Dr. Alan Jackson,
Central Michigan University Professor of Physics.
There is tremendous interest in the properties of matter at the nanometer scale. Profit-minded people in the electronics industry have been pushing for years to make integrated circuits smaller and smaller. Packing more circuits onto a single semiconductor wafer reduces materials costs and makes for faster and more powerful circuitry for computers, cell-phones, video games, etc. The dimensions of integrated circuits have been shrinking rapidly over the past three decades at a rate that would attain the ultimate limit of atomic-scale circuits by roughly 2010. Whether it is possible to design working circuits with single-molecule-sized components is a serious question researchers are actively investigating today. Other areas researchers are interested in the mechanical properties of nanoscale matter. They envision new generations of materials that make use of specific properties of nanoscale components. The most futuristic thinkers envision using sub-micron-scale machines in nano-factories to rearrange atoms from raw materials like sand and water to create useful materials like semiconductor wafers. The nanotechnology area is strongly interdisciplinary, involving scientists from a wide variety of disciplines. It may well be an area offering lots of opportunities to new students of science.
INTRINSIC vs EXTRINSIC PROPERTIES
This section distinguishes between material properties that depend on the amount of material present (extrinsic) vs those that are independent of sample size (intrinsic). This concept is important to have in mind when comparing the properties of samples of vastly different size.
The term "modulus" is foreign to most ears and may be confusing or intimidating to students. A loose definition is "a measure". Youngs modulus is a measure of the intrinsic elasticity, or stretchy-ness, of a material. It may be helpful for students to picture the chemical bonds that bind the atoms of a material together as tiny springs. The springs can be stretched or compressed by applying an appropriate external force. Youngs modulus is simply a measure of how hard it is to stretch the springs.
A lot of interesting practical problems can be posed involving Youngs modulus. The examples give several illustrations.
The nanorods used in Liebers experiment are made from silicon carbide (SiC). A typical rod used in the experiment had a diameter of 20 nm. Macroscopic SiC is a very hard and heat resistant ceramic material, used, for example, on cutting blades. The carbon nanotubes used in Liebers experiment had similar diameters. Nanotubes can be thought of as graphite sheets rolled up into cylinders. See Figure 1 below. Nanotubes are the subject of a tremendous amount of current research, because they are relatively easy to make and because they display an array of potentially useful properties. One can imagine using nanotubes as molecular wires, as structural beams--there are even suggestions that a hollow nanotube would suck atoms into its interior like a nanovacuum cleaner!
Figure 1. A carbon nanotube viewed from the side. See: http://www.physics.purdue.edu/nanophys/
ATOMIC FORCE MICROSCOPE (AFM)
The AFM uses the same type of piezoelectric positioners as the STM and other scanning probe microscopes; however, with an AFM, the force between a tiny tip and the surface is used to determine the tip-surface separation, rather than using a tunneling current. The tip is attached to a thin cantilever that is bent by the contact force. The degree of bending is detected using a reflected laser beam, as indicated in Figure 2. Generally speaking, the horizontal resolution of the AFM is not quite as good as that of the STM, but the AFM has the advantage that it can be used to probe all sorts of surfaces, while the STM is limited to conducting or semiconducting surfaces. (The tunneling current must be able to flow through the surface, otherwise charge accumulates below the STM tip and alters the current-distance relationship used for imaging.)
(The image in Fig. 1 is taken from a paper by G. Y. Liu, S. Xu, and S. Cruchon-Dupeyrat, in Thin Films, Volume 24, p. 81 (1998), with permission of the authors.)
Hookes law (F = - k·x) forces are very important in many areas of physics. Such forces are responsible for oscillatory behavior. A simple harmonic oscillator results from a pure Hookes law force. The hanging spring is an example. It can be shown that the period of oscillation is given by:
T = 2·m/k
Where m is the hanging mass (in kg) and k is the spring constant (in N/m). The spring constant of a coil spring is not an intrinsic property of the spring; it depends, for example, on the length of the spring. An easy way to investigate this is to string two identical springs together and measure the spring constant of the combination.
MEASURING YOUNGS MODULUS
The Lieber experiment is a nifty way to measure the Youngs modulus (E) of a rod. The derivation of the formula given relating E to the bending of a rod comes from a textbook on solid mechanics: Mechanics of Materials by F. P. Beer and E. R. Johnston, Jr. (McGraw-Hill, New York, 1992), p. 486. Students should be able to obtain E accurate to about 10-20% using this technique. One tricky point is being able to accurately determine the amount the rod bends for a given mass. The measurement should be made at the point at which the mass is hung, or very close to it. One method is to mount a meter stick vertically behind the rod and to read the deflection of the rod directly. Error can be introduced with this method if the students are not careful to maintain a level view of the rod and the meter stick. One way to accomplish this is to mount two meter sticks, one behind the other. A level view is attained when the marks on the two sticks are aligned.
It is interesting to show that Youngs modulus is an intrinsic property by comparing E for vastly different samples. Students can measure E for a steel paper clip, for example, and compare the result to that of a rod. A piece of copper wire could also be used.
The following websites may be the starting point for further study.
This is a site that contains many links to a variety of web pages on nanotechnology, from the very general to the very technical. It is nicely arranged.
This is the Purdue University Nanoscience groups homepage. It contains some interesting science as well as many links to related sites.
THE (SMALL) WORLD OF NANOSTRUCTURED MATERIALS
1) Investigating the world at the atomic scale
You have seen that scanning probe microscopes like the STM and the AFM can produce images of individual atoms on surfaces. Such images have revolutionized the study of solid surfaces. An even more spectacular feat is the ability of researchers to manipulate individual atoms using these devices. The image shown in Fig. 1 was made at Purdue University by Mario J. Pannicia. To make this image, individual atoms were dragged across a surface by an STM tip. While applications of this technology have not progressed much beyond the nanoscale graffiti stage, the possibilities are vast.
Figure 1. Intel logo created by arranging individual atoms on a surface using the STM. See: http://www.physics.purdue.edu/nanophys
The ability of scanning probe microscopes (like the STM and AFM) to image and manipulate matter at the atomic level has helped to launch a new area of research into the properties of nanoscale materials, materials that are no larger than roughly 100 atoms across in at least one spatial dimension. At these small sizes, materials can have properties that are different than those of the corresponding macroscopic materials. For example, nanoscale materials can be mechanically stronger than macroscopic samples of the same material. The ability of a material to stretch or bend without breaking depends on the chemical bonds that hold the atoms together. In principle, the bonding is the same in nanoscale and macroscopic materials. But typical macroscopic samples include many bonding defects--atoms out of place, missing atoms, etc.--that limit the mechanical properties. The density of these defects can be much smaller in nanoscale materials, making them stronger. Materials scientists are working on ways to exploit the enhanced properties of nanoscale matter by combining nanoscale pieces together into so-called nanostructured materials, matter that is composed of nanoscale building blocks.
2) Extrinsic vs. intrisic properties
If you wanted to compare the properties of nanoscale materials with corresponding macroscopic materials, you would need to be careful to choose properties that are intrinsic to the material, rather than extrinsic properties that depend on the amount of material present. For example, it would make little sense to compare the mass of a nanometer-sized piece of iron to that of a macroscopic piece, an iron nail, say. Mass is an extrinsic property that depends on the total number of atoms in the sample. On the other hand, the density of a sample, the ratio of the mass of the sample to its volume, is an intrinsic property that is characteristic of the material and not peculiar to a particular sample. Some other extrinsic properties are volume, heat capacity and electrical resistance. Some intrinsic properties include density, specific heat capacity and electrical resistivity.
3) Youngs modulus
The elastic properties of a material depend on how its atoms are arranged and on the nature of the chemical bonds between them. We know that some compounds like rubber are highly elastic and stretch easily when you pull on them. These materials are made of long, chain-like molecules called polymers. The individual polymer molecules tend to be flexible and the molecules are rather weakly bound together in an irregular way, making the macroscopic material very flexible.
All materials are elastic to some degree. Even materials like steel will stretch if you pull on them hard enough or shrink (and finally buckle!) if you compress them hard enough. This is important information if you are an architect designing a building. If you are using steel beams to support your building, for example, you need to know how much weight the beams can support without buckling. Less dramatically, but still very important, you need to plan for the compression of the beams under the weight of the building. You can imagine the headaches caused by one side of a building shrinking by an inch or two more than the other side.
Youngs modulus is a measure of the intrinsic stiffness of a material. Suppose you have a sample of a material in the shape of a cylindrical rod of length L and cross-sectional area A. Applying a load to the rod (for example, stacking bricks on it) causes the rod to compress slightly, i.e. it causes the length to change by an amount DL. For a load F (equal to the weight of the bricks), Youngs modulus, E, is defined in the following way:
F/A = E DL/L
Defined in this way, E is an intrinsic property of the material. It is a measure of how much pressure (F/A) must be applied to the ends of a sample to compress the material by some fractional amount (DL/L). The larger E is, the larger the load required to compress the material. Another important property is the yield strength, Sy. This is defined to be the load the material can withstand before it is permanently deformed--the first step toward buckling. Table I compares elastic properties for a number of materials.
Table I. Elastic properties of some common materials. See: "Fundamentals of Physics" 4th Ed., by Halliday, Resnick and Walker (John Wiley and Sons, New York, 1993) and "American Institute of Physics Handbook" (McGraw-Hill, New York, 1957). See text for definitions.
|Material||Density (kg/m3)||Young's Modulus E (109N/m2)||Yield Strength Sy(106N/m2)|
4) Nanobeam mechanics
One active area in nanostructure research is investigating the properties of nanoscale rods. These superfine whiskers are envisioned as potential building blocks for ultrasmall devices of various kinds. Recent experiments in the laboratory of Professor Charles Lieber at Harvard University (see E. W. Wong, P. E. Sheehan and C. M. Lieber, "Nanobeam Mechanics: Elasticity, Strength and Toughness of Nanorods and Nanotubes," Science, vol. 277, p. 1971 (1997)) examined the properties of silicon carbide (SiC) nanorods and multi-walled carbon nanotubes. The basic idea of the experiment was simple: pin one end of the rod and use the tip of an atomic force microscope (AFM) to push on the other end, causing the rod to bend. The ratio of the amount of bend to the amount of applied force is related to the stiffness of the rod, i.e. to the Youngs modulus of the rod.
Figure 2. A schematic diagram of Liebers experiment to determine Youngs modulus for nanorods. See "Nanobeam Mechanics: Elasticity, Strength and Toughness of Nanorods and Nanotubes,"Science, vol. 277, p. 1971 (1997)
The experiment is shown schematically in Fig. 2. A force F is applied to the nanorod by the AFM tip, causing a deflection y of the rod. E for the rod can be determined from the following formula, in which x is the distance between the point where the force is applied and the point where the rod is pinned, and r is the radius of the cylindrical rod.
???E = 4A x3/(3A pA r4) A F/y
Measurements were carefully made for a number of rods, using many values of x for each rod. Both E and Sy were found for the nanorods and nanotubes. The values for E for the SiC nanobeams were about 660 x 109 N/m2, and for the carbon nanotubes, about 1200 x 109/m2. These values are comparable what would be measured for the corresponding macroscopic samples; on the other hand, the values for Sy found for the nanobeams were significantly larger than for macroscopic samples. Thus, atom for atom, the nanoscale beams are stronger than their macroscopic counterparts. The scientists attribute the strength of the nanobeams to the relative absence of defects in the chemical bonding in these small samples.
1) Spring constants
A coil spring has the familiar property that it stretches when you pull on it, the amount of the stretch being proportional to the stretching force. This relationship is summarized in Hookes Law:
????F = - kA x
Here F is the restoring force of the spring, and x is the amount the spring is stretched or compressed. Note that if the spring is stretched, x is positive and F is negative--the spring opposes the stretch. The k in the equation above is the spring constant of the spring. It is a measure of the springs stiffness. The goal of this experiment is to measure the spring constant of selection of springs and to determine whether the spring constant is an intrinsic property of the spring.
a) Static approach
Materials:?a coil spring, several objects of known mass that can be attached to the spring, a ruler.
Procedure: Mount the spring vertically so that the masses can be attached to the lower end of the spring to stretch it. Attach a mass to the lower end of the spring and measure the amount the spring stretches. Use the table below to record your results. Systematically increase the mass hanging from the spring, measuring the corresponding stretch of the spring. The table below contains a column for the weight of the hanging mass. Remember that the weight is related to the mass by the formula: W = M g, where g = 9.8 m/s2 is the acceleration due to gravity. W is the force that the hanging mass applies to the spring. Since the system is in equilibrium, this must equal the magnitude of the springs restoring force. (Be careful to convert units correctly.)
Mass, M (g)??W = M g (N)??stretch x (cm)
Make a plot of M vs x on a sheet of graph paper. Draw the best straight line you can through your data. The slope of this line is equal to the spring constant in units of N/cm. Standard units are N/m. Convert your result to N/m.
k = ____________ N/m
b) Dynamic approach
Materials: a coil spring, several objects of known mass that can be attached to the spring, a stopwatch.
Procedure: Mount the spring vertically and attach a known mass to the lower end of the spring. Make sure the mass is able to oscillate up and down freely. Pull the mass down slightly and release it, allowing the mass to oscillate. Measure the period, T, of the oscillation, i.e. the time it takes the mass to go up and down once. An accurate way to do this is to measure the time for five oscillations and divide by five. Record the mass, M, you used and the result of your timing below:
Mass = ___________ (g); Time for 5 oscillations = __________ (s)
Analysis: The spring constant can be determined from the mass and the period using the following formula:
k = MA (2A p/T)2
If M is expressed in kg and T in seconds, k is given in units of N/m. Find the value of k for your spring.
k = ________________ N/m
1) The two methods used above should give the same results for k. How close are your values? Which method do you think should be more accurate? Why? Test your dynamic measurement for different masses. Do you get the same result for each?
2) Is the spring constant you measured above an intrinsic or extrinsic property of the spring? Test your hypothesis using the simplest experiment you can think of.
3) Hookes Law says that the restoring force of a spring is proportional to the amount the spring is stretched. This is actually an approximation that breaks down for large stretches. How far must your spring be stretched before the restoring force is no longer a linear function of x? Does the spring become stiffer or softer when it becomes non-linear?
2. Youngs Modulus for a rod: macroscopic analogue of Liebers experiment
Materials: cylindrical rods of various thicknesses, made from different materials; known masses to hang from rods; a clamp; a ruler.
Procedure: Clamp one end of a rod so that the rod extends horizontally. Hang a known mass from the rod at a distance X from the clamp. Measure the displacement of the rod, Y, from its original horizontal position. Record X, Y and the mass M in the table below. Measure Y for several values of M for the same X.
??ROD #: ???Radius (cm):
??X (cm)??Y(cm)??M(g)??W = MA g (N)
?Use your data to determine Youngs Modulus, E, for the rod. First, make a graph of Y vs M for the same value of X. Make the best straight line through your data and determine the slope of the line. The slope is DW/DY. Express this quantity in Newtons/meter. Use the following expression to determine E for your rod. Convert all your length measurements to meters. The units for E will then be N/m2, or Pascal.
E = 4A X3/(3A pA r4) A DW/DY
1) How reliable is your estimate of E? If you repeat your measurements using a different X, how much does your value of E change?
2) Can you identify the material that makes up your rod from its value for E? How could you test this identification? What other intrinsic properties of the material do you know?
3) How could you test to see whether E is an intrinsic property of a material?
Problems: (These problems require use of the data in the table above.)
1) Discuss the factors you might use in deciding whether to use aluminum or steel supports for a construction project.
2) If the femur, the large bone in your thigh, has a cross-sectional area of 5 cm2 and a length of 30 cm, how much weight could this bone support before it yields (i.e. breaks!)?
3) The floor of a new building is supported by four steel beams, each 1 m long and having a cross-sectional area of 10 cm2. How much are the beams compressed if mass of the floor and all the material stacked on it is 1 x 104kg?
James Gormley || Dr. Alan Jackson