NB. Embedded Google book (EB) trial, read on and scroll down for relevent pages.

and first experience with Wolfram Alpha. Both postive, a pleasure.

Having "dived in at the deep end" on a couple of recent posts, only just managing to "scratch the surface" in related post 1_ "Multiscale materials modelling" and related post 2 _ "Universal size/shape-dependent law for characteristic temperatures, phase transformation in nanoparticles," I decided to go back to basics.

It is widely known in the nanotechnology field, and as previously mentioned in my earlier post (2) that the ratio of surface area to volume increases as the size of particles size decreases (roughly as the inverse ratio of the characteristic dimension (x). cf graphs above where r is taken for spherical shapes and L for cubic shapes). At small nanosizes this increase in surface is quite dramatic. This surface to volume relationship is the most basic engineering factor in nanoscience and technology. It underpins the surface dependency of most if not all nanoscience and technology fields, eg. where surface properties and effects at low weight are required such as in catalysis or in fuel-cell applications to mention only two.

Numerous works mostly from teaching nanomaterials sources approach the subject but the best to my mind is the work by Mike Ashby et al in their book "Nanomaterials, nanotechnologies and design: an introduction for engineers ..." By M. F. Ashby, Daniel L. Schodek, Paulo J. S. G. Ferreira (ref. 1, and relevent pages in the google embedded book, read it below ) (*Prof Mike Ashby, FRS, is famous for his ability to reduce complex engineering materials mechanical properties to their simplest expression and to describe them comparitively in his now well-known Ashby diagrammes. This lead Mike to create Grant Design Ltd., in 1994 with Dr D.Cebon both of Cambridge Univ., UK. cf. Related posts )

Not to give Mike a full clean slate,(although most deserved) I have checked, re-calculated and presented Ashby et al's results in the above graphs using the fairly intuitive Wolfram Alpha's Mathamatical Tool.

To get a better grip on nano-things, their book Ch 6.2 also gives a simple numerical example of the number of increasing number of particle when reducing a particle size from 10µm diametre to a group of 10 nm in diametre particle of identical total volume. (N=V10µ/V10nm). This 10nm group is shown to be comprised of 10^9 particles which in turn is shown to to give a 1000 times increase in surface area. (notice the unit 10 is diametre and not radius)

The graph below right follows Ashby et al's eqn. 6.9 cf. embedded book pages below

Next, the authors treat crystalline nanoparticles. They point out that in addition to shape, structure must be taken into account.

There is a brave attempt to show how we get to the above equation by Univ of Wisconsin, Chem 801 Lecture notes (ref 2 below).

No wonder as a student I started to see modulable atomic structures (lego like principle), in the chemistry professors offices or on the lecture hall benches!)

Now thermodynamics imposes a total energy minimisation. For the FCC nanoparticle this given by (surface area X the surface energy), neglecting edge and curvature effects. They present an arguement based on atomic planes of high symmetry; Amoung possible shapes the smallest FCC nanoparticle is the cubo-octohedron. (Ashby fig. 6.20) which is a 14 sided polyhedron (looks almost spherical, doesn't it.) consisting of 12 surface atoms and one bulk atom.

Total nos of surface atoms Ns= 10n2-20n+12 (eqn 6.13 Ashby)

More about this book

To get the most out of this book via author authorized limited preview, the reader will find my blog format too small, so take a squint to judge your degree of interest, but whatever, do check the full sized version by clicking the button "More about this book" on the bottom right hand (RH) corner of my embeded version, it's free. A new page opens which allows you to enlarge to suit any level of reading capacity. You can do your own review there and please leave a comment either on the subject of my post or on any of the themes available in the preview. Thanks in advance

### Multiscale modelling of materials,MMM - Introduction and Explanatory Notes; Refs.,Images, on a Hot Interdisciplinary field

1."Nanomaterials, nanotechnologies and design: an introduction for engineers ..." By M. F. Ashby, Daniel L. Schodek, Paulo J. S. G. Ferreira CH 6 Size effects Surface to volume ratio versus Shape.

2. Chem 801 Lecture Notes, Univ of Wisconsin

and first experience with Wolfram Alpha. Both postive, a pleasure.

Having "dived in at the deep end" on a couple of recent posts, only just managing to "scratch the surface" in related post 1_ "Multiscale materials modelling" and related post 2 _ "Universal size/shape-dependent law for characteristic temperatures, phase transformation in nanoparticles," I decided to go back to basics.

It is widely known in the nanotechnology field, and as previously mentioned in my earlier post (2) that the ratio of surface area to volume increases as the size of particles size decreases (roughly as the inverse ratio of the characteristic dimension (x). cf graphs above where r is taken for spherical shapes and L for cubic shapes). At small nanosizes this increase in surface is quite dramatic. This surface to volume relationship is the most basic engineering factor in nanoscience and technology. It underpins the surface dependency of most if not all nanoscience and technology fields, eg. where surface properties and effects at low weight are required such as in catalysis or in fuel-cell applications to mention only two.

Numerous works mostly from teaching nanomaterials sources approach the subject but the best to my mind is the work by Mike Ashby et al in their book "Nanomaterials, nanotechnologies and design: an introduction for engineers ..." By M. F. Ashby, Daniel L. Schodek, Paulo J. S. G. Ferreira (ref. 1, and relevent pages in the google embedded book, read it below ) (*Prof Mike Ashby, FRS, is famous for his ability to reduce complex engineering materials mechanical properties to their simplest expression and to describe them comparitively in his now well-known Ashby diagrammes. This lead Mike to create Grant Design Ltd., in 1994 with Dr D.Cebon both of Cambridge Univ., UK. cf. Related posts )

Not to give Mike a full clean slate,(although most deserved) I have checked, re-calculated and presented Ashby et al's results in the above graphs using the fairly intuitive Wolfram Alpha's Mathamatical Tool.

To get a better grip on nano-things, their book Ch 6.2 also gives a simple numerical example of the number of increasing number of particle when reducing a particle size from 10µm diametre to a group of 10 nm in diametre particle of identical total volume. (N=V10µ/V10nm). This 10nm group is shown to be comprised of 10^9 particles which in turn is shown to to give a 1000 times increase in surface area. (notice the unit 10 is diametre and not radius)

The graph below right follows Ashby et al's eqn. 6.9 cf. embedded book pages below

Next, the authors treat crystalline nanoparticles. They point out that in addition to shape, structure must be taken into account.

They chose a nanoparticle with a the face centered cubic (FCC) structure, due to its practical importance, eg. Au, Ag, Ni, Al,Cu,Pt have FCC structure. The FCC unit cell, has 14 atoms all on the surface.(cf. image in EB below, ) The general equations for increasing numbers of atoms (n) by increasing unit cell layers are given as:

Total Nos of Surf Atoms Ns=12n^2+2 (eqn 6.10, Ashby et al EB)

Total Nos of Bulk Atoms Nb=4n^3-6n^2+3n-1 (eqn 6.11 Ashbey et al EB)

With (n) as input, resultats are tabulated for n, Ns, Nb, Ns/Nb ratio and percent. in table 6.1 of the EB below and are the so called "structural magic numbers".

There is a brave attempt to show how we get to the above equation by Univ of Wisconsin, Chem 801 Lecture notes (ref 2 below).

No wonder as a student I started to see modulable atomic structures (lego like principle), in the chemistry professors offices or on the lecture hall benches!)

Now thermodynamics imposes a total energy minimisation. For the FCC nanoparticle this given by (surface area X the surface energy), neglecting edge and curvature effects. They present an arguement based on atomic planes of high symmetry; Amoung possible shapes the smallest FCC nanoparticle is the cubo-octohedron. (Ashby fig. 6.20) which is a 14 sided polyhedron (looks almost spherical, doesn't it.) consisting of 12 surface atoms and one bulk atom.

For the cubo-octoherdral nanopartical, the crystal structure is maintained and the eqns giving the structural magic numbers are:

Total nos of surface atoms Ns= 10n2-20n+12 (eqn 6.13 Ashby)

Total nos of bulk atoms Nb= 1/3(10n^3 - 15n^2 + 11n - 3) (eqn 6.13 Ashby)

I have ploted the ratio Ns/Nb from the above eqns with hints on the ease and flexability of wolfram's maths tool. Trial and error is a good guide in this easy to use tool. Enough...

There is more and better in this my first Embedded Google Book.

It is with immense pleasure and priviledge that I am able to present "Nanomaterials, nanotechnologies and design: an introduction for engineers ..." By M. F. Ashby, Daniel L. Schodek, Paulo J. S. G. Ferreira.

More about this book

To get the most out of this book via author authorized limited preview, the reader will find my blog format too small, so take a squint to judge your degree of interest, but whatever, do check the full sized version by clicking the button "More about this book" on the bottom right hand (RH) corner of my embeded version, it's free. A new page opens which allows you to enlarge to suit any level of reading capacity. You can do your own review there and please leave a comment either on the subject of my post or on any of the themes available in the preview. Thanks in advance

Peruse with pleasure and make your own judgement.

**RELATED POSTS:**### Multiscale modelling of materials,MMM - Introduction and Explanatory Notes; Refs.,Images, on a Hot Interdisciplinary field

### Universal size/shape-dependent law for characteristic temperatures, phase transformation in nanoparticles

### Ashby Diagrammes, Granta Design, Materials selection software

### It's not HSLA-Bainite"Nanostructured Steels"-Green Light by Irvine-based Materials Science Co-MMFX Tech Corp - Corrosion and Toughness Themes

**REFERENCES.**1."Nanomaterials, nanotechnologies and design: an introduction for engineers ..." By M. F. Ashby, Daniel L. Schodek, Paulo J. S. G. Ferreira CH 6 Size effects Surface to volume ratio versus Shape.

2. Chem 801 Lecture Notes, Univ of Wisconsin

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