Nanomaterials: Properties

Nguyễn Tuấn Anh

Scale and Properties

• polycrystalline, nanocrystalline, and amorphous structures

• made up of ordered crystals meeting at boundaries where there is disorder

Scale and Properties• two structural length scales: the crystals and

the atoms• fraction of disordered material

• the crystal size shrinks until it becomes of atomic dimensions completely disordered amorphous

• Properties of amorphous state and nanostructured would represent limits

Scale Dependence of Properties

• basic assumption: small piece has the same values as a large

• greatly simplifying the analysis• However, it does not always work

Mechanical Properties

• Nanodispersions• high-strength aluminum alloys: 4% copper and

96% aluminum• alloy is heated to 550oC, the copper dissolves.• cooled rapidly, the copper stays dissolved

Mechanical properties (cont.)

• “aged” by holding it at 150oC• allow atoms to rearrange by diffusion but too

to remain dissolved, the copper precipitates out as a nanodispersion of a compound with the approximate composition CuAl2

Nanocrystalline

• The boundaries of grains act as obstacles to dislocation motion

• measured by the force f*• The number N of dislocations

• Breakdown: N=1

Nanolaminates• multilayers, alternating layers, usually of two

different materials. • the layers have to be thin (few atomic)• Data for the tensile strength of copper-nickel

laminates• are plotted as a function of the bilayer period in

the upper　 part of Figure 7.8. • The properties are remarkable: a nanolaminate

made from two soft metals (copper and nickel, for example) with a bilayer period of a few nm can have a strength of several GPa,

THERMAL PROPERTIES• Melting Point– fundamental point of reference– correlates with the bond strength

• surface effects should be considered

• and Lo is the latent heat of melting, To is the melting point of the bulk, T is the melting point of the extended system, where surface effects are included, and VL is the volume of liquid.

• Γ is the surface tension and ΔA is the increment in surface area.

• At equilibrium, the solid core of radius r has the same chemical potential as the surrounding liquid layer of thickness t, which occurs when the differential ∂ΔGTotal/∂t = 0.

• For a sphere

• the upper melting temperature

Melting Point

• nanoparticles in a matrix (e.g. nanocomposite)• the surface of the nanoparticle is in contact

with a matrix instead of being exposed to the surrounding atmosphere.

• according to Young’s theorem

Melting point

• assuming that θ = 90o, – when γSM > γLM, the melting point of the embedded

nanoparticles should be lower than the bulk– when γSM > γLM, the melting point of the embedded

nanoparticles should be higher than the bulk• The later call superheating, and it has been shown

experimentally.• Due to nanoscale effects, the melting temperature

can be either increased or reduced with respect to the bulk material.

Thermal Transport

• In addition to the melting temperature, many of the current applications of nanomaterials require knowledge about thermal transport.

• In some cases, such as microprocessors and semiconductor lasers, the goal is to transport heat away as quickly as possible,

• whereas for applications such as thermal barriers, the objective is to reduce thermal conduction.

• heat is transported by two different mechanisms: lattice vibration waves (phonons) and free electrons.

• In metals, the electron mechanism is dominent• In nonmetals, phonons are the main • In both metals and nonmetals, as the system

length scale is reduced to the nanoscale, there are quantum confinement and classical scattering effects.

• bulk homogeneous solid nonmetal materials, the wavelengths of phonons are much smaller than the length scale of the microstructure.

• However, in nanomaterials the length scale of the microstructure is similar to the wavelength of phonons.

• In 0-D nanomaterials (e.g. nanoparticles), quantum confinement occurs in three dimensions.

• In 1-D nanomaterials (e.g. nanowires and nanotubes), confinement is restricted to two dimensions.

• In 2-D nanomaterials such as nanofilms and nanocoatings, quantum confinement takes place in one dimension

• the presence of nearby surfaces in 0-D, 1-D, and 2-D nanostructures causes a change in the distribution of the phonon frequencies as a function of phonon wavelength as well as the appearance of surface phonon modes.

• Changes in the velocity with which the variations in the shape of the wave’s amplitude propagate, the so-called group velocity

• a phonon bottleneck occurs in 0-D nanostructures.

• One-dimensional nanomaterials behave as a phonon waveguide similar to optical ones for light.

2D nanomaterial

• Single-layered nanoscale thin films, most results show that the thermal conductivity is less than bulk materials

• multilayered films grains, an interface produces a thermal resistance.

• The interface disturb the regular crystal lattice on which phonons propagate.

• The interface can separate two crystals of the same material with different orientations, such as a grain boundary, across which the two regions have a different distribution of phonons.

• On the other hand, an interface can separate dissimilar materials, such as a multilayer structure, for which the two different materials have different

• densities and sound velocities. • phonon is scattered, thermal conductivity is reduced.

• For nanoporous materials, the nanosize effect is determined by the number and size of the pores.

• Due to the porosity, these materials have low permittivity and thermal conductivity

Heat capacity and coefficient of thermal expansion

• The heat capacity and coefficient of thermal expansion of nanomaterials have been much less studied.

• Different trend• Difficult to explain

Electrical properties

• Conduction• Large size system– Mechanism: delocalized, electrons can move freely in

all dimensions– Scattered by phonons, impurities, interafces, random

walk• Nano-size system– Quantum effect: band is replaced by discrete– Classical effect: mean free-path becomes comparable,

reduce scattering

3-D nanomaterials

• The three spatial dimensions are all above the nanoscale.

• Two aforementioned effects can be neglected. • Bulk nanocrystalline materials exhibit a high

grain boundary area-to-volume ratio• Electron scattering• Reduce the electrical conductivity.

2-D nanomaterials• Thickness at the nanoscale, quantum confinement will occur• carrier motion is uninterrupted along the plane• the wave functions of electrons are limited to very specific values • only electron wavelengths that are multiple integers of the

thickness will be allowed. • All other electron wavelengths will be absent. • Reduction in the number of energy states available for

conduction• The electrons become trapped in potential well of width equal to

the thickness. • In general, the effects of confinement on the energy state can be

written as

• L is the width of the potential well (thickness of 2-D nanomaterial),

• and n is the principal quantum number. • assumes an infinite-depth potential well model. • the carriers are free to move along the plane of the

sheet. • total energy of a carrier has two components– a term related to the confinement dimension – a term associated with the unrestricted motion along the

two other in-plane dimensions.

Unrestricted motion• the z-direction to be the thickness direction • and x and y the in-plane directions in which the

electrons are delocalized. • Characterized by two wave vectors kx and ky,

which are related to the electron’s momentum • px = ћ kx and py = ћ ky. • The energy corresponding to these delocalized

electrons is given by the so-called Fermi energy,

• where kF= (kx2+ky

2)0.5 • At the temperature of absolute zero, all the

conduction electrons are contained within a circle of radius kF.

• The total energy of an electron (due to confinement and unrestricted motion) in a 2-D nanomaterial with thickness at the nanoscale can be given by

• Since the electronic states are confined along the nanoscale thickness,

• the electron momentum is only relevant along the in-plane• directions. • scattering by phonons and impurities occurs mainly in-

plane, leading to a 2-D electron conduction. • However, for 2-D nanomaterials with nanocrystalline

structure, the large amount of grain boundary area• So, the smaller the grain size, the lower the electrical

conductivity of 2-D nanocrystalline materials.

1-D nanomaterials

• quantum confinement occurs in two dimensions, whereas unrestricted motion occurs only along the long axis of the nanotube/rod/wire.

• For a 1-D nanomaterial, the energy of a 2-D confinement depends on two quantum numbers, ny and nz, in the form

Total Energy in 1-D• Considering now the electron free motion along the x-direction (long axis),

• Because of the confinement, the nanoscale dimensions of 1-D nanomaterials act as reflectors, not allowing the electrons to exit the surfaces.

• In addition, scattering by impurities and/or phonons becomes restricted to the long axis of the tube, despite the fact that boundary scattering is more pronounced due to the high surface-to-volume ratio of 1-D nanomaterials.

• the transport of electrons along the tube occurs without significant loss of kinetic energy the transport is ballistic, particularly at low temperatures.

• Metallic carbon nanotubes, the conductivity is extremely high, as much as one billion amperes/cm2, in contrast to 1 million amperes/cm2 for copper.

• In addition to the effects described here, carbon nanotubes also exhibit a low density of defects and high thermal dissipation, reducing even further the chances for scattering.

0-D nanomaterials

• Confined along the three directions Lx, Ly, and Lz.

• Total energy can be given by

• In this fashion, all the energy states are discreet and no electron delocalization occurs.

• Metallic systems can behave as insulators due to the formation of an energy band gap, which is not allowed in the bulk form.

In practical point of view• These materials need to be coupled to external circuits by electrodes. • For 2-D and 3-D nanomaterials, ohmic contacts are possible. However, for 0-D

and 1-D nanomaterials, the contact resistances between nanomaterials and the connecting leads are usually high.

• Thus one mechanism of providing conduction is through electron tunneling. This is a quantum mechanical effect in which an electron can penetrate a potential barrier higher than the kinetic energy of the electron.

• Think of a configuration in which two metals are separated by a thin insulator • For an electron to tunnel from one metal to the other across the insulator, one

of the metals must have unoccupied energy states. • A simple way of achieving this is to apply a voltage V across the circuit to raise

the Fermi energy (the energy of the highest occupied quantum state) of one of the metals.

• In this fashion, electrons can tunnel from the metal with the highest Fermi energy to the metal with the lowest Fermi energy, producing a current I along the circuit.

• The current I = V/R, where R is the resistance. • However, in this case, the resistance is primarily due to

electron tunneling. • As an example, arrays of gold nanoparticles have been

electrically coupled by connecting the nanoparticles to each other by organic molecules. The nanoparticles act as the metal electrodes, whereas the organic molecules play the role of a thin insulator.

• Under these conditions, the conductance C, defined as C = I/V, was shown to increase due to electron tunneling compared to the case in which the nanoparticles were not connected by the organic molecules.

Application electron tunneling• The phenomenon of can also be used to develop field-effect transistors (FET) made from

quantum dots• two electrodes, a source and a drain, are coupled to a quantum dot and connected

through a circuit (see Figure 7.20). In addition, a gate voltage is provided to the quantum dot to control its resistance and ultimately the current I passing between the lead and the drain.

• Due to the discrete nature of the electrical charge, electrons tunnel from the source to the quantum dot and then to the drain, one at the time.

• Therefore the junction, which acts as a capacitor, suffers a raise in voltage V = e/C (e is the elementary charge) when a single electron is added.

• If the change in voltage is large enough, another electron can be prevented from tunneling. This effect is called a Coulomb blockade.

• electrons will not tunnel until a discreet voltage is reached. • To promote electron tunneling, the temperature has to be low enough so that the energy

(e2/C) necessary to charge the junction with one electron exceeds the thermal energy kT. • As the capacitance decreases with the size of the particle, a nanoparticle allows the

Coulomb blockade to be observable at higher temperatures.

dielectric behavior• The large number of grain boundaries in nanocrystalline materials is expected to

increase the dielectric constant. • Nanocrystalline TiO2, a higher dielectric constant was found, compared with

coarse-grained samples. • This is due to the fact that under an applied electric field, the positive and negative

charges that are segregated at the interfaces will lead to some form of polarization. • Since for nanocrystalline TiO2 the volume fraction of grain boundaries is much

larger than in coarse-grained TiO2, the polarization mechanisms will be much more important, leading to a higher dielectric constant

• A similar effect was found for polymermatrix nanocomposites reinforced with nano TiC fillers.

• An increase of the dielectric constant was observed for higher loading levels of TiC, especially when the TiC content was near the percolation threshold. As in the case of nanocrystalline TiO2, the increase of the dielectric constant is due to interface polarization.

Magnetic Properties• In general, for any ferromagnetic material, the total energy can be written

as the sum of various terms, in the form

where Eexc is the exchange energy, Eani is the anisotropic energy, Edem is the demagnetization energy, and Eapp is the energy associated with an applied magnetic field. • For macroscopic magnetic materials, a magnetostrictive energy must also

be included, but for nanoscale materials this energy can be neglected. • This magnetization energy Etotal can then be related to a magnetic field

according to the expression

where M is the magnetization vector and H is the applied magnetic field

• The first term is due to the quantum mechanical interaction between atomic magnetic moments and represents the tendency for the magnetization vectors to align in one direction.

• In other words, if the magnetic moment is sufficiently large, the resulting magnetic field can drive a nearest neighbor to align in the same direction, provided the exchange energy is greater than the thermal energy.

• The second term represents the anisotropy energy that results from the spin’s tendency to align parallel to specific crystallographic axes, called easy axes. Thus, a “soft” magnetic material will exhibit low anisotropy energy, whereas a “hard” magnetic material shows high anisotropy energy.

• Though both the exchange energy and the anisotropy energy try to order the spins in a parallel configuration, the third term, namely the demagnetization energy, which is related to the magnetic dipole character of spins, leads to the formation of magnetic domains. Thus, for macroscopic ferromagnetic materials, all the magnetic moments are aligned in magnetic domains, although the magnetization vectors of different domains are not parallel to each other. Each domain is magnetized to saturation, with the moments typically aligned in an easy direction. Depending on the ratio of anisotropy to demagnetization energy, we can expect open (ratio <1) or closure domain (ratio >1) structures.

• Finally, the energy associated with an applied magnetic field, called Zeeman energy and represented by the last term, results from the tendency of spins to align with a magnetic field.

• Initially, as the magnetic field increases, the magnetization of the material increases.

• However, at some point, a saturation point, called saturation magnetization, is reached, above which an increase in magnetic field does not produce an increase in magnetization.

• The saturation magnetization is material and temperature dependent.

• For nanocrystalline ferromagnetic materials, an important consideration is the interaction among exchange energy, anisotropic energy, and demagnetization energy.

• For very small particles or grain sizes, the exchange forces are dominant due to strong coupling, causing all the spins in neighboring grains to align, superseding in this way the anisotropic and demagnetizing forces.

• Therefore there is a critical grain size, below which the material will be single domain. For spherical grains, the critical diameter is given by

• where γB = 4(AK1)1/2 is the wall energy of the material; A is an exchange constant, also known as exchange stiffness, which is a function of the material and temperature; K1 is the anisotropic constant; μ0 is the permittivity of free space; and Ms is the saturation magnetization.

• Thus if the particle or grain size is below the critical diameter, the material is single domain.

• For example, the critical diameter for Co is around 70 nm, whereas for Fe it is 15 nm.

• If the particle or grain size becomes significantly smaller (typically a few nm) than the critical diameter, the magnetization is likely to become unstable and loss of magnetization occurs due to thermal fluctuations.

• These materials are called superparamagnetic.

• Other magnetic properties are also strongly affected by scale. • To address this point, first recall the magnetic response of a

bulk ferromagnetic material to an applied magnetic field • As previously discussed, the hysteresis shown in Figure 7.24 is

associated with the fact that, on removal of the magnetic field, the magnetic domains do not revert to their original configuration. In other words, there is a remnant magnetization. On the other hand, the coercive field is the applied magnetic field that needs to be applied in the direction opposite the initial magnetic field, to bring the magnetization back to zero. By reducing the particle size or grain size to the nanoscale, the magnetization curve can be altered.

• In general, the coercive field of a ferromagnetic material increases with decreasing particle size or grain size, reaching a maximum within a range around the critical diameter.

• If the particle size or grain size is further decreased below this range, the coercive field will be drastically reduced until the magnetization becomes unstable due to the superparamagnetic behavior.

• Within this regime, the hysteresis can be completely removed at any temperature.