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**Introduction to Nanotechnology**

Module 3 Surface Area to Volume Ratio

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**The “Big Ideas” of Nanoscale Science***

Sense of Scale Surface area to volume ratio Density, force and pressure Surface tension Priority of forces at different size scales Material/Surface properties Understanding of these concepts requires an integration of the disciplines of math, biology, chemistry, physics and engineering This is the second big idea of nanotechnology. Surface area is an extremely important quantity since surface area determines chemical reactions and many of the force interactions between molecules. From a nanotechnology standpoint, the fact that surface area increases with respect to volume as we scale down the size of objects plays a significant role in many of the benefits and applications of nanoparticles or objects. © Deb Newberry 2008

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**Surface Area to Volume Ratio**

This ratio is an important factor in understanding many of the straight-forward, counter intuitive or unusual properties that can be observed at the nanoscale. Surface Area to Volume ratio (SA/V) changes as the size of the material changes – it is not constant!!!! Not convinced? Do the calculation for a die and a Rubics cube. Also – if we keep the total volume of a material constant but divide that volume into smaller and smaller pieces – the SA does not increase in a linear fashion. Summary chart of the objectives and concepts of this module.

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**Surface Area to Volume Ratio**

The SA/V ratio also represents the percentage of the atoms that are on the surface of the material. SA drives chemical, electrical and biological interactions and systems V drives weight, cost, inertia, momentum and other factors Both SA and V are dependent on a linear dimension (length) but SA goes as the square and V goes as the cube – this is “rub” Ratios of this type are found in many equations – this is the first encounter to get familiar with this concept – it can be found in all aspects of nanoscience

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**Pressure, force and density**

Surface area to volume Sugar Cubes Basic algebra Rules of exponents, Units conversion Other shapes Excel optional Surface area to volume Pictorial representation of prereqs module, activities and follow on application of these principles to pressure, force and density. Soap bubbles Pressure, force and density

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**For a cube: V= a3 Surface area =6 a2 Notice the difference in powers**

of the linear dimension in the ratio of surface area to volume (SA/V) a V= a3 Surface area =6 a2 Here we can select a dimension for the side of a cube and calculate the surface area and volume. Then, as shown above break the large cube down into 8 smaller cubes. The total volume will stay the same, but the new total surface area increases significantly. This is a good place to have the sugar cubes available for either a demonstration by you – or an on-class activity for the students. Breaking the large cube into smaller cubes keeps the Total volume the same but Increase the total surface area Impacts: Cell sizes Surface tension Nanotex pants

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This represents how the ratio of SA/V changes as we reduce the overall size of the object. The smaller sphere has much more surface area – compared to the total volume than the larger sphere. This is a direct correlation to the die versus Rubiks cube calculation from slide 3. Ref: NanoInk

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**SA/V represents the percentage of atoms on the surface of an entity**

Let’s assume we have a cube that is 1 cm 3 The SA will then be 6 cm 2 Assume each atom is 1 nm 3 in size and takes up an area on the surface of 1 nm 2 How many atoms in the cube? How many atoms on the surface? What is the percentage? Now break the larger cube into cubes 1 mm on a side Percentage of total atoms that are on the surface Percentage of atoms on the surface for each smaller cube The final aspect of this module is that the ratio of SA to V represents the percentage of atoms that are on the surface of the object. This is the percent of atoms that are available for chemical reactions or bonding. (This aspect is important when discussing purity of nanoparticles and quality control.)

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**Surface area to volume ratio**

Changes for an object as the size of that object changes Impacts percentage of atoms on the surface that are available to participate in reactions Changes non linearly as a large object is broken down into smaller objects Introduces us to thinking about the dependence of different parameters on different powers of the linear dimension Summary Chart

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**Start looking for…. “Hidden” dimensional dependencies**

At first glance pressure only appears to be dependent on the area aspect of the length dimension… But upon closer inspection – see we have a volume dependence in the numerator…… This happens many times in all of the traditional sciences. This critical thing concept extends to other parameters (temperature, material properties etc.) For this module the powers of dependencies on the linear dimension was very obvious. In some cases – like Module 4 – the dependency may be less obvious. Start thinking about where are the parameter dependencies in various equations.

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References Poole, Charles P., and Frank J. Owens. Introduction to Nanotechnology. Hoboken, NJ: J. Wiley, 2003.

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