1. Introduction to Nanotechnology
Module 3
Surface Area to Volume Ratio

2. 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

3. 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.

4. 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

5. 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

6. 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

7. 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

8. 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.)

9. 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

10. 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.

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