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Surface Energy, Surface Stress & Shape of Nanocrystals. Partly bonded surface atoms.
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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 • 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 neededto 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 a unit area of surface (at constant V, T & i) Surface Tension ()→ is the average of surface stresses in two mutually perpendicular directions 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 number of 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
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 the stability of clusters. These reflect the bonding characteristics between atoms. • Stable XeN clusters form with N = 13, 19, 25, 55, 71, 87, 147…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. • 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. Structural/geometric origin Electronic origin
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… XeN (predominantly position ordering) Geometrical 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)N]+ (or [NaN+1ClN]+) with N = 13, 22, 37, 62.. are more abundant corresponding to 333, 335, 355 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 SiN and GeN are difficult to produce and stabilize. Cluster beams with N > 60 for Si and N > 50 Ge have been produced. • When Si12+ is fragmented, SiN+ with N = 10, 8, 6, 4 are produced. • Small clusters of SiN can have metallic or covalent character. Small clusters with higher 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 respectively. 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-VI 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 lengthscales)
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 (TiO2, 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 || [110]Ag Ag nanocrystal Ag Nanocrystal superlattices have been synthesized with ‘tetrakaidecahedral’ shaped crystals arranged in an FCC lattice Lattice point 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.
Assembling particles with multiple sizes/phases • For micron sized particles of two mono-disperse crystals A and B the following is predicted: 0.48 < RA/RB < 0.62 → mixture will form a stable structure 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 (rA/rB) = (3/2)1/2 ~1.225[or usually rA/rB (1.1, 1.6)] • If rA/rB = 1.225 then a high packing density is achieved with the chemical formula AB2 with a average coordination number of 13.3 • Crystal structures: Hexagonal→ MgZn2 (C15), MgNi2 (C36) FCC→ MgCu2 (C14) • 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.
Curvature effects in 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 CuAl2 precipitate of the size 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 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 (ravg) increases with time. Gibbs-Thomson effect
Increasing T • r0→ ravg at t = 0 • D → Diffusivity • Xe → XB (r = ) D is a exponential function of temperature coarsening increases rapidly with T small ppts coarsen more rapidly Volume diffusion rate Rate controlling factor Interface diffusion rate
Precipitation hardening systems employed for high-temperature applications must avoid coarsening by having low: , Xe or D Low Nimonic alloys (Ni-Cr + Al + Ti) • Strength obtained by fine dispersion of ’ [ordered FCC Ni3(TiAl)] precipitate in FCC Ni 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 is due to low value of Xe. • The term DXe 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.
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 to ‘put’ a vacancy in a crystal (i.e. remove an atom!). 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 • N→ number of sites in the lattice • 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 104 (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. • D (Ni in FCC Fe at 1000ºC) = 2 1016 m2/s
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. • → atomic volume • r → radius of curvature • → surface free energy • Convex surface → r positive → Xnanocrystal < Xbulk • Concave surface → r negative → Xnanocrystal > Xbulk
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.
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. • P → pressure difference between inside and outside of droplet • d → diameter of the droplet • → surface energy Gauss-Laplace Formula Compressibility ~1.5% decrease • d a 0 • d a Schematic trend line showing variation in the lattice parameter of Al with decrease in particle size J. Woltersdorf, et al., Surface Science, 106 (1981) 64.
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 ~ 3.2105 /K)→ akin to bulk materials. • Beyond 125 K the thermal expansion becomes negative (with ~ 2.5105 /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 MnF2magnetic 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 TN the usual behaviour of +ve 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 (ZrW2O8) [1]. • 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 ZrW2O8. [1] Mary, T. A., Evans, J. S. O., Vogt T. & Sleight, A. W. Science 272, 90–92 (1996).
