grain boundaries in ceramics. grain boundaries grain boundary: interface between two crystals...
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Grain boundaries in ceramics
Grain boundaries
Grain boundary: interface between two crystals (grains) of the same phase but different orientation.Regions with:
- lower density- different coordination of atoms/ions- relaxation of atomic positions- often different composition (segregation of impurities, dopants, lattice
defects)- different properties (charge and mass transport, dielectric, optical, etc.)
Small angle tilt boundary: misfit accommodatedby formation of dislocations Low energy tilt boundary: coincidence of lattice positions
Dislocations
Edge dislocation
Screw dislocation
Line defects originated by the relative shearing of two parts of a crystal (plastic deformation).
Non-equilibrium defects, can not be treated by thermodynamics.
Grain boundaries
Left. HRTEM image of the 5.4 [001] (010) symmetrical tilt grain boundary in SrTiO3. Right. Strain field around the dislocation cores evaluated from the HRTEM image. The size of the lozenges reflects the unit cell size in the respective area.
Simulated image. Half oxygen atoms were removed from one O column on the g.b. (white arrow). The oxygen deficiency produces a higher brightness in this position ().
O Sr-O
Imaging of oxygen sublattice and grain boundary structure in SrTiO3 by HRTEM
Central dotted lines connect CSL positions(CSL: coincidence site lattice)
Tilt grain boundary of SrTiO3
The intensity profile along the g.b. shows that the intensity of the O column s is variable. Column with oxygen vacancies
110
Observed displacements and grain boundary expansion in agreement with first-principle calculations (0.06 nm expansion). Expansion related to Ti-Ti repulsion and existence of O vacancies on the g.b.
Relaxation of atomic positions near the grain boundary
Expansion compared to perfect lattice
Differences of Ti-Ti spacing along the direction perpendicular to the g.b. The spacing closer to the g.b. is smaller. Increased separation of the two Ti columns facing each other at the g.b.
Differences of Sr-Sr spacing along the direction perpendicular to the g.b. The spacing closer to the g.b. is larger. Decreased separation of the two Sr columns facing each other at the g.b.
Differences normal to the g.b. of Sr-Sr spacings located on CSL sites. The first spacing is increased meaning that there is an expansion of the g.b. (0.043 nm, 1% lattice parameter).
Decreased spacing
Increased spacing
Ti-Ti Sr-SrSr(CSL)-Sr(CSL)
Segregation at grain boundaries
Surface tension or surface energyinPT
rev
A
G
dA
dw
,,
Natural tendency to minimize the surface energy of a system by redistribution of the components. Components which lower the surface/interface energy tend to concentrate at the surface/interface (adsorption, segregation). Oxygen is strongly surface-active in liquid metals and non-oxide ceramics. Segregation at surfaces and grain boundaries in ceramics is determined by the different formation energies of defects at interfaces than in bulk.
Segregation in alumina ceramics as visualized by SIM
250 ppm MgO 1000 ppm La2O3250 ppm CaO
bulkMMseg EEH ,sup, OOAl VOMgMgO2
122 '
OAl OLaOLa 332
Segregation modifies the energy of the different crystallographic surfaces and, consequently, the equilibrium shape of crystals
Segregation in alumina ceramics – atomistic simulation
Equilibrium morphologies of undoped -alumina (a) front and(b) top view. Equilibrium morphologies of 10 ppm Y-doped -alumina at 1600°C seen again from (c) front and (d) top view.
Wulff’s theorem for the equilibrium shape
i
i
hhh
.....
2
2
1
1 i: surface tension of ith facehi: distance from the center
Surface energies of pure and Y-doped -Al2O3
Basal plane
Interfacial energies of pure and Y-doped -Al2O3
.)(int.)(2 surf
Predicted grain boundary structuresFor highly symmetric (left) and more general case (right).
Calculated grain boundary structure showing a regular La pattern resulting from segregation.
Segregation in alumina ceramics – atomistic simulation
Enthalpy of segregation for La in alumina
Segregation in alumina ceramics – atomistic simulation
Calculated and experimental solubility limit of MgO in alumina as a function of grain size
bulkMMseg EEH ,sup,
Grain boundary phases and films
In many cases, solid phases located at grain boundaries result from the solidification of a liquid phase formed during sintering. The grain boundary phase can form a continuous film, pockets at the triple junctions or discrete particles.
Wetting of a liquid on a solid
SLSVLV cos
> 90°: nonwetting < 90°: wetting
= 0°: spreading
Necessary condition for spreading: SVLV
SL θ
LV
SV
Wetting of grain boundaries
2cos2
SLGB
Liquid phase forming additives in ceramic oxides: SiO2, glass, alkaline oxides (Li2O, Na2O, K2O), alkaline-earth oxides (CaO, SrO, BaO), TiO2, B2O3, CuO, ZnO, V2O5
Distribution of liquid/amorphous phase at grain boundaries
(1)
(3)
(4)
(2)
(1)Y2O3:ZrO2
CaO:Si3N4
AlN
(3)Y2O3:ZrO2
(2)Si3N4
(4)AlN
SL
GB
2
1
2cos
“Special” grain boundaries show little segregation and are free of an amorphous grain boundary layer
Criterion for film formation: GBCA 2
Special (A) + random (B) grain boundaries
Random grain boundary
Grain boundaries in SrTiO3 ceramics
Distribution of liquid/amorphous phase at grain boundaries
2A: interfacial energy of a gb containing a wetting amorphous phase
GBC: interfacial energy of a clean gb
No grain boundary layer
GBC
(rotation)
A
A
Distribution of secondary phase at grain boundaries
Ordered grain boundary phase in Ti-rich BaTiO3 ceramics
Segregation and space charge at grain boundaries
The defect formation energies and defect chemistry at the grain boundaries is, in general, different from that of the bulk. Preferential segregation of charged defects in ionic solids leads to net charge at the grain boundary core which is compensated by a space charge cloud of opposite sign adjacent to the boundary, with formation of an electrostatic Schottky barrier.The thickness of the space charge layer is of the order of the Debye length.
220
2 Fzc
RTr
r: relative dielectric constantz: number of charges on defectc: defect concentration in the bulk
Schematic diagram of a positively charged grain boundary (segregation of oxygen vacancies) and compensating space charge (acceptor impurity). The region adjacent to the grain boundary will be depleted in oxygen vacancies.
OSrOTiSrTiO OSrVFeOFeSrO 5222 '
323
2
2
1exp
)(
)(
Lx
c
xc
O
O
V
V L is the width of the space-charge layerL = 2.5 nm in Y-doped ZrO2.
Segregation and space charge at grain boundaries
Transport through a polycrystal. Due to anisotropy of grain boundaries and their specific topology, different situations are encountered: (a) parallel effects, (b) perpendicular effects and (c) e flux constriction.
The effect of grain boundaries on the properties of ceramic
oxides
The segregation of oxygen vacancies in acceptor-doped oxygen conducting electrolytes (Y:ZrO2, Gd:CeO2. Fe:SrTiO3) leads to positively charged grain boundaries cores and a depletion of oxygen vacancies in the adjacent space charge layer. The combined effect of the electrostatic potential barrier (Schottky barrier) and the depletion layer determines a decrease of the oxygen conductivity at gbs (blocking gbs). In doped zirconia ceramics with clean boundaries the resistivity of gbs is at least two orders of magnitude higher than the bulk resistivity. A size effect is expected for grain dimensions in the nanoscale region (grain size <4λ).
Ionic conductivity in oxides: the effect of grain boundaries and grain size
Specific bulk and grain boundary conductivity in 3 mol.% Y2O3 doped ZrO2 (oxygen conductor)
At present, the minimum grain size (30-40 nm) of dense Y:ZrO2 ceramics is still >> 4λ and strong size effects on ionic conductivity are not observed.
OOZrZrO VOYOY 322
32Oxygen conduction in Y-doped ZrO2
Dopant segregation: decreasing effective bulk dopant concentration
Space charge effect
Because of the high density of gbs in nanoceramics, the total conductivity (not shown) is dominated by the resistive grain boundaries.
D
[VO••]
[VO••]∞
+++
+++ D
[VO••]
[VO••]∞
+++
+++
2
D
[VO••]
[VO••]∞
+++
+++
Mesoscopic fast ion conduction in thin-film heterostructures
Parallel ionic conductivity in CaF2-BaF2 thin-film heterostructures with overall thickness L comprising of N layers of thickness d. The overall thickness (L) is approximately the same in all cases.
d = L/NBlack lines: reference single phase films;Green lines: semi-infinite space-charge zones (period >8)Red lines: overlapping space-charge regions (period <8)
σT versus 1/T
Variation of ionic conductivity with the density of interfaces, N/L.
Nanosize effect. Loss of individuality of the single compounds.
interface effect
d < 8λ
d > 8λ
d=50nm
20nm
16nm
430nm
d
d
FMF
F VFF '2
Ionic conductivity in oxides: the effect of grain boundaries and grain size
Impedance spectroscopy
Y axis
Fourier transform
v(t) V(ω)i(t) I(ω)
v(t) = v0sin(ωt)
i(t) = i0sin(ωt+θ)
v: voltagei: current: angular frequencyθ: phase difference
Time domain
V(ω) = I(ω) Z(ω)Z(ω) = 1/(C ω j)
Z: impedanceC: capacitancej = 1
V(ω), I(ω) and Z(ω) are complex quantities
Frequency domain
ωp R1 C1 = 1
Simple RC circuit
ω R1 C1 = 1
R0R0+R1
R1
ω
Solid materials can be described by one (homogeneous single crystal) or more (ceramics, composites) semicircle in the impedance plot. Each semicircle is described by one resistive and one capacitive component.
d
S
d
SC r 0 d
S
Ionic conductivity in oxides: the effect of grain boundaries and grain size
Proton conduction in Y-doped BaCeO3
Sintered 1250°C/2h; gs: 0.38 μm Sintered 1500°C/48h; gs: 5 μm
Trivial size effect. The lower conductivity (left) of the fine grained ceramic is only due to the higher density of resistive grain boundaries. The specific conductivities (right) are the same irrespective of grain size.
OHOHOV
OVYYBaOOYBaCeO
OO
OOBaCeBaCeO
2
52223
3
2
''32
Total conductivity
grain interior
grain boundary
Intrinsic conductivity
Ionic conductivity in oxides: the effect of the grain boundary phase
Impedance spectra
Bulk resistivity
Grain boundary resistivity
freq.(a)
(b)
ZrO2: 3 mol % Y2O3
(a)
Continuous grain boundary phase
ZrO2: 6 mol % Y2O3
(b)
Lenticular grain boundary phase + clean boundaries
Colossal permittivity in CaCu3Ti4O12 : the role of interfaces
TiO6 octahedra
Ca
Cu
• Perovskite-like, non polar structure
• Not a ferroelectric relaxor
• Ab-initio calculations: r = 40
• Processing-dependent properties
tan0 rac tanδDielectric lossDissipation factor
Ceramic
102 Hz
106 Hz
Rel
ativ
e di
elec
tric
con
stan
t
Step-like behaviour of dielectric constant observed in ceramics as well as in single crystals. Strong frequency dispersion
l
S
l
SC r 0 l
S
The dielectric constant (real part of dielectric permittivity) is calculated from the measured capacitance C taking into account the sample geometry:
Single crystal
20 Hz
106 Hz
Rel
ativ
e di
elec
tric
con
stan
t
Colossal permittivity in CaCu3Ti4O12 : the role of interfaces
Colossal permittivity in CaCu3Ti4O12 : the role of interfaces
Insulating gbs
Semiconducting grains
Semiconducting core
Insulating skin
semiconducting ceramic
insulating layer
electrode
- semiconductive grain interiors; - more insulating grain boundaries and related interfacial polarization - insulating layer at the electrode-ceramic interface; - insulating surface skin
} IBLC effect – only for ceramicsMaxwell-Wagner relaxation
} also exist in single crystals
Brick layer model of a ceramic
Apparent colossal dielectric constant is of extrinsic oringin and is associated to the electrical heterogeneity of the samples and the contribution of different interfaces:
Colossal permittivity in CaCu3Ti4O12 : the role of interfaces
The step-like behaviour of the dielectric constant and all other electrical properties can be reproduced by using equivalent circuit models.
l
S
l
SC r 0
l
S
lR
bbR
bbC
gbgbgb xR 3
1
gb
b
gb
gbgb xx
C 33
b
gb
gb
b
C
C
D
d
Cgb>>Cb
Cb depends only on composition
Cgb depends on microstructure
Intrinsicbehaviour
1V
Vx bb
1V
Vx gbgb
D
d
ρgb >> ρb; gb = b
S
Colossal permittivity in CaCu3Ti4O12 : the role of interfaces
Inhomogeneous conduction probed by atomic force microscopy (AFM)
Vceramic
conductingtip
electrode
Current image Topograpicimage
Fracturesurface
Insulting grain boundaries (brown) Strongly nonlinear
current-voltage properties
Influence of grain size on the dielectric constant of ferroelectric BaTiO3 ceramics
Progressive depression of the dielectric constant with decreasing grain size when d1< 1 micron
d1
d2
ε 2
ε 1 >> ε 2
ε 2 ε eff ε 1
ε 1
The microstructure of BaTiO3 ceramics corresponds to ferroelectric grains with high dielectric constant (ε1 = 3000-5000) separated by non ferroelectric (ε 2 100) grain boundaries (“dead layer”). The NFE gbs do not necessarily imply a second phase grain boundary layer.d2 = 1-3 nm depending on ceramic preparation method.
“clean” boundary
2
2
1
11
x
gx
eff
εi ≡ Ki’
grain
grain boundary
Relative dielectric constant (298 K, 10 KHz) of dense BaTiO3 ceramics, 1998-2006
0
1000
2000
3000
4000
5000
6000
10 100 1000 10000
Grain size (nm)
Rel
ativ
e d
iele
ctri
c co
nst
ant
Arlt et al., HPSFrey & Payne, IPRandall et al., CSMRandall et al., HPSTakeuchi et al., SPSZhao et al., SPSBuscaglia et al., SPSDeng at al., SPSZhu et al., SPSWang, 2SS
Dead layer effectDomain size and
mobility effect
HPS: hot pressingIP: pseudo isostatic pressing in a multi-anvil cellCSM:combined sintering methodSPS: spark plasma sintering2SS: two-step sintering
Influence of grain size on the dielectric constant of ferroelectric BaTiO3 ceramics
Dispersion of experimental values related to processing (purity and stoichiometry of powders, sintering method) and microstructure (porosity, second phase grain boundary layer)