iczc - apps.dtic.mil · -table aa 171i thermal expansion in the mgo-wgoa1 203 system 88 xviii...
TRANSCRIPT
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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
___ __ __ _ ---- - - . -
.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
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
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
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'
22 • . , - - ... - 'i.,....... -.- '-t . ...... ---- ------- I - ,--NI : a " -,,.. _ I
I.I,
I R9 I. I I. -I • I ; .
IL -S I . Ito.ooc
* a .* * S -* ' a
I Is.
] ....~~~z ..............- ,-r"...--- -. i....:....i •
01 2 0 0 80 1 00F 2 O ,.SI B
= .. S . . ..
ON KERERS THEOR
I 8 * , , 5 5 5
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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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------. -- --:. ... .... .........b a
76 $
* . ................
ýKENGER3 (ET957)
..................................-......................j-
0IGR 197 5
0 20 '10 60 8010
Al WEI GMT PERCENT 30
Figure 24 THERMAL EXPANSION IN THE ALUMINUM -1 SLI CA SYSTEM4
85 'PI-1273-M-12.
.4A ac ui .cc~ ad
oe ixa AL9 m - L
F - cc - c&
CC -. -1 4 cmO'"cm -= usI~ UjLA dn Cie
9m I0 'II.
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1." 10 -, Ow L - S'DLa
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UiTURNERS METHOD>N. ..
900
70 CAL(DILATOMETER)
S(X-RAY) BEALS A COOK 12
ALESCHEN13
0 20 '10 60 80 100
WEIGHT PERCENT
Figure 25 THERM4AL EXPANSION IN THE MAGNESIA - SPINEL SYSTEM4
87 ?I-1273-M-12
wa 40 C3~a a
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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
HgO A1203 SPINEL 725%$PIXEL*
1 n * I i ! l •'I
18 50% PYREX-50 SriEL
161
1141 1
4J12 §1 .t~:175% PYREX-."... .......... . 25% SPINEL -
.. . .... ...-. - o , E -
'12 ,,4 / so l liL.
~I0 PYREX 1
..............- ".............. i......'
8 . ........... ..-..-4 . ............ ....... ---
------............... ..... .......... t....I.............. ...
- . -1 - ----- 4--- .............. ... ---------------
* I
------------ -
. .............. --..... -... -- -----
*, I I *
2 DATA CORRECTED FOR FUSED QUARTZ EXPANSION 5 x 10- 7 /0C.TEMPERATURE SCALE SHI FTED TO R.T.= 23*C
-u -..- .... .... .. I.... .-...... " .... I " ....... I. .....
I I, :
* : * I **
* 5 1* Ii I * I *
00 ý50 100 150 200 250 300 350
TEMPERATURE -C
Figure 26 DILATOMETER THERMAL EXPANSION MEASUREMENTS FOR PYREX-SPINEL BARS
92 P2-1273-1--12
140 40
32
IL1 28.9~28
Ir Z~24IN
'g: ~20 -KAj 16 .15.5
12
G 9.0
4 5.04 3.58 .43.74
2.23
v0 20 40 60 80 100VIP . A1203 WEIGNT PYREX
PERCENT
I Figure 27 ELASTIC MODULI OF PYREX.-SPINEL BODIES BASED ON KERNER'S ThEORY
93 PI-1273-M--12
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S!Fl
100
88
-s.0
LL" 60 ,-o C
"a a: KERNERM ETEHOD
-0 : . :140
0 CALlOI LATOHETER)
-0
0 0 o60 so 100WgO . A1.O3 WEI GR T PERCENT D EX
Figue 28THERMAL EXPANSION IN THE PYREX-S PINEL SYSTEM
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 --.. ...
..... .......... - .
"" - .... .. ...... ..... ........ + .....I ,
.... --- . .......... 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
56 - acR LUCALOX ( 5656 x 106, j2 3.98) ....56 .....
.. . .. . . . . . . . . . .. . .. . . .
: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
4.. .. .. . . . . . ... .. . . - - - - - ... . . . .Y .... . . .. . .#-a*. . .
v
C *a- - - - -- --
4 0 -- - - - - - - .. . . . . . ... . . . . . . . . . . a-- - -- - - - - - -
. . . . . . . . ... . . . . . . . . . . . . . . . . . . . . .. . . - - -j- - - - . . .
F3 ....................- -- .... .... ..........
'4-0----------- - ---- ......... ......---
2. 1. . I. i
38DIFIE..: S.. . . - - --
------ ------ I---- ---- ----K R E '
32 + SMETHOD
* l
------- ---- ---2 ---- .... ------ ----
-------- ------- --------------- ----.... ---
-- -- -- - - - - - - --. . . . . .20 ---
--- -- - --- - - - -- -- -- - -- -
i ---- -----
---- .... .. ---- --- --12 ------- LAN -
+ .....----- 1---- ---- ---
DESTYI GASCNINE
, ... 6 01I2 53
PECN POROSIT
Fiur YON' MODLU OF POOU MAGE 1I
- C I27 !1 2
2 0
.. . .. . .. . .. . .. . . ... .
.I .--- .... . . ........ . .a4... .
16...............
---- .....3 ..... .. . . .. ... . .. .. . . ... . .. .. . ... .... .. .... .
........ t ......
............ ....
.4...........
.. .... .. .... I- -- ---,. . .
1 ----. ............. ... .
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
T .... 7I~34.. .... ...y ............... I ...... .......
.. . . . . . . . .. . . . . . . . .... .% --- - - - - - .... . . .. . . .
...... ..........
to f
2Kt ftf
22~~~~~~~~~~~~ ~ ~ ~ ~ ~ ----4----- -- --- ----... --- ....------ - ...... M D F E
C2 f4 . fERNER03
---- -t ---- ---- ... ... ... --- ---- I.... M THO
1.. ... . .... .... .... ------------ .....
---- ---- -------- 4.....tA KN I'
----- --------- .... ---- .... ---- ...---- ---- ---- t---
ftft7tf
-------.-.-----.........
3.6 3.. 1 3 0 . . .
Fiur R- 5t YON'SMDLSOfOOUtPNL ( l23
* I2-f-1
-. - I, 'p
. . .. ---I** -+'-*-* * .'I 'I: -.
. .. .... .. .. .. .. . . .- ---- ... .
---.......-- ...... .' ... • ........ ...... ...... . . ..................... ":. ....... :" • ...... ------.
-1 6 ---- ---
II , * - ,.
.. ........ .. ... - .
. 4 ... - - - - -
. . ... ..."- - - .. . *........... . ... .......
.. .. ..... F - --- - - -
,*!
* : . * COLD PR.ESSED -
S i"- * , * I S * ** z I * 5 5 i
1 ---- ---- --- --- --- .. ... ..... ..... ------- --------
.... ...., .. • . . . . , ... .-
- i * e * * , *S5 * I , *
. --- . - -- ... ............ + --- ------
4 10 --- ---..... . ...... . -... -....... I ----- .... I ------ ,, •
12 ----- ---- -----. -- -- .. .+ .. .. .• -.--..-.. -. ' ...... .. . ........ .. ....... --...... ..... .... t...... ..
.. ..... Fe ... .. -- - -. • . I . . . ..-- ---!eI i I i 5
./* : I : , ..
---- ---- --- --- --2F . %'
w ....------ .I.----- - ............-- ....... .;".... ....... 44 '- 4 .. .. ......... t- - ...... ..•
.- ...... . . . . .. . 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
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D-1 PI-1273-M-12
LI.
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D-2 PI-1273-4-12
RB••-
25. W. A. Weyl, "An Interpretation of the Thermal Expansion of the Alkali
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26. R. J. Beals and R. L. Cook, "Directional Dilatation of Crystal Lattices
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28. F. R. Charvat and 7. D. lingery, "Thermal Conductivity: XIII, Effect of
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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,
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D-3 PI-1273-M-12
-
36. S. M. Lang, C. L. Fillmore and L. J. Maxwell, "T1e System Beryllia-
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37. G. H. Johnson, "Influence of Impurities on Electrical Conductivity of
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40. F. A. Mauer and L. H. BoIz, "Measurement of Thermal Fcpansion of CermetComponents by High Temperature X-ray diffraction," WADC TH-55-473,
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D-4 PI-,273-74M-12
v 47. S. M. Lang, "Properties of High-Temperatura Ceramics and Cermets -
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48. W. P. Mason. "Piezoelectric Crystals and Their Applications to
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7 -1273
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D-6 PI-l273-11N-l2
69. E. H. Kerner, "Elastic and Thermoelastic Properties of Composite Media,"
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I ii
*I!
I[I
fl i P -. 3 - 4 1
I