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Size-dependent physical properties of nanostructures: Introduction

Five times stronger than steel, this shiny beast is also twice as strong as any bullet stopping or bomb protection kit in use today. Developed by Israeli firm, ApNano, the material used is referred to as inorganic fullerene-like nanostructures – or buckyballs

can protect you..........can protect you..........can protect you..........can protect you..........

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can add to toxicity

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Lets think about one interesting property of nanostructures

HST researchers have experimented with polymer-coated iron oxide nanoparticlesheld together by DNA tethers to help them create a visual image of a tumor through magnetic resonance imaging.

The researchers are studying DNA sequences to gauge the point at which heat activates the nanoparticles after they have reached tumors in the body. One advantage of a DNA tether, the HST team members say, is that its melting point is tunable—scientists would be able to control when the bonds between the nanoparticles break by creating links of varying lengths with different DNA sequences.

melting gold nanoclusters

They melt at low temps

This can be useful

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The shape of the melting curve as a function of particle size is common

This is for CdSenanoparticles

Melting Points were measured using TEM

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OK: so let's think about melting points

Correlation of Melting Point with Lattice Energy for Cubic Ionic Solids

-6686603.18NaI-7277902.94NaBr-7658012.79NaCl-8829882.31NaF

Lattice Energy(kJ/mol)

Melting Point(Celsius)

Interionic Distance(Angstroms)Compound

You met Lattice Energies in CHEM 1050 and in CHEM 2060

lattice energy is the energy released when 1 mole of a substance is formed from its gas phase ions.

lattice energy is the energy released when 1 mole of a substance is formed from its gas phase ions.

not quite the reverse of melting

Lattice energy gives us an ideaof the cohesive energy:

from my CHEM 2060 NOTES

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Melting Points scale with lattice energies

There is not however a linear relationship WHY?

You should recall that we get LE's from Born -Haber Cycles

N a +( g) e

- C l ( g) + +

N a ( g) C l 2( g) + 12_

N a ( s ) C l 2( g) + 12_

∆ H f N a C l ( s )

U ( L A T T I C E E N E R G Y ) +

1 2 _ C l 2 B E

T E N a E A _ C l -( g) ( N a

+( g) + + N a ( g) C l ( g)

N a ( s ) ∆ Hat

IE: Na(g)

Born Haber Cycle

This is the MAJOR energy term in the understanding ofthe stability of ionic solids

: chemical accounting

DEBITS CREDITS

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As you saw in CHEM 2060 Lattice Energy is coulombic in nature

Recall Coulombs Laws:

Usually given in terms of forces: we want Energies: E = mF.dr

(Energy is Force X dist: added up for all distances)

rcoulombic interaction = r4

qq -

πε+

E = 1 X

important: see in a min

E per ion : see in a min

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For an array of ions as in a nanostructure it gets more complicated

+

- +

-

r

all attractions and reps must besummed up to get total energy

E = -4 (q2/r) + 2 (q2 /(r2)r)

review of 2060: I hopereview of 2060: I hopereview of 2060: I hopereview of 2060: I hope

attractiverepulsive

= 4 – (2) ½ [ -q2/r]

= 2.586 [ -q2/r]

= 0.6465 [ -q2/r] per ion

This number is calledthe Madelung Constant

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For any given structure the Madelung Constant can be determinedBy convention it is quoted for r=1.

this is not trivial!!!NaCl 1.7475

CsCl 1.7626

ZnS (Zincblende) 1.6380

ZnS(wurtzite) 1.6413

CaF2 2.5193

TiO2 (rutile) 2.408

These values you have seen in CHEM 2060: they are for bulk "infinite arrays"

-they are used to determine LE's of bulk crystals

- recall Born –Meyer and Born-Landé equations

In CHEM 2060 you were taught that the MC could be determined byfocusing on one ion and summing all its attractions and repulsions in shells

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This is because for an infinite array of ions each ion is the same

so all you have to do is work out the energy per that ion

recall : MC is "per ion"

For a nanostructure this is WRONG: the array is far from "infinite"

so we must work out all the attractions and repulsions separately

add them up and divide by the number of ions

if you do that for a HUGE particle you get the same MC as with the "single ion method

I know I know I know I know ––––I have done it I have done it I have done it I have done it

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So for a nanoparticle the MC is an "average" over the whole particle

If the MC is less than the bulk value so will be the LE and thus the melting point

Let's try one by hand!

We will look at the "primitive unit cell of NaCl

Cl-Na +

corner ions : 3 attractions at r: 3 repulsions at (r)1/2

: 1 attraction at (r) 1/3

- a very small nanostructure

what is the real value of r ???

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In this case if we count each ions we will get 8 terms

However each is the same

to get the MC per ion we will divide by 8 anyway

so in this case we need only use one of the corner ions Cl-

Na +

E = -3 q2/r + 3 q2/ (2) ½ r - q2/ (r) 1/3

MC = 1.455

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So: Things to note

smaller than bulk

larger than a dipole : MC =1

overall attractive (not a neg number)

Erwin Madelung

May 18, 1881 – August 1, 1972

In 1921 he was appointed head of theoretical physics at the University of Frankfurt-Main, which he held until 1949. His worked specialized in atomic physics and quantum mechanics, and it was during this time he developed the Madelung Equations, an alternative form of the Schrodinger Equation

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Secrets of the Madelung Constant

C l

C l

C l

C l

N a r

N a

N a

N a 2 r

3 r

6 rqq- E ClNa ×⋅

=

12 2r

qq E NaNa ×⋅+=

83rqq- E ClNa ×=

First 6 nearest neighbour Na-Cl (Attractive

Second 12 next nearest neighbor Na-Na(Repulsive)

Third 8 next nearest neighbour Na-Clseries oscillates wildly!!

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Table 1. Nonconvergence of the NaCl Madelung SeriesTerm No. Term Calcd MadelungConstant346,000 1152/. (346,000)1/2 5.277831078346,001 +8832/. (346,001)1/2 9.736997604346,002 4608/. (346,002)1/2 1.903185248346,003 +1920/. (346,003)1/2 5.167269230346,004 5112/. (346,004)1/2 3.523341179

Table 2. Sample Terms from NaCl Madelung SeriesShowing Net Charge on Ion ClusterTerm No. Term Calcd MadelungConstant Charge346,000 1152/. (346,000)1/2 5.277831078 4132346,001 +8832/. (346,001)1/2 9.736997604 4700346,002 4608/. (346,002)1/2 1.903185248 92346,003 +1920/. (346,003)1/2 5.167269230 2012346,004 5112/. (346,004)1/2 3.523341179 3100An In-Depth Look at the MadelungConstantfor Cubic Crystal SystemsRobert P. Grosso Jr., Justin T. Fermann, and William J. Vining*

Journal of Chemical Education • Vol. 78 No. 9 September 2001

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The long-range interaction energy is the sum of interaction energies between the charges of a central unit cell and all the charges of the lattice. Hence, it can be represented as a double integral over two charge density fields representing the fields of the unit cell and the crystal lattice

Until recently special methods were needed: Ewald's Method was the most popular

where the unit-cell charge density field is a sum over the positions

of the charges qk in the central unit cell

and the total charge density field can be represented as a convolution of

with a lattice function

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Since this is a convolution, the Fourier Transform of is a product

where the Fourier transform of the lattice function is another sum over delta functions

where the reciprocal space vectors are defined (and cyclic permutations) where

is the volume of the central unit cell (if it is geometrically a paralleapiped, which is often but not necessarily the case). Note that both and are real even functions

For brevity, we define an effective single-particle potential

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Since this is also a convolution, the Fourier transformation of the same equation is a product

where the Fourier transform is defined

The energy can now be written as a single field integral

Using Parseval's theorem, the energy can also be summed in Fourier space

where

This is the essential result. Once is calculated, the summation/integration over

is straightforward and should converge quickly. The most common reason for lack ofconvergence is a poorly defined unit cell, which must be charge neutral to avoid infinite sums.