Nano-1
Nanoscience I: Downscaling of classical laws makes nano different
Kai Nordlund3.10.2010
Faculty of Science
Department of Physical Sciences
Accelerator Laboratory
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Contents
Introduction
Effects of surface atomsReduced cohesion Landing on surfacesSurface reactivity
Radiative
cooling
Nanoresonators, “nanokantele”
Hall-Petch
resonators
Scattering of light
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Introduction
On this lecture we go through a few examples of taking certain simple basic equations in physics and materials science, and scaling their size parameters downwards to the nanoscale. The results will show that nanomatter
really can be dramatically different from ordinary bulk (macroscale) matter
Of course it is not automatically clear whether laws originally made for the macroscale
give correct results on the nanoscale
But the examples have been chosen such that it is known that these scalings
do work at least qualitatively, or it is known where the limit comes in
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Number
of surface
atoms
0.2 nm
We start by repeating
the surface atom calculation of lecture 1
What fraction of atoms are on the surface of a sphere?
We know one atom layer is about
t=0.2 nm thick
Volume of surface atoms:
Vsurface
= 4 r2 t
Volume of the whole ball:
Vball
= 4 r3/3
Ratio, i.e. fraction of surface atoms:
Vsurface
Vball
= 3 t / r
Consider different values of r:
Macro ball:
r= 1 m => 3 t / r = 6
•
10-10
Micro ball:
r= 1 m => 3 t / r = 6
•
10-4
Nano ball:
r= 1 nm => 3 t / r = 0.6
!!
On the nanoscale the fraction of surface atoms is enormous!
From surface science we know these behave differently from the bulk
=> huge effects on material properties!
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Why is this then so significant?
Qualitatively because it is well known from surface science that
surface atoms behave often dramatically different from bulk ones
But this qualitative statement can even be quantified using simple basic concepts of surface physics
The surface energy of a material is defined as the work W
divided by area A which should be done when a surface is formed from bulk matter
Background: surface energy
AW
2surfWE
A
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Background: surface energy
A few typical surface energies and cohesion energies (= amount of energy/atom by which a material is held together)
Esurface
Ecohesion
(eV/Å2)
(eV/atom)
Cu
0.11 3.54
Ni 0.15
4.45
Au
0.09 3.93
The surface energy tells in essence: Ecoh, surf = Ecoh
– Esurf
Asurf
Using an area/atom of Asurf
≈
10 Å2
we see that the binding energy of surface atoms is ~ 30 % lower than normal
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Reduction in cluster cohesion
Let us now combine these two results: if the surface atom energy is some 30 % lower than normal and 60 % of all atoms are on the surface:
=> The cohesion of the entire cluster is ≈
20 % lower than normal!
In reality the effect may be even stronger because the whole electronic structure of the cluster differs from the usual.
On the other hand the fact that the atoms have more freedom to organize in energetically favourable
configurations may improve
on the situation
But overall the simple 20% estimate is definitely in the right ballpark
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Lowering of melting points of clusters
Because the cohesion of clusters is lower than usual, it is not surprising that the melting point of the clusters is much lower than the bulk melting pointExample: the melting point
of Au clusters as a function of their size
This is also related to
surface meltingSurfaces melt at lower
temperatures than the bulk
Thus with a lot of surface...
(you can figure out the rest yourself)
[Roy L. Johnston: Atomic and Molecular Clusters. Taylor & Francis 2002, via Tenhu
Nano III lecture]
Tm (R)/K = 1336.15 –
5543.65 (R/Å)-1
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Example: cluster landing on surfaces
An even more dramatic result is obtained when we consider what happens when a nanocluster lands on a surface at thermal (very low) kinetic energy/atomOur daily experience from macroscopic systems tells that if a
macroscopic ball made of a hard material softly lands on a flat surface, nothing of interest happens: it just stays there
But if we now consider this on the atomic scale, it is clear that right at the intersection the surface vanishes and new bonds are formed at the interface (atomistic terminology)
Same in continuum terminology:
Surface energy is freed up and
becomes interface energy
Let us now estimate how much
energy is freed up
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Example: cluster landing on surfaces
The range of interatomic interactions
is typically roughly 5 Å
Let us thus assume that when
the atom ball lands on the surface, surface energy is freed from an
h
= 5 Å
cap of a sphere
The area of the cap is (from basic geometry) A=2πrh
The energy freed is Esurf
A
This potential energy is freed up and becomes kinetic energy (heat)
If we assume that half of the freed energy goes initially to the
atom ball, we can estimate how much heat is generated
2 Å
h
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Example: cluster landing on surfaces
Using the kinetic energy equivalence
of energy and the atomic density ρat
we can calculate the heating effect-
Because all this happens very rapidly,
the heating is not an equilibrium process
and this is more suitable than using the heat capacity
The kinetic energy equivalence of temperature gives
and in this case Esurf
2A/2 = Esurf
A becomes kinetic energy:
h
TkrTkrTVkTNkE BatBatBatBkin3
3
23
423
23
23
Bat
surf
Bat
surfBatsurf kr
hETkr
rhETTkrrhE 23
3
22
22
[T. T. Järvi, K. Nordlund et al: Physical Review B 75 (2007) 115422;footnote: this calculation was actually originally done for the
first NanoI
course, and then later published in a scientific journal]
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Example: cluster landing on surfaces
From the result we see that the heating
reduces dramatically with the cluster size, as r-2
Let us now insert h
= 5 Å
and using e.g. the values
for Cu Esurf
= 0.11 eV/Å2 and ρat
= 0.084 1/Å3
we getMacroball:
r= 1 m => ΔT = 7.6e-16 K
Micro ball:
r= 1 m => ΔT = 7.6e-4 KNano ball:
r= 1 nm => ΔT = 760 K
I.e. nanometer sized clusters are heated a lot when they meet the surface, macroscopic ones practically not allCaveat: in the macro scale, surface oxidation also reduces the
heating, but in ultra-high vacuum conditions or for nonoxidizing
materials this calculation is directly relevant.
The melting point of Cu is 1360 K, but considering that the melting point is reduced, the whole cluster may melt on impact!
h
Bat
surf
krhE
T 2
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Example: cluster landing on surfaces
This leads to the observation that the whole cluster can change shape dramatically on impact
For very small clusters the change in shape may be so violent it does not matter what the original shape of the cluster is
This effect makes it possible
even for a fairly large cluster to become fully epitaxial with the surface directly on impact
”Epitaxy”
= lattice planes
match
[Meinander
et al, Thin Solid Films 425 (2002) 297]
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Kai Nordlund, Accelerator Laboratory, University of Helsinki
Example: “Contact epitaxy”
We call this effect contact epitaxy
The largest clusters do not fully change their shape, but also in them the atom layers closest to the surface ’melt’
for a moment and become epitaxialOn the right side an originally
single crystalline Ag cluster
on a Cu surface From the picture we see that the
bottom layers are no longer in the same orientation as the original, top ones
Instead the bottom planes are parallel to the Cu ones, even though the lattice
constant difference between Ag and Cu is 13% !
This has also been experimentally observed [Yeadon et al, J. Elect. Microsc. 48 (1999) 1075]
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Reactivity of surfaces
Surfaces are especially reactive chemically
The basic reason is easy to understand
In the bulk and normal stable molecules
all chemical bonds are saturated
But on a surface a few of the bonds are
‘missing’
i.e. atoms have unpaired electrons,
non-saturated
or dangling bonds
These are highly reactive
But things are really not quite that simple
The surface itself can often partly compensate the lack of surface bonds by rearranging the atoms such that the dangling bonds compensate each other: “surface reconstruction”
But this compensation is seldom
perfect, so additional reactivity remains
Si ‘100’
surface with dangling bonds marked with red short sticks
Same surface reconstructed so that the dangling bonds meet
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Example: radiative
cooling of nanoclusters
Let us consider how a “black”
body cools purely by
electromagnetic radiation The hotter the body, the more it
emits energy by radiation-
This is the reason to e.g. iron
glows when heated
If the body is “black”
i.e. does not reflect light, the total intensity of this radiation is
where σ
is the Stefan-Boltzmann constant (=5.67x108
W/m2K4) and T
the temperatureOn the other hand from basic definitions
where E
is energy, A
surface, t
time, and cV
specific heat capacity
4TI
AdtdE
API
mdTdEcV
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Example: radiative
cooling of nanoclusters
Let us now calculate how long the cooling of a black body sphere
takes starting from some temperature T0 to another temperature T1
solely by this radiative
cooling
Let us solve dE
from both equations:
By setting these two equal (ρ
= density)
Let us use this for a sphere with A = 4πr2
and V = 4πr3/3:
By integrating
this
in the range
T0 -> T1
we get the cooling time t:
dTmcdE VdtATdtIAdE 4
444
TdT
AVc
TdT
AmcdtdTmcAdtT VV
V
442
3
3344
TdTrc
TdT
rrcdt VV
1
0
4 3 3 3 31 0 0 10
1 1 1 1 13 3 3 9
TtV V V
T
c r c r c rdTdt tT T T T T
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Example: radiative
cooling of nanoclusters
We obtained:
The crucial thing here is that the cooling time is directly proportional to the size of the sphere r
!
The smaller the sphere, the faster it cools!
Let us as an example calculate how fast a ball consisting only of gold would cool from the boiling point to the melting point
For simplicity, let’s use the normal bulk values: ρ
= 19.3 g/cm3, T0 = 3129 K, T1
= 1337 K, ja
cV
= 129 J/(kgK)
3 30 1
1 19Vc rt
T T
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Example: radiative
cooling of nanoclusters
Result for spheres of different size: r = 5 mm: 9 s r = 5 µm: 9 ms r = 5 nm:
9 µs !
It is in fact quite questionable to use the bulk values for the boiling and melting points, and in fact the Stefan-Boltzmann law is in a more general form for non-black bodies
where ε
is some number < 1
But the basic argument and order of magnitude is quite correct: experiments do show that nanoclusters cool very rapidly in vacuum, where other cooling
mechanisms are not significant!
4TI
[Elihn
et al, Appl. Surf. Sci. 186 (2002) 573]
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Nanoresonators, “nanokantele”
For instance a xylophone or the Finnish
instrument ”kantele”
produces sounds by the classical resonator principle In cases where the strings are not
under tension the sound frequency is
where L is the length of the resonator, A
is its cross-sectional area and the other constants are properties of the string material
Crucial is that f
is inversely proportional to L2
: The shorter the string, the higher the sound frequency!
Thus if a xylophone or kantele
could be implemented on the
nanoscale, one could obtain very high frequency (ultra-) sound with it!
24.73 EIf L A
[Image from wikimedia
commons]
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
“Nanokantele”
Sort of a nanokantele
has been implemented with
micromechanics: nano and microscale
resonators have been etched into Si The resonance frequency was up to 380 MHz!
[Carr et al, Appl. Phys. Lett. 75 (1999) 920]
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Kai Nordlund, Accelerator Laboratory, University of Helsinki
Hall-Petch
relation: background
There are many measures of the hardness
of materials
One of the most important is the so called
yield strength σy
which is an applied measure of at what pressure a material has been
subject to a significantly large permanent elongation
Most common definition: σy
≈
at what pressure has the material permanently elongated by
0.2%?
The empirical so called Hall-Petch
relation says
that the yield strength of materials is
where σ0
and K
are material-dependent variables and d
is the average crystalline grain size of the materialAll ordinary metals are polycrystalline with grain sizes ~ 10-100 µm
dK
y 0
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Kai Nordlund, Accelerator Laboratory, University of Helsinki
Hall-Petch
relation
Because the fraction is proportional to grain size, this would predict that when the grain size of the material → 0, the yield strength of the material → infinity
In practice this can not of course happen, since the atom size of ~ 0.2 nm is eventually reached If the grain size is the atom size, one has a single-crystalline
material with a known, not so high yield strength The law is empirical, so it has to have a lower limit of validity The crucial question thus becomes, at what grain size is Hall-Petch
no longer valid, and how strong can the material maximally be?
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Hall-Petch
relation
This has been examined systematically with atomistic
simulations(animation gr16KB_T300)
The main result is that Hall-Petch
is no longer valid at a grain size of about 15 nm
Below it a reverse Hall-Petch
effect is observed
But maximally, according to the
simulations, the strength of Cu could be σy
= 2.3 GPa
In ordinary Cu the strength
is only about 0.069 GPa 33 x improvement!
[K.W. Jacobsen; CSC News 1/2005]
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Hall-Petch
relation
The same behaviour
has also been observed experimentally
Youssef
et al reported for a 23 nm grain size
σy
= 0.77 GPaNot quite as high as in
the simulationsBut still about 10x higher than the normal value for Cu!
In addition, according to the same reference, the material still
also has a good ductilityNormally ductility and strength go in opposite directions But the research in the field is very new, so this result is better to be
considered promising than definitive
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Rayleigh scattering
In classical electrodynamics the scattering of light from small particles is described by the so called Rayleigh scattering equations The equations explain for instance why the sky is blue
But in nanoscience we are interested in the dependence of the scattering intensity on the diameter d
When d << λ
we have
So from small particles the scattering is very weak => they are practically transparent
~ d6 !
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Other scattering
On the other hand also so called plasmon
resonances can occur
in metallic nanoparticles Their fundamental nature is too complicated to be described during
this course But in essence it means that a nanoparticle can scatter or absorb
light in a rather narrow range of light wavelengths Also quantum mechanics may lead to similar effects (cf. QM lecture)
But the basic conclusion from all this is:Due to Rayleigh scattering, nanoparticles
made of normally opaque materials become almost transparent on the nano scale
Due to plasmons
and/or quantum effects, they can start absorbing or scattering light in some well-defined colours
The colour
of nanoparticles can be anything
Nano-1
Kai Nordlund, Accelerator Laboratory, University of Helsinki
Summary
During this lecture I have described several types of scaling with size and how these change materials properties dramatically
Summary of these scalings
and how the affect a given property
when going from a scale of, say, 10 mm to 10 nm:
Similar effects can be obtained for any property which scales with particle size d
–
can you think of your own!?
Scaling law Proportionality
with diameter dRelative change from
10 mm to 10 nm
Spontaneous impact heating d-2 1012
Radiative
cooling time d+1 10-6
Nanoresonator d-2 1012
Hall-Petch
relation d-1/2 103
Rayleigh scattering d6 10-36