Size-dependent physical properties of nanostructures: Introduction

Transcription

1 Size-dependent physical properties of nanostructures: Introduction can protect you... 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 fullerenelike nanostructures or buckyballs 16-1

2 can add to toxicity 16-2

3 Lets think about one interesting property of nanostructures melting gold nanoclusters They melt at low temps HST researchers have experimented with polymer-coated iron oxide nanoparticles held together by DNA tethers to help them create a visual image of a tumor through magnetic resonance imaging. This can be useful 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. 16-3

4 The shape of the melting curve as a function of particle size is common This is for CdSe nanoparticles Melting Points were measured using TEM 16-4

5 OK: so let's think about melting points Correlation of Melting Point with Lattice Energy for Cubic Ionic Solids Compound Interionic Distance (Angstroms) Melting Point (Celsius) Lattice Energy (kj/mol) NaF NaCl NaBr NaI You met Lattice Energies in CHEM 1050 and in CHEM 2060 lattice energy is is the the energy released when 1 mole of of a substance is is formed from its its gas phase ions. from my CHEM 2060 NOTES not quite the reverse of melting Lattice energy gives us an idea of the cohesive energy: 16-5

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7 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 16-7

8 Born Haber Cycle : chemical accounting DEBITS CREDITS N a + ( g) + e - + C l ( g) IE: Na(g) T E N a + N a ( g) C l ( g) _ E A C l - ( g) ( N a + ( g) + 1 _ 2 B E C l 2 _ 1 N a ( g) + C l 2 ( g) 2 Hat N a ( s ) _ 1 N a ( s ) + C l 2 ( g) 2 + U ( L A T T I C E E N E R G Y ) H f N a C l ( s ) This is the MAJOR energy term in the understanding of the stability of ionic solids 16-8

9 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) E per ion : see in a min E = r coulombic interaction = 1 X q+ q - 4πεr important: see in a min 16-9

10 review of 2060: I hope For an array of ions as in a nanostructure it gets more complicated + - all attractions and reps must be summed up to get total energy r - + E = -4 (q 2 /r) + 2 (q 2 /(r2)r) = 4 (2) ½ [ -q 2 /r] This number is called the Madelung Constant attractive repulsive = [ -q 2 /r] = [ -q 2 /r] per ion 16-10

11 For any given structure the Madelung Constant can be determined By convention it is quoted for r=1. NaCl CsCl ZnS (Zincblende) ZnS(wurtzite) CaF TiO 2 (rutile) this is not trivial!!! 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 16-11

12 In CHEM 2060 you were taught that the MC could be determined by focusing on one ion and summing all its attractions and repulsions in shells 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 I have done it

13 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 - a very small nanostructure corner ions : 3 attractions at r : 3 repulsions at (r) 1/2 : 1 attraction at (r) 1/3 what is the real value of r??? Cl - Na

14 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 q 2 /r + 3 q 2 / (2) ½ r - q 2 / (r) 1/3 MC =

15 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 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 16-15

16 Secrets of the Madelung Constant First 6 nearest neighbour Na-Cl (Attractive C l N a 2 r N a r C l 3 r N a C l C l N a E E + q = = -q q 2r q r Na Na Na Cl Second 12 next nearest neighbor Na-Na (Repulsive) 12 6 series oscillates wildly!! Third 8 next nearest neighbour Na-Cl E = -q q Na Cl 3r

17 Table 1. Nonconvergence of the NaCl Madelung Series Term No. Term Calcd Madelung Constant 346, /. (346,000)1/ , /. (346,001)1/ , /. (346,002)1/ , /. (346,003)1/ , /. (346,004)1/ Table 2. Sample Terms from NaCl Madelung Series Showing Net Charge on Ion Cluster Term No. Term Calcd Madelung Constant Charge 346, /. (346,000)1/ , /. (346,001)1/ , /. (346,002)1/ , /. (346,003)1/ , /. (346,004)1/ An In-Depth Look at the Madelung Constant Journal of Chemical Education Vol. 78 No. 9 for Cubic Crystal Systems September 2001 Robert P. Grosso Jr., Justin T. Fermann, and William J. Vining* 16-17

18 Until recently special methods were needed: Ewald's Method was the most popular 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 where the unit-cell charge density field of the charges qk in the central unit cell is a sum over the positions and the total charge density field can be represented as a convolution of with a lattice function 16-18

19 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 16-19

20 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 of convergence is a poorly defined unit cell, which must be charge neutral to avoid infinite sums