Structural transformations in nanomaterials

Deepak Varandani

Nanomaterials

Materials with structural elements which have at least one dimension less than 100 nm

Polycrystals with finest grains and extremely high fraction of boundaries

Quantum confinement of charge carriers Larger fraction of surface atoms

• Lead to significantly altered properties

Characteristic features

Definition

Boundary/Interface volume fraction

10 100 10000.0

0.2

0.4

0.6

0.8

1.0

Crystalline component Intercrystalline component

Vo

lum

e fr

acti

on

Grain size (nm)

d= 5 nm

V.F.= 50 %

d Δd6Δ

2dπ

34

Δ2d4π

C 3

2

=

=d= 12 nm

V.F.= 50 %

Structure of boundaries

Adjacent misoriented crystallites separated by grain boundaries

Boundaries carry the crystallite geometric mismatch

Dislocations, Vacancies

Coarse grained polycrystals

Volume fraction extremely low (<1%)

Low angle, high angle, non-equilibrium, amorphous

Structure of…..

Nanocrystals

Volume fraction high. Triple junctions important

Conflicting reports

• Long-range stresses, frozen-gas like behaviour, reduced density, high energy

• Well ordered, low energy, small excess volumes

•Structure mostly non-equilibrium

Boundary structure crucially depends on synthesis conditions

Affects the structure of the crystallites they surround

Structural transformations1. Solids exist in different structural phases depending on temperature,

pressure and other ambient conditions

2. In nanomaterials size is an additional variable controlling structure

amorphization

Nanomaterials

allotropictransformations

latticedistortion

metastable phases

crystallites/grains

Lattice distortion

38 39 40 41 42

2θ (deg.)

Inte

nsity

(a.u

.)

26 nm

14 nm

9 nm

0 20 40 60 802.4

2.8

3.2

3.6

B(2

93

)(Α

)2Grain size (nm)

The variation in Pd (111) peak with the nanoparticle size, showing the size induced lattice contraction

Variation of Debye Waller parameter with grain size in Se

Unit cell dimensions Debye-Waller parameter Debye temperature

Pd nanoparticle layers Se nanoparticle layers

Zhang 1997Aruna 2005

Lattice expansion or contraction

Metastable phase stabilization

0 20 40 60 80 1000.040

0.045

0.050

0.055

0.060

0.065

αθγ

Particle size (nm)

N.U

.C.V

.

50 60 70 80 90 10050

60

70

80

90

Tet

rag

on

al P

ha

se (

%)

Particle Size (nm)

Normalized unit cell volume (N.U.C.V.) as a function of particle size for different phases of Al2O3

Mole percentage of tetragonal phase in BaTiO3

as a function average particle size

Lattice expansion along with structural transformation in Al2O3 nanoparticles

Cubic metastable phase in BaTiO3

Kwon 2006Ayyub 1995

Theoretical considerations

Interface driven structural transformations

Nanocrystallites enveloped by highly non-equilibrium grain boundaries reduced density or excess volume vacancies, vacancy clusters, extrinsic dislocations

Defects generate stress fields

Atoms displaced from equilibrium positions due to stresses

Interface driven…..

)11()2(

2

1distortion lattice Relative 3 −∆+

++= Va

a

d o

o

ζζζ

Square shaped grains with orthogonal boundaries

Stress due to vacancy and vacancy clusters ∝x-3,. x is the distance from the defect center

a0= perfect lattice interatomic separation

ξ= mean grain boundary width

ΔV= excess grain boundary volume

d=crystallite diameter

Lattice distortion depends on a0 and microstructure

Distortion mainly in thin layer near boundary

Qin 1992

Thermodynamic treatment

G=U-TS+PV…………….Volumetric free energy

G=U-TS+(P+ΔP)V+γA …….. Size independent

γA= Surface free energy

U=internal energyT=temperature, P=pressure, V=volume, A=area ΔP=excess internal pressure due to surface stressγ=surface energy density

Free energy G decides which phase is stable

G is modified for small particles

Gilbert 2003

Thermodynamic……

At small sizes metastable phases may have low total G due to low γ

Thus phase inversion at nanodimensions is possible

Size-dependence of structure in Co nanoparticles

Structure Size Surface energy density (J/m2)

hcp (bulk stable)

fcc (metastable)

bcc (metastabe)

10-20 nm

2-5 nm

2.79

2.73

2.73

Ram 2001

Thermodynamic ……..• Co nanoparticles

Sample Lattice parameters (nm)

Lattice area (10−2 nm2)

Lattice volume

(10−3 nm3)

Lattice surface energy

(10−20 J)

Bulk

hcp structure

fcc structure

a=0.2507

c=0.4070

a= 0.3545

93.90

75.40

66.50

44.55

261.98

205.85

bcc structure

fcc structure

a =0.2840

a=0.3540

48.39

75.19

22.91

44.36

132.11

205.27

fcc structure a=0.3535 74.98 44.17 204.70

•Thus lower surface energy ensures that below a critical size fcc or bcc phase is stabilized in preference to hcp phase

Amorphization

Gc<GA+GD

Gc=Free energy of crystalline phase

GA=Free energy of amorphous phase

GD=energy increase due to defects

In nanomaterials the anti-site disorder & anti-phase boundaries increase Gd, resulting in amorphization

Universal thermodynamic approach

Pl-Po=2γ/r

Pl=pressure inside

Po=pressure outside

γ=surface tension/energy

r=radius

Temperature (K)P

ress

ure

(G

Pa

) Metastable phase

Stable phase

Liquid

Phase boundary line

Temperature (K)P

ress

ure

(G

Pa

) Metastable phase

Stable phase

Liquid

Phase boundary line

As size decreases metastable phase region is driven into its strongly unstable region due to the shift in the phase boundary line

In fine particles internal pressure increases due to Laplace-Young effect

Wang 2005

P(r,T)=a+bT-2γ/r

Bond-OLS correlation mechanism

• Atoms at surface suffer bond order loss

• Spontaneous relaxation of rest of bonds: Contraction or expansion

• Reduced binding energy and increase in bond strength

• In nanomaterials effect is significant

]8/)12exp[(1

2)(

)()(

iiii

BmiiB

ii

zzzc

bEcbE

bcb

−+=

=

=−

ci = contraction (<1) or expansion (>1) factor for the ith layer (i≤3)bi, EB(bi) are respectively the bond length and the binding energy of the ith atomic layer of atoms

b, EB(b) are respectively the bond length and the binding energy of the bulk atomszi = coordination number of the ith layer

m = a parameter which varies with the nature of the bond, being equal to 1 for elemental solids and 4 for compounds and alloys.

Sun 2002

Bond-OLS ……..

∑≤

−=∞

∞−=∞

∆3

)1()(

)()(

)(

)(

iii cb

bDb

b

Db γ Δb= lattice distortionγi = Ni/N= weighting factor

2 4 6 8 10 12

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

Lat

tice

co

ntr

acti

on

(%

)

D (Lattice constant)

0.88/0.96 0.90/0.97 Ni Cu Ag

The bond-OLS correlation mechanism as applied to the case of lattice contraction in Ni, Cu and Ag. Agreement is reached by taking z1=4 (c1=0.88, 0.9), z2=6 (c2=0.96, 0.97) and m=1 (Sun 2002).

Conclusions

• Boundary component critically influences structure• Boundary defects generate stress fields, leading to

static distortion of lattice• Metastable phases stabilized below a critical crystallite

size: Explained using free energy considerations• UTA and BOLS mechanism commonly invoked to

explain structural transformations• Need to recognize the role of synthesis conditions

• Size calculated using different techniques. Unambiguous comparison difficult

References

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(9), 6135 (1995)• Soon-Gyu Kwon, Kyoon Choi, Byung-Ik Kim, Materials Letters 60, 979 (2006)• W. Qin, Z. H. Chen, P. Y. Huang and Y. H. Zhuang, Jl. Alloys and Comp. 292, 230 (1999)• Benjamin Gilbert, Hengzhong Zhang, Feng Huang, Michael P. Finnegan, Glenn A.

Waychunas and Jillian F. Banfield, Geochem. Trans. 4, 20 (2003)• S. Ram, Materials Science and Engineering A 204-306, 923 (2001) • C. X. Wang, G. W. Yang, Materials Science and Engineering R 49, 157 (2005) • Chang Q Sun, B. K. Tay, X. T. Zeng, S. Li, T. P. chen, Ji Zhou, H. L. Bai and E. Y. Jiang, J.

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