t00

Iczc

II

."i_____INVESTIGATION OF THEORETICAL AND PRACTICAL1 ASPECTS OF THE THERMAL EXPANSION OF

CERAMIC MATERIALS

1 Prepared for:Department Of The NavyBureau Of Weapons

II FINAL REPORTS .... nCoaftact No. NOrd - 18419

CAL Report No. P1- 1273-M- 125 July 1962

CORNELL AERONAUTICAL LABORATORY, INC.OF CORNELL UNiVERSITY, BUFFALO 21, N. Y,

IC\ i

VA - OU . .I... . .i i - i nI I

I.

NDTICE: 'When government or other drawings, spec1 -

fications or other data are used for any purposeother than in connection with a definitely relatedgovernment procurement operation, the U. S.Govere&t thereby incurs no responsibility, nor anyobligelon whatsoever; and the fact that the Govern-ment may have formulated, furnished, or in any waysupplied the said drawings, specifications, or otherdata is not to be regarded by implication or other-wise as in any manner licensing the holder or anyother person or corporation, or conveying any rightsor permission to manufacture, use or sell anypatented invention that may in any way be relatedthereto.

f '-¶uln copies ofk 'thi

ttpbf from AS~ul

CORNELL AERONAUTICAL LABORATORY, INC.BUFFALO 21, NEW YORK

F INAL REPO RT

INVESTIGATION OF THEORETICAL AND PRACTICALASPECTS OF THE THERM4AL EXPANSION

OF CERAMIC MATERIALS.

* iiREPORT NO. P1-1273-14-12

CONTRACT NO. NO rd- 18'419

KJULY 1962

*PREPARED BY: APPROVED BY:

Howad A.Sche~tzH. P Xirehn . r. Staff Scienti/st* How rd A S~h e.FzComputer Research Department

Harold r. Smwy h

.H.. P., KirchnerProject Engineer Compt.uter Research Derartmfen-t

PREPARED FOR:

DEPARTMENT OF THE N4AVYBUREAU OF WEAPONS

ABSTRACT

!itSSeveral methods were used to predict the thermal expansion coefficientsI of pure, single-phase ceramics. The predictions were extended to cover 1700

pure phases. Based upon these calculations a number of phases, predicted

to have low values of thermal expansion coefficient, were synthesized and the

thermal expansion properties of these materials were measured. Several of

these were found to be low-expansion materials. The thermal expansion

properties of cubic. UP 0 are particularly interesting. This material expands

with increasing temperature up to about 40&0 C. Above this temperature the

crystal contracts, returning to its room temperature length at about 100°0 C.

These properties are believed to be unique.

Variation of the thermal expansion anisotropy by addition of solid

solution atoms to several base crystals was attempted. Addition of vanadium

Sto rutile (TiO2 ) resulted in a significant decrease in thermal expansion

anisotropy. Vanadium additions to cassiterite (SnO2 ) which has the same

F structure as rutile, also caused a significant decrease in thermal expansion

anisotropy.

STwo principal methods were used to predict the thermal expansion of

two-phase ceramic bodies ,nowing the thermoelastic properties of the individual

phases. Kerner's method was applied to ceramic bodies for the first time.

Satisfactory predictions were made for several two-phase systems.

Ii

1 iii P1-!273-1412 1

K

ABSTRACT

Several methods were used to predict the thermal expansion coefficients

of pure, single-phase ceramics. The predictions were extended to cover 1700

pure phases. Based upon these calculations a number of phases, predictedto have low values of thermal expansion coefficient, were synthesized and the

thermal expansion properties of these materials were measured. Several ofr these were found to be low-expansion materials. The thermal expansion

.properties of cubic UP2 0 are particularly interesting. This material expands

F with increasing temperature up to about 400°C. Above this temperature the

crystal contracts, returning to its room temperature length at about O000 C.

These properties are believed to be unique.

Variation of the thermal expansion anisotropy by addition of solid

solution atoms to several base crystals was attempted. Asidition of vanadium

. to rutile (TiO2 ) resulted in a significant decrease in thermal expansion

anisotropy. Vanadium additions to cassiterite (Sno2 ) which has the same

structure as rutile, also caused a significant decrease in thermal expansion

anisotropy.

Two principal methods were used to predict the thernal expansion of

two-phase ceramic bodies knowing the thermoelastic properties of the individual

phases. Kerner's method was applied to ceramic bodies for the first time.

Satisfactory predictions were made for several two-phase systems.

iii PI-1273-N-12

r-:

"FOREWORD

This final technical report covers research performed on a programsponsored by the Department of the Navy, Bureau of Weaponr, under Contract

NOrd-18419. The report includes significant sections from reports previously-submitted for the period June, 1958 to September, 1960 and complete coverage

of research performed during the period September, 1960 to July, 1962.

This research was performed under the general technical direction of iMr. S. J. Matesky of the Bureau of Weapons.

iiI

I ij

iv PI-12 73 -M-12i

.1--"

rTABLE OF CONTENTS

Page

FORWdORD. 0 v . 0 a . 0 0 iv

LEST OFTABLtES . . . . . . . . . . .. .. BB j

I. IhTRODUCTION. *..........*

-I. PREDICTION OF. THE THERMAL EX3ANSION GWEFFICMITS OFWREECERAMIC CRYSTALS 0 ......... 3

A. Introduction.. .

C. Openness of Homologous Series Method . .. . 13

-.Prediction of the Thermal Expanion ofiinBased Upon Atomic Spacing at Minimum Free Energy 22

1. Indlependient Lattice Vibrations . ...... 23

2. Vibrations of' the Lattice as a Whole ..... 34

[73. Method ofAttack k a. .0 0 ........ 34

V S~~. Reciprocal lattice 35 .....v ~ ~6. Expar'sion Coefficin 35. ..

7o Discussion .. . ... . . . ...... 36

I MI THERT4~AL E)XPMSION OF SOLID SOLU¶IIODS ..... .. . ....... 4L

A*Introduction . . *.... .. . .. * * 4.1

- ~B. Excnerixrrntal 4 ....... . L2

1. Specimen Preparation ....... 42

2. Experixndntal Procedures ........ a 43

V. PI-1273-11-121,-

TABLE OF CONTENTS

pageC. Discussion of Results ... * ......... 149

IV. TIM THAL EXPANSION OF TWO-PHASE BODIES . ..... 65

A. Introduction . . *......* • •• * .• 65

B. Relationships Between Elastic Moduli and StressWave Velocities in Isotropic Bodies . . . . . . . 65

C. Applications of Ultrasonic Measurements 70

1. Stress Wave Velocity Measurements ..... . 70

D. Elastic Moduli of Pblycrystalline Ceramics • . . 76

E. Theoretical Thermal Expansion of Multiple-Phase*Ceramic Bodies 7...... e.. ...... 9

F. Comparisons Between Theoretical and EmpiricalThermal Expansion in Two-Phase Materials 841. Aluminum-Silica system .. . . . .. 84o

2. Magnesium Oxide-Spinel System .... • • • 84

3. Pyrex-Spinel System .... 89 1G. Thermoelastic Anisotropy .......... . 95

H. Elevated Temperature Effects .......... 96

I. Conclusions .. . ....... ..0 . . . -1

V. CONCLUSIONS AND REOOICODATIONS • .... .. .. 101

* A. Conclusions .. . . . . . . . .101

APPENDIX A: "OpennessT Ratios of Some Ceramic Phases A-I

APPENDIX B: The Effect of Porosity on the Elastic IProperties of Ceramics ........ B-I

APPENDIX C: Relationships Between Sirgle Crystal and IPolycrystallire Elastic Constants,Including Data for Several Ceramic Phases C-I

APPENDIX D: References ...... . ... .... nD-IIvi PI-1273-M-12

LIST OF TABLES

11Table PgI Methods Used for Prediction of the Thermal Expansion

Coefficients of Cr 7stals 4

II The Openness Ratio and Coefficient of ThermalExpansion of One and Two Component Oxides

III The Openness Ratio and Coefficient of ThermalExpansion of Some Silicates 9

IV Thermal Expansion of Phases Havirg Open Structures 10

V Thermal Expansion of Cubic Eare Earth Oxides 16

VI Thermal Expansion Coefficients for the Alkali Halides

at 270 C;Calculated and Literature Values 33

1VII Therml Expansion Coefficients for the Alkali Halidesat 27 C;Calculated and Literature Values 37

SVIII Data On Preparation of Ruti le Specimens 44

IX Room Temperature Lattice Constants of Rutile SolidUSolutions 47

X Thermal Expansion Data for Rutile Solid Solutions(Room Temperature to 10000G) 20

XI Thermal Expansion Data for Al20 -Cr20 Solid Solutions(00o•C to ooo0 c)- 2 57

:XII Thermal Rxpansion Data for SnO2-V205 Solid Solutions23 - 8o00 60

XIII Shear Wave Measurements for Pyrex-Spinel Specimensat Room Temperature (23'-28'C) 73

XIV Shear Wave Transit Times for Test Specimen Components 75

XV Elastic Properties of Fully Dense PolycrystallineCeramics at Room Temperature 80

XVI Comparison of Thermal Expansion Relations for

1 Composite Media 86

vii PI-1273-M-12t/

1LIST OF TAB3LES (CONT'D)

-Table aa

171I Thermal Expansion in the MgO-WgOA1 203 System 88

XVIII Forming Conditions For Rot-Pressed Iyrex-Spinel Disks 901

XIX Thermal Expansion of ?MgO.Al 03 Spinel 97

B-I Dimensionless Elastic Constant of Oxide Ceramics(Used in Porous Body Computations) B-1

B-II Shear Modulus Dependence on Ptorosi1iv Based onIMackenzie's Relation B-.2

C-I Elastic Constants of Oxide-Ceramic Single Cryst~als C-3

v PI-173-M-1

LIST OF FIGURES

Figures Page

1 Openness Ratio Vs. Linear Thermal ExpansionCoefficient For One and Two Component Oxidesat Room Temperature ............... 6

2 Openness Ratio Vs. Linear Thermal ExpansionCoefficient For Some Silicates at Room Temperature 8

3 ThermalExpansionof 7 ...........

4 Alkali Halides ..... ...... ..... 1%

Fl Thermal Expansion Coefficient Vs. Openness Ratio ofCubic Rare Earth Oxides...... ...... 17

6 Linear Thermal Expansion of Sc ........ 18

7 Linear Thermal Expansion of Yb 2 3 ....... 19

8 Linear Thermal Expansion of Er2 03........ 20

9 Linear Thermal Expansion ofDY2 0 3 ....... 21

10 Locations of the Atoms In A Simple Example UsedFor Calculations of Interatomic Forces ..... 24

11 Lattice Constant c at Various Temperatures Vs.00

Composition for 1200 C Firing Temperature .... 48

12 Linear Thermal Expansion of Pure fintile (TiO2 ). 52

13 Linear Thermal Expansion of TiO2 +(10 Mole %)ZrOO 53

lh Linear Thermal Expansion of TiO2 + 7.7 Mole.%SVanadium .... .. .. ... . . .. 54

• 15 Linear Thermal Expansion of TiO2 + (10 Mole %)V205 55

16 Linear Thermal Expansion of TiO2 +.21.4 Mole %Vanadium 56... ..... ..... ....

17 Linear Thermal Expansion of Paure Cr 0 582 3

18 Percent Linear Expansion of Snr2 . . . ..... 61

ix PI-1273-11-12

LIST OF FIGRES (CONT'D)

19 Percent Linear Expansion of SnO2 + 10 MolePercent V ... .... 62

20 Block Diagram of CAL Ultrasonic Test System ForStress-Wave Pulse-Transmission Measurements . . . 67

21 Ultrasonic Transducer and Test Specimen Assembly * 71

22 Deformation of a Two-Phase Model Subjected to•Compressive Lodn V a .. .. . . 0 77.

23 Elastic Moduli of Magnesia-Spinel Ceramics BasedOn Kerner's Theory.... ........ . . 83

2h Thermal Expansion in the Aluminum-Silica Srstem . 85

25 Thermal Expansion in the Magnesia-Spinel System . 87

26 Dilatometer Thermal E&pansion Measurements ForPyrex-Spinel Bars .• • • • * .......... 92

27 Elastic Moduli of Pyrex-Spinel Bodies Based OnKerner's Theory • 0 0 • * • 0 • • ....... • 9328 Thermal Expansion in the Pyrex-Spinel System . . . 94

1B-I Shear Modulus of Porous Alumina : ....... B-41

B-2 Young's Modulusof Porous Alumina ........ B-8

.--3 Y ngsModulus of Porous Magnesia ....... B-9

3-4 Shear Modulus of Porous Magnesia . ........ B-la

B-5 Young's Modulus of Porous Spinal (MgO.A1 203 ) . . . B-II

3-6 Shear Modulus of Porous Spinal (MgO.A1 203 ) . . . . B-12

B-7 Elastic Moduli of Porous Mullite . . .. . . B-13

3-8 Elastic Moduli of Porous Thoria B-143-9 Bulk Modulus of Pyrex-Spinal Mixtures . . . . . . B-16

X PI-1273-M-12

I

I. IMRODtTION

The thermal expansion of solids is one of several phenomena includedin the general concept of the equation of state of solids. Several aspectsof the thermal expansion of ceramic materials were investigated in this

-program. Among these were the following:

1. Prediction of the thermal expansion properties of pure

ceranic crystals. Crude prediction methods useful in

the search for low -expansion mterials were investigated

Ualong with more sophisticated methods useful for calculating

expansion coefficients.

U2. Investigation of the themal exparsion anisotropy of solidsolutions.

U 3. Prediction of the thermal expansion coefficients of two-phase ceramic bodies, knowirg the thermoelastic properties

Sof the individual phases.

In recent years rapid progress has been made toward understanding

thermal expansion phenomena in terms of atcode vibration mechanisms. The

obJcctive of this research has been to indicate ways in which this knowledge

j can be used to solve practical problems. In addition, several new low-

expansion phases were found and previously unavailable thermal expansion

and elastic property data were determined.

P1

1 PI-1273-M-12

IT. PREDICTION OF THE THERM4AL EXPANSION COEFFICIENTS

OF PURE CERAMIC CRYSTALS

A. introduction

Several methods can be used to predict the thermal expansion coefficients

of pure ceramic crystals. The choice of a method depends to a great extent

on the amount of available information and the accuracy required in the

prediction. Since both of these factors may be subject to wide variations,

it is desirable to have several different prediction methods. The methods

investigated in this program are listed in Table I together with listings of

the information required for prediction. The predictions made in the program

and the evaluations made of these predictions using our measurements and those

of other investigators are described in the following paragrapis.

SB. 2W, enness Method

The relationship between the openness of the crystal structure and the

thermal expansion coefficient has been noted by several authors. -5 Several

measure- of the openness of crystals can be calculated with little prior

knowledge of crystal properties. One such measure is the openness 'ratio (R).

R-VFWrVI

IFFinwbich YFW is the volume of a formula weight of the solid

And V 1 is the volume of a formula weight of ions or atoms

'The openness ratios of a number of materials having known values of thermalexpansion coefficient were calculated as part of the CAL sponsored research

program uhich preceded this contract. The effects of various assumptionsconcerning the degree of ionic binding, on the volume of the atoms, were

6[determined. Calculations using Pauling's electronegativity concept todetermine the ionic radius, with the additional assumption that there is

a linear variation of ionic radius with percent ionic bindirg, seemed to

3 PI-1273-M-12

1A

TABLE IMETHODS USED FOR PREDICTION OF THE THERMAL H

EXPANSION COEFFICIENTS OF CRYSTALS

METHOD INFORMATION REQUIRED

I. OPENNESS CHEMICAL FORMU.AspECIFIC GAVITY,

2. OPENNESS OF HOMOLOGOUS SERIES CHEMICAL FORMULA)SPECIFIC GRAVITYSTRUCTURE TYPE

3. ATOMIC SPACING AT MINIMUM FREE ENERGY CHEMICAL FORMULASTRUCTURE TYPE

ASSUMPTION OF INDEPENDENT VIBRATIONS INTERATOMIC DISTANCE (LIMITED TO IONICCRYSTALS WITH ONE BOND TYPE)

I. ATOMIC SPACING AT MINIMUM FREE ENERGY CHEMICAL FORMULA,STRUCTURE TYPE IUSING EWALD'S METHOD: VIBRATIONS NOT INTERATOMIC DISTANCE (LIMITED TO IONIC

INDEPENDENT CRYSTALS WITH ONE BOND TYPE)

12i3

4 1

1

give Ike best correlation between the openness ratio and the thermal ennui

coefMflet. The results for some ceramic phases are plotted In F•=gw IL

and 2 frtm data given in Tables II and III. At high values of the '

11ratio 1ere is a good correlation. At lower values of openness rati b

factors become important. For example, the phases falling farthemt fk= is"

[1 curre an the high R side are those which have relatively weak bonft I Im

caticms and isolated silicate groups (wollastonite, pseudo-wollatomtta%,

forsterite, etc.). Those falling farthest from~i the curve on the lw R s

are phases such as corundum (A120 3) which have especially strowC hod

Theref, it seems likely that this approach ban be used most s~ c

when akrpU-cd to the search for low-expansion materials.

he more rigorous theoretical approaches to prediction of the f

1 exansi of ceramic phases provide some confirmation for these st*L-f.myth2 and others who have investigated the thermal expcnsion of matertla

H which have low or negative thermal expansion coefficients have attrbat~e

the low expansion of some materials to the greater freedom for tramsvse

vibraticm in materials having open structures. The observed chavges Im

vibratimn frequency for internal oscillations (vibratior3witbin grow_=) an

lattice oscillations described by Ganesan for the case of sodium chlmau

[and s=d= bromate, indicate the greater importance of the lattice

oscil~aions, over that of internal oscillations in determining the thmmvl

expar of crystals with group formation. The importance of le strh

of 1 s is recognizable in expressions derived by Gruneisen MInrO54

Cartz, Borelius mr 12 '13 and others.

Me openness ratios of many phases have been calculated, The resaiheof about 1400 of these calculations were presented previously V eresults of additional calculations are presented in Appendix A of this

report. Based upon the results of the first series of these calculatln=

a number of phases having open structures were selected for synthesis ant

thermal expansion measurement. The results are summarized in Table •o.Ii As a reszlt of these measurements, several new low-expansion phases wre

avail- . Hl

___ __ __ _ ---- - - . -

lEl

mii

C4

Cl

LH~~ I ...

as so

0I127 F-I1

.344 C

- 0 10 6, em0 6

ac

a-a

*J We -....al .

E - -. - - -

2 A -- Al -ton

.a a 0, a...

R~ a J ;4W .

FFl.r 21 .jr

em 3133

PIl273H,1

.. .i ... ..... ...... ........ ... II .. .I..I.......I.... ...

1I

usI

r

. ........ . .... ... ..... ..... . -

-- - €

....Q • .4 ,o . ..... ..... ............... .. G :: Li on

.. . ..7 .

aC.0

S " "mI. ., ... z.z~zz z .. 11. .7'... .411

.......... . -.. .

. ....... . ..... ... . ...........- "..- - . . . -. -. ~.

. ... .......... _ ... ...... ".....

o a a4&a 1

Oliva 29r8Mo .*1

A.-

8PI-1273-Y-12 .'1

Table III

FlTH E OPENNESS RATIO AND COEFFICIENT OF THERMAL EXPANSION OF SOM4E SILICATES

3Azrs~ POLICAIYST., 04noOIsooVC, NULLImT 0. S? 36 1 107 is@e11u111FIEbgetTt (PAUS OF MUINTED uTU(O4

414.41203. ,POI.YCff$?.. 961 tEVAN, ,auCLINIC 0 .63 32 . I@ 25-NOOC

so r SIOV.1 CRYSTa. OEM, NEIA40AL 0.4, 14 1 fK-2o{1C.O.3i02 POLICRIST.. ICVOOLS01tMO- Ss 9aI'? 806.3061CLINIC, 11LA161(3 1m6fhIJ?3)

C41. 3402 WCOeCYST.. PSSUW.WOLLASTONOTE O.o . 67

Iss-me6CLRINIC DOLATWTER 16(i3

POLYCRISt., VOLLASI1uITI I.5 I. I*'? N-S~

1] ::;> NMYCRYST.. IM OA.011.1I'04 4 v: :1;- 0-0OILATUVETE 11"112

Cas.AI203. POLYCRYST.. 111CLIVIC. ANUSTNITE 0.6* Is a, so? 293a234, SMA SIE~tl4T BtLLCIIaf 107

C8.00~. PoijcuvSr., if Imauuc, oloesiee *.Sq SO xo u?10-3ON23i02 *ILATOMME m"P(1173)

CdN. POICSIST.. NTNommWS3C, Morli. 0.Sq *114 1*- ownso.6c$102 CELLITIE, DILAI06ITER 4i0ts (1137)

ZC*.N 1O. POLTCRIT".. TIETIAGOUIL, AN~ftWITE 0."8 84 0-7 0-30ofC

23S02 DILAT06Kt~tSCO. No. PKtTEIYST., * NM INIC. me"w3NITE 0.5%1 0 a0 IO'l 100-3400C2310* OILATO101711 1aegy(gII7)

2 *16.3302 POLYCOIST.. @BTI.OtOMMOIC, FAVALITE 0.63 s 33 ' W-300fP ILATRUFTEN 21611(1173)BLiZO.A.03 - PkVCIVST., NE"GOMA £I3CRYPTITE 0.15 -4 1Q 10.*o100"

"mIt IL312O.A1 2%3.SSL.0 of- p0I~v'C3T*. wom10c PETALITE(?) 0.6a a .r 3Q 0.)O6C[1 1106) NOTE I

Wg.12 POLICRYST., MOWOCLINIt, CL3UOEMSTATITE 0.546 3 of IO-7 oo-oco1

3g.0 POLY:RYST.. OimOROM..eIC, FORSTERITC, *.5S so tr I03" IO2rC

2m1O 2H0A 12 #3 POLYCRIYSS., 303 CO#01E93TI INT(IFER- 0.63 33 10-7 2S-30N'C

Him 2 .Al 203. FOLYCEYST.. TRIILIVIC. CANNEGIEITE, 0.43 110 1 0r- 20-20WC

2S'02 OILATOMETER maLIO)

N42*.C*O.3i0 2 POLYCOYST., C984C. OVLATOMETCE 0.54 ISO I 30? 20-3RVC

Sia VIRIIEOUS SILICA,IELDVICROSCOPt 0.72 7 xO l' 25-300'eJ I0TTENORVE(126S)

I 42,302.i POLYCIYST., TETNIOWAL, 71VCO, 0.59 3? IV'7

2S-300*C______________________________ IWITTEMOREf 1165)

NOTE I

THE LITHIUiM ALUMINUM SILICATES ARE WIDELY INOwN T0 HAVE LOW VWOES OfTHERMAL EXPANSION C(EFEICIElir. ThEY AVE ALSO NOTED FOR LOW TALlIES OfNOSULIIS IF RUFTU1E WIdM hAt THOUGHT TO St .E TO IVTEUAL 01"1A FA*'E9

AS A RESULT OF A9VSOTOOP3C THERMAL EAPAIS ION. THE I-RAI OETEPWIIWATIR0 Ofi~~1 TOE THELRMAL EXANIJSION COCfFICIfEMT4.EIICRYPTITE IS REOMTED I0 THE 101.2.0-L.A ING REFERENICE.

F.". GILLERY ANN C.A. 3USS. TURLCONTRACTION 0F.0-EUCRIYPITIY

(LI O.61203.2S,0 2 ) 3"Al VA AN* OIULIOIETER LTUOOS. J- AM. CESAMISIC. 42 (A) 375-77. (3059).

9 P1-127314-"f12

Tabl e I VTHERM4AL EXPANSION OF PHASES HAVING OPEN STRUJCTURES

THERMAL EXPANSION [FORMULA DESCRIPTION OPENNESS COEFFICIENT TEMP. REFERENCE

A123.A20 P1 DCOMOSD; -RY EIDNCEOF RATIO ac f0 7 RANOEOC

Al A~o AML DCM"E; -AYEIDNE F 0.69 -KIRCHNER ET AL

ON-CRYSTALLINE MATERIAL. (1959)

CdO.ceo2 SAMPLE D ECOMPOSED 0.70 -- KIRCHNER ET At r3CoO. P20S X-RAY ANAL. INDICATED SAMPL.E WAS MAINLY 0.78 80.5 20-700 K IRCHN El ET AL.

COO.P205 (1959)

L12O.2SiO 2 X-RAY ANAL. INDICATED 1.120.23102 PLUS 0.63 105 20-700 K I CHN EE ET. ALLi2o'S'02 AND QUARTZ (1969)

MnO.SI0 2 06 X-RAY DIFFRACTION, ANAL. 0.66 83.3 (090 1KR6NE9) A

Th0 2-Si02 X-RAY ANAL. INDICATED ONLY Th0 2.510 2 9.66 .61.8 25-1015 MERZ ET AL(1960) [

33n0.P205 X-RAY ANAL. INDICATED MAINLY 3S80-P205 0.74 96.7 20-700 KIRCHNER ET &L

3Ba.A103 WITH SMALL AMOUNTS OF SflSOq AND N&AP205 25 2013 (1969)3

38OA23 X-RAY ANAL. INDICATED PRESENCE OF B&O. 0.57, 525 2M13 ERZ ET ALA1203 AND 9.0. NO PATTERN WAS AVAIUABLE (1960)FOR 3B&0.A1203

SrO.2A1203 X-RAY ANAL. INDICATED MAINLY 3rO.2A1203 0.62 67.0 2H15 ERZ ET ALSOME FREE 5,0 AND A1203 ALSO PRESENT. (1960)

SrO.AI203.2S!02 X-RAY PATTERN SIMILAR TO B&0.A1 203. 0.62 52.3 25-1000 M4ERZE AL

2Si.02 HOMOLOG 1O)-

CaO.CuO.'45i02 X-RAY PATTERN INDICATED ONCY CaO.CvO. 0.04* 29.7 39-1015 MERZ ET AL65i0 2 (1960)

SrO.CuO.'15102 X-RAY PATTERN INDICATED ONLY SrO.CuiO. -19.0 26-1013 MERZ ET AL

IS io2 (1960)1

B&O.CuO.01Sj 2 X-RAY PATTERN INDICATED ONLY Sa0.CuO. -36.16 38-1007 MERZ ET ALttiS102 (1960)

U02 '102 05 X-RAY ANAL. INDICATES CUBIC UP207. WITH 0.72 43.8 20-4100 HERZ ET AL1

A PSSBL TAC O (O) P07-32.3 4100-1200 (1960)

A PSSILE RAC OF(uO 2 27 -6.66 29-1200

*THIS COMPOUND WAS SELECTED FOR MEASUJR04ENT BASED UPON THE OPEN SHEET SILICATE STRUCTURE. ITHE OPENNESS RATIO A WAS CALCULATED AFTER THE MEASUREMENT OF TNlE THERMAL. EXPANSION

COEFFICIENT.

10 PI-1273-14-12I

The most unusual low-expansion _phase studied in this program is cubic

-~ U2070Expansion 3nausuts were made using both the x-ray and dilatometer

methods. The results raa- both methods are presented -in Figure 3. The material

[ Go DIUL0ATOE DATA

8 6 ' t o3

[10

[1 TENPERMlURE .OC

Howe 3 NEUIAL EXPANSION OF UP207

expndsup o aout400 C and thncontracts, returning to the roomtemperature dimensions at about 1000C. Agreement between the two methods

of measuring the thenual expansian is good UP to about 406O0C. Above 4~0&C,- the d±Jlatoiueter gives slightly higher results. Between 900 and 100OO 0C the,

curves cross again so that at WOOC the dilatometer measurement is lowerthan the x-ray measurevent. The lack of good agreement at 10000 C and above

may be due to the pres~ene or a small percentage* of (UO)P0 7 The presence

of this material i'ouIA affect the dilatometer measurement since this method

measures the overall expansion of both phases, wihereas the x-ray method is

U1 PI-1273-11-12

sensitive only to changes in the cubic UP2 07 phase and not to other phases,

Comparison of the x-ray patterns for UP207 at room temperature and 1000°C VL(the temperature at which the unit cell of UP207 has almost returned to itsroom temperature dimensions) shows close agreement in both "d" spacings and Lintensities, The pattern taken at 1000°C is excellent and gives no indication

of structural changes. Only two weak lines not characteristic of cubic

UP 2 07 were observed. This type of expansion followed by contraction without

a phase transformation is unique. Curvature of the thermal expansion curve

in the other direction is not uzcmcmon (Si, Ge, InSb and vitreous SiO2 at

low temperatures) but the reverse has not previously been observed.

The thermal expansion of uranium pyrophosphate can be compared with the

results for the similar titanium and zirconium compounds measured by Rarrison

and Hummel.11 6 TiP2 07 has a rather typical thermal expansion curve. The

material expands almost linearly from room temperature to 100°OC and has a

thermal expansion coefficient of approximately 122 x 10/°C. for the

temperature range from 50°C to 10O00 C. These crystals are weakly birefringent.

On the other hand, zirconium pyrophosphate (ZrP2 07 ) expands slightly more

than the titanium compound in tbe range from room temperature to 400°C, but

above this temperature the thermal expansion coefficient decreases so that

little expansion is observed. These crystals are isotropic. Harrison and

Hummell5state that the "normalw zirconium phosphate, ZrP2 O7 , has a reversible

inversion from a low temperature cubic form to a high temperature cubic form

at 300 0 C and dissociates to the zirconyl compound (ZrO) 2P207 at 1550°C.

-The thermal expansion of UP20 seems to be related to that of ZrP 0 in that27 2 7

the material expands to temperatures in the range from 300 to 4060C and then

there is a change in the shape of the thermal expansion curve. In the case

of UP2 07 , the thermal expansion coefficient gradually decreases becoming

negative at temperatures in t!e range from 400-5000C so that the sample

contracts with increasing temperature in the range from 400°C to 12000C.2Smyth has explained the negative thermal expansion of vitreous silica

arn the high temperature forms cC quartz, tridymite, and crystobalite on

the basis of the variation ith tzmperature of the transverse vibrational

frequency of the oxygen atoms vibrating between two silicon atoms. As the

12 PI-1273-M-12

SI I

Hvolume decreases, the transverse vibrational frequency of the oxygen decreases

F1 dne to an increase in the high order repulsion forces vhich resist restoration

of the oxygen to its original position. This increase in the high order

repulsion forces is. only partially offset by an 'increase in the electrostatic

attraction forces which tend to pull the oxygen back into its position.

SAccording to Smyth there is always the possibility of negative expansion if

one or more of the lattice frequencies can decrease in this way as the volume

decreases. Perhaps the negative expansion of UP2 07 can be explained on this

basis. Wyckoff 1 7 describes the structure of ZrP2 07 (which is the same as

the structure of UP207) as made up of an undistorted array of Zr+4 and

F] P 0 ions. The P 0 ion consists of a pair of PO tetrahedra which share

an oqgen ion with a P-O separation of 1.56 angstroms. The other P-0

separations are 1.52 angstroms. Each Zr+4 ion is surrounded by six

octahedrally dis tributed oxygen neighbors at a distance of 2.02 X. Relatively

i large holes run through the structure bordering the oxygens shared by two

posiphorous atoms. Since there are no close neighbors in the transverseI directions, a displaced oxygen atom is drawn back into position by electro-

static attractions of its pair of phosphorous neighbors. This is opposed

somewhat by the high order repulsion of the same phosphorous atoms. As the

lattice shrinks there is a decrease of the restoring force on the transversely

displaced oxygen atoms and a decrease in the transverse vibration frequency.

This ithe condition required for negative expansion.

C. Openness of Homologous Series MethodI191 20 1 2Many authors including Hummel Austin1, Megaw , Rigby and &irth

Shave stressed the importance of considering the crystal structure type in

ax7 attempt to systematize the thermal expansion properties of materials.

SSince structure type is an important factor in determining the thermal

expansion properties, it is likely that variations among the phases within

1 one structure type will be substantially less than the variations observed

"U in groups of phases having several different structure types. Therefore, the

effect of variations other than structure type can be studied in series of

*13 PI-1273-M-12

S~i

compounds having the same structure type. For this reason, an attempt was-

made to correlate thermal expansion coefficients and openness ratios for F]homologous series of compounds. Chemical homologs are compounds having the

same structure and similar di emical formulas; for example., the series of

cubic compounds TiP2O7 , ZrP2O7an 207.

The thermal expansion properties of several homologous series have been

studied. 2 1 ' 2 2 These studies were handicapped by lack of data for many phases.

In same cases 1egaw's rule (x-Ap•" in which p is the coordination number,

t is the ionic charge and A is a constant) was approximately obeyed 1](cubic alkaline earth oxides, oxides with the CaF2 structure, ard carbides

with the NaCl structure). That is, within homologous series the thermal iexpansion coefficients are the same. In other cases, the presence of

displacive transformations or other processes made application of this method

of uncertain validi17 (the cubic pyrophosphates TiP2 07 , ZrP2 07 and UP2 07 ).

In the case of the alkali halides there is a reciprocal relationship between

the thermal expansion coefficients and the openness ratios fbr the alkali

metal and halogen series. This is shown in Figure 4. In these calculations

100% ionic binding was assumed. The thermal expansion decreases with 4increasing openness ratio for the halogen series F-, C13, Br" and 1. This

relationship is also observed for the alkali metal series Li, +Na, K+ , etc. +a

Exceptions occur for the fluoride series LiF, NaF, KF and CsF. The halides

with the cesium chloride structure (CsCl, CsBr, CsI) also show a small Idecrease in thermal expansion coefficient with increasing openness ratio.

The phases with the cesium chloride structure have high values of thermal

expansion coefficient relative to those with the sodium chloride structure adue to an increase in coordination number.

The cubic rare earth sesquioxides represent an interesting homologous

series because variations in chemical bonding are minimized while at the

same time there are substantial variations in the openness ratios. Available

data on the thermal expansion of these crystals are given in Table V. The

openness ratio is plotted vs. thermal expansion coefficient for two temperature

ranges in Figure 5. The thermal expansion curves for four members of this

series measured by the x-ray method are given in Figures. 6 - 9. It is

14 PI-1273-M-12

S• •T . .... ... ... .... ... ..

Ii

* I.,,. .... o.o .. . ...... . -- -........ - - . ..... * ......... .......

S..... ............. ... ....... -- .--.-....... ,.......... ........ --.. ...... ....... i....... 1 ........ 4-......... ...... . ..........

: . ........

XI Cot~t STRUCTURE

4 . . . . ... . . .

* .- S

* ., 2•

RATIO j... \\ 2

ft '. ! M E,-,

....3 -- .. .. .-- .. . ... . .. .: - ' . . .. " " . .. . . . - _ _...... ..... ... ....... .. ......

:,- - ._ I .? .-, .- ......

%tl i, , ,75 t '

..... . ... ............ ... . . .... .......... .

L l F . .

Sj. r I.. S.. ..{l

.2 ... . . . ...... J5........... ............ t ..... ...

....... • ... .. %N .. .... .i . .....

.l , 7 .t .... .. .....S,. i l.I j1 ti

.. ,! ..... •" . .. "'--l---f...... : - I'•- -z4n•, •z••z• ..... • .... -['-%• ........ ÷ ....... .......... ....... .......

. -l - - - . -,- --,i±, ! -el , . -:k

:300 -0i 500 S 800.

THERMAL EXPANISION COEFFICIENT x 1 I0/aC

F gure Ii ALKALI HALIDES

. ...15 1,I t : ;

7- , .i i t

.1 -"2-'- -" ••"' ""•... •".. .... -k .f ... ... v'.,"l....• ... ...

.'•. i | f 8 : % i

Tabl e V L-THERMAL EXPASION OF QIBIC RARE EARTH OXIDES

OXI DE SAMPLE THERMAL EXPANJN T04PEPATURE REFERENCEDESCRIPTION COEFFICIENT O/oC

SC203 CUBIC ps 0-9990C STEOJRA ICAMPBELL C 196)12

Y3 CuslIC 65 O-993*C STECIJRA & CAMPBELL (1961)62

CUBIC 68 20- IOOO*C WARSHAW & Roy (1961)60

02 P3 CUBIC as O-9500C STECURA aCAMPBELL (1941)42

Eu2D3 CUBIC 78 O-10110C STECURA & CAMPBELL (1961)62 LSYS. CUBIC so 0'.1062%C STECURA &.CAMPBELL ( 1961)02

PY2O3 CUBIC 82.9 2S-1200*C BROWN a KIRCHNER (1961)61

CUBIC 62 0-9as0c STECURA & CAMPBELL (1901062

63 301h8I0OC PLOErL, KRY37YNIAX & DUNESI

IHo2 CUBIC 62 0-9490C STECURA & CAMPBELL (1961)82

Er203 ýCUBIC 82.1 25-12000C BROWN a KIRCHNER (1961)61

CUBIC 79 0-1038*C STECURA & CAMPBELL (1961)62

TN203 CUBIC 62 O-10310C STECU RA & CAMPBELL (19 .61)62

Yb7.O3 CUBIC 82.1 25-1.2000C ýBROWN & KIRCHNER (1961)61

CUBI 810-101*C 3TECRA CAMBEL (19111L23 CUBIC 7I 0'.1025 0C STECURA a CAMPBELL (1961)62

16 PI-1273-14-12I

A

... ,.... i........ . ......... r ........ i ........ i ........ ......... r .........0.0.................*...... .... .f....

TN0 , E •r . I !3

ou*3 E 3 2 io ....... .....

* I

Yb-i - ------ ------

t° ........... .... .... .'".........• ....... ........... + .........

a I I I

! ! a. i I

--- -I-- -- - . .. ........ .......... i -..... --- - . .... -- --..... --

- R

a I i

* I .

vbi a---------------..... --.... -i1- - - -. ----.. ------.......... -

[1i "i 1II I. j...4.

o a: : v... ........... :....

o.. +" ... 80. .- -. so..... loo o

OF CUBI RAEERHOIE

I I 1IH .1....~. ........ ....

0.5o 4...... .... . * . p4 ... q. a S ....

*=o ... ...... a .'Aa ... .... 0i - - .......... R RA G ..2O

r, ........ , ~.................... .. .. 2A-TEI4PERAUREEANOE0-1200°C

*-I 8 S O 1001aii j 1

*• LIEA THRA XASO OEFCETXOj

0 6i6 P00 -1203M1

/ a

..... cc.... ..

-- -----•. . ........ .

lea ~ ..... ........ cc~e . . ...... ..ccc .cece .....c. c .... .... .c..... * . cc . ..... .

0 c.4 - - '...... c.c....e. ...... c~l . ...... . ..... ... .cc.....e.e...... c...... .

.... . -- - -,

0. ..2... ., ...... ......... ..... . . ..... ...... ...... . ..... ... .... ........... .......

S.... ..... '.... - ...... iJ. -. •... ... -... .. . .. ... ... ....... ........ ...-... .. I* .5

e• o~l - l111 ,411

0 200 '100 .600 Boo 1000 1200TE"4PERATIRE OC

Figure 8 LINEAR THERI4AL-EXPAIISION OF 5 .V

18 iP

I i1"" 1-23-41O, q --- -- --I,... .... ...... ......... ..... . .......

1. .... .... .... ...... ................... ......... ....

0. .... .

. t1

..... ......... - ..... ........ ............ I

. ... .... ... ............ ..... ...............

0. -------- ---- ..... ...........i . ............... 4 .... .....

19 FI-173-4-1

1.20 I7

... 1. ........ . I . I

. .. .. ... ... .. . .. . . * .- .. .. F'/

* *l

* II

"---- "... ........ ------ ........ -, ... ..... ................... . ........ ....

--- ... ---

. ..... . .. .. .. .. .. .. ...... . .. .. .. .. . ....... -t ---------- ..... --.... -..... -... .. . .. .- .. .. .. .. .. .. ... . .. ...

I.* i i ~ Z* I f

"0 2 .. ........ . .+ . . 100 2•

-... ....... .... . 4* * * I

So~~a ... ............ ........- ÷.....................+.... ..*1, . I : I

Fir 8 LI ! NA I I E O 'o..................... ........ .. ........ P-2.......* . i i I

o. .... ............ .. .." ! 1 . ~ .........

*S * I S ;

S I . , . S 9 . I

.. ,. ,I.. .. 4- - Ii II | -I

0 200 iWo0 600 800 I000 9200 JTEI4PERATURE 0C

*IFigure 8 LINEAR THERMAL EXPANSION OF " O ;

*1

I II I I . .. . . i . . . . . ..... • ... . . ....... .. - ' : . .. . . .. . .... . . ..

1. . .2........ I........ ............ ........ ; ........ •"...... ; ";...............................

1-....................................

0. .. . .. . . . . . . . . . .. .. .. . ..

-- - - - -- - - --- ---

j a

I . - . . .

......... ,.. . .... --... ..... 1 ......... +,... ...... 0t, ..... ----- -------- ---------.... ...... .... •.....

If. a a a

... . .... ..... . . .. ........ ..................1

I~----- .o .........• .... • ..... ...... -. '....... ........ ... ......" ........ ..... ..... i....

~~~~~~~~, --------- .• - .+ . ..-- - ....÷..... . .... .... ....... 4. ..... .- ....o.• .... --- -., -. . ---- .... ... .. -. ..!"........... ------ . . . .... ... .....

a 20 a 0 62 go 0010

1.0...,j.. - ....a. . . . , . . . .a... .. . I--I". ... . I". . . . . . . ." ; a i S

"a i ........ 1 ....... .......

Ii, . • .

a a I ,

a.. .. . I ..... .. [.... ... .. ..

- ..a . a ...,!I 1 2

a ........

SI i ii , '

a I I

I a I'

a 20 a aO 6t 20 100 120

a a-1 21 P1-12 3 a -a

a .... [..I .

a

Ilal

m ~Im

evident that there is little variation in thermal expansion coefficient

among these phases. The thermal expansion coefficient of Sc2 03 increases F-more rapidly with increasing temperature, than that of the other phases,

The conclusion to be drawn from these observations is that the openness

ratio does not provide a method for predicting the thermal expansion within

homologous series except in the case of series like the alkali halides in

which there are very large variations in ionic radius, ionic polarizability

and openness ratio. In series like the cubic rare earth oxides, carbides

with the sodium chloride structure, alkaline earth oxides with the sodiumchloride structure, etc., little variation in the thermal expansion

coefficient is observed. ID. Prediction of the Thermal Expansion Coefficient Based Upon Atomic

Spacing at Minimum Free Energy

Our study of various methods of predicting the thermal expansion of

materials included an investigation of the atomic spacing at minimum free i Ienergy utilizing statistical mechanics. In this research some relationships

-which exist between the atomic structure of solids and thermal expansion are jpointed out. This approach is important becamse our principal approach to

the prediction of the thermal expansion using the openness concept does not

involve direct knowledge of the atomic structure. Therefore, it seems

likely that many of the inconsistencies in the predictiors will have to be

explained in terms of the atomic structure. In the following paragraphs

are presented some explanations of the effect of crystal structure on the

thermal expansion ard a summary of calculations of the thermal expansion i1

values for some alkali halides having the sodium chloride structure. TheSrrhe14 -

complete calculations are presented by Kirchner based upon Smyth's research. F]The purpose of this work is to show how the structure of some crystals

and glasses can be related in a qualitative and even semi-quantitative way F]to the expansion characteristics of the materials. Two different methods

were used. The first method is not mathematically rigorous since the

vibrations of the individual atoms are considered as independent of each other

rather than treating the crystal as a whole and computing the proper modes of

22 PI-1273-14-12

?

vibration. The latter is possible but extremely time consuming while the

treatment of the individual atoms, although not rigorously correct, can be

very helpful in leading to a better understanding of the relations between

expansion and structure. It has been shown 2,8 that a mode of vibration,

the frequency of which increases with the expansion of the lattice, makes a

I negative contribution to expansion and vice-versa. Some of the ways in

which this principle can work will be illustrated below.

1. Independent Lattice Vibrations

In order to compute the frequency of vibration of an atom or ion

about its equilibrium position, it is necessary to know the forces which act

on it when it has been displaced by a known amount from this position.

Since the atom at rest is in a position in which all the forces acting on it

add up to zero, it is necessary only to calculate the amount by which the

[3 force of each neighboring atom is altered when the atom in question is

displaced by a known amount. In order to allow actual calculation of some

of the forces involved, the interactions are treated as purely ionic with

--Coulomb attractions or repulsions modified by higher power repulsions. The

equations for calculating the restoring forces are indicated below. These

apply when the magnitudes of the displacements are small as compared with

the interatomic distances.

If an ion at the origin has a neighbor at x, y, z, attracting it

with a force

U ~ ~e. e. 111

Sin which e, and e are the charges of the 2 ions and r is the distance

between thea, and if the atom at the origin is given a small displacement

tou, V ,w , the change in the X -component of the force exerted on the

displaced ion by virtue of its displacement is

2a(3 (1I-2)

23 PI-1273-4-12

in -hich X -

The y and z components have the same form. If the ions have charges of 1the sý,me sin, the results are the same idth the opposite sign.

If the ions repel each other with a force I

--A (11-3) [1

in which A is a constant

then the chanec in the x -component of the force acting on the displaced atom

will be

A U r X Z} +nh Fl

They and r canponents have the same form. IThe effects of these forces can be illustrated by a simple example.

In Figure 10an ion P is located at the origin between a pair of ions 47 at(-a, 0,0 ) and • at (a,, 0,). If it is assumed that the electrostatic

attractions between ,P and Q are of the form (1) and the higher order repulsions fare of the form (3) then, if P is displaced to ( u, 0, 0), the force actingon P by virtue of its displacement winl be, on applying (2) and (4) I I

Sa. 3

"Figure 10 LOCATIONS OF THE ATOW4t IN A SI4PLE EXAMPLE USED FORCALCULATIONS OF INTERATOMIC FORCES

24, PI-1273-M-12

I-

If the ions were originally at their equilibriium separations, the

first term will predominate giving a positive restoring force. As a

decreases, the first term increases more rapidly than the second giving a

greater restoring force per unit displacement or a higher frequency of

vibration. In this case the restoring force has come from the high order

f Irepulsions diminished somewhat by the Coulomb attractions.L If, on the other hand, the ion at the origin is moved to (o, v, 0 )

which means a trarsverse displacement, the force, now in the y direction, is

-A e e, 2 Y(1-6)Sa77

If the original separation is the equilibrium separation, the second

term will predominate and there will be a positive restoring force. Asna

decreases the first term increases more rapidly than the second, diminishing

Tthe restoring force per unit displacement and decreasing the frequency. Here

the restoring force comes from the Coulomb attractions diminished by the high

order repulsions. This accounts for the different behavior from the previous

case. A vibration of this kind contributes to a negative expansion and one

of the fbrmer kind to a positive expansion.

The conditions for the existence of a vibration of the type leading

to a negative expansion are that in the structure there must be some atoms

connected only to a pair of neighbors about 180 degrees apart and so arranged

- as to have considerable freedom of transverse vibration. These conditionsare met in several of the forms of silica and other materials hhich crystallize

with comparable structures. Vitreous silica shows a negative expansion at

- very low temperatures and very low expansion at room temperature. The high

temperature form of cristobalite shows a negative expansion. The more compact

-i forms, such as quartz, where the more efficient packing inhibits the freedom

of transverservibration, show positive expansions.

If the vibrations of individual atoms or ions are considered as

taking place irdependently of each other, it is possible to develop a fairly

simple quantitative treatment for crystals of cubic symmetry. The examples

V considered will be phases having the sodium chloride structure. Phases

25 PI-1273-M-12

I "

having this structure include most of the alkali halides and crystals of

ceramic interest such as magnesium oxide and other alkaline earth oxides.

in the sodium chloride structure, a cation at the origin is

surrounded by six anions at (qO, 0O)),0 (0,,0O,0), (0,-a-,), (O-oý)

and (,) O a) where a is the closest cation-anion distance and is one-half of

the edge of the unit cube. Equation 2 may be used to find the restoring

force on the ion at the origin when it is given a small displacement to

(u,V,W) where u , V and W are small compared with a , The total

effect is zero and in the same way it can be shown that the complete group

of ions at any one distance of separation from arV one of the cations or anions

makes a zero contribution to the restoring force on that ion ihen it is

displaced in ary direction. This, of course, applies only to that part of

the restoring force arising from the Coulomb interactions and applies only

in the case of surroundings of high symmetry. It applies, for instance, to

the vibrations of a silicon in the cristobalite structure but not to an

oxygen ion in the same structure. Therefore, for independent vibrations in

crystals having the sodium chloride structure, the restoring force on any

displaced ion can be computed entirely in terms of tha high order repulsion iforces.

The repulsion forces fall off very rapidly with increasing separation

and not much error will be introduced by considering only interactions

between nearest neighbors and second nearest neighbors. If the force of

repulsion between any ion at the origin and arn of the six nearest neighbors

at distance a from this, is of the form

then the X -component of the restoring force on the atom at the origin

arising from a displacement to ( V, W, w) can be written using the equation

4. Adding the six equations and simplifying gives

(I1-8)

2II

26 PI-l273-M!-l2

-°

III I II I II I I I I i

If 7 is greater than 2 this force is always negative for a

L positive displacement u indicating that it really is of the nature of a

restoring force. The x -component of the restoring force depends only on a

which means that the vibrations in the X, y and z directions are

independent of each other.

th If the force of repulsion between the ion at the origin and any ofthe twelve ions at a distance d ;•P is of the form

7--• (11-9)

the X -component of the force on the atom at the origin when displaced to

(U, v., W ) can be determined using the equation 4. Adding the twelve equations

and simplifyirg gives

4AZ

Some reasonable estimate now has to be made for A A2 and 77

for the different alkali halides. The method of tr-ýatment implies the

following model. Each ion has three independent modes of vibration, all with

the same frequency. Just enough interaction between the different modes is

assumed to allow the attairment of thermodynamic equilibrium so that the usual

Slaws of statistical mechanics can be employed. This allows the calculation

and use of the free ener•j of the system and the condition that this free

Senergy of the system is a minimum determines the state of expansion or

contraction of the crystal at any given temperature.

The energy per mole can be written in the form

i JN ,LV

FN¢7hP

27 PI-1273-M-12

where • is the quantum number describing the state of vibration of the ith

mode of vibration of the cation, and 7ý is the frequency of vibration of the

cation, n-j and P2_ describe the jth mode of the anion, and e is the

potential energyrof the lattice at rest. and 72_ are all functions

of the cation-anion separation. In the usual method of attacking such a

problem, the partition function Q is set up and this has the form2 ' 2 3

Kr _eer__ 1- 2I- • er (--1)J Il

The Helmholtz free energy is given by iU~ -KT~i 4~(11-13) L

Hence

Inthe absence of stress or external pressure, the cation-anion separation II

will be such as to make U a minimum. The quantities

and®

are the Einstein characteristic temperatures.

E3.has the form

____-I Re NI2(•" P (1r,-125)

28 Fi-1273-M-12

and tay be expanded about ro , the value of ' for which FS is a minimun.

M is the Madelung constant for the lattice and 5 is a constantcharacteristic of the lattice, ro is given by

• Nea -. n o e/Ie (-(11-6)

If "r4 0 ÷ Ar , then 4 takes the following form using the usual Tylor

series expansion and omitting terms containing powers of . r higher than

the second.S_N/V e 0 1117

Here, A r is assumed to be the amount by 'which the value of the cation-anion separation at any temperature varies from what this separation would

be in the absence of thermal motion. The quantity A.P is detcrmined by

choosing that value which minimizes U so that

The substituting of the derivative of 4 with respect to A r and

solving for A," gives

!3--4,1 h 9 (e-n11)e 17 1.e 49-n1

29 PI-1273-M-12

Taking the derivative and simplifying

_____ ___ ~(11-20)1

But1

ard if and can be considered as constants

independent of the state of expansion and the refor e of the temperature., then

77_ d J(1-22)

* where is the Einstein Function of1

In order to compute the quantities ®* and ,it is necessary

to know the frequency of vibration of the cation and the anion. This can be

determined by differentiating the potential energy functions suggested by

Pauling 6to give the force 1

-n (11-23)

30 PI-1273-M-12

1*

for the cation and anion and equations of similar form for the forces between

1 cation anr cation, and anion and anion. Comparing these with equations II-?,

8, 9 and 10, it can be seen that by equating II-7 and the second term of

1 11-23, we can solve for Al . In a similar way we can find A 2 . Then, by

substituting the values of A, and A2 in 11-8 and II-10 and adding the

two terms we have an expression for the restoring force per unit displacement

as follows.

For the cation this is

7, 1- 77 7"-" . (€e"0 * W (11-24)

L• and for the anion

* 77,V (h1-25)

j[L!The frequencies of vibration of the cation and anion are given by

'-- (11-26)

where 2,, and ;z_ are the respective masses of the cation and anion.

v Since

e V E)-= 1 (1 -2'ZZM + ""a

and

dr 2K 1 2-28)

31 PI-1273-M-12

and

/ dý ® _ -z C1 ?7-29 )

d r 2 7K M_ z

®+ dr - _ ® dr - Zr (11-30) 1

Equations 11-24 and 25 can be used to evaluate C.. and C_ . 30 can be ievaluated using equation 13-13 from Pauling . In this equation 6 must

be chosen so that , )�o has'a value of unity for p 0.75. Here p

is the radius ratio. When this is solved, it gives for 30 the value 0.029

for f =9. This choice of '5o allows the use of Pauling's ionic radii i1for ;r and 7'_ and gives the proper value for r (the cation-anion

distance for all of the alkali halidies).

Combining equations 11-22 and 30, the expansion coefficient can be!L

written in the form

Ot 3 RQ(77+2) r, + E®(11-31)2 l= 2 f e,2 Fn 1-) rL)

Using 9

R = 8.314 x 10-7 ergs per mole per degree L1Ml 1.748

e .4.77 x 10"I e.s.u. N

The first factor which is common to all the alkali halides has the

value 711. Using this, the known values of the cation-anion separations I

( r0 ) and the values of the Einstein functions, Table VI was prepared

showing the computed exparsions. The experimental values for the alkali

halides are given for comparison. These are the "best" values from the24,literature as given by Thielke. -1

32 PI-1273-M-12 -I

'.

I1

Table VITHERMAL EXPANSION COEFFICIENTS FOR THE ALKALI HALIDES

AT 270C CALCULATED AND LITERATURE VALUES

F CI Sr I

Li CALCULATED 210 320 360 l -10EXPERIMENTAL. 3410 435 600 590

iNa CALCULATED 270 380 41I0 '50EXPERIMENTAL 380 '00 '1I5 '70

.LK CALCULATED 350 '30 '60 190EXPERIMENTAL 370 3a0 ,00 '30

Rb CALCULATED 390 '160 80 520

EXPERIMENTAL - 360 375 '15

Sr. Ca CALCULATED '20

,. EXPERIMENTAL 320 460 4T0 490

THIELKE'S (1953) BEST VALUES

I3Jr

: 33 P1-1273-14-12

The agreement between the theoretical and experimental values seems

to be quite good considering the assumptions that have been made. It may be

possible to improve the calculated thermal expansion values by using

experimental values for the exponent of the repulsion term rather than the Ininth power function vhich has been assumed..

2. Vibrations of the Lattice As a Whole '1The previous computations of thermal expansion of alkali halide

crystals were based on calculation of the frequemny of oscillation of the

ions urder the assumption that the ions are free to vibrate independentlywithout disturbing neighboring ions. The present studyr is undertaken to [

investigate the feasibility, in calculations of this kind, of treating somewhat 'lmore rigorously the vibrations of the lattice as a whole. The assumptions

are still made that these vibrations are simple harmonic vibrations and the 11

polarization of individual ions is completely neglected. Because of the

great volume of work involved, the calculations were limited to six alkali

halides which are lithium fluoride, lithium chloride, sodium fluoride,

sodium bromide, sodium iodide and potassium fluoride.

3. Method of Attack

The forces between the ions were taken to be those used by Pauling

in his calculation of the interionic separations in the alkali halides. The

high order repulsions were all assumed to vary so that the potential energy Flof a pair of ions varies inversely as the ninth power of the separation and

the constant of proportionality was taken to be proportional to the eighth . I.

power of the sum of the -radii and to have the general form postulated by

Pauling. This completely specifies all the parameters necessary for a

description of the dynamics of the lattice.

4. Ionic Forces F]

The role of the Coulomb forces was handled by the method of Ewald

which gives the value of the vector at any ion location for a wave described Iby any wave number vector in the reciprocal lattice. The method was modified

to give directly the components of electric field instead of the Hertz vector.

34 PI-1273-M-12

The high order repulsions were considered by calculating the forces

arising from the displacements of nearest neighbors and second nearest

neighbors only.

9 The frequency of a wave travelling with a wave number vector described

by a point in the reciprocal lattice was computed by assuming a frequency

for this wave and then writing down the equations of motion for each of the

two ions in each of the x, y and z directions. This gave six equations which,

on elimination of the amplitudes, gave a determinant which was a sixth degree

equation in the square of the frequency. Most of these determinantal equations

could be rather easily broken down into, at the most, second degree equations

L yielding easy solutions. The equations were all solved by electronic computer.

5. Reciprocal Lattice

1] To minimize the number of calculations, Sixty four points uniformly

distributed in one unit cell of the reciprocal lattice were taken to describe

the wave numbers of travelling waves. Because of the high symmetry of the

lattice, this involved only eight different calculations each yielding six

Svalues for the frequerry.

6. Expansion Coefficient

ii If there are f21 modes of vibration idth frequency Al , 727 modes

with frequency P , etc. then for one mole of the alkali halide crystal

.. 6N (11-32)

where A/ is Avogadro's number.

As was shown by Sryth 2 the coefficient of expansion aX of thewhole crystal will be given by

_ I R[1I I~((fTI dr-E r (11-33)d r-2

35 PI-1273-M-12Ilii

Ii'

where R is the gas constant in ergs per mole per degree, Es is the

potential energy of a mole of the crystal when the ions are at rest, , - 1is the cation-anion separation in centimeters and E(X) is the Einstein

function of X -

For each of the modes considered, tho frequency P was computedd Pas described above and -57r- was found by increasing r by one percent,,

keeping all of the force constants the same, and redetermining each frequency

P *dThe value of considered to be given by

Equation 11-17 showed that E3. could be expressed as follows

2Ae (- ) A/Mee( A) 2lSf!7

(11-34&)

where A r is the amount by which the cation-anion separation at ary

temperature varies from the equilibrium value it would have in the absence

of thermal motion. From this

_,_, -- Nre2 (n) (11-35) -

If the cation-anion separation is known, as it is for each alkali

halide, everything is known to determine the value of the coefficient of

expansion for each alkali halide. Table VII gives the values of the

calculated and experimental coefficients of exparsion at 300 K for each of f

the compounds studied.

7. Discussion -

The accuracy of the calculated expansion coefficients depends upon

a number of assumptions including:

1. Negligible ionic polarizability.

2. Simple harmonic vibrations.

3. Degree of interdependence of lattice vibrations.,

4. Choice of the exponent in the term representing the

repuls ion forces.

36 PI-1273-M-12

Table VI ITHERMAL EXPANSION COEFFICIENTS FOR THE ALKALI HALIDES

'AT 270 d CALCULATED AND LITERATURE VALUES

CALCULATED THERMAL EXPANSIONCOMPOUND COEFFICIENT BASED UPON EWALD'S3 METHOD EXPERIMENTAL

or- x to? DATA*

LIP 280 3110

LICI 710 '135

NaF 3410 360

VNsBr 580 '415

Nat 6711 470

KF 338 370

1NHIELKE'S (1953) BEST VALUES

IV37 PI-1271>W4-2

The choice of the Ewald method for allowing some interdependence of lattice

vibrations, requires extensive calculations which, in turn, limits the

number of results. The predicted expansion coefficients are of the correct

order of magnitude and vary in accordance with the experimental values.

However, further calculations of this sort for the alkcali halides do not seem

to be worthwhile.

Variations in ionic polarization seem to account for many of the

differences between the calculated and the experimental expansion coefficients

(Table VI). The influence of polarization on the thermal expansion of the

alkali halides has been studied by Weyl. Using Weyl's approach, we note

first that the theory gives reasonably correct results for NaCl and NaBr. JJSubstitution of Li for sodium leads to calculated results that are too small

whereas substitution of K and Rb leads to calculated results that are too

large. The princip:le forces determining the arrangement of the ions in the

structure are the attractions between anions and cations and the repulsions -

between the anions. Neglecting the polarization of the cations, as a first

approximation we note that polarization of anions results in higher electron

densities between anions and cations. At the same time, the electron density

between neighboring anions is decreased. Therefore, the effective charge is

greater for coulomb forces between anions and cations and less for coulomb

forces between anions. As the temperature increases, the electrons become

more symmetrically arranged around the anions. This results in a decreased

effective charge for the anion-cation attraction forces and an increased

effective charge for the anion-anion repulsion forces. Therefore, the lattice

constants increase more with increasing temperature, than would be expected

without polarization of the anions and the experimental expansion coefficient

is larger than the calculated expansion coefficient. Since the Li÷ polarizes

the anions to a greater degree than the other cations, this polarization

effect increases the experimental values for the lithium halides to a greater

extent than the other compounds. Within the group of lithium halides the

polarization effect increases with increasing anion size and polarizability Iso that LiI has the highest expansion coefficient of the alkali halides. On

the other hand decreasing the polarizing power of the cation, for example

38 PI-1273-M-12

I

by substituting K or Rb for Na results in a decrease in the polarizationeffect and the calculated expansion coefficients are larger than theexperilnentaJ. values.

FT

[I 39 .P1-1273-14-12

III. THE THERMAL EXPANSION OF SOLID SOLUTIONS

A. Introduction

There is a large difference in the thermal expansion coefficients along

different crystallographic directions in many anisotropic materials. 6 27

Single phase ceramic bodies prepared from materials having a high thermal

expansion anisotropy tend to form internal cracks upon cooling. The

hysteresis observed in the thermal expansion curve of TiO is probably

caused by these internal microcracks2. Other polycrystalline ceramic

bodies such as those composed of MgO.2TiO2 , AI 2 03 .TiO2 and Li 2 O.AI 2 03 2SiO2d(, -eucryptite) have low thermal expansion coefficients but are very weak

because of a high expansion anisotropy. Reduction of the thermal expansion

r anisotropy of the crystals used in these bodies should result in stronger,

more useful bodies.

The effect of solid solutiom atoms on the thermal expansion of ceramic11 19materials has been studied in a few cases. Austin has suggested on the

V basis of literature data, that exterded solid solution lowers the expansion

coefficient. For instance Rigby, Lovell and Geen29 showed that the addition

of FeO-SiO2 to CaO-SiO2 reduced the thermal expansion. On continued addition

of FeO-SiO2 , however, the thermal expansion coefficient began to increasesomewhat. Kozu and Ueda 3Ohave found in their study of plagioclase that the

$1relative amount of albite and anorthite strongly influences the magnitude31of the thermal expansion anisotropy. These authors have also studied the

effect of impurities on the thermal expansion of diopside minerals from

various localities. The thermal expansion anisotropy of one of these diopside

crystals 4hich contained a relatively large amount of iron was found to be

considerably less than that of pure diopside (CaMgSi 2 06 ). Ricker and Hummel 3 2

have reported that the solid solution of Si0 in TiO decreases the thermal22

expansion coefficient.The papers cited above indicate that the thermal expansion of crystals

can be varied by addition of solid solution atoms. In the case of one

naturally occuring mineral sample, variation of thermal expansion anisotropy

41 PI-1273-M-12

with composition was noted. The objective of this investigation was to

study the variation of thermal expansion anisotropy with composition for

several ceramic solid solutions. Since this effect has not been studied in

series of prepared solid solutions, it was decided to begin a study of the

influence on the thermal expansion anisotropy of the replacement of Ti÷4

in ratile with ions of differing size and charge such as Zr+4 and FMi 2

TiO2 was chosen for study since there are approximately 40 cations which are iwithin 15% of the size of the Ti+4 ion so that substantial solid solution is

possible with many ions. Also, rutile possesses the relatively simple

tetragonal structure and, therefore, the thermal expansion anisotropy can be

determined by high-temperature x-ray diffraction methods. Later in the

investigation, samples in the SnO 2V 205 an Al 20 3-Cr 203 systems were

investigated to determine whether these solid solutions also obey the rules

that appear to hold for rutile solid solutions.-

B. Experimental I1. Specimen Preparation

N+5 33 n+4,34 35 +2 36Except for N S Zr 4 , and Be 2 there is little

information in the literature on the extent of solid solution in TiO2 . Most

of the additives were selected for the present study on the basis of ion Isize since it is generally considered necessary for the substituting ion to

have an ionic radius within about 15% of the substituted ion in order for '1+4extensive substitutional solid solution to take place. Ti has an ionic

radius of 0.68 • so that the radii of the substituting ions should lie in

the rarge of 0.58 - 0,78 R.Because of the lack of information on the extent of solid solution

in TiO2, 10 mole per cent of the oxide additive (based on the cation) was

usually used. If this amount of substitution takes place on firing thespecimen, a relatively large change in lattice constants is expected. The

samples containing 0.5 and 1.0 mole percent additive were based on Johnson's37research. It should be noted that the actual extent of solid solution was 71

not known in most cases.

42 FI-1273-M-12

. .......... .

A standard firing temperature of 15009 C was selected on the

Tbasis of ZrO2 -Ti 2 and BeO-TiO2 phase diagrams, all of

which show considerable solid solution at this temperature. In the case

of the additives Li 2 CO 3 , Sb 2O 3 2 20 a temperature of 1200 C was used

becaus e of the high vapor pressure of these conpounds at higher temperatures.

These additives were all reagent grade materials. The TiO was2Titanium Pigment Corporation's Titanox TG. The analysis of this material

is < 0.01% CaO, < 0.1% W0, < 0.008% Nb, < 0.007% Sb 203 , 3 0.0o4% Si02 , 0.006%

Fe203 , 10 ppm V, 0.5 ppm Mn, 5 ppm Cu, 4 ppm Cr and 10-20 ppm Pb. In Table VIII

the additives are given together with the relative weights of TiO2 and

Sadditive, and the firing temperature and time.

The rutile solid solutions were prepared by ball-miliing theweighed additive and TiO2 for four hours. The powder was passed through a

"4i mesh standard screen, mixed with 15% binder solution and dried at lUO°c for

an hour. After passing the dried mix through a 16 mesh screen, it was pressed

at 7000 psi into 1" discs about 1/h" thick. These discs were fired to

I• temperature and then cooled to room temperatire in less than 30 minutes to

I avoid possible precipitation. The x-ray powder specimens were prepared by

crushing the discs in an alumina mortar to pass a 325 mesh screen. A samplefl of pure TiO2 for use as a standard was given the above treatment. The SnO2 -

-2 V••5 samples were prepared as described above using a firing temperature of

I0000 C for 16 hours. Also, solid solution specimens in the Cr2 0 3-Al 203

- system were prepared as described above using a firing temperature of 15000Cfor 17 hours. Additional experiments were performed usirg the phases V02 ,

Ge2, MgF2 , MgO2TiO2 a A 2 03 -TiO2 These latter experiments were

unsuccessful for a variety of reasons such as decomposition of the desired

p phases, lack of stoichiometry or inability to get the compound in the desired

crystal form.

2. Experimental Procedures

SThe equipment used for the x-ray therral expansion measurements

. consisted of a G. E. XRD-5 spectrogoniometer =-d a Tem-Pres high temperature

x-ray diffraction furnace. The sample holder was a platinum plate with a

43 PI-1273-M-12

Table. VII I

DATA ON THE PREPARATION OF RIJTILE SPECIMENS

1MOLE % FIRING FIRINGADDITIVE ADDI TI VE WEIGHT OF WEIGHT OF TEM4PERATURE TIME

USED (CATION BASIS) ADDITIVE (g) T10 2 (g) MRS. vzno 10 11.3 100 1500 17

LI 0o 10 5.111 t00 1200 is

3490 10 5.6 ..00 1500 37 1Cr2O3 10 0.57 300. 1500 1

'6203 1020.25 100 1200 Is

14n0 2 30 12.08 t00 1500 17

V25 10 12.614 100 1200 Is I

W03 30 33.2 300 ý1500 17 HW03 3.0 5.7 1964.3 1500 15

Nb2O5 9.2 '42 2110 11400 3

ZrO2 30 17.15 100 1500 17

PbO 30 23.16 76.341 1500 171

ýy2 0.5 0.71 99.29 3500 317

CaO 0.5 0.35 99.65 13500: 17

9.0 3.0 0.33 99.69 1500 17

14003 3.0 3.58 196.41 3500 17

102 s0 '46 352 1500 3

Sno2 10 16.75 80 3500 37

SnO2 20 37.7 s0 1600 17

Sn02 30 614.7 so 3500 371

414 PI-1273-M-12

milled depression 13 x 16 x 0.75 mm deep. Because of the high thermal

corductivity of platinum, the maximum difference in temperature across the

sample holder was found to be 20C over the range 25-1000°C. Using a precision

'potentiometer, it was possible to maintain the temperature within +_O.5C0

during a diffraction peak measurement.

The sample holder mounting was provided with rotational, tilting an:!

translational adjustments so that the specimen surface could be aligned on,

and parallel to the spectrometer axis.

Copper radiation ( A 1.54050 • for K ) was used for all

measurements. The x-ray tube was operated at 50 KV and 16 ma with a 0.0007"

I nickel filter. A 30 beam slit and 0.20 detector slit were used with a

scanning rate of 0.20 (29) per minute and a chart speed of 30 inches per hour.

The peak positions were taken as the midpoint of the diffraction curves at

one-half peak height. Precision in measurement of the peak position was

Swithin + 0.010(28). To attain precision and accuracy in the determinationof lattice constants, use was made of lines in the back reflection region.

These were: for TiO2 (303), 2e 134.7° (521)20 1 138.80 and (213) 20 1-19.7;0

I I~ for SnO2 (323) 20 - 137.70 and (512) 20 = 147.3; for A1203 (0, 4, 10)20 4•5.2°,ard (416)20 = 136.00; and for Cr 2 03 (330) 20 - 137.5%, and (3, 2, 10) = 1149.80.

An analysis was made of the precision and accuracy of the x-raymeasurements. At 240C, considering diffraction lines for 1300 20, or above,

random errors due to angular measurement and specimen centering were estimated

to be within + 0.0004 R. This was checked by making a series of nine

-i measurements on a sample of Fisher precipitated silver (99.99% Ag) usingthe (51) and (422) lines. All of the measurements fell within the estimated

i i precision. The accuracy of the measurements was estimated by comparing the

extrapolated ASTM card value for the a of silver at 24.00C (4.859 X) with

the observed values. Agreement between the values was also within + 0.0004 •.

The accuracy was therefore within the limits of the precision. The effect

of x-ray absorption on accuracy was catimated by comparing the diffraction

peaks of pure silver ard a mixture of silver and graphite in the ratio 1/3.

The mixture had an a value of 0.0001 R greater than pure silver. The0

absorption difference of these two compositions is considered to be much

45 PI-1273-M-12

Ii

greater than those between the oxide solid solutions investigated so that

errors due to x-ray absorption are considered to be negligible, At high

temperatures the data are less precise because of a temperature variation

across the specimen surface of + 2.0°C. This is equivalent to an error of

+ 0.00016 R. Added to the random errors (+ 0.000 •) at room temperature,

the total precision is + 0.0006 • for high angle lines up to 1000°C.

Concerning accuracy at high temperature, systematic errors in

temperature measurement and sintering of the sample or sample movement due

to expansion of the furnace parts must be estimated. Considering the latter,

a displacement of the specimen of 0.001" was found to charge a of rutile0

by 0.00006 R. Assuming the maximum movement of the sample surface to be 5 mils,

an error of + 0.0003 R is introduced which is added to the estimated room

temperature accuracy of + 0.0004 • to give a total high temperature accuracy 1of + 0.0007 .

To obtain an overall estimate of the accuracy of the measurements

including temperature, the a of silver was measured to 6000 C and compared[I0

with the values of Hume-Rothery and Reynolds (1938)0 The observed values

agreed with those of Hume-Rothery within + 0.0003 •.

The room temperature lattice constants were measured for each

rutile solid solution specimen. These are presented in Table IX. If there

was a significant change in the lattice constants, it was assumed that a

considerable amount of solid solution had taken place and x-ray thermal

expansion measurements were made. In the case of rutile, the peak positions

for the (303) and (521) lines were measured as a function of temperature up

to l000°C. These data were then used to obtain the variation of the lattice

constants with temperature.

In order to obtain information on the extent of solid solution of

vanadium in TiO2 , a series of samples was prepared with various percentages

of vanadium. As can be seen from Figure 11, the a lattice constant varies

linearly with composition up to about 17-20 mole % vanadium. The results are

plotted in terms of the compositions originally prepared and do not take into

account changes due to volatilization of V 0 A chemical analysis of the2

14.8 mole percent vanadium sample indicated the presence of 16.1 mole percent

46 PI-1273-14-12

I- Table IX

ROOM T04PERATURE LATTICE CONSTANTS OF RUTILE SOLID SOLUTIONS

ADDITIVE TD4P. LATTICE CONSTANTS A*

SMOLE % co c/a

PURE T10 2 25 41.5937 2.9593 .614•2

I LI+I 2.5916 2.9587 .6140

10 LI+ 1 28 4.59s4 2.9588 .011o

10 1,1142 22.5933 2.9598 .611111

9.2 NbO5 28 4.6128 2.9632 .64211

[ 5 Y+3 29 1.5890 2.959 .6110

0.5 C&+2 26 4.5933 2.9589 .o612

+227 11.6934 2.9689 .64112

10 ,g+2 29 '11.5940 2.9591 .66441 "

I0 Zn+ 2 27 1.5937 2.9589 .6141

10 Sb÷3 27 4.5957 2.9629 .0647

Siv+5 27 4.5869 2.968 .6615

10- Cr÷3 28 11.5989 2.9580 .8432

10 n÷2 27 4.5975 2.9579 .61434

e 28.5 .6940 2.9586 .61139

10 W6 26 1.591"4 2.9598 .611f2

10 Zr+' 21 11.6180 2.9899 .641711

I mo• 28 4.5987 2.9685 .6110

30 S1+4 2.6. 4.5933 2.9583 ,646

10 Snm+' 21 1.6099 2.9817 .8481

20 Sn+ 4 26 411.258 3.0059 .6898

L 30 S 25 4.61106 3.0282 .6525

-! 7 PI-1273-M-12

...... ....... T ..... .r .. .... ......... ...T

a. . ..........- --------I- ---.- -..... ......S

1 1 *

o --------

me I I .

80~4. 0 4 -------- ... ... -. - - -----

ol a

'1.6900 ...........-- i---------H10-0 0 710 50140LE PECNTVMAW

Fiur 11LTIECNTN A AIUSTMEAUE s OPSTOFOR 20oo FIRNG TMPERTUR

48 I-17314

vanadium after heating to form the solid solution. lvnile an increase in the

ratio of vanadium to titanium is unlikely (an error in chemical analysis is

more probable), the analysis does provide assurance that a substantial fraction

[of the vanadium does remain in the material after heating.

C. Discussion of Results

In discussing the results, several things should be kept in mind.

First, except for the case of vanadium and tin in TiO2 , the extent of solid

Ssolution in the titania specimens is not known. Second, the valence of the

additive is not definitely known in some cases. Some work has indicated that

there is a tendency for solid solution atom to assume the oxidation state of

the host lattice.38 This "valence inductivity" effect has been investigated for

the case of low concentrations of Kn in A12 0 3 , where the Mn at first assumes a+3, ~'+3 oxidation state in imitation of Al but then changes to a +4 state at

higher concentrations. Recent electron spin resonance experiments reported

by Gerritsen and Lewis3 9 indicate that vanadium replaces titanium and is

present in the four valent state, at least at small concentrations.

i The question of valence is not only important because of its bearing

on ion size, but also because electrical neutrality must be maintained in

the lattice. For instance, ions with a valence greater than four vould requirethe formation of Ti+3 ions. Since Ti÷3 has an ionic radius of 0.76 R, the

overall effect might be similar to the addition of an ion larger than Ti+4

(0.68 •). Similarly, the addition of ions with valences less than four

requires the formation of oxygen vacancies (0- radius is 1.40 R). Mauer and

Bolz have measured the thermal expansion anisotropy of an oxygen deficient

sample of TO with the composition of TiO1 .9 up to a temperature of 4000C.In such a sample each oxygen vacancy requires the formation of two Ti+3 ions

in order to maintain electrical neutrality. There are. then, two factors

operating; cations of larger size and lower charge than Ti are created and

many anion vacancies are introduced into the lattice. In Table X, data based

on Mauer and Bolz' work are given which shcw that the TiO1 9 7 sample has a

much larger expansion anisotropy (,A c) ard a somewhat decreased volume

expansion coefficient (CLv ) and average expansion coefficient (CO&,) than

49 PI-1273-14-12

I

Tab) e XTHERMAL EXPANSION DATA FOR RUTILE SOLID SOLUTIONS -1

(ROOM TEMPERATURE TO 10000C)

AVERAGEVOLUME LINEAR

LINEAR EXPANSION EXPANSION EXPANSION EXPANSIONCOEFFICIENT ANISOTROPY COEFFICIENT COEFFICIENT

HOLE % Ac /°C °C boCADDITIVE a-o iao Co 0 Co C0 co" a o r AAv •avar

PURE T10 2 83. q x I0o7 107.7 x 10-7 24.3 x 10-7 274.5 X 10-7 91.5 x 10-7

10 V205 89.2 x 10-7 69.8 x I0o7 5.4 x 10-7 273.'.C 10-7 91.0 x 10-7

10 ZrO2 77.7 x 10-7 110.7 x 10-7 33.0 x 10-7 206.0 x 10-7 68.7 x 10-7

10 Sb2 03 81.1 x 0 103.0 I 1- 21.9 x 10-7 285.2 x 10 $ 88.4 x 10-7

9.2 Nb2 05 82.5 x 10-7 111.1 x 10-7 28.6 x 10-7 276.1 x I07 92.0 x 10-7

10 W03 79.4 x 10-7 105.7 x1 26.3 x I0-7 264.5 x 10-7 88.2 x 10-7

30 S102 82.5 x 10-7 104.8 x 1 0-7 22.3 x I0-7 289.8 x 10-7 69.9 x 10-7

I0 NnO2 77.6 x o0-7 111.0x I0-7 33.4 x 10-7 266.2 x I0 7 88.7 x 10-7

10 L12 CO3 82.2 x 10-7 108.0 x 10-7 25.8 x 10-7 272.4 x 10-7 90.8 x 10-7

10 3n0 2 " 75.5 x 10-7 I04. x 10-7 28.1 x 10-7 248.8 x 10-7 85.0 x1-

20 SnO2 70.9 x 10-7 93.0 x 10-7 22.1 x 10-7 234.8 x 10"7 78.2 x 10-7

30 &,02 48.64 x I0'7 92.0 x 10-7 23.6 x I0-7 228.8 x 10-7 76.3 x 10-7

IOCr 2 O3 75.1 x 10-7 109.7 x 10-7 34.6 x I0-7 254.0 x I0-7 84.7 x 10-7

DATA OF HAUER AND BOLZ (1957) 2

PURE Ti02# 78.8 x 10-7 99.2 x 10-7 20.6 x 10-7 268.4 x 10-7 85.6 x 10-7

Ti 01.978 68.0 x 10-7 104.5 x I0-7 36.5 x 10V7 240.5 x 10-7 80.0 x 10-7

PURE T10 2# 84.7 x 10-7 105.0 x 10-7 20.3 x 10-7 274.4•x I0-7 91.4 x 10-7

# ROOM TEM4PERATURE TO O000CROOM TEMPERATURE TO 800 0 C

50 PI-1273-M-.12 .

1..

stoichiometric TiO2 . In Table X the thermal expansion data from room

temperature to 1000°C are listed for the specimens studied. Those ions

which produced relatively small charges in the lattice constants, Li, Si,

Sb and W, also produced little charge in the thermal expansion anisotropy

( 4 O•). Except for Sn, ions which increased the lattice constants (Zr, Nb

Sand Mn) also increased A 0-. And finally, vanadium which decreased the

lattice constants, also decreased A0 to a remarkable degree. It appears

that most of these data can be systematized on the basis of ionic radii; if

-there are no valence changes or vacancies formed, the substitution of an

ion of larger radius increases 4 W , and conversely, an ion of smaller atomic

radius decreases l4 XC. According to this interpretation, Cr3 , with the same

ionic radius as V÷. produces a larger increase in A&. because of the

formation of anion vacancies. The result here is similar to that obtained

by Mauer and Bolz for oxygen deficient rutile. In the case of Sn , the

Sionic radius is only slightly larger than that of Ti÷4 and there is almost

complete solid solution formation between SnO and TiO Figures 122 2

to 16 show the thermal expansion curves obtained for pure TiO and for

zirconium and vanadium additions to TiO2 .

To provide additional evidence for the above explanation for

II changes in AO, further work was done in the Al 20 3-Cr203 system. These

oxides are completely miscible in all proportions. Unfortunately, however,

the addition of only a small amount of one into the other causes a large

broadening and loss in intensity of the diffraction peaks. Besides pureAl 20 3 -Cr 2 03 , only two solid solution specimens were measurable. The resultsare indicated in Table XI and Figure 17. Since the Curie temperature of

Cr 2 03 is just above room temperature and this causes a discontinuity in the

thermal expansion properties41l 42 the thermal expansion anisotropy has been

calculated for the temperature range 400-1000C. The observed changes of

thermal expansion anisotropy are too small to be considered significant.

The fact that the sign of the thermal expansion anisotropy of AI203 is positive,

while that of Cr203 is negative, leads to the conclusion that sijnificrnt

changes in thermal expans ion anisotropy will be found for larger additions

and a composition with zero thermal expansion anisotropy (defired for a

particular temperature range) will be found in the internediate composition

rarge.

51 PI,-1!273-M-12

1.2 -. . .. .. -- - --ii i

. ..... . ..... * .. .. I ---------- ------- 1-" ...... ..i....... , ........ i .... ... ......... .. ........ i......... ........ ... ..........! ........!. . . t ... C i

0.8 ........ ......... .......... . .... . ......... " . .

- -I i . ° ... C.° • ° . ° ° o •

_ _ .I I I i :I

S.......... ... ... ... .........., ..... ....... ....

0 .6 -------- ..... .. ---- -- -................. .........-----' ' -'--J -..... , ....... ......-. -,-.......1I s

0 C AXIS

1 13. AXIS0.82 ...... -F 1 ......... &....... , . ......

I i , , ,= I j i

-.. . . . -- ---- - . . . . . . . -. . . . . . .. . . . . . . . .--- -----

0. _ i C , C

j I CI, I C

0200 400 Soo 800 1O000i

. 1

TE14PERATURE °C

Figure 12 LINEAR THERMAL EXPANSION OF PURE RUTILE (Ji 02

02.2 PI-1?73 M-12

I C C

II

I J , . *2;1 .2 ........ .......... ......... r...... •.............. . ..... " .................. ......• .......... "

I C S- I I-. . ....... . .....". ........ ..1 ... ........ '

.0 ....... .. ......... ...Z ...I.......... [ ....S!o , _ . .. ...... ....... . ..... . ........ . . ... .. .... ... .... ........

0. .-. .............. ......... .... ..- ------- .. ............ ...........

----------- -------- ............. ........... . ........ .... ...,......1 a- ...................... - - ... ..... .......1I

I , 1

[: I" .A i

0 .4 .. . ...... .. ........ .......---------------------------

i!I o ,Io I• ,o-- -------- .......- - ------ --

I I

0 200 '#00 Goo 800 i000

pTEMPERATURE -C

r ~~Figure 13 LINEAR THERMAL EXPANS ION OF r'0,(o2 *9MOL.E' %) Ze .'2

53 PI-1273-1i-12

--------- .......... ......................... r ......... •

I .... ... .......... ......... ,... I"...... :"........ r.......... ......... !......... ........ . ......... t ...... . . -.. . . . . .....

AL -------..... e............. ....... . i.. o......b. ---- ..... ... ... ;o...

I I I *S 5

-- -- - -- - - - - -- -- - ----.

A XIS

..... ....... - ......... -- ......... ------ --

200 0 "01 Boo [000!!!!!!!!!!::::::::::: :I....~~i........1. .....- .--.......... " -.....06.............b.........f........' I 5 ii

o., I.-----/-...---.S..

-I/

Fl"- - - -Ii

0 2 00 'oo 6 00 500 1000 1•TEMPERATURE OC

Figure I4 LINEAR THERMAL EXPANSION OF T;0 2 - 7.7 MOLE % VANADIUM -I

54 HZ-1073-17-1-122

Ii I

- ...... .... . ..... .................. . ........ . .. ............. .

1.0 ...............f..... .... .... .............

S I ii.. ... .... .............. ......... ........ L.......-. .................. o.•

0. , . ... ...... ......... .......... .... ...... ..... ...... i

-1 - ... ...... . ............. .................. .... ....0. o . . . ..... .............. ...................

S................. ..... .......... .. ./ ..... ...0.5 . ......... .................... ..... :..

.... .... . -- -----AXI

0.2 _ .• . -........................ ... .....------- ----

611

17 -1

S IAS, iIi

5511-23-41

1 4I,ia .............. ..... ............... ... !........*. ........, i,.....,

- 0 ! l I SS..

0 200 '100 600 800 1000

Ii ~TENPERATURE -

F!!

•55 P1-1273-14-12 i

Ifi

I j

1.2 ......... .... ...... ........* ....... ....-..... .....-

.......................................... -........ ................. . ... ....

= 1

..................... ... . ... . ...... ............. ..... . . .. ................ .. . . .

..-.-- ] c AXIS

-- - - - - - - -- - - -

0 200 400 600 800 1000

TEMPERATURE *C

Figure 18 LINEAR THERMAL EXPANSION OF 1"o02 + 21.4 MOLE % VANADIUM

56 PI-1273-M-12

LL

ca a

30

'oil

ULLb

11 C3C'a (43,

uM Its ; cl C-

ILb-

eJ. Lb

Eco C3-

Jul #4

57 P-23M1

1.2 -------..---- -----------..... ......................

----- - ----- ...............

4- S

-I.~ ~ --------..- . - --------

S

------ -------------- -- ---- ----- ----- ...... t - --- --- --- --- ---

S. ----- ---- .-..

-- --

.-- --- - --------- I --

1.00 C AI S

------ --------- ---- ..... ----- AXIS

0. --- -I- -----------.......--------

-------------.

F ~ gI u e 1 LIN A TH R A EX A S O OF P R4r2 O

5 8 SI 1 7 - 14

I

In some cases polycrystalline ceramics made from crystals having

substantial thermal expansion anisotropy, show thermal expansion hysteresis.

This effect was shown for rutile by Mauer and Bolz40 and Charvat and Kingery.28

•In s~ite of this complications the thermal expansion coefficient measured by.

anuer and Bolz by the dilatometer method was 9003 x 0-7/oC (O-lO00C) .hich

compares favorably with the average thermal expansion coefficient calculatedfrom the x-ray data.

SThe addition of vanadium to the rutile lattice results in a

progressive decreases in thermal expansion anisotropy with increasing vanadium

content up to 10 mole percent vanadium. At 10 mole percent vanadium, the

thermal expansion anisotropy is substantially zero. Continued increase in

vanadium content results in a reversal of the anisotropy so that at low

ftemperatures the percent expansion is greater in the direction parallel to

a0 than in the c0 direction. An increase in slope of the thermal expansion

curve parallel to the c axis at high temperatures was observed in some cases.

Little change in the volumetric expansion with increasing vanadium content

was observed.I According to Croremeyer43- the structure of TiO 2 can be considered

as made up of Ti-0-0-Ti-O-0-Ti chains oriented normal to the c axis. Therefore,

replacement of titanium by vanadium with resulting decrease in a and slight

decrease in c can be thought of as causing a decrease in the length of the

chains but having little effect on the distance between them. To determine

whether or not vanadium would have the same effect in other crystals of the

rutile structure, the effect of vanadium on the thermal expansion anisotropy

of cassiterite (Sn0 2 ) was determined. In this case, too, the thermal

expansion anisotropy was reduced. The results are given in Table XII and

Figures 18 and 19. The thermal expansion anisotropy of crystals is important,

in part, because of the effect of this anisotropy on local stresses, porosity

and strength of polycrystalline bodies made from these crystals. At high

temperatures during sintering, the stresses between crystals are relieved.

During subsequent cooling, the crystals contract unequally causing local

stresses. Thcse stresses may cause local failures and the formation of pores

that are large in two dimensions. This change in microstructure can result

.59 PI-1273-11-12

--

C2C

a F..

CC~44

C4 -

a -4-a C)ý

60I-17-M1

1.2- - - - - - - -

3.0 -- -

0.1

-J.

.0 200 400 600 300 9000

TEMPERATURE *C

{AFigure 18 PERCENT LINEAR EXPANSION OF SnO2

61 P1-1273-M-12

1.2-

1.0- -

U.1

0.2

o Lc

TEMPERATURE OC

Fi gure 19 PERCENT LINEAR EXPANSION OF SnO2 + 10 MOLE PERCENT V205

62 PI-1273-M-12I

Iin low str~ength and low thermal conductivity. The reduction of the thermal7 expansion anisotrvpy off rutile and other crystals may make it possible to

I prepare ceramics with improved properties.

63 PI.-1273-M-12

ISIV. THE THERMAL EXPAISION OF TWO-PHASE BODIES

A. Introduction

The thermal expansion of composite materials (including multiple-phase

ceramics) can be predicted fro' the thermoelastic properties of the individual

Sphases providing the resulting body is microscopically continuous ar. the

phases do not enter into solid solution. The degree of validity of the theory

Sis dependent, primarily, upon the completeness with which the internal

stresses arising from thermoelastic dissimilarities are described. The case

in which the composite body is homogeneous and macroscopically isotropic is

- of immediate interest and will be treated in this report. The results

reported here are an outgrowth of research reported previously by Merz et a. 2L

I iIn this earlier research the inadequacies of available prediction methods

became apparent. This led to the use of more satisfactory methods which

are described in this report.

In the literature, several simplifying assumptions have been used to

permit computation of the linear thermal expansion of two-phase ceramic

bodies from existing data. These assumptions have been reviewed and a more

complete theory for multiple-phase ceramic bodies is presented.

An important factor in improved correlation between thermal expansion

theory and experiment is the determination of the intrinsic elastic properties

Sof each ceramic phase. Gross reduction in the elastic moduli due to finite

porosity resulted in attempts to produce high-density ceramic bodies.

Apparatus was assembled for induction-heated hot-pressing to obtain high-

density spinel and mixtures of Pyrex glass with spinel. The Pyrex-spinel*1 system has been studied in detail both theoretically and experimentally.

B. Relationships Between Elastic Moduli and Stress Wave Velocities InIsotropic Bodies

The application of a dynamic stress at the boundary of a solid will

generate a stress wave (or waves). For small disturbances, these stress

waves will propagate away from the source with a velocity dependent solely

upon the elastic properties, density, and georetry of the medium. Measure-

65 PI-1273-M-!2

ment of elastic wave velocities in appropriately shaped specimens will

provide data for computation of the elastic moduli. 1Transmission of ultrasonic pulses, consisting of a burst of radio

frequency (R.F.) stress waves, provides a convenient method for simultaneous

measurement of stress wave velocity and attenuation over a range of

frequencies. High frequency compressional waves and high frequency shear :waves in ceramic specimens have been measured using the electronic test

system indicated in Figure 20. The stress wave velocities (C) in

semi-infinite media are given by: L

where L stress wave path length 71Y= stress wave transit time

and by Kolsky.4 1comFressional wave velocity '.('/_2ý fOr a>> ?-

I2

shear wave velocity, C•--[ •- }

where Y Young's modulus (modulus of elasticity)

G shear modulus (modulus of rigidity)

mass density

?' =wavelength of stress wave in medium

a = cross sectional dimension of specimen

I) Poisson's ratio 7These equations along with the relationships between the isotropic elastic [1

constants,45 e. g

2(, ,.-,') y1'-I

K p 3(-2 V)J

66 PI-1273-M-12

-0 N.OSCILLOSCOPECAMERA

III Ii)OU14ONT297

I.VIDEO TERM4INATING CATHODE-RAY

DETECTOR CIRCUIT OSCILLOSCOPELIT lI ~2OATEKTRONIX

535

PULTRSEDI WIDEDAND AMPLIFIERSCL.CI

FRE9. GENERATOR

ULTRASONI WIDEBAJIB AMPLIFIERS CABO

TESOLNATION

WDELAYLNE

67 I-173-N-1

were used to derive the following

G',,_'yoC

- -

k (C,2 A CS)

Note that whiile four elastic constants may be of interest, only two

independent elastic constants exist for isotropic media, hence only two

different stress wave velocities need be measured.

The ultrasonic pulse method also permits convenient use of interferometric

techniques for accurate determination of temperature coefficients of stress Iwave velocity and attenuation. By simultaneous excitation of two similar

specimens maintained at different temperatures, snall changes in propagation Ti

characteristics are readily observed as R. F. phase changes. Since these

specimens may be excited at megacycle frequencies, detection of small phase

changes constitutes a detection of velocity changes of a few parts per

million or less.

* Identical relations were also given by Birch.4

I68 P1-1273-44-12

S .. .. ! i I f~i......... i i i -u , 'I

The attenuation of megacycle stress waves in ceranics will arise from

F• several factors:

1. Elastic anisotropy of the individual grains introduces

scattering losses.

2. Elastic hysteresis within grains introduces irreversible

F •absorption losses.

3. Relaxation effects between grains due to grain boundary

Smotion introduces relaxation-absorption losses.

14. Thermal conduction between regions of compression and

rarefaction introduces thermoelastic losses.

-. The presence of impurities within grains introduces an atomic

relaxation-absorption loss.

6. The presence of pores, voids and microcracks introduces

another form of acous tic scattering loss.

[All together, these various losses are called internal friction losses and

are intrinsic to the material (i.e. interoal friction losses are independent

of specimen geometry). Measurement of internal friction over a range of

temperatures and frequencies provides a method for separating the several

losses. Grain boundary relaxation generally produces an internal friction

peak at slightly elevated temperatures while elastic hysteresis losses

generally increase as a power of the absolute temperature. Scattering from

microcracks, however, will tend to decrease with temperature since the size

of the microcrack decreases. In some ceramics, the microcracks seem to

completely close and reheal, hence the acoustic scattering loss from such

scattering centers would disappear at appropriately high temperatures. If

this loss mechanism contributes significantly to the total internal friction

at lower temperatures, then ultrasonics provides a method for nondestructive

determination of the onset of microcracks. Such information would be

immediately useful in computing the stresses required to induce microcracks.

Private discussion with Professor W. R. Buessem, Pennsylvania State University.

69 PI-1273-14-12

I I

Computation of the thermal stresses -necessarily involves knowledge of

the single crystal elastic constants for each of the ceramic phases along

'with their temperature dependence. Such data are generally obtained by

ultrasonic pulse techniques and have been reported for a few ceramic single-

crystals, e.g. sapphire, ruby, magnesium oxide, and magnetite (see Appendix C).

The number of independent elastic constants increases from 3 in a cubic

crystal to 13 in a monoclinic crystal. In each crystal class, the several

elastic constants may be used to compute the single-crystal bulk modulus.

In multiphase ceramics, the bulk modulus will be a function of the porosity

and the component single-crystal bulk moduli as discussed in detail in

Appendix C. Since each of these properties may be measured independently, ia theory of elasticity for porous ceramic bodies will provide the basis

for computation of composite elastic moduli, thermal expansion and resulting

thermal stresses. A systematic conputation of thermal stresses in multiphase

ceramics should provide the insight toward the design of high-strength

ceramic bodies.

C. Applications of Ultrasonic Measurements

1. Stress Wave Velocity Measurements

The ultrasonic equipment was arranged for use in measurement of the

onset of microcracks in polyphase ceramic bodies. Because of the desire to

measure stress wave losses introduced by the smallest possible crack dimension,

megacycle frequencies were chosen. However, at megacycle frequencies, the

acoustic scattering lcsses due to both elastic anisotropy of the grains and

the presence of voids or pores are sufficiently great that only short path

lengths through the specimens may be used successfully. In order to obtain

sufficient accuracy in the determination of stress wave transit time in bar

type specimens, several variations of ultrasonic interferometer techniques

were employed. The most useful technique utilized a long pulse-length such

that the pulse reflected twice at the fused silica-ceramic specimen interfaces

as indicated on Figure 21. The twice-reflected pulse produced phase

interference with the directly transmitted pulse introducing combined response

nulls and peaks as the carrier frequerny was varied. Observation of the

70 PI-1273-11-12

-iiI I II:IIi Ii I~i -II7-l

JA646

da -1

-J 96 f

usLU LAI

-9

-0 W

71 FI17-M -

frequercies at which maximum phase cancellation occurred, permitted computation

of the phase velocity. The expression derived for stress wave velocity is

similar to that given by McSkimin.h 6

2LF

n .360

where P= frequency at which a phase interference null occurs

•b phase shift at the specimen-buffer block interface

in degrees

27 number of wavelengths in a 2L path length of the specimen

In the pulsed carrier measurement system, the pulse length is always chosen

to include a minimum of several complete R. F. cycles in order to minimize

the frequency spectrum. Thus for a given pulse length, a minimum frequency

of operation exists and the value of 77 cannot be determined by counting

from zero frequency. However, measurement of two consecutive null frequencies

provides a method of evaluating both 71 and C . For specimens whose

geometry is selected to avoid velocity dispersion,

2LP• -- - 2L(P ,-A')

n0. 5tŽ360 360

and

4f 360

where A4P frequency change between successive nulls

For all measurements made using this technique, eleven consecutive null

frequencies were measured and the mean frequency change between nulls

determined. In general, the stress wave velocity was accurate to three jsignificant figures by the long pulse overlap technique 4hich was adequate

for use in thermal expansion computations.

Shear wave velocities were measured with two independent methods:

1. Total pulse transit time, and 2. The long-pulse overlap method. A

sumrmary of the data obtaired with the second method is listed in Table XIIIo

72 PI-1273-4-12

w = co

to X3 t,Lo to n co C. m~ ~ .

L) COC

). W)

9.X

U.OC +14-1 +1+1 +1

UA 40 40zR !C

cc a to

40i -Iaci

IN C --r 4Lnc

r qjC2 I&O W.

LO VUA..

L6

0; Uo N4 94 C

C3 96 a- 0.0

ui 'k lk v %kA k %

ia aIC

73P -17 -M1

A length of Pyrex cane was cut and lapped on both ends to obtain

the 100% Byrex data. For this specimen it was convenient to measure the 1null frequencies between tu pulses, the difference being represented by

(Is P--Is2s-,-27--25,5 ) - (I#I+Ks25+2r-28,) '2P -2

where T= path length through one transducer

path length through one silica block

P= path length through Pyrex specimen

Bt path length through one transducer/silica bond i.8= path length through one transducer/specimen bond

It can be shown that the small effects of the bonds and transducer paths

completely cancel out with the interferoneter technique providing that the

reflection interfaces are properly identifLed. As a check on the accuracy Iof these rneisure' .nts, data for Pyrex from two other investigations were

obtained from the literature. The differences between these three sets of

shear wave velocity measurements were about 1%. The long-pulse null

frequencies were generally determined to only 1% accuracy (frequencies were

read directly from the signal generator dial, higher accuracy may be cbtained

through the use of a frequency counter when necessary).

The total pulse transit time provides a convenient check on the

stress wave velocity, but the accuracy decreases with decreasing specimen

transit time. Typical values of transit times are given in Table XIV for

the shear mode transducer assemblies illustrated in Figure 20.

If the total transit time was determined with + 3% accuracy, the

ceramic specimen transit time would be known to only + 40% for the values

given in Table XIV. In order to improve the attainable accuracy, a fixed -

delay line having a delay time nmar the value to be measured was used

(Figure 20). By incorporating a delayed trigger circuit, the two delayed

pulses could be observed, simultaneously, with an expanded sweep. With this

arrangement, the delay difference could be determined with + 3% accuracy.

Thus, with the fixed delay equal to the time delay of the transducer-buffer Iblock assembly (without the specimen), the delay difference becomes equal to

74 PI-1273-M-12

T '~the specimen t ra nsit tire and n'ay be measured directly wi4th + 3' accuracy..

In general, all thrce m.ethods v:ere used to determine shear wave velocities

in each ceramiic specimen in order to minimize opn~rator errorsý.

F Table XIV:SEAR WAVE TRANSIT TIMES FOR TEST SPECIMEN. COMPONENTS

PATH SHEAR WAVE SHEAR WAVEL EN GTO VELOCI TY TRANSIT TIME

IFSYMBOL MEDIUM INCHES 105 CM/ SEC MI CROSECONDS

S FUSED SILICA BLOCK 2.000 3.760 13.51

Bt INDIUM SOLDER BOND 0.0001 0.71 0.003

Fi14SALOL BOND 0.0001 0.'4 0:006

CCERAMI1C SPECIMEN 0.2505 '4.68 1.36

(100% SPINEL)_______ _____

28 + 28t + 286 + 2T TRANSDUCER 4-BUFFER - 27.08BLOCK ASSEMBLY

[I2S-- 2Bt+ 28X +2T +C TEST SPiCIMEN AssEJ4BLY -- 28.'42

KCorapressional wave measurements wore severely res~tricted because the

transducer assemblies available consisted of' one maegacycle barium titanate

ceramic transducers. The frequency response o'these transducers was poor

at the pulsed oscillator carri-er frequencies (~me-acycles andx above). Large

response ripples due to reverberations within the cerariic transducers precluded

use of' interferometer techniques. The exterded rise t--ime obtained wi-th these

transducers also precluded precise determination of' trarsit times. Preliminarl

measuremints of cariiressiorial wave velocities in the Fyrex-spirel series were

not accurate enotgh to provide comparison irith Kerner's theory for Youn-'s

75 FI-1273-14-12

modulus and bulk modulus of -the composite bodies. Fused silica blocks and

-25 megacycle x-cut quartz crystal transducers prepared with optical polished 1surfaces should be used.

D. Elastic Moduli of Polycrystalline Ceramics 4Multiphase ceramics usually have average thermal expansion coefficients

which lie between the two extreme end members. This occurs because the

internal stresses present in continuous solid-phase bodies prevent the

high-thermal expansion phase from dilating as much as in the isolated free-

particle state. Of course, these same stresses also cause the low expansion

phase to dilate more than in the isolated state. A similar effect occurs in I'simple compression of composite bodies as illustrated by the two-phase model

in Figure 22. The important point is that the internal stresses are present

within and between the individual grains. Hence, it is the elastic properties

of the individual grains which should enter into the internal stress calculations.

Or restated, it is the single crystal thermoelastic constants which d etermine

the composite thermoelastic properties. Aside from the anisotropy considerations,

the thermeelastic properties of available single-phase ceramics bodies differ

from the single-crystal values because of the general inability to producefully dense, pure single-phase bodies. Finite porosity causes each of the

three principal elastic oduli (\/,K,G ) to decrease. Lang4, for example,

found a 70% decrease in Young's modulus for a pure alumina body with 28%

porosity. Because 6f these gross dianges, it becomes important to be able

to predict the porosity dependence. In fact, porosity accounts for mcet of

the variability found between reported values of elastic moduli for single- 4phase ceramic bodies.

However, a comparison of static and dynamic moduli requires consideration

of two additional factors. First, the measurement of elastic moduli under

static conditions gives values for the isothermal moduli while sonic or

ultrasonic measurements give the adiabatic moduli. The difference between

these two moduli is dependent upon the ratio of specific heats C -_- ) and

is generally assumed to be negligible. Mason4• has computed the elastic

compliances of a-quartz under the two conditions and found the largest

76 PI-1273-M-12

Iy

PxO , ij 0X I Z J

a)LINEAR ARRAY OF GRAINS WITH UNIAXIAL LOADING

V* K 1 ý K2 - 4 x

PX= Py A Px Py),0Z*rj A o

hb) PLANE ARRAY OF GRAINS WITH BIAXIAL LOADING

Floura, '2 DEFORMATION OF A TWO)-PHASE 140DEL SUBJECTED TOCOMPRESSIVE LOADING

7? PI-1273-M-12

difference to be less than 2%. In futile,4 9 the differences are less than

0.3%. For isotropic solids, a relation given by Zener 5 0 may be used to

evaluate the difference between the adiabatic Young's modulus ( Vý ,

unrelaxed modulus) and the isothermal or static modulus ( \I )S

7- c I (IV- ) H'is PC,,

where Cv= specific heat at constant volume

Sdensityv"T= absolute temperature

ai linear thermal expansion coefficient

For anisotropic solids, a relation derived by Voigt and reported by Vick

and Hollander 9 may be used

ALk =- (v-2)

where dik oadiabatic elastic compliance - isothermal elasticcompliance

Li, Mk thermal expansion coefficients in the i and * directions

CP specific heat at constant pressure

Using the following values for MgO, Y/-t 3.03 x 1012 dyn 2 T 2880 K

C = 0.20 ca/g/0 C, Y = 135 x 10 7 /°C, and /o = 3.58 g/cm 3, the tyamicmodulus ( ) was computed to be 0.5% greater than the static modulus

( YS ). A cursory review of the range of these physical constants fbr

other single-phase ceramic bodies suggests that the ratio of adiabatic to

isothermal elastic moduli will not differ from unity by more than a few

percent.

At elevated tenperatures the test specimen may deform slightly by !

creep producirn an apparent lowering of the modulus. At sonic frequencies

78 PI-1273-M-12[

and above, the period between cycles is too short to permit stress relaxation

j by creep. Hence, sonic measurements provide an unrelaxed modulus which is

always greater than the static or relaxed modulus. Dynamic elastic moduli

have been used for all computations.

Table XV is a compilation of the elastic properties of fully densev oxide ceramic bodies obtained from space averaging of single-crystal elasticJ51constants and from extrapolation of porous body data to zero porosity.51

E. Theoretical Thermal E xpansion of Multi£ e-Phase Ceramic Bodies

r Turner" considered the thermal expansion of composite bodiesby assuming

J that only isostatic stresses occurred within each phase due to the

dissimilarities in thermoelastic properties. Hence only the modulus of volume

elasticity, or bulk modulus K K ), and the volume coefficient of thermal

expansion (oa,,) were required to define the composite thermal expansion, i.e.,

Scav = ov'M(TV-3)

where V volume fraction of the ith phase

aVj =volume coefficient of expansion of the ith phase

Al = bulk modulus of the ith phase

For isotropic bodies, the linear coefficient of thermal expansion (OG)

is approximately one-third of the volume coefficient (M-v)* so that equation

"(IV-3) remains unchanged when C0 is used. Previous applications of Turner's

V i method223 have resorted to an additional simplifying assumption which

introduces considerable error in mar7 systems, i.e. assuming 2f=Zr,= Zr =Zri.,

then Xi may be replaced by Y because Y--K(/--2v-) where Y Yourg's

modulus (modulus of elasticity), W Poisson's ratio and

0•6 -(v~

* For unrestrained isotropic materials, 1#+4A t 4(I / 41t) -f-(+k3oAt)

for CA t(<f, hence ay---os

79 PI-1273-M-12

M -" 4 (Ic C- ('4 ws ' n o ~ as

aa

C man-LI or 0 C'00 0 0a-

%do V4 'n I I a aus

C, a'. I.. CAC

up I. zr C4 C4 - - - - - .

4a abI~; 0;-_ en 4~ .. a. & a0& . .a ~* .a (4 -

cog an U-

~cU4

to - ac

C, I. i.,I- Cm 0 a

('4

an

C 4J

gn P4 asII

ac 114 w~. bjfl a

ml Ca- a

: z!C3

'ZC- X- IX

8o PI-1273-M-12

It is to be noted that for the entire range of solid materials, Poisson's

ratio is only required to vary from zero to 0.50 (the latter corresponds

to an ideal liquid). Hence a small numerical difference in Poisson's ratio

F may reflect a large difference in solid state properties. The agreement

between values predicted by equation (IV-4) and dilatometer measurements has

Fgenerally been due to an opportune choice of values for the elastic constants.When the gross effects of porosity upon elastic moduli are ignored, the

resulting apparent disparity of elastic moduli data has tended to permit a

selection of those values which gave the best correlation with Turner's method.

An examination of the internal stress model used by Turner reveals thatI shear stresses between different grains and between similar grains with

dissimilar orientations have been neglected. Assuming that shear stresses

.I are negligible is equivalent to stating that at least one phase behaves as

a liquid. Clearly, such a model is not adequate for use with ceramic bodies.

After establishing the necessary conditions for a more realistic

i -moela review of the literature revealed that Kerner6ad

derived expressions for, the thermoelastic properties of composite materials

V based upon such a model. Kerner's model incorporates both the volume dilation

and the grain boundary shear stresses which arise in continuous-solid-phase

composite bodies,* The linear thermal-exparsion coefficient of composite

bodies given by Kerner is

dlII

[ A

* Neither Turner's method nor Kerner's method is applicable to discontinuous-solid-phase bodies. Thus, the presence or occurrence of microcracksdecreases the validity of both approaches.

81 PI-1273-M-12

where Z stands for a summation excluding the index 1

Go iG, I I _I

(1v-6),

and-,Z Ki V4

= T7 V 1. (IV-7)

th--

The shear modulus (Go) and the bulk modulus (KO ) refer to the moduli of

the composite body, Kerner's method requires computation of these two moduli iibefore the thermal expansion of the composite body may be determined. Since

most applications of thermal expansion data also require data on elastic

moduli, these intermediate steps serve dual purposes (Figure 23). Since it

is generally more convenient to prepare bodies on a weight-fraction basis,

the volume fractions may be replaced by where P•- is the weight fraction,Qj

of the ith phase and/oj is the density of the ith phase. Note that phase-

density dces not effect either the internal stresses or the composite thermal

expansion except by defining the fractional composition. Obviously, when

the densities of each phase are equal (e.g., in the MgO-MgO.A 2 03 system),,

the weight fractions become equal to the volume fractions. If the bulk

moduli of each phase were equal, then equations (IV-3) and (IV-5) would

reduce to one equation corresponding to a simple rule of mixtures, i.e.,,

2c-i Vi pC_1 (IV-8)

* Index 1 was used for the annihilated suspending fluid.

I8 2 -273•1

V

20.21 .... ... .:: .L .t ... ..-- ---- ----------. . I.... .......... .... .. .......---

0.2

Hgo NO 90 A' 203.... 22... .. ....... ...... ....

2T ~----- ----.

. .. .. .. .0.2 -- -- .. ......... ....... ------. "I S I "

0.13-S: " " L f

A

16.21 1.4Y14g0 14g0• Ala03

....... ..... ..... I.. ......

.. __... .......... ....... ............ - ---- 31......8

*925C

I aI 0000

lB 1o'1'

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F. Comparisons Between Theoretical and Enpirical Thermal Expansion

in Two-Phase Materials

1. Aluminum-Silica System

Figure 24 illustrates the differences between predictions by

equations (IV-3) and (IV-5) for the aluminum-fused silica system. Measurements

of thermal expansion for two different compositions as reported by Kingery53

are included. The difference between theory and experiment may be due to

stress induced microcracks. Microcracks arise when internal shear stresses

exceed the local shear strength and hence the occurrence of microcracks wuld

cause the empirical data to lie between the values predicted by equations (UV-3)

and (IV-5) as was fouxd for the aluminum-silica system.

Kingery used the oversimplified version of Turner's method (equation IV-h)

to compute thermal expansion coefficients for this system. Since Young's moduli

are equal for these two materials, the assumption of equal Poisson's ratio is

equivalent to assuming equal bulk moduli and equation (IV-4) reduces to a simple

rule of mixtures. The simple rule of mixtures, is valid for a compacted

mixture of pow-ders in which no binder has been added, no sintering accomplished

and hence no internal stresses developed. This model does not seem representative

of the aluminum-silica system unless all aluminum-silica grain boundaries have

fractured. Table XVI summarizes the differences between the several approaches

to theoretical prediction of thermal expansion in composite bodies.

2. Magnesium Oxide-Spinel System

In the magnesium oxide-spinel (MgOAl 2 03 ) system, the difference

between predictions by equations (IV-3) and (IV-5) is small (Figure 25). In

fact, the range of reported values for the linear thermal-expansion coefficient

for the end menbers alone constitutes the greatest uncertainty to the predicted

values for the two-phase ceramic body. Table XVII lists the values used for

this analysis.

The thermal expansion coefficient of several other compositions in

the 14gO-spinel system were measured with a quartz tube dilatometer. However, jthe reproducibility of tU- measurements was unsatisfactory (+ 10%). These

data were omitted in Figure 25 since a precision of + 1% would be required to

84 PI-1273-M-12

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Figure 24 THERMAL EXPANSION IN THE ALUMINUM -1 SLI CA SYSTEM4

85 'PI-1273-M-12.

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Figure 25 THERM4AL EXPANSION IN THE MAGNESIA - SPINEL SYSTEM4

87 ?I-1273-M-12

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validate the theory. Even with the several obvious improvements in the

j dilatometer, it is clear that it will be difficult to obtain adequate

precision to establish the relative validity of equations (IV-3) and (IV-5)

in the MgO-spinel system.

The concept which has been developed suggests two perturbations to

r the thermal expansion coefficient: 1) The thermal expansion of a single-

phase polycrystalline body (having the same chemical ccmposition as the

single crystal) will depend upon porosity if the single crystal is anisotropic

I in thermal expansion and elastic properties, and the pore phase is continuous.*

This dependence should arise from the charges in the internal stress pattern

. surrounding each grain as the continuous-pore phase increases. The investigation

of porous alumina by Coble and Kingery" was made. with discontinuous pore-

B phase bodies in which the expected charge in internal stress would be small.

"They concluded that porosity had no effect upon the thermal expansion in

[H alumina. 2) The thermal expansion of multiple-phase bodies will depend upon

Ii porosity if the elastic moduli and the expansion coefficient are dissimilaramong the phases.

These perturbations would not be present in-powder x-ray measurements.

Dilatometer equipment available for this study did not provide sufficient

accuracy to conclusively validate these predictions. For example, several

100% spinel bars measured during this program produced thermal expansion

'1 curves which appeared acceptable but which were not reproducible. Theporosity-dependent thermal expansion of this elastically amisotropic phase

could account for at least part of the data scatter, if large scale stresses

induced in processing result in local stresses between grains due to elastic

anisotropy (see Section IV-G).

3. Pyrex-Spinel System

H Attempts to produce bars in the magnesia-spinel system with porositiesof five percent or less by cold-pressing and sintering were not successful.

* The ratios of and based on data given in Table XV provideSGQ

a measure of the elastic anisotropy. Isotropic crystals would have a

uni-ty ratio for each modulus.

89 PI-1273-114,12

•*1

In order to expedite development of hot-pressing equipment to produce high-

density bodies at moderate temperatures, a glass-ceramic system was chosen.

Pyrex and spinel were selected as representative of two-phase systems having

grossly dissimilar thermoelastic properties.*I

Glass cane was crushed, ball-milled, sieve-separated, and mixed

with minus 325-mesh** commercial grade spinel. These mixtures were hot-pressed

in an induction-heated graphite mold in the shape of 4-inch diameter disks

about 3/8 inches thick. Forming data for the three disks selected are listed

in Table XVIII.

Tablea XVIIIFORMING CONDITIONS FOR HOT-PRESSED PYREX-SPINEL DISKS

FOR4NINO TIME ATCOMPOSI TION PRESSURE FORMING MAXIMUM - j

SAMPLE NO. BY WEIWOT (GAUGE READING) TEMPE"ATURE TEMPERATURE0S-4 25% PYREX 6000 PSI 10500C 60 MINGS-5 50% PYREX 6000 PI 3090 0C o0 MINL 3-6 75% PYREX 1000 PSI 950 0C 30 MIN J

Fl

A test bar 0.2 x 0.5 x 4 inches was cut from the central portion of each

disk. Densities of the rectargular prism specimens were measured by a

displaced-mercury weight-change method (Table XIII).

--

* Pyrex Code 7740, Corning Glass Works.¶ * Particle size less than 44 microns.

90 PI-1273-M-12

I

Thermal expansion of the Pyrex-spinel series of bars was measured

1. with a quartz-tube dilatometer over the temperature range from 20° to

*300 °C (Figure 26). These specimens had been subjected to no previous thermal

L cycling except for the initial cooling after hot-pressing. No further

annealing was attempted in order to avoid the possibility of increasing the

r number of microcracks. Dilatometer readings were taken while the temperatureL0increased at a nearly steady rate of 3°C per minute. The thermal expansion

was found to be a nearly linear function of temperature over the temperature

range of 1000°to 3000 C. The data below 1000 C contained apparatus-originated

errors and was excluded from further consideration. Contraction measurements

Eduring the cooling cycle produced a hysteresis effect indicative of non-

equilibrium stress relaxation in the glassy phase. It is recognized that

future measurements on glass-containing bodies will necessitate discrete

measurements at stabilized specimen temperatures even when the test temperatures

are well below the glass strain point. The elastic moduli for the Pyrex-spinelFit system as computed from equations (IV-6) and (IV-7) are shown in Figure 27.

Figure 28 is a comparison of the Pyrex-spinel thermal expansion

measurements with the predictions obtained by equations (IV-3) and (IV-5).

The superiority of Kerner's method is clearly demonstrated in this system.

Two end points for spine, are shown. The upper point (Cy, 88 x 107/oC)corresponds to the thermal expansion of powder specimens measured by x-ray

Sdiffraction techniques. The lower point (CY - 81 x 10-7/oC) representsdilatometer measurements on polycrystalline bars. The difference may be due

to elastic anisotropy in the spinel crystal (see Section IV-G). The

appropriate value to be used in prediction of thermal expansion in Pyrex-spinel mixtures is determined by consideration of the origin of the internal

stresses. These stresses arise within individual grains which are surrounded,on the average, with grains having the elastic properties characteristic

[ of the mixture. Hence, the thermal expansion of spinel powder (x-ray technique),

not of the bulk ceramic, should be used in the computations. Improvedfl correlation would be expected if Kerner's model was extended to include elastic

anisotropy. Elastic anisotropy tends to lower the thermal expansion in

If porous bulk specimens if large scale stresses induced by the fabrication

91 PI--2 73 -M-12

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TEMPERATURE -C

Figure 26 DILATOMETER THERMAL EXPANSION MEASUREMENTS FOR PYREX-SPINEL BARS

92 P2-1273-1--12

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93 PI-1273-M--12

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94 PI-1273-Y.-12 -

[Iprocess, are present. This change would be dependent upon the relative

amount of spinal present in the Pyrex-spinel system. It is expected that

t1he change in tlermal expansion predictions due to the addition of

. anisotropy terms would be similar to the differences between the two curveslabeled "Kerner's Method" in Figure 27. As a reasonable approimation (and

one which greatly simplifies the computations), substitution of the thermal

* expansion coefficient for bulk specimens (dilatometer data) could be used in

L *. equation (IV-5) to obtain the theoretical thermal expansion in a two-phase

system, one of which is elastically anisotropic. This simplification would

be much less valid if more than one phase were elastically anisotropic.

It is interesting to note that equation (IV-5) predicts a thermal

expansion in the region of 15% Pyrex-B5% spinel which is slightly higher

K (a- - 88.5 x 1l'7/0C) than for pure spinel ( a-- 88.0 x 107/o0) whiile

- equation (IV-3) predicts a continuously decreasing thermal-expansion

S I coefficient with increasing Pyrex content. This difference serves to

emphasize the importance of shear stresses to the internal stress model and

to computations intended to define composition for purposes of matchingI thermal exparnion coefficients.

G. Thermoelastic Anisotropy

In single-phase polycrystalline bodies, internal stresses may arise

from either thermal anisotropy, elastic anisotropy, structural anisotropy

or combinations thereof. For a cubic crystallite imbedded in a rigid,

Sisotropic body with a matching thermal expansion coefficient, and if norestraints are present at the sintering temperature and no temperature

gradients are rresent during forming, then the internal stresses on the

Ti crystallite reduce to zero. However, if restraints are present during

forming due to temperature, pressure or compaction gradients, then stresses

Smay arise in a polycrystallire medium even with cubic symmetry die to

anisotropy of the elastic restraints.

9 I S i95 Pl-1273-)14.2

For a spinel crystal of the type (MgO'3.5 AI2 03 ) reported by Verma,5

the anisotropy ratio is 2.1 to 1, indicating that elastic restraints in

a polycrystalline body may be ariisotropic to a high degree. In a medium of

this type, the anisotropy superimposes a spatial modulation upon the internal

stresses. Nonromogeneous polycrystalline bodies will generally possess regions

of significant internal stress levels ( e. g., cold-pressed and sintered bodies

in which uniaxial pressure was employed are frequently found to have large-

scale laminar-type cracks). When these internal stresses are significant,

then the effects of elastic arisotropy will be significant even though the 11thermal expansion coefficient is isotropic. Hence, for precision dilatometer

measurements upon polycrystalline bars, it is not sufficient to form a

specimen in a manner which results in inhomogenities.

In general, the spatial stress function for a crystallite within a

pol]ycrystalline body will contain terms involving the direction cosines

up to the fourth power. The space averaged stress obtained by integration

of the cosineh and cosine2 functions does not reduce to zero for elastically

anisotropic crystals. Hence the thermal expansion of single-phase

nonhomogeneous polycrystallime bodies may be expected to differ from the

average thermal expansion for the single crystal. Comparison of x-ray and

dilatometer thermal expansion data for MgOAI 2 03 spinel indicates that

differences are often founi (Table XIX).

H. Elevated Temperature Effects

Each of the internal stress theories requires the composite body to

consist of a continuous solid-phase. The presence of microcracks, from any

cause, seriously impairs the validity of these several predictions. Most

-1

* Elastic anisotropy in a cubic crystal is proportioml to the ratio ofelastic compliance constants ) given by the expression'

,2 (', - S,2)2 ý 2

(Iv-9)

where a unit7 ratio indicates an isotropic body.

96 PI-1273-M-12

F70 :Table XIX

_THERMAL EXPANSION OF MgO.A1 203 SPINEL

AVERAGE LINEAR THERMAL EXPANI10N COEFFICIENT X 107 /0C

POWDER SAMPLE, X-RAY POLYCRYSTALLINE BAR, DILATOMETER

DIFFRACTION TECHNIQUE COARSE GRAIN FINE GRAIN OTHER SOURCES

SEALSSTUTZMANBDEALS ET AL•4

ET AL ZIMMERMAN WI TTEMORE ET ALTEMPERATURE RANGE (1957)26 (1966) 65 ET AL 66 THIS 3JUDY (1950)67

25P - 3000 C 88.2 83 52 76 67

250 - 600oC 87.8 AS 75 9o 73

250 - 9000 C - 89 84. 90 80

250- IO00OC - - - 88 85

i 250 - 12D00 C 88.3 90.5 91 --

"250 - Iqoo0C - -. 86

I j-i250_ 1500 0C - - 96 - -

microcracks arise when the internal stresses exceed the local strength of

Sthe body during cooling from forming temperatures. Hence, prediction of the

elevated-temperature thermal expansion of composite bodies (uhere the

"~ T occurrence of microcracks would be less) should provide a better opportunity

for experimental validation. In order to apply equation (IV-5) to elevated

1 17 temperature predictions, the temperature dependence of both the elastic moduli

and the thermal expansion for each phase must be known. Wachtzan, et al5

have shown that Young's mocdlus for several oxide ceramics (e.g., sapphire,

alumina, magnesia, and thoria) follow an exponential temperature dependence

of the form

Y= "-"o -are T1 I-0

9I97P-17-M1

.1

where T absolute temperature in K

85,To are empirical constants .1Young's modulus at 0K

Based upon Wachtman's data, Young's modulus for alumina ceramics would

decrease by 10% from 25 0 C to l00°0C. Similar data for other ceramics show

a more rapid decrease with increasing temperature. Smiley, et al have

compiled data on the elastic moduli versus temperature for spinel, mullite

and several other ceramic phases.

A few observations related to the temperature-dependent effects can be

made. Young's modulus for spinel decreases with temperature more rapidly

than for MgO. Assuming both the shear modulus and the bulk modulus decrease

in a similar manner, the elevated-temperature thermal-expansion coefficient

for the spinel-MgO system will be everywhere higher than at room temperature.

Both increasing agO (Table XVII) and decreasirg Gspiel would contribute to

this change. An increase of 10 percent was obtaired by high-tenperature

dilatometer measurements for the 50% MgO, 50% MgO'Al 2 specimen (Table XVII).

The elastic moduli of a few materials increase with temperature below

8000C. This is characteristic of high silica glasses, for example. In the

aluminum-silica and Pyrex-spinel systems, the elastic moduli changes may be

expected to produce significant differences between the room temperature and

elevated-temperature thermal expansion.

I. Conclusions

Each of the several methods used to predict thermal expansion of

composite bodies requires data on the elastic properties of the end members.

It has been shown, that the elastic moduli which should be used for ceramics

corresponds to the values for the fully-dense polycrystalline body rather

than the lower values associated with the more readily available porous

ceramics. The greater validity of Kerner's method over Turrer's method for

prediction of thermal expansion for multiple-phase ceramics has been clearly

shown for three different two-phase systems. The completeness of Kerner'smodel for internal stresses is expectcd to reveal the superiority of Kerner's

method for most com-•osite bodies (which remain continuous solids).

98 PI-1273-M-12

i *1

In general, 'he thermal expansion coefficients of two-phase bodies will

lie between the end members for all ranges of composition with one exception.

When the thermoelastic properties are grossly dissimilar, the resulting

thermal expansion may lie slightly outside the bracketed range for compositions

which are predominantly one phase (e.g., 15% Pyrex- 8 5% spinel).

I F Preliminary calculations for polycrystalline specimens composed of

elastically anisotropic crystals indicate that the presence of structural

inhomogenities will introduce internal stresses which will tend to lower

the thermal expansion coefficient. Under these conditions, the bulk thermal

expansion for polycrystalline bodies will differ from the single-crystal orjF the x-ray-measured thermal expansion for powder specimers. These local

stresses may arise even though the single-crystal thermal expansion coefficient

[1 is isotropic, provided that large scale stresses induced during fabrication

are present.

The elastic properties (e. g., 9,G/, zr ) of composite bodies,

including multiple-phase ceramics, may be predicted (with small error for

bodies which remain continuous solids) by a method developed by Kerner. This

Ii study appears to be the first application of Kerner's method to ceramic

bodies.

99 PI1-273-44-12

1F.

1 .

-FV. CONCLUSIONS A.ND RECOMIENDATIONS

A. Conclusions

1. The relationship between the openness ratio and the thermal

expansion coefficient for a number of oxides and silicates, was investigated.

In these groups of materials, phases having high values of the openness ratio

usually have low thermal expansion coefficients.

2. Several phases having high values of the openness ratio were

synthesized and the thermal expansion coefficients were measured. The

"following phases have especially low thermal expansion coefficients:

SrO-Cu04SiO2 Tetragonal sheet silicates[j 2H ~ BaO" CuO'4Si02

U1 ~2O07 (cubic)

3. Cubic UP 0 has unique thermal expansion properties. The crystal

H expands with increasing temperature up to about 400C. Above this temperature

the crystal contracts, returning to its room temperature dimensions at about

1000- 0 C. Since the peak positions and intensities of the x-ray diffraction

patterns are very nearly the same at room temperature and at lOOO°C, these

expansion properties are not the result of a major structural change.

4. Little variation of the thermal expansion coefficient with openness

ratio is observed within homologous series. These observations confirm

Megaw's(rule A

5. The thermal expansion coefficients of.the alkali halides were

calculated based upon the charge in lattice spacing with tempecature at

minimum free energy using the techniques. of statistical mechanics. These

calculations require knowledge of tle atomic structure, ionic charge and

atomic spacings at one temperature. The repulsion exponent was assured to

101 PI-1273-11-12

have one value for the entire group. Comparison of the results of these

calculations with experimental values gave encouraging results. Improvements

can probably be made by taking into account the polarization of the ions and

variation of the repulsion exponent. 16. Mhe thermal expansion anisotropy of crystals can be varied by

addition of appropriate solid solution atoms. Addition of vanadium atoms to

replace titanium in rutile (TiO2 ) results in a marked decrease in the

thermal expansion anisotropy. At an addition of about 10% vanadium the

thermal expansion anisotropy reverses so that the a axis becomes the

direction of highest thermal expansion coefficient. The effect of varadium

on the thermal expansion anisotropy is not restricted to rutile since a

similar reduction in thermal expansion anisotropy was observed when vanadium

was added to cassiterite (Sn0 %hich also has the rutile structure.

227. The thermal expan Ision anisotropy of Cr20 3 is opposite in sign from

that of corundum (A1203). Since corundum and Cr2 03 form a continuous solid

solution series, an intermediate composition must exist for which the thermal

expansion anisotropy is zero.

8. Kerner's method was used to predict the thermal expansion

coefficients of two-phase ceramic bodies. The predictions are satisfactory

and the superiority of Kerner's method, which accounts for shear stresses

was demonstrated.

9. In cases in which the thermoelastic properties of the pure

phases are grossly dissinilar, the th..al expansion coefficients of some

intermediate ccmpositions may be slightly outside the rarte between the

end members. ,

1

102 PI-12 73-H-i273-1 -!

I

APPE OIX A

"Openness" Ratios of Some Ceramic Phases

In the followirg list the ccmpounds are mainly covalent in character

of chemical bindirng. Therefore, the calculation of these openness ratios

is based upon the assumption of 100% covalent binding. This computation

is s impler than those previously made14 since the radius of each atom does

Snot depend upon the other atoms present.

The information in Appendix A is printed directly by the IBM computer.

fThis results in chemical formulae written in an unconventional form. Lower

* case letters were not available. Hence, for example, the abbreviation for

- aluminum is written AL rather than the customary Al. This will not result

in confusion if one notes that the abbreviation for each single element is

always enclosed between two numbers. As an illustration in line 1, one

finds the formula IALIB2. Inasmuch as there is no number between A and L,

these letters designate a single element, namely aluminum, normally written

Al. The number 1 which heads this formula indicates there is one AID2 group

in the formula. The numbers after each element have the meaning of

7 conventional subscripts. In some cases rather unusual groups, such as

INlNl in line 36, appear. The formula for the compound is LiGaN2 . This

N2 is written ININ1 as a matter of convenience in programming. The number

of significant figures should be two numbers in most cases. A larger number

are given mainly for convenience.

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APPENDIX B

THE EFFECT OF POROSITY ON THE ET.ASTIC PR)PERTIES OF CERA.IIOS

The sintering of pure ceramic specimens generally results in densities

rlower than theoretical unless the p~article size,, forming pressure., sintering4- eprtr cce itrn environment., and other factors affecting final.

density are carefully c ontrolled and optimized for each new ceramic bodyb.I:Lowering of density due to the presence of pores. also lowers the elasticF.mioduli values. In view of the difficulty of obtaining suitable ceramic

F1specimens having maxd~im= density, it appeared desirable to investigatejmethods of predicting the elastic properties of bodies having theoretical

density from measurements made on specimens with finite porosity.F1Mackenzie 59derived expressions for the bulk modulus C )and the

vshear modulus (G )for solids containi.g spherical isolated pores:

vi4K 4G, (1QP)

G C 6~(- ZP)

4CGov ~whaere +G) (Table B-I)

P volume fraction of pores )01 -

ii Tabl e 5- 1DIMENSIONLESS ELASTIC CONSTANT OF OXIDE CERAMICS

(USED IN POROUS BODY COMPUTATIONS)

CERAMIC PHASE Z = 6 ~ : :'CORUNDUM 1.97MAGNESIUM OXIDE 2.03SPINEL. 2.014FUSED SILICA 2.02

4-QUARTZ 1.98

B-1 PI-1273-M-12

Table B-2SHEAR NODULUS DEPENDENCE ON POROSITY

BASED ON MACKENZIE'S RELATION 4POROSITY, e o/% Z 0

SZ, -I.93 Z "1.98 Z -'2.05

0 .1.000 1.000 1.000 45% 0.0o4 0.901 0.699

10 0.807 0.802 0.79720 0.614 0.604 0.591130* 0.423 O.40 0.391 01

Mackenzie's model consisted of a spherical pore of radius (AO) surrounded LIby a spherical shell of solid with radius ( wr ) hich in turn as surrounded

by an equivalent homogenious material havirg the same properties as the pore

and shell combined. His solutions were obtained by expanding the function

(p-cia in a power series. However, the higher power terms (i.e. those

above the third power) were neglected in his solution. Coble and Kingery

suggested the next term in Mackenzie's shear modulus expression will be of

the form A? 2 where the constant (A') may be evaluated by setting G 0

at 1P lthus:

G G. (1-2zP*AS*)

A -Z-1

and G Go4(I-ZP (Z-1) Pj I

B-2 PI-1273-M-12

-A

,n m I I I n

ATable B-I lists the values of Z computed for several different ceramics.

I Since Z is approximately two for many ceramics, it can be seen that this

term increases the predicted value of shear modulus, slightly, for porous

I materials over that obtained from Mackenzie's original expression as shown

in Figure B-1.

f Coble and Kingery54 found reasonably good agreement between predictions

using the latter expression and measurements made on specially prepared

J low-density alumina bodies as shown in Figure B-I. However, recent data by

Lang47 does not fit either expression. In order to investigate further the

effects of Mackenzie's approximate solution, an additional term in the form

'I was added to give

Ii G."Go(f-ZP--AP 2 -BP'3)

let G 0 at P 1

"•1and "0 at P 1c/P

then A=2Z--3

15 =Go -Zp,(?Z-,3)P 2+(2-Z)P

The effect of the BP 3 term was small and the fit to Lang's data was

not improved. Clearly, the finite porosity in Lang's specimens was not of

the discontinuous-pore-phase considered by Mackenzie's model. In fact,

with the exception of ceramic bodies purposely made low in density by foaming

I or "burn-out binderit techniques, normal pressing and sintering will not

produce isolated spherical pores except in the final stages of sintering

where the porosity will be under 5%. Mackenzie's relation involved only the

elastic constants of the solid material and the bulk density of the poroussolid. Hence it could not distinguish between continuous solid-phaseoand

,- continuous pore-phase materials.

B-3 PI-!273-M-12

i .14 .1 *l . . 4w . .,

.. GR • ~ ~~~.'......................... . '.... • U A O

23o ------- LUCALOX (G =2.3 1068, _/3 3.98) ......... . . . . . . .o - ----- . . . . . . .. . . . . . ...

4..

""1 ........ .. .. t

.I..

.I ....

. . .19' .. .. .... .. .. ....... " : .. . . .. .: . .. .2 1 ..... ... .J ........................ ,... . . .

"11 ~~~~ ~ ~ ~ ..k: t • 1 ' : , ' : : = , ...... ,.. .. ..... . -'% , -,. . . . . . . ... . . . . .. ' ...

S. ........ ... . ......1 ...... .... ..... .;.1- ...... 4 ...... .- .. . .

15 ....... .. ... .. .. .. .. .. .. .. '...'".. . ..... . .. . .. . .. . .

. . . . . I. . .. ...,, . , ."----- MODIFIED --.. ...

..... .......... - .

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.... --- . .......... 4 .......

..~ . , . , 5t-~ J--9 ... . . .......- -....... - .. ..

------ - ----- . .... t ----. 1. 4.... .] ..x'-MDFE SAKXII METHOD

',.. .... .4. - -- 0.

------ ----- -------......... ---ETHOD - ----

.. .... . . iA L.'

~~~~~~~~~~- --.. .. -- -- .. ..... . ........... .. . .i~

"•.1 .--- �.--::z - --.--- j

COBL AK GEý4 DIC4IUU POEPHS

5-- ...... . .......... .. . .. . ..... - - - - -I

............. .

....... .... ,""...... ...... . ...... ...... ...... + ...... ......

"1..... ..... 5 ..... \ \f •

0. 5 0 D.ENSIT I ., , , ,-----NT4TN....F 8.7 1 S A

B 1 12.,:, ,

"• ::41. -'---i ......... - E!C, .E ,.........................I.i I.4.. 1 o

9EST I NH/ENI4TR

. 1 f--*+._PERCENT DISCOTINUOS ORE ......Fiue.1SERHOUU FPRU LUIA.1 .

f B.E~-l•P-23/I] 6

I

Continuous pore-phase materials could be represented by a simple cubic

if structure of equal-size spherical grains in which an uncompressed model would

have a porosity of 48%. Using G 0 at P = .48 for the boundary conditions

J on Cobles' extension of Mackenzie's relation gives

A ---J.55- •,O9Z

Ian G=Go f-ZP-(4.35-2.0.ZPQ

It can be seen that the predicted shear modulus is lowered by these boundary

.conditions and approaches Lang's data for the continuous pore-phase boundary

conditions. Therefore, if the constant A were treated as an empirical

constant for continuous pore-phase bcdies, Mackenziels relation could be

made to fit the experimental data. In this case, the value of A would

be dependent upon the porosity of the unsintered body which is in turn

dependent upon the particle size, the particle size distribution, the

particle shape, the forming pressure, the percent binder and perhaps other

experimental factors. Another approach to the porous body problem was made

by extension of relations derived by Kerner for elastic moduli of composite

bodies. Kerner derived an expression for the shear modulus of composite

materials (G 0 ) contained within a "suspending fluid" (subscript 1) as

I 17Go GG~F-I/ (7-5z';G, -,Y-f,-Oi,i)G1 I5Y-v-,)

V,_Z (7-_v,1r)_G',7V "i-/Ovrf)Gi 15t V

By considering a porous body as composed of a porous phase (subscript 2)

and a solid phase (subscripts e :* 2), Kerner's equation may be uZed to

"derive an expression for porous solids. For a porous sirgle-phase solid

B-5 PI-1273-M-12

( G2 0) and the shear modulus becomes:

Vf

(7 5 () -

For comparison with Mackenzie's expression. this equation may be written

in terms of the bulk modulus C KO) and shear modulus (G 0 ) of the fuly-

dense body. Collecting terms and substitution of

/013k2,- 4GoZ15 (9K÷6G0 /

gives 2G ýG 0 [_I j:j

Figure B-1 illustrates the dependence of shear modulus upon porosity

in alumina ceramics according to the two approaches. The later expression

is labeled "IModifi. Kerner's Method" althou6-h he did not imply applicability

of his method to the porous body case.

Following a similar line of reasoning, the bulk modulus for porous

single-phase materials as derived from Kerner's expression becomes:

- -oKo ('P)4 Go -,l'oP

Young's modulus for a porous single-phase materials was derived from

the expressions for bulk modulus and shear modulus and their isotropic

interrelationship:

9KG '2

B-6 PI-1273--M-12

I

o,1

Young's modulus for porous media based upon Kerner's expressions becomes:I

Go G(-P)• "Go ZP P

- Based upon Mackenzie's expressions, Young's modulus becomes:

Go (1-P) ZP)0'-~ 4/ yo P2.

T A comparison of these two expressions for the case of porous alumina is

shown in Figure B-2. The deviation of Mackenzie's relation from the

empirical data of Coble and Kingery4 above 30 percent porosity is probably,

S due the higher order terms which Mackenzie neglected daring his series

expansion. As Mackenzie noted, his relations are valid only for small porosity.

F For the application of Kerner's equations, the pore phase was consideredas a simple mixture within the solid phase without regard to pore geometry.

It might have been expected that such an approach would not correlate wellwith empirical data indicated in Figures B-1 and B-2.

The successful prediction of elastic properties of fully-dense ceramic

tI Ubodies from single crystal data described in Appendix C, postponed attempts

to improve the porous-body model by including terms dependent upon poregeometry. Present equations are reasonably valid for porosities of five

percent or less porosity.

L Young's modulus and shear modulus for porous magnesia, spinel, mullite,and thoria'are illustrated in Figure B-3 through B-8. For the examples of

alumina, magnesia and spinel, values of Young's modulus Y&. and Y and

shear modulus GC, and G,, for the fully-dense ceramic body as derived fromsingle crystal elastic constants are plotted along the ordinate. In each

* Iinstance, the extrapolated porous-ceramic value lies between the tm values

- computed from single crystal constants. This denonstrates a good correlation

between two independent methods for predicting elastic properties of fully-

dense single-phase ceramic bodies. The use of single crystal elastic

constants is described in Appendix C.

B-7 PI-1273 -M-12

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.. . .. . . . . . . . . . .. . .. . . .

:L t

........... ...... .... ... .. ......

.......... ... ...

9 32

- - - - - - - - - - -. . . . . . . ..r - - -

C2 24 -- .. . . . . . . .. . . . . .r.... . . .... . . - - - -

--------------------- ...................... ... .................... FIEER a'

0METHOD

16 C .A -- - - - - --; .a.. .- - - - - - - . . . .

ci) S * a7

0= LA14 0r ---------- --- -

COL &aiRGR *MACKEZIE'SMETHO---- a--- ---- ---- a- -----a------

COR AD9

a a4 * ,A,0FIE

0 a' . . . J .

a. 3 . . . .

PT12 I- a1-12

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v

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4 0 -- - - - - - - .. . . . . . ... . . . . . . . . . . a-- - -- - - - - - -

. . . . . . . . ... . . . . . . . . . . . . . . . . . . . . .. . . - - -j- - - - . . .

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2. 1. . I. i

38DIFIE..: S.. . . - - --

------ ------ I---- ---- ----K R E '

32 + SMETHOD

* l

------- ---- ---2 ---- .... ------ ----

-------- ------- --------------- ----.... ---

-- -- -- - - - - - - --. . . . . .20 ---

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i ---- -----

---- .... .. ---- --- --12 ------- LAN -

+ .....----- 1---- ---- ---

DESTYI GASCNINE

, ... 6 01I2 53

PECN POROSIT

Fiur YON' MODLU OF POOU MAGE 1I

- C I27 !1 2

2 0

.. . .. . .. . .. . .. . . ... .

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16...............

---- .....3 ..... .. . . .. ... . .. .. . . ... . .. .. . ... .... .. .... .

........ t ......

............ ....

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10.....j:.......

........ ..... MACKENZIE'SMETHOD

8 -------- I-4....------ ....... ..... .. .. ---- . .... ......

................. ... .. ............... +

--------- --- ---

68 ----- ------

S--- -............. .......j ---- 1---.---1-

0 VYAN 6 (SAiC MOULS 25+.. 30t~....

PECN P.OR.......

B- SHAR 1DLU FPRUSMGEI

B-1 PI1 73-14-1

II

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...... ..........

to f

2Kt ftf

22~~~~~~~~~~~~ ~ ~ ~ ~ ~ ----4----- -- --- ----... --- ....------ - ...... M D F E

C2 f4 . fERNER03

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----- --------- .... ---- .... ---- ...---- ---- ---- t---

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3.6 3.. 1 3 0 . . .

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.- ...... . . . . .. . I +. - -•

-- : o , - . a *

I1 ' ----- ---- -- --- --- ETHOD - ~ -I e I -. ,i

---- ---- .... -------- .... .....-------------- ----

... . .... ............ ..-

------ -- - -. .. .. .... ---- -

--- --- --- -- ... . ............ .. I. .. . . _.. . . .. • . . . . .. ... . .

0 L

- - - .'

31.6 34 3.2 3.0 2.8 2.6 2'

DENSITY IN GRAMS /CEN TI ETER 3

0 5 to .15 20 25 30

"I r~ O PIES [ I

PERCENT POROSITY

Figure B-6 SHEAR MODULUS OF POROUS SPINEL (14g0 A1203)

3-12 ?-23M1

............ ......... +..... +........ ... .• .......... + .... , .... , • :-• ÷ ....

I

* - . w - iJj a. a

a , a ,

32 ,* * i a o

-- 77 28~ . ..... . .. .

.. . ... ... ------ t i

-~ ~~ ---- - ---4 ..... .. .. . .....-1 ....... . ....... •.. .

S* * a a ,

CL~~~~~~~~~ ~ ~ ~ ~ ~ ~ a -~ -4 -.- -------------I -----....... 3to-,

-------- 2- --------------. . . .I

a .. ... a...... - . .. . .1......1

-, ,a i ,ai

.. - - a 12 a4 a-

8 - - ------

I -- -- -.-----... . -------- '12 ------ . .: -------------- -----------

1'"~- ----------- *

: *: : : " 0 .

0 0

S, . .. . .. .

T - :3:. . ... 8

0 a 1 15 0 5 0 15

Fgr B- ELSI MOUL OF POOSMIT. E . ,,

I, .....--------.....--.... [a... , 0 .... -n -:-• XM I,, * ,- , .

-r '6 .... 5 ... .. .. . .. ,, HO aa l * a . a a * ,* a a * o*,

a.a* a a* a a a oS. .. . a..a , -a , ,

S• i a i

.. " a a * * *- o a o

"t I a

4TqO • a* i* a a a I a*.. . * a a S I 5 a a a1-=--.............. a a a'• .... a .... o................................. . . ..1:/' '£ * a a * a a i aai

r + --...Ii IIo 6 __ _ _ _ __ _ __. . ...... a.I _ _ _ __ _ __ _ _ _

77 a .. .

PERCEN POROSITY-1

=[ nQ ~I gure .8- ELSI MoDL OF POROUS Hi'-J"....LL .. IT "...-- ...

1.

ac aa *- a

UAa C

* i .I I

.. a..... ..... .. .... .. a..

C-2

la01

*A c o a ll i

--------. -a - -

-- + - --- - a

* 5 0

-- --- - --- -4 .. ... .. . .- - -

e--- --- - -

----- -- -- -- - - -- - -- -- - - - - -- - -- - -- --

S O Is sO c

------"r .. .... -, ...... • ------ - " ------ . .. .. -- -- -- -- --- --- --- -- -- -- ------- --.. . .--"-.. ..--... .

,S_ ,/ , ,---7---- ---- ----------------...... ..:....... •......,, ,

I sd 91 9 #4

B0 1

: : , . C :

' * a S S~'lO' , V e

-,b1 -j . .. r i eLI.. I I ,, : : *i.l i "

!o- __;--.1

1..... ......... -. . . . . . . . a j; ,

: 1 I I a -a a

-- ' I'a S -,

iS~ .02 9 'mnao Mw39 i

:-l

14.

T- Kerner's formula may also be extended to the case of porous multiple-

I phase ceramic bodies. Fcr example, the bulk modulus for a porous t•-phase

'ceramic wuld be given by:

S, 4 K2 P

,<, (,(3K,-+4 G) A2(J3K 2 "JG,

P,.

'where Vj volume fraction of pores porosity

j subscript 1 refers to the basic matrix into uhich

phase 2 material is added.

j Application of this expression to the aluminum-silica system was illustrated

at the top of Figure 24. In this system, an increase in porosity for arn

I ~composition results in a decrease of the composite bulk modulus as might be

expected. However, the mixture of phases with different compressibilities

pnecessarily introduces shear stresses between the particles durirg compres-

1 - sion of the composite if the mixture has been sintered or otherwise bonded

together. In some systems the presence of finite porosity will not cause

a monotonic decrease in the composite bulk modulus because of the independent

effects of porosity upon the large-scale shear modulus. The Pyrex-spinel

I system, for example, has a compositional range in which small porosity

produces a slight increase in the bulk modulus (Figure B-9). This rather

surprising result server to illustrate the importance of includirg shear

stresses in any model designed to describe the thermoelastic properties of

composite media. It also provides further evidence that Kerner's method

for predicting thermal expansion of composite media should be superior to

Turner's method in which shear stresses were neglected.

Alternatively, the bulk modulus of a porous composite body might have

been computed in two steps. First, the bulk modulus of the solid composite

could be computed from Kerner's equation. Then Mackenzie's relation could

be applied to the composite bulk modulus to obtain the values for finite

porosity. However, Mackenzie's relation always predicts a lowering of the

-8-15 PI-1273-14-12

...........................................1

32 ---- ------------ --------........ .. ...

---- ---- .... .... ... ........ .....

28.9

28 ----- --------- ........... .... .... .......

*...........

CO TE T APROCHS ZER

. . .. . 4.......I............. S

20 --- 0% POOI T - ----------- I ....-------

------- -----.

-- - - -- - - .. . t .. . . .. .. .. .. .I ...... I

-- -- -- - - -- -- -- - - - - - P -- --- ---

------ 4

.. .. ----- ------t U T -- S- PYREX--- -- ------

20 0 60 IG

ISIE WEGH PERCENT PYREX

6~ -16

FSmoduli due to the Presence of pores. Internal stresses ari sing firom -the

dissimilar thermoelastic properties of the several phases would have to

remain constant with increasing porosity if this approach were valid. But

Sin the limit of high porosity, the internal stresses approach zero. Hence,

Mackenzie's relation cannot be correctly applied to multiple-phase ceramics

or composite bodies.

BI7PI17-ie1

Ii

,In

hi

APENDDIX C

RELATIONSHIIS BL71%MN SINGLE CRYSTAL AN) PLCRYSTALLINE

ELASTIC CONSTANTS INCLUDING DATA FCR SEVUMAL CERAMIC PHASES

The elastic properties of fully-dense polycrystalline bodies which are

macroscopically isotropic can be predicted from the elastic constants for

single crystals of each of the phases present by the use of appropriate

f space averaging techniques. Space averaging presumes random orientation

of the grains. The validity of the random grain orientation equations will

Sbe greatest if neither the particle shape nor the forming technique induce

significant amounts of preferred orientation. Voigt took the stiffnesses of

71 the aggregate to be the space average of the compliances of the single

crystal. Alternatively, Reuss took the compliances of the aggregate to be

the space average of the compliances of the single cqystal. These tuo

S. V - approaches are equivalent to assuming uniform strain and uniform stress

respectively, for each spherical grain in the polycrystallire body. 4iile

neither condition will exist in a real body, these approaches are valuable±2

in that they define the two extreme cases. Empirical data for polycrystalline

Smetallic elements generally lie between these extremes according to data

tabulated by Hearmon Similar results have been fourd for sirgle-phase

polycrystalline ceramics during the course of Ihis study.I;V Hooke's law states that stress c a' ) is proportional to strain ( S )

for sufficiently small strains. The generalized statement for an anisotropic

medium may be taken as

. . .j=/

where CG1 elastic stiffness constants or the moduli of elasticity.

Alternatively, the relation may be written in terms of the elastic compliance

constants ( sy ) where

C-"PSiI-,.73jc-.,

iC-1 PI-!273-M-!2

In general,

The elastic stiffness and complairce constants obtained with these relations

hold for a rectangular coordinate system with one preferred crystallographic

orientation. Table C-I lists reported values for oxide crystals. If the

elastic constants for a more general direction are desired, equations for56rotated elastic constants must be used. Hearmons has tabulated the 126

equations required for the most general case. By integration of the ePoation

for rotated elastic constants over all space, i.e.

2 rr 2r 217IC (9) flod (Ye

/0 0 o

the space averaged value will be obtained. This was done by Hearmon

who gives Voigt's results in terms of the elastic stiffness constants as

(A -28 -4c)

~ 2 #(A#-2C)

where

,5A =Cf + Caa+ Cza

,.3C= 0 4 . C5 5 *C

and the overline represents an average value. Since space averaged values

are macroscopically isotropic, the isotropic constants, Young's modulus ( Y ),modulus of rigidity (G ) and bulk modulus (K) may be derived in terms of

C-2 PI-1273-11-12

w P.l La

Go

020~ 0 O~(' ~cs *i

C44 N

t ~ vo N -" =P 0-- a - ID

3-. * j a -* C;*

C-1 C2 aAa -q 8 to

0 3

-i d co C;- .

ace 'A a a C - a -

- ~9to ~ 40 .

cm~~d'. co4 .0 oc

*U w

I.- - cX- A I- p-9K

u-c c - 0 c 0 9

3-.- a 4 .=1 Cd* cm a ý C) 1

1ý72 c L.- U-** -

C-.3 PI-1273-1-1-12

the elastic stif~fness cons tants as given by Heaz-mon:

(A-8i-5 ýA + 28)~2A 38 C

where the suf fix tr denotes the Voigt moduli. Sizrdlarly, Hearmon derived-the expressions based on the Resus approach as

where

Comparison off Youn~gIs modulus or shear modulus as com puted by. the Voigt

and Reuss techniques provides a measure off the elastic anisotropy off the

single crystal. For elastically isotropic bodies, Voigt's approach and

Reuss' approach will give identical values.

For a cubic crystal such as MgO, only three independent elastic stirf-

ness or compliance constants appear

c,11 = C22 -=C,3, 2= Z6 6xf10'dy es/c 2

SICI

C4 PI-127.1-14-1 2

I

HenceK .ji -2.O 6 ne

S1 25 S, 2 -0.95 xAO"' cm1..vne.

. 344 =, 5 5" __ =6.76.0O_'acnjdYne"

KR f2zX06-eKRI -*,2 2 2xlO6ps..

In a cubic system,

I: ZC1 * +ZC2

and herce the bulk moduli ( k and q ) will be equal for all cubic crystals.

Computation of both k', and K4 for cubic crystals provides a measure of the

consistency of the reported values of elastic stiffness (C~j ) and (sj )a

Birch4 has applied these space averaging techniques to several ceramic

crystals to obtain the compressional wave and shear wave velocities for

polycrystalline bodies. For example, the shear modulus ( G ) for polycrystalline

corundum (W10 computed from Birch's space averaged data are:

II

!i:(Voigt) =24.0 x 106 psi

S -(Reuss) =23.2 x 10psi

L Lan47 has measured the dynamic shear modulus of a series of aluminm ceramic

bars with varying densities. Extrapolation of his data to theoretical density

(p = 3.98 g/cm ) gives an experimental value of 23.5 x 10 psi for the""shear modulus -.hich lies between the tw predicted values. Birch' has

: C-5 PT-1273-M-12

IA

tabulated similar data for eleven other ineral or synthetic crystals.

These data have been used to compute the elastic properties of fully dense

ceramics listed in Table XV. However, it is to be noted that several of

the ceramic phases of interest in studies of thermal expansion, internal-

stress due to thermal expansion anisotropy, tensile strength of multiphase

ceramics and other ceramic properties have no sirgle crystal elastic

constant data reported in the literature,

.1

IT

[I1I

C-6 I-123-1"-i2i

APEENDIX D

1. RERCES

1. G. R. Rigby, "X Reversible Thermal Expansion From Theoretical Considerations,"

Trans. Brit. Ceram. Soc. 50, 175-83 (1951).

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5 5. H. P. Kirchner, "A Preliminary Study of the Thermal Expansion of Ceramics,"

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7. S. Ganesan, "Thermal Expansion of Sodium Chlorate and Bromate," J. Indian

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11. G. Borelius, "On the Connection Between the Thermal Expansion and Potential

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of Sci. of irnia, 25A(6) 364-372 (1959).

D-1 PI-1273-M-12

LI.

14. H. P. Kirchner, "Investigation of the Theoretical and Practical Aspects

of the Thermal Expansion of Ceramic Materials," Cornell Aeronautical

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15. D. E. Harrison and F. A. Hummel, "High Temperature Zirconium Fhosphates,"

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16. D. E. Harrison and F. A. Hummel, "Raactions in the System TiO -P205w

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17. H. W. G. Wyckoff, "Crystal Structures," Vol. III, Interscience Publishers,

Inc., New York (153).

18. F. A. Hummel, "Observations on the Thermal Expansion of Crystalline and

Glassy Substances," J. Am. Ceram. Soc. 33, 102-107 (1950).

19. J. B. Austin, "Thermal Exparsion of Nonmetallic Crystals," J. Am. Ceram.

Soc. 35 (10) 243-253 (1952).

20. H. D. Negaw, "The Thernmal Expansion of Crystals in Relation to Their

Structure," Z. Krist. 100, 56-76 (1938).

21. K. f. Merz, "Investigation of the Theoretical and Practical Aspects off

the Thermal Expansion of Ceramic Materials," Cornell Aeronautical

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22. K. M. Herz, H. P. Kirchner and H. T. Srmth, "Investigation of the

Theoretical and Practical Aspects of the Thermal Expansion of Ceramic

Materials," Cornell Aeronautical Laboratory Report No. PI-1273-M-8,

Contract No. NOrd-18419 (September 30, 1960).

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Sors, N'f York (1940).

24. N. R. Thielke, "Refractor. terials for Use in High Temperature Areas

of Aircraft," WADC Technical Report 53-9, Te Pennsylvania State College

(January 1953).

D-2 PI-1273-4-12

RB••-

25. W. A. Weyl, "An Interpretation of the Thermal Expansion of the Alkali

1 Halides and of the Structural Changes Occurring in Glass Under High

Pressure," Central Glass and Ceramic Res. Inst. Bulletin (Inxdia) 6 (W)

Ii 147-74 (1959).

26. R. J. Beals and R. L. Cook, "Directional Dilatation of Crystal Lattices

at Elevated Temperatures," J. Am. Ceram. Soc. LO, 279-84 (1957).

27. F. H. Gillery and E. A. Bush, "Thermal Contraction of A1- Eucryptite2IF (Li 20A103-2Si 2) by X-ray and Dilatometer Methods," J. Am. Ceram.

Soc. _4 , 175 (1957).

28. F. R. Charvat and 7. D. lingery, "Thermal Conductivity: XIII, Effect of

Microstructure on Conductivity of Single-Phase Ceramics," J. Am. Ceram.

Soc. 4, 306-15 (September, 1957).

29. G. R. Rigby, G. H. B. Lovell and A. T. Green, "The Reversible Thermal

I Expansion and Other Properties of Some Calcium Ferrous Silicates," Trans,

* Brit. Ceram. Soc. L4.4, 37-52 (1945).

L 30. S. Kozu and J.,J. Ueda, "Thermal Expansion of Plagioclase," Proc. Imp.

Acad. Tokyo 9_, 262-4 (1933).

L 31. S. Kozu and J. J. Ueda, "Thermal Expansion of Diopside," Proc. Imp.

Acad. Tokyo 7_ 317-319 (1933).

- 32. .. W. Ricker and F. A. Hummel, "Reactions in the System TiO 2-S2;

of the Phase Diagram," J. Am. Ceran. Soc. 34, 271-9 (1951).

33. R. S. Roth and L. W. Coughanour, "Phase Equilibrium Relations in the

Systems Titania-Niobia and Zirconia-Niobia," J. Res. Nat. Bur. Stds.55, 209 (1955).

34. L. W. Coughanour, R. S. Roth and S. Marzullo, "Solid-State Reactions andI I- Dielectric Properties in the System Magnesia-Lime-Tin Oxide-Titania," J.

Res. Nat. Bur. Stds. 54, 149, R. P. 2576 (1955).

- 35. F. H. Brown a:d P. Duwez, "The Zirconia-Titania System," J. Am. Ceram.

Soc. 37 129-32 (1954).

D-3 PI-1273-M-12

-

36. S. M. Lang, C. L. Fillmore and L. J. Maxwell, "T1e System Beryllia-

Alumina-Titania: Phase Relations and General Physical Properties of

Three-Component Porcelains," J. Res. Nat. Bur. Stds., .4 298,

R. P. 2316 (1952).

37. G. H. Johnson, "Influence of Impurities on Electrical Conductivity of

Rutile," J. Am. Ceram. Soc., 3.6, 97-101 (1953).

38. P. W. Selwood, "Magnetochemistry,"t Second Edition, Interscience

Publishers, Inc., New York (1956).

39. H. J. Gerritsen aii He. R. Lewis, "Paramagnetic Resonance of V4 in

Ti02," Phys. Rev. 119, 1.010-2 (1960).

40. F. A. Mauer and L. H. BoIz, "Measurement of Thermal Fcpansion of CermetComponents by High Temperature X-ray diffraction," WADC TH-55-473,

Supplement 1 (June, 1957).

41. J. Jaffray and J. Vilateau, "Sur l'analyse thermique et la dilatoaetrie

du sesquioxide de chrome," Compteo Rendus (Paris) 226 (21) 1701

(may 24, 1948).

42. S. Greenwald, "Changes in Lattice Constants of Cr2 03 Near the Curie

Temperature," Nature 4270 379 (September 1, 1951).

43. D. C. Cronemeyer, "Electrical and Optical Properties of Rutile SingleCrystals," M. I. T. Laboratory for Insulation Research, Technical

Report No. 46 ATI 118352 (August, 1951).

44. H. Kolsky, "Stress Waves in Solids," Oxford at the Clarernon Press,

(1953).

45. F. Birch, "The Velocity -f Compressional Waves in Rocks to 10 Kilobars,Part 2," J. Geophys. Res., _ pp. 2199-2224 (July, 1961).

46. H. J. McSkimin, "Use of High Frequency Ultrasound for Determining the

Elastic Moduli of Small Specimens," IRE Trars. Ultrasonics Ergr.,

PGUE-5, p. 29-43 (August, 1957).

D-4 PI-,273-74M-12

v 47. S. M. Lang, "Properties of High-Temperatura Ceramics and Cermets -

Elasticity and Density at Room Temperature," NBS Monograph 6

(March, 1960).

48. W. P. Mason. "Piezoelectric Crystals and Their Applications to

Ultrasonics,': D. Van Nostrand Co, New York (190).

49. G. L. Vick and L. E. Hollarder, "Ultrasonic Measurement of the Elastic

vi Moduli of Rutile," J. Acoustical Soc. Lm., 32, 947-49 (August, 1960).

50. C. M. Zaner, "ME.-sticity and Anelasticity of Metals," The University

of Chicago Press (1956).

51. H. A. Scheetz, "An Investigation of the Theoretical and Practical Aspects

of the Thermal Expansion of Ceramic Materials," Cornell Aeronautical

Laboratory Report No. PI-1273-N-11 (November, 1961).

52. P. S. Turner, "Thermal Expansion Stresses in Reinforced Plastics,"i I) J. Res. Nat. Bur. Stds., 3 239-250, R. P. 1745 (1946).

53. W. D. Kingery, "Note on Thermal Expansion and N!icrostresses in Two-Phase Compositions," J. Am. Cer. Soc., 40O, pp. 351-52 (October, 1957).

* 54. R. L. Coble and W. D. Kingery, "Effect of Porosity on Physical Properties

of Sintered Alumina," J. Am. Cer. Soc., 39, P. 377 (1956).

[ 55. R. K. Verma, "Elasticity of Some High-Density Crystals," J. Geophysical

Research, , pp. 757-66 (February, 1960).

56. R. F. S. Hearmon, "The Elastic Constants of Anisotropic Materials -II

Adv. in Phys., 5, PP. 323-382 (J'tly, 1956).

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58. W. D. Smiley; L. E. Solon, et al., "Hechanical Property Survey of

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7 -1273

59. J. K. MacKenzie, "The El'astic Constants of a Solid Containing Spherical

Pores," Proc. Phys., Soc. London, B63, p. 2 (1950).

60. J. Warshaw and R. Roy, "Therm~al Expansion !Ldasurcnents from Yornindcxed

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68. J. F. Wygant., "Elastic and Flow Properties of Dense, Pure Oxide .Refractories" J. Am. 'Ceramic Soc. 34, 374-380 (1951).

D-6 PI-l273-11N-l2

69. E. H. Kerner, "Elastic and Thermoelastic Properties of Composite Media,"

Proc. Phys. Soc. London, B69, pp. 808-813, (1956).

70. W. Hume-Rothery a-d P. W. Reynolds, "High Temperature Debye-Scherrer

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' Rare Earth Oxides," KAPL-N-GLP-1, (March 28, 1957).

72. H. B. Huntington, "The Flastic Constants of Crystals" from Solid State

U Physics, Vol. 7, Academic Press, Inc. (1958).

I ii

*I!

I[I

fl i P -. 3 - 4 1

I