Nanotechnology and mathematics

Summary: Nanoscience relies on mathematical modeling to predict the behavior of substances at the nanoscale.

Nanotechnology is a relatively new field of scientific study, the conceptual origins of which are typically credited to a presentation by physicist Richard Feynman in the late 1950s. A nanometer is one-billionth of a meter, and nanoscience focuses on matter with dimensions between 1 and 100 nanometers. For comparison, an ordinary sheet of paper is about 100,000 nanometers thick, a human hair is between 60 and 120 nanometers thick, and the diameter of one atom of gold is about 1/3 of a nanometer. Thus, nanotechnology is concerned with studying materials at a very small scale, ranging from roughly larger than a single atom at the lower end to objects that can be seen with a high-quality optical microscope at the upper end.

Physicists, mathematicians, and other nanotechnologists are often particularly interested in how the physical, chemical, and biological properties of materials may differ at this scale as opposed to properties of the same materials in bulk or at the scale of single atoms or molecules. Feynman discussed the notion that human beings would someday be able to create increasingly smaller and smaller machines, in part through directed, precision arrangement of atoms and molecules. He also introduced the idea that change in scale would affect the mathematical and physical properties of technology and processes. For example, relatively large-scale forces like gravity would begin to diminish in importance as machines grew smaller, while molecular-level van der Waals attractive forces, named for chemist Johannes van der Waals, and other properties would take on more important roles. However, he did not call his own ideas “nanotechnology.” Instead, the first use of the term as it is typically meant in the early twenty-first century is credited to engineer K. Eric Drexler in the 1980s. He also helped spread nanotechnology and molecular manufacturing ideas to a broader audience. There are types of technology that are already being created at nanoscales. Some visions about the future of molecular manufacturing are much like the replicator device in the science fiction franchise Star Trek: human-scale or even larger objects, even complex devices like computers, quickly assembled atom by atom.

Any word with the prefix “nano” means at a nanometer scale (for example, the word “nanofilter” would refer to a filter at the nanometer scale), but there are also some basic classifications that are in common use. Nanomaterials are furthered classified as nanoparticles (if all three dimensions are nanosized), nanotubes (which have a nanosized diameter but greater length), and nanofilms or nanosheets (the thickness is nanosized, but the width and height may be much greater). Nanostructured materials have an internal structure that is nanosized, but the pieces of material may be much larger.

Principles

Nanotechnology draws on many scientific fields, including chemistry, physics, and biology, as well as engineering and materials science, and one common thread among them all is mathematics. Interestingly, the extreme difference in size between usual applications and applications at the nanoscale means that some of the most fundamental laws describing natural processes do not apply. For instance Ohm’s law describes the flow of electrical current as

where I is the current in amps, V is the potential difference in volts, and R is the resistance of a conductor in ohms. This law is based on the free flow of electrons and hence does not describe the movement of electrons through nanowires, which may be so narrow as to allow only one electron to pass through at a time. To take another example, at the nanoscale, heat flow is no longer governed by standard continuity boundary conditions and different assumptions that allow for discontinuities must be used instead. Identifying and quantifying how such fundamental laws and expectations change at the nanoscale is one important field of study within nanotechnology.

Construction of systems at the nanoscale allows researchers great control over the form of the nanoparticles developed as well as the ways they form three-dimensional wholes. One line of research involves devising structures that require the minimum number of molecules for a given construct, while another involves developing self-assembling structures, such as cubes and buckyballs. Nanotechnology also adds new complications to issues of dimensionality. From elementary geometry, humans are accustomed to thinking in terms of one dimension (a line), two dimensions (a plane), and three dimensions (a cube, or any object in space). However, at the nanoscale the picture is not so clear. For instance, quantum dots or “artificial atoms” that contain only one or a few electrons with discrete energy states are zero-dimensional solids, which can function in quantum computers as a binary switch. Fractals, which are described by noninteger dimensionality (for example, a two-and-a-half-dimensional object) are also used to model nanoscale systems.

Applications

Medicine is one of the most promising fields for nanotechnology because many internal processes of the human body take place at nanoscale dimensions. Drug delivery is one promising field: nanoparticles can be used to deliver drugs directly to particular cells, for instance, for chemotherapy that targets cancerous cells but not healthy cells and thus reduces tissue damage. Nanotechnology has also developed ways to use nanoshells to concentrate heat from infrared light to destroy cancer cells with minimal damage to adjacent healthy cells. Nanotechnology promises to allow some drugs now delivered by injection to be taken orally, encapsulated in a nanoparticle, which would help it pass into the bloodstream from the stomach. Nanofibers have been used to repair damaged joints by stimulating the body’s production of cartilage; nanoparticles have been used to increase the speed of blood clotting to prevent blood loss in trauma patients; and nanocrystalline silver is already being used as an antimicrobial agent for wound treatment. Nanocrystal technology is being developed to improve medical imaging, and in the future it may be possible to develop cell repair nanorobots, which could be programmed to repair diseased or damaged cells in a person’s body.

Nanotechnology has many applications in the fields of energy production and pollution control. Nanotechnology has made it possible to create more efficient solar cells at a lower cost (making the technology more likely to be adopted) and provided new forms that make solar technology more convenient. For instance, solar cells created by embedding nanoparticles in plastic film can be incorporated into mobile phones and portable computers. Batteries created using nanotechnology can be made lighter and more powerful and can also be charged more quickly than conventional batteries, increasing the efficiency of hybrid automobiles. Nanofilters are increasingly being applied in food production, water filtration, and air pollution control, and nanoparticles are also used in some applications to absorb contaminants.

In manufacturing and construction, nanotechnology has led to the development of new materials that are lighter, stronger, and possess more desirable properties than their conventional analogues. For instance, nanomolecular structures are already being used to make concrete and asphalt more resistant to water, and nanomaterials added to light-emitting diode (LED) lighting makes them more resemble standard lighting, allowing the incorporation of more efficient LED lights in home and industrial use while retaining the look of traditional lighting. Nanocoatings are commercially available that resist corrosion, offer insulation and UV protection, and can remove pollutants from a building’s atmosphere.

Bibliography

Foster, Lynn E. Nanotechnology: Science, Innovation and Opportunity. Upper Saddle River, NJ: Prentice Hall, 2006.

Garcia-Martinez, Javier, ed. Nanotechnology for the Energy Challenge. Weinheim, Germany: Wiley-VCH, 2010.

Matthews, Miccal T., and James M. Hill. “Micro/Nano Thermal Boundary Layer Equations with Slip-Creep-Jump Boundary Conditions.” IMA Journal of Applied Mathematics 72 (2007).

“Nanotechnology.” Scientific American. http://www.scientificamerican.com/topic.cfm?id=nanotechnology.

“Nanotechnology News.” Science Daily. http://www.sciencedaily.com/news/matter‗energy/nanotechnology.