s i n t e r i n g . To e v a l u a t e its use, c o n s i d e r t h e following e x a m p l e . A l u m i n a , A1203, is a v e r y r e f r a c t o r y c o m p o u n d . It m e l t s above 1 9 5 0 ~

and

is n o t v e r y r e a c t i v e w h e n h e a t e d . If we a t t e m p t to s i n t e r it at 1730~

we

f'md t h e following v a l u e s to apply: 5.2.15.-

~P/Po y

=- 0. I

-= 2 0 0 0 d y n e / c m .

V = M/d = 25.4 cc./mol R

= 8.3 x 10 -7 e r g / m o l

r

= 6x

10 - 6 c m

=0.06micron

W h a t t h i s m e a n s is t h a t ff t h e a l u m i n a p a r t i c l e s are s m a l l e r t h a n a b o u t 0 . 0 6 m i c r o n , t h e y will n o t s i n t e r .

E v e n t h o u g h t h e above is a n a p p r o x i m a t i o n , t h e

specific c a s e for a l u m i n a h a s b e e n c o n f i r m e d e x p e r i m e n t a l l y . THUS, WE HAVE SHOWN THAT IF T H E PARTICLES ARE TOO SMALL, T H E Y WILL NOT

SINTER.

NORMAL NUCLEI

GROW22-1 T H R O U G H

DIFFUSION

P R O C E S S E S REMAINS T H E NORM UNTIL T H E PARTICLES G E T LARGE ENOUGH TO S I N T E R . Let u s n o w e x a m i n e w h y t h i s m e c h a n i s m

might

be t r u e .

In t h e

c a s e of

s i n t e r i n g of s p h e r e s , w e c a n define two c a s e s as before, t h a t of no s h r i n k a g e a n d t h a t of s h r i n k a g e , b o t h as a f u n c t i o n of v o l u m e . If we h a v e two s p h e r e s in

201

direct c o n t a c t , we c a n define c e r t a i n p a r a m e t e r s , as s h o w n in t h e following diagram:

We s t a r t w i t h s p h e r e s of radius, r, in direct contact. The two c a s e s s h o w n are: 1) n o ~ll-lnkage

2} w i t h s h r i n k a g e , Actual s i n t e r i n g o c c u r s by flow of m a s s from e a c h s p h e r e to the m u t u a l p o i n t of contact, w h i c h gradually t h i c k e n s . We cmn e s t i m a t e t h e v o l u m e of m a s s , V, at the c o n t a c t area, A, in t e r m s of the foUov~mg p a r a m e t e r s : r, t h e r a d i u s of the s p h e r e s ; p , the t h i c k n e s s of the layer buildup; a n d x, the r a d i u s of c o n t a c t of the b u i l t - u p layer. If we have s h r i n k a g e , t h e n we m u s t also evaluate h, the a m o u n t of s h r i n k a g e , s h o w n above as t h e h e i g h t of i n t e r l i n k i n g layer. This m o d e l b r i n g s u s to an i m p o r t a n t point, vis: If t h e s p h e r e s are too small, t h e r e c a n be little m a s s flow to t h e a r e a of joining (sintering) of t h e s p h e r e s . Actually, it is the r a t e of flow of m a s s to t h e joining a r e a t h a t is i m p o r t a n t a n d the a r e a of t o u c h i n g of t h e s p h e r e s vail d e t e r m i n e this.

202

It is therefore logical that a size l i m i ~ should apply to the case of s i n t e r i n g a n d its m e c h a n i s m s . If the s p h e r e s are too small, t h e n t h e r e is n o t e n o u g h t o u c h i n g area a n d v o l u m e for the s i n t e r i n g m e c h a n i s m to o c c u r . The actual values of the s i n t e r i n g p a r a m e t e r s have b e e n found to be: 5.2.17.VALUES OF SINTERING PARAMETERS V h A No S h r i n k a g e

nx21 2 r

0

n2x21r

x2/2r

n2x3 / 2r

x 2 / 4r

,

With Shrinkage

nx 2 / 2r

x2 / 2r

M o s t s y s t e m s exhibit s h r i n k a g e in s i n t e r i n g a n d it h a s b e e n f o u n d t h a t t h e

following e q u a t i o n applies: AL / L o = h / r

5.2.18.-

andx=ct

m

w h e r e t is the time of sintering, h is a c h a r a c t e r i s t i c s i n t e r i n g c o n s t a n t for shrinkage,

and m

is an e x p o n e n t

dependent

upon

the

mechanism

of

sintering. We have i n d i c a t e d t h a t flow of m a s s is i m p o r t a n t in sintering. T h e r e are several operative m e c h a n i s m s w h i c h are d e t e r m i n e d by the type of m a t e r i a l involved. Actually, the s t u d y of s i n t e r i n g d e s e r v e s a s e p a r a t e c h a p t e r , b u t w e will only s u m m a r i z e the m a j o r m e c h a n i s m s t h a t have b e e n o b s e r v e d for t h e c o n s t a n t s of 5.2.18. T h e s e are given in 5.2.19. on the next page. Note t h a t the values of h d e p e n d s u p o n the m a t e r i a l a n d the m e c h a n i s m of sintering. Also, t h e s e values d e p e n d u p o n r a n d x, the radii of t h e p a r t i c l e s a n d the r a d i u s of joining. T h e s e

e q u a t i o n s apply only to s p h e r o i d s a n d a

s h a p e factor m u s t be c o n s i d e r e d as well.

203

5.2.19.-

SINTERING MECHANISMS Mechanism Viscous flow .

.

.

.

.

.

Evaporation-Condensation Grain b o u n d a r y diffusion Vacancy F o r m a t i o n Volume diffusion

1

1/ 3

~./~

by ~/4

Grain b o u n d a r y diffusion by 1 / 6 .

.

.

.

Glass

1/2

1/ 5

Intersitial F o r m a t i o n

Material

h

m

o

= .

.

.

.

.

NaCI

1/ 2

.

.

2/5 .

.

.

Cu, Ag

1/ 3

.

Surface diffusion

1/7

5L/Lo = 0

Ice

The derivation b e c o m e s m u c h m o r e difficult a n d is beyond the scope of this chapter. The reader is referred to specific references on sintering and sintering m e c h a n i s m s . 5.3.-

PARTICLE SIZE

Most solids t h a t we actually deal with are powders. The powder is c o m p o s e d of discrete particles, a n d each particle m a y be single crystal or contain grain boundaries. The size of the particles are of i n t e r e s t to us because m a n y of their chemical a n d physical p r o p e r t i e s are d e p e n d e n t u p o n particle size. For example, the "hiding power" of p i g m e n t s is d e p e n d e n t u p o n the size of t h e p i g m e n t particles, a n d in particular the distributions of sizes of the particles. There is an o p t i m u m particle size distribution to obtain m a x i m u m h i d i n g power, d e p e n d i n g u p o n the specific application. P h a r m a c e u t i c a l s are a n o t h e r p r o d u c t w he r e particle size (PS) a n d particle size distribution (PSD) are i m p o r t a n t . The effectiveness of dosage and ingestion d e p e n d s u p o n t h e p r o p e r PSD. Particles are usually defined by their d i a m e t e r since m o s t are spheroid. We use the m i c r o n as a base, i.e.5.3.1.-

1 ~ (micron) = 10 -6 m e t e r

= 10 "4 c e n t i m e t e r .

The following defines particle r a n g e s t h a t we usually e n c o u n t e r in solid state chemistry:

204

5.3.2.PARTICLE RANGES Range

.Centime.ter .s

~

Description

Macro

1.0 - 0 . 0 5

104 - 5 0 0

Gravel

Micro

0.01- 0.0001

I 0 0 - 1.0

"Normal"

Sub-micro

0.0001-

1.0 - 0 . 0 0 1

Colloidal

It is e a s i e r to d e s c r i b e

10 -7

PS in t e r m s

of m i c r o n s

rather

than meters

or

centimeters. In t h e r e a l world, we e n c o u n t e r p a r t i c l e s in a v a r i e t y of forms, a l t h o u g h w e m a y n o t r e c o g n i z e t h e m as s u c h . In t h e following d i a g r a m , given as 5.3.3. o n t h e n e x t p a g e , are s u m m a r i z e d m a n y of t h e p a r t i c l e s of i n t e r e s t . At t h e top of t h e d i a g r a m , t h e p a r t i c l e d i a m e t e r s in m i c r o n s is s h o w n . I m m e d i a t e l y b e l o w are s t a n d a r d s c r e e n sizes, i n c l u d i n g b o t h U. S. a n d Tyler s t a n d a r d m e s h . S c r e e n s are m a d e b y t a k i n g a m e t a l w i r e of specific

diameter

and cross-

w e a v i n g it to f o r m a s c r e e n w i t h specific h o l e sizes in it. T h u s , a 4 0 0 m e s h s c r e e n will p a s s 37~ p a r t i c l e s or s m a l l e r , b u t h o l d u p all t h o s e w h i c h

are

larger. A 6 0 m e s h s c r e e n will p a s s u p to 250~ p a r t i c l e s , etc. (U. S. S c r e e n Mesh). T h i s d i a g r a m p l a c e s a n d d e f i n e s t h e size of m o s t of t h e p a r t i c l e s t h a t we a r e likely to e n c o u n t e r

in t h e

real world.

On the

left

a n g s t r o m s (/~) a n d m i c r o n s u p to 1 ~ = 10, 0 0 0 A .

is a c o m p a r i s o n In t h e m i d d l e

of

of t h e

s e c t i o n , " E q u i v a l e n t S i z e s " , is a c o m p a r i s o n of sizes a n d " t h e o r e t i c a l m e s h " . W h y t h e l a t t e r t e r m is s o m e t i m e s u s e d will b e d i s c u s s e d later. In " T e c h n i c a l Definitions", size r a n g e s a r e given for v a r i o u s t y p e s of solids in g a s e s , l i q u i d s in g a s e s , a n d t h o s e for soft. In a d d i t i o n , we h a v e a s e p a r a t e c l a s s i f i c a t i o n for w a t e r in air (fog). Finally, t h e r e products

and

by-products

we

is a c l a s s i f i c a t i o n for v a r i o u s c o m m e r c i a l might

encounter,

including

viruses

and

b a c t e r i a . Note t h e v a r i e d sizes of p a r t i c l e s t h a t we n o r m a l l y e n c o u n t e r . T h i s l i s t i n g is n o t m e a n t to b e all- i n c l u s i v e b u t o u t l i n e s m a n y a r e a s of t e c h n i c a l i n t e r e s t w h e r e t h e sizes of p a r t i c l e s are i m p o r t a n t .

205

206

Now let u s b e g i n to m o r e clearly define t h e a g g r e g a t e p r o p e r t i e s of p a r t i c l e s . Let u s t a k e a 1.00 cm. c u b e a n d c u t it into f o u r t h s , viz5.3.4.-

NUMBERS

mmmmele

mUll(),

SURFACE

- 43

_ 64

= ( 6 4 ) ( 6 )( I/4 )2 = 0 . 0 0 9 6 sq. m .

mmmm

Now we t a k e t h e s a m e c u b e a n d divide it into 1.0 ~ c u b e s , w i t h t h e f o l l o w i n g result: 5.3.5.-

I~ c u b e s = 10 4 c u t s in 3 - d i m e n s i o n s D i a m e t e r = 1.0 x i 0 -6 m 3 for e a c h p a r t i c l e Number

= ( 104) 3 = 1.0 x 1012 p a r t i c l e s

Surface

= (6) (1.0 x 1012) (1.0 x 10-6) 2 - 6 . 0 sq. m e t e r s

We h a v e d i s c o v e r e d two facts a b o u t p a r t i c l e s : a) T h e r e

are large n u m b e r s p r e s e n t

in a relatively s m a l l a m o u n t of

m a t e r i a l (we s t a r t e d w i t h a 1-cm. cube). And: b) T h e total s u r f a c e a r e a c a n be q u i t e large a n d d e p e n d s u p o n t h e size of t h e p a r t i c l e s . S i n c e t h e r a t e of solid s t a t e r e a c t i o n d e p e n d s u p o n s u r f a c e area, we c a n s e e t h a t very s m a l l p a r t i c l e s o u g h t to r e a c t m u c h f a s t e r t h a n large p a r t i c l e s .

207

A n o t h e r o b s e r v a t i o n is t h a t solid s t a t e r e a c t i o n s involving very fine p a r t i c l e s o u g h t to be very fast in t h e b e g i n n i n g , b u t t h e n will slow down as the p r o d u c t p a r t i c l e s b e c o m e larger. T h e r e are o t h e r p r o p e r t i e s

of p a r t i c l e s w h i c h w e

c a n t h i n k of, as follows: 5.3.6.-

PROPERTIES OF PARTICLES

Size

Surface a r e a

Porosity

Aggregates

Agglomerates

Shape

Density

Pore size

Numbers

Size d i s t r i b u t i o n

At this point, we are m o s t i n t e r e s t e d in t h e size of p a r t i c l e s a n d how t h e o t h e r factors r e l a t e to the q u e s t i o n of size. T h e n e x t m o s t i m p o r t a n t factor is s h a p e . Most of the p a r t i c l e s t h a t we will e n c o u n t e r are s p h e r o i d a l or o b l o n g in s h a p e , b u t ff we discover t h a t we have n e e d l e - l i k e (acicular) particles, h o w do we define t h e i r average d i a m e t e r ? Is it a n average of the s u m of l e n g t h plus c r o s s - s e c t i o n , or w h a t ? 5.4.-

PAR~CLE DISTRIBUTIONS

F u n d a m e n t a l to p a r t i c l e t e c h n o l o g y is t h e f a c t o r , s / z e distribution. By this, w e m e a n t h e n u m b e r of e a c h size in a given collection of particles. F r o m a close e x a m i n a t i o n of 5.3.3., it is evident t h a t we m u s t deal w i t h large n u m b e r s of particles, even w h e n we have a small s t a r t i n g s a m p l e weight. This p r o b l e m b e c o m e s c l e a r e r ff we c o r r e l a t e n u m b e r s of p a r t i c l e s w i t h size, s t a r t i n g w i t h a cube, one (1) c m in size, as in 5.3.4. , a n d m a k e the c u t s i n d i c a t e d in 5.4.1, given on the n e x t page. Having done so, we m i x t h e seven (7) s e p a r a t e p r o d u c t s . We could t h e n p l o t the p a r t i c l e d i s t r i b u t i o n as s h o w n in 5.4.2.,

also given on the n e x t page,

w h e r e we have p l o t t e d the log of t h e size of p a r t i c l e s in m i c r o n s vs: the log of the n u m b e r of p a r t i c l e s c r e a t e d . Obviously, t h e r e is a linear r e l a t i o n s h i p b e t w e e n t h e two variables. T h e o t h e r factor to n o t e is t h a t this d i s t r i b u t i o n c o n s i s t s of specific (discrete) sizes of p a r t i c l e s .

208

5 . 4 . 1 . - V a r i o u s Sized C u t s of a O n e C e n t i m e t e r Cube

5.4.2.-

Size 1 ~

Number / cm 3 1.0 x 1012

I0 ~

1.0 x 109

50/J

8.3 x 106

100 ~

1.0 x 106

2 5 0 la

6.4 x 104

500 ~

8.3 x 103

1000 ~

1.0 x 103

A D I S C R E T E P O P U I ~ T I O N OF PARTICLES to

L.

12

E c

13

0

~ 0

4

0

1

2

log p

3

In N a t u r e , h o w e v e r , w e always h a v e a c o n t i n u o u s d i s t r i b u t i o n of p a r t i c l e s . T h i s m e a n s t h a t we h a v e all sizes, even t h o s e of f r a c t i o n a l p a r e n t a g e , 18.56V,

18.57V,

differences). formation,

The i.e.-

18.58

~, etc.

reason

(supposing

for t h i s

precipitation,

is

embryo

that and

that we the

can measure

mechanisms

nucleation

for

growth,

i.e.-

0.01 particle Ostwald

r i p e n i n g , a n d s i n t e r i n g , are rm~_d_c~n p r o c e s s e s . T h u s , w h i l e we m a y s p e a k of t h e " s t a t i s t i c a l v a r i a t i o n of d i a m e t e r s " , a n d w h i l e we u s e w h o l e

n u m b e r s for

t h e p a r t i c l e d i a m e t e r s , t h e a c t u a l i t y is t h a t t h e d i a m e t e r s a r e f r a c t i o n a l in n a t u r e . Very few * p a r t i c l e - s i z e " s p e c i a l i s t s s e e m to r e c o g n i z e t h i s fact. S i n c e the processes

are r a n d o m in n a t u r e , we c a n u s e s t a t i s t i c s to d e s c r i b e

the

209

properties

of a p o p u l a t i o n of p a r t i c l e s ,

and the particle

size d i s t r i b u t i o n .

S t a t i s t i c a l c a l c u l a t i o n is well s u i t e d for t h i s p u r p o s e s i n c e it w a s o r i g i n a l l y d e s i g n e d to h a n d l e large n u m b e r s in a p o p u l a t i o n . It s h o u l d b e c l e a r t h a t w e do h a v e a p o p u l a t i o n of p a r t i c l e s in a n y given p a r t i c l e d i s t r i b u t i o n . 5.5,-

PARTICLE D I S T R I B U T I O N S AND T H E BINOMIAL T H E O R E M

To d e s c r i b e p a r t i c l e d i s t r i b u t i o n s , we will u s e o u r o w n n o m e n c l a t u r e ,

but

will s o o n find t h a t it is r e l a t e d to t h e s c i e n c e of s t a t i s t i c s a s well. We b e g i n b y t h e u s e of t h e P r o b a b i l i t y Law. If we are given n - t h i n g s w h e r e w e c h o o s e x v a r i a b l e s , t a k e n r at a t i m e , t h e i n d i v i d u a l p r o b a b i l i t y of c h o o s i n g (1 + x) will be: 5.5.1.-

Pi = (I + x ) n

T h i s is t h e BINOMIAL T H E O R E M . U s i n g a T a y l o r E x p a n s i o n , w e c a n find t h e t o t a l p r o b a b i l i t y for i t e m s t a k e n r a t a t i m e as: 5.5.2.-

P(r) =

In}

pr (I - p ) n - r

----

nCr = n ! / r ! ( n - r ) !

where5.5.3.

{ n~

{ r} If we n o w let n a p p r o a c h infinity, t h e n w e get5.5.4.-

P(r)

= E Pi = I / [ 2n n p ( I -p)] I/2 . exp (- x 2 / 2 n p ( I -p)

One will i m m e d i a t e l y r e c o g n i z e t h i s as a f o r m of t h e BOLTZMANN e q u a t i o n , or t h e GAUSSIAN LAW. We c a n m o d i f y t h i s e q u a t i o n a n d p u t it into a f o r m m o r e s u i t a b l e for o u r u s e b y m a k i n g t h e following d e f i n i t i o n s .

210

m

First, w e d e f i n e a m e a n (average) size of p a r t i c l e s in t h e d i s t r i b u t i o n as d , a n d t h e n d e f i n e w h a t w e call a " s t a n d a r d d e v i a t i o n " as o, for t h e d i s t r i b u t i o n of p a r t i c l e s .

From

statistics,

we

know

that this means

that

68%

of t h e

p a r t i c l e s a r e b e i n g c o u n t e d (34% o n e i t h e r s i d e of t h e m e a n , d), i.e. m

5.5.5.where

Z(d

+do)

=0.68

do is t h e d i a m e t e r

measured

to give + 3 4 % of t h e t o t a l n u m b e r

of

p a r t i c l e s . In a l i k e m a n n e r : m

5.5.6.-

Note

that

2o

----

0.954

=

Z(d • d2o)

3o

----_

0.999

=

Z(d +d3o)

these

standard

deviations

only

apply

if w e

have

a Gaussian

d i s t r i b u t i o n . In t h i s w a y , w e c a n s p e c i f y w h a t f r a c t i o n w e h a v e of t h e t o t a l d i s t r i b u t i o n , or l o c a t e p o i n t s in t h e d i s t r i b u t i o n . By f u r t h e r d e f i n i n g : m

5.5.7.-

npi

~-

o

---- ( n p ( 1 - p ) 1/2 = s t d . dev.

fl

r

~

np+x

= d

+x

=

(d-d)

w e c a n o b t a i n e x p r e s s i o n s to d e s c r i b e a p a r t i c l e d i s t r i b u t i o n w h e r e w e h a v e u s e d x as a d e v i a t i o n f r o m t h e m e a n . T h i s t h e n b r i n g s u s to t h e e x p r e s s i o n

for t h e GAUSSIAN PARTICLE S I Z E

D I S T R I B U T I O N , as s h o w n in t h e f o l l o w i n g : m

5.5.8.-

Pr =

1

exp ( - [ d -

(2n) II2 o

2 02

d ]2)

if: { _ o o < d < + o o } Let u s n o w e x a m i n e a G a u s s i a n d i s t r i b u t i o n . A p l o t is s h o w n as follows:

211

W h a t we have is the familiar "Bell-Shaped" curve. T h i s d i s t r i b u t i o n h a s b e e n variously called: 5.5.10.Gaussian Log N o r m a l Boltzmann Maxwell - B o l t z m a n n We shall u s e the t e r m "Log Normal" for r e a s o n s w h i c h will clear later. It s h o u l d be now a p p a r e n t t h a t we u s e t e r m s b o r r o w e d from statistics a n d a statistical

approach

to

describe

a

distribution

of

particles.

The

two

disciplines are well s u i t e d to e a c h o t h e r since s t a t i s t i c s is easily capable of h a n d l i n g large a s s e m b l a g e s , a n d the solid s t a t e p r o c e s s with vchich we deal are r a n d o m g r o w t h p r o c e s s e s w h i c h p r o d u c e large n u m b e r s of p a r t i c l e s . Let

us

now

give

some

examples

of

the

log-normal

distribution.

A

r e p r e s e n t a t i o n of several t y p e s of t h e s e d i s t r i b u t i o n s is given on the n e x t page as 5 . 5 . 1 1 . In this diagram, the p a r a m e t e r s of the various log n o r m a l d i s t r i b u t i o n s are

212

c o mp ar ed . Here, d is the average particle size, is the s p r e a d of the p a r t i c l e size distribution and A is the area, i.e.- the total n u m b e r of particles p r e s e n t in the particle distribution. Note t hat each distribution can look quite different from the others, d e p e n d i n g u p o n the values of the p a r a m e t e r s involved. Thus, in Case A, the three distributions differ in n u m b e r s only, t h e average d i a m e t e r s of the distribution being equal as well as the s p r e a d of t h e distribution, i.e.- o. But in Case B, the two distributions have the s a m e n u m b e r s of particles b u t not the s a m e s p r e a d of particles. Yet, in all of t h e s e cases, each particle distribution r e m a i n s a log-normal distribution and conforms to the definition of a Gaussian distribution. Thus, it s h o u l d be clear t h a t one c a n n o t d e t e r m i n e if a particle distribution is truly log-normal j u s t by the a p p e a r a n c e of plotted data. That is only decided by plotting the p a r t i c l e data on a log-normal plot and d e t e r m i n i n g if the distribution c o n f o r m s .

213

5 . 6 , - MEASURING PARTICLE D I S T R I B U T I O N S O u r n e x t t a s k is to d e t e r m i n e h o w to m e a s u r e a PSD. We k n o w t h a t we c a n specify a m e a n d i a m e t e r , b u t h o w do we o b t a i n it? T h e following is a s i m p l e e x a m p l e i l l u s t r a t i n g o n e m e t h o d of d o i n g so. S u p p o s e w e o b t a i n a s a m p l e of b e a c h s a n d . F r o m 5 . 3 . 3 . , we c a n see t h a t t h e p a r t i c l e s are liable to r a n g e f r o m a b o u t 8la to 2 4 0 0 ~. In o r d e r to g e n e r a t e a PSD, we m u s t s e p a r a t e t h e p a r t i c l e s . One w a y we c a n s e p a r a t e s u c h p a r t i c l e s is b y seiving t h e m . But w e find t h a t seives are only available in c e r t a i n sizes, as s h o w n in t h e following: 5 . 6 . 1 . - COMPLETE LISTING OF ALL SEIVE SIZES AVAILABLE- (U.S. STD.) Seive No.

Microns

Permissible Variati0n{%}

Seive N0.

Microns

.........................

Permissible Variationf%)

3.5

5660

• 3

40

420

• 5

4

4760

• 3

45

350

• 5

5

4000

+ 3

50

297

• 5

6

3360

• 3

60

250

• 5

7

2830

• 3

70

210

• 5

8

2380

+_ 3

80

177

• 6

10

2000

+ 3

100

149

•

,,,

,

,,

.

.

.

.

12

1680

• 3

170

88

+

14

1410

• 3

200

74

•

16

1190

• 3

230

62

•

18

I000

• 5

270

53

•

20

840

• 5

325

44

+ 7

25

710

• 5

400

37

• 7

30

590

+_ 5

35

600

+ 5

214

There

a r e two t y p e s of s t a n d a r d s c r e e n s , t h e U.S. a n d t h e Tyle r s t a n d a r d

s c r e e n s . We h a v e given t h e U.S. s c r e e n values. T h o s e of t h e Tyle r a r e v e r y s i m i l a r , e.g. - #5 Tyler = 3 9 6 2 ~ , # 20 = 8 3 3 ~, a n d # 4 0 0 = 38 p. To s c r e e n o u r s a n d s a m p l e , w e c h o o s e to e m p l o y t h e following seives: #10, # 18, # 2 0 , #30, # 4 0 , #50, #80, # 1 0 0 , #170, #200, #270, #325 & #400. Now ff w e s c r e e n t h e s a n d w i t h t h e # I 0

screen,

all p a r t i c l e s l a r g e r t h a n

2 0 0 0 + 6 0 p will b e r e t a i a e d u p o n t h e s c r e e n . T h e n e x t s t e p is to r e s c r e e n t h e p a r t t h a t I m a ~ d t h r o u g h t h e # 10 s c r e e n , u s i n g t h e # 18 s c r e e n , to o b t a i n a f r a c t i o n w h i c h is > 1 0 0 0 ~ + 50p. We t h e n r e p e a t t h i s p r o c e d u r e to o b t a i n a s e r i e s of f r a c t i o n s . T h e r e s u l t s a r e given in t h e following: m

5.6.2.-

Fractions Obtained fo > 2 0 0 0 p

Mean diameter d do

l O001a < fl < 2 0 0 0 la

1500

dl

8401a < f2 < 1 0 0 0

920

d2

590ja

< f3 < 8 4 0

715

d3

4 2 0 p < f4 < 5 9 0 p

505

d4

2 9 7 p < f5 < 4 2 0 p

358

d5

177ja < f6 < 2 9 7 la

237

d6

1491a < f7 < 177

163

d7

88

Ja
< 1491a

118

d8

77 la < f9 < 88

83

d9

53 ~ < flO < 77

68

dlO

44ja
< 53

49

dll

37 p < f12 < 4 4

41

d12

f13 < 37 lu

do

215

We n o w h a v e 12 f r a c t i o n s t h a t we c a n use. T h e m e a n d i a m e t e r is a p p a r e n t f r o m t h e s c r e e n s u s e d to s e p a r a t e t h e fractions. In t h e last c a s e t h a t we c a n use, d'-12 = 4 1 + 2 . 8 ~ . We c a n n o w w e i g h e a c h f r a c t i o n a n d c a l c u l a t e the % w e i g h t in e a c h fraction, p r o v i d i n g we u s e t h e total w e i g h t of all 14 f r a c t i o n s . K n o w i n g t h e d e n s i t y of the s a n d , a n d a s s u m i n g t h e p a r t i c l e s to be s p h e r e s , we c a n t h e n o b t a i n t h e

n u m b e r of p a r t i c l e s in e a c h fraction. (Note t h a t w e

have a s s u m e d t h a t all p a r t i c l e s s m a l l e n o u g h to p a s s t h r o u g h a given s c r e e n have d o n e so. In m a n y cases, t h i s is n o t true, a n d we h a v e to be c o g n i z a n t of this e~or,

i.e.- t h i s factor is t h e m a i n s o u r c e of

error

in t h e

SIEVE

METHOD of p a r t i c l e size analysis). We n o w find t h a t t h e r e are two g e n e r a l m e t h o d s for d a t a - r e p o r t i n g , n a m e l y : 5.6.3.-

METHOD I = % of total w e i g h t METHOD II = C u m u l a t i v e w e i g h t - %

E a c h h a s its a d v a n t a g e s . In t h e first m e t h o d , we c a l c u l a t e the total w e i g h t a n d a s s i g n e a c h f r a c t i o n a %-value. M e a n d i a m e t e r s , d', a r e e a s y to o b t a i n in the first m e t h o d b e c a u s e we k n o w t h e s c r e e n d i a m e t e r s u s e d to s e p a r a t e t h e fractions. In the s e c o n d , we a d d f2 to f l , t h e n f3 to f2 + f l , t h e n f4 to f3 + f2 + fl, etc. To o b t a i n d', w e t a k e t h e a v e r a g e of t h e a d d e d fractions, s h o w n as follows: m

5.6.4.-

d

fl

1 5 0 0 ~I

f2 + fl

1210

f3+f2

+ fl

f4 + f 3 + f 2 + f l

963 734

We c o n t i n u e w i t h all t h e f r a c t i o n s we have. If we n o w plot t h e data, we g e t t h e d i a g r a m , given as 5.6.5. o n t h e n e x t page.

216

We c a n see the s p r e a d of p a r t i c l e s in M e t h o d I, b u t c a n n o t d e t e r m i n e

the

m e a n accurately, since d" is a guess. M e t h o d II allows u s to o b t a i n the m e a n from t h e 50% point, t h a t is, t h e point w h e r e

50%

of t h e p a r t i c l e s

are

s m a l l e r t h a n , a n d 50% are larger t h a n . Actually, the d a t a of M e t h o d I are b e t t e r p l o t t e d as s h o w n in t h e following diagram:

217

M e t h o d III

is called a "Histogram Plot" while M e t h o d s I & II are called

" F r e q u e n c y Plot" a n d "Cumulative F r e q u e n c y Plot", respectively. T h e r e is o n e i m p o r t a n t p o i n t w h i c h n e e d s to be e m p h a s i z e d . T h a t is: "ALL METHODS FOR PRESENTING DATA FROM T H E MEASUREMENT OF PARTICLE SIZE DISTRIBUTIONS, W H E T H E R INSTRUMENTAL, SEDIMENTATION, OF

THE

TOTAL

SENSITIVE,

THE

SEIVING,

OR PHOTOMETRIC METHODS, MEASURE FRACTIONS PARTICLE

DISTRIBUTION.

FRACTION-SEGMEN~

IF

CAN BE

THE

METHOD

SMALL,

AND

IS THE

MEASURED PARTICLE DISTRIBUTION WILL BE CLOSE TO THE ACTUAL ONE.

IF THE

MEASUREMENT

IS LESS SENSITIVE,

THERE

MAY BE

SIGNIFICANT DEVIATIONS FROM THE CORRECT PSD. The d a t a for the f r a c t i o n - s t e p s c a n be in t e r m s of n u m b e r s of p a r t i c l e s , weight,

%-weight

or even

a packed

volume.

We shall n e x t

investigate

p a r a m e t e r s of d i s t r i b u t i o n s as a function of m e t h o d of r e p o r t i n g data. 5.7, -

AN/~YSIS OF PARTICLE DISTRIBUTION PARAMETERS

E a c h of the m e t h o d s of d a t a p r e s e n t a t i o n h a s its own special p r o b l e m s in t e r m s of the a m o u n t of e x t r a c t a b l e i n f o r m a t i o n one c a n obtain. We will a d d r e s s e a c h of t h e s e in t u r n , s t a r t i n g vcith t h e least c o m p l i c a t e d one, w h i c h is the " H i s t o g r a m " . A. THE HISTOGRAM A typical h i s t o g r a m is s h o w n in t h e following diagram, given as 5.7.1. on t h e n e x t y page, along w i t h an analysis of the p a r a m e t e r s w h i c h c a n be calculated. The first step is to calculate the individual a r e a s of t h e steps, Ai. T h e s e are s u m m e d a n d t h e n divided by the n u m b e r of individual areas, a n d t h e n m u l t i p l i e d by the s u m of (d2 - d l ) ! ni to find the m e a n , d. Admittedly, t h e m e t h o d is c u m b e r s o m e , b u t s o m e t i m e s this is all the d a t a we have.

218

5.7.1.-

Ai

!00

- ~;l:d2_ d I )

d = IAiln.

% of

9I ( d 2 - d

t )/ni

m

d

Total 5 0 Weight

B. FREQUENCY PLOTS Many p a r t i c l e - m e a s u r i n g m e t h o d s u s e STOKE'S LAW to d e t e r m i n e

particle

d i s t r i b u t i o n s . By suitable m a n i p u l a t i o n { s e e below), we o b t a i n a n e q u a t i o n r e l a t i n g the S t o k e s d i a m e t e r , M, w i t h the particle density, P I ' a n d t h e liquid density, P2' n a m e l y : 5.7.2.-

M = 18 vl h / c~ t (p~ - P2 )g

w h e r e ~ is t h e liquid viscosity, h the d i s t a n c e settles, a is a s h a p e factor (a = 1 for

a spherical

particle),

and

g

is

the

gravitational

constant.

p h o t o m e t r i c m e t h o d (which we shall d i s c u s s in m o r e detail below) at a specific

particle

diameter

and

hence

can

be

directly

The

gives %

plotted

as

frequency, as given in the following d i a g r a m w h i c h is labelled 5 . 7 . 3 . on t h e n e x t page. As s h o w n in the d i a g r a m , do c a n be c a l c u l a t e d from the m e a n d i a m e t e r if the d i s t r i b u t i o n is s y m m e t r i c a l , b u t th~-~r r a r e l y are. Note also t h a t o, w h i c h is defined as the d i a m e t e r limits e q u a l to 68% of the total particle p o p u l a t i o n , c a n n o t be obtained, only d o .

219

C. CUMULATIVE FREQUENCY The

most

popular

method

of d a t a

presentation

is

that

of

cumulative

f r e q u e n c y . An e x a m p l e of t h i s is s h o w n as follows:

m

Note t h a t b o t h d a n d o are easily o b t a i n e d . But no o t h e r i n f o r m a t i o n c a n b e derived. W h a t w e n e e d is a m o r e v e r s a t i l e m e t h o d . T h i s c a n b e a c c o m p l i s h e d

220

by c o n s i d e r i n g a m e t h o d r e l a t e d to the statistical n a t u r e of n e a r l y all p a r t i c l e distributions. The

most

versatile

method

for

d i s t r i b u t i o n s is by plotting t h e m

displaying

and

analyzing

particle

size

as a log n o r m a l d i s t r i b u t i o n . W h a t t h i s

m e a n s is t h a t we u s e log n o r m a l probability p a p e r to plot the data. It is t h i s m e t h o d t h a t we shall investigate in detail since it p r o d u c e s i n f o r m a t i o n about the

distribution

distinguish

not

between

obtainable by any o t h e r natural

and

artificial

method.

PSD's

and

In fact, one make

can

inferences

c o n c e r n i n g the origin of t h e PSD. Also, we c a n easily d i s t i n g u i s h the b i m o d a l c a s e , i.e.- two PSD's p r e s e n t at the s a m e time. T h i s is i m p o s s i b l e o t h e r m e t h o d s of d a t a p r e s e n t a t i o n . D. I.J3G NORMAL PROBABILITY METHOD C o n s i d e r t h e log n o r m a l (Gaussian) distribution, of w h i c h an e x a m p l e is given as follows:

Our a p p r o a c h is to a s s u m e t h a t all PSD's have a n origin in g r o w t h t h a t t e n d s to p r o d u c e a log n o r m a l population. This is r e a s o n a b l e since n e a r l y all, if n o t all, particle g r o w t h m e c h a n i s m s are r a n d o m in n a t u r e . Hence, we e x p e c t to

221

see a log n o r m a l PSD as the usual case. W h a t we t h e n do is to l o o k for d e v i a t i o n s f r o m log n o r m a l i t y . It is t h e s e a b e r r a t i o n s w h i c h s u p p l y a d d i t i o n a l i n f o r m a t i o n about the PSD. To plot a distribution, we u s e log probability paper. ,an s a m p l e is given in t h e following diagram:

A log n o r m a l d i s t r i b u t i o n will give a s t r a i g h t line w h e n p l o t t e d on this type of paper. This m e a n s t h a t the PSD is not l i m i t e d , i.e.- all sizes of particles are p r e s e n t from - ~o to +~o. However, ff the PSD is g r o w t h - l i m i t e d , it will r e a d i l y apparent

from

the

graph.

Ostwald ripening,

a mechanism

where

large

222

p a r t i c l e s grow at the e x p e n s e of small ones, is one m e c h a n i s m t h a t c a n give rise to g r o w t h limits. G r o w t h limits are d e n o t e d by: 5.7.7.-

LL

._..~

"""T

I

Xo

Upper Growth

Limit

Lower

Limit

Growth

- U p p e r D i s c o n t i n u o u s Limit - L o w e r D i s c o n t i n u o u s Limit

The d i s c o n t i n u o u s limit is t h a t in w h i c h all particles b e y o n d a specific size have b e e n removed, or do not exist. The d i a m e t e r of the p a r t i c l e s is f o u n d on the y-axis a n d p l o t t e d at the p r o p e r p o i n t on the x- axis as "% less than" (see M e t h o d II of 5.6.3.

a n d 5.7.4., given above).

5.8 - TYPES OF LOG NORMAL PARTICLE DISTRIBUTIONS We shall p r e s e n t

some

examples

of particle

distributions

and

how

to

i n t e r p r e t t h e m in o r d e r to s h o w the versatility of the m e t h o d . As you will see, the m e t h o d of plotting particle d i s t r i b u t i o n s via a l o g - n o r m a l m e t h o d allows one to i n t e r p r e t

particle

size in a m a n n e r

not feasible by o t h e r

m e t h o d s . Yet you will find t h a t m o s t particle size specialists do not t a k e advantage of the m e t h o d . A. UNLIMITED PARTICLE D I S T R I B U T I O N S The following d i a g r a m , given as 5.8. I. on the next page, s h o w s a typical PSD w h e r e the d i s t r i b u t i o n does not have limits.

223

Following t h i s e x a m p l e are d i s t r i b u t i o n s t h a t do h a v e l i m i t s to t h e " n o r m a l " d i s t r i b u t i o n . W h a t t h i s m e a n s is t h a t t h e d i s t r i b u t i o n s c o n f o r m to t h e l i m i t s d e f i n e d in 5 . 7 . 6 . Note t h a t in 5 . 8 . 1 . , a s t r a i g h t line is evident. T h i s is t h e t y p e of d i s t r i b u t i o n usually f o u n d as a r e s u l t of m o s t p r e c i p i t a t i o n p r o c e s s e s . But as we

s h a l l see,

t h i s is n o t t r u e for t h e

other

t y p e s of l o g - n o r m a l

distributions. B. LIMITED PARTICLE D I S T R I B U T I O N S If a p r e c i p i t a t e is allowed to u n d e r g o O s t w a l d r i p e n i n g , or is s i n t e r e d , or is c a u s e d to e n t e r i n t o a solid s t a t e r e a c t i o n of s o m e k i n d , it will often d e v e l o p into a d i s t r i b u t i o n w h i c h h a s a size limit to its g r o w t h . T h a t is, t h e r e is a maximum, distribution

or m i n i m u m approaches.

limit The

(and s o m e t i m e s distribution

both)which

remains

the

continuous

particle as

it

a p p r o a c h e s t h a t limit. T h e l o g - p r o b a b i l i t y plot t h e n h a s t h e f o r m s h o w n in 5.8.2. o n t h e n e x t p a g e . In t h i s c a s e , b o t h u p p e r a n d l o w e r l i m i t s are s h o w n . However, o n e m a y h a v e o n e or t h e o t h e r , or b o t h , in t h e

general c a s e . O s t w a l d r i p e n i n g t e n d s to u s e

224

u p all of t h e s m a l l p a r t i c l e s , w i t h o u t l i m i t i n g t h e u p p e r size. T h e n w e w o u l d h a v e t h e l o w e r limit b u t n o t t h e u p p e r . C. PARTICLE D I S T R I B U T I O N S WITH D I S C O N T I N U O U S L I M I T S Suppose we

u s e a 2 0 0 - m e s h s c r e e n to r e m o v e p a r t i c l e s l a r g e r t h a n 74~ a n d

a 4 0 0 - m e s h s c r e e n to r e m o v e t h o s e s m a l l e r t h a n 37V. T h e PSD is t h e n :

225

Note t h a t the d i s t r i b u t i o n is linear b e t w e e n 37 a n d 74 ~ b u t t h a t it abruptly shifts to + oo at t h e s e points. This is a particle d i s t r i b u t i o n for w h i c h it is i m p o s s i b l e to obtain an a c c u r a t e p i c t u r e by any o t h e r m e a n s . A

frequency plot

of the s a n e d a t a w o u l d look like t h e n look like this:

If we w e r e u s i n g the f r e q u e n c y plot m e t h o d to p r e s e n t this data, we w o u l d t h i n k t h a t the curve w a s j u s t a s s y m m e t r i c a l a n d t h a t the d i s t r i b u t i o n w a s n o t log-normal. Yet, it is obvious from 5.8.3. t h a t it is l o g - n o r m a l . D. MULTIPLE PARTICLE D I S T R I B U T I O N S Log probability plots are p a r t i c u l a r l y useful w h e n the d i s t r i b u t i o n is bimodal, t h a t is, w h e n two s e p a r a t e d i s t r i b u t i o n s are p r e s e n t .

S u p p o s e we have a

d i s t r i b u t i o n of very small particles, say in s u s p e n s i o n in its m o t h e r liquor. By an

Ostwald

ripening

mechanism,

the

small

particles

redissolve

and

r e p r e c i p i t a t e to form a d i s t r i b u t i o n of l a r g e r particles. T h i s w o u l d give u s t h e d i s t r i b u t i o n s h o w n in 5.8.5. on the next page.

226

In this illustration, the size distribution p a r a m e t e r s of b o t h distributions are readily apparent. A similar d e t e r m i n a t i o n is almost impossible with o t h e r types of data p r e s e n t a t i o n m e t h o d s .

Although a large d i ~ o a t i a u o u s

gap

b e t w e e n the two distributions is s how n in 5.8.5., it is rarely the case. As examples

of the

usefulness

of particle

distributions,

consi der

the

following. Two actual cases e n c o u n t e r e d in I n d u s t r y are described in t h e following examples: CASE I - FLUORESCENT LAMP PHOSPHOR PARTICLES Fluorescent la mps are m a n u f a c t u r e d by squirting a s u s p e n s i o n of p h o s p h o r particles in an ethyl cellulose lacquer u p o n the inner surface of a vertical glass tube. Once the lacquer drains off, a film of particles is formed. T h e lacquer is t h e n b u r n e d off, leaving a layer of p h o s p h o r particles. E l e c t r o d e s are sealed on; the tube is evacuated; Hg a n d inert gas is added; a n d the l am p e n ds are added to finish the lamp. Lamp b r i g h t n e s s and lifetime are d e p e n d e n t u p o n the particle size distribution of the p h o s p h o r particles. T h e n u m b e r of small particles is critical since they are low in b r i g h t n e s s o u t p u t

227

b u t high in light s c a t t e r i n g a n d a b s o r p t i o n . A l a m p b r i g h t n e s s i m p r o v e m e n t of 10% is easily achieved by r e m o v a l of t h e s e particles. In 5.8.6. is s h o w n t h e PSD t h a t r e s u l t e d w h e n a c e r t a i n m e c h a n i c a l m e t h o d w a s u s e d to r e m o v e the "fines", viz-

In this application of the l o g - n o r m a l plot, note t h a t the " m e c h a n i c a l " s e p a r a t i o n of "fines" h a s c r e a t e d a new p a r t i c l e d i s t r i b u t i o n w i t h d 2 = 2 p. Even the value of 02 differs from t h a t of the m a j o r p a r t i c l e distribution. In the "fines" fraction, it a p p e a r s t h a t the l a r g e s t particle does not e x c e e d about 5 p. N e e d l e s s to say, l a m p s p r e p a r e d from this p h o s p h o r w e r e inferior in b r i g h t n e s s . A r m e d w i t h this i n f o r m a t i o n , one could t h e n r e c o m m e n d

that

t h e m e t h o d of "fines" r e m o v a l be c h a n g e d . A n o t h e r c a s e is p r e s e n t e d on t h e n e x t page. CASE II. - T U N G S T E N METAL POWDER The lifetime of a t u n g s t e n filament in a n i n c a n d e s c e n t light bulb d e p e n d s a g r e a t deal on t h e g r a i n size of the wire u s e d to m a k e the filament. W - m e t a l p o w d e r is p r e s s e d into a b a r at p r e s s u r e s w h i c h e n s u r e t h a t the d e n s i t y is as

228

close as possible to the theoretical density. The bar is si nt ered in an i n e r t a t m o s p h e r e a n d t h e n "swaged" , i.e.- "hammered" into a long rod. This rod is t h e n d r awn into a fine wire (~ 2-5 mil) w hi ch is w o u n d into the filament form. Metal powder particle size has a major effect u p o n wire quality since fine particles form small grs.hm in the wire while large particles will f o r m

ulrge grains. When the filament is o p e r a t e d at 3 2 5 0 ~

to p r o d u c e light, the wire is h o t

e n o u g h t h a t vacancy migration will occur in the wire. Simultaneously, grain growth plus an increase in grain size also occurs. The grain b o u n d a r i e s b e t ween large grains will "pin" vacancies, b u t grain b o u n d a r i e s bet w een small grains do not appear to do so. If they do, the effect is at least an order of m a g n i t u d e smaller. As the vacancy c o n c e n t r a t i o n builds up at the large grain b o u n d a r i e s (see the Kirchendall effect given in Chapt er 4), local r e s i s t a n c e to electron flow increases. Eventually, a "hot spot" develops and the t u n g s t e n wire melts at that point, with c o n s e q u e n t failure of the filament a n d t h e lamp. If only small metal particles are us ed to m a k e the wire, t h e n the small grains p r o d u c e d m u s t grow into large grains before "hot spots" cause its eventual f a i l u r e . Therefore, i n c a n d e s c e n t lamp m a n u f a c t u r e r s exercise very close control of the t u n g s t e n - m e t a l powder PSD and the sintering p r o c e s s e s u s ed to p r o d u c e the wire to eliminate, as m u c h as possible, pores and control actual density. The di agr a m s how n as 5.8.7. on the next page illustrates a typical PSD for a t u n g s t e n metal powder where the large particles have been removed. In this case, the PSD is suitable a n d p a r t i c l e s larger t h a n about 5 ~ appear to have successfully r e m o v e d . This c o m p l e t e s the types of particle distributions t h a t we m i ght e n c o u n t e r . It is now time to show how particle size c o u n t i n g - d a t a are used. To do this, we m u s t select an i n s t r u m e n t t h a t p r o d u c e s c o u n t s of size of p a r t i c l e s correlated with n u m b e r s of particles in each size range. There are several types of s u c h i n s t r u m e n t s w hos e n a t u r e will be delineated below. But, first, we m u s t show how this is done. Let us now examine a m e t h o d of calculating a particel size distribution.

229

5 . 9 - : A TYPICAL PSD CALCULATION

Let us s u p p o s e t h a t we have a particle c o u n t i n g i n s t r u m e n t w h i c h s o r t s a n d c o u n t s the n u m b e r of p a r t i c l e s at a given particle size. The e x p e r i m e n t a l d a t a t h a t we collect are:

Having o b t a i n e d the d a t a as given in 5.9.1., o u r n e x t step is to calculate t h e a ~ e

size in e a c h particle intea-wal. For p a r t i c l e s less t h a n 2.2 ~, we have 2

230

p a r t i c l e s . At 3 . 0 la, w e h a v e 4 p a r t i c l e s . T h e r e f o r e ,

i n t h e r a n g e 2 . 2 - 3 . 0 ~,

w e h a v e t w o (2} p a r t i c l e s w h o s e a v e r a g e size is 2 . 6 p. In t h e r a n g e 3 . 0 - 5 . 0 tJ, w e h a v e f o u r (4} p a r t i c l e s w h o s e a v e r a g e size is 4 . 0 ~. We t h e r e f o r e c o n t i n u e to c a l c u l a t e An, as s h o w n in T a b l e 5 - I . for e a c h size r a n g e t h a t w e h a v e m e a s u r e d : TABLE 5 - 1 A TYPICAL PARTICLE SIZE D I S T R I B U T I O N C A I ~ U I A T I O N Diameter

n-Counts

2 . 2 ~1

0

4

5.0

7.0

9.0

10

15

An

d

o%

z%

2

2 . 6 tl

0. I

0. I

24

4.0

1.2

204

6.0

10.2

11.5

370

8.0

18.5

30.0

300

9.5

15.0

45.0

500

12.5

25.0

70.0

500

20.0

25.0

95.0

80

27.5

4.0

99.0

18

35.0

0.9

2 1.3

28

232

602

902

1402 ,,

25

30

4O

1902

1982

20OO

99.9

231

We have a total of 2 0 0 0 p a r t i c l e s t h a t we have m e a s u r e d a n d we c a n calculate the p e r c e n t (%) of total c o u n t s for e a c h size range. T h i s gives the following l o g - n o r m a l PSD plot, viz-

Here, the p a r a m e t e r s of t h e d i s t r i b u t i o n are given. T h i s m a t e r i a l a p p e a r s to have b e e n p r e c i p i t a t e d , or it m a y have b e e n o b t a i n e d by a g r i n d i n g p r o c e s s , since t h e p a r a m e t e r s i n d i c a t e a Gaussian u n l i m i t e d distribution. P a r t i c l e s g r o w n by solid s t a t e reaction, i n c l u d i n g Ostwald ripening, g e n e r a l l y have PSD c h a r a c t e r i s t i c s of the original PSD from w h i c h t h e y w e r e formed. T h a t is, t h e r e will be Emits a c c o r d i n g to the original d i s t r i b u t i o n from w h i c h it h a s arisen. If the original PSD h a d limits, t h e n the p r o g e n y PSD ,;viii also have s u c h limits. T h e basis for this b e h a v i o r lies in the fact t h a t ff one s t a r t s w i t h a c e r t a i n size r a n g e of p a r t i c l e s as t h e basis of particle r e a c t i o n a n d g r o w t h , one will e n d with t h e s a m e size r a n g e of p a r t i c l e s in the PSD of t h e p a r t i c l e s p r o d u c e d by the solid s t a t e r e a c t i o n . S u c h a case is s h o w n in the following d i a g r a m , given as 5.9.3. on the n e x t page. Here, the oxalate w a s p r e p a r e d by addition of oxalic acid to Gd(NO3)3 solution to form a p r e c i p i t a t e .

232

T h e r e is a n u p p e r limit of about 23 m i c r o n s in size. T h i s m a y be d u e to t h e fact t h a t the p r e c i p i t a t i o n w a s a c c o m p l i s h e d at 90 ~

or from the fact t h a t

r a r e e a r t h oxalates t e n d to form very small p a r t i c l e s d u r i n g p r e c i p i t a t i o n w h i c h t h e n grow via Ostwald r i p e n i n g ones. N e v e r t h e l e s s ,

and agglomeration

to form l a r g e r

it is clearly evident t h a t w h e n the oxalate is h e a t e d at

elevated t e m p e r a t u r e (~ 9 0 0 ~

the oxide p r o d u c e d rctaln-q the s a m e PSD

c h a r a c t e r i s t i c s of the original p r e c i p i t a t e . 5.10.-

METHODS OF MEASURING PARTICLE D I S T R I B U T I O N S

Having defined p a r t i c l e s a n d p a r t i c a l d i s t r i b u t i o n s , we now e x a m i n e m e t h o d s by w h i c h we c a n m e a s u r e p a r t i c l e s a n d p r o p e r t i e s four (4) p r i m a r y m e t h o d s

used

to obtain

of p o w d e r s .

data concerning

distributions. These include: OPTICAL

ELECTRICAL

SEDIMENTATION

ABSORPTION

PERMEATION

There

particle

are size

233

A..The M i c r o s c o p e - Visual C o u n t i n g of Partic,!es At the p r o p e r m a g n i f i c a t i o n a n d with a special eye piece, one c a n d i r e c t l y m e a s u r e particle

diameters.

The

eye-piece

m u s t have i n t e r n a l l y - m a r k e d

c o n c e n t r i c circles so t h a t a given particle will fall w i t h i n one or m o r e of t h e circles. The d i a m e t e r t h e n c a n t h e n be r e a d a n d / o r e s t i m a t e d directly as s h o w n in the following:

In this case, t h r e e p a r t i c l e s are shown, a 40 ~, 20 ~ a n d a 10 ~ particle. T h e m o s t i m p o r t a n t step is s a m p l e p r e p a r a t i o n on the m i c r o s c o p e slide, s i n c e only a p i n c h of m a t e r i a l is u s e d , one m u s t be s u r e t h a t the s a m p l e is u n i f o r m and

representative of the material. Also, since m o s t m a t e r i a l s t e n d to

a g g l o m e r a t e d u e to a c c u m u l a t e d surface c h a r g e in a dry state, one a d d s a few d r o p s of alcohol a n d w o r k s it with a spatula, s p r e a d i n g it o u t into a t h i n layer which

dries.

Too

much

working

breaks

down

the

original

particles,

234

p a r t i c u l a r l y the larger ones, a n d too little w o r k i n g leaves a g g l o m e r a t e s . With a little practice, the p r e p a r a t i o n of a p r o p e r slide of p a r t i c l e s b e c o m e s easy. One t h e n p r o c e e d s to c o u n t particles. It h a s b e e n f o u n d t h a t ff one w i s h e s to d e t e r m i n e an a c c u r a t e value of the m e a n d i a m e t e r , one m u s t c o u n t at least 4 5 0 particles, as the n u m b e r of c o u n t s i n c r e a s e s , the m e a n c o m e s closer a n d closer to the t r u e value, as s h o w n in the following:

To calculate the m e a n d i a m e t e r , we c o u n t the n u m b e r of p a r t i c l e s at d I, t h e n t h o s e at d2, etc., s u m t h e s e a n d average the value, i.e.m

5.10.3.-

I! = X n i d i

/Xni

A n o t h e r m e t h o d is to c o u n t p a r t i c l e s b e t w e e n a given r a n g e a n d t h e n s u m the c o u n t s as in 5.9.1. Alternately, one c a n p u r c h a s e an a u t o m a t i c p a r t i c l e c o u n t i n g i n s t r u m e n t for about $ 5 0 - 6 0 , 0 0 0 . microscope,

The i n s t r u m e n t c o n s i s t s of a

a s c a n n i n g device (usually a flying-spot s c a n n e r ) ,

a television

display a n d a p r e - p r o g r a m m e d m i c r o p r o c e s s o r . All of the p a r t i c l e s w i t h i n a

235

given frame can be c o u n t e d a n d g r o u p e d automatically. Thus, several fram es per m i n u t e can be c o u n t e d by an operator, or the i n s t r u m e n t can be set to operate automatically according to a p r e s e t program. Since the i n s t r u m e n t scans imagca of the particles, a p r o b l e m of orientation arises. This is s h o w n in the following diagram:

The m i c r o p r o c e s s o r is p r o g r a m m e d to recognize length and width, b u t t h e question

of the

correct

diameter

of irregular

particles

has

remained

controversial. For example, Martin's d i a m e t e r is defined as the s h o r t e s t line which divides the area of the image in half. A depiction of ~.is is shown as follows:

Yet, it is obvious t h a t this is not the correct d i a m e t e r for the t hese particles. Another school of t h o u g h t would use the a ~ e of the length and width, but for the above particles, the actual value of the length t h a t ought to be u s e d

236

r e m a i n s questionable. This p r o b l e m has r e m a i n e d the m o s t serious b a r r i e r to obtaining PSD by optical microscopy until recently. With the advent of c o m p u t e r s which can process data at the rate of 7 0 0 billion bits per second, this p r o b l e m has been solved satisfactorily. This PSD i n s t r u m e n t {Malvern I n s t r u m e n t s , 10 Southville Road, Southborough, Ma. 0 1 7 7 2 - Sysmex FPIA-2100) is a fully a u t o m a t e d particle size mad ~hape analyzer. It uses CCD, i.e.- "charge-coupled device", technology (the optical basis of a digital video camera) to c a p t u r e a series of images of p a r t i c l e s s u s p e n d e d in a liquid m e d i u m . Particles are s a m p l e d from a dilute s u s p e n s i o n which is forced t h r o u g h a "sheath-flow" cell. This i n s u r e s that the largest area of the particle is oriented toward the video c a m e r a a n d that all of the particles are in focus. The cell is illuminated via a stroboscopic light source and images are c a p t u r e d at 30 Hz. per second. A c o m p u t e r p r o g r a m t h e n p r o c e s s e s the images in real time by the following steps: digitization, edge highlighting, binarization, edge extraction, edge t r a c i n g a n d finally storage. The following diagram shows how this is a c c o m p l i s h e d :

All this requires a high speed c o m p u t e r with large m e m o r y storage capacity and RAM (something not possible until recently). Image analysis software

237

t h e n calculates the area and p e r i m e t e r of each of the c a p t u r e d Lnaages and t h e n calculates the particle d i a m e t e r a n d circularity. The particle shape is easily recognized in this diagram. The area of the circle is equal to t hat of the digitized particle so t h a t the particle circularity (shape) can be classified. Circularity, C, is d e t e r m i n e d by: 5.10.7.-

C = p e r i m e t e r of circle / p e r i m e t e r of p a r t i c l e

Once the m e a s u r e m e n t is complete, the particle size a n d circularity can be displayed in b o th graphical a nd table form. A typical r e p o r t includes: p a r t i c l e size distribution, circularity distribution and a s c a t t e r g r a m of particle size vs: circularity. All t h e s e can be displayed and p r i n t e d in h a r d copy. Individual particle images can be displayed including uni-particle, hi-particle,

and t h e n classified into cat egori es a n d agglomerate, This r a t h e r n e w

i n s t r u m e n t has solved the p r o b l e m of particle shape and c o r r e s p o n d i n g size t h r o u g h optical digitization m e t h o d s . B. SEDIMENTATION METHODS There are several particle sizing m e t h o d s , all b a s e d u p o n s e d i m e n t a t i o n and Stokes Law. If a particle is s u s p e n d e d in a fluid (which m a y be gas, or any liquid),

the

force

of r e s i s t a n c e

to m o v e m e n t

by the

particle

will

be

proportional to the particle's velocity, v, a n d its radius, r, vis5.10.8where

f = 6n r ~ v ~ is the viscosity of the fluid, providing the particle is spherical. If

the particle settles u n d e r the influence of gravity, t h e n we can write: 5.10.9.-

f = 4/3/nr3

(Ps - P l ) g

= 6nr~iv

where ps

and pl are the densities of the solid particle

respectively. Since the distance settled is: h = v t, t h e n :

and the liquid,

238

5.10.10-

d={

18~l h / a t ( p s

-Pl)g}l/2

This is S t o k e s Law for s e d i m e n t a t i o n w h e r e we have a d d e d a , a s h a p e factor, j u s t in case we do not have s p h e r i c a l particles.

For s p h e r e s ,

r = 1. It is

fractional o t h e r w i s e . One way to m e a s u r e p a r t i c l e s is to weigh

them

as t h e y settle

s u s p e n s i o n , s u c h an a p p a r a t u s is called a " s e d i m e n t a t i o n

out from

balance" a n d is

d e s i g n e d as s h o w n in the following diagram:

The weight of the p a r t i c l e s builds u p with time a n d is p r o p o r t i o n a l to I / d . If we a s s u m e s p h e r i c a l p a r t i c l e s , t h e n we c a n c o n v e r t

the above curve to

particle d i a m e t e r from S t o k e s Law. Although we have a d d e d the p a r t i c l e s u s p e n s i o n to a "water cushion" as s h o w n above, it m i g h t not s e e m t h a t t h e settling of the particles w o u l d strictly a d h e r e to S t o k e s Law, w h i c h a s s u m e s the t e r m i n a l velocity to be c o n s t a n t . But, as s h o w n in the following Table, the a p p r o x i m a t e d i s t a n c e a p a r t i c l e n e e d s to travel to r e a c h t e r m i n a l velocity in a given liquid is v e r y s h o r t .

239

Table 5- 1

Rate of Fall i n W a t e r at 25 ~

Size ( m i c r o n s l

for Particles Having a Specific Gra,~ity of 2 . 0

A p p r o x i m a t e D i s t a n c e in c m .

Approximate

T r a v e l e d prior to R e a c h i n g

Terminal

T e r m i n a l Velocity

Velocity_ ( c m / s e c l

2000

1.2 x 102

240

200 20

1.2 x 10 -2 1.2 x 10 -6

2.4 2.4 x 10 -2

2

1.2 x 10 -I0

2.4 x 10 -4

0.2

1.2 x 10 -14

2.4 x 10 .6

0.02

1.2 x 10 -16

2.4 x 10 -8

T h e s e d a t a s h o w t h a t t h e p a r t i c l e s s e p a r a t e very fast d u e pri_mm-ily to t h e i r differing t e r m i n a l velocities. T h u s , in t h e g r a p h s h o w n in 5.10.9., t h e initial w e i g h t is d u e to t h e large particles, while the u p p e r p a r t of t h e curve is d u e to the s m a l l e r sizes. For p a r t i c l e s less t h a n about 2.0 m i c r o n s , settling t i m e s u n d e r gravity b e c o m e s e x t r e m e l y long a n d this size r e m a i n s a lower limit for s e d i m e n t a t i o n m e t h o d s of d e t e r m i n i n g particle size. 2 0 0 0 ~ is p r o b a b l y t h e u p p e r limit since one c a n n o t a d j u s t the b a l a n c e to o p e r a t e quickly e n o u g h before t h e s e large p a r t i c l e s are lost. Major p r o b l e m s e n c o u n t e r e d with t h e sedimentation m e t h o d include: P r o p e r Design of the A p p a r a t u s Using the C o r r e c t Liquid with the Specific P o w d e r P r o p e r d i s p e r s i o n of the P a r t i c l e s Operator Technique The

latest

commercial

apparatus

incorporates

a

microprocessor

to

a u t o m a t i c a l l y plot the size d i s t r i b u t i o n as a f r e q u e n c y or c u m u l a t i v e plot. A n o t h e r s e d i m e n t a t i o n m e t h o d u s e d involves the ANDREASEN PIPETTE. A typical design is s h o w n in the following d i a g r a m :

240

This glass a p p a r a t u s is i n e x p e n s i v e a n d c o n s i s t s of a bottle having a n i n t e r n a l s a m p l i n g t u b e a n d c a l i b r a t e d s a m p l i n g v o l u m e (5 ml). One d r a w s a s a m p l e a n d t h e n expels it into an e x t e r n a l 5 ml b e a k e r . Using the t i m e s c a l c u l a t e d for s u c c e s s i v e d i a m e t e r s to settle p a s t the orifice, one w i t h d r a w s s a m p l e s c o r r e s p o n d i n g to t h a t d i a m e t e r . A series of s a m p l e s are o b t a i n e d at t 1 t 2 , etc., w h i c h are dried a n d weighed. The actual s a m p l e ,

o b t a i n e d is a n a r r o w particle d i s t r i b u t i o n r a t h e r t h a n one Sl~cUlc size. T h u s , the d i a m e t e r m e a s u r e d (by calculation) does not exactly c o r r e s p o n d to t h e true Stoke's diameter, Andreasen

Pipette

sedimentation

b u t to the p e a k of a n a r r o w distribution.Yet,

method

method

since

continues

to

be

the

it involves i n e x p e n s i v e

most

widely

equipment,

the used

and

is

r e a s o n a b l y a c c u r a t e if one will t a k e the time to develop a c o r r e c t e x p e r i m e n t a l t e c h n i q u e . The r e p r o d u c i b i l i t y c a n be excellent, t h e a c c u r a c y less so.

241

Another sedimentation method

u s e d is the so-called MSA-analyzer. If t h e

value of "g" in 5.10.8. is i n c r e a s e d (such as the u s e of a centrifuge) one c a n analyze the very small p a r t i c l e s in tony given d i s t r i b u t i o n in a s h o r t time. T h e p r o b l e m of c o u r s e lies in a c c u r a t e d e t e r m i n a t i o n of the w e i g h t a c c u m u l a t e d at a given time u n d e r a specific c e n t r i p e t a l force. T h i s p r o b l e m h a s b e e n n e a t l y solved by careful design of the s e d i m e n t a t i o n tube, as s h o w n in t h e following d i a g r a m : 5.10.13.A S E D IM E N T A T TUBE

IO N

Calibraled Stem

The p a r t i c l e s build u p b y l a y e ~ b e c a u s e it h a s b e e n f o u n d t h a t all m o n o s i z e d p a r t i c l e s c a n be r e m o v e d from s u s p e n s i o n by r o t a t i n g at a specific

speed.

T h u s , one r u n s the i n s t r u m e n t at a series of r o t a t i o n a l s p e e d s , m e a s u r i n g t h e w e i g h t of t h e b u i l d - u p layers in b e t w e e n e a c h r u n . T h e overall analysis is r u n at

specified

rpm's

which

correspond

to

selected

particle

diameters,

r e s u l t i n g in d a t a sufficient to c h a r a c t e r i z e the p a r t i c l e d i s t r i b u t i o n . C. ELECTRICAL RESISTIVITY - THE COULTER C O U N T E R P e r h a p s t h e m o s t useful m e t h o d for d e t e r m i n i n g p a r t i c l e d i s t r i b u t i o n s is t h a t of electrical

conductivity,

the m o s t widely u s e d i n s t r u m e n t

is t h e

Coulter C o u n t e r ( n a m e d after t h e Inventors), a l t h o u g h t h e r e are now o t h e r similar i n s t r u m e n t s on the m a r k e t . Originally, this i n s t r u m e n t w a s d e s i g n e d to m e a s u r e blood c o r p u s c l e s w h i c h are 2-8 p in size. It h a s p r o v e n to be v e r y

242

suited to m e a s u r e m i c r o n and s u b m i c r o n particles easily, accurately, and reproducibly. Consider a conductive solution consisting of water with a soluble salt, i.e.1% NaCI, an d a dispersing agent u s e d to prevent agglomeration of p a r t i c l e s in s u s p en s io n . Two electrodes are placed in solution with a n o n - c o n d u c t i n g orifice b e t w e e n them, as s h o w n in 5 . 1 0 . 1 4 , given on the next page:

A ceramic block s u c h as alumina is generally u s e d because of its c h e m i c a l i n e r t n e s s a n d the orifice is bor ed or drilled to precise

di m ensi ons. T h e

block is placed so t h a t two (2) c h a m b e r s result. T h e n ff a DC voltage is applied across the electrodes, a c u r r e n t will flow r t h r o u g h the orifice, a n d an effective resistance arises w hi ch d e p e n d s u p o n the voltage applied a n d the size (volume of the c o n d u c t i n g solution) of the orifice. Particles are added to one side of the two c h a m b e r s created by the ceramic block, and the s u s p e n s i o n is p u m p e d t h r o u g h to the other side. There

will be an

electrical pulse as each particle p a s s e s t h r o u g h the orifice. A depiction of the pulses obtained is shown in the following diagram, given as 5.10.15. on the next page. Each pulse will be proportional to the

size of

the particle because the particle volume displaces part of the c o n d u c t i n g solution d u r in g its passage t h r o u g h the orifice. If we have p r o p e r l y established steady state c o n d i t i o n s ,

243

5.10.15.-

IDisplayed Pulses - COulter CounterJ Counting

l

TMOv ab ie hreshold

Non-counting

we

will

obtain

a drop

in

current

as

each

particle

passes

through.

Electronically, this c a n be c o n v e r t e d to a positive p u l s e w h o s e i n t e n s i t y is p r o p o r t i o n a l to e a c h particle volume. We a s m i m e s p h e r i c i t y of t h e p a r t i c l e s , If V is c o n s t a n t , t h e n : 5.10.16.-

I = V /AR

and:AR=roVp

/A 2 -

1/ {1/1- (ro/re)} - a / A

w h e r e ro is the actual particle radius, re is t h e effective r a d i u s (incase t h e particle is not spheroid}, Vp is the v o l u m e of t h e particle, A is the c r o s s sectional a r e a of the orifice, a n d a is a c o n s t a n t w h i c h d e p e n d s

upon the

solvent a n d solute u s e d to m a k e the c o n d u c t i n g solution. The size of the orifice m u s t be fitted to the r a n g e of p a r t i c l e s p r e s e n t . If t h e p a r t i c l e s are too large, t h e y will n o t p a s s t h r o u g h t h e orifice. Too small a n d the p u l s e is not easily d e t e c t e d , present,

the

powder

content

Since large n u m b e r s of p a r t i c l e s c a n be m u s t be controlled.

c o n c e n t r a t i o n - d e n s i t y will r e s u l t in "coincidence

Too

high

counting".

a particle

T h a t is, two

small p a r t i c l e s c a n be c o u n t e d as one larger one. T h i s p h e n o m e n a h a s b e e n

244

a d d r e s s e d a n d tables are p r o v i d e d to c o r r e c t electronic

for i n c i d e n c e

p a r t of the Coulter C o u n t e r h a s several h u n d r e d

counting. T h e "channels" to

a c c e p t pulses. The "threshold" is electrical, a n d mowable, so t h a t one c o u n t s the large p a r t i c l e s ffrrst (large pulses) a n d t h e n m o v e s the t h r e s h o l d lower to c o u n t the s m a l l e r particles. The t h r e s h o l d is actually a p u l s e h e i g h t analyzer w h i c h c o n v e r t s h e i g h t s to n u m b e r s . The d a t a o b t a i n e d are easily c o n v e r t e d to a f r e q u e n c y plot or a c u m u l a t i v e plot. Since fur r e s p o n s e is linear, one c a n readily calibrate the i n s t r u m e n t at one point, u s i n g m o n o s i z e d p a r t i c l e s s u c h as pollen or specially p r e p a r e d

plastic particles. The only r e q u i r e m e n t

is

t h a t the calibrating p a r t i c l e s o u g h t to be in the r a n g e of the p a r t i c l e s to be measured. The Coulter C o u n t e r is p a r t i c u l a r l y useful a n d u n i q u e in t h a t s t e p s as small as 1.0

micron

m a y be

used

if so

desired.

Results

by

this

author

have

d e m o n s t r a t e d t h a t s o m e d i s t r i b u t i o n s w h i c h a p p e a r to Gaussian m a y actually c o n s i s t of m o r e t h a n one subdivision w h e r e the p e a k s are s e p a r a t e d by only a few m i c r o n s , an e x a m p l e is given as follows: 5. I 0 . 1 7 . - A PARTICLE DISTRIBUTION WITH S U B - D I S T R I B U T I O N S

This

result

was

achieved

with

a

selected

particle

distribution

which

originally a p p e a r e d to be a single Gaussian distribution. W h e n the p a r t i c l e

245

m e a s u r i n g steps were c h a n g e d so t h a t a range of about 1-2 m i c r o n s was covered, the above result was obtained. Even so, the m e a s u r e m e n t m u s t be done carefully to reveal the s u b s t r u c t u r e . To the aut hor's knowledge, no other m e t h o d has the capability of s u c h precision in particle d i a m e t e r m e a s u r e m e n t . The latest Coulter C o u n t e r instanm~ents i n c o r p o r a t e a m i c r o p r o c e s s o r and the oper a t or can specify the form of the o u t p u t data to be obtained. The i n s t r u m e n t converts c o u n t - d a t a to practically any form at desired, with the exception of log-probability plots. D. OTHER METHODS OF MEASURING PARTICLE SIZE P e rmeab ility is a n o t h e r m e t h o d

for obtaining i nform at i on about p a r t i c l e

diameters. If one p a c k s a tube with a weight of pow der exactly equal to its density, a n d applies a calibrated gas p r e s s u r e t h r o u g h the tube, the p r e s s u r e drop can be eq u a t e d to an average particle size. The i n s t r u m e n t based on this principle is called the "Fisher Sub-Sieve Sizer TM'. Only one value can be obtained b u t the m e t h o d is fast a nd reproducible. The i n s t r u m e n t itself is not expensive a n d the m e t h o d can be applied to quality control p r o b l e m s of powders. P e r m e a m e t r y is useful in t h e particle range of 0.5 to 50 p. Gas a d s o r p t i o n is one ot he r m e t h o d s o m e t i m e s u s e d for d e t e r m i n i n g average particle size. In this case, one is usually i n t e r e s t e d in the surface area of a powder a n d calculates the average size of the particles

secondarily. T h e

m e t h o d is called the B E T - m e t h o d after its developers, Brunauer, E m m e t t and Teller. The p r o c e d u r e is t i m e - c o n s u m i n g a n d is a c c o m p l i s h e d as follows. A gas analysis train is u s e d in which gas volumes can be recycled and m e a s u r e d very accurately. A w e i g h e d sample is placed in the sam pl e tube and allowed to come to equilibrium. Since all solid materials have a m o n o lay er of water on the surface of the particles (we live in a wet world), the sample is h e a t e d to 300 ~ to expel the water, in a nitrogen gas flow. Next, it is cooled to liquid nitrogen t e m p e r a t u r e . At this point, the s a m p l e will adsorb a m o n o l a y e r of N2-gas molecules on the surface of each particle. By allowing the t e m p e r a t u r e of the s a m pl e to r e t u r n to r o o m t e m p e r a t u r e ,

246

w h i l e m e a s u r i n g t h e v o l u m e of n i t r o g e n

gas released,

o n e o b t a i n s a value

w h i c h c a n b e c o n v e r t e d to a s u r f a c e a r e a of t h e p a r t i c l e s b e i n g m e a s u r e d . I n practice, one recycles the s y s t e m several times, m e a s u r i n g adsorption and d e s o r p t i o n v o l u m e s s u c c e s s i v e l y so as to o b t a i n a n a v e r a g e value. It is f o u n d that the

unoccupied f r a c t i o n

of t h e s u r f a c e ( l - y) a p p r o a c h e s a c o n s t a n t value

w h e r e it is a s s u m e d t h a t y = 0. N i t r o g e n g a s is e s s e n t i a l s i n c e it is t h e only g a s w h i c h f o r m s m o n o l a y e r s easily w i t h o u t t h e t e n d e n c y to f o r m m o r e t h a n o n e l a y e r of g a s m o l e c u l e s on t h e s u r f a c e . If t h e gas p r e s s u r e is k e p t b e t w e e n s t r i c t limits, t h e B E T e q u a t i o n is: 5.10.18.where: the

pg / ( V g [ 1 - p g ] = 1/ {Vmono " C )

+ {C- 1) p g / { V m o n o " C )

C = exp { (FAds. - FCond.) / RT} , a n d pg is t h e g a s p r e s s u r e . Vg is

gas volume

measured;

Vmono is t h e

gas volume

of t h e

monolayer

c a l c u l a t e d f r o m t h e k n o w n d i m e n s i o n s of t h e n i t r o g e n - g a s m o l e c u l e s ; FAds. is t h e Gibbs free e n e r g y condensation,

of a d s o r p t i o n ,

pg is m e a s u r e d

a n d FCond. is t h e

free

e n e r g y of

as t h e r a t i o of t h e e q u i l i b r i u m p r e s s u r e

at

specific t e m p e r a t u r e s , i.e.- r o o m a n d liquid n i t r o g e n t e m p e r a t u r e s . It h a s b e e n f o u n d t h a t a plot of pg / {Vg [1-pg] vs: pg is l i n e a r for t h e p r e s s u r e r a n g e of 0 . 0 5 to 0.4, w i t h a slope of (C - 1) / (Vmono

9 C ) a n d i n t e r c e p t of

1/ (Vmono " C ). Let u s n o w do a s i m p l e c a l c u l a t i o n u s i n g B E T d a t a o b t a i n e d . S u p p o s e we h a v e a 2 0 gm. s a m p l e h a v i n g a d e n s i t y of 2.0. We m e a s u r e t h e s u r f a c e a r e a as 6 m 2 . F r o m t h e a r e a of a s p h e r e , A = n D 2 , a n d t h e v o l u m e of a s p h e r e , V = 4 / 3 n D 3 ,, we find t h e total v o l u m e of n s p h e r e s to b e 10 cc, i.e.- n { 4 / 3 n D3} = 10. T h e s u r f a c e a r e a of n{nD 2} s p h e r e s is 6 m 3. T h e t o t a l n u m b e r of s p h e r e s p r e s e n t , n, is t h e s a m e in b o t h f o r m u l a s . T h e r e f o r e ,

by

s u b s t i t u t i o n , we find D= 10 la. If we o b t a i n a p a r t i c l e d i a m e t e r b y s o m e o t h e r m e t h o d a n d find t h a t it is m u c h s m a l l e r t h a n t h a t of t h e B E T m e t h o d , w e infer t h a t t h e p a r t i c l e s a r e p o r o u s . We t h u s s p e a k of t h e p o r o s i t y a n d n e e d to c o r r e c t for t h e p o r e s u r f a c e a r e a if we are to m a k e a r e a s o n a b l e e s t i m a t e of the true d i a m e t e r by the BET m e t h o d .

247

Particle Size by Laser R e f r a c t o m e t r y is b a s e d u p o n Mie scat t eri ng of p a r t i c l e s in a liquid m e d i u m . Up until about 1985, the power of c o m p u t e r s s u p p l i e d with laser diffraction i n s t r u m e n t s was not sufficient to utilize the rigorous solution for h o m o g e n e o u s spherical particles formulated by Gustave Mie in 1908.

Laser

particle

instrument

manufacturers

therefore

used

a p p r o x i m a t i o n s conceived by Fraunhofer. The h y p o t h e s e s m a d e in Mie T h e o r y include: I. The particle i s a n optically h o m o g e n e o u s s m o o t h sphere w hose real a n d imaginary refractive indices are b o t h k n o w n . 2. The sphereical particle is illuminated by a plane wave of infinite extent an d of defined wavelength. 3. The real a n d imaginary refractive

indices

of

the

medium

s u r r o u n d i n g the particle are bot h k n o w n . The Fraunhofer a p p r o x i m a t i o n includes: 1. The in c i de nt light-wave is plane, of infinite extent and of k n o w n wavelength. 2. The scattering is from a circular a p e r a t u r e in a t h i n opaque s c r e e n . 3. The extinction coefficient for all sizes is 2. 0 Fraunhofer rules do not include

the

influence

of refraction,

reflection,

polarization and o t h e r optical effects. Early laser particle analyzers u s e d Fraunhofer a p p r o x i m a t i o n s because the c o m p u t e r s of t h a t time could n o t h a n d le the storage a n d m e m o r y r e q u i r e m e n t s of the Mie m e t h o d . For example, it has b e e n found t h a t the F r a u n h o f e r - b a s e d i n s t r u m e n t a t i o n c a n n o t be u s e d to m e a s u r e the particle size of a s u s p e n s i o n of lactose {R.I. = 1.533) in iso-octane (R.I. = 1.391) b e c a u s e the relative refractive index is 1.10, i.e.- 1 . 5 3 3 / 1 . 3 9 1 . This is due to the fact t h a t diffraction of light passi ng t h r o u g h the particles is nearly the s a m e as t h a t passi ng a r o u n d the particles, creating a c o m b i n e d interference p a t t e r n vchich is not indicative of the t r u e

248

scattering in the far field where the detector is located. The Mie solution anticipates this. A laser is required to provide a single wavelength so that photons passing through or around any given particle is diffracted and scattered, free form any other optical interference. The Mie theory is rigorous and is used to predict the scattering from particles whose size range from Angstroms to centimeters. Mie originally devised his theory to better define the properties of "fog", i.e.a dispersion of very small circular drops of water in the atmosphere. It has now been adapted to particle counting and particle distributions. Mie t h e o r y requires that the real and imaginary refractive be known or be m e a s u r e d . This may require additional work initially and the choice of liquid m e d i u m , dispersing agents the like may be subject to experimentation initially. This may be especially true if one is trying to determine the PSD of pharmaceutical agents such as new drugs. Laser diffraction particle size i n s t r u m e n t s have now become one of the major p r o c e d u r e s for particle size determination, particularly for the Pharmaceutical Industry, i.e.- organic based particles. Suggested Readin.g 1. R.R. Irani and Clayton F. Ca!lis, "Particle Size- M e a s u r e m e n t , Interpretation and Application" - J. Wiley & Sons, New York (1963). 2. J o h n Wulff et al, "The S t r u c t u r e and Properties of Materials - Volumes I, II, III, & IV" - J. Wiley & Sons, New York (1964). 3. Polakowski and Ripling, "Strength and S t r u c t u r e of E n g i n e e r i n g Materials" - Prentice-Hall, Englewood Cliffs, NJ (1966). 4. R.E. Newnham, "Structure-Property Relations" - Springer-Verlag, New York (1975).

249

p r o b l e m s for C h a p t e r 5 1. For ZrO 2, given t h a t y -= 1 8 0 0 d y n e / c m . ; MP = 2 , 7 1 5 ~

d = 5.89 gm/cc,

c a l c u l a t e t h e m i n i m u m d i a m e t e r at w h i c h z i r c o n i a p a r t i c l e s will n o t a i ~ t e r . 2. Given a c u b e of 1.00 c m a c r o s s in size, Give t h e n u m b e r of c u b e s p r o d u c e d and

their

size

in m i c r o n s

d i m e n s i o n s is m a d e ? ;

produced

if:

100 c u t s a c r o s s ? ;

10

cuts

across

each

of t h r e e

1 0 0 0 c u t s a c r o s s ? Also, give t h e

total s u r f a c e a r e a p r o d u c e d in e a c h case. 3. Given t h e following w e i g h t f r a c t i o n s in % for a s a m p l e of s a n d , plot y o u r c a l c u l a t e d d a t a in t e r m s of: 1. H i s t o g r a m 2 . " F r e q u e n c y Plot" 3. "Cumulative F r e q u e n c y Plot" using:

#10, # 18, #20, #30, # 4 0 , #50, #80, # 1 0 0 , # 1 7 0 , #200, #270, #325 & #400 screens. S a m p l e size w a s 2 0 0 g m . S c r e e n Data= % R e t a i n e d

# 1 0 = 1.5%; # 1 8 = 3.0 %; # 2 0 = 4.5%; # 3 0 = 6 . 0 % ; #40=8.0%;

# 50=

10%; # 8 0 =

16%; # 1 0 0 = 18%;

# 1 7 0 = 15%; # 2 0 0 = 8%; # 2 7 0 = 5%; # 3 2 5 = 3.5%; # 4 0 0 = 1 . 5 % .

250

4. Plot y o u r s c r e e n - d a t a via log-probability;

are t h e r e

any limits

to t h e

distribution? 5. You have o b t a i n e d the following particle data: Particle Count

Measured Diameter

4

2.8

12

3.2

84

3.7

804

7.7

2404

9.3

4824

15.7

6428

20.3

7610

76.7

7930

29.3

8002

36.0

A. Calculate the p a r a m e t e r s of t h e d i s t r i b u t i o n . B. Classify the d i s t r i b u t i o n as to its p r o b a b l e origin. C. Plot the d a t a as: I. H i s t o g r a m ; II. F r e q u e n c y ; III. Cumulative F r e q u e n c y . Make s u r e t h a t you specify all of the d i s t r i b u t i o n p a r a m e t e r s esch method.

C o m m e n t on t h e efficiacy of the m e t h o d s

s t u d y of particle size analysis.

available for

as a p p l i e d to t h e

251

Chapter 6

.............

Growth of Crystals Having e x a m i n e d particle size and particle growth, i.e.- powders, in s o m e d ep th in the last chapter, let us now t u r n to an investigation of the g r o w t h of single crystals. Many of the devices t h a t we use on a daily basis employ a single crystal e l e m e n t as the h e a r t of the m e c h a n i s m . A good example is the "quartz-crystal" w a t c h t h a t you wear on your wrist. Here, a quartz crystal is m a d e to vibrate u n d e r an applied voltage. Its vibrations are coupled to a s ens i ng circuit a n d t r a n s l a t e d into seconds, m i n u t e s and h o u r s by using and count i ng the k n o w n r e s o n a n t frequency of vibration of the crystal. The r e s o n a n t vibrational frequency is d e t e r m i n e d by the angle of "cut" (or angle in relation to the crystallographic axes) and the size of the crystal. There are m a n y o t h e r s ens i ng devices b a s e d u p o n a crystal c o m p o n e n t s u ch as a device w h i c h controls heating, t r a n s l a t e s force into electrical voltage, m o d u l a t e s light, or regulates motion. We use t h e m as electrical heaters, strain gauges, laser controllers or piezoelectric devices. We will explore some of these later in this chapter, particularly single crystal silicon u s e d to fabricate i nt egr at ed circuits u s e d in c o m p u t e r s . As we s tate d previously, particles usually grow from a nucl eus to w h i c h a t o ms are added in a regular m a n n e r to form a t h r e e - d i m e n s i o n a l structure. S u c h crystals cease growing w h e n the "nutrient" (the m a t e r i a l which serves to form the particle) b e c o m e s depleted. S u c h particles are k n o w n as "crystallites". Each will consist of several grains, having a differing orientation of the crystal lattice, within each individual particle, namely:

!]

252

Grain b o u n d a r i e s form j u n c t i o n s b e t w e e n grains within the particle, due to vacancy an d line-defect formation. This situation arises because of t h e 2nd Law of T h e r m o d y n a m i c s (Entropy). Thus, if crystaUites are formed by precipitation from solution, the p r o d u c t will be a powder consisting of m a n y small particles. Their actual size will d e p e n d u p o n the m e t h o d s u s e d to form them. Note t hat each crystallite can be a single-crystal but, of necessity, will be limited in size. However, if one desires a large ~ l a ~ e - ~ , t h e n the m e t h o d s n e e d e d to grow it differ considerably. This is the subject of this Chapter. We will examine crystal-growing m e t h o d s a n d the e q u i p m e n t n e e d e d to do so. We will t h e n examine the different type of crystals u s e d b o t h as s e n s o r s and as the basis of specific devices u s e d in the electronics industry. 6.1- METHODS OF GROWTH OF CRYSTAI~ Obviously, the major difference in the single-crystal and polycrystalline (crystaUite) state is a m a t t e r of size. For the single-crystal, the size is large (> 10 cm), w h e r e a s in the polycrystalline state, the size of the crystals is small (10 ~ n = 0.001 cm.) The m e t h o d s for obtaining one or the o t h e r differ considerably. They include formation from: 6. I. 1-

Single Crystal

Crystallites [ p o w d e r s )

Liquid solvent

Vapor

Vapor Melt Molten salt

Melt Flux Precipitation from solution

The m e t h o d of choice for growing a single crystal d e p e n d s u p o n m a n y factors. Most relate to the physical p r o p e r t i e s of the c o m p o u n d w h o s e single crystal we desire. Some of the m ore i m p o r t a n t p r o p e r t i e s of such a c o m p o u n d are given in 6.1.2. on the next page. For the m o s t part, single crystals are grown from a melt of the c o m p o u n d , provided t h a t m e l t i ng the c o m p o u n d does not cause it to decompose, i.e.-

253

6.1.2.-

Melting p o i n t Partial v a p o r p r e s s u r e of m e l t T h e r m a l stability of solid p h a s e Reactivity of liquid or g a s e o u s p h a s e T h e r m a l c o n d u c t i v i t y of b o t h liquid a n d solid p h a s e s .

m e l t incongr~aently. Before we c o n s i d e r crystal g r o w t h in detail, let us e x a m i n e the h a r d w a r e n e e d e d to obtsin a m e l t . T h e r e a s o n for this is t h a t m a n y of the f u r n a c e c o m p o n e n t s

available have t e m p e r a t u r e

limitations,

a n d m a n y of the c r y s t a l s we m i g h t w i s h to grow have high m e l t i n g p o i n t s , i.e.- > 1 6 0 0 ~

The first a p p a r a t u s we m a y n e e d is a furnace, w h i c h is

s i m p l y a closed space, h e a t e d by electrical e l e m e n t s , w h e r e i n t h e i n t e r n a l t e m p e r a t u r e is c o n t r o l l e d . 6.2. FURNACE C O N S T R U C T I O N T h e r e are m a n y ways to build a furnace. Basically, a f u r n a c e c o n s i s t s of a few e s s e n t i a l p a r t s , e a c h of w h i c h c a n be varied a c c o r d i n g to t h e final o p e r a t i n g r e q u i r e m e n t s n e e d e d for the furnace. T h e following s h o w s t h e e s s e n t i a l s r e q u i r e d for the p r o p e r design of a furnace: 6.2. I.- E l e m e n t s of F u r n a c e Design a. a power s o u r c e , b. h e a t i n g e l e m e n t s c. t h e r m a l insulation, d. a crucible to hold the m e l t e. a s e n s o r for t e m p e r a t u r e - f e e d b a c k f. a t e m p e r a t u r e s e t - p o i n t c o n t r o l l e r g. a relay or o t h e r p o w e r - c o n t r o l device for the p o w e r s o u r c e . The p o w e r source, relay, s e t - p o i n t c o n t r o l l e r

and sensor need

n o t be

d i s c u s s e d here. However, t h e insulation, h e a t i n g e l e m e n t s , a n d c r u c i b l e s involve m a t e r i a l s , a n d n e e d to be e x a m i n e d in m o r e detail. H e a t i n g e l e m e n t s c a n be m e t a l w i r e s or c e r a m i c r o d s w h i c h b e c o m e

254

incandescent when

an electrical

current

flows t h r o u g h

them.

Their

c o m p o s i t i o n s are critical and have been developed over m a n y years to optimize their p e r f o r m a n c e as heaters. Insulation is u s e d to retain t h e heat g e n e r a t e d and to disperse the heat uniformly t h r o u g h o u t the h e a t e d space, namely the internal cavity of the furnace. Table 6-1 s u m m a r i z e s t h e t h e r m a l p r o p e r t i e s of Insulation and Heating E l e m e n t s for furnaceconstruction.

TABLE 6-1 INSULATION

HEATING ELEMENTS

Max. T e m p . Usabl.e

In Air ........

Max. Temp.

Power N e e d e d Voltage C u r r e n t

Glass Wool

600 ~

Nichrome Wire

med.

med.

Fiberfrax

1350

"Globars"TM(SiC) 1475

low

high

Kanthal Wire

1300

reed.

med.

MoSi2

1700

low

high

Alumina(foam& beads) 1850~ Zirconia (ZrO2) 2400

Pt (40% Rh) ZrO2 :Y

1800 1900

low low

high high

Magnesia

LaCh~ 3

1900

10w

high

Quartz wool Fire Brick

.

.

.

.

.

.

1100 1100 to 1650~ ,,

900 ~

,,

Those

2800

heating e l e m e n t s

given in Table 6-I

are generally u s e d in air

a t m o s p h e r e , up to about 1800 ~ If one needs to p r o d u c e a melt above 1800 ~ it is n e c e s s a r y to use refractory metals which m u s t be u s e d in an inert a t m o s p h e r e . Table 6-2 lists some of these heating e l e m e n t s : Table 6-2 E l e m e n t s For NON- .AIR Usage Mo or W wire

Voltage C u r r e n t 2400 to 2800~ low high

Iridium w i r e

2400

low

hi gh

Graphite R.F. C u r r e n t

3400 2800

low low

high hi gh

4000

NA

NA

Oxy-hydrogen flame

The t e m p e r a t u r e s given are approximate, and are p r e s e n t e d solely for

255

comparison

purposes.

In

furnace

insulation,

Fiberfrax TM

(fibers

of

a l u m i n u m silicate) and "fire-brick" (bricks m a d e from insulating silicate c o m p o u n d s ) are the two m o s t c o m m o n l y u s e d materials. For very high t e m p e r a t u r e work, Zirconia (= ZrO2) in the form of beads, "wool", and b o a r d s p r e s s e d from fibers are u s e d for t e m p e r a t u r e s above 1600 ~ Alumina, i.e.- A1203 , is m u c h cheaper, b u t does not have the t h e r m a l shock, or very high t e m p e r a t u r e capability (i.e.- > 2 0 0 0 ~

Zirconia.

Wire-wound e l e m e n t s are u s e d m o s t frequently. They are usable in air up to about 1200 ~

and consist of a heat i ng wire or coil, w o u n d u p o n an

i n s u l a t i n g - p r e f o r m . They are cheap and will last for considerable l e n g t h s of time, especially if they are r u n at < 1200 ~ ( i.e.-Kanthal TM wire). On the o th er hand, wire m a d e from precious metal, i.e.- 60% Pt - 40% Rh = {Pt(Rh)}, can be o p e r a t e d up to 1800 ~ in air, and will o p e r a t e continuously at 1700 ~ for long periods. However, it is expensive and special care is r e q u i r e d w h e n w r a p p i n g a furnace core to form the furnace element. Another idiosyncrasy of this type of heating el em ent is t h a t t h e p o w e r - s o u r c e n e e d s to be especially designed. The Pt(Rh) wire has a very low r e s i s t a n c e degrees,

at r o o m t e m p e r a t u r e .

As it w a r m s

to a few h u n d r e d

its resistivity c h a n g e s considerably. Thus, a cttrr 9169

is

n e e d e d in the power control circuit during start-up, or the {Pt(Rh)} w i r e will m e l t . "Globars T M

(silicon carbide rods)

are the

next m o s t

frequently u s e d

furnace h eatin g elements. They will operate continuously at 1450 ~ and i n t e r m i t t e n t l y at 1500 ~ A newer type of element, Mo -~re coated w i t h silicide, i.e.- MoSi2 (to p r o t e c t the Mo wire against oxidation), has b e c o m e c o m m o n . S u c h heating e l e m e n t s have a p p e a r e d as "hairpins" and will operate at 1750 ~ in air on a c o n t i n u o u s basis. Another heat i ng e l e m e n t coming into use is the defect conduct or , Z r 0 2 , doped with Y203 to m a k e it conducting. However, this type of heating e l e m e n t has special o p e r a t i n g c h a r a c t e r i s t i c s w hi c h h i n d e r its w i d e - s p r e a d usage. It does not b e c o m e conductive until 600 ~ is reached. But, it can be o p e r a t e d continuously at 1800 ~ C. in air for long periods of time, with a m a x i m u m of 1900 ~ I t does require very high c u r r e n t s a n d low voltage to operate satisfactorily. Still a n o t h e r type of heat i ng e l e m e n t

is t h a t of l a n t h a n u m c h r o m i t e ,

256

LaCrO3. These are available in the form of rods and will operate at 1 8 0 0 ~

in air for long p e r i o d s .

Most heating e l e m e n t s fail due to the d e v e l o p m e n t of internal flaws in their s tr u ctu r e. For example, a "globar" is formed by c o m p r e s s i n g SiC particles to form a rod, a nd t h e n sintering it. During operation (especially if it is o p e r a t e d at the u p p e r end of its t e m p e r a t u r e - o p e r a t i n g range), it develops "hot-spots". These are due to oxidation a n d formation of localized resistive areas within the rod (due to diffusion and collection of vacancies at grain boundaries). These areas dissipate power locally, so t h a t the rod eventually fails, i.e.- melts locally at the h o t - s p o t and b e c o m e s non-conductive. Wire-wound e l e m e n t s fail in a similar m a n n e r ,

except

t h at formation of vacancies within the metallic s t r u c t u r e is the m o s t prevalent m e c h a n i s m w hi c h causes "hot-spots" in the operational h e a t i n g coil. These vacancies also migrate to grain boundaries. The grainb o u n d a r y - j u n c t i o n decreases in conductivity, due to vacancy d e f e c t formation. T h e r e u p o n , h o t - s p o t s form within the wire during operat i on, causing ultimate failure of the heating e l e m e n t . Among those heating e l e m e n t s which require the use of neutral, r e d u c i n g a t m o s p h e r e s , or vacuum, M o - w i r e or W-wire in the form of heat i ng coils, graphite in the form of rods or semi-cylinders, are m o s t often used. Iridium wire is also u s e d but it is very expensive. Both Mo and W wire are usable up to 2800 ~ while Ir can be u s e d only to 2 4 0 0 ~ G raphi t e heating e l e m e n t s can be u s e d above 3000 ~ We have also included R.F. (radio-frequency) c u r r e n t as a heating e l e m e n t , although it is only a heating m e t h o d w h e n employed with a suitable succeptor. Finally, one ot her m e t h o d is listed for the sake of c o m p l e t e n e s s , t h a t of the oxy-hydrogen flame. It generat es c o m b u s t i o n p r o d u c t s (H20) but the RF- m e t h o d can be u s e d in any a t m o s p h e r e including vacuum. The m o s t critical e l e m ent in m e l t - g r o w t h of single crystals is t h e container, or crucible. The first r e q u i r e m e n t for selection of a suitable crucible is t h a t the crucible does not react with the melt. The second is

257

t h a t it be t h e r m a l l y s h o c k temperature

resistant.

capability. A f o u r t h

A third

is t h a t

it be

is t h a t of o p e r a t i o n a l stable

in

the

chosen

atmosphere. These requirements eliminate many potential crucible m a t e r i a l s for a given application. For u s e in air, t h e silica crucible (Si02) h a s no p e e r w h e n u s e d with o x i d e - b a s e d m a t e r i a l s u p to 1200 ~ a l m o s t n o n - r e a c t i v e , t h e r m a ~ y s h o c k - r e s i s t a n t , a n d inexpensive.

It is

Mullite,

i.e.- a l u m i n u m silicate, is m o r e reactive, less s h o c k - r e s i s t a n t b u t c h e a p e r t h a n silica. It is u s e d in m a k i n g g l a s s - m e l t s for the m o s t part. Table 6 - 3 lists the

crucible

materials

most

often

used

and

their

temperature

capabilities. TABLE 6- 3 COMPOSITIONS SUITABLE FOR CRUCIBLES FOR U S E IN AIR

FOR U S E IN NON-OXIDIZING A T M O S P H E R E

MATERIAL MAX. T E M P .

MATERIAL

IVIAX. T E M P .

Silica

1200 ~

Iridium

2400 ~

Alumina

1700

Zirconia*

2400

Platinum

1750

Magnesia*

2800

Mullite

1400

Carbon*

2800

Boron

1400

Platinum*

1700

Nitride * Not in h y d r o g e n Most m e t a l s c a n be u s e d as crucible m a t e r i a l s , b u t only the p r e c i o u s m e t a l s s e e m to p o s s e s s the n o n - r e a c t i v i t y r e q u i r e d for m e l t s at t h e h i g h e r t e m p e r a t u r e s . A l u m i n a is less s h o c k - r e s i s t a n t t h a n silica b u t c a n be u s e d at h i g h e r t e m p e r a t u r e s . S h o c k - r e s i s t a n c e r e l a t e s to how fast one c a n h e a t the crucible a n d its c o n t e n t s u p to the m e l t i n g point of the m a t e r i a l w i t h o u t c r a c k i n g the crucible. Z r 0 2 a n d BN are m o s t useful for m e t a l melts. Pt is u s e d for m e l t i n g in air a n d r e m a i n s c h e m i c a l l y i n e r t to m o s t melts. For p r e p a r a t i o n of m e l t s in i n e r t a t m o s p h e r e ,

Ir s t a n d s alone. It is n o n -

reactive, h a s a very high t e m p e r a t u r e limit, a n d is not s u b j e c t to t h e r m a l

258

s h o c k or stress. Pt c a n also be u s e d b u t at lower t e m p e r a t u r e s . Ir is about 4 t i m e s as expensive as Pt. A good rule of t h u m b is to u s e a m e t a l c r u c i b l e for oxide m e l t s a n d an oxide crucible for m e t a l melts, w h e n e v e r possible. Next, we n e e d to d e t e r m i n e exactly how one goes about o b t a i n i n g a single crystal. 6 . 3 . - S T E P S IN GROWING A SINGLE CRYSTAL Materials t e n d to grow polycrystaUine. This behavior is r e l a t e d to the 2 n d Law of T h e r m o d y n a m i c s a n d the E n t r o p y of the s y s t e m . W h a t h a p p e n s is t h a t a large n u m b e r of nuclei begin to a p p e a r as the melt t e m p e r a t u r e a p p r o a c h e s the freezing point (but before it freezes). All of t h e s e n u c l e i grow at about the s a m e rate, a n d at the freezing p o i n t p r o d u c e a large n t t m l m r of small crystals. If we w i s h to r e s t r i c t

the g r o w t h to j u s t o n e

crystal, we w o u l d like a single n u c l e u s to grow

preferentially

at t h e

e x p e n s e of the o t h e r s . But usually, it does not. However, if we u s e a "seed" crystal a n d set up c o n d i t i o n s so t h a t it will grow, t h e n we c a n obtain o u r d e s i r e d single crystal. One m e t h o d

we

might

u s e is to cool

the

melt

to i n c i p i e n t

nuclei-

formation, toss in the seed-crystal, a n d allow the m e l t to freeze into a single crystal. This is the KYROPOUI~S m e t h o d w h i c h we will d i s c u s s in detail later. Alas, this m e t h o d only w o r k s for a few s y s t e m s , n o t a b l y alkali halides (cubic) a n d the like. We find t h a t we c a n u s e a s e e d - c r y s t a l to grow

single

crystals, b u t

only ff we

use

it

under

carefully

defined

c o n d i t i o n s . A modified Kyropoulos m e t h o d h a s b e e n u s e d for m a n y y e a r s to form single-crystal s a p p h i r e u p to 13.0 i n c h e s in d i a m e t e r . Plates c u t from s u c h crystals are u s e d as w i n d o w s a n d s u b s t r a t e s for all s o r t s of i n t e g r a t e d circuits, as well as w a t c h "crystals". The p r o b l e m s of o b t a i n i n g a s e e d - c r y s t a l are not simple. We c a n f r e e z e the m e l t to a p o l y c r y s t a n i n e state. W h e n cool, we e x a m i n e the boule (after first r e m o v i n g the crucible) to try to find a single crystal large e n o u g h for a seed (* 3 - 6 ram.). We could also c a s t the m e l t into a m o l d a n d t h e n look for seeds. We could also freeze a polycrystalline rod by pulling it

259

vertically from the melt. We w o u l d use a small loop of Pt wire to c a t c h p a r t of the m e l t by surface tension. By r o t a t i n g the wire loop while pulling vertically, we find t h a t a polycrystalline rod of small d i m e n s i o n builds up. Once we have the polycrystalline rod, we c a n r e h e a t it next to the m e l t surface so t h a t it r e m e l t s diagram:

and

"necks-in"

as s h o w n

in the

following

What h a p p e n s is t h a t t h e crystaUites m e l t a n d fuse into a small tip. If w e do this carefully, we will have o u r "seed". The

tip's small size l i m i t s

r e g r o w t h of the r e m e l t e d p a r t to t h a t of a single crystal. T h e n , w h e n w e r e t u r n the seed to the melt, we c a n initiate the g r o w t h of a m u c h l a r g e r single crystal, p r o v i d e d t h a t g r o w t h - c o n d i t i o n s are suitable. A n o t h e r way to obtain a s e e d is to dip a capillary t u b e into the m e l t . Surface t e n s i o n c a u s e s the t u b e to fill a n d w h e n it freezes inside, a small seed results. The difficulty w i t h this m e t h o d is t h a t it is difficult to obtain a t u b e of p r o p e r d i a m e t e r , m a d e of the p r o p e r material. Glass softens at too low a t e m p e r a t u r e a n d q u a r t z m e l t s a r o u n d 1 4 0 0 - 1 5 0 0 ~

Usually, w e

are r e s t r i c t e d to m e t a l s a n d even then, we m u s t be able to cut the t u b e to obtain the seed, since it is c o n f i n e d w i t h i n the tube. Once in a while, w e c a n u s e the tube directly a n d obtain g r o w t h directly u p o n the seed, even t h o u g h it h a s r e m a i n e d w i t h i n t h e tube. T h e r e is one o t h e r m e t h o d t h a t c a n be e m p l o y e d to o b t a i n a seed. We use a m e t a l crucible having a small tip a n d c a u s e a m e l t to form. The tip acts

260

in the

s a m e way to form t h e seed.

Nevertheless,

we have the

same

p r o b l e m as w h e n we u s e the m e t a l capillary. We n e e d to o b t a i n the s e e d free from its h o l d e r .

T h u s , the crucible m u s t be sacrificed, or else the whole m a s s m u s t be e x t r a c t e d from the crucible with the s e e d in the tip intact. If t h e r e is a sufficient difference in c o n t r a c t i o n b e t w e e n the m a s s a n d the crucible, p e r h a p s t h e n we c a n obtain the whole m a s s intact, i n c l u d i n g o u r s e e d . 6.4.- CZOCHRALSKI GROWTH OF SINGLE CRYSTAI~ The m o s t c o m m o n m e t h o d for growing single crystals, w h e t h e r t h e y be a m e t a l or a c o m p l e x m i x t u r e of oxides, is to pull a single crystal from a melt.

This

is the

so-called

C

~

Method,

using

the

method

i n v e n t e d in 1918 by t h e Polish s c i e n t i s t J a n Czochralski, w h i c h he called crystal pulling. Large crystals c a n be g r o w n rapidly from the liquid f o r m e d by m e l t i n g

any given m a t e r i a l

(providing

t h a t the m a t e r i a l s

does

not

d e c o m p o s e u p o n melting). One a t t a c h e s a seed crystal to the b o t t o m of a vertical a r m s u c h t h a t the s e e d is barely in c o n t a c t with the m a t e r i a l at the surface of the m e l t a n d allows the crystal to slowly form as the a r m is lifted from the melt.

Let us e x a m i n e

this m e t h o d

in m o r e

detail

to

illustrate its versatility in crystal growth. As an example, we will grow a crystal from a m e l t

of oxides w h o s e m e l t i n g

points

exceed

1800

C.

Automatically, we are limited to u s e of an i r i d i u m (Ir) crucible a n d we will u s e an R . F . - g e n e r a t o r for the p o w e r source. In 6.4.1., (see n e x t page), w e illustrate the typical s e t u p for the crucible in the Czochralski a p p a r a t u s .

261

W e p l a c e t h e Ir c r u c i b l e c o n t a i n i n g t h e m i x t u r e o f o x i d e s o n a Z r O 2

262

platform. This acts as a t h e r m a l b a r r i e r for the b o t t o m of the a p p a r a t u s . We m a y t h e n place a larger Z r 0 2 cylinder a r o u n d t h e crucible for f u r t h e r insulation. An R.F. coil is p l a c e d in position a r o u n d the o u t s i d e of t h e insulation. Finally, an o u t s i d e wall of i n s u l a t i o n is p u t into place a n d a t o p cover plate is p u t into position. At 2 0 0 0 ~ the outerwall t h i c k n e s s of Z r 0 2 n e e d s to be at least 2.5 - 3.0 cm. The whole is t h e n covered by a bell-jar having a hole at the top. The d o m e a n d its c o n t e n t s are f l u s h e d for several m i n u t e s w i t h an i n e r t gas before the R.F.- g e n e r a t o r is t u r n e d on. Gas flow is m a i n t a i n e d t h r o u g h o u t the e n t i r e operation. As t h e c r u c i b l e h e a t s ups, it's entire c o n t e n t s degas. As the m a t e r i a l s melt, t h e y c o n t r a c t so t h a t it is usually n e c e s s a r y to a d d m o r e m a t e r i a l to the m e l t until t h e crucible is full before actual c r y s t a l g r o w t h is a t t e m p t e d . This m a y t a k e 3 4 crucible v o l u m e s of m a t e r i a l . T h u s , the top-hole

needs

to be large

e n o u g h to a c c o m m o d a t e this o p e r a t i o n . a g o o d single c r y s t a l is n o t easy. T h e r e are m a n y factors involved, e a c h of w h i c h i m p o s e s r e s t r a i n t s u p o n the others. S o m e d e p e n d u p o n t h e crystal pulling s y s t e m design, while o t h e r s d e p e n d u p o n t h e n a t u r e of t h e material

being

grown.

These

factors,

for

a given

system,

become

parameters, i.e. t h e y are i n t e r - r e l a t e d . T h e p a r a m e t e r s for CZOCHRALSKI GROWTH are listed in t h e o r d e r of their importance

9

6.4.2.- CZOCHRAI~KI CRYSTAL GROWTH PARAMETERS I. Melt t e m p e r a t u r e a n d its t h e r m a l c o n d u c t i v i t y 2. C h e m i c a l stability of t h e m e l t 3. Degree t h a t m e l t will "super-cool" 4. T e m p e r a t u r e control a n d t e m p e r a t u r e g r a d i e n t achievable 5. Interface angle w h i c h forms b e t w e e n crystal a n d m e l t 6. Degree of d i a m e t e r control p o s s i b l e 7. Rotation r a t e u s e d 8. Rate of p u l l i n g 9. Melt level m a i n t a i n e d

263

All of t h e s e

factors

are

dependent

upon

the

nature

of the

crystal

c o m p o s i t i o n being grown i n c l u d i n g its t h e r m a l , physical a n d c h e m i c a l properties.

Because of the

importance

of R.F. g e n e r a t o r s

to

crystal

growth, let us n o w e x a m i n e t h e m in m o r e detail. A r a d i o - f r e q u e n c y g e n e r a t o r is essentially a gigantic R.C.-taruk circuit (R.C. = r e s i s t a n c e - c a p a c i t a n c e ) w h i c h u s e s m e r c u r y - p o o l - d i o d e s as oscillators. The f r e q u e n c y is fixed a n d the g e n e r a t o r is self-regulating in p o w e r o u t p u t . As a result, c h a n g e of power- o u t p u t , w h i c h affects t e m p e r a t u r e regulation, is slow. The actual o u t p u t of p o w e r o c c u r s t h r o u g h the R.F. coil. The coil itself is a carefully-wound c o p p e r t u b e helix w h i c h has cooling w a t e r r u n n i n g t h r o u g h it, as s h o w n in t h e following:

If a s u s c e p t o r s u c h as a m e t a l crucible is placed w i t h i n the coil, The R.F. power i n d u c e s " e d d y - c u r r e n t s " in t h e crucible,

c a u s i n g it to h e a t up.

E d d y - c u r r e n t s are circular electrical c u r r e n t s i n d u c e d w i t h i n the m e t a l by the R.F. field of the coil.

It is essentially a "skin" effect, a n d the d e p t h of

p e n e t r a t i o n , i.e.- d e p t h of e d d y - c u r r e n t g e n e r a t i o n w i t h i n the crucible is defined by: 6.4.4.-

depth =

1 / 2 ~ [ p / ~tf x 10 9 ]1/2

w h e r e ~t is a p e r m e a b i l i t y , p is a resistivity of the s u s c e p t o r (crucible) mad f is t h e f r e q u e n c y of the R. F. field. The s p a c i n g of the coils is e x t r e m e l y critical. One usually s p e a k s of the "coupling ratio" for a given crucible- R.F. coil c o m b i n a t i o n . This is the ratio of p o w e r t r a n s m i t t e d to t h e c r u c i b l e divided by the power delivered to the coil. R. F. g e n e r a t o r s various sizes, as follows on the n e x t page:

come

in

264

6.4.5.-

10kc-

20KW

50kc-

100 kc - 50 KW

25KW

2 meg. - 75 KW

The size to u s e d e p e n d s u p o n the s u s c e p t o r b e i n g used, the t e m p e r a t u r e of o p e r a t i o n

desired

a n d the h e a t losses w i t h i n the s y s t e m . For an I r

crucible, the I 0 k c - 2 0 KW g e n e r a t o r w o r k s b e t t e r t h a n the 2 meg.- 7 5 KW g e n e r a t o r . R e t u r n i n g to Czochralski growth,

of the 9 p a r a m e t e r s

given in 6 . 4 . 1 .

above, the first 3 are f u n c t i o n s of the m a t e r i a l w h o s e single crystal we are trying to grow. #4 r e l a t e s b o t h to the m a t e r i a l a n d t h e physical design of the

melt-furnace

p l u s t h a t of the

coil.

Parameter

#5

relates

almost

entirely to the o p e r a t i o n of the s y s t e m . As we have i n d i c a t e d , t h e s t e p s in o p e r a t i n g the a p p a r a t u s are: 6.4.6.-

OPERATION OF THE CZOCHRALSKI APPARATUS 1. Load crucible a n d s t a r t melt. times.

Maintain i n e r t a t m o s p h e r e

at all

2. Add m a t e r i a l to m e l t until crucible is full. 3. W h e n the crucible is full, gradually lower the m e l t t e m p e r a t u r e to achieve s u p e r c o o l i n g a n d i n c i p i e n t n u c l e a t i o n . 4. Allow s y s t e m to equilibrate a n d dip s e e d into melt. Rotate s e e d at predetermined speed. 5. Allow s e e d to grow o u t w a r d to d e s i r e d d i a m e t e r . 6. Begin

pulling

incipient

crystal o u t of m e l t

at a r a t e

which

m a i n t a i n s c r y s t a l size a n d g r o w t h . We find t h a t the r o t a t i o n r a t e affects b o t h the "interface angle", i.e.- t h e angle b e t w e e n c r y s t a l a n d melt, a n d control of crystal d i a m e t e r , as s h o w n in the following d i a g r a m , given as 6.4.7. on the n e x t page. Note t h a t we have defined

"interface angle" as the angle b e t w e e n

the

growing crystal a n d the r e s i d u a l melt. Rate of pulling also affects t h e quality of the c r y s t a l as well as the actual n u m b e r of i n t r i n s i c d e f e c t s w h i c h m a y a p p e a r in t h e final crystal. In t h e u p p e r left of 6.4.7., a fiat-

265

interface between the crystal and melt surface has been maintained by regulating the rate of rotation. If it is not maintained correctly, the edges of the crystal may grow too fast (resulting in a hollow crystal- u p p e r right) or the edges may r o u n d (lower left) or facet (lower fight). If faceting is allowed to continue, we find that the crystal so p r o d u c e d is not usable because the facets produce a polycrystalline rod r a t h e r t h a n a single crystal. Diameter control, P a r a m e t e r #6, is i m p o r t a n t to the final quality of t h e obtained crystal. The sides of the crystal need to be straight because they reflect the regularity of the lattice planes within the crystal. Effects of deviation from "correct" growing conditions on the quality of the crystal so-produced are shown in the following diagram, given as 6.4.8. on t h e next page. When growth conditions are not "optimum", serious defects can appear in the growing crystal as a result of physical c i r c u m s t a n c e s of the crystal growth apparatus. Because of this, we need to be very careful while t h e crystal is growing.

266

It is this factor w h i c h m a n d a t e s the very careful design of the Czochralski Apparatus. A p r o p e r design c o n s i s t s of a heavy base, to m i n i m i z e effects of e x t e r n a l vibrations on the surface of the melt, a p r e c i s i o n screw driven by a c o n t r o l l e d r e v e r s i n g - m o t o r (so as to control rate of pulling precisely), a n d a precision m o t o r controlling rate of rotation. S u c h a design is s h o w n on the s u c c e e d i n g page as 6.4.9. Note t h a t we have n o t s h o w n the R.F. g e n e r a t o r . However, we have s h o w n one design of the pulling a p p a r a t u s (there are many). The next m o s t i m p o r t a n t p a r a m e t e r s in Czochralski g r o w t h of crystals are: the h e a t flow a n d hcmt lo~m~ in the system. Actually, all of t h e parameters

(with the possible exception

of #2

a n d #9)

are s t r o n g l y

affected by the h e a t flow w i t h i n the crystal-pulling s y s t e m . A typical heatflow p a t t e r n in a Czochralski s y s t e m involves b o t h the crucible a n d t h e melt. The p a t t e r n of heat-flow is i m p o r t a n t b u t we will n o t e x p a n d u p o n this topic here. Let it suffice to point o u t t h a t heat-flow is set u p in t h e melt by the direction of r o t a t i o n of the crystal being pulled. It is also affected by the u p p e r surface of the m e l t a n d h o w well it is t h e r m a l l y i n s u l a t e d from its s u r r o u n d i n g s . The circular h e a t flow p a t t e r n c a u s e s t h e surface to radiate heat. The crystal also absorbs h e a t a n d r e - r a d i a t e s it

267

f u r t h e r u p on the s t e m . If the crystal is k e p t stationary, we see t h a t t h e h e a t flow p a t t e r n is u n e v e n . T h u s , it is m a n d a t o r y t h a t the crystal be r o t a t e d to control crystal d i a m e t e r , so as to obtain a defect-free crystal. It s h o u l d be clear t h a t the m o s t irnportaxtt p a r t of the a p p a r a t u s is tile "pullrod ~. In m o s t cases, this is f o r m e d from a p r e c i s i o n screw w h i c h is t u r n e d to raise the rod while the whole a s s e m b l y is being r o t a t e d . The above d e s c r i p t i o n

applies to the s y s t e m w h e r e

only the g r o w i n g

crystal is rotated. T h e r e is at least one o t h e r way to "stir" the m e l t so as to control h e a t flow. This is i l l u s t r a t e d as follows: 6.4.11.-

+_ c r y s t a l r o t a t i o n = +_ crucible r o t a t i o n + crystal r o t a t i o n ,

•

crucible r o t a t i o n

268

We c a n r o t a t e the crucible by itself, or in c o n j u n c t i o n with the crystal. But a n o t h e r c o m p l e x i t y arises,

n a m e l y w h a t d i r e c t i o n of r o t a t i o n a n d w h a t

relative s p e e d of r o t a t i o n s h o u l d we u s e for both, or e i t h e r ? If we r o t a t e the c r y s t a l clockwise a n d the crucible c o u n t e r - c l o c k w i s e , flow p a t t e r n s b e c o m e

complex

indeed.

These

then the heat

complexities

have b e e n

s t u d i e d in detail b u t will not be e n u m e r a t e d h e r e . The b e s t c o m p r o m i s e s e e m s to be fast- r o t a t i o n for the crystal a n d slow o r no r o t a t i o n for t h e crucible. Of all the possible m e t h o d s

of s t i r r i n g t h e

melt, the s t a t i c - c r u c i b l e m e t h o d s e e m s to be the best, a n d this is t h e m e t h o d u s e d by m o s t m ' y s t a l - ~ e r s .

T h e n e x t b e s t m e t h o d s e e m s to be

r o t a t i n g the crucible at a slow rate, c o u n t e r to the d i r e c t i o n of the c r y s t a l rotation. It is clear t h a t crystal- r o t a t i o n n e e d s to d o m i n a t e the s t i r r i n g p a t t e r n so t h a t m i x i n g of the m e l t c o n t i n u e s while the crystal is g r o w i n g . The

Czochralski M e t h o d

method

for

complexities control

obtaining

has remained single

the m o s t f r e q u e n t l y

crystals.

The

involved have b e e n c o n q u e r e d

of the c r y s t a l - g r o w i n g

Parameters.

seemingly

by t h e

employed formidable

u s e of c o m p u t e r -

One s u c h s y s t e m ,

available

c o m m e r c i a l l y , is s h o w n in 6.4.12. (on the n e x t page). Here, we s h o w only a b a r e outline of t h e individual c o m p o n e n t s

in t h e

overall s y s t e m . This SYSTEM is capable of o p e r a t i o n in i n e r t a t m o s p h e r e or

vacuum.

A

slave-processor

controls

both

crystal

diameter

and

m e n i s c u s - c o n t a c t of the growing crystal. As we have stated, this is m o s t i m p o r t a n t ff we w i s h to obtain a crystal essentially free from i n g r o w n defects.

This t a k e s

separate monitor observing

the

the

form

is p r o v i d e d

of a m e l t / s e e d for the o p e r a t o r

crystal d i a m e t e r

as it grows.

contact

monitor

and a

as a C C T V c a m e r a for There

is

also

a crystal

a n n e a l i n g f u r n a c e to r e m o v e any crystal s t r a i n t h a t m a y have b e e n i n d u c e d by the c r y s t a l - g r o w i n g

conditions.

A base heater

h e l p s to m a i n t a i n

a

u n i f o r m t e m p e r a t u r e g r a d i e n t in the m e l t - c r u c i b l e d u r i n g crystal g r o w t h . Note t h a t b o t h t h e crucible a n d crystal r o t a t i o n c a n be controlled.

In

o r d e r to control the h e a t - c o n v e c t i o n p a t t e r n s w h i c h n o r m a l l y a p p e a r in t h e melt, a n e x t e r n a l c r y o m a g n e t c a n be s u p p l i e d . Its m a g n e t i c field h e l p s control h e a t l o s s e s a n d m a i n t a i n s a b e t t e r control of t h e c r y s t a l growth.

269

This h a s n o t b e e n s h o w n b u t i n c l u d e s a s o u r c e of liquid n i t r o g e n for t h e

270

c r y o m a g n e t as well as a c o n t r o l l e r to m a i n t a i n the c o r r e c t m a g n e t i c field n e e d e d for defect-free g r o w t h of the crystal. A final c o m m e n t on the Czochralski Method: GaAs h a s b e c o m e i m p o r t a n t in c o n s t r u c t i o n

of i n t e g r a t e d

circuits

for

computers

because

p r o m i s e of s p e e d of r e s p o n s e a n d d e n s i t y of c o m p o n e n t s ,

of t h e

c o m p a r e d to

silicon wafers. Both As a n d Ga t e n d to oxidize in air as t h e y a p p r o a c h t h e melt stage a n d As203 sublimes. Both e l e m e n t s are toxic to m a n . In t h e melt stage, they have high partial p r e s s u r e s as well, so t h a t t h e u s e of an inert a t m o s p h e r e is n o t sufficient to allow g r o w t h of a single crystal. One m e t h o d t h a t h a s b e e n u s e d for this case h a s e m p l o y e d a "liquid e n c a p s u l a t i n g agent". As s h o w n in the following diagram, this c o n s i s t s of a lower m e l t i n g a g e n t u s e d to form a liquid barrier, floating on t h e surface of the m e l t w h i c h serves as the source for the crystal (in this case, Ga-As).

The arrow indicates the liquid b a r r i e r layer. This u s e of a b a r r i e r m e l t illustrates t h a t t h e r e are several ways to grow crystals w h i c h w o u l d be difficult to obtain u n d e r "ordinary" m e a n s of crystal growth, i.e.p r e v e n t i o n of oxidation a n d evaporation of GaAs d u r i n g crystal g r o w t h . 6 - 5 - T h e B r i d g e m a n - S t o c k b a r g e r M e t h o d for Crystal G r o w t h W h e r e a s the Czochralski m e t h o d r e q u i r e s r a t h e r elaborate e q u i p m e n t to obtain single crystals of good quality, the B r i d g e m a n ( S t o c k b a r g e r ) m e t h o d uses a fairly f~mple a p p a r a t u s . This is s h o w n in the following diagram, given on the next page as 6.5. i.

271

The major c o m p o n e n t s n e e d e d to grow single crystals by this m e t h o d are: A two (2) zone furnace Two (2) set point t e m p e r a t u r e - c o n t r o l l e r s A crucible with "seed" tip A constant-speed elevating a n d lowering the crucible.

device

for

The p r o c e d u r e involves obtaining a full crucible of melt w h o s e c omp o s itio n m i r r o r s t h a t of the crystal we wish to grow. We t h e n l ow er the crucible c o n t a i n i n g the melt t h r o u g h a baffled zone within the furnace

272

at a slow rate. The baffles within the furnace p r o d u c e s a u n i f o r m t e m p e r a t u r e decrease b e t w e e n the two zones, resulting in a t e m p e r a t u r e proi~fle like th at given on the fight in the Figure. The t e m p e r a t u r e setpoint of the u p p e r zone is j u s t above the melting point of the material to be grown, and the lower zone is set j u s t below the melting point of t h e material. This r e s ul t s in a s m o o t h t e m p e r a t u r e prot~fle, above and below t h at of the crystal fusion point. Thus, if we can induce a seed to form in the tip of the crucible, then, as the crucible is lowered t h r o u g h t h e freezing p ar t of the t e m p e r a t u r e zone, the rest of the crucible will freeze as a single crystal. The p a r a m e t e r s for the lh'idgemml M e t h o d are: 6.5.2.- pARAMETERS INVOLVED IN THE BRIDGEMAN METHOD I. T e m p e r a t u r e Profile within Furnace 2. Rate of Crucible Raising or Lowering 3. Melt T e m p e r a t u r e 4. Supercooling of Melt 5. Annealing T e m p e r a t u r e 6. T h e r m a l Conductivity of Melt and Crystal 7. T h e r m a l E xpans i on of Crystal 8. Crucible Material U s e d 9. Chemical Stability of Material Being Grown P a r a m e t e r # 1 is a m a t t e r of furnace design. There shoul d be provision for several baffles so as to adjust the t e m p e r a t u r e - gradient distance. Then, it is a m a t t e r of adjusting the two set-point t e m p e r a t u r e s to achieve t h e p ro p er gradient. For some crystals, the gradient needs to be sharp, w h e r e a s for o th er s it can be m or e gradual. The two set-point p a r a m e t e r d e p e n d s primarily u p o n the degree of supercooling experi enced in t h e melt, prior to nucleation. This in t u r n m a y be affected by purity of t h e c o n s t i t u e n t s . The raising and lowering m e c h a n i s m can be simple, t h a t of a gear-driven m o t o r with a c o u n t e r w e i g h t for the crucible-melt m ass. It is convenient to use the lower set- point as the annealing t e m p e r a t u r e for the crystal. But, for very high melting points, this m a y not be possible. Then, we need to add a lower annealing furnace. As the crystal freezes, it m a y not do so uniformly. If so, i n t e r n a l strain resul t s (a polariscope will

273

reveal this). F u r t h e r m o r e , ff the expansion coefficients of the crystal and crucible are too disparate, t h e n external strain on the crystal will g e n e r a t e internal strain. The crystal will t h a n have to be annealed to relieve this strain. The B r i d g e m a n M e t h o d is useful for m a n y types of crystals. T h e key to getting this m e t h o d to w o r k is to induce a seed to form in t h e "seed-tip". If the crucible "seed- tip" is not properly designed, the m e t h o d will not w o r k . A similar way to grow crystals is the Kyropoulos M e t h o d . In this m e t h o d , we first form a melt and t h e n i n t r o d u c e a seed. By raising the m e l t t h r o u g h the t e m p e r a t u r e gradient, a single crystal will grow from t h e point w h e r e follows:

the seed engages the melt.

The

apparat us is shown as

Note t h a t we have r e v e r s e d the t e m p e r a t u r e gradient within the furnace and th at the top is cooler t h e n the bottom, w here the melt is first f o r m e d . In a variation of this m e t h o d , the seed is i n t r o d u c e d after the m e l t t e m p e r a t u r e has been stabilized and t h e n b r o u g h t to incipient nucleation. In this case, the whole m a s s can be m a d e to freeze nearly instantaneously. It is this m e t h o d w h i c h is c u r r e n t l y being u s e d to m a n u f a c t u r e s a p p h i r e boules as large as 12 inches in diameter. The boule is t h e n cut into slabs

274

w h i c h are p o l i s h e d a n d u s e d for UV t r a n s m i t t i n g w i n d o w s , s u b s t r a t e s for various e l e c t r o n i c devices a n d even n o n - s c r a t c h i n g faces on y o u r w a t c h . Note

that

transmitting

sapphire

is

about

as h a r d

as d i a m o n d

and

capability a l m o s t equal to m a n y metals,

has

while

a heat-

remaining

essentially c h e m i c a l l y i n e r t a n d electrically n o n - c o n d u c t i v e . In o t h e r variations of the Kyropoulos m e t h o d , we raise the m e l t t h r o u g h the freezing point of the melt, by raising it p a s t the baffles of t h e furnace. A n o t h e r possible m e t h o d

is one w h e r e we u s e a single- zone furnace,

stabilize the melt, lower the f u r n a c e t e m p e r a t u r e to i n c i p i e n t n u c l e a t i o n a n d t h e n cool to form t h e crystal. However,

this w o r k s only for a few

s y s t e m s . T h e r e a s o n is p r o b a b l y supercooling. T h e r e are only a few m e l t s w h e r e a large d e g r e e of s u p e r c o o l i n g occurs. Addition of a s e e d will t h e n c a u s e very r a p i d g r o w t h of the single crystal. But, it is usually s t r a i n e d a n d m u s t be carefully a n n e a l e d to obtain the m a s s intact.

If the a p p a r a t u s

s h o w n in 6.5.3. is u s e d a n d the crucible is exactly p o s i t i o n e d so t h a t a vertical t e m p e r a t u r e

g r a d i e n t is m a i n t a i n e d

e z a c t l y along t h e c r u c i b l e

h e i g h t , t h e melt, w h e n seeded, will grow into a single crystal m o r e slowly. T h e n , the degree of s u p e r c o o l i n g is not as i m p o r t a n t . As a m a t t e r of fact, m e l t s w i t h little or no d e g r e e

of s u p e r c o o l i n g c a n be i n d u c e d to f o r m

single c r y s t a l s by following this p r o c e d u r e . 6.6.-

ZONE MELTING AS A MEANS FOR FORMING SINGLE CRYSTALS

T h e r e is n o t h i n g to say t h a t we m u s t m e l t all of the m a t e r i a l at one t i m e . If we have a long rod of polycrystalline m a t e r i a l , a n d m e l t a p a r t of it, i.e.a "zone", we c a n sweep this zone down the l e n g t h a n d u l t i m a t e l y m e l t a n d freeze all of the rod. The m a j o r p r o b l e m is, again, i n d u c i n g a "seed" of single- c r y s t a l m a t e r i a l to form (and to continue). In t h e forward d i r e c t i o n of travel, t h e zone is m e l t i n g w h e r e a s at the b a c k of t h e zone, f r e e z i n g occurs. If we c a n i n d u c e the b a c k p a r t to freeze as single- crystal, t h e n w e can transform

the

whole

rod.

This

means

that

we

polycrystalline m a t e r i a l a n d e n d u p with a single- crystal: 6.6.1.-

P o w d e r ~ P o l y c r y s t a l l i n e ~ Single Crystal

can

start

with

275

For the case of Z o n e - M e l t i n g to f o r m a s i n g l e between

crystal, we c a n d i s t i n g u i s h

two s e p a r a t e cases, i.e.- h o r i z o n t a l a n d vertical zone m e l t i n g .

This is s h o w n as follows:

Our choice a m o n g h e a t i n g e l e m e n t s is limited to the R.F.- coil, a h o t - w i r e coil or p e r h a p s the MoSi2 "hairpin" element. The r e a s o n is t h a t we m u s t r e s t r i c t the melt zone as m u c h as possible, while m o v i n g the zone t h r o u g h the material. Heating by r a d i a t i o n is not very effective since it o c c u r s in a 4n direction. One possible solution is to u s e a h e a t reflector a r o u n d t h e wire

heating element

to limit

the

h e a t i n g length_ of the

melt

zone.

A n o t h e r limitation of the m e t h o d is t h a t the p o w e r losses in the s y s t e m are large b e c a u s e we m u s t u s e a protective t u b e or o t h e r i n s u l a t i o n a r o u n d the m a t e r i a l so as to be able to move the h e a t source. Unless we are willing to s q u a n d e r power, we c a n n o t r e a c h the very high t e m p e r a t u r e s of m e l t i n g r e q u i r e d for s o m e m a t e r i a l s . Therefore, the z o n e - m e l t i n g m e t h o d of single crystal g r o w t h h a s usually b e e n confined to m a t e r i a l s of m o d e r a t e to low m e l t i n g t e m p e r a t u r e s s u c h as silicon. The vertical m e t h o d s h o w n above h a s s o m e t i m e s b e e n called the "floating zone" m e t h o d .

The

thickness

of polycrystaUine

rod

determines

the

276

m a x i m u m l e n g t h of m e l t zone t h a t c a n be u s e d successfully. T h i s r e l a t i o n c a n be c a l c u l a t e d f r o m : 6.6.3.-

THIN RODS ZMax -

THICK RODS_

2 (3r}I/2

ZMax = 2.8 [ y / p g ]

w h e r e r is t h e r a d i u s in cm., 7 is the s u r f a c e t e n s i o n of t h e m e l t , p is its d e n s i t y , g is the g r a v i t a t i o n a l c o n s t a n t , a n d ZMax is t h e m a x i m u m l e n g t h of the floRtlng zone. 6.7.- ZONE R E F I N I N G W h e n a c r y s t a l freezes f r o m t h e melt, it t e n d s to reject i m p u r i t i e s . If z o n e - r e f i n i n g is a p p l i e d to s u c h a crystal, the r e s u l t i n g single c r y s t a l eml b e

p u r e r t h a n t h e originaL If we m a k e m o r e t h a n one p a s s , we c a n a p p r o a c h a h i g h d e g r e e of purity. S u c h a p r o c e d u r e is called "zone- refining". It h a s b e e n d e t e r m i n e d

that there

is a d i s t r i b u t i o n coefficient

for t h e

i m p u r i t i e s b e t w e e n c r y s t a l a n d m e l t w h i c h favors t h e melt. We c a n s e e h o w this a r i s e s w h e n we reflect t h a t i m p u r i t i e s t e n d to c a u s e f o r m a t i o n of i n t r i n s i c defects w i t h i n the c r y s t a l a n d lattice s t r a i n as a r e s u l t of t h e i r p r e s e n c e . In the melt, no s u c h r e s t r i c t i o n applies. Actually, e a c h i m p u r i t y h a s its o w n d i s t r i b u t i o n coefficient.

However, o n e c a n a p p l y a n average

value to b e t t e r a p p r o x i m a t e t h e b e h a v i o r of t h e m a j o r i t y of i m p u r i t i e s . One c a l c u l a t e s t h e c o n c e n t r a t i o n of i m p u r i t i e s left in t h e single crystal, as a f u n c t i o n of a single p a s s , f r o m : 6.7.1.-

Ci

--

Co

[1 - (1-ko) exp {- k o L / Z }l

w h e r e Co is the o H g l m ~ c o n c e n t r a t i o n of i m p u r i t i e s , L is the l e n g t h t h e zone travels (in cm.), a n d Z is t h e m e l t - z o n e

length.

Note t h a t we are

t a k i n g a ratio of L to Z. We n e e d to m i n i m i z e Z in o r d e r to m a k e t h e p r o c e s s efficient in s e g r e g a t i o n of i m p u r i t i e s , ko is the d i s t r i b u t i o n coefficient, defined by:

277

6.7.2.-

ko = ci {solid) / ci {melt)

We plot c i / Co vs: L / Z, t h e n u m b e r of z o n e - l e n g t h s solidified. T h i s is i l l u s t r a t e d in t h e following d i a g r a m :

278

At t h e top of this d i a g r a m , we s h o w t h e effects of o n e p a s s on various s e l e c t e d values of t h e d i s t r i b u t i o n coefficient, k o . Note t h a t at values of ko > 1, we do not o b t a i n any significant i m p r o v e m e n t in i m p u r i t y c o n t e n t . For e x a m p l e , we s t a r t at k = 5.0, b u t e n d up at 1.0. Even at ko = 0.5, t h e f u r t h e r we move t h e m o l t e n zone, t h e less we purify t h e m a t e r i a l . T h i s h a s b e e n called t h e i m p u r i t y - l e v e l i n g effect. At t h e very low values of ko, one still c a n n o t i m p r o v e t h e i m p u r i t y level to any significant degree, u s i n g j u s t one

pass. We find

that multiple

passes

are

required

to

obtain

purification. At t h e b o t t o m of 6 . 7 . 3 . is s h o w n t h e effect of m u l t i p l e p a s s e s u p o n t h e degree of p u r i f i c a t i o n o b t a i n e d . Here, we have used: 6.7.4.-

ko

=0.5 andL/Z=8

a n d have r e c a l c u l a t e d t h e d a t a given before. Note t h a t as t h e n u m b e r of p a s s e s r e a c h e s 10, we o b t a i n a c h a n g e of s o m e 90 t i m e s in t h e i m p u r i t y c o n t e n t a t t h e front of t h e rod. But, at t h e back, i.e.- L / Z = 8, we a p p r o a c h an u l t i m a t e d i s t r i b u t i o n w i t h 20 p a s s e s (the d o t t e d line). If we could s e e t h e i n t e r n a l d i s t r i b u t i o n of i m p u r i t i e s after t h e rod h a d b e e n z o n e - r e f i n e d by 20 p a s s e s , we w o u l d see:

where

t h e d e n s i t y of dots r e p r e s e n t s

the

concentration

of i m p u r i t i e s .

W h a t h a s h a p p e n e d is t h a t we have m o v e d m a n y of t h e i m p u r i t i e s f r o m t h e front to t h e b a c k , b u t not all. a n d we finally e n d up w i t h an i m p u r i t y d i s t r i b u t i o n we c a n n o t c h a n g e . To u n d e r s t a n d this, we m u s t e x a m i n e t h e i m p u r i t y - l e v e l i n g factor in m o r e detail.

279

6 . 8 . T H E IMPURITY LEVELING F A i R When

a melt-zone

is

moved

through

a long

crystal,

an

impurity

c o n c e n t r a t i o n b u i l d s u p in t h e m e l t z o n e d u e to r e j e c t i o n b y t h e c r y s t a l as it resolidifies. We c a n also say t h a t t h e d i s t r i b u t i o n coefficient p u r i f i c a t i o n p r o c e s s , i.e.- k<< are c o n c e r n e d )

favors a

1. A n o t h e r r e a s o n (at l e a s t w h e r e

is t h a t a s o l i d - s o l u t i o n b e t w e e n

metals

impurity and host ions

exists. It h a s b e e n o b s e r v e d t h a t t h e following s i t u a t i o n , as s h o w n in t h e following d i a g r a m , o c c u r s :

T h e i m p u r i t y , x , b u i l d s u p at t h e solid- liquid i n t e r f a c e as t h e liquid z o n e m o v e s a n d t h e solid forms. We c a n w r i t e for t h e d i s t r i b u t i o n c o e f f i c i e n t : 6.8.2.where

kx = CxL / CxS CxL is t h e i m p u r i t y - c o n c e n t r a t i o n

in t h e liquid, a n d C~s is t h e

i m p u r i t y c o n c e n t r a t i o n in t h e solid. We t h e n e s t i m a t e k x f r o m : 6.8.3.-

k x NI / N MX exp - ( EM - EX ) / k T

w h e r e I is t h e i m p u r i t y , MX is t h e c o m p o u n d u n d e r c o n s i d e r a t i o n , a n d E M , EX a r e t h e a c t i v a t i o n e n e r g i e s for f o r m a t i o n of solid s o l u t i o n of MX I X . Note t h a t we h a v e a s s u m e d t h a t I is a c a t i o n i m p u r i t y in t h e c r y s t a l , MX. We c a n d i f f e r e n t i a t e

between

2 separate

cases,

as s h o w n

in t h e

following d i a g r a m , given as 6 . 8 . 4 . o n t h e n e x t p a g e . Here, w e s h o w two c a s e s for i m p u r i t y s e g r e g a t i o n

between

melt

and

c r y s t a l as it g r o w s in t i m e . Note t h a t a n initial p u r i f i c a t i o n o c c u r s in b o t h c a s e s b u t t h e d i s t r i b u t i o n coefficient for t h e c a s e o n t h e r i g h t is s u c h t h a t t h e a m o u n t of i m p u r i t y a c t u a l l y i n c o r p o r a t e d into t h e c r y s t a l , ki - Co.

280

6.8.4.- Behavior of I m p u r i t i e s as a F u n c t i o n of Type of Solid Solution Formed

ICase I- CXL> Cxs

I

ko<1

I~ k x " ~ Liquid rlI~ i ~.~ (Melt) I ~. kx2 ir~~ Sol

.

.

I ! i

I I i

XI rl

ih._ iil,'-

Li quid

ko<1

~

kx2

(Melt) j ~ . ~kx, [2

/

.,/!

~i

X2

L.-

1.1"!

SoJil

9

9

I I

I I

9

X

]

9

J

x

I

x2

1

Note t h a t it rapidly a p p r o a c h e s the orlglm~ i m p u r i t y c o n c e n t r a t i o n , c o . At t~, it actually in~re, m ~ over the original c o n c e n t r a t i o n at s o m e p o i n t w i t h i n the solid crystal. It s h o u l d be clear t h a t if the i m p u r i t y freezes into the solid from the melt, t h e r e m u s t be a certain a m o u n t of solid solution formation, even t h o u g h the i m p u r i t y c o n t e n t be less t h a n 0.01%. Case I & II s h o w simple solid solution b e h a v i o r for k < 1. For CxL > CxS , we will get an i m p r o v e m e n t in p u r i t y in the crystal. But for o p p o s i t e effect is seen. T h u s , two different

CxL < CxS

impurities

,

the

could m a n i f e s t

o p p o s i t e behaviors, d e p e n d i n g u p o n w h a t h o s t t h e y were in. The following diagram, given as 6.8.5. on the next page, i l l u s t r a t e s f u r t h e r p h e n o m e n a r e g a r d i n g zone refining. The s a m e s i t u a t i o n s e e n in 6 . 8 . 4 . occurs

for the

distribution Nevertheless,

case

at the

coefficient simple

bottom

is s u c h t h a t

right the

of 6.8.5. impurity

except buildup

that is

the

slower.

solid-solution for i m p u r i t y s y s t e m s is rarely t h e

n o r m . The m o s t p r e v a l e n t case is t h a t of Case III of 6.8.5. Limited solid solution occurs, a n d we get a t w o - p h a s e s y s t e m .

281

This

illustrates

the

fact

that

impurity

segregation

and

purification

p r o c e s s e s are d e p e n d e n t u p o n the type of i m p u r i t y involved and its individual segregation coefficient. As we illustrated above in 6.8.1., t h e p r o b l e m is th at the i m p u r i t y is initially rejected from the solid, but its c o n c e n t r a t i o n builds up in f l o a t of the g r o w i n g crystal, The s e g r e g a t i o n coefficient, ki , t h e n operates on t h a t i n c r e a s e d c o n c e n t r a t i o n and t h e product, ki Co, i ncr eas es . If we use individual zone lengths, as shown in 6.7.3. , we would have: 6.8.6.-

ZONE I~

2. 3.

IMPURITY CONCENTRATION (Co - k Co) (Co-kco) + k(co-kco) (Co - k c o ) +k(co -kco) +k2(co -kco),

etc.

This is the r eas on for the behavior s how n in 6.8.4. The i m p u r i t y front builds up until its c o n c e n t r a t i o n s u r p a s s e s the original c o n c e n t r a t i o n

282

( s o m e t i m e s by manyfold). T h e r e u p o n , the i m p u r i t y c o n c e n t r a t i o n levels out (but not n e c e s s a r i l y to the s a m e Co we have u s e d for i l l u s t r a t i o n in 6.8.4.).

It is for t h e s e

reasons

t h a t zone-refining

has found limited

application, since actual purification by this m e t h o d does not p r o d u c e a crystal with a c o m p l e t e l y u n i f o r m i m p u r i t y d i s t r i b u t i o n . The c o n c l u s i o n we m u s t d r a w is t h a t we s h o u l d purify o u r r a w m a t e r i a l s b e f o r e we a t t e m p t to grow o u r single crystal, not after. Zone refining has b e e n found to be effective only in a few limited cases. It is not applicable to the g e n e r a l case or to high m e l t i n g crystals. But if we w i s h to do z o n e l e v e l i a g of a n a d d e d i m l m f i t y , t h e n we have a very useful t e c h n i q u e .

A

good e x a m p l e w o u l d be to a d d a small a m o u n t of Ga to Ge . We c a n a d d a small v o l u m e of Ga to the front zone a n d will fund t h a t it h a s b e c o m e evenly i n c o r p o r a t e d into the Ge crystal after several p a s s e s , u s i n g zonerefining. 6.9.- THE VERNEUIL METHOD OF CRYSTAL GROWTH This m e t h o d

h a s also b e e n called the "flame-fusion" m e t h o d

of c r y s t a l

growth. If a p o w d e r is blown t h r o u g h a flame, it will m e l t if t h e flame is hot e n o u g h . T h e only flame hot e n o u g h to do this is the o x y - h y d r o g e n flame.

However,

a specially d e s i g n e d

burner

is r e q u i r e d

so t h a t t h e

crystaUites will m e l t d u r i n g the s h o r t time t h a t t h e y p a s s t h r o u g h t h e flame-front. The original w o r k w a s a c c o m p l i s h e d by Verneufl (1931)

and

the a p p a r a t u s is n a m e d after this investigator. T h e tip of t h e t o r c h is i m p o r t a n t . One d e s i g n u s e d for s u c h a t o r c h is given as follows:

283

The center tube is used for oxygen gas which t r a n s p o r t s the powder. T h e b u r n e r is designed so that the outer tubes contain only hydrogen gas, with the interstiee~ between the H 2 - tubes t r a n s p o r t i n g additional oxygen gas. This design will melt Y203 powder which has a melting point of 2380 ~ The overall Verneuil apparatus, shown in 6.9.2. on the next page, consists of a sealed hopper to contain the powder, the TORCH itself, a refractory pedestal to hold the growing crystal, and an after-furnace to anneal t h e crystal. The procedure for using the apparatus shown in 6.9.2. is given as follows: 6.9.3.-

Operating Instructions for Verneuil Apparatus I. Fill powder hopper, close and attach to burner. 2. Light b u r n e r and adjust flame to oxidizing. 3. Adjust pedestal to proper height and begin rotation. 4. Slide after-furnace into position. 5. Turn on and adjust oxygen flow in powder h o p p e r . 6. Begin tapping. 7. As melt begins to build up, adjust both height and speed of rotation so that crystal begins to form in a regular m a n n e r . 8. When the proper crystal size has been reached, stop powder flow by shutting off the oxygen flow to the h o p p e r . Continue flame operation and pedestal rotation. 9. Lower the crystal into the after- furnace, and anneal crystal, Gradually lower furnace t e m p e r a t u r e temperature.

and let cool to r o o m

Thermal stability is important in this m e t h o d because the high t e m p e r a t u r e s r e a c h e d may be sufficient to cause decomposition of t h e material. The Verneuil Method of crystal growth is not generally applicable to all types of crystals. There are serious deficiencies in t h e method. For example, there is a large t e m p e r a t u r e drop of h u n d r e d s of degrees over a few millimeters within the crystal. This causes a large difference in thermal expansion within a limited space, and c o n s e q u e n t

284

strain. Many crystals are not refractory e n o u g h to w i t h s t a n d the s t r e s s

285

buildup an d so crack into m a n y smaller parts. This m a k e s it very difficult to obtain a single crystal of any size. Thus, if a crystal is grown by this technique, it m u s t be annealed carefully in order to obtain it intact. Generally, the m e t h o d is r e s t r i c t e d to crystals like A1203 w hose t h e r m a l conductivity is high and w hos e refractive n a t u r e m a k e s it possible to obtain a crystal. For the m o s t part, one is r e s t r i c t e d to growing s i m p l e oxides by this m e t h o d s . Complex oxides s u c h as Ca2Si04 or Ca2P207 generally c a n n o t be g r o w n easily an d the Czochralski Method b e c o m e s the m e t h o d of choice. The p a r a m e t e r s involved in the Verneufl m e t h o d are: 1. Melt T e m p e r a t u r e 2. T h e r m a l Stability at Melt T e m p e r a t u r e 3. T e m p e r a t u r e Gradient p r e s e n t in l~arnace

and

after-

furnace. 4. Sintering Volume Losses 5. T h e r m a l Conductivity of Crystal 6. Annealing T e m p e r a t u r e R e q u i r e d 7. Effects of Reducing Flame on Material 8. Chemical Stability of Material being U s e d 6-10: MOLTEN FLUX GROWTH OF CRYSTALS This m e t h o d employs a m o l t e n flux which dissolves the material and redeposits it u p o n a selected substrate. T hat is, the m o l t e n flux acts as a t r a n s p o r t m e d i u m . The t e m p e r a t u r e of the flux can be varied to suit t h e material an d to p r o m o t e high solubility of the solute material in t h e m o l t e n solvent. One example is '"fiG", y t t r i u m iron garnet, i.e.- Y3Fe5012. This material is u s e d in the Electronics I n d u s t r y as single crystals for microwave method.

generating

devices.

It can be grown via the m o l t e n

A typical molten-flux a ppa r at us is s how n as 6. I0. I. on the next page.

flux

286

The six steps involved in using this m e t h o d are as follows: 1. A flux s u c h as lead borate is melted. PbB204 is useful in this m e t h o d because it will under g o supercooling r a t h e r easily. 2. The material w hi ch is to form single crystals is dissolved in the m o l t e n flux to near saturation (Note t h a t this r e q u i r e s prior knowledge regarding solubility of c o m p o u n d in m o l t e n flux). 3. The crucible, w hi ch is usually platinum, is rot at ed to obtain a

uniform

temperature

distribution

within

the

melt-

c o m p o u n d solution. 4. The solution t e m p e r a t u r e is gradually lowered to i n c i p i e n t nucleation. At this point, because of the physical a r r a n g e m e n t

287

of the h e a t i n g e l e m e n t s ,

a temperature

g r a d i e n t will e x i s t

along the l e n g t h of the crucible, from top to b o t t o m . 5. Single c r y s t a l s will begin growing along t h e b o t t o m of t h e crucible, a n d s o m e t i m e later along t h e e d g e s . 6. T h e crucible is k e p t r o t a t i n g to m a i n t a i n a u n i f o r m m i x i n g a n d h e a t flow while the c r y s t a l s are g r o w i n g . T h i s m e t h o d h a s s e r i o u s deficiencies for u s e as a g e n e r a l m e t h o d . We find t h a t if we dissolve MX in a BX 2 flUX a n d g r o w all MX c r y s t a l , it is likely to be c o n t a m i n a t e d w i t h B, or even KK2. The c r y s t a l - g r o w i n g p a r a m e t e r s for this m e t h o d are: 6.10.2.-

Molten F l u x G r o w t h P a r a m e t e r s 1. Flux m e l t t e m p e r a t u r e 2. Solubility of MX in t~K2 3. Degree of s u p e r c o o l i n g 4. T e m p e r a t u r e g r a d i e n t a c h i e v e d 5. Rate of r o t a t i o n u s e d

If t h e s e are not w i t h i n the c o r r e c t

range, one does not obtain single

crystal growth. This m e t h o d h a s b e e n u s e d in t h e p a s t only b e c a u s e of its relative simplicity of apparatlas a n d m a t e r i a l s . crystals so-produced

h a s b e e n r a t h e r poor.

However,

the quality of

Crystals p r o d u c e d

by t h i s

m e t h o d are suitable for s t r u c t u r e deterwAnations, b u t are poor in o p t i c a l quality a n d are n o t a t all s u i t e d for e l e c t r o n i c a p p l i c a t i o n s . T h e r e is one a r e a w h e r e m o l t e n flux g r o w t h h a s b e e n u s e d to form an epitaxial g r o w t h of a single c r y s t a l • m on a s u b s t r a t e . In this case, we c a n u s e m o l t e n lead t e t r a b o r a t e (PbB407), w h i c h m e l t s at about 9 6 0 ~ T h e m e l t is r a i s e d to 9 8 0 ~ The t e m p e r a t u r e

a n d t h e r e q u i s i t e oxides are dissolved t h e r e i n .

is lowered

to n e a r 9 6 0 ~

a n d s u p e r c o o l i n g begins.

Since the substra_te is cooler, a g a r n e t • m

will grow on its surface u n d e r

t h e s e conditions. T h e surface of the ~

slice will have b e e n carefully

p o l i s h e d to m i n i m i z e surface d e f e c t s .

288

It takes about 6 m i n u t e s to grow a single-crystal film about 5o - IOO thick. The trim grows in an epitaxial m a n n e r , that is- it builds up on t h e crystallographic planes of the substrate itself. It should be obvious that the degree of supercooling that the melt will undergo before the dissolved oxides precipitate out depends upon the amount of oxides dissolved. This is generally d e t e r m i n e d by trial a n d e r r o r . We lower the t e m p e r a t u r e to just above that point. A substrate slice of single- crystal ~ (gadolinium gallium garnet = Gd3Ga5 O12) is then s u b m e r g e d in the molten flux, while being rotated. Expitaxial growth was used in the supplanted by plasma deposition and the like.

past but has b e e n

6.11.- HYDROTHERMAL GROWTH Single crystal growth by h y d r o t h e r m a l m e a n s utilizes water as t h e m a t e r i a l - t r a n s p o r t medium. The m e t h o d is most often used for growing single crystal quartz. Quartz (SiO2) is not very soluble in water, but its solubility increases considerably at higher t e m p e r a t u r e s . Thus, growth is accomplished at high p r e s s u r e s in a sealed autoclave. These quartz crystals are used as resonant frequency "tuning forks" for timing applications in digital watches. The following diagram, given as 6.11.1 on the next page, shows a typical apparatus. Seed crystals are h u n g within an autoclave on a revolving hanger. Nutrient (high purity sand or natural crystal quartz) is contained at t h e

289

b o t t o m a n d dissolves as the autoclave is heated. A t e m p e r a t u r e g r a d i e n t is m o s t often u s e d so t h a t the n u t r i e n t dissolves a n d is t r a n s p o r t e d to t h e cooler area w h e r e it is r e d e p o s i t e d

as single crystal material. Note t h e

t e m p e r a t u r e g r a d i e n t at t h e baffle. The solubility of the solute (in this case, quartz) is a function of b o t h p r e s s u r e a n d t e m p e r a t u r e . P r e s s u r e could be in t h e o r y be u s e d as t h e controlling p a r a m e t e r r a t h e r t h a n t e m p e r a t u r e . However, it is difficult to design an a p p a r a t u s with a p r e s s u r e gI'adient, w h e r e a s o b t a i n i n g a t e m p e r a t u r e g r a d i e n t is fairly easy. Actually, w h a t is c o n t r o l l e d is the critical v o l u m e from the e q u i l i b r i u m : 6.11.2.-

H20(g)

r

H20(1)

w h e r e 1 a n d g refer to the liquid a n d g a s e o u s state, respectively. T h e p h a s e d i a g r a m for w a t e r is s h o w n at the lower left of the following diagram, given as 6.11.3. on the n e x t page.

290

Above the critical t e m p e r a t u r e , water exists as a gas at all pressures. But, if we fLx the volume of both gas and liquid (as in an autoclave), then no liquid is possible below the critical volume (-- 30 % by volume). By filling the available volume greater t h a n 30% with water, we can go to very high p r e s s u r e s (i.e.- > 3000 atmospheres) and still maintain a liquid volume.

In hydrothermal growth, the materials ~ grown as single crystals are t h o s e classified as in-qoluble at standard t e m p e r a t u r e and pressure (STP). For quartz, growth is usually accomplished a t m o s p h e r e s at 350 ~

at

85%

fill

and

2000

The p a r a m e t e r s for h y d r o t h e r m a l growth of single crystals are: 6.11.4.-

Parameters Controlling Hydrothermal Growth of Single Crystals. 1. Operating P r e s s u r e 2. Operating T e m p e r a t u r e 3. Solubility in super-critical w a t e r 4. Degree of supersaturation 5. Degree of supercooling 6. Purity of N u t r i e n t

291

The last t h r e e factors r e q u i r e s o m e f u r t h e r c o n s i d e r a t i o n . We will d i s c u s s t h e s e in light of t h e g r o w t h of quartz, since it is this crystal for w h i c h t h e m o s t e x p e r i e n c e h a s b e e n gained. The r a t e of crystal g r o w t h is a f u n c t i o n of s e e d - c r y s t a l orientation. The r a t e of growth, r h k l , m a y vary several orders

of m a g n i t u d e ,

depending

upon

the

{hkl}

plane

orientation.

However, a c e r t a i n degree of s u p e r s a t u r a t i o n is m a n d a t o r y as the n u t r i e n t dissolves, Otherwise,

passes the

through nutrient

the would

baffle,

and

moves

to

precipitate

before

reaching

the

seed

area.

the

seed

crystals. It h a s b e e n d e t e r m i n e d t h a t t h e r a t e of crystal g r o w t h is a d i r e c t function of the degree of s u p e r s a t u r a t i o n , A Sat: 6.11.5.-

rhkl

where ~hkt

=

(x - k h k t

" A Sat

is the s e e d o r i e n t a t i o n factor a n d a is a c o n s t a n t d e p e n d e n t

u p o n the crystal s y s t e m . However, it h a s b e e n found t h a t the d e g r e e - o f s u p e r c o o l i n g factor is c o n t r a - i n d i c a t i v e to t h e g r o w t h rate. To u n d e r s t a n d this, i m a g i n e the face of a growing crystal. Rejection of i m p u r i t i e s at t h e growing interface c a u s e s a localized i n c r e a s e of i m p u r i t y c o n c e n t r a t i o n . This is s h o w n in t h e following d i a g r a m :

If the s u p e r i m p o s e d t e m p e r a t u r e g r a d i e n t is Ta, i.e.- d T / d l is a c o n s t a n t , a n d Te

is the e q u i l i b r i u m curve, i.e.- d C / d T e , t h e n s u p e r c o o l i n g will

292

occur. The rate of d e p o s i t i o n t h e r e b y c h a n g e s . If it b e c o m e s slower, crystal g r o w t h slows w h e r e a s if it s p e e d s up, t h e n d e n d r i t i c g r o w t h o r faceting occurs. Note t h a t the impurity- rejection m e c h a n i s m , o b s e r v e d for the case of Molten Flux growth, also o c c u r s h e r e . For all t h e s e r e a s o n s , p u r i t y of m a t e r i a l s is one of, if n o t the, i m p o r t a n t of the p a r a m e t e r s controlling h y d r o t h e r m a l g r o w t h . 6-12:

most

VAPOR METHODS USED FOR SINGLE CRYSTAL GROWTH

If the m a t e r i a l w h o s e single crystal we w a n t is volatile or sublimable, t h e n we m a y choose a v a p o r - m e t h o d of crystal growth. T h e s e m e t h o d s have b e e n u s e d for a variety of crystals i n c l u d i n g ZnS a n d CdS. In this m e t h o d , a carrier- gas is m o s t often u s e d for m a t e r i a l t r a n s p o r t a n d for t h e sulfides, H2S is the gas of choice. The following s h o w s a simple apparatus:

In this m e t h o d ,

we s t a r t the gas flow a n d t h e n begin heating.

It is

i m p o r t a n t to h e a t the m a t e r i a l to j u s t below the p o i n t of s u b l i m a t i o n (volatilization) a n d let the s y s t e m come to equilibrium. Gas flow n e e d n o t be rapid b u t n e e d s to be sufficient to carry the volatiles to the cooler p a r t of the furnace. The t u b e u s e d in the furnace is m o s t often a silica-tube, a l t h o u g h m e t a l t u b e s have s o m e t i m e s b e e n used. The choice u p o n the n a t u r e of the m a t e r i a l b e i n g s u b l i m e d a n d crystallized.

depends

293

For ZnS an d CdS, it is i m p o r t a n t to exclude all traces of oxygen since these materials are easily oxidized: 6.12.2.-

2 ZnS + 3 02

= 2 ZnO + SO2

The operating p a r a m e t e r s for the vapor p h a s e m e t h o d of growing crystals are s h o wn in the following: 6.12.3. - Operating P a r a m e t e r s for Vapor Phase Growth of Crystals i. T e m p e r a t u r e of sublimation (volatilization) of m a t e r i a l 2. A m o u n t of gas flow u s e d 3. Degree of furnace- t e m p e r a t u r e set- point above material sublimation t e m p e r a t u r e . 4. T e m p e r a t u r e gr a di e nt at crystal-growing j u n c t i o n of tube. Obviously, w h e t h e r the material has a low (high vapor pressure) or a hi gh t e m p e r a t u r e of sublimation is i m p o r t a n t because of furnace- c o n s t r u c t i o n material considerations. In general, we c a n n o t use this m e t h o d for materials wh ich are volatile above about 1200 ~

because m o s t m a t e r i a l s

of c o n s t r u c t i o n c a n n o t w i t h s t a n d the corrosive n a t u r e of the vapors at this high t e m p e r a t u r e . If we set the t e m p e r a t u r e of the furnace above t h e sublimation point of the material, it will volatize all at o n c e . This w oul d n eces s itate

setting

the

gas-

flow

such

that

all of the

material

is

t r a n s p o r t e d by the gas. It would be b e t t e r to set the t e m p e r a t u r e j ust below the vaporization (sublimation) point. There will be enough m a t e r i a l t r a n s p o r t e d to begin growth of single crystals. In fact, it has b e e n determined obtained.

th at the

slower

the

growth,

the b e t t e r

are the

crystals

There is a n o t h e r m e t h o d t h a t has been s o m e t i m e s employed in the vapor p h a s e growth of crystals. This m e t h o d uses an evacuated capsule as s h o w n in 6.12.4., given on the next page. The capsule is generally m a d e f r o m quartz, although p l a t i n u m is s o m e t i m e s used. T h e capsule n e e d s to b e e~-acuatcxl t o r e m o v e any r e s i d u a l g a s l ~ f o r e h e a t i I ~ i s s t a r t e d . Otherwise, the internal p r e s s u r e would build until the capsule would explode.

294

Even t h o u g h it is evacuated, c a p s u l e s have b e e n k n o w n to explode b e c a u s e the q u a r t z (metal) walls could not c o n t a i n the i n t e r n a l vapor p r e s s u r e of the m a t e r i a l being grown as single crystal. Care m u s t be e x e r c i s e d n o t t o h a n d l e t h e h o t eJ~am~le before a n d after crystal g r o w t h . In addition

to s u b l i m a t i o n or vaporization, we c a n

also u s e c h e m i c a l

t r a n s p o r t as a m e t h o d of single crystal growth. For example, we could use e i t h e r of the a p p a r a t i of 6.12.1. or 6.12.4. to grow a crystal of ZnCI2 by the following r e a c t i o n s : 6.12.5.-

ZnO+3HI

= ZnI2 il + H 2 0

Zn + I2

= ZnI2

Metal c a r b o n y l s are also c o n v e n i e n t for g r o w t h of c e r t a i n METAL single crystals, vis6.12.6.-

M(s} + x OD(g}

~

M(CO(g})x

M(CO{g})x

~

M(s} + x CO(g}

In t h e s e cases, one is limited

to the g r o w t h

of single c r y s t a l s of t h e

t r a n s i t i o n m e t a l s since it is t h o s e m e t a l s w h i c h form volatile c a r b o n y l c o m p o u n d s . The alkali m e t a l s , alkaline e a r t h m e t a l s a n d c e r t a i n of t h o s e e l e m e n t s w h i c h are allotropic in n a t u r e are not at all s u i t e d for this t y p e of crystal g r o w t h .

295

6.13.- EDGE DEFINED CRYSTAL GROWTH Many times, the form of the crystal o b t a i n e d is n o t s u i t e d for the e n d use c o n t e m p l a t e d . For example, a fiat plate of ~ - A1203 is often u s e d as a base for i n t e g r a t e d circuits (IC's) b e c a u s e of its high t h e r m a l c o n d u c t i v i t y a n d its low electrical conductivity. Si is vapor d e p o s i t e d on t h e surface of t h e A l u m i n a plate, a n d t h r o u g h various p h o t o g r a p h i c t e c h n i q u e s a n d s e l e c t i v e etching

(with

integrated

suitable

circuit

additives

diffused

is built up. T h e

into

IC relies

the

silicon

upon

the

layer),

high

an

thermal

c o n d u c t i v i t y of the A l u m i n a b a s e for its long life since "hot-spots" c a n quickly d e s t r o y the IC. One w a y to m a k e t h e s e p l a t e s c o n s i s t s of pulling an ( z - AI203 crystal by Czochralskd m e a n s , slicing it into wafers, p o l i s h i n g b o t h sides of the wafers a n d t h e n c u t t i n g e a c h one into s m a l l e r fiat p l a t e s of the r e q u i r e d size. cr

A1203 ( c o r u n d u m ) is e x t r e m e l y h a r d (9.5 on the Mohs scale) a n d is

difficult to w o r k with. It w o u l d be m u c h easier if we could grow (~ - •1203 directly as a fiat plate. By u s i n g e d g e d e f i n e d g r o w t h , one c a n do this. A typical a p p a r a t u s is s h o w n as follows:

The a p p a r a t u s c o n s i s t s of a n o r m a l Czochralski m e l t w i t h an anvil at t h e surface of the melt.

Once the crystal h a s s t a r t e d

to grow, vce pull it

t h r o u g h the anvil, t h u s defining its size. Once it is in the form of a s t r i p ,

296

as s h o w n below, it c a n be d r a w n a n d w o u n d over a large wheel. The s t r i p is later cut directly into plates, with no p o l i s h i n g r e q u i r e d . Because of t h e high m e l t i n g point of ~-A1203 (M.P. = 1920 ~ an i r i d i u m crucible a n d anvil are needed.

But, the m e t h o d

is m o r e

versatile

t h a n we

might

s u p p o s e . We c a n grow crystals in the form of tubing, r o d s or strip, as s h o w n by the anvils at the right.

In fact, we c a n grow in n e a r l y any

configuration we m i g h t wish. T u b i n g in the form of a - Al203 is u s e d for the c o n s t r u c t i o n of the familiar high- p r e s s u r e s o d i u m - vapor l a m p s u s e d for street- lighting. End- c a p s of n i o b i u m m e t a l are sealed on the tubing. The c a p p e d t u b i n g is evacuated, s o d i u m m e t a l is a d d e d a n d the whole sealed off a n d m o u n t e d . O p e r a t i o n of the l a m p o c c u r s a t - -

800

~

a n d about

15 a t m o s p h e r e s

internal

p r e s s u r e of s o d i u m vapor. T h e s e o p e r a t i n g c o n d i t i o n s m a n d a t e the u s e of a t r a n s p a r e n t , chemically- a n d t h e r m a l l y - stable t u b i n g s u c h as a - A1203. In fact, no o t h e r m a t e r i a l is k n o w n t h a t will successfully w i t h s t a n d t h e s e operating conditions. 6 . 1 4 . - MELTING AND STOICHIOMETRY Although we have d e s c r i b e d the u s e of m e l t s to grow single crystals, w e have said little

concerning

how

such

melts

are

obtained

and

their

s t o i c h i o m e t r y . In general, one a d d s the c o r r e c t ratio of oxides a n d m e l t s t h e m to obtain a m e l t of the

compound desired.

We m a y also p r e - r e a c t t h e

oxides to form the c o m p o u n d a n d t h e n melt it. W h e n a crystal is g r o w n from either

melt,

little

difference

can be n o t e d

in individual crystal

p r o p e r t i e s . However, this applies only to c o n g r u e n t l y meltimg c o m p o u n d s . What this m e a n s is t h a t the m e l t r e t a i n s the s a m e c h e m i c a l s t o i c h i o m e t r y as the crystal u s e d to m a k e the melt. As a m a t t e r of fact, it is possible to use an excess of cations, or a n i o n s (as oxides), in the m e l t a n d still obtain a

stoichiometric

crystal.

There

are

some

cases

where

a

non-

s t o i c h i o m e t r i c c o m p o s i t i o n will be o b t a i n e d even t h o u g h the c o m p o u n d is c o n g r u e n t l y - melting. An e x a m p l e of s u c h a p h a s e - s y s t e m is s h o w n in 6.14.1., p r e s e n t e d on the next page. Even t h o u g h this p h a s e d i a g r a m is c o m p l i c a t e d , it is easy to u n d e r s t a n d .

297

6.14.1-

I"Phase .................. Diagram For a Non-Stoichiometric Composition I Congruently Melting Composition

Liquid

r T

L o w Temperature Stoichiometric Composition

i' Change in Composition We s t a r t with c o m p o u n d s

"A" a n d "B", one on e a c h e n d of the p h a s e

diagram, i.e.- 100% A or 100% B. T h e r e are two regions in b e t w e e n t h e s e limits w h e r e solid solutions of A & B occur. More i m p o r t a n t l y , a r e g i o n exists w h e r e the c o m p o s i t i o n freezes a n d m e l t s c o n g r u e n t l y (that is- t h e m e l t does n o t d e c o m p o s e . W h e n it melts, however,

the m e l t t h e n d e c o m p o s e s

into two s e p a r a t e

p h a s e s . On e i t h e r side of t h e c o n g r u e n t l y - m e l t i n g c o m p o s i t i o n is a p h a s e

298

region of p a r t i a l - solid- solution, In t h o s e regions, one solid freezes b e f o r e the other, so t h a t it is i m p o s s i b l e to grow a crystal from a m e l t of t h e s e m i x e d oxides. Note t h a t even at the eutectic c o m p o s i t i o n

(there are two

of them), one still obtains a m i x t u r e s of oxides. However,

t h e r e is a c o n g r u e n t l y -

compositions

a s s o c i a t e d with

melting

it. The

composition

a n d a r a n g e of

low t e m p e r a t u r e

stoichiometrie

c o m p o s i t i o n m e l t s i n c o n g r u e n t l y b u t it is not easily achieved from t h e melt. It shifts c o m p o s i t i o n u p o n m e l t i n g or freezing! It t u r n s o u t t h a t t h i s behavior is m o r e p r e v a l e n t t h a n one m i g h t realize. It is, in fact, typical for m a n y s y s t e m s a n d s o m e of t h e s e are listed in Table 6-3, s h o w n on t h e n e x t page. The m a x i m u m c o m p o s i t i o n -

existence

r a n g e is given along with t h e i r

d y s t e e t i e c o m p o s i t i o n , t h a t is, the m o s t usual c o m p o s i t i o n . A closer e x a m i n a t i o n of Table 6-3 reveals a r a t h e r s t a r t l i n g conclusion. It c a n be s e e n t h a t m o s t of t h e s e

compounds

are s u b j e c t to s t r u c t u r e -

v a c a n c y defects. Certainly all of t h e p n i c t i d e s fall into this class. We are even given the defect c o n c e n t r a t i o n . T h u s in GaP, we have: 6.14.2.where freezing

6 pp 5

--7

is usually 0 . 0 0 0 0 5 . behavior

stoichiometry.

of

a

However,

defects in the crystal

PGa + ( 1 + 6 ) Vp + (xVi We c a n c o n c l u d e

compound

has

a

t h a t the

significant

melting effect

of

and its

it a p p e a r s to m a n i f e s t itself m a i n l y as p o i n t

produced.

The e s s e n c e of this d i s c u s s i o n is t h a t

a l t h o u g h we m e l t the c o m p o n e n t s , w h e t h e r t h e y be oxides or o t h e r w i s e , to f o r m a d e s i r e d c o m p o u n d or c o m p o s i t i o n , t h e r e are m a n y c a s e s w h e r e we do not obtain the s t o i c h i o m e t r y we e x p e c t to get. This is g r a p h i c a l l y i l l u s t r a t e d w h e n we c o n s i d e r the t y p e s of defects t h a t are actually o b t a i n e d in single c r y s t a l s as t h e y are b e i n g g r o w n .

299

T A B L E 6- 3 N o n s t o i c h i o m e t r i c N a t u r e of S o m e B i n a r y C o n g r u e n t l y Melting Compounds COMPOUND

MAX. E X I S T E N C E R A N G E

DYSTECTIC COMPOSITION

G ~ . 5 + x Po.5-x

+ 0.00025

to - 0 . 0 0 0 0 5

+ 0.00006

Gao.5+x As0.5-x

+ 0.00023

to - 0 . 0 0 0 1 8

+ 0.00004

Pb0.5+xS0.5-x

+ 0.0005

Pb0.5+x Sc0.5-x

to - 0 . 0 0 0 0 5

NOT KNOWN

+ 0.0005 - 0.00005

Pbo.5+x T eo.5-x

+ 0 . 0 0 0 5 to - 0 . 0 0 0 1 3

- 0.00012

Sno.5+xSo.5-x

+ 0

- 0.000005

Cdo.5+xTeo.5-x

+ 0.000002

to - 0 . 0 0 0 0 0 8 to - 0 . 0 0 0 0 0 8

NOT K N O ~

Mgl+xA12-2xO4-2x + 0 . 4 7

to

Gd3+xGa5_x O12

+ 0.30

to - 0 . 0 0 2

Lil_5xNb l+xO3

+ 0 . 0 3 1 to O-

+ 0.0092

Lil-5xTal+xO3

+ 0 . 0 2 5 to O

+ 0.0066

6.- 15:

- 0.42

- 0.000005

+ 0.05

ACTUAL I M P E R F E C T I O N S IN S I N G L E C R Y S T A I ~

Now we can summarize

all of t h e i m p e r f e c t i o n s

l i k e l y to a p p e a r in s i n g l e

c r y s t a l s . S o m e o f t h e s e , p a r t i c u l a r l y s t a c k i n g - fmalts, w e r e d i s c o v e r e d o n l y w h e n s i n g l e c r y s t a l s w e r e g r o w n l a r g e e n o u g h so t h a t t h e d e v i a t i o n f r o m long range order became apparent. T h e i n t r i n s i c d e f e c t s f o u n d in s i n g l e c r y s t a l s i n c l u d e v a c a n c i e s ,

300

interstitials, i m p u r i t i e s a n d i m p u r i t y c o m p e n s a t i o n s , reverse order, a n d c o m b i n a t i o n s s u c h as V- S a n d I- S, etc. T h e i r n u m b e r s are well d e s c r i b e d by: 6.15.1.-

Ni

= No exp - A Ei / k T

w h e r e i refers to the intrinsic defect. In addition, we have dislocations, b o t h edge (line) a n d spiral (screw), b o t h of w h i c h are three- d i m e n s i o n a l . Their n u m b e r s are well d e s c r i b e d by: 6.15.2.-

A Gi =

n AHi

-

T (ASconfig +

n ASvib)

This e q u a t i o n arises b e c a u s e b o t h of t h e s e e ~ ' i n a i c

defects

affect t h e

e n e r g y of the crystal. We can also have grain b o u n d a r i e s w h i c h m a y be: c l u s t e r i n g of line defects or m o s a i c blocks. The latter m a y be r e g a r d e d as very large g r a i n s in a crystallite. Another

imperfection

in crystals

is called

"twinning".

This

usually

h a p p e n s w h e n e m m t i o m o r p h s are p r e s e n t , or possible. A good e x a m p l e is quartz, i.e.6 . 1 5 . 3 .-

a-quartz

"---v

~- q u a r t z

573 ~ In this case, two crystals grow a n d are j o i n e d at a given plane, each b e i n g a m i r r o r image of the o t h e r . The o t h e r crystal i m p e r f e c t i o n we have not covered is "stacking faults". A good e x a m p l e is SiC. Here, we have two mxblattices, one b a s e d on Si a n d the o t h e r on C, each of w h i c h is hexagonal. In s t a c k i n g a l t ~ t e

layers of

identical atoms, we first s t a c k Si a n d t h e n C. If we refer to "A" as the 1st layer, "B" as the 2 n d layer, a n d "C" as the 3rd layer, t h e n the n o r m a l s t a c k i n g s e q u e n c e for the h e x a g o n a l lattice is: 6.15.4.-

AI~-

AI~-

AI~

301

w h e r e we have s h o w n t h r e e s e q u e n c e s . For SiC, "A" in

t h e first s e q u e n c e

is Si, "B" is C, a n d "C" is Si. We still h a v e t h e s a m e p a c k i n g s i n c e o n l y "A" a t o m s are over "A" a t o m s , as s h o w n in t h e following diagrmm: 6.15.5.-

JSTACKING SEQUENCES IN THE HEXAGONAL LATTICE I

.LAYERS

A

.....

C

'

B A

,

C

....

B

A

A

A C

-

C

oTA

A

w

C A

A '

C

,

B A

B A

B

'

C

B A

B

A

C

B A

B

A

C

B A

....

A

C A

A

B

C

......... A

C

B A

8o

Here, we have a r r a n g e d t h e l a y e r s o n a t w o - d i m e n s i o n a l s t r u c t u r e , e v e n t h o u g h t h e layers are a r r a n g e d in t h r e e d i m e n s i o n a l o r d e r . Note t h a t only two c r y s t a l l o g r a p h i c a x e s are i n d i c a t e d , We call t h i s t h e n a t u r a l s t a c k i n g s e q u e n c e b e c a u s e of t h e n a t u r e of t h e h e x a g o n a l close- p a c k e d l a t t i c e . However, SiC also e x h i b i t s o t h e r s t a c k i n g s e q u e n c e s , as s h o w n in 6 . 1 5 . 6 . , given o n t h e n e x t page. T h e s e a r r a n g e d l a y e r s are called "polytypes" a n d are p r e v a l e n t w h e r e s i m p l e c o m p o u n d s s u c h as SiC a n d SiN are involved. In m a n y cases, t h e p r o p e r t i e s

of s u c h c o m p o u n d s

depend,

to a l a r g e

e x t e n t , u p o n t h e specific s t a c k e d layers o b t a i n e d d u r i n g f o r m a t i o n . 6.15.6.

p - s~c

-

ABC A t ~

SiC - 4H

=

ABCA ABCA

SIC - 6 H

=

ABCACB AI~ACB

S i C - 15R

= AI~BACABACBCACBABCBACABACBCACB

For SiC, w e c a n also h a v e "polytypes" w h e r e two s t a c k i n g s e q u e n c e s l i k e 4H - 6H c a n c o m b i n e to f o r m a u n i t . A n o t h e r " p o l y m o r p h " is 4H - 1 5 R .

302

This p h e n o m e n o n h a s b e e n t h o r o u g h l y s t u d i e d a n d p o l y m o r p h s of

87R

a n d 2 7 0 R have b e e n r e p o r t e d . A n o t h e r t y p e of s t a c k i n g fault is called

"polystructure".

A good e x a m p l e is

ZnS, w h i c h is d i m o r p h i c (has two forms). The cubic form of ZnS is called sphalerite, w h e r e a s the h e x a g o n a l form is called w u r t z i t e (These are t h e i r mineral

names,

after

the

first

geologist w h o

discovered

them).

The

s t a c k i n g s e q u e n c e for s p h a l e r i t e is: AB or ABBA; t h a t for w u r t z i t e is: ABC. A polystructure

sometimes

results

when

sphalerite

is c o n v e r t e d

to

wurtzite: ~

6.15.7.-

ZnS cubic

1200 o c.

=

Z n S hexagonal

The p o l y s t r u c t u r e s e q u e n c e is: ABC ...... AB .......A I ~ .....AB ......ABC. This m a y be r e g a r d e d as cubic- close- p a c k i n g w i t h i n a hexagonal- close- p a c k e d s t r u c t u r e . We m a y also note t h a t ZnS is subject to the s a m e "polytype" s t a c k i n g faults as t h o s e given for SiC above. T h u s , the

s t a c k i n g fault

p a t t e r n s n o t e d are a function of the type of lattice involved, not on t h e c h e m i c a l c o m p o s i t i o n of the material. For the m o s t part, t h e s e s t a c k i n g faults are found only in the high s y m m e t r y lattices

s u c h as h e x a g o n a l

c l o s e - p a c k e d a n d cubic c l o s e - p a c k e d s t r u c t u r e s . 6.16-

ELECTRONIC PROPERTIES OF CRYSTAI~

Although we have d e s c r i b e d the g r o w t h of crystals in s o m e detail, we have not c o n s i d e r e d the behavior of e l e c t r o n s in either crystallites or single crystals. It is the electronic p r o p e r t i e s of s u c h m a t e r i a l s t h a t are useful in i n d u s t r i a l applications. Therefore,

it w o u l d behoove us to c o n s i d e r t h e

factors t h a t c a u s e c h a n g e in electronic p r o p e r t i e s of solids as a function of s t r u c t u r e a n d b o n d i n g . It is evident t h a t this aspect is very i m p o r t a n t , since the electrical n a t u r e of solids varies a c c o r d i n g to b o t h c o m p o s i t i o n a n d a r r a n g e m e n t of a t o m s in the s t r u c t u r e . S o m e m a t e r i a l s

are s e m i -

c o n d u c t i n g a n d o t h e r s are dielectric insulators. Still o t h e r s are g o o d c o n d u c t o r s of c h a r g e (metals). Still o t h e r s have u n i q u e optical p r o p e r t i e s , due to a r r a n g e m e n t of e l e c t r o n s a n d a t o m s w i t h i n the solid state.

303

One e x a m p l e of the l a t t e r is the crystal, ~- BaB204. T h i s crystal b e l o n g s to the trigonal s y s t e m w i t h point syrrm~etry of C3v. It is a negative uniaxial crystal a n d its m a i n u s e is to double the frequency of a laser b e a m . T h e r e is a high t e m p e r a t u r e form, a - B a B 2 0 4 {transition t e m p e r a t u r e of the ~ to a - f o r m = 925 ~ b u t the a - f o r m is not optically active. It is n a t u r a l to a s k w h y this s h o u l d be so. In the case of ~- BaB204, we k n o w t h a t w h e n photons

from a l a s e r b e a m t r a v e r s e

the

crystal, m a n y are

absorbed,

c a u s i n g local a t o m s to b e c o m e e n e r g i z e d {i.e.- go into an excited state). B e c a u s e of the p r e v a l e n c e of a d e n s e p h o n o n s p e c t r u m (i.e.- vibrational s t a t e s of the

lattice),

the

photon

emitted

by e a c h

excited

atom

is

e n h a n c e d by e x t r a e n e r g y a d d e d by the p h o n o n states. T h e s e p h o n o n s t a t e s are c o u p l e d to t h e atomic excited s t a t e s in t h e ~ - f o r m b u t not in the ~- form (probably due to the difference

in lattice

symmetry and

s t r u c t u r e ) . The e m i t t e d p h o t o n t h u s is twice as e n e r g e t i c the fact t h a t its f r e q u e n c y is twice

(2X) t h a t of the

absorbed

wavelength

by the

lattice

while

the

is

as evinced by

original

half

photon

(1/2X).

This

p h e n o m e n o n o c c u r s only w i t h c e r t a i n c l a s s e s of lattice s y m m e t r y a n d c e r t a i n c o m p o s i t i o n s of m a t t e r . A m o n g t h e s e are: L i N b 0 3 , K T i P 0 4 , LIB02 a n d K2TiO3. In s e m i - c o n d u c t i n g c o m p o u n d s , we k n o w t h a t s o m e of the e l e c t r o n s f o r m b o n d s b e t w e e n t h e cation a n d t h e anion, e i t h e r as covalent or ionic b o n d s (or s o m e w h e r e in between). W h a t h a p p e n s to the r e s t ? Do t h e y r e m a i n a r o u n d t h e p a r e n t a t o m ? Why are s o m e solids c o n d u c t i v e while o t h e r s are not? T h e following d i s c u s s i o n a d d r e s s e s t h e s e q u e s t i o n s . Obviously, w e c a n n o t be e x h a u s t i v e b u t we c a n e x a m i n e

the m a i n

features

of e a c h

p h e n o m e n o n to s h o w w h a t h a p p e n s in the solid. We will not derive t h e equations

associated with

each

subject.

This

aspect

is left

to

more

a d v a n c e d studies. A. Conductivity..'.m I o n i c C o m p o u n d s We have a l r e a d y d i s c u s s e d diffusion in solids to s o m e degree. While b u l k p r o p e r t i e s s u c h as h e a t c a p a c i t y are not sensitive to defect c o n c e n t r a t i o n , m a n y o t h e r p r o p e r t i e s s u c h as c o n d u c t i v i t y are. T h u s , the m e t h o d of p r e p a r a t i o n b e c o m e s i m p o r t a n t if one w i s h e s to obtain a c o n d u c t i v e o r

304

even a s e m i - c o n d u c t i n g solid. Most materials in I n d u s t r y are polycrystalline r a t h e r t h a n single-crystal because of cost and difficulty of m an u f actu r e. However, only the single-crystal state suffices for s o m e applications. The degree of conductivity of solids is d e p e n d e n t

u p o n at

least two factors, s t r u c t u r e and ionicity. Ceramic materials are notoriously low in conductivity b e c a u s e their b a n d gaps bet w een the valence and c o n d u c t i o n b a n d are relatively large. This t ransl at es to conductivities of 10 -6 or lower. Contrast this to metals where conductivities of 102 are common. Conductivity in solids d e p e n d s u p o n m o v e m e n t of charge t h r o u g h t h e lattice and includes bot h types of charge. In o x y g e n - d o m i n a t e d m a t e r i a l s s u ch as solid ceramics, the diffusion of charge can only involve e l e c t r o n s and the cation t h r o u g h the s t r uct ur e . We have already c o n s i d e r e d

this

aspect in Chapter 3. In certain s t r u c t u r e s , the electrical conductivity associated with ion m ot i on can exceed conductivity from electrons by several orders of m a g n i t u d e . Diffusion and ionic conductivity are r e l a t e d t h r o u g h the N e a r s t - E i n s t e i n equation, vis: 6.16.1.-

o = Dnq 2 kT

w h er e D is the diffusion coefficient, n is the n u m b e r of charge c a r r i e r s per u n it volume, q is the charge per ion, k is Boltzmann's c o n s t a n t and T is the absolute t e m p e r a t u r e . D generally refers to the more rapidly m o v i n g species. However, there are m a n y ambiguities in the use of this equation to characterize the degree of conductivity, especially w h e n a d i r e c t c u r r e n t is involved. Diffusion usually occurs by the m o v e m e n t of ions to a neighboring crystal defect. In a salt crystal, the m e a s u r e d c u r r e n t m ay be i n d u c e d by m o v e m e n t of interstitial ions (Frenkel) or vacancies (Schottky). The position of a defect c h a n g e s w h e n a neighboring ion moves in to Fill it. S c h o t t l ~ defects p r e d o m i n a t e in KCI where bot h cation and anion vacancies occur. In AgCI, some Ag ions occupy interstitial sites p r o d u c i n g positive Frenkel effects and negative S c h o t t k y defects. Nonetheless, for stoichiometric c o m p o u n d s , the vacancy c o n c e n t r a t i o n and ionic conductivity is very low. If one adds certain "dopants", t h e

305

conductivity will c h a n g e a n d t h a t is w h a t is done to p r o d u c e solids w i t h suitable c o n d u c t i v i t y for c e r t a i n electronic devices. To illustrate exactly w h a t we are discussing, c o n s i d e r the following diagram, w h i c h is b a s e d o n a solid array of w a t e r molecules, i.e.- "ice":

Here, we have a t w o - d i m e n s i o n a l c h a i n of w a t e r m o l e c u l e s w h i c h

are

b o n d e d by van der W a ~ s forces b e t w e e n positive a n d negative c h a r g e s of the individual w a t e r dipoles. If an electrical field (via a voltage) is a p p l i e d to ice, c h a r g e c a n move by diffusion of p r o t o n s . E a c h w a t e r m o l e c u l e in ice is s u r r o u n d e d by four (4) n e i g h b o r i n g molecules. H y d r o g e n a t o m s lie n e a r lines joining a d j a c e n t oxygen a t o m s a n d are closer to one t h a n t h e other. Two n e u t r a l i t y c o n d i t i o n s are satisfied: 1) Water m o l e c u l e s are i n t a c t a n d electrically n e u t r a l 2) The integrity of the h y d r o g e n - b o n d i n g n e t w o r k r e m a i n s intact. One a n d only one h y d r o g e n directly c o n n e c t s to two n e i g h b o r i n g oxygen ions. Violation of either n e u t r a l i t y rule p r o d u c e s an electrically-active defect. A positive b o n d i n g defect w o u l d look like this, as s h o w n in t h e following d i a g r a m as 6.16.3. on the next page.

306

If a p r o t o n is t r a n s f e r r e d to a n o t h e r molecule, a positive h y d r o n i u m ion is c r e a t e d along with a negative hydroxyl ion, i.e.: 6.16.4.-

2H20r

H3 O+ + O H -

The d o t t e d line s h o w s the e x t e n t of the positive defect due to p a c k i n g of the w a t e r m o l e c u l e s in a lattice s t r u c t u r e w h i c h violate the h y d r o g e n b o n d i n g n e t w o r k rule. The s a m e c a n be s e e n for the negative b o n d i n g defect:

Note t h a t the actual b o n d i n g c o n d i t i o n is t h r e e - d i m e n s i o n a l .

In b o t h of

t h e s e cases, r o t a t i o n of a w a t e r m o l e c u l e r e s u l t s in either a positive or negative b o n d i n g defect w h i c h is electrically active. In the first case, a

307

positive defect results w h e n 2H are on the same b o n d and a negative defect w h e n no h y d r o g e n s are present. Note t h a t this type of defect d i s r u p t s the s t r u c t u r e to a m u c h greater degree t h a n the b o n d i n g type of defect.

It is possible for the two types of ions, i.e.- h y d r o n i u m and

hydroxyl, to be n e a r e s t neighbors in the lattice s t r u c t u r e (although w e have not s h o wn t h e m as s u c h here). This gives rise to bot h positive and negative ionic defects in the s t r uct ur e, as show n in the following diagram:

The motion of h y d r o n i u m a n d hydroxyl ions leaves molecules o r i e n t e d against the applied electrical field. The molecular dipole m o m e n t is t h e n anti-paraUel to the field. If b o n d i n g defects are p a s s e d by either of t h e h y d r o n i u m and hydroxyl ions, the molecular dipoles are left parallel to t h e field. Bonding defects dom i nat e the polarization in pure ice because t h e y

308

are mo r e n u m e r o u s t h a n ionic defects. C o n c e n t r a t i o n s of b o n d i n g defects average about 10 .6 per mole of ice. Although ionic defects are fewer in n u m b e r t h a n b o n d i n g defects, they have higher mobility (compared to t h e diffusion m e c h a n i s m of b o n d i n g defects). Thus, the t r a n s p o r t of c h a r g e due to ionic defects is about equal to t ha t actually t r a n s p o r t e d by b o n d i n g defects, w h e n a direct c u r r e n t is applied to ice. Our motive in p r e s e n t i n g this analysis of the m ot i on of charge in ice w h e n an electrical voltage is applied has been designed to acquaint you with t h e complexities of m ot i on of charge in a crystalline solid. Motion of charge in an a m o r p h o u s solid s u c h as glass is even slower t h a n m o s t crystal solids. Solids are divided into three groups bas e d u p o n their conductivity, as w e have already discussed. They are: metals, semi-conductors and insulators.Aqueous solutions are the ot her class of c o n d u c t o r s w hi ch can be u s e d to carry a c u r r e n t of electrons. Metals carry charge because t h e i r valence and c o n d u c t i o n b a n d s overlap, allowing electrons to be easily t r a n s p o r t e d . S e m i - c o n d u c t o r s have a n a r r o w b a n d gap bet w een the two energy b a n d s in the solid and a slight electrical

field is all t hat is

n e c e s s a r y for t h e m to t r a n s p o r t c u r r e n t (electrons). 6.17.- Silicon Single Crystals and I nt e gr at ed Circuits in C o m m e r c e We have already referred to the use of quartz crystals as u s e d in "quartzcrystal" w a t c h e s and timepieces. However, the single crystal grown in largest

quantity

is

silicon

(Si).

Single-crystal

silicon

is

endemic

t h r o u g h o u t our culture and is responsible for the c u r r e n t "InformationAge" e n g e n d e r e d by the use of c o m p u t e r s and ot her e l e c t r o n i c e q u i p m e n t . However, growing defect-free silicon crystals is not an easy task. A. Silic0n Silicon (Si) is a nonmetallic chemical e l e m e n t of the carbon family (Group IVa of the periodic table) and m a k e s up 27.7 p e r c e n t of the E a r t h ' s crust. It is the s eco n d m o s t a b u n d a n t el em e nt in the crust, being s u r p a s s e d only by oxygen.

309

Its e l e c t r o n i c p r o p e r t i e s are: Atomic n u m b e r = 14; atomic w e i g h t = 2 8 . 0 9 Melting p o i n t = 1410 ~ boiling p o i n t = 2 3 5 5 ~ D e n s i t y = 2.33 g / c m S ; oxidation s t a t e = - 4 & +4 E l e c t r o n c o n f i g u r a t i o n = 1 s 2 2 s 2 2 p 2 3 s 2 3 p 2. Silicon w a s first isolated a n d d e s c r i b e d as an e l e m e n t in 1824 b y J 0 n s J a c o b Berzelius, a S w e d i s h c h e m i s t . Silicon does n o t o c c u r u n c o m b i n e d in n a t u r e , i.e.- as a n e l e m e n t . It is f o u n d in p r a c t i c a l l y all r o c k s as well as in sand, clays, a n d softs, c o m b i n e d e i t h e r w i t h oxygen as silica (SiO2= silicon dioxide) or w i t h oxygen p l u s o t h e r e l e m e n t s (e.g., a l u m i n u m , m a g n e s i u m , calcium,

sodium,

p o t a s s i u m , or iron)

as silicates.

Its c o m p o u n d s

also

o c c u r in all n a t u r a l w a t e r s , in t h e a t m o s p h e r e (as siliceous dust), in m a n y plants, a n d in the s k e l e t o n s , tissues, a n d b o d y fluids of s o m e animals. P u r e e l e m e n t a l silicon is a h a r d , d a r k g r a y solid w i t h a metallic l u s t e r a n d with a crystalline s t r u c t u r e the s a m e as t h a t of the d i a m o n d

form of

carbon. For this r e a s o n , silicon s h o w s m a n y c h e m i c a l a n d p h y s i c a l similarities. T h e r e is also a b r o w n , p o w d e r y form of silicon h a v i n g a m i c r o c r y s t a l l i n e form. The e l e m e n t is p r e p a r e d c o m m e r c i a l l y by r e d u c i n g the oxide by r e a c t i n g it w i t h c a r b o n (as coke) in electric f u r n a c e s . On a small scale, silicon h a s b e e n o b t a i n e d from the oxide by r e d u c t i o n w i t h aluminum metal. Silicon, like carbon, is relatively inactive at o r d i n a r y t e m p e r a t u r e s .

But,

w h e n h e a t e d , it r e a c t s vigorously w i t h the h a l o g e n s (fluorine, c h l o r i n e , bromine,

a n d iodine)

to form halides a n d with c e r t a i n m e t a l s to f o r m

silicides. It is u n a f f e c t e d by all acids e x c e p t hydrofluoric. At r e d heat, silicon is a t t a c k e d by w a t e r v a p o r or by oxygen, forming a surface layer of silicon dioxide. W h e n silicon a n d c a r b o n are c o m b i n e d at electric f u r n a c e t e m p e r a t u r e s of 2 , 0 0 0 to 2 , 6 0 0 ~ carbide (Carborundum reacted wi~-hydrogen,

TM

( 3 , 6 0 0 to 4 7 0 0 OF), t h e y form silicon

= SIC), w h i c h is a n i m p o r t a n t

abrasive. Vv"nen

silicon forms a series of h y d r i d e s ,

the sflanes.

Silicon also f o r m s a series of organic silicon c o m p o u n d s called silicones, w h e n r e a c t e d with various organic c o m p o u n d s .

310

Sflicon's

atomic

semiconductor.

structure

makes

Highly purified

silicon,

it

an

doped

extremely with

important

such elements

as

boron, p h o s p h o r u s , a n d arsenic, is the basic m a t e r i a l u s e d in c o m p u t e r chips, t r a n s i s t o r s , silicon diodes, a n d various o t h e r e l e c t r o n i c circuits a n d electrical-current

s w i t c h i n g devices. Silicon of l e s s e r p u r i t y is u s e d in

m e t a l l u r g y as a r e d u c i n g a g e n t a n d as an alloying e l e m e n t in steel, brass, and bronze. The m o s t i m p o r t a n t c o m p o u n d s of silicon are the dioxide (silica) a n d t h e various silicates. Silica in the form of s a n d a n d clay is u s e d to m a k e c o n c r e t e a n d b r i c k s as well as r e f r a c t o r y m a t e r i a l s for h i g h - t e m p e r a t u r e applications. T h e m i n e r a l quartz, i.e.- Si02, m a y be s o f t e n e d by h e a t i n g a n d s h a p e d into glassware. Silicates, m o s t of w h i c h are insoluble in w a t e r , are e m p l o y e d in malOalg glass as well as in the fabrication of e n a m e l s , pottery, china, a n d o t h e r c e r a m i c m a t e r i a l s . S o d i u m silicates, c o m m o n l y known

as w a t e r glass, or silicate of soda, are u s e d

in soaps,

in t h e

t r e a t m e n t of wood to p r e v e n t decay, for the p r e s e r v a t i o n of eggs, as a cement,

a n d in dyeing.

Silicones

are

synthetic

organo-silicon

oxides

c o m p o s e d of the e l e m e n t s silicon, oxygen, carbon, a n d h y d r o g e n . T h e y are

used

as

lubricants,

hydraulic

fluids,

waterproofing

compounds,

v a r n i s h e s , a n d e n a m e l s b e c a u s e , as a class, t h e y are c h e m i c a l l y i n e r t a n d unusually stable at high t e m p e r a t u r e s . T h r e e stable isotopes of silicon are known: silicon-28, w h i c h m a k e s up 92.21 p e r c e n t of the e l e m e n t in n a t u r e ; silicon-29 =4.70%; a n d silicon-30 = 3.09 %. Five r a d i o a c t i v e isotopes are k n o w n . B. Silicon as a S e m i - C o n d u c t o r In s e m i c o n d u c t o r s s u c h as silicon, e a c h a t o m in the s t r u c t u r a l lattice has four o u t e r electrons, e a c h of w h i c h covalently p a i r s w i t h an e l e c t r o n f r o m one of the four n e i g h b o r i n g a t o m s to form the i n t e r a t o m i c b o n d s , i.e.- t h e "diamond"

structure.

Completely

pure

silicon t h u s h a s e s s e n t i a l l y no

e l e c t r o n s available at r o o m t e m p e r a t u r e for e l e c t r o n c o n d u c t i o n , m a k i n g it a v e r y p o o r c o n d u c t o r . However, the key is g e t t i n g the silicon p u r e e n o u g h . Originally, silicon w a s t h o u g h t to be a n a t u r a l s e m i - c o n d u c t o r until really p u r e silicon b e c a m e available.

311

Growth of silicon crystals begins with the p r e p a r a t i o n of extremely p u r e polycrystalline silicon having fewer t h a n 1 d o p a n t a t o m per 10 billion silicon atoms. This

silicon is m e l t e d

in a quartz-lined

furnace. T h e

t e m p e r a t u r e of the m o l t e n silicon is r e d u c e d to j u s t above the m e l t i n g point, an d a small bar (the seed)

of silicon in single-crystal form is

i n t r o d u c e d into the surface of the melt. The m o l t e n silicon freezes slowly onto the seed with a crystalline s t r u c t u r e t h a t is contiguous with t h e s t r u c t u r e of the seed (For m o r e details, see the Czochralski m e t h o d given above). The seed is slowly w i t h d r a w n , while rotating u n d e r carefully controlled conditions, to form a single-crystal cylindrical ingot of silicon. This ingot is greater t h a n 200 m i l l i m et ers in d i a m e t e r a n d weighs up to 100 kilograms (220 pounds). After growth, the silicon crystal is g r o u n d to a s m o o t h cylinch'ical s h a p e and sliced into thin wafers, using d i a m o n d tools. These wafers are usually about 0.6 millimeter thick. The surfaces of the wafers are polished fiat by a series of successively finer abrasives until one side has a perfect m i r r o r finish. For the m a n u f a c t u r e of i n t e g r a t e d circuits, the m e t h o d s u s e d to achieve planarity and s m o o t h n e s s of t h e individual discs are extremely i m p o r t a n t . We will discuss these below. If an a t o m from c o l u m n V of the periodic

table, s u c h as e l e m e n t a l

p h o s p h o r u s , is s u b s t i t u t e d for an atom of silicon, four of its five o u t e r electrons are us e d for b o n d i n g in the t e t r a h e d r a l lattice, b u t the fifth is free to move within the crystal (which t h e n b e c o m e s a "negative-type" semi-conductor). If the r e p l a c e m e n t a t o m comes from c o l u m n III of t h e periodic table- s u c h as boron, it will have only three out er electrons, i.e.one too few to complete the four i n t e r a t o m i c bonds. Since the crystal a t t e m p t s to become electrically neut r a l were

this b o n d complete,

the

vacancy acquires a positive charge because an electron is m i s s i n g in t h e overall electronic s t r uct ur e . A neighboring electron can move into t h e vacancy, leaving a n o t h e r vacancy in the space previously occupied by t hat electron. This -vacancy, with its positive charge, is t h u s mobile and is called

a "hole."

Holes in

semiconductors

move

about as readily

as

electrons do, but because they are positively charged, they obviously m o v e in a direction opposite to the m o t i o n of electrons. This form of silicon is called a "positive-type" s e m i c o n d u c t o r .

312

The p r o c e s s of s u b s t i t u t i n g e l e m e n t s

for the silicon is called d o p i n g ,

while the e l e m e n t s are r e f e r r e d to as d o p a n t s . The a m o u n t of d o p a n t t h a t is r e q u i r e d dopant

in practical devices is very small, r a n g i n g from about 1 0 0

a t o m s per

million

silicon

atoms

downward

to

1 per

billion.

D o p a n t s are usually a d d e d to the silicon after the crystal g r o w t h p r o c e s s , w h e n a n i n t e g r a t e d circuit is b e i n g f o r m e d on the surface of the wafer. C. S e m i c o n d u c t o r Devices The fact t h a t p u r e silicon is not a c o n d u c t o r b u t is "doped" to m a k e it a good s e m i - c o n d u c t o r

might

s e e m odd,

p a r t i c u l a r l y since c o n s i d e r a b l e

effort goes into o b t a i n i n g very p u r e silicon. N o n e t h e l e s s ,

the u n d o p e d

silicon is n o t conductive. This allows one to form d i s c r e t e a r e a s of n - t y p e a n d p-type silicon o n the s a m e wafer a n d to position t h e s e in c o n j u n c t i o n with e a c h o t h e r in a m a n n e r so t h a t a c u r r e n t amplifying device r e s u l t s . An e x a m p l e of s u c h a device is s h o w n in the following diagram:

Here, we s h o w a so-called " c o m m o n - e m i t t e r "

configuration of a p - n - p

t r a n s i s t o r , i.e.- the e m i t t e r s t a t e s are c o u p l e d together. In general, s u c h a

313

configuration w o u l d be f a b r i c a t e d by forming the p- a n d n - t y p e s of silicon t h r o u g h injection

of the p r o p e r

i m p u r i t y at n e i g h b o r i n g

sites on t h e

surface of the silicon wafer. As s h o w n in the diagram, a voltage is a p p l i e d between

the p o i n t s E & B (where

E is electrically c o n n e c t e d

to t h e

e m i t t e r p o r t i o n of the p - t y p e a r e a s a n d B is c o n n e c t e d to the b a s e (or ntype) area). The c o n n e c t i o n s are usually c o n d u c t i v e lines p r i n t e d on t h e surface of t h e n o n - c o n d u c t i v e Si-wafer. One p - t y p e a r e a is u s e d as the e m i t t e r ( h o l e - c o n d u c t i o n ) while the o t h e r is u s e d as the collector (of c u r r e n t ) w h e n the i n p u t voltage is a p p l i e d b e t w e e n p o i n t s B & E at t h e t r a n s i s t o r . T h i s r e s u l t s in a c u r r e n t , I B , f r o m the b a s e as well as a c u r r e n t ,

IC , from t h e collector.

T h u s two local

circuits are at work, one t h r o u g h the e m i t t e r - b a s e a n d t h e o t h e r t h r o u g h the base-collector. The i m p o r t a n t p a r t to notice is t h a t a l t h o u g h t h e c u r r e n t , I B , is in m i c r o - a m p s , the c u r r e n t , I c , is in m i l l i a m p s , an amplification of 1000 x. T h u s , the p - n - p c o n f i g u r a t i o n on the surface of a silicon

wafer

results

in

an

amplifier

of c u r r e n t

(also

known

as

a

transistor). A n u m b e r of d i s c r e t e s t e p s are r e q u i r e d to m a k e s u c h a t r a n s i s t o r a n d passive p a r t s s u c h as r e s i s t e r s a n d c o n d e n s o r s

are also a d d e d to m a k e

w h a t is called an " i n t e g r a t e d circuit" or IC. D. I n t e g r a t e d C i r c u i t s An i n t e g r a t e d circuit (IC) is a m o n o l i t h i c a s s e m b l y of electrically i s o l a t e d circuit e l e m e n t s . W h a t this m e a n s is t h a t e a c h circuit e l e m e n t is f o r m e d on top o f , or b e n e a t h , o t h e r circuit e l e m e n t s to f o r m a c o m p a c t assembly. E a c h c o n d u c t i v e layer is s e p a r a t e d by a n o n - c o n d u c t i n g layer, usually c o m p o s e d of an oxide s u c h as silicon dioxide, SiO2. T h e a s s e m b l y i n c l u d e s b o t h active s e m i c o n d u c t o r

devices ( t r a n s i s t o r s a n d diodes)

a n d passive

c o m p o n e n t s ( c a p a c i t o r s a n d resistors). T h e s e are c o n n e c t e d

t o g e t h e r by

m e a n s of electrically c o n d u c t i n g i n t e r c o n n e c t i o n s s e p a r a t e d b y i n s u l a t i n g layers w i t h holes fabricated

for layer-to-layer i n t e r c o n n e c t i o n .

in situ by an iterative

process

AU e l e m e n t s

of l i t h o g r a p h i c

are

definition,

deposition, a n d e t c h i n g on a c o m m o n s u b s t r a t e (in m o s t cases, silicon) in

314

s u c h a m a n n e r t h a t the resulting i n t e r c o n n e c t e d e l e m e n t s p e r f o r m t h e desired electrical circuit function. Lithographic refers to the p h o t o g r a p h i c process u s e d w her e a p h o t o r e s i s t (a material w h i c h f o r m s either a water-soluble or water-insoluble area u p o n exposure to the p r o p e r wavelength of light) is u s e d to define the areas exposed. It is these areas which are t h e n subjected to definition, deposition a n d etching. These are a m o n g the m a n y p r o c e s s e s to which each wafer is subjected during t h e process to m a k e IC's. Many ICs, typically about 5.5 m m long or up to 33 square millimetres in area, are fabricated c o n c u r r e n t l y on silicon wafers up to 3 0 c e n t i m e t e r s (12 inches) in d i a m e t e r and s u b s e q u e n t l y sawed into individual chips (dies) prior to packaging. An IC chip t h u s p r o d u c e d is usually sealed in a plastic package with

electrical

leads

that

are

internally c o n n e c t e d by f'me wires to o u t p u t pads on the silicon die t hat p e r m i t the p a c k a g e d IC to be a t t a c h e d to a circuit card. The integration of a large n u m b e r of s e m i c o n d u c t o r devices on a single die of silicon is m a d e possible by the high operational efficiency of t h e individual devices. Because of this, power dissipation is m i n i m a l as are t h e r e q u i r e m e n t s for he a t removal. The result is an IC of high dependability, which is further e n h a n c e d by the process of integration itself. It is t h e m e t h o d of m a n u f a c t u r i n g the electrical i n t e r c o n n e c t i o n s b e t w e e n t h e devices and circuit e l e m e n t s by metal d e p o s i t i o n and etching t h a t yields an extremely dependabl e circuit s t r uct ure. The n u m b e r of devices i nteg r ated on a single tiny chip has i n c r e a s e d from an initial few to nearly 1 0 , 0 0 0 , 0 0 0 individual circuit e l e m e n t s per die with ever-smaller features in the d e v e l o p m e n t stage. This has, in turn, led to a progressive i n c r e a s e in complexity, as exemplified by the m e t a l - o x i d e - s e m i c o n d u c t o r (MOS), the dynamic random-access memory (DRAM), and the MOS microprocessor. Since m a n u f a c t u r i n g cost is pr opor t i onal to chip area, the use of s m a l l e r features has not only led to an increase in complexity a n d r e s u l t i n g functionality b u t also has p r o d u c e d a decrease in cost per t r a n s i s t o r and an increase in circuit p e r f o r m a n c e . There are two r e a s o n s for the latter:

315

(i) the p e r f o r m a n c e of the individual active devices improves as their internal d i m e n s i o n s become smaller. (2) the quality of the p e r f o r m a n c e of the entire circuit improves as the active devices are positioned closer to one a n o t h e r . Large-volume

applications

for

IC

dies

have

further

lowered

the

m a n u f a c t u r i n g costs per IC. The cumulative result is t h a t the i n t e g r a t e d circuit has become the m o s t pervasive technology of the 20t h century. It has provided

the

cornerstone

of m o d e r n

microelectronics

and

has

p r o m o t e d the d e v e l o p m e n t of the so-called information society. Applications of ICs include their use in c o m p u t e r s as well as o t h e r i n s t r u m e n t s . It is t h e s e applications t h a t are bringing about revolutionary advances in medical

diagnosis, biotechnology,

aeronautical and space

e ng in eer in g of our age. Other technologies affected (and vastly i m p r o v e d ) are t e l e - c o m m u n i c a t i o n s , and military defense systems. IC's have fueled the d e v e l o p m e n t of new c o n s u m e r p r o d u c t s capable of bringing new types of services an d information to the h o m e

and office e n v i r o n m e n t

that

otherwise would not have b e e n possible. Our next step will be to describe the steps r e q u i r e d in the m a n u f a c t u r e of Ics. As you will see, they have becom e quite c o m p l i c a t e d . E. Manufacture of I n t e g r a t e d Circuits v

The p r o ces s of fabricating s e m i c o n d u c t o r devices is a complex series of m o r e t h a n 600 sequential steps, all of w hi ch m u s t be done with u t m o s t precision in an e n v i r o n m e n t cleaner t h a n a hospital operat i ng room. If particles

are present,

they can cause electrical

shorting

among

the

conductive lines et ched on the surface of the silicon wafer. Thus, no impurities which m i g h t affect the wafer surface either physically or chemically can be tolerated. A series of layers is t h u s formed. This reality m a n d a t e s the m a n u f a c t u r e of IC's in a closed "clean-room". Since a h u m a n being emits a r o u n d 5 0 0 0 particles per m i n u t e (notably skin particles), it is n e c e s s a r y t h a t all oper a t i ng p e r s o n n e l involved in the m a n u f a c t u r i n g p ro ces s wear special clothing and m a s k s to prevent c o n t a m i n a t i o n of t h e wafer during the various 600 o p e r a t i o n s r e q u i r e d to m a k e a finished IC.

316

The objective is to a d d the c o r r e c t d o p a n t s in the p r o p e r a m o u n t s in t h e right places on the silicon wafer to form t r a n s i s t o r s a n d c o n d u c t o r s at t h e right place on t h e surface layer, w h i c h t h e n m a y be covered

u p by a

s u c c e e d i n g layer. It is usual to form a n u m b e r of s e p a r a t e i n t e g r a t e d circuits on e a c h wafer, to form an isolated a r e a on t h e "die" or "chip". As many

as several

thousand

are

formed

at

the

same

time.

This

is

a c c o m p l i s h e d by u s e of an applied coating w h i c h is sensitive to light, i.e.a "photoresist". By u s e of a carefully crafted film, m u c h like a p h o t o g r a p h i c negative, the outline a n d details of the i n t e g r a t e d circuit are f o r m e d by e x p o s i n g only c e r t a i n p a r t s of the p h o t o r e s i s t layer or film to t h e light as modified by the " p h o t o m a s k " u s e d in e x p o s u r e . E a c h die is f o r m e d by a machine

called

a

"stepper".

The

photo-outline

desired

is

formed

s e q u e n t i a l l y a c r o s s t h e e n t i r e surface of the silicon wafer, as s h o w n in 6.17.2., given on the n e x t page. Note t h a t lines of 0.13 p, i.e.- 0.13 m i c r o n s , in w i d t h are b e i n g f o r m e d r o u t i n e l y in the m a n u f a c t u r e of IC dice

t o d a y ( J u s t 3 y e a r s ago, t h e

n u m b e r w a s 0.18 p. It is e x p e c t e d to go to 0.08 ~ by 2010).

Once t h e

basic circuit is formed, t h e n s u b s e q u e n t s t e p s on e a c h individual die f o r m b o t h n- a n d p-layers, electrically isolated from e a c h o t h e r by a d i e l e c t r i c s u c h as S i O 2 . Both r e s i s t o r s a n d c o n d e n s e r s are r o u t i n e l y f o r m e d in p l a c e as well. Note t h a t f o r m a t i o n of s o m e of the layers is a c c o m p l i s h e d by use of g a s e s s u c h as ASH3, BBr3, or P H3. In o t h e r cases, a thin-trim p r o c e s s is u s e d to form the layers. T h i n - f i l m p r o c e s s e s u s e d in the m a n u f a c t u r e of IC's i n c l u d e : Physical v a p o r d e p o s i t i o n

Chemical vapor deposition

Electron-beam evaporation

Plasma etching

Electroplating

Photoresist

Reactive ion e t c h i n g Wet e t c h i n g

Photolithography

Molecular b e a m e p i t a x y

Ion i m p l a n t a t i o n

Chemical-mechanical polishing Rapid t h e r m a l p r o c e s s i n g Vacuum sealing

S p i n - o n glass d e p o s i t i o n Cathodic arc Ion p l a t i n g

Resistance heating

317

In practice, the s t e p p e r consists of a m a c h i n e i n c o r p o r a t i n g a light source, a p h o t o m a s k holder and a lens for focusing the p a t t e r n on t h e p h o t o r e s i s t layer on the silicon wafer. The p a t t e r n is r e p e a t e d clear across the wafer, step by step, hence the name. The lens s y s t e m m u s t be of highest quality so t h a t definition of lines and areas r e m a i n accurate and do not overlap each o t h e r . Note t h a t there are two types of p h o t o r e s i s t s - positive a n d negative. One is polymerized (made insoluble in water) by exposure to light while t h e

318

o t h e r b e c o m e s soluble ( d e - p o l y m e r i z a t i o n ) u p o n e x p o s u r e to light of t h e p r o p e r w a v e l e n g t h . T h e u s e of one or the o t h e r d e p e n d s u p o n t h e actual step in IC m a n u f a c t u r e b e i n g a c c o m p l i s h e d . simple

washing

w a s h i n g or m o r e

procedure, strenuous

whereas means

the such

One c a n be r e m o v e d by a

other

is r e m o v e d

as b a k i n g or

removal of the n o n - e x p o s e d p a r t s of the p h o t o r e s i s t ,

by e i t h e r

ashing.

Before

the n e x t step is to

expose the silicon surface to w h a t e v e r r e a c t a n t is r e q u i r e d to form t h e layer u n d e r c o n s t r u c t i o n on e a c h IC wafer. T h i s is done r e i t e r a t i v e l y until the basic IC circuit is formed. Once this is done, t h e layers are covered to form the final form of the wafer.

T h e n the p o r t i o n c o n t a i n i n g e a c h circuit is c u t into a small s q u a r e called a "die", m a n y being called "dice". The scale of line d i m e n s i o n s , i . e . - w i d t h , in i n t e g r a t e d circuits in 1997 w a s 0 . 2 5 m i c r o n s (2.5 x 10 v m e t e r s ) . It c o n t i n u e s to d e c r e a s e y e a r by year. A p p r o x i m a t e l y

1,850

dice

c a n be

o b t a i n e d from a 2 2 0 m m wafer, if all of the dice t e s t "good". T h u s , t h e d i a g r a m given above is not a c c u r a t e b e c a u s e the details of m y d r a w i n g cannot

be

made

precise

enough

to

accurately

detail

all

of

the

c h a r a c t e r i s t i c s of dice f o r m a t i o n on the surface of the silicon wafer a n d still illustrate the point. A h i g h - p o w e r s e m i c o n d u c t o r device for i n d u s t r i a l use, on the o t h e r h a n d , m a y be so large as to r e q u i r e a slice of silicon m e a s u r i n g well over 125 m i l l i m e t e r s in d i a m e t e r . Of late, the c u r r e n t size {or c r o s s - s e c t i o n ) for a s e p a r a t e line or c o n d u c t o r in an i n t e g r a t e d c i r c u i t is 18 m i c r o n s . R e s e a r c h h a s d e c r e a s e d this size to 13 m i c r o n s in 2 0 0 3 with 8 m i c r o n s (~rn) in the offing. Obviously, this i n c r e a s e s t h e yield in the n u m b e r of dice from one wafer by d e c r e a s i n g the actual size of e a c h individual die b e i n g formed. T h i s year, the first 3 0 0 m i l l i m e t e r c a m e on s t r e a m in p r o d u c t i o n

units making integrated

circuits.

wafer This

e n l a r g e m e n t in size w a s d e s i g n e d to i n c r e a s e the yield of dice by 1 4 5 % over w a f e r s only 2 0 0 m m in d i a m e t e r . A m a j o r c h a n g e in e q u i p m e n t was n e e d e d to h a n d l e the larger w a f e r s a n d the total cost to do so w a s in t h e billions of dollars. F. S t e p s in t h e M a n u f a c t u r e of I n t e g r a t e d Circuits The fabrication of ICs b e g i n s w i t h the p r e p a r a t i o n of silicon of very h i g h

319

purity. Single c r y s t a l b o u l e s w i t h a d i a m e t e r as large as 30 c e n t i m e t e r s are p r o d u c e d . T h e b o u l e s are sliced into w a f e r s h a v i n g a specified c r y s t a l o r i e n t a t i o n . This m e a n s t h a t t h e d i r e c t i o n of s i n g l e - c r y s t a l g r o w t h m u s t be controlled. W h e n t h e i r s u r f a c e s have b e e n p o l i s h e d fiat to a m i r r o r l i k e finish t h a t is free of defects, the silicon wafers are r e a d y for fabrication into

IC devices.

The

supreme importance formed.

For example,

methods

of p l a n a r i z a t i o n

and

polishing

are

of

to t h e overall quality of t h e c h i p s or dice b e i n g the following lists s o m e of t h e factors t h a t a r e

specified for a c c e p t a b l e silicon wafers w h i c h are u s e d to form 0.13 v m lines in the IC: Critical P r o c e s s S t e v Oxygen c o n t e n t > 30 p p m

Crystal g r o w t h

F l a t n e s s = 1.5 lma (side to side)

Lap ping, G r i n ding,Po l i s h i n g

W a r p = 25 ~ m

Slicing

Particles = < 1 0 0 / w a f e r

Polishing, Cleaning,

Non-critical m e t a l s = < 16 p p m

Cleaning

Critical m e t a l s = < 45 p p m

Cleaning

Crystal g r o w t h

Note t h a t the oxygen c o n t e n t is c o n t r o l l e d in the c r y s t a l growing p r o c e s s . The p r e s e n c e of m e t a l s on the wafers is d u e to t h e slicing a n d p o l i s h i n g s t e p s w h e r e the c l e a n i n g p r o c e s s

did not r e m o v e all of t h e s e e m ~ m a l

m e t a l particles. T h e difference b e t w e e n critical a n d n o n - c r i t i c a l m e t a l s is whether

t h e y w o u l d affect t h e c o n d u c t i v i t y of the silicon if t h e y w e r e

a c c i d e n t a l l y c a u s e d to c o m b i n e w i t h the silicon c o m p r i s i n g the wafer. T h e o t h e r factor not yet a d d r e s s e d by u s is the d a m a g e c a u s e d by t h e various p r o c e s s e s u s e d to form the finished wafer. A m o n g the d a m a g e c a p a b l e of b e i n g evoked on t h e surface of t h e silicon is the f o r m a t i o n of vacancies, s c r a t c h e s a n d dislocations in t h e silicon s t r u c t u r e .

For t h e s e

reasons,

e a c h m a n u f a c t u r e r of wafers h a s developed its own p r o p r i e t a r y p r o c e s s . S o m e of t h e s e p r o c e s s e s are slow and, of course, c o n t r i b u t e to t h e overall cost of t h e s o - p r o d u c e d wafers. T h i s is s h o w n in t h e follo~qng table, p r e s e n t e d on t h e n e x t page.

320

Table 5-4 Comparison of Some Wafer Manufacturing Operations Operation

Slicing (ID saw)

Depth of Damage

Flatness of Processed Wafer

Rough, two-side polish

poor high poor medium good low-med good medium zero good zero very good

Finish polishing

zero

good

Plasma chemical e t c h i n g

zero

Plasma torch & ADP*

zero

excellent good

Slicing (wire saw) Lapping(multiple wafer) Grinding(single wafer) Wet chemical e t c h i n g Rough, one-side polish

Relative Speed of Process

slow medium fast fast medium slow slow very slow slow medium

Ease of Automation

good medium poor good good good poor good good good

ADP = Atmospheric Downstream Plasma Note that some processes produce little or no surface damage but that t h e process is slow. What this m e a n s is that the m a n u f a c t u r e r m u s t choose between finishing processes and acceptable damage during processing. When some damage is left on the surface of the wafer or if the planarity is not good, then the IC processes used result in defects in the IC circuit a n d / o r in the individual transistors formed. This results in a "bad" die or chip, i.e.- it does not function properly. It is for this reason that t h e n u m b e r of "good" dice or chips vary from wafer to wafer. Note that electronic testing is required to determine ff a die is good or not. T h e typical IC manufacturing process involves a sequence of more t h a n 6 0 0 operations. These operations may be categorized according to function: 1. Deposition of thin films 2. Introduction of impurities (doping) 3. Lithographic patterning of IC features corresponding to those of the physical layout 4. Etching to define the features of individual circuit e l e m e n t s

321

5. Annealing to cause a chemical reaction or to form t h e desired m i c r o s t r u c t u r e between films

of

deposited

films

or

interfaces

6. Polishing to level and s m o o t h the surface 7. Cleaning to pr e ve nt c o n t a m i n a t i o n a n d the c o n s e q u e n t i n t r o d u c t i o n of defects into circuit e l e m e n t s by p a r t i c u l a t e matter. These o p er ations are r e p e a t e d

again a n d again in different

sequential

order until the IC fabrication is completed. At the sam e time, t hey are n o t the actual steps n e e d e d to form the individual dice. We will a d d r e s s t h e s e later on after we

have described

the

individual operat i ons

used

in

p e r f o r m i n g each step in the IC m a n u f a c t u r i n g process. It is very important to distinguish be t w een manufacturing operations and m a n u f a c t u r i n g steps. The actual m a n u f a c t u r i n g steps involved include: I. Wafer cl eani ng 2. Lithography 3. Planarization 4. Ion i m p l a n t a t i o n 5. S p u t t e r i n g 6. C V D - c h e m i c a l vapor d e p o s i t i o n 7. Resist s t r i p 8. Wafer cl eani ng 9. T h e r m a l p r o c e s s i n g I0. Metrology The following diagram, given as 6.17.3 on the next page, shows the cyclic n a t u r e of the steps u s e d to form the i n t e r m e d i a t e layers of an e l e c t r o n i c device. In this case, we have s h o w n at least 8 individual steps n e e d e d to fabricate this s t r u c t u r e . Note t h a t the use of p h o t o r e s i s t is not i n d i c a t e d in the CVD steps, although it plays a significant role in the overall p r o c e s s . Note t h a t these s a m e 8 steps can be r e p e a t e d so as to build up a series of electrical an d isolated layers w h i c h c o m p r i s e the overall IC d e s i g n

322

6.17.3.-

[I

p,oCe,,, ,o,uo.oo- II INPUT WAFER

+

IWAFER CLEANING (PRE-TREATMENT)]'~------l OXIDATION

--

+

[ETCHING ILIONIMPLANTATION] + [ WAFER CLEANING (POST-TREATMENT)

1

.~

circuits. However, even this diagram does not show all of the cyclic s t e p s and m a c h i n e r y needed to accomplish the 600 individual steps referred to above in the m a n u f a c t u r e of an integrated circuit on a silicon wafer. As shown in 6.17.4. given on the next page, the m a n u f a c t u r e begins with a polished wafer. Study this outline well. It gives m o s t of the 600 steps involved in IC manufacture. The first step is usually formation of an oxidized Film layer, followed by microlithography steps. We have indicated the individual steps involved in lithography, as well as those involved in m a s k making. Note t h a t the n e x t step involves etching and stripping steps. Inspection is the next major step, followed by doping and t h e n diffusion and oxidation ( d e p e n d i n g upon the specific layer being formed). Reinspection t h e n takes place. If

323

[!SEMICONDUCTORFABRICATION 'STEPS I[

6.17.4.-

Equipment Includes: Wafer Processing- Test - Assembly

IR ~

Silicon & Wafer Making[

Includes: Crystal growing, grinding to diameter, wafer slicing & polishing Circuit Design and Mask Fabrication

Includes: Pattern generator, photo-repeater, electron -beam lithography and contact printer

inc1 d :

I or reactive ion etch I'~ Dry or wet strip I or both. I Ist

LInspecti~

Includes either optical or electrical inspection or both

I ~, i~ r

2nd

ASSEMBLY

[~i~olithog.~phy[ Includes: applying photoresist, scrub & dei~0rclrate,mask align & expose, develop, rinse & bake. Equipment: Contact/proximity scan projection alignment, wafer stepper

Ne~:t

Cy~ Ie

.....~................

l_ A,

Ist

IW~-~R PROBE & TEST I includes: Automatic test equipment: KS[ logic I_SI m e m o r y Other testers Probing and handling equipment

I [Film Deposition] Includes: Chemical (oxide films) " and Physical (like "spin-on" glass or metallization films) Chemical Physical Equipment Equipment Normal press C V D Spin -On table Lo-~ press C V D we~uum Plasma C V D evaporator, Epitaxial sputtering

~I

I DopinI~ I Includes: Ion implanterboth m e d i u m & high current; Furnacesdiffusion annealing oxidation

IDiffusion & Oxidation I Includes: Thermal or chemical processes or both

+ I M A S K FINISHEI3-? I y~I

I

I

h._

i

1.Dici.ni~-here x v ~ r is cut in dice

2. Die Attach

3.Wire B o n d - each die mc~anting is bonded, both electricallyand physically

Here each die 4. Mold & Seal is attached to Here, the a mounting mounting is sealed Equipment: Scribers & saws, die bonders, molds & presses, Other.

324

the m a s k lithography is finished, t h e n wafer-probe testing t h e n t a k e s place. If not, t h e n the cycle begins anew at the f'flm deposition stage. After wafer probe an d testing, assembly of the cut dice into a m o u n t i n g t a k e s place by attaching wires to appr opr i at e points of the i n t e g r a t e d circuit, the wires are t h e n bonded, and the whole is sealed before final t e s t i n g and s h i p m e n t . In order

to illustrate

exactly w h a t we

are discussing,

the

following

diagram of a CMOS design u s e d in 1997 is p r e s e n t e d on the following diagram. That is, a p-type silicon source is formed on an n-type well, and vice-versa.

"Sources" are formed in the first step and t hen "Gates" are formed.

A

silicide is t h e n u s e d to lower the contact resistance bet w een the silicon of the gate, source a n d drain a n d the cont act i ng plug. The t h i c k n e s s of gate oxides is only about 4 0 - 5 0 A. They are t herm al l y grown r a t h e r t h a n

325

deposited by CVD. Nitrides, s o m e t i m e s c o m b i n e d with oxides, are b e i n g u s e d to s u p p r e s s bor on p e n e t r a t i o n from the gate. Silicides are formed by depositing a metal s u c h as titanium, t u n g s t e n or cobalt, followed by r a p i d t h e r m a l p r o c e s s i n g to form the silicide. The silicide in greatest usage was t h at of Ti, wh ere the silicide was essentially a TiOxSi type of c o m p o u n d . Ti/TiN is mostly u s e d now. The d e p t h of the s o u r c e / d r a i n s for 0.25 ~ m line CMOS designs is about 60 nm., i.e.- quite shallow. Elemental boron is i m p l a n t e d by p l a s m a implantation, using about 5 kV acceleration of B3+ ions. Polysilicon gate t r a n s i s t o r s are formed separately as p-type and ntype. Diffusion barriers are u s e d to prevent diffusion of materials to or from the r e t r o g r a d e wells already formed. The s m o o t h n e s s and t h i c k n e s s of these films are of particular concern. S m o o t h films help s u b s e q u e n t viahole filling by hot or w a r m a l u m i n u m deposition. Thin films are desirable to keep the r es i s t a nc e of the plug low. T u n g s t e n is u s e d for electrical contact and via-plugs. I n t e r c o n n e c t m e t a l s u s e d in 1997 were a l um i num , s o m e t i m e s c o m b i n e d with copper. T hi s was called "damascene", b u t in 2003, copper is being u s e d al m ost exclusively. Once the m e t a l c o n n e c t is deposited by evaporation or CVD, metal etching is u s e d to remove those p a r t s of the m et al film not n e e d e d . I n t e r c o n n e c t dielectrics include the use of "spin-on-glass" (which is a slurry of l o w - t e m p e r a t u r e - s o f t e n i n g glass particles) or silicon oxide films formed by h i g h - d e n s i t y p l a s m a processes. shown in the

above figure,

are n e e d e d

Note t h a t at least 8 steps, to form the

finished

CMOS

t r a n s i s t o r s tr u c t ur e. The outline given above is the fabrication t e c h n i q u e u s e d to m a n u f a c t u r e this s i m p l e CMOS device on a silicon wafer. R e m e m b e r t h a t the s t r u c t u r e given a nd described above was fabricated and duplicated about 1800 times on a specific silicon wafer. Following t h e f'mal step, the wafer was cleaned and t h e n cut into dice. Each die is t h e n m o u n t e d and electrical c o n n e c t i o n s m a d e prior to its use in a given device. What we have not described adequately is the complexity of achieving j u s t one of the features, i.e.- layers, s h o w n in 6.17.5. C o n s i d e r the actual fabrication steps n e e d e d to m a k e j u s t one of the features s h o w n therein, i.e.- the "gates" of the CMOS s t r u ct ure. At level 3 of 6.17.5., t h e gates are shown. The separate steps are illustrated in the following diagram, p r e s e n t e d as 6.17.6. on the next page.

326

The

major

steps

are

labeled

1 to

5.

However,

there

are

several

i n t e r m e d i a t e s t e p s also involved. At "1", the shallow t r e n c h e s are f o r m e d , s o u r c e a n d d r a i n i m p l a n t s w e r e a c c o m p l i s h e d , diffusion b a r r i e r s w e r e f o r m e d from Ti/TiN a n d a d u m m y gate m a d e from Si3N4 w a s d e p o s i t e d on a d u m m y oxide gate. Five s e p a r a t e p h o t o m a s k s were u s e d to a c c o m p l i s h t h e s e i n t e r m e d i a t e steps. At "2", a p r e - m e t a l dielectric film, S i O 2 , was deposited

by low

pressure

CVD a n d

planarized

by CMP

("Chemical

M e c h a n i c a l Planarization") in o r d e r to u n c o v e r the top of the d u m m y g a t e (2 steps). At "3", wet a n d dry e t c h i n g w a s u s e d to r e m o v e the d u m m y g a t e a n d the gate groove w a s fabricated. A gate oxide of T a 2 0 5 w a s t h e n g r o w n in the groove. (4 steps). At "4", a W / T i N layer w a s d e p o s i t e d (2 steps) a n d then

planarized

to

its

final form

in

"5".

Note

that

a total

i n t e r m e d i a t e s t e p s were r e q u i r e d in the f o r m a t i o n of t h e gates.

of

14

327

Metailization of contacts, formed and c o n n e c t e d .

vias and i n t e r c o n n e c t

metals

still need

to

A significant n u m b e r of different m a c h i n e s are n e e d e d to accomplish t h e above fabrication steps. Such m a c h i n e s includeI. Lithographic- Optical wafer steppers are the most expensive and critical m a c h i n e s used in the Fabrication Laboratory. 2. Planarization- Chemical mechanical planarization m a c h i n e s are used to level the topology of the films deposited on the wafer. Chemical-mechanical smoothing is the process used to improve process uniformity, repeatability, removal rate, planarity and defect reduction. Thus, planarization is requisite to producing defect-free CMOS circuits. 3. Ion Implantation- these plasma m a c h i n e s are used to form t h e various parts of the coupled transistors at voltages of 5 kV to 2 MV. These include retrograde well formation, gates, drains and sources. 4. S p u t t e r i n g - Deposition of copper doped (damascene) aluminum interconnects, as well as diffusion barriers, and anti-reflective stacks of Ti/TiN is done by this technique. The metal is heated by an electron b e a m and "sputters" onto the target. Nitrogen gas is used w h e n TiN is r e q u i r e d . 5. Chemical Vapor Deposition- Deposition of silicon oxide films is accomplished by CVD equipment. Either plasma CVD or ozone oxidation is used. Blanket tungsten films are also deposited by CVD equipment to create contact and via plugs. Polysilicon and silicon nitride films are deposited in hot-waU furnaces. TiN diffusion barrier films are deposited by either sputtering or CVD, the latter giving superior step coverage. 6. E t c h i n g - m e t a l and oxide etch e q u i p m e n t are used to r e m o v e excess parts of the deposited film. High density plasma sources

328

provide a m or e etch in g .

rapid removal of material

and p r o d u c e

cleaner

7. Resist s t r i p p i n g - Both wet and dry removal p r o c e s s e s are u s e d at this step

in processing.

Dry ashing removes

the

b u l k of t h e

p h o t o r e s i s t and wet s t r i p p i n g removes r e m a i n i n g resi dues. 8. W.afer cleaning- wafers are cleaned in a u t o m a t e d wet b e n c h e s using the RCA wet clean m e t h o d (which uses a c o m b i n a t i o n of dilute acids, peroxide and bases in a particular order). E n v i r o n m e n t a l c o n c e r n s has p r o m p t e d the search for a new m e t h o d of cleaning. 9. Th er mal p r o c e s s i n g - Diffusion furnaces are u s e d not only for t h e anneal of i m p l a n t e d d o p a n t s but for growing high quality t h e r m a l oxides,

depositing polysilicon nitride

films (SiN x) and for r a p i d

t h e r m a l p r o c e s s i n g of deposited films. I0. M e t r o l o g y - The goal of m o s t metrology m a c h i n e efforts is to keep

the

pr oc e s s

measurements

under

of physical

control, size

whether

it

involves

of individual features

making and

film

thickness, or m a k i n g electrical m e a s u r e m e n t s of p a r a m e t r i c t e s t s t r u c t u r e s . Defects are also m e a s u r e d and estimated, i n c l u d i n g excess particles and m i s p l a c e d features in the c o m p o s i t e . Let us now discuss in more fabricate IC's. These include:

detail several of the operat i ons

u s e d to

G. Film.. Deposition. Several kinds of thin films are deposited on a silicon wafer by different m e t h o d s (see above) dur i ng various stages of the fabrication process. T h e initial step following cleaning is the formation of a silicon dioxide film. This film is grown by placing the silicon wafer in an oxidizing e n v i r o n m e n t at high t e m p e r a t u r e as, for example, in a quartz-walled furnace tube. This operation m a y be followed by the deposition of a film of silicon nitride. Later in the fabrication process, m et al (e.g.- t u n g s t e n

329

metal) a n d dielectric (e.g.- glass) films are deposited by m e a n s of sputtering. In this technique, a cathode m a d e of the material to be deposited is b o m b a r d e d with electrons, resulting in the ejection of a t o m s and molecules f r om the cathode. This material is deposited on the wafer surface an d forms the desired film. Metal and polysilicon films are formed by a chemical-vapor d e p o s i t i o n process using organometallic gases t h a t react at the surface of the IC s t r u ctu r e. Various metal silicide films m a y also be deposited in this m a n n e r by reaction with the surface of the silicon wafer to form m e t a l silicides. Glass and polymer films are deposited or spin cast or both, as are p h o t o r e s i s t films (those of a photosensft_ive material). T h i s process is a c c o m p l i s h e d by applying a liquid polymer onto a rapidly rotating wafer. The exact m e t h o d u s e d varies from m a n u f a c t u r e r to m a n u f a c t u r e r and usually r e m a i n s p r o p r i e t a r y . H. I m o u ~ W Doping, Selected imp u ri t i e s (e.g.- bor on a n d p h o s p h o r u s ) a r e i n t r o d u c e d into t h e silicon s u b s t r a t e to control its conductivity in a selective m a n n e r . This is a c c o m p l i s h e d by the c o m b i n a t i o n of two m e t h o d s : ion i m p l a n t a t i o n and t h e r m a l diffusion. In the ion i m p l a n t a t i o n process, the silicon wafer is exposed to a b e a m of energetic particles (i.e.- ions accelerated t h r o u g h a potential of several h u n d r e d kilovolts) of the s u b s t a n c e s t h a t are to be i n c o r p o r a t e d into the silicon. These i m pu ri t i es are driven into the silicon wafer via kinetic energy to a d e p t h ranging from a few h u n d r e d a n g s t r o m s to several m i c r o m e t e r s , d e p e n d i n g on the energy a n d the m a s s of the ions in the beam. The wafer is a n n e a l e d after i m p l a n t a t i o n of the i m p u r i t y ions in order to remove damage c a u s e d dur ing the ion i m p l a n t a t i o n p r o c e s s and to diffuse the impurities to the desired positions in the silicon wafer. I. Lithograohic p a t t e r n i n g . The p r o ces s of lithographic p a t t e r n i n g d e t e r m i n e s the geometric features specified by the layout a n d p a t t e r n i n g as the i n t e g r a t e d circuit is fabricated layer by layer. A p h o t o m a s k , containing p a t t e r n information in

330

the form of an e t c he d c h r o m i u m layer on glass, is p r e p a r e d for each layer. Note t h a t a specific p h o t o m a s k is r equi r e d for each layer of the IC. Thus, m a n y differing p h o t o m a s k s are r e qui r ed for the overall p r o c e s s to p r o d u c e all of the layers c o m p r i s i n g individual c o m p o n e n t s .

the

integrated

circuit

and t h e

An image of the p h o t o m a s k is projected onto the surface of the silicon wafer. This is a c c o m p l i s h e d with a very high-precision optical p r o j e c t i o n p r i n t e r after the wafer has been coated with a thin layer of photoresist. A positive p h o t o r e s i s t is a material w hi ch is m o n o m e r i c to begin with and which is cau s ed to polymerize w h e n a light b e a m from the p h o t o m a s k strikes selected por t i ons of the p h o t o r e s i s t so as to becom e insoluble in either water or a selected solvent. There are b o t h positive a n d negative p h o t o r e s i s t s , so-called because the major portion of the p h o t o r e s i s t is c a u s e d to become either insoluble or soluble by the action of bei ng exposed to selected wavelengths of light. The light is focused down to a point. Thus, to p r o d u c e a 18 m i c r o n line in the photoresist, the focus c a n n o t exceed this value. At this point in time, the focus has a p p r o a c h e d the focal-limit of the natural wavelength of visible light and even ultraviolet light c a n n o t be focused beyond a certain point. Thus, x-ray sources are being developed to p r o d u c e the 0.13 m i c r o n lines r e q u i r e d in advanced IC circuits. We will not fully explore the chemical n a t u r e of t h e p h o t o r e s i s t s r equi r e d in this technology except to say t h a t c o n s i d e r a b l e effort has gone into their d e v e l o p m e n t . The exposure is done one die at a time, st eppi ng from each die to t h e next to complete the exposure of the wafer. So-called "steppers" are an i m p o r t a n t part of the overall process for m a k i n g IC circuits on a silicon wafer. The resist containing the p a t t e r n image is developed (usually by exposure to light), and the exposed photosensitive material is removed by w a t e r - w a s h i n g from the areas t h a t have not been exposed to light (in t h e case of a positive photoresist). The desired p a t t e r n is thereby t r a n s f e r r e d from the p h o t o m a s k to the p h o t o r e s i s t on the surface of the partially fabricated IC. This p h o t o r e s i s t p a t t e r n serves to define the areas of t h e wafer where, d u r i n g a s u b s e q u e n t process step, film is to be d e p o s i t e d , i.e.- in a via (passage), material is to be removed by etching, or i m p u r i t i e s

331

are to be introduced. Once the selective deposition or removal of material has been accomplished, the remaining photoresist is cleared off t h e wafer. J. Etching. During this process, material is selectively removed from the wafer surface as defined by the p a t t e r n e d photoresist in order to define t h e structure of the previously deposited layer. The etching process is accomplished by exposing the wafer to a gas plasma, which both chemically reacts with the material to be removed and physically ablates it. At the completion of etching, the remaining photoresist is cleared from the wafer. K. A Final ~ k

at the IC Man llfacturing Process

Although the basic fabrication process of an CMOS circuit has b e e n illustrated above for the process used in 1997, the exact materials and sequence of steps has changed considerably in j u s t two years. T h e following illustrates how the process steps described above have b e e n modified. The details of the components used in manufacturing an CMOS circuit, having a line width of 0.18 ~m, is shown in the following diagram~ given as 6.17.7. on the next page. Contrast this diagram to 6.17.5. In this case, we have shown a n u m b e r of layers comprising the CMOS device and have n u m b e r e d some of the layers. At "1", the m e t a l interconnects are shown. These are composed of t u n g s t e n or c o p p e r metal and are either evaporated or applied by electrolysis in an inert gas atmosphere. The use of "damascene" has decreased b u t is still used for some types of DRAM's. The use of "diffusion barriers" continues, connecting the Cu metal to the active semi-conductors. Inter-level dielectrics being used are fluorocarbon polymers and low-k glasses, both in the form of spin-on suspensions. Siloxane glasses (which are a form of silico-organic materlals) are also in use. For purely electrical "connects ~, t u n g s t e n is still used since its tendency to diffuse through the dielectric layers is almost nil.

332

If a diffusion barrier is required, t h e n a t i t a n i u m l n i t r i d e (Ti/TiN) is used. This c o u n t e r a c t s the t e n d e n c y of m o s t met al s to diffuse t h r o u g h a given structure, particularly if the layers are c o m p o s e d of several different types of metals, c o n n e c t i o n s . After the p h o t o r e s i s t has been developed and the isolation o x i d e / n i t r i d e m a s k defined by etching, the wafer is i o n - i m p l a n t e d to electrically isolate the areas where the MOSFETs are to be formed, and oxide is grown over the i m p l a n t e d regions. The o x i d e / n i t r i d e film is t h e n removed, and a t h i n gate oxide is regrown in these regions, which are the sites for the MOS transistors. Polysilicon m ay be deposited and p a t t e r n e d in the n e x t p h o t o l i t h o g r a p h y p r o c e s s using a second p h o t o m a s k to define the gate s t r u c t u r e of the NMOS transistors. A s u b s e q u e n t i o n - i m p l a n t a t i o n p r o c e s s i n t r o d u c e s the r e q u i r e d i m pur i t i es into the s o u r c e / d r a i n regions of t h e

333

MOSFET's, and silicon oxide is deposited. The s o u r c e / d r a i n p o s i t i o n s where electrical cont act is to be m a d e to the MOSFET's are defined, using the oxide-removal m a s k and an etch process. For shallow t r e n c h isolation, anisotropic mechanical

silicon etch, t h e r m a l oxidation, oxide fill and c h e m i c a l leveling are the processes employed. For shallow

s o u r c e / d r a i n s formation, ion i m p l a n t a t i o n t e c h n i q u e s are still be used. For raised s o u r c e / d r a i n s (as s h o w n in the above diagram) cobalt silicide is being u s e d in st e a d of Ti/TiN

silicides.

Cobalt metal is deposited

and

reacted by a rapid t h e r m a l t r e a t m e n t to form the silicide. Capacitors w e r e m a d e in 1997 from various oxides and nitrides. The

use of t a n t a l u m

pentoxide in 1999 has proven superior. P l a t i n u m is u s e d as the plate material. Metal is t h e n deposited into the o p e n e d vias (openings) in the oxide layer and over its surface. During the s u b s e q u e n t p h o t o l i t h o g r a p h y process, it is p a t t e r n e d to form the desired electrical i n t e r c o n n e c t i o n s . These two steps are r e p e a t e d for each s u c c e e d i n g level to p r o d u c e additional levels of i n t e r c o n n e c t i o n s .

Finally, a protective

overcoat of o x i d e / n i t r i d e

is

applied (passivation), and vias are opened so t hat the wires c o n n e c t i n g the IC chip to its carrier package can be b o n d e d to o u t p u t pads. The m o s t i m p o r t a n t p a r t of the overall p r o c e s s for building or layering an i n t e g r a t e d circuit involves the use of dielectrics to isolate the electrically c o n d u c t i n g t r a n s i s t o r s built into the s t r u ct ure. Diagram 6.17.8., given on the next page, shows how these insulating layers are used. Here, it is easy to see the various layers a n d steps n e c e s s a r y to form t h e IC. We have already e m p h a s i z e d the formation of the n- and p-wells and the individual process steps n e e d e d for their formation. Note t h a t an epitaxial layer is u s e d in the above model. There are isolation b a r r i e r s p r e s e n t wh ich we have already discussed. However, once the polysilicon gate t r a n s i s t o r s are formed, t h e n m et al i n t e r c o n n e c t s m u s t t h e n be placed in p r o p e r position with p r o p e r electrical isolation. This is t h e function of the dielectric layers p u t into place as succeedi ng layers on t h e IC dice. Once this is done, t h e n the wafer is t e s t e d .

334

The a s s e m b l y p r o c e s s b e g i n s with an electrical test of the ICs die by die on e a c h wafer prior

to s e p a r a t i o n .

This

locates

and enumerates

n u m b e r a n d location of the good units. The wafer is t h e n

the

sawed into

individual dies, a n d e a c h good u n i t is installed in a plastic or c e r a m i c p a c k a g e by a t t a c h i n g the die to a lead frame a n d s o l d e r i n g wires b e t w e e n the o u t p u t p a d s on the die a n d the i n t e r n a l leads of the package. A final test is p e r f o r m e d on the p a c k a g e d die to d e t e r m i n e

whether

the u n i t

o p e r a t e s w i t h i n the specified s t a n d a r d s . The following diagram, s h o w n as 6.17.9. on the n e x t page, s h o w s an IC chip m o u n t e d u p o n a circuit board, with a t t e n d a n t r e s i s t o r s , c o n d e n s e r s

335

a n d solder 'l)umps" (the b l a c k spots). The i n t e r c o n n e c t i o n s

(shown as

b l a c k lines on the diagram) are usually copper. T h e y are also f o r m e d by a p h o t o - m i c r o - l i t h o g r a p h i c p r o c e s s similar to the l i t h o g r a p h y p r o c e s s u s e d in m a n u f a c t u r i n g IC circuits. Here, the lines are in mils, i.e.- t h o u s a n d s of an inch, r a t h e r t h a n in m i c r o n s ,

i.e.- millions of a m e t e r .

Note t h a t a

n u m b e r of c o n n e c t i o n s are h i d d e n w i t h i n the device m o u n t i n g a n d c a n n o t

336

be seen directly. L. Crystal Growth and Crystal Defects Affecting IC's Under conventional crystal growth conditions of silicon boules, vacancies agglomerate to form small, low-density octahedral voicls, commonly called "D-defects", i . e . - d i s l o c a t i o n defects. Interstitials can form distributed dislocation clusters. Both have serious effects upon the process of forming integrated circuits. The void defects have been clearly associated with dielectric breakdown failures in integrated circuits, while the dislocation defects have been associated with controlling the rate of the electrical defect producing reactions due to integrating the m a n a g e m e n t of defects throughout the entire growth process. In other words, the dislocation defects t e n d to cause the formation of defects in the integrated circuit at the point of the dislocation defect. Additionally, such a defect may not appear until the IC is being operated. Such defects are known to be related to certain classes of leakage current. The void defects have recently become a particular concern throughout the industry in spite of their extremely low density. In the growth of the single-crystal silicon, the crystal boule is pulled from the melt and is allowed to cool as it is being pulled. By annealing the upper part of the boule and finally subjecting it to a controlled cooling rate, this "zone engineering" ensures that the reactions that p r o d u c e either the void or dislocation cluster defect are completely s u p p r e s s e d . The result is crystals that are completely free of both void (vacancies in our terminology) and dislocation cluster defects. Eliminating one or the other can be readily achieved. Crystals have been made that are completely free of both classes of defects and have earned this material the n a m e "perfect silicon,". At present, defect-flee silicon crystals have been achieved at only at diameters of 200 mm. Comparisons of crystal quality were m a d e among three techniques: a typical conventional Czrochralski crystal g r o w t h technique, a slow-cooled controlled reaction and the "perfect silicon" process. The quality levels achieved in D-defect levels of the material is

337

mirrored in the gate oxide integrity of these materials. This technique is extendible to 300 m m growth where the propagation of in-grown defects is a major challenge. Engineering work on newly developed 300 m m crystal pulling systems has already begun and is e x p e c t e d to be complete in 2003. As device geometries s h r i n k and active gate areas increase, the probability of one of these defects occurring in a sensitive area increases. This can result in a crystal-related device yield limiting factor that can become critical in certain advanced IC structures. Thus various processes that reduce the density of these void defects are being developed to improve yield potential. This has been achieved primarily by controlling the crystal cooling rate through certain important t e m p e r a t u r e ranges that have the effect of controlling the defect g e n e r ating reaction. The effect is fewer, but not zero, voids. *Blow-cool" silicon is a c o m m o n expression used for this general type of approach to the problem. But as device generations progress, it is less clear that such an approach will be successful in producing sufficiently low void densities to produce acceptably profitable yields. Epitaydal silicon provides an alternative but is problematic in certain cost sensitive applications, particularly DRAMs. 6.18.- Future Methods for Manufacture of Integrated Circuits At this point, we have described how integrated circuits have b e e n designed and manufactured. At the dawn of the DIGITAL AGE (which we as a society have already entered), Gordon Moore (who founded Intel Corporation which m a n u f a c t u r e s nearly a~ of the chips used in computers) predicted that the n u m b e r of transistors on an i n t e g r a t e d circuit, and hence its speed, would double every 18 months. He has b e e n accurate and this has become known as "Moore's Law". Since 1965, this law has been the guiding light of the Semi-conductor Industry and has been good for the Industry as well as the c o n s u m e r since c o m p u t e r s have dropped drastically in price while rising dramatically in speed of computing, i.e.- rate of handling bytes. Speed is m e a s u r e d now in megabytes per second w h e r e a s in 1965, it was in t e r m s of kilobytes p e r

338

second. However, Moore's Law will soon collide w i t h a m u c h less flexible set of laws, t h o s e of t h e laws of physics. By 2004, the optical l i t h o g r a p h y u s e d to m a k e i n t e g r a t e d

c i r c u i t s will r e a c h its limits b e c a u s e of t h e

diffraction limit of visible light.

Although we

have d e s c r i b e d

current

d i m e n s i o n s in t e r m s of 0 . 1 8 V, lines less t h a n 0 . 1 0 V will not be possible. However,

t h e solution s e e m s

to be t h a t e i t h e r very s h o r t w a v e l e n g t h

ultraviolet will be u s e d or t h a t a b e a m of e l e c t r o n s will be e m p l o y e d to etch the silicon circuits in a wafer. Both m e t h o d s will e m p l o y a m u c h different a p p r o a c h to t h e s a m e p r o b l e m . For far-UV at 193 n m . , q u a r t z optics c a n be used. However, t h e c a p a c i t y for finer lines is n o t as g r e a t as for 157 nrn. Light s o u r c e s for t h e s e w a v e l e n g t h s do exist b u t the optics r e q u i r e d involve CaF 2. T h i s m a t e r i a l is not stable in air a n d is a t t a c k e d by w a t e r vapor. One solution for t h i s optical p r o b l e m h a s b e e n found to be the u s e of reflecting m i r r o r s . T h e following d i a g r a m s h o w s a c o n v e n t i o n a l s y s t e m for l i t h o g r a p h y :

339

Here, we s h o w n a "conventional" system. It c o n s i s t s of a light source, an etched, t r a n s p a r e n t circuit d i a g r a m a n d a lens to focus the p a t t e r n u p o n the wafer w h i c h h a s a p h o t o r e s i s t layer. Upon e x p o s u r e , the u n e x p o s e d p a r t of the p h o t o r e s i s t is w a s h e d off. S o m e t i m e s , the e x p o s e d area is removed, d e p e n d i n g u p o n w e a t h e r the p h o t o r e s i s t is positive or negative. C o n t r a s t this simple s y s t e m w i t h a m e t h o d u n d e r d e v e l o p m e n t is t h e "Extreme Ultraviolet (EUV) s y s t e m , u s i n g r a d i a t i o n from excited Xe gas, i.e.- 157 rim. The EI_W s y s t e m is s h o w n in the following:

It h a s similar c h a r a c t e r i s t i c s to the above s y s t e m s s h o w n in 6.17.2. a n d 6.18.1. But, ff the EUV m e t h o d is successful, it will r e q u i r e a n e a r v a c u u m since oxygen a n d w a t e r vapor strongly absorb 157 n m radiation. 6.19.- P u s h i n g the Um~its of S e m i - C o n d u c t o r T e c h n o l o g y For the p a s t 30 years, the s e m i c o n d u c t o r i n d u s t r y h a s followed M o o r e ' s law, w h i c h s t a t e s t h a t t r a n s i s t o r p e r f o r m a n c e a n d d e n s i t y double every 3 years {1). Although n o t truly a law, G o r d o n Moore's s t a t e m e n t h a s yet to be violated. But n o w it s e e m s to be in serious danger. F u n d a m e n t a l t h e r m o d y n a m i c limits are b e i n g r e a c h e d in critical areas, a n d u n l e s s new,

340

innovative solutions are found, the c u r r e n t rate of i m p r o v e m e n t c a n n o t b e m a i n t a i n e d . The d o m i n a n t electronic device u s e d today in i n t e g r a t e d circuits is the silicon-based metal oxide s e m i c o n d u c t o r (MOS) t r a n s i s t o r , which consists of a source, drain, channel, and gate region (see t h e figures given above). The source region provides a supply of mobile c h a r g e carriers, enabling c u r r e n t to flow from the source to the drain w h e n t h e t r a n s i s t o r is t u r n e d "on". The source and drain regions are electrically isolated from one a n o t h e r by an oppositely charged channel region. A controlling gate e l e c t r ode is s e p a r a t e d from the c h a n n e l by an insulating oxide material. By applying a voltage across the insulating material, an electric field is created. If the applied voltage repels the c h a n n e l c h a r g e and attracts the source and drain charge, a c o n d u c t i n g layer is f o r m e d , and c u r r e n t can flow from the source to the drain. In contrast, if t h e applied voltage attracts the c h a n n e l charge and repels the source and drain charge, no c o n d u c t i n g layer between the source and drain can form, and the t r a n s i s t o r is "off"' The t r a n s i s t o r t h u s acts as a digital switch t h a t is t u r n e d on or off by applying a voltage to the gate. Individual s w i t c h e s are c o m b i n e d to form the building blocks for m i c r o p r o c e s s o r a n d m e m o r y chips. For more t h a n 30 years, the switching speed of the MOS t r a n s i s t o r has been i n c r e a s e d by r e d u c i n g the size of the device. For proper operat i on, MOS t r a n s i s t o r scaling theory (i.e.- size r e q u i r e m e n t s ) requires all vertical and lateral d i m e n s i o n s to be scaled simultaneously (2). At t h e s a m e time, the total a m o u n t of charge in the source, drain, and c h a n n e l regions m u s t not decrease in order to m a i n t a i n low device r e s i s t a n c e . Charge in the source, drain, and c h a n n e l regions is c r e a t e d by locally adding d o p a n t atoms to the silicon lattice. In silicon, each silicon a t o m is covalently b o n d e d to the four n e a r e s t neighbors in the lattice. Adding a d o p a n t ato m concentration three valence silicon atom;

with five valence electrons increases the mobile c h a r g e by donat i ng an u n b o u n d electron. A d o p a n t a t o m with only electrons needs to accept an extra electron to s u b s t i t u t e a the localized t r a p p i n g of the extra electron effectively

creates a positive free charge referred to as a "hole." T r a n s i s t o r scaling requires an increase in the c o n c e n t r a t i o n of these donor and a c c e p t o r a t oms to m a i n t a i n a c o n s t a n t total charge in the source and drain regions.

341

The d o p a n t c o n c e n t r a t i o n s in MOS t r a n s i s t o r s have i n c r e a s e d m o r e t h a n 100-fold over the p a s t 20 years a n d are on the order of 1% of the silicon lattice density for c u r r e n t device technologies. The maximum t h e r m o d y n a m i c a l l y stable c o n c e n t r a t i o n

of a t o m s in silicon,

or

solid

solubility, varies for different d o p a n t atoms. It is a f u n d a m e n t a l t h e r m o dynamic p r o p e r t y a nd is not d e p e n d e n t on the m e t h o d of i n c o r p o r a t i o n of the d o p a n t atoms. Above the solid solubility limit, the d o p a n t a t o m s begin to interact with each o t h e r as a r e s u l t of e l e c t r o c h e m i c a l i n t e r a c t i o n s and the strain fields c a u s e d by the atomic size m i s m a t c h of the d o p a n t a t o m s a nd the silicon lattice. This leads to the formation of clusters of dopmnt atoms, wh ich are not located on silicon lattice sites a n d do not i n c r e a s e the mobile charge density (3). Unfortunately, the charge c o n c e n t r a t i o n s n e e d e d for c u r r e n t process technologies are at the solid solubility limit for the d o p a n t a t o m s c u r r e n t l y in use. New d o p a n t at om s have b e e n evaluated, b u t none have yet been found to create higher c o n c e n t r a t i o n s of mobile charge. Thus, unl e s s new m e t h o d s are developed, future scaling of the

t r a n s i s t o r will result

in

a loss of total charge,

an increase

in

resistance, and a potential decrease in p e r f o r m a n c e . Scaling of the gate oxide insulating material is facing an equally critical f u n d a m e n t a l limit. The gate oxide-silicon s y s t e m can be t h o u g h t of as a parallel plate capacitor. By applying a voltage on the gate e l e c t r o d e c h a r g e is a t t r a c t e d or repelled at the silicon interface. T h i n n i n g the insulating oxide material increases the

electric

field

and resul t s in a s t r o n g e r

coulombic force, which increases' the charge density in the silicon and leads to lowered r e s i s t a n c e a nd improved device c h a r a c t e r i s t i c s . However, increasing the electric field in the insulating oxide can cause the material to b r e a k down, resulting in device failure. For previous technology generations, this has d e t e r m i n e d the m i n i m u m oxide t h i c k ness t h a t could be used. As supply voltages are scaled with each generation, the oxide t h i c k n e s s has been scaled to m a i n t a i n the s a m e m a x i m u m electric field. However, within the last few technology g e n e r a t i o n s , a new f u n d a m e n t a l limit to t h e scaling of the insulating oxide has emerged. The oxide layer has b e c o m e so thin t h a t q u a n t u m m e c h a n i c a l t u n n e l i n g of electrons f r o m the silicon

342

s u b s t r a t e to the gate electrode is now possible (4). The probability of an electron t u n n e l i n g t h r o u g h a potential barrier d e p e n d s exponentially on the t h i c k n e s s and potential energy of the barrier. State-of-the-art gate oxide t h i c k n e s s e s are c u r r e n t l y between 1.5 and 2.0 n m (5). T hi s r e p r e s e n t s 3 to 4 atomic layers of oxide. At these dimensions, c u r r e n t flow t h r o u g h the gate oxide as a result of electron t u n n e l i n g b e c o m e s substantial. The t u n n e l i n g process does not appear to damage the oxide, but the resulting gate leakage can close circuit failures b e c a u s e circuit designs a s s u m e no appreciable gate current. Even ff circuit t e c h n i q u e s can be designed to deal with the leakage, the a m o u n t of power c o n s u m e d will b eco me unacceptably large. If the insulating gate dielectric c a n n o t be scaled, MOS device p e r f o r m a n c e will be severely degraded. Scaling of the gate dielectric is r eq u ir ed not only for the capacitive coupling of the gate to the c h a n n e l t hat decreases device resistance; it is also critical for scaling t h e t r a n s i s t o r length. I nc r eas e d coupling of the gate to the c h a n n e l allows a higher doping density to be u s e d in the c h a n n e l while m a i n t a i n i n g a low r e s i s t a n c e w h e n the t r a n s i s t o r is switched into the c o n d u c t i n g state. T hi s increase in the c h a n n e l doping density increases the c h a n n e l barrier, thereby improving the isolation between source and drain w h e n t h e t r a n s i s t o r is t u r n e d off. This p e r m i t s the lateral distance bet w een t h e source and drain regions to be scaled. Thus, d e c r e a s e d capacitive coupling and inability to scale lateral d i m e n s i o n s m a y result ff oxide t h i c k n e s s c a n n o t be scaled. Statistical fluctuation is also a potentially f u n d a m e n t a l limit for c o n t i n u e d t r a n s i s t o r scaling (6). The t h a t only about a h u n d r e d trical characteristics. As a distribution of the a t o m s

t r a n s i s t o r d i m e n s i o n s have become so small d o p a n t atoms are n e e d e d to control the elecresult, small c hanges in the exact n u m b e r and can cause appreciable c h a n g e s in the device

behavior. The statistical n a t u r e of the d o p a n t distribution is i n h e r e n t to the fabrication process a nd c a n n o t easily be changed. For very large i n t e g r a t e d circuits t h a t can use more t h a n 10 million transistors, this statistical variation can cause serious design problems. Unless ways for reducing statistical variation are found, it m a y not be possible to scale

343

dimensions to the point where characteristics.

tens of atoms d e t e r m i n e

the

device

Solutions for these problems have not yet been found (7). It has b e e n proposed that the semiconductor material m u s t be changed to continue transistor scaling. Alternate semiconductor materials such as GaAs and SiGe have been evaluated for more t h a n 20 years, but although t h e s e materials have found a niche for certain applications, neither has b e e n able to solve the problems of silicon without causing even more c o m p l e x problems. Alternate insulating materials for the gate dielectric are also u n d e r evaluation. By using an insulating material with a dielectric c o n s t a n t m u c h larger t h a n that of Si02, the t h i c k n e s s of the material can be increased while still increasing the capacitive coupling. The increase in thickness would strongly decrease the electron tunneling c u r r e n t and would permit continued scaling of the transistor. Unfortunately, no material with a substantially increased dielectric c o n s t a n t that is also compatible with MOS transistors has yet been found. A substantial effort is being m a d e to increase charge concentrations by creating dynamic equilibrium. Processes such as laser annealing and epitaxial growth have been proposed for creating ultrahigh mobile c h a r g e concentrations. Unfortunately, these carrier densities are fragile, and the metastable states are extremely difficult to maintain during processing of the device. These fundamental issues have not previously limited the scaling of transistors and r e p r e s e n t a considerable challenge for the s e m i c o n d u c t o r industry. There are currently no known solutions to these problems. To continue the performance trends of the past 20 years and maintain Moore's law of i m p r o v e m e n t will be the most difficult challenge t h e semiconductor industry has ever faced. 6.20.- The.Solar Cell In one way or the other, the Sun is responsible for most of the Earth's energy. Green plant (chlorophyll) photosynthesis has provided the basis of

344

conversion of light energy into coal and crude oil in past eons. Heat from the Sun is also responsible indirectly for other forms of energy used by Man. Examples are wave and wind power (Here, energy is converted into electrical c u r r e n t by use of generators that are rotated by either form of energy). Thus, solar power usage by Man has evolved into two distinct technologies: i) Direct use of the Sun's heat energy for supplying hot water for homes by special designs of rooftop units (Also known as Solar Thermal) 2) Photovoltaics,

defined

as the conversion of light energy into

electrical energy. It is the latter with which we will be concerned, a technology which has been touted as the answer to pollution abatement. Additionally, it has b e e n recognized that World crude oil supplies will dwindle to zero within t h e next 100 years, and that a suitable energy r e p l a c e m e n t will be r e q u i r e d . The Solar Energy industry traces its origin back about 150 years. Development of solar-cell technology stems from the work of the F r e n c h physicist Antoine-Cesar Becquerel in 1839. Becquerel discovered t h e photovoltaic effect while experimenting with a solid electrode in an electrolyte solution. He observed that voltage developed w h e n light tell upon the electrode. About 5 0 years later, Charles Fritts c o n s t r u c t e d t h e first true solar cell using junctions formed by coating the s e m i c o n d u c t o r , selenium, with an ultrathin, nearly t r a n s p a r e n t , layer of gold. Fritts' devices were very inefficient converters of energy as they t r a n s f o r m e d less t h a n 1% of the absorbed light-energy into electrical energy. T h o u g h inefficient by today's standards, these early solar cells fostered among some a view of abundant, clean power. In 1891, R. Appleyard wrote of "The blessed vision of the Sun, no longer pouring his energies u n r e q u i t e d into space, but by m e a n s of photoelectric cells which would cause t h e total extinction of s t e a m engines and smoke". ....... By 1927, another metal semiconductor junction solar-cell (In this case m a d e of copper and the semiconductor copper oxide), had b e e n

345

d e m o n s t r a t e d . By 1935, bot h the s e l e n i u m cell and the copper oxide cell were being employed in light-sensitive devices s u c h as p h o t o m e t e r s for use in photography. However, these early solar cells stiU had an e n e r g y conversion efficiency of less t h a n 1 percent. This i m p a s s e was finally overcome with the d e v e l o p m e n t 1941.

Thirteen

years later

of the silicon solar cell by Russell Ohl in

three

other

American

researchers,

G.L.

Pearson, Daryl Chapin, and Calvin Fuller, d e m o n s t r a t e d a silicon solar cell capable

of a 6%

energy-conversion

efficiency

when

used

in

direct

sunlight. By the late 1980's, silicon cells, as well as those m a d e of galliumarsenide, with efficiencies of m o r e t h a n 15% h a d been fabricated. In 1989, a c o n c e n t r a t o r solar-cell, a type of device in w hi ch sunlight is c o n c e n t r a t e d onto the cell surface by m e a n s of lenses achieved an efficiency of 37%, be c a us e of the i n c r e a s e d intensity of the c o l l e c t e d energy. In general, solar cells of widely varying efficiencies a n d cost are now available. At present, solar-energy voltaics are b a s e d u p o n m o n o c r y s t a n i n e silicon (Si) discs, similar to the i n t e g r a t e d - c i r c u i t industry. As a m a t t e r of fact, reject-discs from t h a t i n d u s t r y are u s e d to m a k e solar-cells w h e r e i n t h e discs are cleaned and the reject

IC circuits are removed

before

the

silicon-disc is p r o c e s s e d to m a k e a solar-cell. However, solar cells based on the III-V transition e l e m e n t s (such as GaAs alloys, where the valence state of the two m e t a l s is Ga 3+ and As 5+) can also be fabricated. But t hey are, at present,

too costly. Cost is a major c o n s i d e r a t i o n in solar-cell

technology since any electricity g e n e r a t e d by solar power m u s t be p r i c e d similar to the actual cost of us i ng coal or off to generate electrical p o w e r (what we call "electricity"). At present, solar-power electricity g e n e r a t e d from silicon discs costs approximately twice t h a t of coal- or off-based g en er atin g plants. (Factored into this is the cost of prevent i ng pollution from the g e n e r a t i n g plants). Since solar power is non-polluting, it has, potentially, a large advantage ff the actual energy efficiency can be rai sed from about 15% to higher values. Thus, the major factors t h a t m u s t be a d d r e s s e d by solar cell m a n u f a c t u r e r s are the efficiency and cost of conversion of light energy into electrical energy. Although other types of solar cells b a s e d u p o n transition e l e m e n t s s u c h as GaAs and the like exhibit higher efficiencies, the m u c h lower cost and availability of Si discs

346

m a k e it the m a t e r i a l of choice w h e n cost c o n s i d e r a t i o n s are t a k e n into account. A solar cell r e q u i r e s

a p-n junction

(positive

a n d negative)

which

is

activated w h e n p h o t o n s c o m e into c o n t a c t w i t h it. T h e p - n j u n c t i o n t h e n acts as a "diode" to g e n e r a t e a voltage a n d r e s u l t i n g electrical c u r r e n t . In solar cells b a s e d on crystalline silicon, the p - s t a t e is achieved by diffusing a trivalent ion s u c h as Ga 3+ into t e t r a v a l e n t silicon, Si 4+, while As 5+ c a n be u s e d to form the n-layer. C o n d u c t i o n in the p-layer involves t h e m o v e m e n t of "holes" t h r o u g h the electrically n e u t r a l silicon while t h e n - l a y e r c o n d u c t i v i t y involves e l e c t r o n s . Because Si 4+ h a s four b o n d s , it h a s a low conductivity

because

of

its

diamond-like

structure,

i.e.-

it

forms

t e t r a h e d r a l b o n d s t h r o u g h o u t the s t r u c t u r e . If a trivalent ion is diffused into the s t r u c t u r e , an electrical defect r e s u l t s in f o r m a t i o n of a "hole", i.e.lack of an electron. Likewise, t h e diffusion of a p e n t a v a l e n t ion r e s u l t s in an "extra" electron, r e s u l t i n g in a n-layer w i t h i n the b u l k of the silicon. A. C r y s t a l l i n e ~ ~ s t a l l i n e

a n d A m o r p h o u s Solar Cells

Solar cells b a s e d on crystalline silicon discs g e n e r a l l y have c o n v e r s i o n efficiencies

of about

15%,

or less,

depending

upon

the

manner

of

fabrication. The efficiency of silicon solar-cells is limited to about 2 0 % because

the

long-wave

emission,

i.e.-

near

infra-red

and

infra-red

radiation, from the S u n is not a b s o r b e d in the silicon m a t r i x , or is n o t a b s o r b e d in the silicon close e n o u g h to the p - n j u n c t i o n for the p h o t o g e n e r a t e d c h a r g e c a r r i e r s to be collected

by the j u n c t i o n . To d e c r e a s e

costs, s o m e m a n u f a c t u r e r s have t u r n e d to poly-crystalline silicon w h e r e melted

silicon is c a s t in a m o l d

correctly,

a large-grained

a n d cooled.

polycrystalline

If this is a c c o m p l i s h e d

silicon

results.

Another

less

obvious benefit results. The silicon c a n be c a s t in a s q u a r e m o l d a n d t h e resulting

square

discs will fit t o g e t h e r

more

tightly t h a t the

round

m o n o c r y s t a l l i n e discs, grown by the Czochralski c r y s t a l - p u l l i n g t e c h n i q u e . Efficiencies of about 16% have b e e n

achieved. A n o t h e r

i n n o v a t i o n has

b e e n the d e v e l o p m e n t of a m o r p h o u s , i.e.- glassy (not crystalline), silicon solar-cells. T h e s e are m a d e by e v a p o r a t i n g (or s p u t t e r i n g ) a thin-film of silicon u p o n a suitable s u b s t r a t e . Both p- a n d n-silicon c a n be e v a p o r a t e d

347

sequentially to form a suitable p - n j u n c t i o n . The advantage of t h i n - f i l m solar-cells lies in efficient u s e of silicon material. But, the d i s a d v a n t a g e lies in lower energy c o n v e r s i o n efficiency a n d m o s t a m o r p h o u s silicon solar cells only have a n efficiency of 11-12%. However, if the silicon is s u b j e c t e d to h y d r o g e n gas d u r i n g m a n u f a c t u r e , a h y d r o g e n a t e d film results, said to have i m p r o v e d efficiencies of u p to 16%. T h u s , t h r e e (3) types of silicon have b e e n u s e d to fabricate solar cells: m o n o c r y s t a l l i n e , polycrystalline a n d a m o r p h o u s films. Silicon solar-cells are fabricated u s i n g silicon discs in a m a n n e r similar to t h a t d e s c r i b e d previously for m a n u f a c t u r e of i n t e g r a t e d circuits. T h e basic external s t r u c t u r e u s e d in solar cells is s h o v m in the following diagram:

Several differences from t h a t of an i n t e g r a t e d circuit c a n be noted. First of all, two (2) electrical c o n t a c t s m u s t be e s t a b l i s h e d across the b u l k of t h e silicon wafer. %nnen light s t r i k e s the surface of the solar cell, its a b s o r p t i o n w i t h i n the silicon b u l k releases e l e c t r o n s w h i c h are c o l l e c t e d as a c u r r e n t . Also s h o w n is the p - n j u n c t i o n . However, the actual silicon disc is only about 3 5 0 vm. in t h i c k n e s s . Diffusion p r o c e s s e s are used, as a m a t t e r of practicality, to form b o t h the p-layer a n d the n-layer. T h u s , t h e

348

actual p h y s i c a l a n d electrical c o n f i g u r a t i o n is b e t t e r visualized as s h o w n in the following d i a g r a m :

Once the silicon disc is cleaned, the first step is diffuse ions into e i t h e r side of the silicon disc to first form e i t h e r the p-layer or the n-layer. S o m e m a n u f a c t u r e r s like to have the n-layer closer to the light source, as s h o w n in t h e above d i a g r a m , while o t h e r s prefer the opposite. At any r a t e , ions like B3+ a n d p5+ are generally u s e d to form the active electrical layers. A n u m b e r of differing p r o c e s s e s have b e e n developed to do this, t h e e x a c t n a t u r e of w h i c h d e p e n d i n g u p o n the specific m a n u f a c t u r e r of solar cells. Sputtering,

v a p o r - p h a s e a n d e v a p o r a t i o n are used. T h e

most

common

p r o c e s s u s e s a volatile b o r o n or p h o s p h o r o u s c o m p o u n d to c o n t a c t t h e surface. Obviously, the o t h e r side must be protected d u r i n g the p r o c e s s to f o r m one or the o t h e r active layer. Clearly, the silicon disc n e e d s to be h e a t e d as well d u r i n g the p r o c e s s to aid the diffusion p r o c e s s .

Note t h a t t h e

surface will be rich in diffusing species a n d t h a t t h e d e n s i t y of s p e c i e s declines w i t h i n t h e interior, forming a diffuse layer w h i c h is d e n s e n e a r t h e top a n d t h i n n e r in the i n t e r i o r of the silicon. W h a t h a p p e n s is t h a t

349

once t h e ion c o n t a c t s t h e silicon surface, it "hops" from site to site into the i n t e r i o r of the b u l k of t h e silicon m a t r i x . Since diffusion p r o c e s s e s are n o t o r i o u s l y slow, t h e e l e c t r i c a l - l a y e r s will be only a few m i c r o n s

(~M)

thick. The silicon disc is about 3 5 0 pM t h i c k so t h a t the b u l k of t h e silicon t h i c k n e s s is n o t electrically (but is optically) active. This h a s a beneficial effect on t h e solar-cell p e r f o r m a n c e since the t h i c k n e s s l a b e l e d "Absorber layer" is actually t h e p - n j u n c t i o n . The p h o t o n s s t r i k i n g t h e surface of t h e solar cell are a b s o r b e d w i t h i n t h e i n t e r i o r layer, having t r a v e r s e d t h e "anti-reflection" layer. The function of this layer is to t r a p the light falling u p o n t h e solar cell a n d to p r o m o t e the t r a n s m i s s i o n of p h o t o n s into t h e e n e r g y c o n v e r s i o n layers below. Materials s u c h a s silicon oxides a n d titanitu-n oxide are e m p l o y e d

as the a n t i - r e f l e c t i o n layer in

m o s t s i l i c o n - b a s e d solar cells. T h e t r i c k is to form an a n t i - r e f l e c t i o n layer w h o s e refractive i n d e x is e q u a l or j u s t slightly above t h e refractive i n d e x of the m a t r i x , h e r e the silicon disc. T h e b a s e c o n t a c t is a l m o s t always t h a t of a c o n d u c t i v e m e t a l like silver or a l u m i n u m . The top c o n t a c t is m a d e via m e t a l s t r i p e s or c o n t a c t fingers w h i c h are p a t t e r n e d a c r o s s the top of t h e silicon solar cell. A b u s b a r is u s e d to c o n n e c t all of t h e collection f i n g e r s to t h e electrical o u t p u t . T h e collection fingers allow m o s t of t h e surface a r e a of t h e disc to b e c o m e active. Only about 0.5% of t h e top surface is usually covered by the m e t a l stripes.

The

rear aluminum contact

also

serves as a highly reflective layer as well. In a c o n v e n t i o n a l solar cell b a s e d on a m o n o - c r y s t a U i n e disc, t h e s t r i p e s or c o n t a c t fingers are

screen-priated,

u s i n g a s i l v e r - b a s e d alloy, in a r e s i n -

b a s e d ink. ,after firing to r e m o v e t h e organic resin, t h e b u s b a r c o n t a c t s are s c r e e n - p r i n t e d

and

then

fired.

However,

some

of t h e

current-

g e n e r a t i n g surface is still s h a d e d b y t h e b u s b a r a n d a t t e n d a n t stripes. T h e m o s t r e c e n t i n n o v a t i o n h a s b e e n t h e u s e of a l a s e r to scribe t h e s e g r i d lines as grooves. T h e collection grid is t h u s b u r i e d in the surface of t h e disc,

reducing

cell

shading

and

making

possible

energy

conversion

efficiencies of u p to 18%. T h e top c o n d u c t o r is a l m o s t always in a g r i d - p a t t e r n to allow as m u c h a r e a as possible o p e n to c o n t a c t w i t h light p h o t o n s . Note t h a t m e t a l s are n o t light-transparent.

T h e b o t t o m c o n t a c t does n o t n e e d

to be p a t t e r n e d .

350

Hence the top grid p a t t e r n is usually widely spaced but not the extent t hat the electrical contact layer will have difficulty in collecting the c u r r e n t p r o d u c e d by the cell's ot her active layer. Clearly, the silicon disc n e e d s to be h e a t e d as well during the process to aid the diffusion process. Note t hat the surface will be rich in diffusing species and t h a t the density of species declines within the interior What h a p p e n s is t h a t once the ion c o n tacts the silicon surface, it "hops" from site to site into the interior of the b u lk of the silicon matrix. When different materials are placed in contact with another, an electrical field exists at the interface, or junction, between the materials. It is this fact t h a t allows the formation of an optical-diode s u c h as a solar ceil. T h e role of the n- and p-layers is to establish this built-in electrical field. T hi s field is n e e d e d since it dictates the direction of electrical current . W h e n light is absorbed within the absorber layer, free electrons result. In o t h e r words, specific Si a t o m s within the bul k layer are excited and e l e c t r o n s can move. The

absorption of light within the

interior

absorber layer

results in energetic free-electrons, causing a c u r r e n t w hi ch is collected by the c o n d u c t o r s for use in an external circuit to do useful work. Silicon is particularly advantageous since its absorption s p e c t r u m lies is st rong in the red an d near infrared portion of the Sun's s p e c t r u m , a region w h e r e m u c h of the Sun's energy lies. The latest innovation has been the use of "layered" solar-cells in w hi ch t h i n - t r a n s p a r e n t trims are t r a n s p o s e d u p o n the s ame substrate. The following shows the types of s e m i - c o n d u c t o r being used: 6.20.3Cell T y p e .

Spectral. Regio n

G e r m a n i u m , Ge

Near I n f r a r e d

Silicon, Si

Red or Near I n f r a r e d

Gallium Arsenide, GaAs

Orange and R e d

GaUimTi I n d i u m Phosphide, GaInP2

Green and Yellow

Gallium Nitride, GaN

Ultraviolet and Blue

or Gallium Phosphide, GaP Projected Efficiently

48.8%

351

In this Chart, the individual semi-conducting layers are p r e s e n t e d as well as the portion of the solar s p e c t r u m each one is supposed to absorb. Obviously, if each layer is not t r a n s p a r e n t , t h e n part of the Sun's energy is wasted. Thus, the 4 8 % efficient of such solar cells may not be achievable. Note that afire-layer solar-cell is proposed. However, c o s t - c o n s i d e r a t i o n s may m a k e this type of solar cell impractical since the cost of electricity will be too much. A three-layer type of solar-cell has been built which had an efficiency of about 39% (consisting of Si, GaAS and GaN). But until t h e price of other energy sources mount, or until the cost of gallium-based materials drops, a layered solar-ceU will not be economically feasible. B.-Thin Film Solar Cells We have already m e n t i o n e d a m o r p h o u s silicon solar cells. New p r o c e s s e s have been developed to m a n u f a c t u r e solar cells based upon deposition of very thin films of photosensitive materials. Such processes have a distinct cost advantage since once t h e films are deposited, little f u r t h e r processing is needed to form the final solar cell module. Amorphous silicon is the only thin film process used for m a s s p r o d u c t i o n of solar ceils. Initially, such cells possessed very low efficiencies of less t h a n 10%, due to their i n h e r e n t degradation problems. To overcome this deficiency, extra layers have been added. In some cases, a triple junction is formed to increase the internal electrical field and hence the collection efficiency. Additionally, extra layers w h i c h capture light at o t h e r wavelengths because of change in their bandgap energy have b e e n employed. All of this increases costs but improves conversion efficiency. As m e n t i o n e d above, one of these modifications has included formation of the silicon hydride, S i H x , which has a slightly modified absorption spectrum. A prime c o n t e n d e r for leading thin Fflrn technology as applied to solar cells is c a d m i u m telluride {CdTe). Its bandgap is almost ideal for use as a solar cell for energy conversion from the Sun's spectrum. Here, CdTe and c a d m i u m sulfide (CdS) are used to produce a low cost thin film solar cell

352

without the i n h e r e n t stability and degradation problems of a m o r p h o u s silicon. The process uses electrodeposition upon a substrate like glass. Obviously, a conductive coating like tin oxide is needed. Efficiencies over 25% are c o m m o n for the CdTe solar cell. In 1998, a full scale p r o d u c t i o n effort was started by British Petroleum Corp., using their "Apollo" process. 6.2 I.- Piezo-Electric Materials and T h e i r Uses Piezoelectricity involves the creation of an electric charge (or voltage) by applying p r e s s u r e to a material. Discovered by Pierre and J a c q u e s Curie in 1880, this p h e n o m e n o n occurs in m a n y non-metallic crystals with noncentrosymetric crystal s t r u c t u r e s and some degree of ionic bonding. Upon the application of stress, a polarization charge (P) per unit area is c r e a t e d which equals do, where o is an applied stress and d is the piezoelectric modulus. The constant, d, which defines piezoelectric behavior, is a t h i r d r a n k tensor (which m e a n s that the applied force generates voltage in varying degrees, depending upon the direction in the piezoelectric crystal structure).. Thus, piezoelectric behavior is highly d e p e n d e n t on the direction(s) along which a crystal is stressed. For example, in a quartz crystal, s t r e s s i n g along the [100] direction will cause polarization (a voltage), while stressing along the [001] will not. While some single crystal ceramics, such as quartz, are still used in piezoelectric devices, the emergence of polycrystalline barium titanates (BaTI03) in the 1940s, and lead zirconate titanates (PbZrTi03), known as PZT, in the 1950s, led to most m o d e r n applications of piezoceramics. Both materials (and modifications of t h e m ) have high values of d, and thus can create a relatively large voltage for a given Stress m called a d i r e c t piezoelectric effect. Alternatively, t h e materials produce a "large" strain if an electric field is applied--called a c o n - v e r s e piezoelectric effect. For polycrystaUine piezoceramics to work, the electrically charged dipoles within the entire piezoelectric c o m p o n e n t m u s t be aligned by placing t h e piezoceramic within a high electric field--a process known as poling. T h e ability of piezoceramics to almost instantaneously convert electrical

353

c u r r a n t to m e c h a n i c a l d i s p l a c e m e n t a n d vice v e r s a m a k e s t h e m useful t r a n s d u c e r s . The efficiency of conversion b e t w e e n m e c h a n i c a l and electrical energy (or the converse) is m e a s u r e d by a p a r a m e t e r called t h e coupling coefficient. PZT b a s e d materials are widely u s e d b e c a u s e of t h e i r high coupling coefficients. I. Applications Sonar: The use of piezoceramics (converse mode) to g e n e r a t e a pulsed under~vater acoustic wave and t h e n receive a reflect(~l e c h o (direct m o d e ) - - s o n a r - - w a s the first practical application of piezoelectrics. By the end of World War I, F r e n c h engineers could m e a s u r e the d e p t h of s u b m e r g e d objects using quartz t r a n s d u c e r s . C o n t e m p o r a r y s o n a r devices generally have a pulse emitter of PZT b a s e d material o p t i m i z e d for g e n e r a t i n g strong pulses. The "receiver" consists of arrays of h y d r o p h o n e s will ceramic compositions optimized to p r o d u c e t h e m a x i m u m electrical signal from a minimal echo. Modern h y d r o p h o n e t r a n s d u c e r s are composite s t r u c t u r e s with piezoceramic rods m o u n t e d in a polymer. This e n h a n c e s the o u t p u t voltage per unit u n d e r p r e s s u r e . Key to these systems are very sophisticated electronics t h a t can s e p a r a t e physical a n d i n s t r u m e n t a l noise from the actual r e t u r n e d signal. Medical Ultrasound: Many of us have first seen our u n b o r n c h i l d r e n or g r a n d c h i l d r e n t h a n k s to medical u l t r a s o u n d technology. This application is s o m e w h a t related to sonar in t h a t both use pulse e c h o m e t h o d s to send a n d receive signals. However, in this s h o r t - r a n g e application, t r a n s d u c e r arrays can be designed to form images. Micropositioning a n d M i c r o - m o t o r s . The converse p i e z o e l e c t r i c effect p r o d u c e s solid state motion. Piezoceramics are u s e d as actuators in m a n y micro-positioning applications, s u c h as fiber optic alignment, m i r r o r tilting for optical c o m m u n i c a t i o n a n d imaging, wafer m a s k alignment in the s e m i - c o n d u c t o r industry, a n d hydraulic servo valves. In exceptional cases, positioning can be controlled to a n a n o m e t e r or less. Piezoelectric . T r a n s f o r m e r s :

If a two-layer t r a n s d u c e r is designed so

354

that a low voltage applied to the first layer to produce a strain, a larger strain is imposed on the second layer. This causes the second layer to generate a higher voltage t h a n the one applied to the first layer. Thus, we have a step-up transformer. When operated at the a p p r o p r i a t e frequencies, voltage increases from 3 V to 1500 V are attainable. Such transformers (inverters) are used to power cold-cathode fluorescent lamps for notebook c o m p u t e r displays. Active noise and vibration damping: Noise and vibration are p r o d u c e d by waves pulsing through the air or through structural materials. Creating an "anti-wave" of the same frequency spectrum, but opposite in amplitude, will cancel out or at least decrease the objectionable noise or vibration. Piezoceramic t r a n s d u c e r s coupled with sophisticated signal detection and generation electronics are used for such d a m p i n g applications. Suggested Reading I. "The Growth of Single Crystals"- R.A. Laudise, Prentice-Hall, New York (1970). 2. "The Art and Science of Growing Crystals"- Ed. by J.J. Gilman, Wiley, New York (1963). 3. "Crystal Growth"- Ed. by S. Peiser, Pergamon, New York (I 967). 4. "Phase Diagrams-Materials Science & Technology"- Ed. by A.M. Alper, Academic Press, New York (1970). 5. "The Metal-Insulator Transition in Selected Zandt, Ann. Rev. Mat. Sci., 5 , 2 2 5 - 2 7 8 ( 197 5).

Oxides" - Honig & Van

R e f e r e n c e s on S i l i c o n D e v i c e s I. Gordon Moore, /EDM Tech. Dig. (1975), p. I I. 2. RH Denard et al, IEEE J. Solid-State Circuits SC-9 256 (I 9 7 4 ) 3. KS Jones, S Prussin & ER Weber, Appl. Phys., A45 1 (1.988)

355

4. J Maserjian & GP Peterson, Appl. Phys. Lett., 25 50 (I 9 7 4 ) 5. HS Momose et al,/E/~M 2"ech,/~g. (1994), p. 5 9 3 6. B Hoeneisen & CA Mead, Solid-State Electron., 15 819 ( 197 2) 7. S T h o m p s o n , P P a c k a n & M Bohr, Intel. Tech., Q3 1 ( 1 9 9 8 ) P r o b l e m s for Chapt.,er 6 I. From 6.2.1. "Elements of Furnace Design", p r e p a r e a drawing s h o w i n g how these

components

are c o n n e c t e d

together,

both

physically and

electrically, to form a furnace suitable for heat i ng a crucible containing a powder for solid state reaction in air, up to 1200 ~

Be sure to use t h e

right insulation, heat er and crucible. 2. Do the s ame for a furnace designed to heat a crucible in a r e d u c i n g a t m o s p h e r e , up to 1600 ~ 3. Design a furnace capable of m el t i ng AI203 in air; do the sam e for m e l t i n g YAG. Be sure to use the right crucible. 4. List the ways t h a t one can obtain a "seed" crystal. Is it possible to use one type of crystal-growth m e t h o d to obtain a seed crystal for Czochralski growth? 5. We wish to grow crystals of CaS and SrS, activated by 0.01 mol of Ce2S ~. These crystals will be u s e d to evaluate the l u m i n e s c e n t p r o c e s s e s w h i c h take place in the Ce ~+ activator center, particularly the polarized e m i s s i o n properties. Det e r m i ne which one (or more) crystal growing m e t h o d s is m o s t suitable for obtaining single crystals in this cubic system. Note t hat we wish to grow the crystal along the {1,1,1} plane since this is t h e direction of highest a t o m - d e n s i t y a nd will give the best crystals for our optical study. Note t h a t a trivalent cation is being situated u p o n a divalent lattice site. 6. MgO has a wide infra-red t r a n s m i s s i o n s p e c t r u m , s p a n n i n g 0.2 ~rn to 9.0 ~rn of usable range. Outline a crystal growing m e t h o d to obtain single crystal material suitable for this purpose. What m e t h o d s are not suitable?

356

7. Ti02 is a m a t e r i a l w h i c h Coatings

h a s f a r - r e a c h i n g u s e s as a UV m o d i f i e r .

on steel have b e e n s h o w n to inhibit bacterial g r o w t h a n d e v e n

kill b a c t e r i a on its surface. We w i s h to grow a single crystal in o r d e r to clarify a n d define this anti-bacterial action. Outline one or m o r e m e t h o d s for o b t a i n i n g a single crystal of TiO2. Since this action m a y be one of a defect-structure, obtaining

i.e.- oxygen vacancies, include at least one m e t h o d for

a controlled

defect

structure

of TiO2.x, w h e r e

x

can

be

controlled. 8. We w i s h to test a new type of c e r a m i c t u b e to the AI20s t u b e n o r m a l l y u s e d to fabricate h i g h - p r e s s u r e s o d i u m l a m p s in o r d e r to c o m p a r e l a m p qualities a n d life-time o p e r a t i o n . Select a m e t h o d w h i c h w o u l d p r o d u c e the d e s i r e d r e s u l t s a n d d e s c r i b e h o w this w o u l d be a c c o m p l i s h e d . N o t e t h a t the c e r a m i c t u b e r e q u i r e s b o t h s t r e n g t h a n d a high m e l t i n g p o i n t .

357

.......

Chapter 7

M e a s u r e m e n t of Solid State P h e n o m e n a W h e n solids react, we w o u l d like to k n o w at w h a t t e m p e r a t u r e the solid state reaction t a k e s place. If the solid d e c o m p o s e s to a different composition, or p h a s e , we would like to have this knowledge so t h a t we c a n predict a n d u s e t h a t knowledge in p r e p a r a t i o n of desired materials. Sometimes, i n t e r m e d i a t e c o m p o u n d s form before the final p h a s e . In this chapter, we will detail some of the m e a s u r e m e n t s u s e d to characterize the solid state a n d m e t h o d s u s e d to follow solid state reactions. This will consist of v a r i o u s t y p e s of t h e r m a l analysis (TA), including differential t h e r m a l analysis (DTA), t h e r m o g r a , ~ n e t r i c analysis (TGA) a n d m e a s u r e m e n t s of optical properties. 7.1.- Methods of M e a s u r e m e n t of Solid State Reactions If we wish to characterize a solid s t a t e reaction, from initial compound(s) to final product(s), there are only a few m e t h o d s we c a n use. Of p r i m a r y i m p o r t a n c e is x-ray identification since we m u s t k n o w w h a t we s t a r t e d with, a n d w h a t we e n d u p with. We have already d i s c u s s e d the x-ray m e t h o d in some detail in the s e c o n d c h a p t e r a n d how one goes a b o u t u s i n g t h a t m e t h o d . If a solid is stable at r o o m t e m p e r a t u r e , it will r e m a i n in t h a t state until s o m e form of energy is applied. In general, it is the application of h e a t t h a t c a u s e s s u c h change. We find t h a t two effects c a n occur s i m u l t a n e o u s l y , a t h e r m a l c h a n g e a n d a weight c h a n g e (but not always). As an example, consider CaC03. W h e n it is h e a t e d to a b o u t 8 4 0 ~ reaction, vis: 7. I . I . The arrow

CaC03

+ heat

it forms c a l c i u m oxide, CaO, b y solid state

~

CaO

+ C02

indicates t h a t a reaction h a s t a k e n place. The gas formed, C02, is

volatile a n d the final p r o d u c t h a s a lower m o l e c u l a r weight t h a n the star-ling material. T h u s , a w e i g h t l o s s occurs. The o r t h o r h o m b i e s t r u c t u r e of C a C 0 3

358

c h a n g e s to the cubic form of CaO. T h e r m a l energy is r e q u i r e d to r e a r r a n g e the atoms. W h a t h a s actually h a p p e n e d is t h a t we have exceeded the b o n d i n g energy of one c o m p o u n d (CaC03) b y i n c r e a s i n g the vibrational energy of the a t o m s to the point w h e r e chemical b o n d s are b r o k e n (by a d d i n g heat). This o c c u r s at a b o u t 8 4 0 ~ a n d C 0 2 gas is formed w h i c h is stable (and volatile). W h e n we cool the product, we find t h a t we have CaO. B e c a u s e this c h a n g e requires h e a t to be absorbed, the overall p r o c e s s is called e n d o t h e r m i c . If it h a d released h e a t d u r i n g the change, it would be called e x o t h e r m i c . M e a s u r e m e n t of the weight c h a n g e is called t h e r m o g r a v i m e t r i c a n a l y s i s (TGA) whereas

measurement

of the

thermal

change

accompanying

a structural

m e t a m o r p h o s i s is called t h e r m a l analysis (TA). Some c o m p o u n d s do not lose weight b u t m e r e l y c h a n g e their s t r u c t u r a l form. A good example of this involves zinc sulfide, ZnS. This m a t e r i a l is o b t a i n e d b y precipitation from solution a n d is cubic

at r o o m t e m p e r a t u r e .

When

heated

to a b o u t

1100 ~

cubic ZnS

(sphalerite) c h a n g e s to a h e x a g o n a l (wurtzite) form. If wurtzite is t h e n cooled to room t e m p e r a t u r e , it r e m a i n s hexagonal. Only ff wurtzite is cooled very slowly t h r o u g h the t r a n s i t i o n t e m p e r a t u r e does it revert to the cubic sphalerite form. AIk, CHANGES IN PHASE involve a release or a b s o r p t i o n of calories. One r e a s o n for this is t h a t e a c h s o l i d h a s its own h e a t c a p a c i t y . T h a t is, there is a c h a r a c t e r i s t i c h e a t c o n t e n t for e a c h m a t e r i a l w h i c h d e p e n d s u p o n the a t o m s c o m p o s i n g the solid, the n a t u r e of the lattice vibrations within it, a n d its s t r u c t u r e . The total h e a t content, or enthalpy, of each solid is defined by: 7.1.2.-

AH -- f Cp dT

, w h e r e - Cp -

(/~I/~r

w h e r e q is a h e a t quantity. T h u s , as we go from one solid to another, we see a c h a n g e in caloric content. One w a y to m e a s u r e this involves the u s e of DTA. A. DIFFERENTIAL THERMAL ANAI.YSIS (DTA) In 182 I, S e e b e c k discovered t h a t b y joining two different m e t a l wires together to

359

form a loop (two junctions), a direct c u r r e n t (DC) w o u l d flow in the circuit. A r e p r e s e n t a t i o n of this is given in the foUowing:

S e e b e c k u s e d a n t i m o n y a n d copper wires a n d found the c u r r e n t to be affected b y the m e a s u r i n g i n s t r u m e n t (ammeter). But, he also found t h a t the v o l t a g e

generated (EMF) w a s directly proportional to the d i f f e r e n c e in t e m p e r a t u r e of the two j u n c t i o n s . Peltier, in 1834, t h e n d e m o n s t r a t e d t h a t ff a c u r r e n t w a s i n d u c e d in the circuit of 7.1.3., it g e n e r a t e d h e a t at the j u n c t i o n s . In o t h e r words, the S E E B E C K E F F E C T w a s found to be reversible. F u r t h e r w o r k led to the development of the t h e r m o c o u p l e , w h i c h t o d a y r e m a i n s t h e p r i m a r y m e t h o d for m e a s u r e m e n t of t e m p e r a t u r e .

Nowadays, we k n o w t h a t the S E E B E C K

E F F E C T arises b e c a u s e of a difference in the electronic b a n d s t r u c t u r e of the two m e t a l s at the junction. This is illustrated as follows:

In this diagram, we s h o w the b a n d model s t r u c t u r e at the j u n c t u r e of two metals, e a c h of w h i c h h a s its own Ferrrd Level. The Ferrrd Level is the energy level of the electrons c o n t a i n e d in the metal. T h a t is- w h e n m e t a l a t o m s (each

360

having its own set of electrons) a s s e m b l e a n d / o r c o n d e n s e to form a metal, the electrons form a *cloud" a r o u n d all of the atoms. This cloud of electrons h a s a n average energy a n d the top of the energy level is called the Fermi Level. Using s u c h a concept helps one to m e n t a l l y conceive how electrons form energy b a n d s in a n y given solid. In c o n t r a s t to a metal, the electrons in m o s t inorganic solids form well defined energy b a n d s w h i c h are d e t e r m i n e d b y the atomic positions of the a t o m s c o m p o s i n g the s t r u c t u r e . The m a i n difference b e t w e e n inorganic solids a n d m e t a l s is t h a t the c o n s t i t u e n t electrons are confined to allowed energy zones, i.e.- Brillouin zones, in the former, b u t not in the latter. It is for this r e a s o n t h a t m o s t m e t a l s are conductive, w h e r e a s m o s t inorganic solids are not. Referring to the above diagram, flow of electrons is indicated b y the arrow. Since the height of the Fermi Level is proportional to t e m p e r a t u r e , t h e n the EMF g e n e r a t e d is a function of t e m p e r a t u r e also.

It is t h u s a p p a r e n t t h a t a

t h e r m o c o u p l e (TC) will consist of a negative a n d a positive "leg". The c o m m o n t h e r m o c o u p l e s in u s e t o d a y are listed in Table 5-1 along with the t e m p e r a t u r e range over w h i c h t h e y are useful. Also listed is the a p p r o x i m a t e EMF g e n e r a t e d over this range, as well as the n a t u r e of each "leg", i.e.- positive or negative. TABLE 7- I USEFUL TEMPERATURE RANGES FOR COMMON THERMOCOUPLE Composition

Code

O u t p u t Range

Useful Temp.

Useful

(millivolts)

Range, ~

Atm.*

(+)Copper-Constantan!-)

T

-5.28to 20.81

- 300 to 750

A,N, R

(+)Iron-Constantan(-)

J

-7.52to 50.05

-300 to 1600

R

(+)Chromel-Alumel(-)

K

-5.51 to 56.05

-300 to 2 3 0 0

A, N

(+)Chromel-Constantan(-)

E

0 to 75.12

32 to 1800

A,N, R

(-)Platinum-Pt(lO%Rh)(+),

S

0 to 15.979

32 to 2 9 0 0

A, N

0 to 18.636

32 to 3 1 0 0

A, N

0 to 38.45

32 to 5 0 0 0

N,R

(-) Pt- Pt (13% Rh)(+) (-)Tungsten(5%Re) -

R

C

W(26%Re) (+) * A = air or oxidizing; N = n e u t r a l 9R --- r e d u c i n g

361

The c o m p o s i t i o n s of the alloys u s e d to m a k e the t h e r m o c o u p l e s listed in Table 7-1 are given as follows: 7.1.5.- C o m p o s i t i o n s Used to Make T h e r m o c o u p l e s CHROMEL:

90% Ni - 10% Cr

ALUMEL:

95% Ni - 5% AI, S i , Mn

CONSTANTAN:

57% Cu - 43% Ni

In differential t h e r m a l analysis, i.e.- DTA, we u s e one t h e r m o c o u p l e 'q~ucked" a g a i n s t the voltage o u t p u t of a n o t h e r of the s a m e composition to p r o d u c e a "net" EMF. W h a t this m e a n s is t h a t either the positive (or negative) legs of b o t h t h e r m o c o u p l e s are electrically c o n n e c t e d so t h a t the n e t EMF at a n y given t e m p e r a t u r e of the two is zero. Only if one t h e r m o c o u p l e t e m p e r a t u r e differs from t h a t of the o t h e r does one o b t a i n an EMF r e s p o n s e . 7.1.6.- A Thermocouple Used for Differential T h e r m a l Analysis TC(I) = TC(2)

If we p u t a s a m p l e next to one t h e r m o c o u p l e a n d a "standard" or "reference" next to the other, we c a n follow a n y t h e r m a l c h a n g e s t h a t m a y take place as b o t h are h e a t e d since each TC g e n e r a t e s its own EMF as the t e m p e r a t u r e changes. T h u s , ff we p u t a reference material, R, directly in c o n t a c t with the "IE(1)" t h e r m o c o u p l e j u n c t i o n (hereinafter, we will refer to this t h e r m o c o u p l e j u n c t i o n as "R") a n d a sample, S, at TC(2), i.e.- "S", t h e n we c a n detect a n y t h e r m a l c h a n g e t h a t m a y o c c u r if either R or S u n d e r g o e s a t r a n s f o r m a t i o n as it is heated.

362

In this configuration, we can detect a n y t h e r m a l c h a n g e s t h a t o c c u r in the s a m p l e as c o m p a r e d to the reference. Note t h a t if b o t h TC-1 a n d TC-2 of 7.1.6. are at the s a m e t e m p e r a t u r e , n o EMF is g e n e r a t e d . Actually w h a t we are m e a s u r i n g are c h a n g e s in h e a t flow as related to Cp (see 7.1.2.). For inorganic materials, the b e s t reference material to u s e is r capacity r e m a i n s r

A1203. Its h e a t

even u p to its melting point (1930 ~

W h a t this

m e a n s is t h a t no t h e r m a l c h a n g e s o c c u r in R so t h a t a n y c h a n g e detected will be t h a t of the sample, S. In DTA, we w a n t to m e a s u r e n C p , b u t find t h a t this is actually: 7.1.7.-

[Cp(sf) - Cp(si} ]dTs

+ [ Cp ]dTR

w h e r e Si is the initial state a n d Sf is final state for a given solid state reaction of the sample, S (No reaction o c c u r s for R). It s h o u l d be a p p a r e n t t h a t we m u s t m a i n t a i n a n equal h e a t f l o w i n t o b o t h R & S s i m u l t a n e o u s l y ,

at a u n i f o r m

rate. If we raised the t e m p e r a t u r e b y steps, we would find t h a t the a c t u a l h e a t flow in b o t h R & S lags b e h i n d the furnace t e m p e r a t u r e considerably, as s h o w n in the following diagram: 7.1.8.-

Programmed (1)

Actual Temperature in Sample

E

CD

Time

IF-

363

The critical p a r a m e t e r s a s s o c i a t e d with the DTA Method have b e e n d e t e r m i n e d tobe: 7.1.9.-

FIVE PARAMETERS ASSOCIATED WI22-1 THE DTA ME22-IOD I. d T l d t = k 2. S a m p l e Size 3. Rate of Heating 4. Degree of Crystallinity of S a m p l e 5. Effects of E x t e r n a l A t m o s p h e r e

The rate of h e a t i n g generally u s e d for m o s t inorganic m a t e r i a l s r a n g e s b e t w e e n a b o u t 2 ~ to 20 ~ to 1 0 0 ~

in. while t h a t for organic c o m p o u n d s lies b e t w e e n a b o u t 15 ~

In the following diagram, a typical t h e r m o g r a m is shovm:

364

In this diagram, the a r r a n g e m e n t of the sample, S, a n d the reference, R, a c r o s s the differential TC is shown, a n d a typical DTA analysis is also given. Note t h a t at low t e m p e r a t u r e s , the DTA p e a k s are e n d o t h e r m i c . T h a t is, h e a t is absorbed. S u c h p e a k s are similar to t h o s e o b t a i n e d w h e n w a t e r - o f - h y d r a t i o n is lost, or w h e n the solid state reaction u n d e r g o e s a loss of water. The above DTA curve is similar to t h a t of the following reaction: 7. I. 1 I.-

CaHP04 - 2 H20 ~

CaHP04 + 2 H20

The first two p e a k s involve the loss of 2 w a t e r s of hydration. The broad e n d o t h e r m i c p e a k following in the above d i a g r a m is similar to t h a t we m i g h t see for a c h a n g e in composition s u c h as: 7.1.12.-

2 CaHPO4

~

Ca2 P2 0 7

+ H20

The e x o t h e r m l c p e a k m a y be a c h a n g e in s t r u c t u r e s u c h as: 7.1.13.-

~-Ca2P207

~

r

In general, solid s t a t e decomposition r e a c t i o n s o c c u r as e n d o t h e r m i c p e a k s (AH is negative a n d h e a t is absorbed) while p h a s e c h a n g e s , i.e.- c h a n g e s in s t r u c t u r e , o c c u r as exothermic p e a k s (AH is positive a n d h e a t is evolved). Note t h a t t e m p e r a t u r e c h a n g e is p r o g r a m m e d to be linear with time, in the above diagram. The c o m p o n e n t s

of a simple DTA A p p a r a t u s

are s h o w n in the

following diagram, given on the n e x t page as 7. I. 14. The first thing to note is t h a t the f u r n a c e s u r r o u n d s

the s a m p l e - h o l d e r

c o n t a i n i n g the differential t h e r m o c o u p l e s . A s e p a r a t e control t h e r m o c o u p l e controls the f u r n a c e t e m p e r a t u r e a n d s h o u l d be placed as close as possible to the position of the s a m p l e holder. Some c o m m e r c i a l m a n u f a c t u r e r s u s e the Reference leg of the differential t h e r m o c o u p l e

to control the t e m p e r a t u r e .

However, if you were to build a DTA u s i n g the c o m p o n e n t s as s h o w n in 7.1.14,

365

the separate furnace control thermocouple works j u s t as well (even t h o u g h it is not in exact juxtaposition with the sample a n d reference. A power supply is controUed by the t e m p e r a t u r e programmer. The temperature-prograrraner can be either mechanical or electrical in nature. A DC to DC amplifier is, in general, necessary to amplify the thermocouple signal to the recorder. It is possible to obtain a recorder with suI~cient sensitivity to directly record the TC signal. However, it is generally better to a m p l ~ j the TC signal in order to avoid s p u r i o u s electrical "noise" t h a t sometimes occurs. A two-pen recorder is

366

superior to a o n e - c h a n n e l recorder. In the latter case, it is n e c e s s a r y to switch from the differential TC signal to the t e m p e r a t u r e TC signal. However, in c a s e s w h e r e the 2 - c h a n n e l recorder is not available, a suitable DTA c h a r t be still be obtained. You will notice, on the above DTA g r a p h shown, t h a t the low t e m p e r a t u r e DTA p e a k s are endothermic. T h a t is, h e a t is absorbed. S u c h p e a k s are similar to t h o s e o b t a i n e d w h e n water-of-hydration is lost, or w h e n the solid state reaction u n d e r g o e s a loss of water. To u s e this a p p a r a t u s , one follows a simple procedure. The m e t h o d t h a t one follows to obtain a DTA r u n is given as follows: 7. I. 15.-

GENERAL PROCEDURE FOR OPERATING A DTA APP_ARATUS

a. Load b o t h s a m p l e a n d s t a n d a r d m a t e r i a l into the DTA holder. b. Put the DTA holder in place in the furnace. c. Set rate for t e m p e r a t u r e p r o g r a m m i n g . d. Set DC Amplifier gain. e, Set recorder gain a n d time drive f. Begin t e m p e r a t u r e program. g. Record T e m p e r a t u r e c h a n g e s The s a m p l e size (a function of its crystallinity) is u s u a l l y d e t e r m i n e d b y trial a n d error. About 500 milligrams is u s u a l l y sufficient. It h a s b e e n d e t e r m i n e d t h a t the p r o g r a m m e d t e m p e r a t u r e rate h a s a m a j o r effect u p o n the s h a p e of the p e a k s observed, as s h o w n in the following diagram, given as 7.1.16. on the next page. The r e a s o n for this is practical. One m u a t m a i n t a i n a c o n s t a n t h e a t flow a c r o s s the DTA head. For inorganics, a rate of 2 ~

s p r e a d s o u t the p e a k w h e r e a s a

rate of 20 ~ C. per m i n u t e s e e m s to be a b o u t correct for m o s t s y s t e m s .

367

7.1.16.-

] dT/dtl 2~ 5~ IO~

20 C o

One c a n p r o g r a m at higher rates, even u p to 100 o C./ rain. However, the inorganic s y s t e m c a n n o t a d j u s t fast e n o u g h to m a i n t a i n a c o n s t a n t h e a t flow. This is, of course, a m a t t e r of s y s t e m - d e s i g n . However, for organics, a rate of 100 ~

in. m a y be required. Most corranercial i n s t r u m e n t s have the correct

h e a t i n g r a t e p r o g r a m m e d into the control circuits a n d one u s u a l l y does not have to a d j u s t h e a t i n g - r a t e s . Only w h e n the DTA p e a k s are not s h a r p does one n e e d to a d j u s t h e a t i n g r a t e s to s e p a r a t e a n y near-lying p e a k s from one another. One c a n quantify the h e a t flow involved within the s y s t e m in t e r m s of the DTA p e a k p r o d u c e d . Consider the following, given as 7.1.17. on the next page. In this diagram, the sample, S, is within the f u r n a c e w h i c h is at a t e m p e r a t u r e , T S. The h e a t flow is d Q / d t a n d r is the t h e r m a l r e s i s t a n c e . The s a m p l e u n d e r g o e s a n ent_halpy c h a n g e , M-I, at its solid s t a t e reaction t e m p e r a t u r e .

368

T h u s , t h e h e a t flow is a f u n c t i o n of t h e differences in t e m p e r a t u r e of t h e s a m p l e a n d t h a t of t h e f u r n a c e , i.e.- T > T s . Then: 7.1.18.-

dQ/dt

= T-Ts/r

w h e r e t is t h e time. A n d t h e e n t h a l p y c h a n g e is t h e difference b e t w e e n t h a t of t h e s a m p l e a n d t h e h e a t flow is: 7.1.19..-

dH / dt = Cp(S) d T / dt - dQ / dt = Cp(S) d T / d t - [T- TS / r]

If we s e t u p t h e s a m e e q u a t i o n s for t h e reference m a t e r i a l , c o m b i n e t h e s e e q u a t i o n s a n d r e a r r a n g e , we o b t a i n : 7.1.20.-

I

II

HI

r[dH / dtl = [Ts - TR ] +[r{Cp(s) - Cp(R) )l dTR / d t + [rCp(S){d(Ts - TR) /dt}] Note t h a t we h a v e divided t h e e q u a t i o n into t h r e e (3) p a r t s , e a c h s u r r o u n d e d b y a b r a c k e t , i.e. []. This allows u s to i n t e r p r e t a DTA p e a k a s s h o w n in 7.1.2 I., given o n t h e n e x t page. T h e deviation f r o m t h e b a s e line (at 0) is a f u n c t i o n of b o t h I a n d II, t h a t is- t h e difference b e t w e e n s a m p l e a n d reference t e m p e r a t u r e s (I) a n d differences in h e a t c a p a c i t i e s of s a m p l e a n d reference.

369

T h u s , if the a p p a r a t u s is properly designed, I is not a problem, b u t II c a n n o t be controlled. It is II t h a t c a u s e s the deviation from linearity w h i c h r e s u l t s in a peak. The slopes are a function of the h e a t c a p a c i t y differences b e t w e e n ~ m p l e

plus reference and product plus reference. The slopes (III) o b t a i n e d are a function of differences b e t w e e n TS a n d T R . Note t h a t at the top of the peak, we still have a p p r o x i m a t e l y 1 / 2 s a m p l e (as r e a c t a n t ) a n d 1 / 2 product. Referring to 7.1.20. a n d 7.1.21., it would s e e m t h a t t h e a r e a of the DTA p e a k s h o u l d be proportional to z~I. This is indeed the case a n d a c o m p a r i s o n of p e a k a r e a s does yield a n e x p e r i m e n t a l value for ~-H(s), vis: 7.1.22.-

~-I(s)

= ~Std.

"i(s} /i(Std)

This e q u a t i o n a s s u m e s a relation b e t w e e n A a n d AH. Actually, this is not too h a r d to prove, as we c a n show in t h e followi_ng. Consider a h e a t s i n k as a block c o n t a i n i n g b o t h S a n d R (sample a n d reference). This is illustrated in t h e following diagram, given as 7.1.23. on the next page. In this diagram, we define t h r e e t e m p e r a t u r e s , TB , TS & T R , w h e r e e a c h s u b s c r i p t refers to block, s a m p l e a n d reference, respectively. We also have two h e a t capacities, C p ( s ) a n d Cp(R).

370

w h e r e a is t h e t h e r m a l d i f f u s i v i t y . For c o n v e n i e n c e , we define t h e total h e a t flow in t e r m s of h e a t flow b e t w e e n S & B, R & B, a n d R & S. We also u s e a STANDARD SAMPLE a n d r u n it a g a i n s t t h e reference, so t h a t we c a n d e t e r m i n e h o w m a n y calories p e r g r a m are r e q u i r e d for a given t r a n s i t i o n so a s to c a l i b r a t e the system. The following table, given o n n e x t page, s h o w s s o m e m a t e r i a l s s u i t a b l e for s u c h calibration, along w i t h t h e t e m p e r a t u r e at w h i c h t h e solid s t a t e c h a n g e o c c u r s . Note t h a t t h e reference m a t e r i a l , r 3, is t h e r m a l l y i n e r t w h e r e a s t h e s t a n d a r d reference m a t e r i a l s a r e n o t . A p p r o p r i a t e v a l u e s of ~

are available for

t h e s t a n d a r d reference m a t e r i a l s given in Table 5-2 o n t h e n e x t page. R e t u r n i n g to o u r d e s c r i p t i o n of a n a n a l y s i s of t h e h e a t flows p r e s e n t in DTA, as s h o w n in 7.1. 17., this gives u s t h e following e q u a t i o n s (Note t h a t -- is "defined as")

9

7.1.25.-

d T / d t -KS --

K (dQ/dt)sB

KR

--

(dQ / dt)RB

k

--

(dQ / dt)RS

371

TABLE 5 - 2 T H E R M O M E T R I C FIXED POINTS FIXED POINTS *

TEMPERATURE oC. o F.

B. P. of 0 2

- 183.0

- 297.3

S u b l i m a t i o n P o i n t of C O 2

- 87.4

- 109.2

F.P. - H g

- 38.9

- 38.0

Triple P o i n t of W a t e r

0.01

32.0

Ice P o i n t

0.00

B.P. - W a t e r

100.0

212.0

32.0

Trk)le P o i n t of B e n z e n e I B. P. of N a p h t h a l e n e

122.4

252.4

218

F.P. o f S n

231.9

B.P. of B e n z o p h e n o n e

305.9

582.6

F.P. of C d

321.1

610

F.P. o f Pb

~3 2 7 . 5

i 621.5

F.P. o f Z n

419.6

787.2

,

1424.3 449.4 [

il

B.P. o f S

.

444.7

i

832.4

F.P. o f S b

630.7

1167.3

F.P. of A1

660.4

1220.7

F.P. o f Ag

961.9

1763.5

F.P. o f A u

1064.4

1948

F.P. of C u

1084.5

1984.1

F.P. o f P d

1554

.

.

.

.

.

.

.

.

F.P. o f Pt

1772

i

2829 3222

* F.P. = F r e e z i n g Point, M.P. = M e l t i n g P o i n t , a n d B.P. = B o i l i n g P o i n t W e c a n i m m e d i a t e l y write( s e e 7 . 1 . 2 . , 7 . 1 . 1 8 . & 7 . 1 . 1 9 . ) 9 7.1.26.-

Cp(S) d T s [ d t = KS (TB - TS) + k {(TR- TS) + d ( A H ) / d t Cp(R) d T R / d t = KR (TsB- TR) + k (Ts - TR)

372

The e q u a t i o n for R is simplified b e c a u s e R is t h e r m a l l y i n e r t a n d t h e r e is no c h a n g e in e n t h a l p y involved. To simplify m a t t e r s f u r t h e r , we define: 7.1.27.-

(I)s

~

CptS) / KS

(I)R

~

Cp(R) / KR

HS

--

k/Ks

HR

--=

k/KR

M a k i n g t h e s e s u b s t i t u t i o n s , we get: 7.1.28. -

(I)s d T s / dt + ( 1+ Hs)Ts - HSTR = TB + ( 1 / KS) d(td-I} / d t

and: (I)R

dTR / dt + ( 1 +HR) TR - HR TS = TB

Now if k = 0, i.e.- t h e r e is no h e a t e x c h a n g e b e t w e e n R & S {as in a p r o p e r l y designed apparatus), and if7.1.29.-

TS = To + t {dT/dt)

i t will b e if we are p r o g r a m m i n g t h e t e m p e r a t u r e . T h e n we c a n define To a s b e i n g e q u a l to zero, so a s to o b t a i n t h e following e q u a t i o n s : 7.1.30.-

(I)s dTs / d t + T s = t (dT/dt) + ( 1 / Ks} d(AH) / d t

and:

(I)R

dTR / d t + TR = t (dT/dt)

If we are n o t in a region w h e r e a solid s t a t e r e a c t i o n is t a k i n g place, t h e n d(AH)/dt = 0. The e l u m g e in b a s e l i n e t e m p e r a t u r e (see above) is now: 7.1.31.-

ATB = d T / dt ((I)s - (I)R) = TS - TR

373

In o t h e r w o r d s , the c h a n g e in b a s e l i n e t e m p e r a t u r e is c a u s e d b y a difference in the relative t e m p e r a t u r e s of s a m p l e a n d reference, w h i c h is r e l a t e d to their relative h e a t capacities. T h u s , one n e e d s to c h o o s e the reference m a t e r i a l very carefully. If we s u b t r a c t t h e e q u a t i o n s in 7.1.30., we c a n get: 7.1.32.-

(I)s d{Ts - TR} / d t + (Ts - TR) - (dTR / dt) = ( 1 / KS) d(AH) / d t

This c a n be r e a r r a n g e d to: 7.1.33.-

r

d ATs I dt + AT - {dT/dt) = (I l KS) d(AH} I dt

AT is t h e difference in t e m p e r a t u r e b e t w e e n the reference a n d the sample. In o t h e r w o r d s , it is AT t h a t c r e a t e s the DTA peak. Since we are n o t m e a s u r i n g

a b s o l u t e v a l u e s of the t e m p e r a t u r e s , we c a n define a r ~ l a t i v e t e m p e r a t u r e : 7.1.34.-

ATRel = AT- ATB = AT- (Ts - T R)

Using t h i s relation, we get: 7.1.35.-

~ s d A T R e l / dt + ATR + { d T / d t - d T s / dt)(CI~s-CI)R} = (1/Ks) d{AH}/ dt

If we c h o o s e a s u i t a b l e reference m a t e r i a l ( s u c h as A1203 for inorgardcs), t h e n dTR / d t will be e q u a l to d T / d t . O u r e q u a t i o n is t h u s simplified to: 7.1.36.-

(I)s f (d ATRel / dt) + ATRel = (1 / K S ) f d All / d t

However, the first t e r m is e q u a l to zero, so: 7.1.37.-

f d td-I = KS f ATRel dt

This is w h a t we s t a r t e d o u t to prove, i.e.- AH e q u a l s the a r e a of the p e a k t i m e s the total h e a t flow to the sample.

374

B. DIFFERENTIAL SCANNING CAIX)RIME2~Y Since AH is proportional to the a r e a of the DTA peak, one o u g h t to be able to m e a s u r e h e a t s of reaction directly, u s i n g the equation: 7.1.22. Indeed we c a n and

such

is the b a s i s

of a related m e t h o d

called Differential S c a n n i n g

Calorimetry (DSC), b u t only ff the a p p a r a t u s is modified suitably. We find t h a t it is difficult to m e a s u r e the a r e a of the p e a k o b t a i n e d b y DTA accurately. Although one could u s e an integrating recorder to convert the p e a k to a n electrical signal, there is no w a y to u s e this signal in a control-loop feed-back to p r o d u c e the desired result. A m o r e practical w a y to do this is to control the rate of heating, i.e.- d T / d t , a n d provide a s e p a r a t e signal to o b t a i n a h e a t i n g differential. One s u c h w a y t h a t b e c a m e the b a s i s of DSC is s h o w n in the following diagram:

The a p p a r a t u s

c o n s i s t s of

a D S C - h e a d within a furnace,

like the DTA

a p p a r a t u s . However, there is also a silver block w h i c h encloses the DSC h e a d as well. This e n s u r e s complete a n d even h e a t dispersion. There are individual

375

heaters

for b o t h

measured

is t h e

the

reference

current

(R) a n d

required

sample

to k e e p

the

(S) p a n - h o l d e r s .

What

is

differential t h e r m o c o u p l e

b a l a n c e d , i.e. AT = 0. This signal cmn be amplified a n d r e c o r d e d . We u s e t h e s a m e a p p r o a c h for DSC a s we did for DTA. We s t a r t w i t h t h e t h e r m a l h e a t flow e q u a t i o n w h i c h is s i m i l a r to O h m ' s Law : 7.1.39.

d Q l d t = TB - TS I r

We c a n define t h e h e a t c h a n g e involved w i t h t h e s a m p l e a s d h / d t so a s to get the equation: 7.1.40.-

d h / (it = Cp (S) [dTs / dtl - dQ / dt = Cp {s) [dTs / d t l - ITs - TB] / r

We are u s i n g t h e s a m e t e r m i n o l o g y for DSC a s we did for DTA. Following t h e m e t h o d s given above, w e arrive at: 7.1.41.-

d q / d t = (Cp(s) - Cp(R}) d T B / d t + 1/r{dTB/dt - t}

We c a n t h u s "interpret" a DSC p e a k in t e r m s of t h i s e q u a t i o n , as we did for t h e DTA peak, as s h o w n in t h e following d i a g r a m :

376

As c a n be seen, the initial p a r t of the reaction u p to (dq/dt)max

is linear

w h e r e a s the curve b e c o m e s exponential p a s t the peak. This is d u e to the difference in Cp b e t w e e n the r e a c t a n t a n d product. T h u s , A I * A2. Actually, the reaction p e a k in 7.2.5. is an idealized curve since the baseline is a function of the d i f f e r e n c e in h e a t capacities between reference to s a m p l e a n d reference a n d product. Usually, we have a different baseline, vis:

This p r e s e n t s a p r o b l e m since it is difficult to estimate the a r e a of the peak. One c a n n o t simply extend the baseline as in Case I. A m u c h better solution is t h a t s h o w n in Case II, w h e r e i n the two very a s y m m e t r i c a l peaks, i.e.- the initial a n d final p a r t s of the overall t h e r m a l reaction t a k i n g place, are delineated. This p r o b l e m h a s not b e e n satisfactorily a n s w e r e d as yet a n d r e p r e s e n t s a challenge to a n y o n e u s i n g DCS m e t h o d s to characterize a solid state reaction. 7.2. - U22LIZATION OF I:YI'AAND DSC A. Applications o.f I:YI'A One of the m a j o r u s e s of DTA h a s b e e n to follow solid-state reactions as t h e y occur. All decomposition r e a c t i o n s floss of h y d r a t e s , w a t e r of constitution, decomposition of inorganic anions, e.g.- c a r b o n a t e to c a r b o n dioxide gas, etc.)

are endothem~ic a n d irreversible. Likewise are the s y n t h e s i s r e a c t i o n s s u c h as

377

CaO reacting with A1203 to form c a l c i u m a l u m i n a t e , CaAI204. P h a s e c h a n g e s , o n the o t h e r h a n d , are r e v e r s i b l e , b u t m a y be e n d o t h e r m i c or exothermic. Thus, if we follow a solid state reaction by DTA a n d o b t a i n a series of reaction p e a k s , it is e a s y to d e t e r m i n e w h i c h are p h a s e c h a n g e s b y recording the p e a k s o b t a i n e d d u r i n g t h e c o o l i n g cycle. W h e r e a s DTA d a t a are qualitative, t h o s e from DSC are q u a n t i t a t i v e a n d give information c o n c e r n i n g the h e a t c h a n g e (change in enthalpy) a c c o m p a n y i n g the exothermic or e n d o t h e r m i c reaction. For example, o n e c a n o b t a i n a value for melting of a solid state reaction p r o d u c t in t e r m s of calories / g r a m or Kcal. / mole. DTA is especially s u i t e d in t h e c o n s t r u c t i o n of unkno,ma p h a s e d i a g r a m s of b i n a r y c o m p o u n d s . A hypothetical p h a s e d i a g r a m a n d the DTA c u r v e s w h i c h w o u l d be u s e d to c o n s t r u c t it are s h o w n in the following diagram, given as 7.2.1. o n the next page. In this diagram, the endotherrrdc (Endo) p e a k s point to the fight while the exothermic (Exo) p e a k s point to t h e left. C o n s i d e r a s y s t e m with two c o m p o n e n t s , A a n d B (see 7.2.1.). They form a n incongruenfly melting c o m p o u n d , AB. The c o m p o u n d , AB, forms only a limited solid solution with A. Most of the c o m p o s i t i o n range is a taro-phase region, with a eutectic. The DTA r u n s are s u p e r i m p o s e d o n the specific c o m p o s i t i o n p o i n t s of the diagram. T h u s at (1) o n the d i a g r a m (about 10% A a n d 90% AB), we see o n e Exo a n d two E n d o peaks. The Exo p e a k is the point where the t w o - p h a s e mixture, A + AB, c h a n g e s to a single p h a s e , i.e. - a solid state solution of A in AB. F u r t h e r on, a n E n d o p e a k indicates the melting point of AB, a n d finally t h a t of A. At (2), we see only the two melting points, first t h a t o f A B a n d t h e n A. B u t at (3), only the melting point of t h e eutectic is seen, t h a t is, b o t h A a n d AB melt at the s a m e temperature. In o u r P h a s e Diagram, the c o m p o u n d , AB, m e l t s i n c o n g r u e n t l y , t h a t is - it d e c o m p o s e s at its melting point. Therefore. at (6), a double p e a k is seen r e p r e s e n t i n g the d e c o m p o s i t i o n of AB a n d the melting of A. However, B m e l t s at

378

a later time. Note t h a t one c a n pinpoint c h a n g e s in the p h a s e d i a g r a m quite accurately by r u n n i n g a DTA t h e r m o g r a m at specific composition points. For the most part, the thermal c h a n g e s observed are specific, b u t it is wise to cool reversibly, while observing the DTA p e a k s in cooling so as to be sure exactly w h a t the original p e a k represents.

379

Still a n o t h e r u s e to w h i c h DTA h a s b e e n employed is the c h a r a c t e r i z a t i o n of a m o r p h o u s materials. The following s h o w s a typical DTA t h e r m o g r a m o b t a i n e d w h e n a p o w d e r e d s a m p I e of glass is r u n as s h o w n in the following diagram: 7.2.2.-

IDiirentisiThermal

200

400

Analysis Of a Typical Glass I 600

1000

800

Tsp - 582 o C.

r

I

200

I

400

T M= 942 Oc. 600 -800 Teml~rature in ~

Note t h a t n e a r l y all of the c h a r a c t e r i s t i c "glass

1000

I

I

points" c a n be d e t e r m i n e d , i.e.-

Tg

= Glass transition temperature

TSp

= G l a s s softening p o i n t t e m p e r a t u r e

TD

= G l a s s devitrification t e m p e r a t u r e

TM

= Melting t e m p e r a t u r e of crystallized p r o d u c t

The o n l y o n e t h a t is not readily accessible b y DTA is the e x p a n s i o n coefficient. It is d e t e r m i n e d b y u s e of a t h e r m a l e x p a n s i o n a p p a r a t u s , i.e.- a dilatometer. Methods a n d u s e s of t h e r m a l e x p a n s i o n will be d e s c r i b e d in a s u c c e e d i n g section.

380

B. Uses of DSC The g r e a t e s t u s e for DSC h a s t u r n e d o u t to be for c h a r a c t e r i z a t i o n of organic polymers. It h a s b e e n found t h a t m o s t polymers are a m o r p h o u s a n d have a c h a r a c t e r i s t i c T g , i.e.- a "glass" t r a n s i t i o n t e m p e r a t u r e w h i c h leads to a "crystalline" p h a s e . O t h e r c o m m o n u s e s include d e t e r m i n a t i o n points, boiling points, T g , % crysta!linity a n d oxidative stability.

of melting

In obtaining boiling points b y DSC, it is n e c e s s a r y (14) to u s e a closed p a n having a n extremely small hole to allow the v a p o r to escape. The hole is m a d e b y u s e of a laser a n d s h o u l d not be m o r e t h a n 50-80 V in diameter. W h e n a semi-volatile m a t e r i a l is heated, a n e q u i l i b r i u m will be e s t a b l i s h e d b e t w e e n m a t e r i a l in the gas p h a s e a n d in the c o n d e n s e d p h a s e . As the m a t e r i a l is being heated, the p r e s s u r e exerted b y the volatile p h a s e , i.e.- the v a p o r p r e s s u r e , increases. The rate of h e a t i n g is i m p o r t a n t a n d s h o u l d be k e p t b e t w e e n a b o u t 6-10 ~

per m i n u t e . It is i m p o r t a n t to have the two p h a s e s in e q u i l i b r i u m as

the s a m p l e i n c r e a s e s in t e m p e r a t u r e , h e n c e the u s e of a n e s c a r p m e n t in the s a m p l e p a n to r e t a r d e s c a p e of the volatile material. This a r r a n g e m e n t allows the v a p o r p r o d u c e d to sweep o u t the air a n d replace it. At the t e m p e r a t u r e w h e r e the vapor p r e s s u r e of the s a m p l e exceeds the total p r e s s u r e of its s u r r o u n d i n g s , the m a t e r i a l boris. If the outside p r e s s u r e is kept c o n s t a n t , there will be a n e n d o t h e r m i c h e a t flow a s s o c i a t e d with c o n d e n s e d p h a s e m a t e r i a l entering the v a p o r p h a s e . As the t e m p e r a t u r e increases, the rate of boiling also increases. W h e n all of the m a t e r i a l is in the v a p o r stage, it r e m a i n s in t h a t s t a t e until all of the m a t e r i a l h a s boiled off. If the hole is not small enough, t h e n all of the m a t e r i a l will be evaporate a n d be lost before the e q u i l i b r i u m condition is attained. If a n equilibrium b e t w e e n v a p o r a n d m a t e r i a l is not achieved, t h e n the boiling point m e a s u r e d will not be the t r u e boiling point. As a n example of how the d a t a are obtained, the following d i a g r a m is p r e s e n t e d as 7.2.3. on the n e x t page. This d i a g r a m s h o w s the behavior of w a t e r at 1.0 a t m o s p h e r e w h e n it is s u b j e c t e d to the above conditions. Note the design of the p a n u s e d to h e a t the water. It c o n s i s t s of a n oval s h a p e , s o m e t h i n g like a n "egg"

381

7.2.3.-

[Boiling Point of Water by DSC[

i00

[Pan Usec~ Hole

6~ 4.5 mg.

80 -,-4

Onset = 100 ~

6o ID

m 40 IlililillliNNllimllli~iil

40

60 80 Temperature, ~

100

120

with a small hole on the top to limit the a m o u n t of water escaping at the boiling point. Keep in mind that the heat flow (which is related to the degree of vapor change achieved) is low in the beginning, b u t rises rather fast as the boiling point is reached. The fiat lead/rag edge of the endotherm represents the point where the sample temperature is constant at the boiling point. If one m e a s u r e s the boiling points at several pressures, including that of atmospheric pressure, one can then extrapolate to obtain the vapor pressure of a material at ambient temperature. This is done using the Clausius-Clapeyron equation, i.e.7.2.4.-

Eo -- RT 2 { d l n k l Eo = AH* + RT

/dt)

7.3.- Thermo_~ravimetry Thermogravimetric analysis fleA) m e a s u r e s changes in weight of a sample being heated. A typical Thermogravimetric analysis fleA) a p p a r a t u s is shown in the following diagram:

382

This i n s t r u m e n t c o n s i s t s of a n analytical b a l a n c e having a w e i g h t - c h a n g e detector on one side of the b a l a n c e . Although c u r r e n t c o m m e r c i a l TGA designs no longer u s e s u c h a balance, the original ones did so. C u r r e n t TGA designs employ

piezoelectric

crystals

or

similar

types

of

crystals

sensitive

to

gravitational force to m e a s u r e c h a n g e s in weight as the t e m p e r a t u r e is raised. As in the DTA design, a t e m p e r a t u r e p r o g r a m m e r is n e e d e d along with a f u r n a c e t e m p e r a t u r e TC. A power s u p p l y is controlled b y the t e m p e r a t u r e programmer.

The

temperature-programmer

can

be

either

mechanical

or

electrical in n a t u r e . It is possible to obtain a recorder with sufficient sensitivity

383

to directly record the TC signal. However, it is generally better to amplify the TC signal in order to avoid spurious electrical "noise" that sometimes occurs. A two-pen recorder is superior to a one-channel recorder where it would be necessary to switch from the differential TC signal to the temperature TC signal. However, in cases where the 2-channel recorder is not available, a suitable TGA chart be still be obtained. The actual weight is monitored in real time. Changes in weight, either gains or losses, are evident immediately. The a p p a r a t u s itself consists of the sample situated within a crucible, which is enclosed within a temperature-controlled furnace. The sample, plus crucible, is counterbalanced on a sensitive balance. Weight changes are directly plotted on a two-pen recorder. Weight readout is usually accomplished by one of two methods, a linear t r a n s d u c e r or a capacitance change between two fiat plates, one of which is free to move with the balance swing. A resistance-capacitance t a n k circuit completes the electronics, producing a readable voltage. We usually employ a crucible to hold the powder sample, although fiat p a n s are also suitable. Furnace temperature is controlled in a linear m a n n e r and recorded. In some cases, the TC is mounted directly at the sample position. It is important that sample and furnace temperatures be nearly equal, so as to record accurate weight losses and gains. Operational parameters for TGA are: 7.3.2.-

I. 2. 3. 4. 5.

Sample Size (buoyancy) Sample Closure Heating Rate Heating Mode External Atmosphere

The SAMPLE SIZE is important because most balances have a limited range of weighing, as well as a limited sensitivity, i.e.- milligrams per gram of weight detectable. Many balances feature automatic counter-weight loading. If a sample is fluffy and a large crucible is used, then the buoyancy factor m u s t be accounted for. At high temperatures, i.e.- > 800 ~ the density of air within

384

the furnace is sufficiently l o w e r t h a n t h a t of the outside, so t h a t the a p p a r e n t weight of the sample plus crucible appears lower t h a n it actually is. And, the larger the crucible, the more air is displaced within the furnace. For 10.000 g r a m s of total weight, the b u o y a n c y factor will be a b o u t 0.002, enough t h a t a correction needs to be m a d e for precision work. SAMPLE CLOSURE is i m p o r t a n t since it affects the rate of solid state reaction. Consider the following solid state reaction7.3.3 9-

CaC03

-7

CaO

+

C02

If the gas is restricted from escaping, t hen the equilibrium is shifted t o t h e left a n d the CaC03 does not decompose at its u s u a l temperature. A h i g h e r t e m p e r a t u r e is required to effect decomposition. atmosphere, s u c h as CO2 in the above reaction,

Likewise, a n external restricts the a p p a r e n t

t e m p e r a t u r e of decomposition, the rate of reaction, and sometimes, the mode of decomposition {depending u p o n the n a t u r e of the c o m p o u n d u n d e r investigation}. As an example, consider the following: CaCO3 normally decomposes at a b o u t 860 ~

In a 1.0 a t m o s p h e r e p r e s s u r e of C02, the

decomposition t e m p e r a t u r e is raised to a b o u t 1060 ~ HEATING RATE is i m p o r t a n t from a practical aspect. Usually, we are m e a s u r i n g furnace t e m p e r a t u r e a n d c a n n o t pr ogr a m the t e m p e r a t u r e too fast, for fear t h a t the sample t e m p e r a t u r e will lag the furnace t e m p e r a t u r e by too great a degree. It is also for this reason t h a t we us e as small a sample as is practical to obtain a weight loss or gain which the balance can discriminate. This again depends u p o n the sensitivity of the balance a nd the n a t u r e of the reaction we are examining. In general, weight gains (due to oxidation) require more sensitivity and larger sample sizes t h a n those of decomposition. Usually, we restrict heating rates to > 15 ~ and use a heating rate of a b o u t 6-8 ~ min. at most. The HEATING MODE to be u s e d depends u p o n the results we wish to achieve. A typical TGA r u n is given in the following diagram:

385

7.3.4.-

I A Typio l Thermogravimetrio

A n a l y a i s , Run

!

A ,i

o

w,

h

: olid S t a t e S Reaction

I I I

,,i.k

.......... J,,,.',

Weight Loaa

L

B

A -~B -~C

II ". ! ;

L.

a ]

C

IIIIiIII

IIIIngIIlII

!

;

I I ,

,: .:

I

.

.

gilman . .

I

T 1T~. Temperature,

!

"

I I ,

9

,,,

|

T8T4 ~

In the solid state reaction depicted, A begins to decompose to B at T1 a n d the reaction t e m p e r a t u r e for decomposition is T 2 , with a weight loss of WI. Likewise, the reaction of B to form C begins at T3 a n d the reaction t e m p e r a t u r e (where the rate of reaction is m a x i m u m ) is T4 . Note that the weight loss becomes c o n s t a n t as each reaction product is formed a n d the individual reactions are completed. If we p r o g r a m the t e m p e r a t u r e at 6 ~ in., we would obtain the results in 7.3.4. This is called dymamte t h e r m o g r a v i m e t r y . However, if we set the furnace t e m p e r a t u r e j u s t slightly greater t h a n T2 , we would obtain a reaction limited to t h a t of A - B, a n d t h u s could identify the intermediate reaction product, B. This technique is called i s o t h e r m a l t h e r m o g r a v i m e t r y , Thus, we can foUow a solid state reaction by first surveying via dynamic TGA. If there are a n y intermediate products, we can isolate each in turn, a n d after cooling (assuming each is stable at room temperature) can identify it by x-ray analysis. Note t h a t we can obtain an a ~ a y easily:

386

7.3.5.-

A s s a y =- final w e i g h t / original weight

If there is a g a s e o u s product, we c a n also identify it b y converting weight loss to m o l s / m o l of original reactant. In 7.3.3. above, 1.00 mol of C 0 2 is expected to be lost per mol of reacting C a C 0 3 .

The a c t u a l n u m b e r of mols lost d e p e n d s

u p o n the original s a m p l e size. The m o s t recent TGA a p p a r a t u s includes w h a t is called EGA, i.e.- effluent gas analysis. Most often, this c o n s i s t s of a small m a s s s p e c t r o g r a p h capable of identifying the v a r i o u s g a s s e s m o s t often e n c o u n t e r e d in TGA. G a s e o u s weight losses c a n be classified according to the n a t u r e of the effluent g a s e s detected. These include the following: 7.3.6.-

G a s e o u s P r o d u c t s = C02 , N 2 0 4 , H 2 0 , S 0 2 , S O 3 , CO. Water - of h y d r a t i o n - of c o n s t i t u t i o n

- adsorbed Regardless of how well a s a m p l e h a s b e e n dried, it will always have a n a d s o r b e d m o n o l a y e r of w a t e r on the surface of the particles. If we r u n the DTA carefully, we will see a small e n d o t h e r m i c p e a k a r o u n d 100 ~

Additionally, ff we r u n the

TGA properly, we will see a small loss p l a t e a u before the m a j o r losses begin. As a m a t t e r of fact, if w e

do

not

see

the

loss

of adsorbed

water,

either the

a p p a r a t u s is not o p e r a t i n g properly, or we do not have sufficient sensitivity to observe the reactions t a k i n g place. The different types of w a t e r w h i c h c a n be p r e s e n t d u r i n g a n y inorganic solid state reaction is easily illustrated by the following example. Dibasic calcium o r t h o p h o s p h a t e forms a dihydrate: 7.3.7.-

C a H P 0 4 - 2 H20

- brushite

CaHPO4

- monetite

B r u s h i t e r e a c t s to form m o n e t i t e w h i c h t h e n r e a c t s to form p y r o p h o s p h a t e : 7.3.8.-

2 CaHPO4- 2 H 2 0 = 2 C a H P 0 4 + 2 H20 2 CaHP04

= Ca2 P2 0 7 + H2 0

387

We c a n illustrate the type of calculations n e e d e d in order to determine the p a r a m e t e r s involved in TGA r u n s . In the foUowing discussion, we p r e s e n t how one confronts this p r o b l e m a n d the calculations needed to p r o d u c e the desired results. The following Table, p r e s e n t s a typical p r o b l e m t h a t one e n c o u n t e r s in TGA a n d the calculations n e e d e d to p r o d u c e the desired results. Tab!e 7- 3 A TYPICAL PROBLEM IN TGA FIRED PRODUCT ORIGINAL Ca2P207 brushite

METHOD X-ray Analysis: Weight:

17.311 g r a m

12.705 g r a m

Molecular Weight:.

172.09

254.11

.....

In this case, we s t a r t with a k n o w n material for which we have already u s e d xray analysis to determine the n a t u r e of the fired product. We s t a r t with the reactions given in 7.3.8. for the reactions of calcium p h o s p h a t e , since this also illustrates h o w a s s a y s are calculated. The s t e p s include: I. Determine a s s a y 2. By s u b t r a c t i n g actual a s s a y from theoretical assay, obtain a m o u n t of w a t e r actually a d s o r b e d on particle surfaces 3. Determine losses i n c u r r e d by stages, ff one w i s h e s to determine the actual reactions o c c u r r i n g d u r i n g the solid state reaction O u r first step is to mnalyze the solid state reaction by m e a n s of the values d e t e r m i n e d in the TGA analysis r u n . T h e reaction p r o d u c t s are given above, along with the requisite m o l e c u l a r weights. The next p a r t required in the TGA analysis is given as follows (this is a c o n t i n u a t i o n of Table 7-3):

388

OVERAIk, SOLID STATE REACTION: 2 CaHP04 - 2 H20 = Ca2 P2 0 7 + 3 H 2 0 Molecular Weight:

2(136.059) + 2(18.015)

254.11

WEIGHT:

17.311 g r a m

12.705 g r a m

+ 54.045

(Assay = 73.39%; Theor.--- 73.83%) ANALYSIS:

12.705 g r a m Ca2 P2 0 7

- 17.208 g r a m of 2 CaHPO4- 2 H20

By c a l c u l a t i o n , we find t h a t we h a v e 9 9 . 4 1 % of 2 CaHPO4 - 2 H 2 0 a n d 0 . 5 9 % a d s o r b e d w a t e r (by s u b t r a c t i o n ) . We c a n also d e t e r m i n e f r o m t h e TGA r u n t h e a m o u n t of w a t e r loss b y s t a g e s in t h e overall reaction: L o s s e s b y Stages:

I st loss = a d s o r b e d H 2 0 2 n d loss =

= 0.102 gram = 3 . 5 3 2 g r a m I~20

3 r d loss = m o n e t i t e to

= 3 . 9 2 tools = 0.972 gram H20

pyrophosphate

= 1.08 tools

Total wt. of w a t e r lost

= 4.606 gram

b r u s h i t e to m o n e t i t e

Note t h a t t h e s e c o n d loss c o r r e s p o n d s to 3 . 9 2 m o l of w a t e r p e r m o l of r e a c t a n t , w h e r e a s t h e 3 r d loss is 1.08 mol. T h i s i l l u s t r a t e s a s e r i o u s p r o b l e m t h a t c a n b e e n c o u n t e r e d in d y n a m i c TGA, If t h e

rate

of heating

is too

fast and not

e n o u g h t i m e o c c u r s d u r i n g p r o g r a m m i n g to achieve t r u e e q u i l i b r i u m b e t w e e n s u c c e s s i v e solid s t a t e r e a c t i o n s , t h e n t h e loss of w a t e r f r o m o n e r e a c t i o n c a r r i e s o v e r into t h e n e x t s u c c e e d i n g r e a c t i o n .

7.4.- D e t e r m i n a t i o n of R a t e P r o c e s s e s in Soh'd S t a t e R e a c t i o n s We h a v e p r e s e n t e d two m e t h o d s u s e f u l in following solid s t a t e r e a c t i o n s . In o r d e r to c o m p l e t e l y classify a r e a c t i o n , we n e e d to o b t a i n a n e s t i m a t e of t h e r e a c t i o n k i n e t i c s a n d o r d e r of t h e solid s t a t e reaction. B o t h DTA a n d TGA h a v e b e e n u s e d to o b t a i n r e a c t i o n r a t e kinetics. B u t first, we m u s t r e a x a m i n e kinetic

389

t h e o r y in light of solid s t a t e reactions, w h i c h differ from t h o s e involving g a s e o u s c o m p o n e n t s u s u a l l y q u o t e d in b o o k s dealing with reaction r a t e kinetics, the following d i s c u s s i o n r e i t e r a t e s w h a t we have already p r e s e n t e d , b u t is given again h e r e to r e e m p h a s i z e its i m p o r t a n c e in solid s t a t e chemistry. A. TYPES OF SOLID STATE REACTIONS In general, we c a n classify solid s t a t e r e a c t i o n s as being either h o m o g e n e o u s or h e t e r o g e n e o u s . The former involves r e a c t i o n s b y a single c o m p o u n d w h e r e a s the latter involves r e a c t i o n s b e t w e e n two different c o m p o u n d s . T h e r e are at least four (4) types of solid s t a t e r e a c t i o n s (as we have a l r e a d y p r e s e n t e d in a prior chapter): 7.4. i.-

Types of Solid S t a t e Reactions I. decomposition:

A ~B+C

2. s y n t h e s i s -

A+B~C

3. s u b s t i t u t i o n a l :

A+B~C+D

4. consecutive:

A~B~C

All four m a y be h e t e r o g e n e o u s ,

b u t only #1 (and s o m e t i m e s #4) will be

h o m o g e n e o u s . Rate p r o c e s s e s are defined in t e r m s of a rate, r , a n d a volume, V, u s u a l l y a m o l a r volume. T h u s , we have: 7.4.2.-

Homogeneous:

r = llVt - dnildt

Heterogeneous:

r = I/V f-

dVt/dt

w h e r e n i is initial mols, Vt is volume at time, t, a n d Vf is final volume. The fraction d e c o m p o s e d at a n y time is: 7.4.3.-

x ----" Vt / Vf

a n d this gives us:

390

7.4.4.-

d x / d t = k l {T}. f{x}

w h e r e kl is the rate c o n s t a n t for the reaction in 7.4.4. This e q u a t i o n is the general form for kinetics equations. F r o m the Kinetic Theory, w h e n a g a s e o u s s y s t e m goes from a n initial state to a final state, the r e a c t i n g species m u s t come close e n o u g h to react. At the m o m e n t of "joining", we have the "activated complex". By u s i n g this concept, we c a n obtain some general e q u a t i o n s useful to us. The c o n c e p t of the "activated complex" is illustrated in the following diagram:

The free e n e r g y of the activated complex, G*, is higher t h a n t h a t of the initial state. K* is the e q u i l i b r i u m c o n s t a n t of the activated complex, a n d k is the B o l t z m a n n c o n s t a n t , then, we c a n write t h e r m o d y n a m i c e q u a t i o n s as follows: 7.4.6.-

AG = RT In K* kl

= k T l h exp (-AG* IRT} = k T l h exp {AS* IR) exp (-AH*/RT)

w h e r e the s t a r (*) refers to the activated species. These e q u a t i o n s o u g h t to be familiar to Physical C h e m i s t r y s t u d e n t s .

391

F r o m t h e C l a u s i u s - C l a p y e r o n e q u a t i o n , we have: 7.4.7.-

Eo-

RT 2 d l n k l / d t

Eo

= AH*

+ RT

w h e r e Eo is a n i n t e r n a l energy. By defining a f r e q u e n c y factor as: 7.4.8.-

Z

= k T / h exp (-Eo*/RT)

we c a n s i m p l y b y m e a n s of 7.4.6. to: 7.4.9.-

kl

= Z exp (-Eo*/RT}

w h i c h is t h e AR_RHENIUS EQUATION for t h e r a t e t r a n s f o r m t h e above e q u a t i o n s into

general

constant,

k l. We c a n

e q u a t i o n s a s well, a n d u s e t h e m to

defme t h e v a r i o u s t y p e s of r e a c t i o n given above. This is given in t h e following table7.4. I0.-

RATE EQUATIONS FOR SOLID STATE REACTIONS

1. Simple n t h O r d e r kl nA ~

B+ C

2. Parallel R e a c t i o n s kl

Rate E q u a t i o n -dXA / d t = k l xn A Rate Equation..

A~B - d x A / d t = k lXA + k 2 XA

k2 {A~C} 3. C o n s e c u t i v e k~ k2 A~B

~C

-dxA/dt

= k l XA - k2 XB

392

These rate e q u a t i o n s c a n be u s e d for quite complicated reactions, b u t a specific m e t h o d or a p p r o a c h is needed. M a n y a u t h o r s have tried to devise m e t h o d s for o b t a i n i n g rate c o n s t a n t s a n d o r d e r s of reaction for given solid state reactions. None have b e e n wholly successful, except for F r e e m a n a n d Carroll (1948). The F r e e m a n a n d Carroll m e t h o d h a s b e e n s h o w n by F o n g a n d C h e n to be the only one w h i c h gives satisfactory a n s w e r s to k n o w n reactions, w h e t h e r zero order, 1st order, 2 n d order, or even higher. Even fractional o r d e r s of reaction m a y be determined. This m e t h o d c a n be u s e d with either DTA or TGA data. B. THE FREEMAN-CARROLL ME12-IOD APPLIED TO DTA DATA The following is a description of the F r e e m a n - C a r r o l l m e t h o d applied to DTA data. Consider the following DTA p e a k in which AT is plotted vs: time, t,

a

r e p r e s e n t a t i o n of w h i c h given as follows: 7.4.11.-

A DTA Peak

A

t

At a n y time, a

,,.

IF-

is the fraction d e c o m p o s e d while a 9 is the fraction w h i c h h a s

yet to react. We can set u p e q u a t i o n s as before: 7.4.12.-

da / d t

= k l (1- a )n

The first p a r t of the e q u a t i o n is the

= Z / + exp - E / R T (1- a )n

general

kinetic equation, from w h i c h it is

393

e a s y to o b t a i n t h e l a s t p a r t (as w e h a v e s h o w n ) . I f w e p e r f o r m t h e m a t h e m a t i c a l operations

of:

1) t a k e

the

naperian

log;

2) d i f f e r e n t i a t i o n ,

and

then

3)

i n t e g r a t i o n , w e o b t a i n t h e following e q u a t i o n : 7.4.13.-

=n(AIn(1-a)-E/RA(1/T)

Ainda/dt

If w e n o w set: A T / A = d(~ / d t a n d a 9 = A - a ( w h e r e A is t h e a r e a u n d e r t h e peak}, we can then obtain: 7.4.14.-

A(lnAT) = n(Alna,)

T h i s a l l o w s u s to plot:

- E/R

A(1/T)

A (In AT) / A (in a *)

vs: A ( 1 / T In a *) so a s to o b t a i n

a s t r a i g h t line. T h e s l o p e is E / R a n d t h e i n t e r c e p t is n ,

the o r d e r of reaction.

H a v i n g t h u s o b t a i n e d t h e a c t i v a t i o n e n e r g y a n d o r d e r directly, w e c a n t h e n c a l c u l a t e t h e r e a c t i o n rate. C. T H E F R E E M A N - C/~fl:(ROLL M E T H O D APPLIED TO TGA DATA In t h i s c a s e , w e d e f i n e o u r e q u a t i o n in t e r m s o f w e i g h t , w : 7.4.15.-

-dw/dt

= k l f(w)

As in t h e DTA m e t h o d , w e d e f i n e w e i g h t r e a c t e d i n t i m e , t, a s w t

a n d final

w e i g h t , w f . T h i s a l l o w s u s to d e f i n e w e i g h t t o b e r e a c t e d as: w = wt - wf . O u r w e i g h t f u n c t i o n is t h e n to b e d e f i n e d : 7 . 4 . 1 6 . - f(w) = w n w h e r e n is a s i m p l e i n t e g e r , r e p r e s e n t i n g t h e o r d e r of r e a c t i o n . T h u s , w e o b t a i n in t h e s a m e m a n n e r a s for t h e DTA d a t a : 7.4.17.-

- dw/dt

= Z e x p (-E/RT) w n

394

We now do the m a t h e m a t i c a l m a n i p u l a t i o n s in the order given above to obtain: 7.4.18.-

A (in (-dw/dt)) - E / R (A (l/T)) + n A In w

This allows u s to plot, as before:

A (In - d w / d t ) / A In w vs: A(I/T) / A In w .

However, in m a n y cases, it is easier to u s e a, the original weight, a n d calculate: (a - x) a n d d x / d t , w h e r e x is the fraction d e c o m p o s e d at time, t. T h e n we plot: 7.4.19.-

h (In dx/ht) / A {In {a-x)) vs: h (1 IT) / A In {a-x)

T h u s if we take a well defined curve from either DTA or TGA a n d find points on the curve, we can calculate all of the kinetic p a r a m e t e r s . Note t h a t the m a i n concept is to u s e the a m o u a t l e f t t o r e a c t at a n y given instant. In DTA, this w a s a* while in TGA, it w a s {a- x). Both t h e r m a l m e t h o d s give equally satisfactory results. 7.5.- DILATOME]~Y M e a s u r e m e n t of the t h e r m a l e x p a n s i o n of solids is called "dilatometry'. W h e n energy is applied to a solid, one of the r e s u l t s is t h a t the solid e x p a n d s in space. S u c h e x p a n s i o n r e s u l t s b e c a u s e the energy generally e n d s u p as "heat'. T h a t is- the lattice of the solid acquires i n c r e a s e d vibrational m o d e s w h i c h e x p a n d s the lattice. On a practical scale, dilatometry is u s u a l l y o b t a i n e d in linear fashion w h e r e i n the c h a n g e in u n i t length of a material c a u s e d b y one-degree c h a n g e in t e m p e r a t u r e is m e a s u r e d . Both volume c h a n g e a n d linear c h a n g e is e x p r e s s e d by: 7.5.1.-

(xv = I { 0 V / 0 T ) p

Vo

and

(XL = 1 { 0 L / 6 r ) p

Lo

where V is the volume a n d L is the length a n d V0 is the original volume a n d L0 is the original length. (~v is the volume t h e r m a l e x p a n s i o n coefficient a n d aL is

395

the linear thermal expansion coefficient. Obviously, 3 (1L = (IV (but only ff the material is cubic. That is, the s t r u c t u r e of the material is uniform in three dimensions). Both coefficients are ordinarily linear functions of temperature, t h a t is- a plot of expansion vs: t e m p e r a t u r e yields a straight line. By usi ng the average value of the linear coefficient of expansion, we get: 7.5.2.-

(IL = I .

L0

AT

where t~L an d AT are changes in length, L, a n d temperature, T, and L0 is the original length of the sample being m e a s u r e d . If a volume expansion is required, then m e a s u r e m e n t s in three s i m u l t a n e o u s dimensions are needed, a result experimentally difficult to achieve, to say the least. Even a slab of a single crystal does not completely solve the problem since thermal expansion in three dimensions is needed for the volume thermal expansion coefficient. The crystal h a s three (3) crystallographic axes and m a y have three (3) linear coefficients of expansion. Only if the crystal is cubic does one have the case where all three values of (1L are equal. Thus,

most

thermal

expansion m e a s u r e m e n t s

use

the

linear

expansion

coefficient method. This requires a rod of solid material and powders c a n n o t be m e a s u r e d by this method. Either the material is melted a n d cast into rod form or it is cut from a slab of material. Glass a n d metals are ideally suited to s u c h m e a s u r e m e n t , w h e r e a s inorganic and organic c o m p o u n d s are difficult to m e a s u r e at best. Single crystals of inorganic or organic c o m p o u n d s are required for the m o s t part, b u t the work of obtaining s u c h is sometimes a d a u n t i n g task. A suitable a p p a r a t u s is shown on the following following page as 7.5.3. Note t h a t only a m e a n s of holding one end of the sample b a r is needed while the sample is being heated. Most materials expand only a b o u t a small fraction of their actual dimensions. Because they expand less t h a n a micrometer, i.e.- one milh'onth of a meter, with a one-degree increase in temperature, m e a s u r e m e n t s

396

m u s t be m a d e by u s e of a highly sensitive device as the detector. Suitable m e a s u r e m e n t s are u s u a l l y m a d e by the following m e t h o d s : 7.5.4.- Suitable Detecting S y s t e m s for Dilametric M e a s u r e m e n t of Solids a. Microscope b. Dial gauge c. Telescope with Micrometer d. Strain gauge e. Interferometer f. Mirror reflection g. Lattice c o n s t a n t s via x-ray diffraction h. Sample density The MICROSCOPE m e t h o d {a) entails the u s e of a hot-stage w h i c h fits on the microscope. A calibrated eyepiece is needed

as well. By setting u p

and

controlling the t e m p e r a t u r e of the b a r u n d e r investigation, a series of lengths c a n be o b t a i n e d u s i n g the e q u a t i o n of 7.9.2. W h e n these p o i n t s are plotted a g a i n s t t e m p e r a t u r e , a straight line will be obtained. The slope of the line will be the linear coefficient of expansion, aLBoth the DIAL GAUGE {b) a n d TELESCOPE WI22-t MICROMETER (c) m e t h o d s are v a r i a n t s of the m i c r o s c o p e - m e t h o d {a). A t e m p e r a t u r e - c o n t r o l l e d furnace is needed a n d a physical s e t - u p where the DIAL-GAUGE or TELESCOPE WITH

397

MICROMETER can observe the change of length as a function of several settings of temperature. (~L is obtained as described before. Both the INTERFEROMETER m e t h o d (d) a n d the MIRROR REFLECTION (e) m e t h o d s use optical m e a n s to detect changes in linear expansion of the sample u n d e r test. The i n s t r u m e n t a t i o n is more complex a n d will not be described here. The other two methods, X-RAY LATI~CE CONSTANTS (f) a n d SAMPLE DENSITY (g) have not been employed to any great extent for determination of ~L a n d are only included for the sake of completeness. M o d e m commercial dilatometxic i n s t r u m e n t s use the electrical o u t p u t from a "strain-gauge". The he a r t of a strain gauge consists of a crystal (or sometimes a disc pressed from a powder) which is piezoelectric. The term, piezoelectric, is defined as the appearance of positive electric charge on one side of certain n o n c o n d u c t i n g crystals a n d negative charge on the opposite side w h e n the crystals are subjected to mechanical pressure. Piezoelectricity was discovered in 1880 by Pierre an d P a u l - J a c q u e s Curie, who found t h a t w h e n they c o m p r e s s e d certain types of crystals (including those of quartz, tourmaline, a n d Rochelle salt) along certain axes, a voltage w a s p r o d u c e d on the surface of the crystal. The next year, they observed the converse effect, the elongation of s u c h crystals u p o n the application of an electric current. Pressure on certain electrically ne ut r al crystals--those not having a center of s t r u c t u r a l symmetry- polarizes t h e m by slightly separating the center of positive charge from t h a t of the negative charge. Equal and unlike charges on opposite faces of the crystal result. This charge separation m a y be described as a r e s u l t a n t electric field a n d m a y be detected by an appropriate voltmeter as a potential difference, or voltage, between the opposite crystal faces. This p h e n o m e n o n , also called the piezoelectric effect, h a s a converse, i.e.- the production of a mechanical deformation in a crystal across which an electric field or a potential difference is applied. A reversal of the field reverses the direction of the mechanical deformation. This effect h a s been u s e d in microphones.

398

The advantage of us i ng a strain gauge w h e n m a k i n g thermal expansion m e a s u r e m e n t s lies in the fact t h a t a c o n t i n u o u s electrical signal can be obtained as the sample bar is being heated. A typical dilatometric r u n for a glass sample is shown in the following: 7.5.5.-

ii

200

I

400 ,ll

I

600 i

I

800

'--

1000

I

I

Glass Softening Point

L~I.~SS

Transition

Temperature

t

I

I

I

200

400

600

600

,

I I000

TEMPERATURE IN o C.

Note t h a t the linear coefficient of expansion, otL is obtained from the slope of the straight line. The glass softening point is also easily observed as is the glass transitional t e m p e r a t u r e (which is the point where the a m o r p h o u s glassy p h a s e begins its transition to a crystalline phase. These glass-points can also be u s e d to cross-check values obtained by the DTA method. It is possible for a glass to have more t h a n one linear coefficient of expansion. One s u c h case is shown in the following diagram, given as 7.5.6. on the next page.

399

7.5.6.-

1"4[ I

IThermal

!

Expansion of [Ba(PO 3 )2]n

1.2--

(-

~-0

1.0--

O .fi (J E

0.8

"-

0.6

--

LL .li, .1-:>.li

or) E EL X Ill

I~

0.4-0.2

- 115x

I 0 -7

in/in/~ C

I

--

I

0 0

I 100

i

I ZOO

I

I 300

i,,

i 400

!

i 500

I 600

Temperature in ~ This glass is polymeric a n d c o n s i s t s of b a r i u m m e t a - p h o s p h a t e units, i.e.[Ba(P03)2] n , polymerized to form long c h a i n s in the a m o r p h o u s state. Note t h a t three v a l u e s are shown. The first is the expected linear expansion, typical of m o s t glasses. In this case, there is a t e m p e r a t u r e r a n g e w h e r e the glass ceases to e x p a n d (between 3 0 0 a n d 3 8 0 ~ a n d is fiat. Above a b o u t 3 8 0 ~ a third value of ~L c a n be seen. Although this behavior is not typical for m o s t glasses, it does illustrate the fact t h a t a n a m o r p h o u s p h a s e c a n have m o r e t h a n one coefficient of expansion. Note also t h a t the softening point is not indicated in the diagram. Its value lies above 5 5 0 ~ well.

The s a m e is t r u e for o t h e r solids as

To illustrate this point, consider a typical metal. W h e n a b a r of s u c h m e t a l s are heated, t h e y e x p a n d in a linear m a n n e r , b u t m a y u n d e r g o a p h a s e t r a n s i t i o n as they r e a c h a critical t e m p e r a t u r e . This is s h o v m in the following diagram:

400

7.5.7.

IExpa,sio:O f various.MetalS withllTemperature,,, .... , ...... ] I

i

flll

l

I

aid

i

i

iii

i

i III

I i[11111

i i

Temperature i

i

ii

iiii

I

ii

i

ii

I

"

w-

In this case, a c h a n g e in s t r u c t u r e occurs. M a n y m e t a l s are elemental in n a t u r e a n d w h e n refined to a p u r e state have a cubic s t r u c t u r e . At s o m e criticai t e m p e r a t u r e (def'med b y the n u m b e r of m e t a l electrons per a t o m p r e s e n t a n d the type of metallic bonding), a c h a n g e to a hexagonal form occurs. This is s h o w n in the following diagram. Note t h a t a simple shift in u n i t cell d i m e n s i o n is all t h a t is required for the crystal s t r u c t u r e c h a n g e to take place. 7.5.8.-

ICubic 'to hexagonal Transformation I

401

Observe also that increased thermal energy is all that is required for the shift to take place. Such change occurs well below the melting point of the metal. Dilatometric m e a s u r e m e n t is the only way that such change was originally discovered and the use of x-ray analysis was needed to confirm the exact n a t u r e of the change measured. 7.6.- Th,ermometry Thermometry is the science of measuring the temperature of a system or the ability of a system to transfer heat to another system. Temperature m e a s u r e m e n t is important to a -,vide range of activities, including manufacturing, scientific research, and medical practice. Thermometry relates to the dilatrometric m e a s u r e m e n t in that the expansion of a gas, liquid or solid is used to determine temperature. The accurate m e a s u r e m e n t of temperature has developed relatively recently in h u m a n history. The invention of the thermometer is generally credited to Galileo. In his instrument, built about 1592, the changing temperature of an inverted glass vessel produced the expansion or contraction of the air within it, which in t u r n changed the level of the liquid with which the vessel's long, openmouthed neck was partially filled. This general principle was perfected in succeeding years by experimenting with liquids such as mercury and by providing a scale to m e a s u r e the expansion and contraction brought about in such liquids by rising and falling temperatures. By the early 18th century as m a n y as 35 different temperature scales had been devised. The German physicist, Daniel Gabriel Fahrenheit, in 1700-30 produced accurate mercury thermometers calibrated to a s t a n d a r d scale that ranged from 32, the melting point of ice, to 96 for body temperature. The unit of temperature (degree) on the Fahrenheit scale is 1 / 1 8 0 of the difference between the boiling (212) and freezing points of water. The first centigrade scale (made up of 100 degrees) is attributed to the Swedish astronomer Anders Celsius, who

402

developed it in 1742. Celsius u s e d 0 for the boiling point of w a t e r a n d 100 for the melting point of snow. This w a s later inverted to p u t 0 on the cold e n d a n d 100 on the hot end, a n d in t h a t form it g a i n e d w i d e s p r e a d use. It w a s k n o w n simply as the centigrade scale until in 1948 the n a m e w a s c h a n g e d to h o n o r Celsius. In 1848 the British physicist William T h o m p s o n (later Lord Kelvin) p r o p o s e d a s y s t e m t h a t u s e d the degrees t h a t Celsius u s e d , b u t w a s keyed to a b s o l u t e zero (-273.15 ~ i.e.- ~

the u n i t of this scale is now k n o w n as the kelvin,

The R a n k i n e scale employs the F a h r e n h e i t degree keyed to absolute

zero (-459.67 ~

i.e.- ~

Any s u b s t a n c e t h a t s o m e h o w c h a n g e s with alterations in its t e m p e r a t u r e c a n be u s e d as the basic c o m p o n e n t in a t h e r m o m e t e r . G a s t h e r m o m e t e r s w o r k b e s t at very low t e m p e r a t u r e s . Liquid t h e r m o m e t e r s are the m o s t c o m m o n type in use. T h e y are simple, inexpensive, long-lasting, a n d able to m e a s u r e a wide t e m p e r a t u r e span. The liquid is a l m o s t always m e r c u r y , sealed in a glass t u b e with nitrogen gas m a k i n g u p the rest of the volume of the tube. Electrical-resistance t h e r m o m e t e r s characteristically u s e p l a t i n u m a n d operate on the principle t h a t electrical r e s i s t a n c e varies with c h a n g e s in t e m p e r a t u r e . T h e r m o c o u p l e s are a m o n g the m o s t widely u s e d i n d u s t r i a l t h e r m o m e t e r s . T h e y are c o m p o s e d of two wires m a d e of different m a t e r i a l s joined together at one e n d a n d c o n n e c t e d to a v o l t a g e - m e a s u r i n g device at the other. A t e m p e r a t u r e difference b e t w e e n the two e n d s c r e a t e s a voltage t h a t c a n be m e a s u r e d a n d t r a n s l a t e d into a m e a s u r e of the t e m p e r a t u r e of the j u n c t i o n end. The bimetallic strip c o n s t i t u t e s one of the m o s t trouble-free a n d d u r a b l e t h e r m o m e t e r s . It is simply two strips of different m e t a l s b o n d e d together a n d held at one end. W h e n heated, the two strips e x p a n d at different rates, r e s u l t i n g in a b e n d i n g effect t h a t is u s e d to m e a s u r e the t e m p e r a t u r e change. O t h e r t h e r m o m e t e r s operate b y s e n s i n g s o u n d waves or m a g n e t i c conditions a s s o c i a t e d with t e m p e r a t u r e

changes.

Magnetic t h e r m o m e t e r s

increase

in

efficiency as t e m p e r a t u r e decreases, w h i c h m a k e s t h e m extremely useful in m e a s u r i n g very low t e m p e r a t u r e s with precision. T e m p e r a t u r e s c a n also be

403

mapped, u s i n g a technique called t h e r m o g r a p h y t h a t provides a graphic or visual representation of the t e m p e r a t u r e conditions on the surface of an object or land area. P h o s p h o r s are now being u s e d in this application. 7.7.- APPLICATION OF DILATOMETRY TO PLASTIC MATERIAI~ The ter m "plastic" refers to t h a t class of materials which is organic in nature. That is, plastics are composed of hydrogen, carbon, nitrogen and sometimes fluorine, i.e.- they are hydrocarbons. S u c h materials are actually "polymers" composed of monomeric u n i t s which are c a u s e d to join to form r a t h e r large uniform molecules having molecular weights as high as 100,000 or more. Plastics or polymers have found a variety of e n d - u s e s in our economy. In addition to containers s u c h as bottles for beverages, plastics are u s e d in all sorts of p r o d u c t s a n d polymers have been developed t h a t are stronger a n d more flexible t h a n m a n y metals. For the m o s t part, plastics are m a n - m a d e since very few plastics are natural, i.e.- n a t u r e - m a d e . Natural plastics include large molecular-weight proteins a n d similar molecules. Man-made plastics can be classified as either thermoplastic or thermosetting. Each class derives its physical properties from the effects of application of heat, the former becoming "plastic" (that is- it becomes soft a n d tends to flow) while the latter becomes less "plastic" and tends to remain in a softened state. This difference in change of state derives from the actual n a t u r e of the chemical b o n d s in the polymer. Thermoplastic pol:yzners generally consist of molecules composed of m a n y monomeric units. A good example is t h a t of polyethylene where the monomeric unit is: -(CH2-CH2)- . The molecule is linear a n d the polymer consists of m a n y units tied together in one long string. In contrast, thermosetting polymers consist of c r o s s - l i n k e d u n i t s where the crosslinking is three-dimensional. That is, the molecules are linked together in three dimensional-space:

-(CH-CH)I I -(CH-CH)-

404

This difference in spatial characteristics has a profound effect upon the polymer's physical and chemical properties. In thermoplastic polymers, application of heat causes a change from a solid or glassy (amorphous) state to a flowable liquid. In thermosetting polymers, the change of state occurs from a rigid solid to a soft, rubbery composition. The glass transition temperature, Tg, and the coefficient of expansion have a profound effect on the performance and reliability of m a n y polymer applications. Although Tg is usually quoted a n d accepted as a single value, the transition usually occurs within a range of temperatures. Factors such as intra-chain stiffness, polar electromagnetic forces and co-polymer compatibility (when two or more polymers are blended together to improve physical properties) can affect the size of the glasstransition region. As shown in the following diagram, property changes occur throughout a temperature region which depends upon the type of polymer(s):

It should be clear that the point where a polymer shifts from a glassy, hard state to a soft, rubbery one is not well defined but occurs within a b a n d of temperatures. In contrast, it is easy to define Tg as a single temperature point

405

for glasses a n d m o s t ceramic materials. We have already described how Tg is m e a s u r e d . Similar m e t h o d s are u s e d w h e n plastics or organic m a t e r i a l s are involved. DSC is the m e t h o d m o s t often employed since it gives b o t h e n t h a l p y a n d h e a t c a p a c i t y d a t a from the s a m e analysis. 7,8- OPTICAL MEASUREMENTS OF SOLIDS In t e r m s of their optical properties, all solids fall into one of two classes. Either t h e y are t r a n s p a r e n t to light (here we are restricting the t e r m "light" to visible radiation) or t h e y are opaque. In the latter case, all of the r a d i a t i o n m a y be reflected. However, m o s t solids reflect some w a v e l e n g t h s a n d a b s o r b others. This is the condition t h a t we call "color". If all visible w a v e l e n g t h s are absorbed, the solid is said to be "alack" while reflectance of all visible w a v e l e n g t h s r e s u l t s in a "white" solid. We i n t e n d to s h o w how "color" is m e a s u r e d b u t first m u s t define the n a t u r e of "light". A. DEFINING LIGHT Light is c o m p o s e d of p h o t o n s

(which are individual energy b u n d l e s ) t h a t

p r o p a g a t e t h r o u g h space. The correct t e r m for light is electromagnetic radiation. We u s e the t e r m "light" to refer to t h o s e p h o t o n s w h i c h we c a n see. "Dark" refers to the a b s e n c e of-visible p h o t o n s (You m a y be i n t e r e s t e d to k n o w t h a t a p h o t o n is now r e g a r d e d as a e n e r g y carrier b e t w e e n f u n d a m e n t a l particles, i.e.leptons s u c h as p r o t o n s a n d neutrons). Light travels at a c o n s t a n t speed, i.e.3.0 x10 I ~ m e t e r s / s e c o n d , t h r o u g h a v a c u u m (a s p a c e w h e r e no m a t t e r exists). W h e n m a t t e r is present, its speed is diminished, b u t is still c o n s t a n t . Since the speed of light is c o n s t a n t , individual p h o t o n s c a n v a r y only in energy, a s t a t e w h i c h r e s u l t s in differences in their wavelength. T h a t is, a p h o t o n ' s energy is m a n i f e s t e d as a specific wavelength. O u r m a i n c o n c e r n will be t h a t of "color", w h i c h is the science of m e a s u r i n g w h a t type of p h o t o n s are reflected a n d t h o s e t h a t are a b s o r b e d b y a solid. "What this m e a n s is t h a t color is d e t e r m i n e d b y w h i c h p h o t o n s , in a s t r e a m of p h o t o n s h a v i n g v a r i o u s energies, are either reflected or a b s o r b e d b y the solid. This m e c h a n i s m implies s o m e sort of

406

interaction between e a c h individual p h o t o n a n d the electrons c o m p o s i n g the solid. P h o t o n s (i.e.- electromagnetic energy) c a n vary in wavelength from: 7 - r a y s = 1 0 12 m e t e r s x-rays = 1 0 I 0 m e t e r s ultraviolet = 10 8 m e t e r s visible = 10 .6 m e t e r s infrared radiation = 10 -4 m e t e r s microwaves = 10 .2 m e t e r s radio - 102 to 104 m e t e r s where the values given are averages of the s p r e a d of wavelengths. Note t h a t radio waves can be several miles long. But, since light travels at 186,000 m i l e s / s e c . , t r a n s m i s s i o n to a n y point on E a r t h is nearly i n s t a n t a n e o u s . The frequency

varies

from

about

105

cycles/second

(radio

waves)

to

1020

cycles / s e c o n d (7 -rays). The relationship b e t w e e n

electromagnetic

radiation

and matter

(solids) is

intertwined in the so-called "space-time" p h e n o m e n o n . All solids em/t p h o t o n s , even yourself. The concept of "absolute zero" lies in the fact t h a t no p h o t o n s are emitted at 0 ~ K. As the t e m p e r a t u r e rises, a s p e c t r u m of p h o t o n energies is emitted, as s h o w n in the following diagram, given as 7.8.1. on the next page. This

diagram

shows

the

radiation

emitted

by

"black-bodies"

at

specific

t e m p e r a t u r e . A b l a c k - b o d y is one t h a t h a s a u n i f o r m t e m p e r a t u r e over all of its surface. One w a y to m a k e a b l a c k - b o d y is to form a n hollow e n c l o s u r e a n d to h e a t it to a given t e m p e r a t u r e . If a small hole is m a d e in the side of the enclosure, radiation characteristic of the t e m p e r a t u r e will be emitted. It s h o u l d be clear t h a t all bodies (even y o u r own) radiate p h o t o n s in the infrared r a n g e of energies. Yours is similar to t h a t of the e a r t h a n d p r o b a b l y p e a r s n e a r to 10.0 m i c r o n s or 10,000 A. If you place y o u r h a n d on y o u r face, you feel w a r m t h b e c a u s e the emitted p h o t o n s are r e a b s o r b e d b y y o u r h a n d .

407

Nonetheless, o u r p r i m a r y interest lies in the 0.4 to 0.7 m i c r o n range, w h i c h we call the visible p a r t of the electromagnetic s p e c t r u m . Note t h a t even bodies at liquid-air

temperatures

wavelength, i.e.-

emit

I00,000 and

t e m p e r a t u r e of 300 ~

photons

between

10

and

100

microns

in

106 A in wavelength. The e a r t h itself at a

h a s a n errdssion b e t w e e n a b o u t 2 0 , 0 0 0 a n d 3 0 0 , 0 0 0 A

408

in wavelength, i.e.- 2 p. a n d 3 0 0 p. Light itself h a s a specific wavelength m e a s u r e d in A n g s t r o m s (1]~ = 0 . 0 0 0 0 0 0 0 0 0 1 m e t e r s = 0 . 0 0 0 0 0 0 0 0 0 0 0 3 feet). Sir Isaac Newton first d e m o n s t r a t e d the s e p a r a t i o n of wavelengths (colors) of s u n l i g h t in 1675 b y u s e of a glass prism. The wavelengths to w h i c h the h u m a n eye r e s p o n d s is called the "visible spectrum". We see each of these wavelengths of light as a perceived color. Additionally, it w a s found t h a t "white" is a compilation of all of the colors while "black" is an a b s e n c e of said colors. P h o t o n s having the highest energy are violet in color while the lowest energy ones are those we call deep-red. The intensity, or "brightness" of light is simply the n u m b e r s of p h o t o n s r e a c h i n g the eye at a n y given m o m e n t . On either side of the visible s p e c t n u n are "ultraviolet" a n d "infrared" wavelengths. The S u n itself r a d i a t e s t h r o u g h o u t the visible spectaum~ a n d far into the ultraviolet a n d infrared r a n g e s as well. The

following diagram,

shows

the

Sun's

radiation,

including

the visible

s p e c t r u m which e n c o m p a s s e s the 4 0 0 0 A to 7 0 0 0 A range of wavelengths. 7.8.2.i00

IThe . Visible . Spectrum . .

and . the Sun's

RadiationI

Ultraviolet Violet Blue BluegreenGreenYellowOr~leRed DeepRed Infrared

~ r.----7--i i

75

~B

i I

i i i i i i l i..i

'~ 50 --,-.-4

25 0

--"

i i i

8000

I

:1

i i

..

Broadband Emission of the Sun Not Attenuated by the Earth's Atmosphere, i.e.- 10,000 ~ -

I

I

Visible Spectrum

l

I

I

I

~-' . I

I

4000 5000 6000 Wavelength of Photons in Angstroms

I

7000

I

8000

It is this range t h a t we u s u a l l y call "daylight". Note the progression of h u e s as

409

we move from violet to deep red colors. Although the S u n radiates wavelengths as small as 2 0 / i and as large as 25,000 A, most of the energy is concentrated between 3000 and 10,000 A. Attenuation by the Earth's atmosphere (ozone in particular) causes the energy to peak near to 6500/k. (What this m e a n s is that if we have a hot "oody" at 6500 ~ (~ = ~ + 273) it would look like the S u n at noon).The actual color temperature of the S u n is about 10,000 ~ but it appears to us to be about 6500 ~ due to scattering of light within the atmosphere. It is n o t h a p p e n s t a n c e that the h u m a n eye responds exactly to the same wavelength intensities as those of the noonday Sun. In other words, m a n k i n d was born, and has evolved, u n d e r sunlight and his eyes have adapted to sunlight intensity, also known as "Daylight". The relative response of the h u m a n eye is shown in the foUowing diagram: 7.8.3.IThe Human Eye Sensitivity Curve i i i

Ul.~tra~Jolet Violet Blue Bluegreen Green Yellow Orange Red Deep Red Infrared

100 75

-

50

" f

25

-

i i

I

TM

o

aooo

I

//

Spectrum .

~'

4000 5000 6000 7000 Wavelength of Photons in Angstroms

I

1

8000

Note that the eye response is greatest in the green and yellow regions of the spectrum, and that response to blue and red wavelengths is m u c h lower. B. MEASUREMENT OF COLOR In order to m e a s u r e color, we first need to define how color is perceived, what is

410

perceived, a n d the factors involved in perception of color. First of all, "color" is a result of p h o t o n s striking the h u m a n eyeball, specifically the retina, a n d c a u s i n g a r e s p o n s e which we interpret as color. Color arises b e c a u s e each p h o t o n h a s its own energy, i.e.- its own wavelength. Color is perceived w h e n p h o t o n s strike the r e t i n a of the eye a n d c a u s e electrical activity in the optic nerve. In order to define the n a t u r e of a photon, we need to consider exactly w h a t light is a n d how it interacts with matter. This p r o b l e m h a s been s t u d i e d for m a n y centuries by various investigators a n d c o n t i n u e s to be s t u d i e d by c o n t e m p o r a n e o u s scientists. C.-THE NATURE OF LIGHT Nowadays, we k n o w t h a t light is c o m p r i s e d of "photons", w h i c h are quantized waves having some of the properties of particles. The concept of p h o t o n s with wave properties h a s its roots in the s t u d y of optics a n d optical p h e n o m e n a . Until the middle of the 17th century, light w a s generally t h o u g h t to consist of a s t r e a m of some sort of particles or corpuscles e m a n a t i n g from light sources. Newton a n d m a n y other scientists of his day s u p p o r t e d the idea of the c o r p u s c u l a r theory of light. It w a s Newton in 1703 w h o s h o w e d t h a t "ordinary" s u n l i g h t could be dispersed into its c o n s t i t u e n t colors by a prism, b u t the p h e n o m e n o n w a s not clearly g r a s p e d at t h a t time. Significant e x p e r i m e n t s on the n a t u r e of light were carried o u t by: I) Fresnel a n d T h o m a s Young (1815) on interference a n d diffraction respectively 2) Maxwell in 1873 who p o s t u l a t e d t h a t an oscillating electrical circuit s h o u l d radiate electromagnetic waves 3) Heinrich Hertz in 1887 who u s e d a n oscillating circuit of small d i m e n s i o n s to p r o d u c e electromagnetic waves which h a d properties of light waves

all of the

4) Einstein in 1905 who explained the photoelectric effect (He did so by extending a n idea p r o p o s e d by Planck five years earlier to p o s t u l a t e t h a t the energy in a light b e a m w a s c o n c e n t r a t e d in "packets" or photons..

411

The wave picture w a s r e t a i n e d in t h a t a p h o t o n w a s c o n s i d e r e d to have a f r e q u e n c y a n d t h a t the energy of a p h o t o n w a s proportional to its frequency). We c a n s u m m a r i z e all of the above r e s e a r c h carried o u t over the last two c e n t u r i e s in t h a t a p h o t o n is a q u a n t u m of r a d i a t i o n a n d a carrier of force b e t w e e n particles, w h e r e a s a n electron is a q u a n t u m of m a t t e r . Now, let u s e x a m i n e the m o r e m u n d a n e

a s p e c t s of light m e a s u r e m e n t including color

measurement. D.- ABSORBANCE, REFLECTIVITY AND TRANSM!'ITANCE. W h e n a b e a m of p h o t o n s strikes a solid, specific i n t e r a c t i o n s t a k e place w h i c h c a n be related to the Q u a n t u m Theory. T h e s e i n t e r a c t i o n s have b e e n m e a s u r e d in the p a s t a n d certain f o r m u l a s have b e e n found to apply. According to HUYGHEN'S principle of electromagnetic r a d i a t i o n s c a t t e r i n g , w h e n p h o t o n s come into close c o n t a c t with a solid, the electric a n d m a g n e t i c field vectors of the incident p h o t o n s couple with those of the electrons a s s o c i a t e d with the a t o m s c o m p r i s i n g the solid. The following p r e s e n t s s o m e of t h e s e f o r m u l a s applicable to c o n c e p t s given here: 7.8.4.- TERMINOLOGY APPLICABLE TO OPTICAL PROPERTIES a. A b s o r b a n c e :

A = logl/T=

b. T r a n s m i t t a n c e :

T = I / Io

c. Absorptivity:

II -- A / bc (A is m e a s u r e d absorption; b is the

logIo/ I ( I is m e a s u r e d intensity; Io is original intensity) optical p a t h - l e n g t h ; c is the m o l a r concentration)

d. Reflection:

Reflectivity -- spectral reflection, i.e.- at specific angles Reflectance = diffuse reflection, i.e.- s c a t t e r e d radiation

412

e. Intensity:

I is defined as the energy / u n i t a r e a of a b e a m of electromagnetic radiation.

This interaction leads to at least four (4) c o m p o n e n t s , namely: R- the radiation

reflected, A- the radiation a b s o r b e d , T- the radiation t r a n s m i t t e d , a n d S - the radiation s c a t t e r e d . A depiction of these interactions is given in the following diagram:

The original intensity of the radiation is defined as Io. A p a r t of the intensity is absorbed, a n o t h e r p a r t is t r a n s m i t t e d , still a n o t h e r p a r t is scattered, a n d a p a r t of the total intensity is reflected. The c o m p o n e n t s , S a n d T, are p r o c e s s e s which are independent of the wavelength (frequency) of the incident p h o t o n s , w h e r e a s R a n d A are primarily wavelength dependent. It is here t h a t the factor of "color" arises. The e x a c t a m o u n t of energy extracted from Io by each process is a complex set of variables d e p e n d i n g u p o n the type a n d mrrm~gement of a t o m s c o m p o s i n g the solid.

413

To simplify t h e s e concepts, consider a n optically h o m o g e n e o u s thin film. By optically h o m o g e n e o u s , we m e a n one t h a t is thin e n o u g h so t h a t no s c a t t e r i n g c a n occur. If a b e a m of p h o t o n s is incident to the surface at a given angle (but less t h a n t h a t where all of the b e a m is t r a n s m i t t e d - the so-called Brewster angle), p a r t of the b e a m will be reflected a n d p a r t will be absorbed, as s h o w n in t h e following:

The reflectance, R, is a c o n s e q u e n c e of the difference in refractive indices of the two media, (1} - air, a n d (2}- the s e m i - t r a n s p a r e n t t h i n film. The a m o u n t of a b s o r p t i o n is a function of the n a t u r e of the solid. Obviously, the a m o u n t t r a n s m i t t e d , T, is d e t e r m i n e d b y b o t h R a n d A. In this case, A, the absorbmaee, is defined as: 7.8.7.-

A ~ {1- R} - T

a n d the original b e a m intensity, Io ~ 1.00, is d i m i n i s h e d according to the Beer-

Lambert law, vis7.8.8.-

A = lnlo / I = Ecl

414

w h e r e 1 is t h e p a t h l e n g t h (depth of film traversed), E is a m o l a r e x t i n c t i o n

coefficient

of t h e

absorbing

species,

and

c is t h e

concentration

of t h e

a b s o r b i n g species. A l t h o u g h we h a v e s h o w n b u t one reflection a t X, a n o t h e r is e q u a l l y likely to o c c u r a t Y a s well. T h i s p r o b l e m of m u l t i p l e reflections w a s w o r k e d o u t b y Bode (1954). We will s h o w o n l y t h e r e s u l t s of his w o r k a n d u s e it to prove t h e validity of t h e e q u a t i o n of 7.8.7. for t h e c a s e of t h e h o m o g e n e o u s t h i n film. Consider the energy interactions which occur when a p h o t o n

s t r i k e s a solid.

The t i m e f r a m e of i n t e r a c t i o n is a b o u t 10-18 s e c o n d s : 7.8.9.-

S p e e d of p h o t o n = v -- 1010 cm. / sec. D i s t a n c e b e t w e e n lattice p l a n e s = d = 10 -8 c m . = ~ 1 A Time for p h o t o n to t r a v e r s e lattice = d / v = ~ 10-18 s e c o n d s

The fact t h a t t h e p h o t o n does t r a v e r s e t h e lattice p l a n e s does n o t m e a n t h a t t h e p h o t o n will b e a b s o r b e d or even s c a t t e r e d b y t h e solid. T h e r e f l e c t a n c e of t h e photon

is a f u n c t i o n of t h e n a t u r e

of t h e c o m p o s i t i o n a l s u r f a c e , w h e r e a s

a b s o r p t i o n d e p e n d s u p o n t h e interior c o m p o s i t i o n of t h e solid. A " r e s o n a n c e " condition

must

exist before

the photon

can

transfer

e n e r g y to t h e solid

( a b s o r p t i o n of t h e photon). In t h e following, we s h o w t h i s r e s o n a n c e c o n d i t i o n in g e n e r a l t e r m s of b o t h R & A. 7.8. I0.- ENERGY TRANSFER TO A SOLID BY A PHOTON ENERGY EXAMPLES

R

A

High

Moderate

"Colored" solid

Low

High High

Very High

"Black" solid

High

Low

Nfl

"White" solid

Low

Low

Nil

TRANSFER .

.

.

.

.

Heat transfer

.

.

.

.

.

.

415

This r e s u l t

defines a b s o r p t i o n of light in t e r m s

of b o t h reflectance

and

absorption. It is well to note t h a t either one or the o t h e r (or both) are required p h e n o m e n a in order for a p h o t o n to interact with a solid. T h u s , w h e n b o t h R & A values are high, we s a y the solid is colored. If either R or A is high, a n d the o t h e r value is low, t h e n we have either a white solid or a b l a c k solid. As a n example of controlled absorption, consider the case of a pigment. It is quite c o m m o n to a d d controlled a m o u n t s of a t r a n s i t i o n m e t a l to a t r a n s p a r e n t solid to form a n inorganic pigment. In one s u c h case, we a d d ~ 1% of c h r o m i u m oxide to a l u m i n u m oxide to o b t a i n a p i n k solid, i.e.- "ruby". The Cr3+ ion in the A120 3 lattice a b s o r b s b l u e a n d green light a n d reflects or t r a n s m i t s m o s t l y the red wavelengths. It s h o u l d be clear, then, t h a t b o t h processes, R & A, are wavelength d e p e n d e n t . T h a t is, t h e y d e p e n d u p o n the energy of the photon(s) striking the solid. E.- MEASUREMErr

OF COLOR

Having defined the n a t u r e of a p h o t o n a n d how it i n t e r a c t s with m a t t e r , we c a n now proceed to describe a m e t h o d to m e a s u r e color. We find t h a t even t h o u g h o u r vision is sufficient to categorize color as a gross feature, we still n e e d to be able to m e a s u r e small differences of color. One m e t h o d , u s e d in the past, w a s to a s s e m b l e a series of s a m p l e s having small b u t observable differences in color, a n d t h e n c o m p a r e an u n k n o w n to these. B e c a u s e the h u m a n eye is a s u p e r b color i n s t r u m e n t , this a p p r o a c h w a s perfectly feasible. This m e t h o d u s e d the so-called the Munsell Color Tree as the s t a n d a r d samples. But, it w a s b a s e d on being able to provide reproducible color s w a t c h e s from reproducible pigments. You c a n imagine the p r o b l e m s a s s o c i a t e d in doing so. However, the r e t i n a does not r e s p o n d equally to all wavelengths. For equal energies, a yellow-green light p r o d u c e s a m u c h stronger r e s p o n s e in the h u m a n eye t h a n a red or blue light. T h u s , we s a y t h a t the yellow-green light is '"orighter" t h a n the red or blue lights. This is called the l u m i n o s i t y r e s p o n s e of the eye. However, the d a r k - a d a p t e d h u m a n eye does n o t r e s p o n d to light in the

416

s a m e w a y t h a t it does for bright light. The former is called "scotopic" a n d the latter, "photopic". The following d i a g r a m s h o w s b o t h the photopic a n d scotopic r e s p o n s e curves for the h u m a n eye7.8.11.-

I LUMINOSCITY RESPONSE

c-

= (1) 1 0 0 (1) :p-

~:

9 80

--

-

Q

r 0 r~

"v"

OF THE HUMAN EYEI

STANDARDI

PHOTOPIC

THE OBSERVER

60SCOTOPIC ~ - ~

4020-

rv

0 -

000

4500

5000

5500

6000

6500

Wavelength, Apparently, photopic vision relates to "sunlight", to w h i c h the h u m a n

had

a d a p t e d t h r o u g h evolution, while scotopic vision related to "moonlight", t h a t is, s u n l i g h t modified b y reflection from the Moon's surface. However, it w a s soon discovered t h a t the color r e s p o n s e s of individuals were not exactly the same. E a c h individual "sees" a color slightly differently from a n y o n e else. We have l e a r n e d to discriminate b e t w e e n colors b u t no one k n o w s exactly w h a t a n y o n e else actually sees. Once this fact w a s realized, it w a s u n d e r s t o o d t h a t a n average of w h a t e a c h p e r s o n s a w w o u l d have to be m a d e if a s t a n d a r d s y s t e m w a s to be formed a n d p r o m u l g a t e d . This led to the c o n c e p t of the " S t a n d a r d Observer". T h u s , the r e s e a r c h required to define a n d m e a s u r e color took a completely different p a t h from the original m e t h o d s s u c h as the Munsell Color Tree.

417

I. The S t a n d a r d Observer By m e a s u r i n g a n u m b e r of individual observers, we c a n obtain w h a t we call a " S t a n d a r d L u m i n o s i t y Curve". Photopic vision p e a k s at 5 5 0 0 A w h e r e a s scotopic vision p e a k s at 5 2 0 0 A. In this case, the relative r e s p o n s e of the o b s e r v e r s are s u m m e d into a r e s p o n s e called "I'HE STANDARD OBSERVER" a n d is normalized for easier usage. Now, let u s e x a m i n e the effects of colors (chroma) as perceived b y the h u m a n eye. Keep in m i n d t h a t e a c h p e r s o n perceives "color" s o m e w h a t differently from other persons.

II. The N a t u r e of C h r o m a The visible s p e c t r u m e x t e n d s from a b o u t 4 0 0 0 A to 7 0 0 0 A. We find t h a t the eye acts as a n i n t e g r a t i n g instlnament. T h u s , two colors m a y a p p e a r equal to the eye even t h o u g h

one is m o n o c h r o m a t i c

light a n d

the o t h e r h a s

a band

of

wavelengths. This is s h o w n in the following diagram: 7.8.12.-

Monochromatic Green Lines

.II=

C--

nV

4000

5000

6000

In this case, we have plotted a *spectxum" of colors w h i c h s h o w s the intensity, or n u m b e r s , of p h o t o n s p r e s e n t in a n y given "color" or wavelength (given in A). In the above diagram, we m a y see the s a m e color, b u t the p h o t o n w a v e l e n g t h s

418

are m u c h different. T h u s , we n e e d a m e t h o d t h a t c a n d i s c r i m i n a t e b e t w e e n s u c h cases. It w a s Newton, u s i n g a glass p r i s m plus slits, who first d e m o n s t r a t e d t h a t s u n l i g h t c o n s i s t e d of colors or c h r o m a . S u b s e q u e n t w o r k t h e n s h o w e d t h a t colors could be duplicated b y mixing the three primaries, red, green a n d blue to o b t a i n the v a r i o u s c h r o m a , including s h a d e s of "white". Actually, t h e s e s h a d e s involved the l u m i n o u s intensity, t h a t is- the a m o u n t of light falling u p o n a surface. Let u s first define s o m e of the t e r m s of m e a s u r e m e n t of l u m i n o s i t y a n d t h e n proceed to d e t e r m i n e how to m e a s u r e the c h r o m a , or chromaticity. The a c t u a l t h e o r y is complicated a n d long so t h a t we shall p r e s e n t only a simplified version of the whole theory. III. Intensity a n d S c a t t e r i n g Consider a t h i n fiat plate w h i c h h a s no absorption,

only reflectance a n d

t r a n s m i t t a n c e . We will find t h a t light falling u p o n its surface with a certain i n t e n s i t y h a s a l u m i n a n c e , L, while the l i g h t t r m a a m i t t e d is d e f i n e d a s H, the exittance, or emittance. H will e q u a l L if no a b s o r p t i o n t a k e s place or if there is no s c a t t e r i n g at the surface of the thin plate. distances, a certain a m o u n t of ~ a t t e r h a g

However, even at atomic

does take place. We fund t h a t two

types of s c a t t e r i n g are possible. If the surface is perfectly s m o o t h on a n atomic level, t h e n a light wave would be b a c k - s c a t t e r e d along the s a m e exact path, a n d we w o u l d have a p e r f e c t diffuser, w h i c h we call So . However, t h e r e is always a n angle a s s o c i a t e d with the scattering, w h i c h we call Sa (where (~ is defined as the s c a t t e r i n g angle), a n d we have a n imperfect diffuser. This is s h o w n in the following diagram, given as 7.8.13. on the next page. If we view the thin plate from the left w h e r e it is illuminated with intensity, L, w h a t we see is the b a c k - s c a t t e r e d light, or light diffusion from the surface. If the plate is a perfect diffuser, t h e n we will see a n exact a m o u n t of L s c a t t e r e d b a c k along the s a m e p l a n e as L as a diffuse c o m p o n e n t . Note t h a t we are not s p e a k i n g of reflection (which is a n entirely different m e c h a n i s m w h e r e the

419

wavelength of the light is affected) b u t of s c a t t e r i n g (where the light is a b s o r b e d , t h e n r e e m i t t e d at the s a m e wavelength). For s c a t t e r i n g b y a perfect diffuser, Lo s h o u l d equal Io / S o . However, this is never the case. W h a t we fund is t h a t there is a n a n g u l a r d e p e n d e n c e of scattering, a n d that: 7.8.14.-

I =

la/Sa

= I a / (So cosr

w h e r e r is the angle of scattering. Therefore, H, the light t r a n s m i t t e d , does n o t equal L. If we define ~ as the flux of light, i.e.- the n u m b e r of p h o t o n s incident per second, we Fund: 7.8.15.-

H=

r

Sa = r

coscz

where: ~ = 4n I (point source) = n I (fiat surface)

The i n t e n s i t y u n i t s for ~ are related to a p r i m a r y r a d i a t i o n source, the candela, Cd. The definition of a c a n d e l a is: 7.8.16.-

CANDELA: a unit of Itm~nous intensity, defined as 1 / 6 0 of the

luminous tnter~stty per square centimeter of a black-body radiator operating at the temperature of freezing platinum (1772 ~ Formerly k n o w n as a candle. The unit is abbreviated as: Cd

420

This gives u s the following i n t e n s i t y units: 7.8.17.-

INTENSITY UNITS

1.0 Cd (at a one-foot distance)

L/n

(foot-lamberts)

H / n (lumens) T h u s , we have two u n i t s of m e a s u r e m e n t

of intensity.

One is related to

s c a t t e r i n g from a surface, L, i.e.- in foot-lamberts a n d the o t h e r is related to emittance, H, i.e.- in l u m e n s per s q u a r e foot. Although we have a s s u m e d "white" light u p to now, either of t h e s e two c a n be wavelength d e p e n d e n t . If either is w a v e l e n g t h d e p e n d e n t , t h e n we have a p i g m e n t (reflective- b u t m o r e properly called scattering) with i n t e n s i t y in foot-lamberts, or a n emitter s u c h as a l a m p or p h o s p h o r (emittance) with i n t e n s i t y in l u m e n s . IV. Color P r o c e s s e s a n d Color M a t c h i n g S y s t e m s In t e r m s of color, we c a n have one of two processes: 1. Additive p r o c e s s e s (emittance) 2. S u b t r a c t i v e p r o c e s s e s (reflectance). If we w i s h to m a t c h colors, the p r i m a r y colors are q u i t e d i f f e r e n t , as s h o w n in the following: 7.8.18.-

ADDITIVE PRIMARIES

SUBTRACTIVE PRIMARIES

red

magenta

green

yellow

blue

cyan

Let u s now consider how to set u p a color m a t c h i n g system. One w a y to do so is to average the eye r e s p o n s e s of a large n u m b e r of individuals so as to eliminate

421

the individual "quirks" of the h u m a n eye (some people c a n "see" into the violet or ultraviolet region w h e r e a s o t h e r s cannot). The r e s u l t a n t average is t h a t we call "The S t a n d a r d Observer". F. The S t a n d a r d Observer a n d The First C o l o r - C o m p a r a t o r Since color m a t c h i n g is m e a n t for h u m a n s , it is n a t u r a l to define color in t e r m s of an average, or " S t a n d a r d Observer". O u r first step is to build a n i n s t r u m e n t which

contains

three

colored l a m p

sources,

a place

for the

individual

observer, i n t e n s i t y detectors, a n d a m o n o c h r o m a t o r , as s h o w n in 7.8.19. on the next page. Essentially, s u c h a n i n s t r u m e n t is called a color-comparator. The design is simple a n d w a s u s e d m a n y y e a r s ago to o b t a i n the experimental r e s u l t s n e e d e d to fully define the t r u e n a t u r e of color m e a s u r e m e n t s . In this i n s t r u m e n t , there are two (2) s o u r c e s of light to be compared. One is from a set of three l a m p s w h o s e emission is modified b y m e a n s of suitable Filters to give a red b e a m ,

a

green b e a m a n d a b l u e b e a m . T h e s e are mixed at the screen to form a single spot (although we have not illustrated it in t h a t way, so as to be m o r e discernible). The o t h e r source c o m e s from a m o n o c h r o m a t o r so as to o b t a i n a m o n o c h r o m a t i c b e a m of light. There are controls to a d j u s t the individual b e a m s of red, green a n d blue light, as well as t h a t of the m o n o c h r o m a t i c b e a m . In this way, the mixed b e a m s of light c a n be directly c o m p a r e d to the m o n o c h r o m a t i c spot. In the b a c k of the screen are photoelectric detectors to m e a s u r e the energy i n t e n s i t y of the b e a m s of light being c o m p a r e d . We n e e d a b o u t 5 0 0 0 observers to obtain a satisfactory average, b o t h for the d a r k - a d a p t e d a n d the light-adapted h u m a n eye. There are three (3) t h i n g s t h a t n e e d to accomplished: 1. Define eye r e s p o n s e in t e r m s of color at e q u a l energy 2. Define s h a d e s of '~r

in t e r m s of % red, % green a n d % blue,

at equal energies of t h o s e "whites". 3. Define "color" in t e r m s of % red, % green a n d % blue, as c o m p a r e d to m o n o c h r o m a t i c radiation

422

The difficulty in setting u p the initial s y s t e m for color c o m p a r i s o n s c a n n o t be u n d e r e s t i m a t e d . The p r o b l e m w ~ e n o r m o u s . Q u e s t i o n s as to the suitability of various lamp sources, the n a t u r e of the filters to be used, a n d the exact n a t u r e of the p r i m a r y colors to be def'med occupied m a n y years before the first a t t e m p t s to specify color in t e r m s of the s t a n d a r d observer were started. As we said previously, the S u n is a b l a c k - b o d y r a d i a t o r having a spectral t e m p e r a t u r e of a b o u t I 0 , 0 0 0 ~

(as viewed directly from space). Scattering a n d reflection

423

w i t h i n t h e E a r t h ' s a t m o s p h e r e is sufficient to lower t h e effective b l a c k - b o d y r a d i a t i o n perceived to 6 5 0 0 ~

T h u s , t h e S u n is a 6 5 0 0 ~

s o u r c e w h i c h we

call "daylight". The direct viewed b r i g h t n e s s of t h e S u n at t h e E a r t h ' s s u r f a c e is about 165,000 candela/crn 2

, t h a t of t h e Moon - 0 . 2 5 c a n d e l a / c m 2 , a n d a

clear s k y is a b o u t 0 . 8 c a n d e l a / c m

2.

For t h e s e r e a s o n s , DAYLIGHT h a s b e e n defined as: ' T h e n o r t h e r n skylight at 11:30 am. at G r e e n w i c h , E n g l a n d o n O c t o b e r 31, definition of ILLUMINANT- C.

1931". This is also t h e

The o t h e r s t a n d a r d i l l u m i n a n t t h a t w e u s e is

ILLUMINANT- A, w h i c h is t h e r a d i a t i o n e m i t t e d f r o m a n i n c a n d e s c e n t t u n g s t e n f i l a m e n t o p e r a t i n g at 3 2 5 0 ~

T h e s e are s h o w n in t h e following d i a g r a m :

7.8.20.-

ISpectral Distribution of Standard Illuminating 200

160 120

sources I

I]11uminant' A I.... Illuminant - C

80 40 0 4000

5000

6000

7000

Wave length, Referring b a c k to o u r Color C o m p a r a t o r of

7.8.19., we u s e t h e s e c o n c e p t s to

c a l i b r a t e o u r l a m p s in t e r m s of s p e c t r a a n d relative e n e r g y in t e r m s of t h e s e standard

s o u r c e s . I l l u m i n a n t - B , b y t h e way, w a s originally defined a s

424

"average sunlight" b u t it w a s soon d e t e r m i n e d t h a t "average" is not the s a m e at all p a r t s of the E a r t h ' s globe. O u r next step in u s i n g the Color C o m p a r a t o r is to set u p p r o p e r filters so as to obtain a n d u s e "primary" color l a m p sources. We find t h a t b y u s i n g the m o n o c h r o m a t o r of the Color C o m p a r a t o r , we c a n a p p r o x i m a t e the wavelength r e s p o n s e of

o u r so-called " S t a n d a r d Observer", in the red region, the green

region a n d the b l u e region of the visible s p e c t r u m . B u t we also find t h a t we n e e d a b a n d of w a v e l e n g t h s for e a c h color, since a m o n o c h r o m a t i c b e a m is not at all suitable. This is w h e r e the choice b e c o m e s subjective, since we are relying u p o n the perceived r e s p o n s e of individuals. We fund t h a t b y choosing a b l u e filter p e a k i n g at 4 4 0 0 A, a green filter p e ~ g

at 5 2 0 0 A, a n d a r e d filter

p e a k i n g at 6 2 0 0 A (but having a lesser p e a k in the blue), we have a "Blue" blue, a "Green"-green a n d a "Red"-red w h i c h wiU satisfy m o s t observers. Note t h a t the original single color w a s c h o s e n b y a p p r o x i m a t i n g the h u m a n

eye

r e s p o n s e in the three-color regions, u s i n g m o n o c h r o m a t i c light to obtain a brightness

response,

and

then

properties of the three filters of the

adjusting

the

broad-band

transmission

l a m p s to obtain the p r o p e r colors. A final

c h e c k of t h e s e l a m p filters w o u l d be to m i x all t h r e e colors additively, a n d t h e n to evaluate the "white" t h e r e b y produced. We c a n t h e n s u b s t i t u t e a S t a n d a r d L a m p for the m o n o c h r o m a t o r a n d see if we c a n r e p r o d u c e its exact color t e m p e r a t u r e . If not, t h e n we n e e d to modify the t r a n s m i s s i o n c h a r a c t e r i s t i c s of o u r Filters u s e d on the s o u r c e l a m p s . Once we have done this, we now have o u r three p r i m a r y colors in the form of s t a n d a r d l a m p s , a n d c a n proceed to d e t e r m i n e Items 1,2 & 3, given above on page 421. To do this, we v a r y the wavelength of the m o n o c h r o m a t i c light, a n d d e t e r m i n e relative a m o u n t s of red, green a n d blue light required to m a t c h the m o n o c h r o m a t i c color. This is done, as s t a t e d before, for a b o u t 5 0 0 0 observers. The r e s u l t is finalized r e s p o n s e c u r v e s for the S t a n d a r d Observer, also called "Tristimulus R e s p o n s e curves", a n d is s h o w n as follows:

425

7.8.21.-

i

of the H u m a n Eye l

fill

fill

R-- Red respons'e ' g -Green response -Blue response

1.5 -~>

I

1.0

I

0.5

0

i 4000

5000 6000 Wavelength,

7000

We finally arrive at t h e r e s u l t we w a n t , since w e c a n n o w set u p ' T r i s t i m u l u s Filters" to u s e in defining colors. We c a n n o w define -Y as o u r s t a n d a r d l u m i n o s i t y c u r v e for t h e h u m a n eye (photopic vision). Note t h a t ~ , t h e r e d t r i s t i m u l u s value, h a s a c e r t a i n a m o u n t of b l u e in it in o r d e r to d u p l i c a t e t h e r e s p o n s e of t h e r e d p r e c e p t o r in t h e retina. I. T r i s t i m u l u s Coefficients O u r n e x t s t e p is to define colors in t e r m s of t r i s t i m u l u s r e s p o n s e s . We k n o w t h a t we c a n define t h e e n e r g y of a n y s p e c t r a l c u r v e a s a s u m m a t i o n of i n t e n s i t i e s t i m e s w a v e l e n g t h s , i.e.7.8.22.-

ER

= Z (IdX)R

EG

= Z (IdX)G

EB

= Z (Id~)B

426

Therefore ff w e t a k e t h e s p e c t r a l curve, a n d m u l t i p l y it b y t h e o v e r l a p of e a c h t r i s t i m u l u s r e s p o n s e curve, we get TRISTIMULUS VALUES, i.e.7.8.23.-

Reflective {scattering]

Emittance

(subtractive)

(additive)

x= ~ EIR dX Y =Y ZIGdX

X =~ Y

= -~ ~ I G R G

Z

Z

=z" ~ IBRB dX

= ~ EIBdk

However, we find t h a t t h e s e v a l u e s

EIR RRdX dX

are difficult to u s e since e a c h color give a

set of t r i s t i m u l u s v a l u e s , b u t e a c h set does n o t h a v e a specific r e l a t i o n to a n y other. T h e r e a s o n for t h i s is t h a t t h e i n t e n s i t y of IR ~ IG ~ IB . Therefore, we define a s e t of c h r o m a t i c i t y c o o r d i n a t e s , a s s h o w n in t h e following, w h e r e t h e tristhnulus

values

are

used

to

defme

what

we

now

call

"Chromaticity

z =

Z /X+Y+Z

Coordinates". 7.8.24.-

CHROMATICITY COORDINATF_~: x + y + z--

x =

X/X+Y+Z

1.00

y =Y/X+Y+Z

Note t h a t we n o w h a v e t h r e e

(3) different s e t s of v a l u e s r e l a t e d to color

specification, a n d t h r e e different t y p e s of s y m b o l s . In o r d e r n o t to get c o n f u s e d , we r e i t e r a t e t h e m again. T h e y are: 7.8.25.-

VALUES RELATED TO COLOR SPECIFICATION

Color T r i s t i m u l u s R e s p o n s e

Tr_-i. " s t i m u l u s Value.s

Chromaticity Coordinates

red:

~

X

X

green:

Y

Y

Y

blue:

~

Z

Z

427

T h e a d v a n t a g e of c h r o m a t i c i t y c o o r d i n a t e s is t h a t we n o w h a v e a set of normalized

v a l u e s w h i c h w e c a n u s e to c o m p a r e colors h a v i n g different

i n t e n s i t y v a l u e s (and t h u s different e n e r g y v a l u e s a s well). F u r t h e r m o r e , we n e e d o n l y specify • a n d y s i n c e x + y monochromatic

+

z = 1.00. T h i s allows u s to specify

r a d i a t i o n in t e r m s of o u r c h r o m a t i c i t y c o o r d i n a t e s . Actually,

w e c a n plot a t h r e e - d i m e n s i o n a l v a l u e o n a fiat ( 2 - d i m e n s i o n a l ) s u r f a c e . !I. Cb_romati'city C o o r d i n a t e Diagram.s S i n c e m o n o c h r o m a t i c r a d i a t i o n is a b o u n d a r y of color-mLMng, t h e n w e c a n c o n s t r u c t a CHROMATICITY COORDINATE DIAGRAM in t e r m s of x a n d y7.8.26.-

O.Y ,4.,m

=

mlmI

L 0

0.8

0.7

o

0.6

.~

0.5

tIE chromaticiiyl.. Diagram I

s30 540

510

oo. ....

;i~i~5.60.:Iby Kelly

Green

(1940) _

500

580"".......

mmIm

E

0.4

ca

0.3

0 L J=

m.

600

Green

Green

White

0.2

700 ~

d~S'-"

".

9

0.I

0.I

0.2 0.3 0.4 Chromaticity

0.5 0.6 0.7 Coordinate

.

9

428

Note t h a t the d i a g r a m is b o u n d e d , as we have already stated, b y the v a l u e s of m o n o c h r o m a t i c light. T h u s , we c a n find a n y color, be it m o n o c h r o m a t i c or polychromatic, in t e r m s of its • a n d y coordinates. We have delved into the m e t h o d s u s e d b y previous investigators in a n effort to quanticize color m e a s u r e m e n t a n d u s e d the s a m e m e t h o d s t h a t t h e y did. Once this w a s done, color specifications b e c a m e s t a n d a r d i z e d a n d were n o t subject to vagaries of the color-method u s e d or the deviations c a u s e d b y the h u m a n eye. It would be m o r e d r a m a t i c to p r i n t the v a r i o u s h u e a r e a s in color b u t it is difficult to a c c u r a t e l y p r i n t reflectance h u e s . It is b e t t e r to n a m e the colors directly. This w a s done b y Kelly (1940). Note also t h a t we do not u s e the t e r m "color" a n y m o r e b u t u s e the t e r m "hue". Any h u e c a n be specified by x a n d y. For example, we c a n specify the locus of b l a c k - b o d y h u e s a n d even I l l u m i n a n t s A, B & C. This is s u m m a r i z e d as follows: 7.8.27.-

xan.d y C h r o m a t i c i t y C o o r d i n a t e s x

_y._.

6000 A

0.640

0.372

5200

0.080

0.850

4800

0.140

0.150

Illuminant A (3250 A}

0.420

0.395

I l l u m i n a n t B (4500 A}

0.360

0.360

I l l u m i n a n t C (6500 A}

0.315

0.320

T h e s e points on the c h r o m a t i c i t y d i a g r a m are s h o w n in the following diagram, given as 7.8.28. on the next page. Nonetheless, while d e t e r m i n i n g color b y the color- c o m p a r a t o r m o d e w a s a distinct i m p r o v e m e n t over prior m e t h o d s of color c o m p a r i s o n , the m e t h o d w a s

429

slow a n d not e a s y to accomplish. W h a t w a s n e e d e d w a s an i n s t n n n e n t a l m e t h o d of color m e a s u r e m e n t a n d m a t c h i n g .

The r e q u i r e m e n t s for a n i n s t r u m e n t a l m e t h o d of specifying reflected color include a light source, the colored object a n d a detector. W h a t this m e a n s is t h a t all we need is a source, a n object a n d a detector. However, since the r e s p o n s e c h a r a c t e r i s t i c s of t h e s e optical c o m p o n e n t s are n o t linear, nor flat, we n e e d a n a n a l o g u e s y s t e m in o r d e r to be able to m e a s u r e color. The analogue s y s t e m simply corrects for the non-linearity of the s o u r c e a n d detector. The reflectance a n a l o g u e system, given as 7.8.29. on the next page,

430

s h o w s s c h e m a t i c a l l y how the s p e c t r a l power d i s t r i b u t i o n of a CIE source, the spectral reflectance, R, of a n object, a n d the spectral c o l o r - m a t c h i n g functions, x, y a n d z, combine b y multiplication (each wavelength b y e a c h wavelength), followed b y s u m m a t i o n a c r o s s the s p e c t r u m , to give the CIE t r i s t i m u l u s values.

However, t h e s e a n a l o g u e s are actually hypothetical. The r e a s o n for this is t h a t it is n e a r l y impossible to obtain optical m e a s u r e m e n t c o m p o n e n t s , s u c h as the s o u r c e a n d the detector, w h o s e r e s p o n s e to light a c r o s s the visible s p e c t r u m is fiat (or n e a r l y so). However, this is not a n impossible t a s k a n d we f'md t h a t a n excellent m a t c h c a n be o b t a i n e d to the t r a n s m i s s i o n functions of 7.8.21., i.e.those of the S t a n d a r d Observer. This is typical for commercially available i n s t r u m e n t s . Now, we have a n i n s t r u m e n t , called a Colorimeter, capable of m e a s u r i n g reflective color. The a c t u a l a n a l o g u e v a l u e s we n e e d to m e a s u r e reflectance are given on the next page as 7.8.30. as follows. Note t h a t the optical r e s p o n s e c u r v e s of the m e a s u r i n g parts, i.e.- the n o n - l i n e a r i t y of the s o u r c e a n d detector, are now corrected in the r e s p o n s e of the overall i n s t r u m e n t .

431

7.8.30.-

We find t h a t a n excellent m a t c h c a n be o b t a i n e d to the t r a n s m i s s i o n f u n c t i o n s of 7.8.20. This is typical for commercially available i n s t r u m e n t s . Now, we have a n i n s t r u m e n t , called a Colorimeter, capable of m e a s u r i n g reflective color. A s y s t e m for e m i t t a n c e color m a t c h i n g is given in the following :

This d i a g r a m s h o w s the energy s p e c t r u m of a given source, coupled with a triter of defined t r a n s m i t t a n c e , w h i c h is e s t a b l i s h e d b y a detector of k n o w n spectral response, as modified b y a s t a n d a r d s o u r c e a n d modified to t h a t of a S t a n d a r d Observer. Once a n i n s t r u m e n t h a s b e e n set u p properly with the p r o p e r optical

432

r e s p o n s e modifications, it will p r o d u c e c h r o m a t i c i t y coordinates t h a t a c c u r a t e l y reflect t h o s e n e e d e d to define the "color" of the emitter or reflector being measured. The p r o c e s s for actually m e a s u r i n g emissive color is s o m e w h a t different arid m o r e challenging. First, we m u s t obtain a n emission s p e c t r u m b y m e a n s of a spectrofluorimeter. We c a n now integrate I dk to o b t a i n the energy a n d t h e n specify this in t e r m s of x a n d y. There are special m e t h o d s w h i c h have b e e n developed to do so w h e r e i n selected w a v e l e n g t h s are used, d e p e n d i n g u p o n the n a t u r e of the emission s p e c t r u m . We will not delve f u r t h e r into this m e t h o d o t h e r t h a n to state t h a t it does exist. 7-9 C O ~ R

SPACES

If we have a certain color, a c h a n g e in i n t e n s i t y h a s a m a j o r effect on w h a t we see (in b o t h reflectance a n d emittance). For example, if we have a blue, at low i n t e n s i t y we see a b l u i s h - b l a c k , while at high i n t e n s i t y we see a bluish-white. Yet, the h u e h a s not c h a n g e d , only the intensity. This effect is p a r t i c u l a r l y significant in reflectance since we c a n have a "light-blue" a n d a "dark-blue", w i t h o u t a c h a n g e in c h r o m a t i c i t y coordinates. The concept of "lightness" involves the reflective power of m a t e r i a l s . If the reflectance a p p r o a c h e s 100%, we s a y the m a t e r i a l is "white", w h e r e a s complete a b s o r p t i o n (0.00% reflectance) p r o d u c e s a "black" material. Let u s e x a m i n e exactly w h a t is m e a n t b y t h e s e terms, p a r t i c u l a r l y those of s u b t r a c t i v e color mixing. W h e n colors are p r e p a r e d b y mixing dyes or p i g m e n t s , the r e s u l t a n t reflective h u e is controlled b y a s u b t r a c t i v e p r o c e s s of the three (3) primaries, w h o s e reflectance s p e c t r u m is given above. W h e n t h e s e are mixed, the r e s u l t i n g h u e is t h a t w h e r e the curves o v e r l a p ,

as is easily s e e n in 7.9.1.,

given on the next page. W h e n we mix t h e s e p r i m a r y colors, their reflectances remove m o r e of the incident light, a n d we see the p a r t w h e r e the reflectances are reinforced. T h u s ,

433

we c a n get red, green a n d blue, b u t t h e y are not p r i m a r y colors in the s u b t r a c t i v e system. I n t e r m e d i a t e h u e s c a n be o b t a i n e d also in this p r o c e s s w h e n the s u b t r a c t i v e p r i m a r i e s are u s e d in less t h a n full c o n c e n t r a t i o n . T h a t is, they

are

"lightened".

Although

we

can

explain

this

effect

on

a

s p e c t r o p h o t o m e t r i c basis, we do not have a w a y of specii~ying h u e s in t e r m s of

saturation, u s i n g the CIE s y s t e m . A....The MunseU Color Tree One of t h e first a t t e m p t s to specify reflective colors a n d color mixing w a s a c c o m p l i s h e d b y Munsell (1903). He devised a color s y s t e m b a s e d on factors he

434

called

hue,

ehroma

mad

value.

MunseU

set

up

a

three-dimensional

a r r a n g e m e n t b a s e d u p o n m/n/mum perceptual color difference steps. He b a s e d these u p o n direct observation since he did not have the i n s t r u m e n t a l m e a n s to do so. Therefore, his r e s u l t s are not the s a m e as the s y s t e m t h a t we u s e today. The a d v a n t a g e of the MunseU s y s t e m w a s t h a t in addition to being able to specify hue, we could also specify C h r o m a , w h i c h is the degree of s a t u r a t i o n of a specific hue. Value is the relative a m o u n t of "blackness" or "whiteness" (equal p a r t s of b l a c k a n d white p r o d u c e gray) in the p a r t i c u l a r reflective color being specified. It is this s y s t e m t h a t gives u s access to the degree of "grayness" a n d / o r s a t u r a t i o n of a n y hue. On a practical basis, if we wish to set u p this system, we w o u l d a s s e m b l e a set of "color-chips". E a c h color-chip w o u l d be specified b y two factors, H = hue, a n d V/C, w h i c h is value (grayness) modified b y c h r o m a (saturation). The actual n u m b e r of layers in the MunseU Color Tree w a s d e t e r m i n e d b y " m i n i m u m p e r c e p t u a l difference". T h a t is, the m i n i m u m c h a n g e t h a t p r o d u c e s a visual perceptible difference. This a r r a n g e m e n t specifies all light colors as well as the d a r k ones. To u s e s u c h a system, one would choose the color-chip closest to the h u e a n d s a t u r a t i o n of the test color a n d t h u s obtain v a l u e s for H a n d V/C. However, it w a s s o o n discovered t h a t the s y s t e m w a s not perfect. R e a s o n s for this include the facts t h a t the h u e s defined b y Munsell are not t h o s e of the p r i m a r i e s of the h u m a n eye. F u r t h e r m o r e , MunseU w a s s o m e w h a t subjective in his definitions of h u e s . In 1920, Priest s h o w e d t h a t if the Munsell-Chips were viewed on a whiteb a c k g r o u n d , the "brighmess", i.e.- lightness as viewed b y the h u m a n eye, could be related to the MunseU s y s t e m by: 7.9.3.-

V = 10 Y I/2

w h e r e Y w a s defined as "brighmess" or eye response. Over the years, the Munsell S y s t e m h a s b e e n modified to agree with actual h u m a n perception,

435

using a "middle-gray" b a c k g r o u n d . Nevertheless, it is well to note t h a t it was the subjective observation of the lack of correction for luminosity in the Munsell Syst em t h a t gave i m p e t u s to the development of the CIE Color System. The major problem with the Munsell system was t h a t each person attempting to m a t c h colors did not produce the exact same result. So color m a t c h i n g became dependent u p o n the person. B. Color Matching a n d MacAdam Space When MunseU devised his color space, he did so on the basis of m i n i m u m observable color perception steps. But the problem with the MunseU System_ w a s one of reproducibility, which the CIE S t a n d a r d Observer cured. In formulating a color match, one w a n t s to be able to predict the correct concentration of colorants required, whose scattering a n d absorption properties are known, i.e.- the lightness, so as to m a t c h the sample submitted, starting with their spectrophotometric curves. In practice, this is not so simple, since two colors m u s t have identical spectrophotometric curves to be e x a c t l y equal. It t u r n s out t h a t the h u m a n eye will identify the two colors to be equal ff their spectrophotometric reflectances are reasonably close. Two colors m a y appear to be equal u n d e r Daylight illumination, b u t quite different u n d e r i n c a n d e s c e n t lamp illumination. These colors are know n as "metamers" a n d the p h e n o m e n o n "metamerism". It was soon determined t h a t the 1931 CIE chromaticity diagram, and l u m i n a n c e function, Y, are not representative of e q u a l visual spacing, t h a t is, equal changes in Y do not represent equal changes in visual perception for all values of Y. Nor do equal increments of x a n d y represent the s a m e visual effect for all locations on the chromaticity diagram. In other words, there is a m i n i m u m perceptual difference on b o t h x a n d y (i.e.- Ax and Ay). But, the size of Ax a n d Ay is not the s a m e at all p a r t s of the chromaticity diagram. This is the same problem t h a t Munsell encountered a n d is due to the fact t h a t the h u m a n eye is

436

a n i n t e g r a t i n g i n s t n m l e n t , n o t a dispersive one. M a c A d a m (1942) s t u d i e d t h i s problem

and

established

limits

of M i n i m u m

Perceptual

Difference

steps

(MPD) in t h e form of ellipses o n t h e CIE d i a g r a m . This is s h o w n a s follows:

It t u r n s o u t t h a t a l i n e a r t r a n s f o r m a t i o n of x a n d y c o o r d i n a t e s to a n e w set of c o o r d i n a t e s , u a n d v, is sufficient to do t h e job. M a c A d a m ' s t r a n s f o r m a t i o n e q u a t i o n s were:

437

7.9.5.-

u v

=

4 x / (-2x+ 1 2 y + 3 )

= 6y / ( - 2 x + 1 2 y + 3 )

Plotting t h e s e v a l u e s gave the following c h r o m a t i c i t y diagram:

The t r u e test is how well the Munsell h u e s plot o u t on the CIE diagram. As c a n be s e e n on the right h a n d side of the diagram, the Value 5 - C h r o m a 8 h u e s do c o n s t r u c t a n e a r l y perfect circle. T h u s , the M a c A d a m t r a n s f o r m a t i o n is a definite i m p r o v e m e n t over the 1931 CIE system. In 1960, the CIE a d o p t e d the MacAdam System, h a v i n g defined t h e e q u a t i o n s (with MacAdam's help): 7.9.7.-

u= v=

4 X / (X+ 15Y+ 3Z) 6Y/

(X+ 1 5 Y + 3Z)

438

The following diagram, shows MacAdam's Uniform Chromaticity Scale, a n d h a s Minimum Perceptible Color Difference steps plotted as ellipses:

It should be clear that the CIE Color Chromaticity diagram, adopted in 1960 is a superior m e t h o d for delineating color differences. Indeed, it is the one specified for j u s t s u c h usage. Recommended Reading v

1. Robert E. Reed-Hill, "Physical Metallurgy Principles"Princeton, New J e r s e y (1964). 2. A.J. Dekker, "Solid State Physics" J e r s e y (1958).

Van Nostrand,

Prentice-Hall, Englewood Cliffs, New

3. Perelomova & Tagieva, "Problems in Crystal Physics"Eng. Transl. - (I 983).

MIR Publ., Moscow,

439

4. W.W. W e n d l a n d t , Thermal Methods of Analysis, Interscience-Wiley, New York (1964). 5. P.D. Garn, Thermoanalytical Methods of Investigation, Academic Press, New York (1965). 6. W.J. S m o t h e r s a n d Y Chiang, Handbook of Differential Thermal Analysis, Chemical Publishing Co., New York (1966). 7. E.M. Barral a n d J.F. J o h n s o n , in Techniques and Methods of Polymer Evaluation, P. Slade & L. J e n j i n s - Ed., Dekker, New York (1966). 8. W.W. Wendlandt, Thermochirra Acta 1 , (1970) 9. D.T.Y. Chem, J. ThermaIAnal., 6 109 (1974) - Part I 10. C h e n & Fong, loc. cit. 7 295 (1975) - Part II. 11. Fong & C h e n , loc. cit., 8 305 (1975) - Part III 12. C. Duval, Inorganic Thermogravimetric Analysis, A m s t e r d a m (1963).

2nd

Ed.,

Elsevier,

13. C. Keatch, An Introduction to Thermogravimetnj, Heydon, L o n d o n (1969). 14. "Vapor P r e s s u r e D e t e r m i n a t i o n u s i n g DSC", A. Brozens, R.B. Cassel, C.W. S c h a u m a n n & R. Seyler, Proceeding. N. Am. Therrra Analysis Soc. 2 2 n d Conf. Denver, Colo. (1993). Problems for Cha~)ter 7 _

1. You have a d d e d a solution of c a l c i u m chloride to a solution of s o d i u m metaborate. A crystalline precipitate resulted. In a t t e m p t i n g to analyze y o u r product,

440

an u n k n o w n x-ray pattern w a s obtained. You then u s e d DTA and TGA to attempt to characterize this product. Using the following DTA and TGA data,

--J........ 2 rrla.loss 54 mg loss

Iso=!,

o~

E L

r 100

m.l

109 mg loss

pi.

c~

co 200

go

0 ..J

|TGA Thermogram of Unknown j

.C:

I

I

200

,

I

,

400

,I

600

,,

I

8OO

,I , ,I

~ooo

J E

L a)

e--

o x LIJ

unI(now nl I-DTATherm-ogra ....... ~ o'f .....

a) e-

e-

200

400

600

Temperature in ~

800

1000

1200

441

identi_t~y all of the p r o d u c t s formed d u r i n g the r e a c t i o n s b y writing the r e a c t i o n s w h i c h o c c u r d u r i n g heating. Be s u r e to characterize b o t h the initial a n d final product. Write also the precipitation reaction. As a n aid, x-ray diffraction analysis w a s able to identify a p r o d u c t o b t a i n e d b y calcination at 1000 ~ as r c a l c i u m m e t a b o r a t e . Be s u r e to identify all of the p e a k s in the t h e r m o g r a m s . 2. C o n s t r u c t y o u r own DTA a p p a r a t u s , u s i n g a '%uilding-block" a p p r o a c h . The desired p a r a m e t e r s for y o u r DTA are: m a x t e m p = 1200 ~

Atm = A, N & R. Be

s u r e to specify the p a r t i c u l a r s of y o u r a p p a r a t u s : e.g.- t e m p p r o g r a m m e r = 1 to 30 ~ per m i n u t e , etc. 3. Using y o u r DTA, d r a w the expected curve for decomposition of:

a. SrHPO4 b. CaCOa C. Y(NOa)a 9 H20 You m a y w i s h to look u p decomposition t e m p e r a t u r e s in a H a n d b o o k of Chemistry. 4. Draw a n expected TGA curve for all 3 c o m p o u n d s , u s i n g the t e m p e r a t u r e s t h a t y o u u s e d in y o u r DTA curves. 5. Given the color c o o r d i n a t e s of: x = 0.34 & y = 0.5 I. Identify the color having this property. If x = 0 . 3 4 & y = 0.61, i d e n t ~ / t h e difference b e t w e e n t h e s e two colors. Identify the a m o u n t s of red, green a n d blue in e a c h of these. 6.

Do

the

same

for

the

following chromaticity

c o m b i n a t i o n s possible}: X = 0 . 2 5 , 0 . 4 5 , 0.65 y = 0 . 2 0 , 0 . 4 0 , 0.60

coordinates

(use

all

6

442

T h e r e are n i n e c o m b i n a t i o n s to be m a d e . Identify e a c h of t h e t h r e e at a time in t e r m s of i n c r e a s i n g green, or red.

SUBJECT INDEX

443

Abrams Eq. Absorbance Activated c o m p l e x Additive p r i m a r i e s Agglomerates Aggregates

188 413 390 420 196 195

Charged vacancies Charged vacancy

152 173

Charged vacancy Chem. Planarization Chromaticity diagram CIE c h r o m a t i c i t y

90 326 427 435

Amorphous solar cells Andreasen p i p e t t e Anion sub-lattice Anion vacancy Anti-Frenkel Anti-structure Arrhenius equation

351 239 78 81 94 110 140

Assay of materials Associated defects Avagadro's n u m b e r Balck and white color

386 102 12 408 41 344 413 245 226

CIS(1960) Color Clean r o o m CMOS design Color c e n t e r s Color Comparator Color m a t c h i n g Color spaces Color specification Common TC's Conductivity in solids Congruently m e l t i n g Coulter C o u n t e r TM

438 315 324 93 422 431 432 426 360 304 296 242

Critical radius Crucibles Cpjstallites Cumulative f r e q u e n c y

183 257 251 219

Czrochralski m e t h o d D - defects in silicon Damascene Defect equilibria Defect m o v e m e n t Defect reactions Defect symbolism Defect t h e r m o d y n a m i c s Defect types Defined vapor p r e s s u r e Degrees of f r e e d o m Diameter control Dice Differential TC

260 336 325 101 152 148 98 102 73 13 9 262 318 361

Band m o d e l s Becqueral Beer-Lambert law BET m e t h o d Bimodal distribution Binomial t h e o r e m Black body Bragg equation Bravais lattices Bridgeman m e t h o d BriUoin zones Buoyancy factor in TGA Calorie Candela Cation sub-lattice Cation vacancy Changes of state Charge c o m p e n s a t i o n Charged interstitials

209 406 35 46 271 40 383 4,5 419 78 81 1,3 79 91

444 SUBJECT INDEX

Diffraction file Diffusion b a r r i e r Diffusion g r o w t h Diffusion in spinel Diffusion r e a c t i o n s Dilatometric d e t e c t i o n Dflatometry Discontinuous limit Dislocation line Dispersion and m i x i n g Doping IKSC DTA DTA heating rate DTA t h e r m o g r a m DTA-glass points DTA-phase d i a g r a m Dynamic TGA Dystectic c o m p o s i t i o n Edge defined g r o w t h Embryo formation E m i t t a n c e analogue Enantiomorphs Encapsulating agent Endothermic Entropy effect Epitaxial g r o w t h Eukaryotic cells Eutectic Eutectic p o i n t Evacuated capsule Evaporation Ewald s p h e r e Exothermic E x t r e m e UV Extrinsic d e f e c t

58 332 144 159 154 396 394 224 87 132 93 374 361 367 363 379 367 385 299 295 142 430 300 270 358 71 288 63 66 25 293 13 38 358 339 86

Eye r e s p o n s e - l i g h t Eye sensitivity curve Eye sensitivity curve

416 408 409

F-Center Fermi level in solids Fick's first law Fine p a r t i c l e s Flame fusion Floating zone Flux Formation of nuclei Freeman-CarroU Frenkel d e f e c t Frenkel pairs Frequency plot Furnace d e s i g n Gas adsorbtion m e t h o d Gaussian distribution Gibbs-Thompson Glass expansion Glass points Globars TM Grades of purity Grain b o u n d a r i e s Growth d e f e c t s Growth limits Halophosphate Hancock a n d S h a r p e Heat Heat capacity Heat factors Heating e l e m e n t s H e r m a n n - Mauguin Heterotype Homotype Hopping m e c h a n i s m

94 359 149 171 282 275 170 191 392 80 161 217 253 245 210 144 398 379 255 111 251 266 222 197 156 4 3 3 254 52 96 96 134

SUBJECT INDEX

Hostogram 217 Huyghen's principle 411 Hydrothermal g r o w t h 2 9 0 IC m a c h i n e s 327 Impingment 192 Impurity distribution 277 Impurity doping 329 Impurity effect 72 Impurity leveling effect 2 7 8 Interface angle 264 Interface energy 186 Interface s t r u c t u r e 180 Internal heat 6,7 Interstice 22 Interstitial sites 90 Intrinsic defect 86 Inversion s y m m e t r y 50 Ionic defects in ice 307 Isothermal TGA 385 J o h n s o n - M e h l Eq. 188 Kirchendall effect 153 Kyropoulos m e t h o d 273 Laser r e f r a c t o m e t r y 247 Lattice angles 44 Lattice Directions 33 Lattice i n t e r c e p t s 44 Lattice points 32 Lattice tension 83 Line defect 82 Lithographic p a t t e r n s 3 3 0 Lithographic p r o c e s s 314 Log normal distribution 2 1 2 Log normal plot 220 M-center 95 MacAdam color space 4 3 6 Martin's d i a m e t e r 235

Mean free p a t h Metal carbonyls Metal expansion Metasilicate Metrology Mie m e t h o d Miller indices Molten flux Monolayer Muncell color MunceU Color Tr e e MXS binary c o m p o u n d

445

11 294 400 168 328 247 37 286 245 434 415 104

n-type 95 Negative bonding defect 3 0 5 Negative p h o t o r e s i s t 317 Newton and colors 408 Nitrides 325 Nuclear g r o w t h 145 Nucleation 141 Octahedron 21 Optical lithography 338 Optical p r o p e r t i e s 411 Ostwald r i p e n i n g 192 P-n-p transitor 312 p-type 95 Pairs of defects 105 , Parabolic law 147 Particle orienation 235 Particle p r o p e r t i e s 207 Peltier effect 359 Permeability 245 Phase boundary 133 Phase boundary g r o w t h 144 Phase relationships 64 Phonon m o d e s 16 Phosphors 100

153

446 SUBJECT INDEX

Photomask Photon s c a t t e r i n g Photon wavelengths

316 418 406

Photons in solids 414 Photopic r e s p o n s e 416 Photresist 316 Piezoelectricity 352 Point defect 74 Point groups 49 Polystructure 302 Polytypes 301 Pore g r o w t h 194 Positive bonding defect 3 0 5 Positive hole 92 Positive p h o t o r e s i s t 317 Propagation m o d e l s 17 Pure silicon 310 Quartz g r o w t h 291 Radiation interactions 4 1 2 Rate equations 138 Rate equations-solids 3 8 8 Rate of particle fall 239 Reciproval lattice 38 Reflectance 413 Reflectance analogue 4 3 0 Reliability factor 57 RF g e n e r a t o r 263 ROntgen x-rays 34 Rotaional s y m m e t r y 49 Schoenflies 51 Schottky defect 80 Scotopic r e s p o n s e 416 Screw dislocation 85 Sealed autoclave 289 Sedimentation balance 2 3 8 Sedimentation tube 241

Seebeck effect 358 Seed crystal 258 Self interstial 75 Shrinkage 194 Shrinkage 202 Silicon crystal g r o w t h 311 Silicon hydride cells 351 Silicon voids 336 Sintering 193 Sintering m e c h a n i s m s 201 Sintering T h e r m o . 198 Size range-particles 205 Slow nucleation 188 Solar cell 347 Solid solution 23 Solid state m e c h a n i s m s 130 Solid s y m m e t r y 35 Sonar 353 Space groups 51 Space lattices 47 Spinel 157 Spiral g r o w t h 87 Stacking faults 299 S t a n d a r d Observer 417 Standard sources 423 Static crucible 268 Stirling's approximation 125 237 Stoke's Law 57 Structure factor Subtractive p r i m a r i e s 4 2 0 Surface energy 198 Surface nucleation 142 Surface sites 90 Symmetry distribution 61 Symmetry e l e m e n t s 36 Symmetry operations 50

SUBJECT INDEX

T e m p e r a t u r e Scales Tetrahedron TGA

2 21 382

TGA-sample closure The plane n e t Thermal analysis Thermal capacity Thermal r e s i s t a n c e T h e r m o m e t r i c points

384 89 357 3 367 371

Thermometry Thin film p r o c e s s Translation vectors Tristimulus coefficient Tristimulus r e s p o n s e Twinning defects Tyler s c r e e n s U.S. s c r e e n s Unit cell volume Unit vector Vacancies Vacancy Vacancy-structure Vapor phase g r o w t h Verneuil g r o w t h Visual counting Volume defect Volume nucleation Wafer fabrication Wafer operations Wafer s t e p s Zone m e l t i n g Zone refining

401 316 34 426 425 300 213 213 33 33 90 75 110 292 282 233 74, 85 143 323 320 322 275 276

447

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