Nanoscience I: Downscaling of classical laws … Nanoscience I: Downscaling of classical laws makes nano different Kai Nordlund 3.10.2010 Faculty of Science Department of Physical

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


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Of course it is not automatically clear whether laws originally made for the macroscale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


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


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


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


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


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


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



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



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



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



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]

Nano-1


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


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


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


Example: radiative


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


Example: radiative


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


Example: radiative


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


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


“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]

Nano-1


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

Nano-1


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


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


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


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


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


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

Documents

Nanoscience I: Downscaling of classical laws … Nanoscience I: Downscaling of classical laws makes nano different Kai Nordlund 3.10.2010 Faculty of Science Department of Physical