NPTEL – Electrical & Electronics Engineering – Semiconductor Nanodevices
Melting points, mechanical properties of
nanoparticles and Hall Petch relationship for
nanostructured materials
R. John Bosco Balaguru
Professor
School of Electrical & Electronics Engineering
SASTRA University
B. G. Jeyaprakash
Assistant Professor
School of Electrical & Electronics Engineering
SASTRA University
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Table of Content
1. MELTING POINTS OF NANOPARTICLES...........................................................3
1.1 LATTICE CONSTANT OF NANOPARTICLE......................................................................................6
2. MECHANICAL PROPERTIES OF NANOMATERIAL.......................................8
2.1
2.2
2.3
2.4
DIFFUSION............................................................................................................................................9
CHARACTERISING MECHANICAL PROPERTIES OF LOW DIMENSIONAL MATERIALS..10
MECHANICAL PROPERTIES OF CNT.............................................................................................10
MECHANICAL PROPERTIES OF SILICON AND ZNO NANOWIRE…………………………...11
3. HALL-PETCH RELATIONSHIP FOR NANOSTRUCTURED MATERIALS..12
3.1 HALL-PETCH RELATION……………………………………………………………………….....12
3.2 GRAIN SIZE EFFECT AND HALL-PETCH RELATION……………………………………..…...13
4. QUIZ AND ASSIGNMENT......................................................................................14
4.1 SOLUTIONS..........................................................................................................................................15
5. REFERENCES...........................................................................................................15
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1 Melting point of nanoparticles
This lecture provides you the basis of size and shape dependent melting temperature
change in the metal nanoparticle
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 depression 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 degrees lower than bulk materials.
However as the dimensions of a material
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
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
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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 S 0 of the
particle is given by
S0 = πD 2
(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 E n we can write
3
4 D
4 d 
π   = n. π  
3 2
3 2
3
Where
D3
d3
the surface area of n atoms is
n=
(2)
S = nπd 2
(3)
when the particle is changed to n atoms the surface area variation is
∆S = nπd 2πD 2
(4)
Let E n be the cohesive energy of n atoms and equals to the surface energy of the solid whose
surface area is ∆S. The surface energy per unit area at 0 K is γ 0 then En = ∆S ⋅ γ 0 .
(
)
(i.e) E n = πγ o nd 2 − D 2
(5)
then the cohesive energy per atom is

D2 

E = πγ o  d 2 −
(6)
n 

from Eqn. (2)
d

(7)
E = π ⋅ γ o ⋅ d 2 1 − 
D

The lattice parameters can be determined for three different structures bcc, fcc and hcp are
written as
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 (3 π )1 3 a
bcc

13
d =  (3 2π ) a
fcc
(8)
 3 3a 2 c 2π 1 3 hcp

Eqn. (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
d

E ' = k ⋅ π ⋅ γ o .d 2 .1 − 
D

For metals
(9)
d
is about 10-7. Eqn. (9) can be rewritten as
D
E b = k ⋅ π ⋅ γ o .d 2
(10)
Where E b is the cohesive energy of the bulk material. When the particle is small, the size of D is
in nanometers or smaller d/D is in the range of 10-2 to 10-1. Rewriting Eqn. (9) as
d

E p = E b ⋅ 1 − 
D

(11)
where E p is the cohesive energy of nanoparticle. Using Eqn(11) the cohesive energy of
nanoparticle can be obtained.
To account for the particle shape difference, let the shape factor be α, which is define by
the equation
α=
S'
S
(12)
where S is the surface area of the spherical nanoparticle and S = 4πR2 (R is its radius). S’is the
surface area of the nanoparticle in any shape, whose volume is the same as spherical
nanoparticle.
From Eqn.(12), the surface area of a nanoparticle in any shape can be written as
S ' = α ⋅ 4πR 2
(13)
Assuming the atoms of nanoparticles are ideal spheres then the contribution to the particle
surface area of each surface atom is πr 2 . The number of the surface atoms N is the ratio of the
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(
)
particle surface area to πr2, which is simplified as N = 4α R 2 r 2 . The volume of the
nanoparticle V is the same as the spherical nanoparticle, which equals to 4 3πR 3 . Then the
number of the total atoms of the nanoparticle is the ratio of the particle volume to the atomic
volume 4 3πr 3 that results to
n=
R3
r3
(14)
The cohesive energy of metallic nanoparticle is the sum of the bond energies of all the atoms.
Considering Eqn. (11), the cohesive energy of metallic crystal in any shape (E p ) can be written
as
Ep =
 R3
1 1
R2
R 2 

 E bond
+
−
4
4
β
α
β
α

 r3
2 4
r2
r 2 

(15)
where E bond is the bond energy. The value ½ is due to the fact that each bond belongs to two
atoms. We can write Eqn. (15) as
Ep =
r
1

nβE bond 1 − 6α 
D
2

(16)
where D is the size of the crystal and D = 2 R . Rewriting Eqn. (16) as
r

E p = E 0 1 − 6α 
D

(17)
where E0 = (1 2)nβEbond and E 0 is the cohesive energy of the solids.
The well empirical relation of the melting temperature and the cohesive energy for pure metals
are given as
Tmb =
0.032
Eo
kB
(18)
where T mb is the melting temperature of bulk pure metals. Replacing the cohesive energy of
solids E 0 by more general for E p , then
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Tm =
r
0.032 
E o 1 − 6α 
kB
D

(19)
Eqn.n (19) can be rewritten as
r

Tm = Tmb 1 − 6α 
D

(20)
Eqn. (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 is
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 difference between the
spherical and the non-spherical nanoparticles
α=
S'
S
(21)
Where S ' is the surface area of the spherical nanoparticle and S = 4πR 2 . S ' is the surface area of
thenanoparticle in any shape, whose volume is the same as the spherical nanoparticle. For
sphericalnanoparticle, we have aα=1, and for non-spherical nanoparticle, α > 1. Eqn. (21) can be
rewrittenas
S ' = αS
(22)
The increased surface energy after being moved out a nanoparticle from the crystal is
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∆γ = α ⋅ 4πR 2 γ
(23)
where R is the radius of the particles, and γ is the surface energy per unit area at the temperature
T (0 ≤ T < Tm ) ,T m is the melting temperature of metals γ can be obtained from the equation given
below
γ = γ 0 +T ⋅
dγ
dT
(24)
Where γ 0 is the surface energy per unit area at 0 K, 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,
f ' = 8πGR 3ε 2
(25)
Where G is the shear module, considering the expression S = 4πR 2
We have
f =π
'
−
1
2
3
2
GS ε 2
(26)
The elastic energy of a nanoparticle in non spherical shape is difficult to calculate. However, we
can give an approximate estimation by Eqn. (26). The parameter ε is the variable, which can be
regarded the same for nanoparticles in any shapes. To account for the elastic energy ( f ) of a
nanoparticle in non-spherical shape, we should replace S with S ' in Eq. (26), then
3
f =α 2 ⋅π
−
1
2
3
GS 2 ε 2
(27)
Eqn. (27) can be rewritten as
3
2
f = α ⋅ 8πGR 3 ε 2
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However, the contraction will make the increased surface energy decrease. Considering
contractioneffect, the effective increased surface energy is
∆γ = α ⋅ 4π [R(1 − ε )] γ
2
(29)
The total energy variation F is the sum of the increased surface energy and the increased elastic
energy, which can be written as
3
F = α ⋅ 4π [R(1 − ε )] γ + α 2 ⋅ 8πGR 3 ε 2
2
(30)
(i.e.)
F = Aε 2 + Bε + C
where
A = 4πγR α + 8πGR α
2
3
3
2
(31)
B = −8πγR α
2
C = 4πγR 2α
In equilibrium, the total energy is minimum, i.e.,
ε=
1
 2G 
 ⋅ R ⋅ α 2
1 + 
γ


1
dF
= 0 , so we have
dε
(32)
For an ideal crystal lattice, the lattice parameter contraction is proportional to the radius of the
nanoparticle.
∆a a p − a (1 − ε )R − R
=
=
a
a
R
(33)
Where a p and a are the lattice parameters of the nanoparticle and the corresponding bulk
material.
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Inserting Eqn. (32) into Eq. (33), we have
∆a
1
=−
a
1+ K ⋅ D
(34)
1
2
Where D (= 2 R ) is the diameter of the nanoparticle, and K = α G 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 (34) is the basic
relation for the size and shape dependent lattice parameters of metallic nanoparticles.
2 Mechanical properties of nanomaterial
This lecture provides you about the fundamental, measurement challenges and mechanical
properties of nanotubes and nanowires
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
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2.1 Diffusion
Diffusion is a key property determining the suitability of nanocrystalline 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
nanocrystalline materials differ from conventional grain boundaries (GBs). Interface diffusion
process in polycrystalline materials can be classified as follows,
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 (GB) 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 (such as discontinuous
precipitation, discontinuous coarsening, etc.), recrystallization, and grain growth.
2.2
Characterising
mechanical
properties
of
low
dimensional
materials
Characterizing the mechanical properties of individual nanotubes/nanowires/nanobelt
[called one-dimensional (1-D) nanostructure] is a challenge to many existing testing and
measuring techniques because of the following constrains. First, the size (diameter and length) is
rather small, prohibiting the applications of the well-established testing techniques. Tensile and
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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 for 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.
Therefore 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 (NWs)
including resonance in scanning or transmission electron microscopes (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 NWs enabled by
microelectromechanical systems has attracted 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 direction such as armchair, zigzag
and chiral. The bond between carbons is similar to
that of graphite and the mechanical property is
closely related to the bond nature between the carbon
atoms. The electronic structure of carbon is 1s2 2s2
2p2 and when carbon atoms combine to form graphite, sp2 hybridization will occurs. In this
process, one s-orbital and two p-orbitals combine to form three hybrid sp2 -orbitals at 120° to
each other within a plane. This in-plane bond is referred to as a σ-bond (sigma–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
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contributes mainly to the interlayer interaction and is called the Π-bond (pi–bond). These out-ofplanes, 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 the defects in nanotubes is presumably less and therefore the
strength is presumablysignificantly higher at the nanoscale.
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
beenconducted, however not clear results were obtained. Especiallythe Young’s modulus of ZnO
nanowires are on debate inthe literature. For instance, Chen et al.[16] showed that the Young’
modulus of ZnO nanowirewith diameters smaller than about 120 nm is signifi cantly higher than
that of bulk ZnO. However, the elastic modulus of vertically aligned [0001] ZnO nanowireswith
an average diameter of 45 nm measured by atomicforce microscopy was found to be far smaller
than that of bulk ZnO. Also the effectivepiezoelectric 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 [22]showedthat the
piezoelectric coefficient for ZnO nanopillar withthe diameter about 300 nm is smaller than the
bulk values. They suggested that the reduced electromechanical response might be due to
structural defects inthe pillars[22]. The fundamental studies on these issues were under research.
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3 Hall-Petch relationship for nanostructured materials
This lecture gives the Hall-Petch relations and size effect on it for low dimensional
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
allimperfections, 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 HallPetch
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 micrometers to few
hundred micrometers). Metals typically
follow the Hall-Petch relation when the
average grain size is 100nm 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.
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3.2 Grain size effect and Hall-Petch relation
Decreasing the grain size is effective in both increase strength and also increase 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:
σ y = σ i + k y D −1 2
Where, k y is the Hall-Petch coefficient, a material constant; D is the grain diameter and σ y is 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
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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 Quiz and Assignment
1.
2.
3.
4.
5.
6.
7.
8.
9.
The metal particle size decreases, the melting temperature also _______
Why nanoscale material have low melting point as compare to the bulk?
What is cohesive energy?
The particle size is greater than ____, the melting temperature of the particles equals to
the corresponding bulk materials
The particle size increases , the cohesive energy is _________(a) Increases
(b) Decreases
(c) No change
(d) None of the above
The carbon nanotube (CNT) is a rolled-up sheet of _______ and has three types
depending upon the rolling direction such as ___ ,___,____
What is a sigma bond in CNT?
Young’ modulus of ZnO nanowire with diameters smaller than about _____ is
significantly higher than that of bulk ZnO
The ________ is describing the relationship between a metal’s yield strength and its
average grain size.
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10. Metals follows the Hall-Petch relationship at average grain size of ________
11. What is Inverse Hall-Petch Relationship?
12. What are the two type of Grain Boundary sliding?
4.1 Solutions
1. Decreases
2. Nanoscale materials have a much larger surface-to-volume ratio than bulk materials,
drastically altering their thermodynamic and thermal properties. So nanoscale materials
have low melting point.
3. It is the energy required to divide the metallic crystal into individual atoms.
4. 100nm.
5. (a) increases
6. Graphene, armchair, zigzag and chiral
7. Sigma bond is a strong covalent bond that binds the atoms in the plane, and results in the
high stiffness and high strength of a CNT.
8. 120nm
9. Hall-Petch equation
10. 100nm or larger.
11. The yield strength of the material decreases as the grain size decreases below 100nm.
12. 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.
5 References
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).
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5. H. J. Fan, W. Lee, R. Hauschild, M. Alexe, G. L. Rhun, R. Scholz, A. Dadgar, K.
Nielsch, H. Kalt, A. Krost, M. Zacharias, and U. G¨osele, Small 2, 561 (2006).
6. W.H. Qi, M.P. Wang, Size and shape dependent lattice parameters of metallic
nanoparticles, 7 (2005), 51-57.
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