RECENT DEVELOPMENTS IN THE ANAELU-MPOD SOFTWARE SYSTEM FOR - - PowerPoint PPT Presentation

RECENT DEVELOPMENTS IN THE ANAELU-MPOD SOFTWARE SYSTEM FOR POLYCRYSTAL CHARACTERIZATION

• L. E. Fuentes-Cobas1, E. E. Villalobos-Portillo1,
• D. C. Burciaga-Valencia1, L. Fuentes-Montero2,
• M. E. Montero-Cabrera1, D. Chateigner3

1 Centro de Investigación en Materiales Avanzados (CIMAV), Chihuahua, Mexico 2 Diamond Light Source, Didcot, UK. 3 Université de Caen Normandie, Caen, France

http://mpod.cimav.edu.mx http://cimav.edu.mx/investigacion/software/

Outline

1) Recap on Texture Analysis. The program ANAELU (Analytical Emulator Laue Utility) 2) Structure-Properties: MPOD (Material Properties Open Database) a) Single Crystals. b) Textured Polycrystals ANAELU

a) Random distribution of orientations b) Texture AURIVILLIUS POLYCRYSTALS (COURTESY J. A. EIRAS, UFSC, BRASIL)

Rolling Texture

Crystallographic Texture: Preferred Orientation

a b

The “classical” description of textures: (Direct) Pole Figures

https://www.researchgate.net/figure/Figure-Initial-pole-figures-for-single-crystal-FCC-cube- texture-simulations_fig6_319446954

Pole figures in axial symmetry (fiber) textures

• D. Chateigner, J. Ricote. Ch 8 of

Handbook “Multifunctional polycrystalline ferroelectric materials”. Eds: L. Pardo y J. Ricote, Springer-Verlag (2011)

Direct pole figures follow the sample symmetry

Inverse Pole Figure (IPF)

001

Inverse pole figures follow the crystal symmetry

Cúbico Hexagonal Trigonal Tetragonal

1,1,1 0,0,1 “Structural” IPF 4mm point group Diffraction (Laue) IPF 4/m 2/m 2/m

Model IPFs for poled BaTiO3

SAMZ-Poly program

Euler space

http://aluminium.matter.org.uk/content/html/eng/default.asp?catid=100&pageid=1039432491

Frequent ODFs in cubic phases

The Orientation Distribution Function (ODF)

dV/V = f(g) dg f(g) = f(Gs•g •Gc)

Bunge (1982) Texture Analysis in Materials Science: Mathematical Methods

0,0,1 1,0,0 0,0,1 1,0,0 0,0,1 1,0,0

Texture Measurement DRX - Bragg-Brentano

I = [I0 K |F|2 p (LP) A T / v2 ]⋅R(φ, β)

Textured BaTiO3 ceramic

) ( ) (

2 1 2 2

1

h h

G exp G G R φ − + =

2 3 2 1 2 2 1

1

/ −

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + =

h h h

sin G cos G R φ φ

• M. Betzl, L. Fuentes, J.

Tobisch: "Texture study of rolling conditions for zinc‑based alloys". JINR

• comm. E14‑85‑473, Dubna

1985.

Texture Measurement Texture goniometer (ideally with neutrons)

g = [ϕ1, Φ, ϕ2] = g(r)

Polycrystal aggregate function

Focus on “stereography”

g(r) investigated by means of Kikuchi lines at the SEM (BSED, OIM)

Texture Measurement

2-D position sensitive detector, BL11-3 SSRL

Texture Measurement

Nano-systems texture analysis by 2D - XRD

“Fibre textures” (axial symmetry): A frequent case in nano-structured functional materials

Nano-islands ↑ Nano-rods ↑ Nano-plates à If a sample shows fibr ibre text xtur ure, then the inverse pole

figure (IPF) of the

symmetry axis plays the ODF role. ß Direct Pole Figure

ANALYTICAL EMULATOR LAUE UTILITY

• J. Appl. Cryst. Vol. 44, pp. 241-246 (2011)

http://cimav.edu.mx/investigacion/software/ https://www.iucr.org/resources/other-directories/software/anaelu http://www.esrf.eu/computing/scientific/ANAELU/Anelu_Page.htm

• A. Sáenz-Trevizo, M. Miki-Yoshida et al

Materials Characterization 98 (2014) 215–221

Preferred growth direction: [001] Distribution width Ω = (20 ± 2)°

Combined 2D grazing incidence XRD + Electron microscopy texture analysis of ZnO thin layers

Observed Calculated

• Friendly GUI
• Background modeling
• Quantitative semi-

automatic refinement

• f parameters

Paraelectricity: P = ε0χP·E Paramagnetism: µ0M = µ0χM·H Elasticity: S = s · T Thermal expansion S = η·Δθ Piezoelectricity: P = d · T

S = d · E

Magnetoelectricity: P = α · H µ0M = α · E

Physical properties: Y = K · X “Principal” and “Coupling” Interactions.

Some effects and their constitutive equations:

• L. Fuentes: Magnetic Coupling Properties in Polycrystals

Textures and Microstructures 30: 167-189 (1998).

THERMO-ELASTO-ELECTRO-MAGNETIC EQUILIBRIUM PROPERTIES

Pà POLAR; Aà AXIAL; r = Tensor rank

Property Related magnitudes Tensor Heat capacity C Entropy (P0) / Temperature (P0) P0 Elasticity s Strain (P2) / Stress (P2) P4

• Electr. susceptibility χP

Polarization (P1) / Elec. Intensity (P1) P2

• Magn. susceptibility χM

Magnetization (A1) / Magn. Intensity (A1) P2 Thermal expansion η Strain (P2) / Temperature (P0) P2 Pyroelectricity p Polarization (P1) / Temperature (P0) P1 Pyromagnetism i Magnetization (A1) / Temperature (P0) A1 Piezoelectricity d Polarization (P1) / Stress (P2) P3 Piezomagnetism b Magnetization (A1) / Stress (P2) A3 Magnetoelectricity α Magnetization (A1) / Elec. Intensity (P1) A2

Tensor ranks: m, n, m+n

HIPERVECTOR

MATRIX NOTATION

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡

• ⎥

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡

3 2 1 6 5 4 3 2 1

P P P T T T T T T d d d d d d d d d d d d d d d d d d

36 35 34 33 32 31 26 25 24 23 22 21 16 15 14 13 12 11

Aij (for example): strain or sress tensor

MATRIZ 3X3

PIEZOELECTRICITY

ó

d · T = P

ELASTO-PIEZO-DIELECTRIC MATRIX

S = s⋅T + d⋅E D (≈ P) = d⋅T + ε⋅E

S S S S S S D D D = s s s s s s d d d s s s s s s d d d s s s s s s d d d s s s s s s d d d s s s s s s d d d s s

1 2 3 4 5 6 1 2 3 11 12 13 14 15 16 11 12 13 21 22 23 24 25 26 21 22 23 31 32 33 34 35 36 31 32 33 41 42 43 44 45 46 41 42 43 51 62 53 54 55 56 51 52 53 61 62 63

⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ s s s s d d d d d d d d d d d d d d d d d d d d d T T T T T T E E E

64 65 66 61 62 63 11 12 13 14 15 16 11 12 13 21 22 23 24 25 26 21 22 23 31 32 33 34 35 36 31 32 33 1 2 3 4 5 6 1 2 3

ε ε ε ε ε ε ε ε ε ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

CRYSTALLOGRAPHIC ELASTO-PIEZO- DIELECTRIC MATRICES, IEEE

CRYSTALLOGRAPHIC ELASTO-PIEZO- DIELECTRIC MATRICES, IEEE

MAGNETOELECTRIC MATRICES

THE NEUMANN PRINCIPLE Ø Effect’s symmetry is always -at least- equal to cause’s symmetry

Cause Effect Electromagnetism Charges and currents E and B fields Crystal Physics Structure Properties

Scalars, polar and axial vectors

2

4 ˆ r dq d πε r E =

Axial (or “pseudo-”) vectors (B) transform almost like polar

• vectors. Except…they ignore

the inversion transformation

m*

E Polar vectors (E) transform as position vectors

2

ˆ 4 r d i d r l B × = π µ

Scalars (Q) are invariant under symmetry operations

x m

• B

m E

Q

p µ

L.Fuentes, R. Font (1993) Rev. Esp. Fís. 7 (2), 49

• L. Fuentes, Ma. E. Fuentes: “La Relación

Estructura-Simetría-Propiedades en Cristales y Policristales”. Reverté, México D.F. (2008) ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡

• ⎥

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡

6 5 4 3 2 1 3 2 1

T T T T T T d d d d d d d d d d d d d d d d d d P P P

36 35 34 33 32 31 26 25 24 23 22 21 16 15 14 13 12 11

The irreps approach: Piezoelectricity in C2v

A selection of material properties databases and representation tools:

• The classical: Landolt-Börnstein

(http://materials.springer.com/)

• The materials project. UC Berkeley

(https://www.materialsproject.org/)

• WinTensor. Univ. Washington

( http://cad4.cpac.washington.edu/ wintensorhome/wintensor.htm)

• MPOD. UniCaen, CIMAV et al (

http://mpod.cimav.edu.mx)

THE REPRESENTATION OF COUPLING INTERACTIONS IN THE MATERIAL PROPERTIES OPEN DATABASE (MPOD)

http://mpod.cimav.edu.mx

Dielectric constant Piezoelectric constant d Elastic compliance s

BaTiO3 4mm

Young modulus

Dielectric constant

∞⁄mmm

Piezoelectric charge constant d à ∞mm Elastic compliance s

4⁄mmm

BaTiO3 4mm

Young modulus

4⁄mmm

Olivine structure, magnetic space group: Pnma’ Magnetic point group: mmm’= D2h:C2v

Magnetoelectricity in LiCoPO4.

Data from Vaknin et al. (2002).

∝=[█0&15&0@30&0&0@0&0&0 ]

Single-crystal ME tensor (T = 10 K)

x = 0 and y = 0 à symmetry planes; z = 0 à anti-symmetry plane

y x

Olivine structure, magnetic space group: Pnma’ Magnetic point group: mmm’= D2h:C2v

Magnetoelectricity in LiCoPO4.

Data from Vaknin et al. (2002).

∝=[█0&15&0@30&0&0@0&0&0 ]

Single-crystal ME tensor (T = 10 K)

x = 0 and y = 0 à symmetry planes; z = 0 à anti-symmetry plane

y x

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡

3 2 1 33 32 31 23 22 21 13 12 11 3 2 1

H H H P P P α α α α α α α α α

Magnetic coupling: Magnetoelectricity

LiCoPO4

Rivera, Ferroelectrics 161, 147 (1994)

3D printing of longitudinal properties surfaces

050 – Elasticity 088 - Elasticity 304 - Piezoelectricity 095 - Elasticity PZN-PT Au BaTiO3 PIN-PMN-PT

Polycrystals Properties

∫ ∫

⋅ = = dg g f g K KdV V K ) ( ) ( 1 ~

Polycrystal properties ≈ single crystal properties, modulated by texture.

≈ means that:

• Physical interactions among grains,
• Stereography,
• Porosity and other heterogeneities…

are factors that play possibly important roles, depending on the case. ​𝑒 ↓𝑗𝑘𝑙 =​1/8​𝜌↑2 ∭↑▒∑𝑛=1↑3▒​∑𝑜=1↑3▒∑𝑝=1↑3▒​a↓𝑗𝑛 a↓𝑘𝑜 ​ a↓𝑙𝑝 ​𝑒↓𝑛𝑜𝑝 𝑔(​ 𝜒↓1 , 𝜚, ​𝜒↓2 )​𝑡𝑗𝑜𝜚 𝑒𝜚 𝑒​𝜒↓1 d​φ↓2

DIELECTRIC CONSTANT - TEXTURED AURIVILLIUS CERAMICS

Inverse pole figure: PbBi4Ti4O15 . Single crystal dielectric constant: ε11 = ε22 = 18300; ε33 = 426

2

) (

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Ω −

=

φ

e R h R

Single crystal Polycrystal Ω = 30° Polycrystal Ω = 60° Random polycrystal

Longitudinal piezoelectric module. Quartz polycrystals

e f = f(g)

) / (

• j

j 2

Ω Ω

∑

1]

• +

) + ( ) + [(1 2 1 =

1

φ ϕ ϕ φ cos cos cos cos

2 1

Ω Ω(°) 10 à 30 à Tridimensional texture. ODF: Euler space

ME coefficient for textured LiCoPO4 polycrystal

L.E. Fuentes-Cobas, J.A. Matutes-Aquino, M.E. Botello-Zubiate, A. González-Vázquez,

M.E. Fuentes-Montero, D. Chateigner: “Advances in Magnetoelectric Materials and their Application”. Ch 3, Vol 24, “Handbook of Magnetic Materials”. Editor: K.H.J. Buschow. Elsevier (2015).

A closer look at THE PROBLEM OF AVERAGING

∫

= XdV V X 1 ~

∫

= YdV V Y 1 ~

X K Y ~ ~ > =<

∫

Δ ⋅ Δ + ⋅ = XdV K V X K Y 1 ~ ~ ~

Bunge (1982) Texture Analysis in Materials Science: Mathematical Methods

Average or “effective” value of a property: Mean values of magnitudes:

K K ~ >= <

http://www.cimav.edu.mx/investigacion/software

∫ ∫

⋅ = = dg g f g K KdV V K ) ( ) ( 1 ~

• nly if the

independent variable is constant in the sample volume.

The Reuss approximation

S = sD⋅T + g⋅D E = -g⋅T + βT⋅D

Series configuration: T, D, B constant Averaging sD, βT, g,..: OK

T = cE⋅S - e⋅E D = e⋅S + εS⋅E

The Voigt approximation

Parallel configuration: S, E, H constant Averaging cE, εS, e,..: OK

CONCLUSIONS

Ø The Reuss, Voigt and Hill treatments allow the approximate prediction of textured polycrystals physical properties. Ø “Coupling” interactions and properties (for example, piezoelectricity) require careful consideration of sample stereography and of dependent/independent variables. Ø The software system ANAELU-MPOD facilitates: Ø The interpretation of 2D-XRD patterns, providing input data for MPOD Ø The representation of single-crystal tensor properties Ø The calculation of polycrystal properties estimates

THANKS FOR YOUR ATTENTION!

http://cimav.edu.mx/investigacion/software/ http://mpod.cimav.edu.mx

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