First-Principles study of Thermal and Elastic Properties of Al 2 O 3 Bin Xu and Jianjun Dong,...
1
First-Principles study of Thermal and Elastic Properties of Al 2 O 3 Bin Xu and Jianjun Dong, Physics Department, Auburn University, Auburn, AL 36849 1. Introduction 2. Computational Methodologies 3. Results 4. Conclusions References Acknowledgements 2007 Alabama EPSCoR Annual Meeting, University of Alabama in Huntsville, February 13, 2007 0 1 (,) () () ln(1 ) 2 B kT static B FTV E V d g kT e 0 200 400 600 800 1000 F re q u e n cy: (cm -1 ) A Z D (a ) P h on o n d isp ersio n (b ) D e nsity of S tate a rb. u n its -A l 2 O 3 P ressure:0G P a 300 600 900 1200 1500 1800 -8.7 -8.6 -8.5 -8.4 -8.3 -8.2 -8.1 -8 .0 -7.9 7.0 7.5 8.0 8.5 9.0 9.5 10.0 1 0.5 0 500 1000 1500 2000 0.0 5.0x10 -6 1.0x10 -5 1.5x10 -5 2.0x10 -5 2.5x10 -5 3.0x10 -5 3.5x10 -5 4.0x10 -5 -A l 2 O 3 Therm ale xp a nsio n co e fficie n t: (K -1 ) T em perature:T (K ) Theory E xp t. W a ch tm a n e ta l.(1 9 6 2 ) S ch a u e r (1 9 6 5 ) A m atunietal.(1976) A ld e b e rt& T ra ve rse (1 9 8 4 ) F iq u e te ta l.(1 9 9 9 ) W h ite a n d R oberts (1983) P=0GPa 0 500 1000 1500 2000 0 .86 0 .88 0 .90 0 .92 0 .94 0 .96 0 .98 1 .00 1 .02 B S (0)=242.47G P a B S (0)=253.85G P a B S (0)=256.85G P a B S (0)=251.97G P a -A l 2 O 3 T em perature:T (K ) N o rm alize d a d ia b a tic B u lk M odulus:B S /B S (T = 0K ) Theory E xpt. G o to e ta l.(1 9 8 9 ) C hung & S im m o n s (1 9 6 8 ) T effet(1966) P=0GPa 2 1 1 P V G V T V TP V F S T P P S C T T P P S T T V V C C P B B V C C V Figure 1. Crystal structure of alumina: (a) The side view of a ball-and-stick model of α-Al 2 O 3 , with the v ertical direction along the hexagonal-close-pack axi s. (b) Al atoms are 100% octahedrally bonded. (c) An d O atoms are 100% tetrahedrally bonded. Bulk crystalline α-Al 2 O 3 Structure Optimization and Total Energy Cal culation First-Principles Quantum Mechanics Theory: Plane w ave, Pseudo-potential, Density Functional Theory (PW-PP-DFT) Thermodynamic Potentials at finite temperatur es Statistical Quasi-Harmonic Approximation (QHA) Figure 4. Calculated Helmholtz free energies per atom of α- Al 2 O 3 as a function of temperature and volume per atom. Figure 3. LDA calculation of (a) phonon dispersion relations, (b) vibrational density of states of α- Al 2 O 3 at zero pressure. Lines denote theoretical spectrum and discrete squares denote experimental data [1] . Figure 5. Comparison of the present theoretical calculation with measured bulk thermal expansion coefficients [2-8] of α- Al 2 O 3 as a function of temperature at zero pressure. Figure 6. Comparison of calculated isobaric heat capacity and entropy of α- Al 2 O 3 with experimental data [9] as a function of temperature at zero pressure. Figure 7. Comparison of the theoretical normalized adiabatic bulk modulus (at T=0K) of α-Al 2 O 3 with measurements [10] as a function of temperature. Figure 8, 9. Calculated elastic constants of α-Al 2 O 3 and Rh 2 O 3 (II)-Al 2 O 3 as a function of hydrostatic pressure. Symbols denote the calculated data at a certain pressure and lines are from linear fitting. Excellent material Excellent material properties and properties and extensive technology extensive technology applications: applications: •Large elasticity •High strength and hardness •Chemically inert •Coating as thin-film on devices •Wear applications and cutting tools Blue color denote C ij tha t is not independent. For rhombohedral symmetry: C 22 =C 11 ; C 55 =C 44 ; C 66 =(C 11 -C 12 )/2; C 23 =C 13 For Orthorhombic symmetry: C 14 =0 Table 1. Linear pressure dependence of C ij from the fit to calculated elastic constants. [1] H. Shober, et al, Z. Phys. B: Condens. Matter 92, 273 (1993) [2] J. Hama, et al, Phys. Chem. Mi nerals 28, 258 (2001) [3] Wachtman Jr JB, et al, J. Am. Ceram. Soc. 45, 319 (1962) [4] Schauer A, Can. J. Phys. 43, 523 (1965) [5] Amatuni AN, et al, High Temp- High Pressure 8, 565 (1976) [6] Aldebert P, et al, High Temp- High Pressure 16, 127 (1984) [7] Fiquet G, et al, Phys. Chem. Miner. 27, 103 (1999) [8] White GK, et al, High Temp-Hi gh Pressure 15, 321 (1983) [9] Furukawa GT, et al, J. Res. N atl. Bur. Stand. 57, 67 (1956) [10] Goto T, et al, J Geophys. Re s. 94, 7588 (1989) 0 500 1000 1500 2000 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0 1 2 3 4 5 6 7 -A l 2 O 3 Iso b a ric h e a tca p a city:C P (k B /atom ) T e m pe rature:T (K ) E ntropy:S (k B /atom ) Theory C P S E xpt. F u ru ka w a e ta l.(1 9 5 6 ) F u ru ka w a e ta l.(1 9 5 6 ) P=0GPa 1k B /atom = 41.5 7J/m o l/K Phonon dispersion Phonon spectrum is computationally challenging. We have developed new codes to optimize the calcul ation. It is proved that the codes are efficient and general for super cell model as large as 160 atoms of any crystal structure. Our calculation is in agreement with experimental data. Thermal properties Our theoretical thermal expansion coefficient, hea t capacity, entropy and bulk modulus agree well with measured results. The agreement ensures the validity of our calculat ion. Elasticity of α and Rh 2 O 3 (II) phase The high strength of Al 2 O 3 is associated with the l arge elastic constants. The newly theoretically predicted Rh 2 O 3 (II) phase i s only 2% larger in density than α phase and this is in consistency with the similarity of calculated ela stic constants of these two phases. This work is supported by National Science Foundation (Grant No. EPS- 0447675 and HRD-0317741). Elasticity of α and Rh 2 O 3 (II)-Al 2 O 3 Thermal properties Alumina (α-Al 2 O 3 ) nanoparticles Primary particles have a size of 13 nm. They stick together and form agglomerates in the size of some microns. Application of ceramic nano Application of ceramic nano particle in polymer based c particle in polymer based c omposite materials: omposite materials: Small ceramic particles are known to enhance the mechan ical and tribological prope rties. F 0 10 20 30 40 50 0 100 200 300 400 500 600 700 Pressure:P(G Pa) E lastic C onstants:C ij (G Pa) C 11 C 12 C 13 C 14 C 33 C 44 -A l 2 O 3 0 20 40 60 80 100 120 140 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 C 11 C 22 C 33 C 44 C 55 C 66 C 12 C 13 C 23 E lastic C onstants:C ij (G Pa) Pressure:P(G Pa) Rh 2 O 3 (II)-A l 2 O 3
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Transcript of First-Principles study of Thermal and Elastic Properties of Al 2 O 3 Bin Xu and Jianjun Dong,...
First-Principles study of Thermal and Elastic Properties of Al2O3 Bin Xu and Jianjun Dong, Physics Department, Auburn University, Auburn, AL 36849
1. Introduction
2. Computational Methodologies
3. Results 4. Conclusions
References
Acknowledgements
2007 Alabama EPSCoR Annual Meeting, University of Alabama in Huntsville, February 13, 2007
0
1( , ) ( ) ( ) ln(1 )
2Bk T
static BF T V E V d g k T e
0
200
400
600
800
1000
Fre
qu
en
cy:
(cm
-1)
A Z D
(a) Phonon dispersion (b) Density of State
arb. units
-Al2O
3
Pressure: 0GPa300
600900
1200
1500
1800
-8.7
-8.6
-8.5
-8.4
-8.3
-8.2
-8.1
-8.0
-7.9
7.07.5
8.08.5
9.09.5
10.010.5
0 500 1000 1500 20000.0
5.0x10-6
1.0x10-5
1.5x10-5
2.0x10-5
2.5x10-5
3.0x10-5
3.5x10-5
4.0x10-5
-Al2O
3
The
rmal
exp
ansi
on c
oeffi
cien
t: (
K-1)
Temperature: T(K)
TheoryExpt.
Wachtman et al. (1962) Schauer (1965) Amatuni et al. (1976) Aldebert & Traverse (1984) Fiquet et al. (1999) White and Roberts (1983)
P=0GPa
0 500 1000 1500 2000
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
1.02
BS(0)=242.47GPa
BS(0)=253.85GPa
BS(0)=256.85GPa
BS(0)=251.97GPa-Al
2O
3
Temperature: T(K)
Nor
mal
ized
adi
abat
ic B
ulk
Mod
ulus
: BS/B
S(T
=0K
)
TheoryExpt.
Goto et al. (1989) Chung & Simmons (1968) Teffet (1966)
P=0GPa
21 1
P
V G
V T V T P
V
FS
T
PP
SC T
T
P P
S TTV V
C C PB B V
C C V
Figure 1. Crystal structure of alumina: (a) The side view of a ball-and-stick model of α-Al2O3, with the vertical direction along the hexagonal-close-pack axis. (b) Al atoms are 100% octahedrally bonded. (c) And O atoms are 100% tetrahedrally bonded.
Bulk crystalline α-Al2O3
Structure Optimization and Total Energy Calculation First-Principles Quantum Mechanics Theory: Plane wave, Pseudo-potential, Density Functional Theory (PW-PP-DFT)
Thermodynamic Potentials at finite temperatures Statistical Quasi-Harmonic Approximation (QHA)
Figure 4. Calculated Helmholtz free energies per atom of α-Al2O3 as a function of temperature and volume per atom.
Figure 3. LDA calculation of (a) phonon dispersion relations, (b) vibrational density of states of α-Al2O3 at zero pressure. Lines denote theoretical spectrum and discrete squares denote experimental data[1].
Figure 5. Comparison of the present theoretical calculation with measured bulk thermal expansion coefficients[2-
8] of α-Al2O3 as a function of temperature at zero pressure.
Figure 6. Comparison of calculated isobaric heat capacity and entropy of α-Al2O3 with experimental data[9] as a function of temperature at zero pressure.
Figure 7. Comparison of the theoretical normalized adiabatic bulk modulus (at T=0K) of α-Al2O3 with measurements[10] as a function of temperature.
Figure 8, 9. Calculated elastic constants of α-Al2O3 and Rh2O3(II)-Al2O3 as a function of hydrostatic pressure. Symbols denote the calculated data at a certain pressure and lines are from linear fitting.
Excellent material Excellent material properties and extensive properties and extensive technology applications:technology applications:•Large elasticity•High strength and hardness•Chemically inert•Coating as thin-film on devices•Wear applications and cutting tools
Blue color denote Cij that is not independent.For rhombohedral symmetry:C22=C11; C55=C44;C66=(C11-C12)/2;C23=C13
For Orthorhombic symmetry:C14=0
Table 1. Linear pressure dependence of Cij from the fit to calculated elastic constants.
[1] H. Shober, et al, Z. Phys. B: Condens. Matter 92, 273 (1993) [2] J. Hama, et al, Phys. Chem. Minerals 28, 258 (2001)[3] Wachtman Jr JB, et al, J. Am. Ceram. Soc. 45, 319 (1962)[4] Schauer A, Can. J. Phys. 43, 523 (1965)[5] Amatuni AN, et al, High Temp-High Pressure 8, 565 (1976)[6] Aldebert P, et al, High Temp-High Pressure 16, 127 (1984)[7] Fiquet G, et al, Phys. Chem. Miner. 27, 103 (1999)[8] White GK, et al, High Temp-High Pressure 15, 321 (1983)[9] Furukawa GT, et al, J. Res. Natl. Bur. Stand. 57, 67 (1956)[10] Goto T, et al, J Geophys. Res. 94, 7588 (1989)
0 500 1000 1500 20000.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0
1
2
3
4
5
6
7-Al2O
3
Iso
ba
ric
he
at c
ap
aci
ty: C
P(k
B/a
tom
)
Temperature: T(K)
En
tro
py:
S(k
B/a
tom
)
Theory C
P
SExpt.
Furukawa et al. (1956) Furukawa et al. (1956)
P=0GPa1k
B/atom=41.57J/mol/K
Phonon dispersion Phonon spectrum is computationally challenging. We have developed new codes to optimize the calculation. It is pr
oved that the codes are efficient and general for super cell model as large as 160 atoms of any crystal structure.
Our calculation is in agreement with experimental data. Thermal properties
Our theoretical thermal expansion coefficient, heat capacity, entropy and bulk modulus agree well with measured results.
The agreement ensures the validity of our calculation. Elasticity of α and Rh2O3(II) phase
The high strength of Al2O3 is associated with the large elastic constants.
The newly theoretically predicted Rh2O3(II) phase is only 2% larger in density than α phase and this is in consistency with the similarity of calculated elastic constants of these two phases.
This work is supported by National Science Foundation (Grant No. EPS-0447675 and HRD-0317741).
Elasticity of α and Rh2O3(II)-Al2O3
Thermal properties
Alumina (α-Al2O3) nanoparticles
Primary particles have a size of 13 nm. They stick together and form agglomerates in the size of some microns.
Application of ceramic nano particlApplication of ceramic nano particle in polymer based composite mate in polymer based composite materials:erials:Small ceramic particles are known to enhance the mechanical and tribological properties.
F
0 10 20 30 40 500
100
200
300
400
500
600
700
Pressure: P(GPa)
Ela
stic
Con
stan
ts: C
ij (
GP
a)
C11
C12 C13 C14 C33 C44
-Al2O
3
0 20 40 60 80 100 120 1400
100200300400500600700800900
100011001200
C11
C22
C33
C44
C55
C66
C12
C13
C23
Ela
stic
Con
stan
ts: C
ij (G
Pa)
Pressure: P(GPa)
Rh2O
3(II)-Al
2O
3
