5 Basic properties of nanoparticles -IV
Dr. Anchal Srivastava
Contents of this Unit
1. Melting points of nanoparticles
1.1 Lattice constant of nanoparticle
2. Mechanical properties of nanomaterial 2.1 Diffusion
2.2 Characterising mechanical properties of low dimensional material
2.3 Mechanical properties of CNT
2.4 Mechanical properties of silicon and ZnO nanowire
3. Hall-Petch relationship for nanostructures materials
3.1 Hall-Petch relation
3.2 Grain size effect and Hall-Petch relation
4. Summary
Learning Outcomes
- After studying this module, you shall be able to understand
- Changes in the melting points of nanoparticles Mechanical properties of nanomaterials
- Hall-Petch relation Grain size effect
1. MELTING POINTS OF NANOPARTICLES
Many physical properties of materials, especially the melting point, change when the physical size of the material approaches the micro and nano scales. Melting point depressions is a term referring to the phenomenon of reduction of the melting point of a material with reduction of its size. This phenomenon is very prominent in nanoscale materials which melt at temperatures hundreds of degree lower than bulk materials.
However as the dimensions of a materials decrease towards the atomic scale, the melting temperature scales with the material dimensions. Melting-point depression is most evident in nanowires, nanotubes and nanoparticles, which all melt at lower temperatures than bulk amounts of the same material. Changes in melting point occur because nanoscale materials have a much larger surface-to-volume ratio than bulk materials, drastically altering their thermodynamic and thermal properties. As the metal particle size decreases, the melting temperature also decreases.
Figure 4.1. Variation of melting point of gold nanoparticle as a function of the size of the particle.
Let we analysis the size and shape dependent of metal nanoparticle. Since the melting temperature depression results from the large surface-to-volume ratio, the surface areas of nanoparticles in different shape will be different even in the identical volume, and the area difference is large especially in small particle size. Therefore, it is needed to take the particle shape into consideration when developed models for the melting temperature of nanoparticles.
Melting temperature relates to cohesive energy refers of the materials. It is the energy required to divide the metallic crystal into individual atoms. It also refers to heat of sublimation that can be determined experimentally or can be calculated using cellular method and density function theory. All these methods will calculate only for bulk material. The properties of nanoparticle vary due to the size effect. A simple method to calculate the cohesive energy of nanoparticle was discussed below. The cohesive energy increases with the increase in the particle size. When the particle size is large the cohesive energy will approach bulk material. Let a metallic particle has a diameter of D and is composed of n atoms. The surface area S0 of the particle given by
| S0 = πD2 | (4.1) |
Assuming when the particle is separated into n identical spherical atoms and let the diameter of the atoms are d without changing its volume by exerting energy En we can write
Let En be the cohesive energy of n atoms and equals to the surface energy of the solid whose surface area is ΔS. The surface energy per unit area at 0K is γ0 then En = ΔS γ0.
The lattice parameters can be determined for three different structures bcc, fcc and hcp written as
Equation (4.2) is the expression to calculate cohesive energy for ideal case. It is necessary to introduce a factor k to account for the difference. Therefore
For metals d/D is about 10-7. Equation (4.9) can be written as
Eb = kπγ0d2
Where Eb is the cohesive energy of the bulk material. When the particle is small, the size of D is in nanomater or smaller d/D is in range of 10-2 to 10-1. Rewriting equation (4.9) as
Where Ep is the cohesive energy of the nanoparticle. Using equation (4.11) the cohesive energy of nanoparticle can be obtained.
To account for the particle shape difference, let the shape factor be α, which is defined by the equation
α = S’/S (4.12)
where S is the surface area of the spherical nanoparticle nanoparticle and S = 4πR2 (R is the radius). S’ is the surface area of the nanoparticle in any shape, whose volume is the same as spherical nanoparticle.
From equation (4.12), the surface area of a nanoparticle in any shape can be written as
S’ = α 4πR2 (4.13)
Assuming the atoms of nanoparticles are ideal spheres then the contribution to the article surface area of each atoms is πr2. The number of the surface atoms N is the ratio of the particle surface area to πr2, which is simplified as N = 4α(R2/r2). The volume of the nanoparticle V is the same as the spherical nanoparticle, which equals to 4/3 πR3. Then the number of the total atoms of the nanoparticle is the ration of the particle volume to the atomic volume 4/3 πr3 that results to
The cohesive energy of metallic nanoparticle is the sum of the bond energies of all the atoms. Considering equation (4.11), the cohesive energy of metallic crystal in any shape (Ep) can be written as
Where E0 = (½) nβEbond and E0 is the energy of the solids.
The well empirical relation of the melting temperature and the cohesive energy for pure metals are given as
Where Tmb is the melting temperature of bulk pure metals. Replacing the cohesive energy of solids E0 by general for Ep, then
Equation (4.19) can be rewritten as
Equation (4.20) is the general equation for the size and shape dependent melting temperature of crystals. The melting temperature of nanoparticles is apparent only when the particle size os smaller than 100 nm. If the particle size is larger than 100 nm, the melting temperature of the particles approximately equals to the corresponding bulk materials, in other words, the melting temperature of nanoparticles is independent of the particle size.
1.1 LATTICE CONSTANT OF NANOPARTICLE
Lattice constant of nanoparticle depends on size and shape and we will arrive an expression for it. A shape factor α will be considered to modify the shape different between the spherical and the non-spherical nanoparticles
where S’ is the surface area of the spherical nanoparticle and S = 4πR2. S’ is the surface area of the nanoparticle in any shape, whose volume is the same as the spherical nanoparticle. For spherical nanoparticle, we have aα =1, and for non-spherical nanoparticle, α > 1. Equation (4.21) can be rewritten as
The increased surface energy after moved out a nanoparticle from the crystal is
Where R is the radius of the particle, and γ is the surface energy per unit area at the temperature T (0 ≤ T < Tm), Tm is the melting temperature of metals γ can be obtained from the equation given below
Where γ0 is the surface energy per unit area at 0K and dγ/dT is the coefficient of surface free energy to temperature. For most solids, we have dγ/dT < 0.
The surface energy will contract the nanoparticle elastically. This kind of contraction is very small compared with the whole particle size. Suppose the small displacement εR results from this elastic contraction, where ε << 1. For spherical particles, the elastic energy f’ can be written as follows
The elastic energy of a nanoparticle in non-spherical shape is difficult to calculate. However, we can give an approximate estimation by equation (4.26). The parameter ε is the variable, which can be regarded the same for nanoparticle in any shapes. To account or the elastic energy (f) of a nanoparticle in no-spherical shape, we should replace S with S’ in equation (4.26), then
Equation (4.27) can be rewritten as
However, the contraction will make the increases surface energy decrease. Considering contraction effect, the effective increased surface area is
The total energy variation F is the sum of the increased surface energy and the increased elestic energy, which can be written as
For an ideal crystal lattice, the lattice parameter contraction is proportional to the radius of the nanoparticle.
Where ap and a are the lattice parameters of the nanoparticle and the corresponding bulk material. Inserting equation (4.32) into (4.33), we have
Where D (= 2R) is the diameter of the nanoparticle and K = a1/2G/ γ Generally, both of the shear module and the surface energy are positive; therefore, the lattice parameter of the metallic nanoparticles will decrease with decreasing of the particle size. Equation (4.34) is the basic relation for the size and shape dependent lattice parameters of metallic nanoparticles.
2. MECHANICAL PROPERTIES OF NANOMATERIAL
Mechanical properties of solids depend on the microstructure, i.e. the chemical composition, the arrangement of the atoms (the atomic structure) and the size of a solid in one, two or three dimensions. The most well-known example of the correlation between the atomic structure and the properties of a bulk material is the variation in the hardness of carbon when it transforms from diamond to graphite. The important aspects related to structure are:
- Atomic defects, dislocations and strains
- Grain boundaries and interfaces
- Porosity
- Connectivity and percolation
- Short range order
2.1 DIFFUSION
Diffusion is a key property determining the suitability of nano-crystalline materials for use in numerous applications, and it is crucial to the assessment of the extent to which the interfaces in nanocrystalline samples differ from conventional grain boundaries. Emphasis is placed on the interfacial characteristics that affect diffusion in nanocrystalline materials, such as structural relaxation, grain growth, porosity and the specific type of interface. Diffusion is a determining feature of a number of application oriented properties of nanocrystalline materials, such as enhanced ductility, diffusion-induced magnetic anisotropy, enhanced ionic mass transport, and improved catalytic activity. Moreover, diffusion in nanocrystalline materials is also relevant to the basic physics of interfaces. Since interface diffusion is highly structure sensitive, diffusion studies can provide valuable insight into the question of the extent to which interfaces in nanorystalline materials differ from conventional grain boundaries. Interface diffusion process in polycrystalline materials can be classified as follow,
1. Rapid diffusion in the crystallite interface or grain boundary diffusion coefficient
2. Diffusion from the interfaces and specimen surface into the volume of the crystallites
Grain boundary diffusion plays an important role in many processes taking place in engineering materials at elevated temperatures. Such processes include Coble creep, sintering, diffusion-induced grain boundary migration, discontinuous reactions, recrystallization and grain growth.
2.2 CHARACTERISING MECHANICAL PROPERTIES OF LOW DIMENSIONAL MATERIALS
Characterizing the mechanical properties of individual nanotubes/nanowires/nanobelts is a challenge to many existing testing and measuring techniques because of the following constraints. First, the size is rather small, prohibiting the applications of the well-established testing. Tensile and creep testing require that the size of the sample be sufficiently large to be clamped rigidly by the sample holder without sliding. This is impossible or 1-D nanomaterials using conventional means. Secondly, the small size of the nanostructure makes their manipulation rather difficult, and specialized techniques are needed for picking up and installing individual nanostructure. Therefor new methods and methodologies must be developed to quantify the properties of individual nanostructure.
A number of methods have been developed for mechanical testing of nanowires including resonance in scanning or transmission electron microscope (SEM/TEM), bending or contact resonance using atomic force microscopy (AFM), uniaxial tension in SEM or TEM and nanoindentation. In particular, in situ SEM/TEM tensile testing of nanowires enabled by microelectromechanical system has attached a lot of recent attention. However each technique has its own merits and demerits in invoking the mechanical parameters of nanowires.
2.3 MECHANICAL PROPERTIES OF CARBON NANOTUBE
The carbon nanotube (CNT) is a rolled-up sheet of graphene and has three types depending upon the rolling directions such as armchair, zigzag and chiral. The bond between carbons in similar to that of graphite and the mechanical property is closely related to the bond nature between the carbon atoms. The electronics structure of carbon is 1s2 2s2 2p2 and when carbon atoms combine to form graphite, sp2 hybridization will occur. In this process, one s-orbital and two p-orbital combine to form three hybrid sp2 orbital at 1200 to each other within a plane. This in plane bond is referred to as a σ-bond. This is a strong covalent bond that binds the atoms in the plane, and results in the high stiffness and high strength of a CNT. The remaining p-orbital is perpendicular to the plane of the σ-bonds. It contributes mainly to the interlayer interaction and is called the π-bond. These out-of-planes, delocalized π bonds interact with the π-bonds on the neighbouring layer. This interlayer interaction of atom pairs on neighbouring layers is much weaker than a sigma bond. Also unlike bulk materials, the density of defects in nanotubes is presumably less and therefore the strength is presumably significantly higher at the nanoscale.
Figure 4.2. Different types of CNTs.
2.4 MECHANICAL PROPERTIES OF SILICON AND ZnO NANOWIRE
Silicon nanowires deform in a very different way from bulk silicon. Bulk silicon is very brittle and has limited deformability, means that it cannot be stretched or warped very much without breaking.” However the silicon nanowires are more resilient, and can sustain much larger deformation. Other properties of silicon nanowires include increasing fracture strength and decreasing elastic modulus as the nanowire gets smaller and smaller. Many studies on mechanical properties of zinc oxide (ZnO) nanowires have been conducted, however not clear results were obtained. Especially the Young’s modulus of ZnO nanowires are on debate in the literature. For instance, Chen et al.[16] showed that the Young’ modulus of ZnO nanowire with diameters smaller than about 120 nm is significantly higher than that of bulk ZnO. However, the elastic modulus of vertically aligned [0001] ZnO nanowires with an average diameter of 45 nm measured by atomic force microscopy was found to be far smaller than that of bulk ZnO. Also the effective piezoelectric coefficient of individual (0001) surface dominated ZnO nanobelts measured by piezoresponse forcemicroscopy was reported to be much larger than the value for bulk wurtzite ZnO. In contrast, Fan et al. showed that the piezoelectric coefficient for ZnO nanopillar with the diameter about 300 nm is smaller than the bulk values. They suggested that the reduced electromechanical response might be due to structural defects in the pillars. The fundamental studies on these issues were under research.
3. HALL-PETCH RELATIONSHIP FOR NANOSTRUCTURED MATERIALS
3.1 HALL-PETCH RELATION
The basic principle in materials science is that the properties can be deduced from knowledge of the microstructure. The microstructure refers the crystalline structure and all imperfections, including their size, shape, orientation, composition, spatial distribution, etc. The types of imperfections or defects in generally were of:
• point defects (vacancies, interstitial and substitutional solutes and impurities)
• line defects (edge and screw dislocations) and
• planar defects (stacking faults, grain boundaries),
For more than half a century, materials engineers have used the Hall-Petch equation to describe the relationship between a metal’s yield strength and its average grain size. The Hall-Petch relation predicts behaviour accurately in metals with ordinary grain sizes (ie. Few micrometres to few hundred micrometers). Metals typically follow the Hall-Petch relation when the average grain size is 100 nm or larger, But Hall-Petch behaviour breaks down at smaller grain sizes. Indeed, an “Inverse Hall-Petch relationship” appears to exist at very small grain sizes, with yield strength actually decreasing as the grain size decreases.
Figure 4.3. Graph showing the Hall-Petch relation
3.2 GRAIN SIZE EFFECT AND HALL-PETCH RELATION
Decreasing the grain size is effective in both increase strength and also increases ductility and as such, is one of the most effective strengthening mechanisms. Fracture resistance also generally improves with reductions in grain size, because the crack formed during deformation, which are the precursors to those causing fracture are limited in size to the grain diameter. The yield strength of many metals and their alloys has been found to vary with grain size according to the Hall-Petch relationship:
Where, ky the Hall-Petch coefficient, a material constant; D is the grain diameter and σy the yield strength of an imaginary polycrystalline metal having an infinite grain size. At this regime, it is suggested that the yield stress of nanocrystals decreases with decreasing grain size and finally it reaches a lowest limit correspondingly to the yield stress of amorphous materials. Grain boundary play a critical role in the yield stress of materials in that there can be several different deformation modes associated with different grain size, grain shape, temperature, stress state and Grain boundary structures. There are four major deformation modes for crystalline materials.
1. Grain boundary sliding (GBS) caused by the atomic shuffling of the BD interface.
2. Collective BG migration
3. Stacking faults
4. Dislocations from the interface to the grain.
The first two modes correspond to GB- mediated deformation and the last modes correspond to dislocation mediated deformation. These deformation modes works together to finally determine the overall plastic behaviour and yield stress of crystalline materials.
For a coarse grain materials, where Hall-Petch relation holds, the plastic deformation is mainly attributed to dislocation mediated deformation such as full and partial dislocations evolution annihilations, in which GBs act as a barrier of dislocation movement, sinks and sources of dislocations. This may be understood by considering the sequence of events involved in the initiation of plastic flow from a point source (within one grain) in the polycrystalline aggregates. The strengthening provided by Grain boundary depends on Grain boundary structure, disorientations and interaction between dislocations and grain boundaries. For crystalline materials with grain size of several nanometers, plastic deformation is mainly attributed to the GB-mediated deformation, such as GBS and GB migration. There are generally two different types of GBS.
• Rachinger sliding-It is accommodated by some intragranular movement of dislocations within adjacent
grains.
• Lifshitz sliding -It is refer to the boundary offset that develop as a direct consequences of the stress-
directed diffusion of vacancies.
This type of sliding is due to thermal activation process, such as diffusion and atom shuffling, although some MD simulations indicate that GB sliding may also happen at 0K, which indicates that GBS also contains an a thermal component.
4. SUMMARY
- Size reduction leads to the reduction in melting point of the nanostructures due to their large surface to volume ratio.
- Cohesive energy is the energy required to divide the metallic crystal into individual atoms. It increases with the increase in the particle size.
- A general equation for the size and shape dependent melting temperature of crystals is given by:
- Tmb = Tmb (1 – 6 )
- Hall-Petch relation which describes the metal’s yield strength and its average grain size.
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Learn More:
1. Niels Hansen, Hall–Petch relation and boundary strengthening, Scripta Materialia 51 (2004) 801–806.
2. J.M. Martinez-Duart, R.J.Martin-Palma, F. Agullo-Rueda, Nanotechnology for Microelectronics and
Optoelectronics, Elsevier, 2006.
3. Michael J.O’Connell, Carbon nanotubes properties and applications, Taylor & Francis, New York, 2006.
4. C. Q. Chen, Y. Shi, Y. S. Zhang, J. Zhu, and Y. J. Yan, Phys. Rev. Lett. 96, 075505 (2006).