SurfaceSurface Energy,Energy, SurfaceSurface StressStress && ShapeShape ofof NanocrystalsNanocrystals Partly bonded surface atoms

. When the calculation of the lowering of the energy of the system on the formation of the condensed state was done all the atoms were taken into account

(assumed to be bulk atoms) → i.e. an over-counting was done . The ‘higher energy’ of the surface is with respect to the bulk and not with respect to the gaseous (non-interacting) state

. Hence the reference state for the surface is the bulk and not the gaseous state

 Hence, it costs energy to put an atom on the surface as compared to the bulk → origin of Surface Energy ()

 The surface wants to minimize its area (wants to shrink) → origin of Surface Tension ()

Let us look at the units of these two quantities Force F []N Energy EJ[] [Nm] [N]    Length L []m Area A []m22[]m [m]

. Dimensionally  and  are identical → Physically they are different type of quantities .  is a scalar while  is a second order tensor

LIQUIDS Surface Energy  Surface Tension

SOLIDS Surface Energy  Surface Tension + Surface energy is Anisotropic Except in certain circumstances

A comparison of the solid and liquid surfaces

 Surface Energy  Surface Tension LIQUID SURFACE Characterized by one number → the surface density

. Liquids cannot support shear stresses (hence use of the term surface tension)

 Surface Energy  Surface Stress (Tensor)  Surface Torque

SOLID SURFACE Has a structure and hence more numbers may be needed to characterize a solid surface

 Crystalline surfaces → all the lattice constants will be required  Amorphous surfaces → Density + a Short Range Order parameter

. In the case of solids the term surface tension (which actually should be avoided) refers to surface stresses

Surface Energy () → is the reversible work required to create an unit area of surface (at constant V, T &  ) i Surface Tension () → is the average of surface stresses in two mutually perpendicular directions

 x  y    2

Surface entropy (Ssurface) → surface atoms have higher freedom to move and have a higher entropy

. Surface stress at any point on the surface is the force acting across any line on the surface which passes through this point in the limit the length of the line goes to

zero

The definition of surface tension in 2D is analogous to the definition of hydrostatic pressure in 3D  Liquid surfaces are characterized by a single parameter: the density (atoms / area)  The short range order in liquids (including their surfaces) is spatio-temporally varying → hence no structure (and no other characteristic) can be assigned to the

surface  Crystalline solids have a definite structure in 3D and hence additional parameters are required to characterize them

 The order at the surface of a crystal can be different from the bulk  Surface relaxation

 Surface reconstruction

 Amorphous solids have short-range order, but NO long-range order. Under low temperature conditions and short times (i.e. low atomic mobility regimes) the

atomic (entity) positions are temporally fixed

From -plot to EQUILIBRIUM SHAPE OF CRYSTAL → the Wulff construction

 Draw radius vectors from the origin to intersect the Wulff plot (OA in Figure)  Draw lines  to OA at A (line XY)

 The figure formed by the inner envelope of all the perpendiculars is the equilibrium shape

 Wulff plot → Equilibrium shape  From the equilibrium shape → it is not uniquely possible to construct a Wulff plot

 Wulff plot with sharp cusps  equilibrium shape = polyhedron  Width of the crystal facets  1/(surface energy) → largest facets are the ones with lowest energy

Size regimes of aggregate of atoms

 Three regimes can be differentiated with respect to number of atoms forming an aggregate: (i) small clusters → the structure and properties do not vary in a monotonic way (ii) Large clusters → this is intermediate between a cluster and a particle (iii) Nanoparticles → here with increasing size the properties slowly approach the bulk value Clusters to Nanocrystals

 Clusters are aggregates of atoms (or molecules or ions), usually with less than a few thousand atoms.

 Typically when the term ‘cluster’ is applied to a collection of a small number of atoms.  The structure and properties of clusters are often very different from the constituent atoms

and their bulk counterparts.

 At this level there are not sufficient atoms to classify the collection as a crystal (which is characterized by long range periodic order).

 If an atom is added to a small cluster of atoms- reconstruction usually takes place- i.e. rearrangement of structure takes place. This implies that the properties of two clusters neighbouring each other in the number of atoms (or entities) could be very different.  At a certain larger size- the Critical Size- the bulk structure will be stabilized.  E.g. small metals clusters on substrates may assume icosahedral, bipyramidal etc. shapes/structures → in many cases when the size reaches about 150 nm crystalline structure is obtained (this crystal structure could be different from the bulk crystal structure).

Shape of Clusters & Nanocrystals

 The shape of nano-clusters and nanocrystals can be very different from their bulk counterparts.

 In fact large clusters may adopt a different crystal structure as compared to the stable bulk crystal structure.

 Given the large surface to volume ratio, equilibrium shape can be attained quickly in nanocrystals.

 Certain Magic Number of atoms may be stabilized in clusters.  Clusters and Nanocrystals may further serve as motifs for ‘higher order’ crystals.

 Hierarchical construction.

Multiply twinned Au nanoparticle (~35 nm)

Mimics 5-fold symmetry

Two ‘variants’ of the twin related by a 72 rotation

Gold nanoparticles (~20nm) can form with icosahedral shape

Schematic of a rotation twin with a combined 5-fold symmetry

Note: In practical examples of such 5-fold twinning the angles may all not be equal to each other (& hence  72) e.g. (i) in twinning of nanocrystalline Cu*, the angles found are 71.2, 70.9, 73.4, 71.5 & 73 (ii) in twinning in twinning in a diamond film**, the angles are : 71.1, 71.4, 72.9, 73.1 & 71.5

* P. Huang, G. Q. Dai, F. Wang, K. W. Xu, Y. H. Li, Applied Physics Letters 95, 203101 (2009) ** Hidetaka Sawada, Hideki Ichinose, Diamond & Related Materials 14 (2005) 109– 112

Size ranges of clusters: alternate classification

 Four size ranges may be identified as in the figure below.

Molecules

Micro-clusters Size ranges Small Particles

Micro-crystals

Molecules Micro-clusters Small particles Micro-crystals

Number of atoms ~1 ~102 -103 ~104 –105 ~106

Radius (nm) > 0.1 ~1 ~10 <50

Surface and interior Interior atoms are ~ atoms have behaviour Surface still ‘active’ Atomic arrangement Usual rules of bonding bulk, surface atoms different from bulk site behave differently atoms Valence electrons exhibit shell structure Quantum size effects Surface plasmons play Electronic properties Discrete energy levels with magic numbers dominate a dominant role (cluster analogue of atom/molecule) Cluster synthesis and characterization

 Clusters may be synthesized by Sputtering, supersonic jet or gas condensation, LASER ablation and vaporization etc.

 Characterization of clusters carried out by: mass spectrometry, TEM, XPS, etc.

Stability and magic numbers

 Magic numbers are usually associated with nucleons in nucleus of an atom and electrons in the orbitals (arising from symmetry and interaction potentials).

 Analogously atomic clusters have their own magic numbers. Mass spectroscopy shows peaks corresponding to certain number of atoms in a cluster → the magic numbers for stability of clusters. These reflect the bonding characteristics between atoms.   Stable Xe clusters form with N = 13, 19, 25, 55, 71, 87, 147… Structural/geometric origin N These numbers correspond to the Mackay Icosahedral clusters. Ca & Mg also show such clusters. This kind of ordered arrangement is incompatible with crystalline symmetry. This arrangement is different from multiply twinned crystals which may be icosahedrally packed. Electronic origin

  Stable NaN clusters form with N = 8, 20, 40, 58, 92 … This stability can be understood as arising from valence electrons of Na moving in a spherical potential.  Filled electron shells → spherically symmetric charge density → van der Waals type interaction between atoms  E.g. Ne, Ar, Kr, Xe

 Mackay icosahedral clusters (Ca, Mg also form icosahedral clusters)  Icosahedral symmetry is not compatible with translational symmetry  Micron sized icosahedral clusters have also been synthesized (  Under high pressures

(~5 Gpa) B6O clusters with icosahedral shape have been synthesized)

 Bond lengths of the atoms inside the cluster are smaller than the bond lengths of the

surface atoms (→ surface relaxation) → interior of the cluster is under higher pressure

 Stable XeN clusters form with N = 13, 19, 25, 55, 71, 87, 147… Geometrical XeN (predominantly position ordering) Reason for magic numbers Electronic NaN (Predominantly electronic shell structures)

 Stable NaN clusters form with N = 8, 20, 40, 58, 92 …

 This stability can be understood as arising from (shell structure of) valence electrons of Na moving in a spherical potential.

 The cluster can be thought of as a ‘Super-atom’

Some issues

 When are there sufficient atoms to call a cluster a crystal?  When does a collection of metal atoms (i.e., behaves like a metal in bulk) show metallic

character?

 When does the energy levels of a semiconducting cluster become continuous? (Energy levels remain discrete up to 7nm for CdS and 14 nm for GaAs)

Alkali Halide Clusters

 Examples are: LiF, NaCl, CuBr, CsI.  Mass spectrum shows an apparent irregularity in cluster abundance.

 For CsI: [Cs(CsI) ]+ (or [Na Cl ]+) with N = 13, 22, 37, 62.. are more abundant N N+1 N corresponding to 333, 335, 355 structures.

+  Large clusters (with N = 171, 364, 665…) of [Na(NaI)N] are very symmetric and start to resemble the bulk structure: with FCC lattice.

Note that the stoichiometry usually cited (e.g. NaCl) is for the bulk crystal + Cluster: [Na14Cl13]

Cl

Actual structure will be ’relaxed’/’distorted’ with respect to this ‘ideal structure’ Na+ Semiconductor Clusters Overview of the richness of possibilities

 Semiconductor clusters are those which are semi-conducting in bulk form (i.e are made of atoms which are semi-conducting in bulk form) → it is usually not implied that the clusters are semi-conducting  Examples: Si, Ge etc.

 Si and Ge prefer maximum coordination number in clusters which is unlike Carbon.

 Very small clusters (N < 10) of Si and Ge are difficult to produce and stabilize. Cluster N N beams with N > 60 for Si and N > 50 Ge have been produced.

 When Si + is fragmented, Si + with N = 10, 8, 6, 4 are produced. 12 N  Small clusters of Si can have metallic or covalent character. Small clusters with higher N coordination number than that in crystalline state (with CN = 4) imply a metallic character

(e.g. in Si7)

 Among small clusters it has been observed experimentally that N = 7 has pentagonal bipyramidal structure.

 Medium sized clusters (N ~ 20-25) exhibit cage like structures essentially with sp3

character (like in bulk). Minimization of dangling bonds on the surface leads to compact

spherical shapes.  For large clusters N > 500 the bulk structure (diamond cubic) is retrieved.

Metallic Clusters

 We have seen that in Metallic clusters with Na, clusters with N = 8, 20, 40, 58, 92… are stabilized. Li, K, Cs, Cu, Ag, Au also show a similar behaviour.

 The stability of metallic clusters is determined by the quantization of electron orbitals. These clusters are referred to as quasi-atoms or giant atoms.

 A ‘jelliium potential’ model is used for the calculation of the energy of the neutral Na clusters. Energy shows a minimum for N = 2, 5, 8, 13, 18, 19, 20…  N = 2, 8, 18, 20 correspond to closed shell configuration of 1s, 1p, 1d and 2s shells resepctively.  While, N = 5, 13, 19 correspond to half-filled shells.

 Electronic shell structures have a role to play in clusters with even 100s of atoms.

 In summary, small clusters are stabilized by electronic shell structures while large clusters

(N > 1000) are stabilized by atomic shells. The shell structure is not compatible with bulk structure which is retrieved with N about 20,000 atoms.  Zn+ and Cs+ have magic numbers N = 10, 20, 28, 35, 46, 54…(due to electronic shell structures).

 Clusters of Ca, Mg, Ba have magic numbers with N = 13, 19, 26, 29, 32 which arise from compact packing of atoms.

Semiconductor Nanoparticles

 Examples of nanoparticles sythesized include: CdS, CdSe, CdTe, ZnS, Mn doped ZnS,

TiO2, ZnO, SnO2 etc

 For basic studies and applications the size, shape and surface characteristics of the particles needs to be controlled.

 III-V semiconductor nanoparticles (GaN, GaP) are more covalent as compared to II-V semiconductor nanoparticles (ZnS)

 Characterization of the particles is done via  Spectroscopy (UV-visible, fluorescence, Raman, X-ray photoelectron (XPS) etc.)  Microscopy (STM, AFM, TEM)  Diffraction (XRD)

Self-Assembled Ordered Nanostructures

 Self-assembling process is a method that organizes/orders units (molecules, nanocrytals, nanoparticles etc.) via non-covalent interactions like hydrogen bonding, van der Waals forces, electrostatic forces etc.

 Under suitable processing conditions ordering can take place of the units without external intervention. (Optimization of process conditions being one of the important tasks in such a process).  Kinds of Self-assembles structures:  Ordered self-assembled nanocrystals  Ordered mesoporous materials  Hierarchically ordered materials (these have ordering at different lenghscales)

Ordered Self-Assembled Nanocrystals

 Each nanocrystal serves as the basic building block (an “super atom”).  The ‘Nanocrystal Solid (NCS)’ can have translational or orientational order.

 Semiconductor (CdSe, CdTe, InP, CdS), Metallic (Au, Ag, Ni, Pt, Co) and Oxide (TiO , 2 CoO, Fe2O3) nanocrystal solids have been synthesized.

 As bare nanocrystal surfaces (especially metallic) are very reactive they may have to be capped before self-assembly can take place.  E.g. Au nanocrystals when put together coalesce to form larger (highly twinned) crystals (and hence the identity of the individual nanocrystals is lost). If surfactant molecules are applied to the surface of Au nanoparticles they can retain their size and shape.  Hence, with the coating of surfactant molecules (monolayer, passivation molecules) the surface of the nanocrystal becomes hydrophobic and can be dissolved in non-polar solvents, forming a stable colloid. If the solvent is evaporated the nanocrystals do not coalesce but can rearrange themselves to form assemblies. The evaporation has to be slow to allow for the rearrangement of the molecules.

Self-assembly is considered and ‘entropy driven’ crystallization process

 Single sized (monodispersive) nanocrystals can be crystallized more easily as compared to nanocrystals with a range of sizes. (Similar to confusion principle for glasses).

 The forces responsible for holding the capped layers together are the weak (non-covalent) interactions (typically van der Waals forces).

 To summarize, the favourable conditions for the formation of ordered self-assembly of

nanocrystals are: i) single size of nanocrystals, ii) passivation layer, iii) slow evaporation rate of the solvent.

 Ag nanocrystals have been assembled in an FCC lattice and the resulting self-assembled structure has orientational and positional order.

 The orientation relation between the particles and the lattice

is as follows: [110]lattice || [110]Ag, [001]lattice || [110]Ag

Ag nanocrystal

Ag Nanocrystal superlattices have been synthesized with ‘tetrakaidecahedral’ shaped Lattice point crystals arranged in an FCC lattice occupied by Ag nanocrystal

Au nanoparticles synthesized by inverse micelles route with a diameter of about 4.5 nm organize into a FCC lattice. Other synthetic methods have led to the formation of ABAB.. type packing.  In spite of the analogy used it should be remembered that 3D packing of atoms is different from 3D packing of nanocrystals:  Atoms (of a single species) have same size but even ‘mono-disperse’ crystals have a small variation is size  Nanocrystals can be faceted (capping layer may alter the ‘polyhedrality’)  Interatomic spacing is fixed by the type of bonding. Interparticle bonding is variable based on the characteristics of the passivation molecule (and can be varied).  Interparticle bonding is not of covalent or metallic or ionic type.

 Particle size may play an important role in the self-assembly process. E.g. in CdSe

dispersions the interactions became ‘repulsive stabilizing’ from attractive as the particle size was increased.

 CdSe nanocrystals have been self-organized into 3D quantum dot NCS (often wrongly termed as superlattices). (the context of the use of the term ‘superlattice’ should be carefully noted). Dipole-dipole inter-dot interactions have been shown to play a role in the ordering process.

 Clusters of TiO2 have been agglomerated into spheres which then form the motif for a ‘superlattice’  two level ordering.

start here Assembling particles with multiple sizes/phases

 For micron sized particles of two mono-disperse crystals A and B the following is predicted:

 0.48 < R /R < 0.62 → mixture will form a stable structure A B  RA/RB ~ 0.58 → AB2 phase will form

 0.458 < RA/RB < 0.48 → phase separation will occur

 For the case of thiol-stabilized gold nanoparticles with two sizes (size ratio RA/RB ~ 0.58)

the prediction for micron sized particles was confirmed for the case of nanocrystals as the

building block → like a binary intermediate phase (AB2).

This is reminiscent of Laves phases in ‘atomic binary compounds’

Size Factor compounds: (i) Laves phases (ii) Frank-Kasper Phases D(i) Laves Phases (1/1.225) = 0.816

 These phases have a formula: AB2  Laves phases can be regarded as tetrahedrally close packed (TCP) structures with an

ideal ratio of the radii (r /r ) = (3/2)1/2 ~1.225 [or usually r /r  (1.1, 1.6)] A B A B  If r /r = 1.225 then a high packing density is achieved with the chemical formula AB A B 2 with a average coordination number of 13.3

 Crystal structures:  Hexagonal → MgZn (C15), MgNi (C36) 2 2  FCC → MgCu (C14) 2  There are more than 1400 members belonging to the ‘Laves family’

 Many ternary and multinary representatives of the Laves phases have been reported with

excess of A or B elements. Some ternary Laves phases are known in systems with no

corresponding binary Laves phases.  The range of existence of the three phases (C15, C36, C14) in ternary Laves phases is influenced by the e/a ratio

Properties/applications of self-assembled nanocrystals

 The properties of self-assembled nanocrystals are different from either the nanocrystals themselves or bulk material (of the same material). In some sense these structures are a

composite of nanocrsytals and the passivating agent (surfactant).

 Hetero-structured diodes have been made with metallic self-assembled nanocrystals connected via molecular nanowires- idea being to reduce the dimension of one diode below

10nm.

 A superlattice of gold nanoclusters (3.7nm diameter) was assembled by organic interconnects (on a silica substrate) showed nonlinear Coulomb charging behaviour.

Ordered Self-Assembled Mesoporous materials

 Porous materials have low density (can be as low as 10-30% of the bulk density), large surface area and low dielectric constant and find applications in catalysis, filtration,

sensors, structural members etc.  Porous materials can broadly be classified into:  Nanoporous materials

(e.g. zeolites with pore size ~1.5nm, Metal organic frameworks (MOF))

 Mesoporous materials

(typically inorganic materials with pore size in the range of 2-50nm)

 Macroporous materials

(with visible pores ~ mm, e.g. Al foams)

 Template assisted method has emerged as a versatile method to form ordered mesoporous materials: Usually mono-dispersed Silica or Polystyrene nanometer sized spheres are used

for the formation of a colloidal suspension → ordered structures is formed when the

suspension is dried. Ordered porous oxides, graphite, organic materials etc. have been synthesized using this technique.  Other techniques of synthesis included: self-assembling mechanism, supra-molecular templating etc.

Incorporation of highly dispersed gold nanoparticles into pore channels of mesoporous silica thin films and their ultra fast non-linear optical response Gu et al, Adv. Mater. 17 (2005) 557.

Eight units surround the pore in the metal-organic framework called MOF-5. Each unit contains four ZnO4 tetrahedra and is connected to its neighboring unit by a dicarboxylic acid group.

Hierarchically Structured Nanomaterials

 A hierarchically ordered structure has ordering at more than one lengthscale.  Spheres of mesoporous material can be ordered with voids in-between the spheres to

produce a material with pores in two lengthscales: one within the spheres with pore size ~

10s of nm and other between the spheres with pore size of the ~ 100s of nm. E.g. ordered mesoporous silica spheres.

 This concept of hierarchical organization can be applied to other structures as well.

Core Shell Nanostructures

 In core-shell structures the core of one material is covered by another material this is a kind of hybrid.

 The shell may be: (i) intentionally designed to impart special properties like exchange anisotropy

(ii) enhance a pre-existing property

(iii) introduced to protect the core from the environment (e.g. superparamagnetic Fe particles carrying a drug is coated with a polymer and then introduced into the bloodstream for targeted drug delivery), (iv) a natural consequence of oxidation/reaction.  The shape of the nanoparticle is usually spherical but can be irregular, rod or wire-like as well.

 Silver nanowires have been coated with amorphous silica → length of the wires > 1000

nm and diameter ~ 50nm

 Notation used: core@shell

 The core usually shows the relevant property while the shell may (i) stabilize the core, (ii) make the core compatible with the environment, (iii) change the charge, reactivity or behaviour of the core surface.  Synthesized by: (i) formation of core followed by shell, (ii) in-situ method (i.e. core and shell formed together)  Some common methods of synthesis are: The shell can be formed by surface chemical reactions, by simple adsorption of molecules on small nanoparticles or the whole core-shell nanoparticle. Can also be formed by self- assembly and cross linking of macro molecules.

Core Shell Metallic Metal, semiconductor, insulator Semiconductor  Metal, metal oxide Organic (polymer, molecules) Cross linked polymer polymer, molecules  Inorganic shell Inorganic core Biomolecules (DNA)

CurvatureCurvature EffectsEffects inin NanocrystalsNanocrystals Curvature effects nanomaterials

 In bulk materials curvature effects can be ignored and the Gibbs Free Energy (G) has no dependence on any geometrical aspect of the material.

 In the case of nanomaterials- say nanocrystalline precipitates in a fine matrix- the free energy (G) of the precipitate is dependent on the size of the precipitate (assumed to be

spherical for now)

 Free energy of small precipitates is curvature dependent (smaller size  higher curvature)

 This is also true for free-standing nanocrystals. This comes about as the atoms on the

surface are ‘less bonded’ as compared to bulk atoms → leading to an increase in the

energy of the system.

(We have already noted that the reference state for the surface is the crystalline bulk state and not the isolated atoms state).  The phenomenon of coarsening (of precipitates) is because of the free energy dependence on curvature.

Particle/precipitate Coarsening

 There will be a range of particle sizes due to time of nucleation and rate of growth.  In the initial stages the precipitate size is in the nanoscale (e.g. GP zone are couple of

atomic layers thick, aging treatment can be controlled to get CuAl precipitate of the size 2 of ~100nm).  In age hardenable Al-4% Cu alloy, low temperature aging: GP zone → ’’ → ’ → 

 Aging  Precipitate type changes

 Size increases

 Interface goes from coherent to Semicoherent



Distorted FCC  '' (001) || (001) 10nmthick ,100 nm diameter [100] '' ||[100]

[100] || [100] (001) ' || (001)  '   ' Tetragonal

~ composition CuAl2  BCT  As the curvature increases the solute concentration (XB) in the matrix adjacent to the particle increases.

 Concentration gradients are setup in the matrix → solute diffuses from near the small particles towards the large particles

 small particles shrink and large particles grow.

  with increasing time * Total number of particles decrease * Mean radius (r ) increases with time. avg

Gibbs-Thomson effect Gibbs-Thomson effect 33 . r0 → ravg at t = 0 rrktavg  0  In . D → Diffusivity c re a s . Xe → XB (r = ) in kDX  ravg g e T

 Q  D is a exponential function of temperature     kT  r  coarsening increases rapidly with T  0 eDD 0 t

dravg k  small ppts coarsen more  Volume diffusion rate 2 rapidly dt ravg Rate controlling 3 factor Linearrtavg versus relation may break down due to diffusion short-circuits or if the processisinterface controlled

Interface diffusion rate Rateof coarsening depends on DX e (diffusion controlled)

. Precipitation hardening systems employed for high-temperature applications must avoid coarsening by having low: , X or D e

Low 

Nimonic alloys (Ni-Cr + Al + Ti) . Strength obtained by fine dispersion of ’ [ordered FCC Ni (TiAl)] precipitate in FCC Ni 3 rich matrix

. Matrix (Ni SS)/ ’ matrix is fully coherent [low interfacial energy  = 30 mJ/m2] . Misfit = f(composition) → varies between 0% and 0.2%

. Creep rupture life increases when the misfit is 0% rather than 0.2%

Nimonic 90: Ni 54%, Cr 18-21%, Co 15-21%, Ti 2-3%, Al 1-2% Low Xe

ThO2 dispersion in W (or Ni) (Fine oxide dispersion in a metal matrix) . Oxides are insoluble in metals. . Stability of these microstructures at high temperatures due to low value of X . e . The term DXe has a low value.

Low D

ThO2 dispersion in W (or Ni) (Fine oxide dispersion in a metal matrix). . Cementite dispersions in tempered steel coarsen due to high D of interstitial C. . If a substitutional alloying element is added which segregates to the carbide → rate of

coarsening ↓ due to low D for the substitutional element.

H C  m  kT DDe 0 Case Study (curvature effects): vacancy concentration & sintering of nanoparticles

 Surface atoms have a higher internal energy and a higher entropy.  Due to ‘Broken bonds’ it costs energy put a vacancy in a crystal. However, there is a

benefit in the configurational entropy. This implies that at any temperature there is an equilibrium concentration of vacancies in a crystal.

. n → number of vacancies nbulk H f . N → number of sites in the lattice exp . NkT Hf → enthalpy of formation of a vacancy [J/atom]  . k → Boltzmann constant . T → Temperature [K] User R instead of k if Hf is in J/mole  Close to the melting point in FCC metals Au, Ag, Cu the fraction of vacancies is about 104 (i.e. one in 10,000 lattice sites are vacant)

 These vacancies (thermal vacancies in equilibrium concentration in the bulk) are responsible for substitutional (or self) diffusion.

HHfm   kT . D (Ni in FCC Fe at 1000ºC) = 2  1016 m2/s DDe 0

 Curvature effects are important in nanocrystals and the vacancy concentration below a curved surface is altered from value derived for a flat surface.

 The sign of the curvature determines if there is an excess or depletion of vacancies with respect to the value for a flat surface.  Below concave surfaces there is an excess concentration of vacancies and the opposite is true for convex surfaces.

 This (the vacancy concentration) further influences the diffusivity below convex and concave surfaces.  below concave surfaces there is an enhanced diffusivity  below convex surfaces there is reduced diffusivity

 As curvature increases with reduction in size the effect becomes pronounced in smaller

particles.

nanocrystal .  → atomic volume X vacancy 1  exp . r → radius of curvature bulk .  → surface free energy XrkTvacancy 

 Convex surface → r positive → Xnanocrystal < Xbulk nanocrystal bulk  Concave surface → r negative → X > X

 An illustrative example of this effect is the sintering of nanoparticles.  Atoms diffuse from convex regions to concave regions, leading to sintering of

nanoparticles at lower temperatures than that employed for large sized particles.

 This is the reason that nanoparticles (typically of metals) have to be kept isolated from

each other by a passive layer (e.g. thiol over gold nanoparticles)- even at room

temperature.

LatticeLattice ParameterParameter ofof NanocrystalsNanocrystals Lattice Parameter

 As the size of the nanocrystal is reduced surface tension effects tend to dominate.  This leads to a reduction in the lattice parameter of the nanocrystal.

Gauss-Laplace Formula . P → pressure difference between inside and outside of droplet . d → diameter of the droplet 4 .  → surface energy P d Compressibility 1 V  K    VP0   T Va 3

 3 a ~1.5% decrease  • d a  0 dKa4 • d a

Schematic trend line showing variation in the lattice

J. Woltersdorf, et al., Surface Science, 106 (1981) 64. parameter of Al with decrease in particle size Thermal expansion of metallic nanomaterials Some startling observations!

 Thermal expansion of nanoparticles is expected to be dominated by . surface tension effects and . reduced geometrical constraint on the particles.

(Some shape dependence may also be expected in this regard).

 We take up couple of startling examples here.

 Bulk metallic materials normally expand on heating (two reasons!).  It has been observed in 4 nm gold nanoparticles that below 125 K the thermal expansion is

positive (with coefficient of thermal expansion  ~ 3.2105 /K)→ akin to bulk materials.

 Beyond 125 K the thermal expansion becomes negative (with  ~ 2.5105 /K) [1]. The

lattice parameter of the particle is ~0.2% smaller than bulk gold.

 The physics of the effect is not fully understood yet, though the valence electron potential on equilibrium lattice separations are expected to play an important role.

[1] W.-H. Li, S.Y.Wu, C. C. Yang, S. K. Lai, and K. C. Lee, H. L. Huang and H. D. Yang, Phys. Rev. Lett. 89 (2002) 135504-1.  In CuO and MnF2 magnetic ordering on cooling leads to expansion of the material → via magnetostriction effect (change in the dimensions of a material when a magnetic field

is applied- caused by shift in domain boundaries and the rotation of domains).

 This effect is observed below the Néel temperature (above T the usual behaviour of +ve N TE is observed).

 This negative thermal expansion (NTE) effect can be amplified by making the particle size in the nanoscale (~5nm)

 In micron sized particles the NTE effect barely overcomes the usual vibrational effects- but in the nanoscale the NTE effect dominates.

Néel temperature

[1] Zheng, X. G. et al. Nature Nanotech. 3, 724–726 (2008).  Rare example of a Bulk material with negative thermal expansion coefficient: zirconium tungstate (ZrW O ) [1]. 2 8  Usually explained in by either geometric flexibility in the linkages or low-energy

phonons.

 In ZrW2O8 the Zr–O–W linkages in this material become increasingly linear at low temperatures as the amplitude of the oxygen vibrations decreases [1].

 The NTE effect in CuO is about 4-5 times that of ZrW O . 2 8

[1] Mary, T. A., Evans, J. S. O., Vogt T. & Sleight, A. W. Science 272, 90–92 (1996). PhasePhase TransformationsTransformations inin NanocrystalsNanocrystals PHASE TRANSFORMATIONS Based on Mass transport Diffusional Diffusionless

Martensitic

PHASE TRANSFORMATIONS

Based on order 1nd order 2nd (& higher) order nucleation & growth Entire volume transforms

1nd order Trasformation Nucleation = of + nucleation & growth  →  Growth till  is exhausted  phase

Homogenous Nucleation Heterogenous

Bulk Gibbs free energy ↓

Interfacial energy ↑ Energies involved Strain energy ↑ Solid-solid transformation

New interface created

Volume of transforming material Liquid → Solid phase transformation

. On cooling just below Tm solid becomes stable . But solidification does not start

. E.g. liquid Ni can be undercooled 250 K below Tm

↑ t

Solid stable Liquid stable

Gv Crude schematic! Solid (GS)

G → ve G → T Liquid (GL) G → +ve

T “For sufficient m T → Undercooling” T - Undercooling

Neglected in L → S Homogenous nucleation transformations Free energy change on nucleation  Reduction in bulk free energy  increase in surface energy  increase in strain energy ΔG  (Volume).( G)   )((Surface).

 4 3  2 ΔG     v ).(  rGr ).(4  3 

 G  f (T ) r3 v f ()r r2

r  1  4 3  2 ΔG     v ).(  rGr ).(4  3  . By setting dG/dr = 0 the critical values (corresponding to the maximum) are obtained (denoted by superscript *)

. Reduction in free energy is obtained only after r0 is obtained

* 2 As Gv is ve, r is +ve Gd * r*   0 r1  0 2 dr Gv Trivial 2 * Gd r   0 Gv dr 3 G  0 * 16  G   2 3 Gv * → r r0 3 G  Embryos Supercritical nuclei G  0 r0  Gv r → v  TfG )( The bulk free energy reduction is a function of undercooling

Turnbull approximation Tm 2  16 3 Tm G  22 * 3TH G  T  ng si ea cr In Decreasing

Decreasing r* → G 

r →

In the nanoscale r* is typically in the range of 1-10 nm

Melting point  Melting point of a material is an indicator of the bond strength of a material (though boiling point is better reference as it is structure independent).

 In bulk systems surface to volume ratio is small and curvature effects can be ignored.  In nanocrystals the number of surface atoms is a considerable fraction of the total

number of atoms. These atoms have a higher energy and higher freedom for vibrations.  This implies that surface atoms are expected to melt below the melting point of the bulk.

 In 0D and 1D nanocrystals curvature effects are important.  As the size of a free-standing crystal is reduced below about 100nm its melting point

is reduced with respect to the melting point of bulk material.  E.g. Au crystals of about 10nm size melt 100C below the bulk melting point (1064C)

 surface 2 SL . TM → melting point of the bulk material TTMM1 surface  . TM → melting point of the surface (first layer) Hrf .  → solid liquid interfacial energy SL . Hf → enthalpy of fusion surface . r → radius of the particle r decreases TM decreases

Other materials like Pb, Cu, Bi, Si show similar trend lines

 If the crystal is embedded in a matrix, its melting point may decrease or increase with respect to the bulk depending on the relative magnitudes of the particle-matrix ( ) and PM liquid-matrix ( ) interfacial energies: LM  If  >   formation of interfacial liquid lowers the energy. PM LM → Depression in the melting point

 If  <  → Superheating (embedded crystal melts above its bulk melting point) PM LM (In nanoparticles in a Al-In alloy matrix)

 This is unlike free-standing nanocrystals where only depression in melting point is possible

Where does the melting start?

 A simple minded guess would be that the melting starts at the surface of the nanocrystal.

 In an experiment by Wang et al (Surface Sci. 440 (1999) L809) on gold nanorods heated by laser

beam. It is seen that the melting occurs simultaneously both inside the nanocrystal and on

the surface. Solid → solid phase transformation in nanocrystals

 Phase transformation in nanoparticles (including nanocrystals) is expected to be different from bulk materials.

 Heterogeneous nucleation at the surface is expected to play a dominant role (due to its proximity).

 Activation energies for the phase transformation are also expected to be different.

 Kinetics of phase transformation is expected to be highly enhanced due to: (i) smaller lengthscale for diffusion involved, Diffusional transformations (ii) dominance of (faster) surface diffusion, (iii) lesser constraint on the system (phase transformation typically involve volume

changes- if the amount of constraining material around the transformed volume is less the

stresses caused are lower- transformation can occur will less impediment).

Smaller strain energy term Bulk Gibbs free energy ↓

Interfacial energy ↑ Energies involved Strain energy ↑ Solid-solid transformation  In bulk materials, multiple nucleation events lead to solid to solid phase transformations.

 In small nanocrystals the size of the transformation nucleus may become comparable to the volume of material  single nucleation event can lead to solid to solid phase

transformation.

 Work done on CdSe and CdS nanocrystals (2.3-4.3 nm size) shows that for pressure induced (3 GPa) transformation from wurtzite to rock-salt structure, a small size (cluster

of 102 to 103 atoms) may be smaller than the critical nucleus size and hence a single

nucleation event can lead to the transformation of the whole volume [1].

3 . CdSe (rock salt) → NaCl type, a = 0.549 nm, Vunit cell = 0.041 nm /motif 3 . CdSe (Wurtzite) → ZnO type, a = 0.429 nm, c = 0.7011Å, Vunit cell = 0.056 nm /motif

[1] Chia-Chun Chen, A. B. Herhold, C. S. Johnson, A. P. Alivisatos, SCIENCE, 276 (1997) 398. [2] Sarah H. Tolbert, Amy B. Herhold, Louis E. Brus, and A. P. Alivisatos, Phys. Rev. Lett. 76 (1996) 4384.

 Nucleation could occur in bulk or on the surface (heterogeneous nucleation).  Activation energy for the transformation is seen to increase with size in the regime of

sizes considered.

 At larger crystallite sizes a single nucleation event is followed by growth for transformation to be complete.

 Additionally, smaller crystallites requires higher pressure for transformation.

 In (SiO2 coated) Si (pressure induced transformation from diamond cubic to primitive hexagonal structure) → it is seen that crystals as large as ~50 nm (with ~106atoms)

transform by a single nucleation event [2].

. As expected the shape of the crystals also change on phase transformation (by homogeneous deformation).

 Bulk behaviour (i.e. multiple nucleation events, followed by growth) is expected to take over after about 100 nm.

[1] Chia-Chun Chen, A. B. Herhold, C. S. Johnson, A. P. Alivisatos, SCIENCE, 276 (1997) 398. [2] Sarah H. Tolbert, Amy B. Herhold, Louis E. Brus, and A. P. Alivisatos, Phys. Rev. Lett. 76 (1996) 4384.

Oxide nanoparticles

 Phase transformation from γ-Fe2O3 (maghemite) to α-Fe2O3 (hematite) [1]  Transition pressure increases with decreasing size (in a certain regime)

 7 nm → 27 GPa

5 nm → 34 GPa

3 nm → 37 GPa

3 . -Fe2O3 → Al2O3 type, hR30, R-3ch, a = 0.504 nm, c = 1.37 nm, Vuc = 0.3005 nm 3 . -Fe2O3 (metastable wrt  form at RT)→cP56, 4132, a = 0.83 nm, Vuc = 0.5814 nm

[1] S.M.Clark, S.G.Prilliman, C.K. Erdonmez, A.P. Alivisatos, Nanotechnology 16 (2005) 2813. Thermal conductivity Summary of some results

 Thermal conduction occurs due to electrons or phonons (Quantized normal modes of lattice vibrations).

 In nanomaterials the phonon wavelength can be comparable to a lengthscale in the microstructure.

 Very high thermal conductivities (~ 3000 W/m/K → 7.5 times that for Cu) are expected due to phonon mechanism in Carbon nanotubes along the length.

 In the case of nanosized Pt films the in-plane conductivity decreased with layer thickness.

 Additionally, the conductivity increased with temperature (in the range 70-340 K).

h h s E  pphonon  phonon   In plane thermal conductivity of Pt thin films Phonon wavelength: -10 λphonon ≈ a0 ≈ 10 m

[1] X. Zhang, H. Xie, M. Fujii, H. Ago, K. Takahashi, T. Ikuta, H. Abe, T. Shimizu, Appl. Phys. Lett. 86 (2005) 171912. Catalysis

 Catalysts play an important role in reactions like: oxidation of hydrogen, hydrogenation of hydrocarbons, oxidation of CO, propylene epoxidation, etc.

 There are only 12 metals (belonging to groups VIII and 1B) which are elemental catalysts.  Most widely used of these are:

(a) 3d metals (Fe, Co, Ni, Cu)

(b) 4d metals (Rh, Pd, Ag) and (c) 5d metal (Pt)  Cu is used as a catalyst in the production of methanol (by hydrogenation of CO) and Ag is used in the production of ethylene oxide (by reaction of ethylene with oxygen).

 Group VIII metals owe their catalytic activity to an optimum vacancy in the d-band (e.g. Ni, Pd).

 Cu, Ag and Au have fully occupied d-band; but, Cu (7.73 V) and Ag (7.58 V) by virtue of their low ionization potential, lose electrons from the d-band and can function as

catalysts in bulk form.  Au on the other hand has a high ionization potential (9.23 V) and performs poorly as a catalyst in bulk form.

 Nanoparticles have better catalytic properties due to two obvious effects: (i) increases surface area per unit volume,

(ii) increased surface activity due to higher degree of unsaturated bonds.

Gold nanoparticles as a catalyst

 As stated before Au has a high ionization potential, thus giving it a poor affinity for molecular hydrogen and oxygen (and would perform poorly as a catalyst for

hydrogenation and oxidation reactions under ambient conditions). (It should be noted that this 'poor catalytic activity' is for smooth bulk gold surfaces at low

temperatures).  Hence, it is very surprising that Au can perform the role of a catalyst in nano form. Au displays a drastic change in behaviour on reduction of size. Au nanoparticles supported on

substrates like Fe2O3 and NiO can be used as a catalyst for CO oxidation even at 473C.

 Factors which play an important role in the catalytic properties of Au nanoparticles are: . size (size distribution and shape) of the particles and

. support substrate related aspects (type of support, interface with support).

 As nanoparticles melt at a lower temperature as compared to bulk materials (5 nm Au crystals melt ~200K below the bulk melting point), the coagulation tendency is high if

nanoparticles are in contact (we have already noted how they can sinter at low temperatures). Au nanoparticles will tend to coagulate to form larger particles and hence a substrate is required to isolate these particles. As we shall see the substrate can have additional beneficial effects, with respect to catalysis.  The substrate used as support for the Au nanoparticles plays a profound role in the catalytic properties of the system (Au + substrate).

First of all, the substrate determines the contact angle which the Au particles make with the substrate (which may be size dependent). E.g. Au on TiO : particles 2 nm particles 2 tended to wet the surface, while larger particles (~5 nm), had a contact angle > 90. This further determines the amount of interface that the Au particles have with the substrate,

which in turn plays a profound role in catalysis. Hemispherical particles with more

interface perimeter show a higher catalytic efficiency than spherical particles attached to the substrate. It is understood that reacting species adsorb on surface and interface defects (like ledges, kinks and interface lines).

 At room temperature and low temperatures (< 300 K) Au nanocrystals beat Pt and Pd hands down by four orders of magnitude in their activity (e.g. in CO oxidation reaction).

Pt and Pd show increased activity at higher temperatures only. This is due to the low activation energy (~30 KJ/mole or less) of Au nanocrystals.