In the Classroom
Introducing the Core Concepts of Nanoscience and Nanotechnology: Two Vignettes
W
Karl Sohlberg Department of Chemistry, Drexel University, Philadelphia, PA 19104;
[email protected] With the advent of a new field of science comes the need to instruct the next generation of scientists in its fundamentals. Quantum mechanics, for example, was born of a desire to understand the results of several experiments that defied explanation by classical physics. Planck’s successful description of thermal emission spectra by invoking the quantization of energy was one of the very earliest steps in the development of quantum mechanics. Since the flowering of quantum mechanics, the Planck radiation law has served in many introductory quantum mechanics texts as an example both to motivate the study of quantum mechanics and to illustrate its core concepts. Today, the new field of nanoscale science engineering and technology (NSET) is emerging—a field concerned with the study and use of materials whose size scale falls into the nanometer regime for at least one spatial dimension. NSET is a natural extension of the drive for ever-greater miniaturization and is currently experiencing tremendous growth because recent advances in technology have made possible the measurement and manipulation of nanometer-scale objects. The corresponding pedagogical challenge is to motivate the study of NSET and illustrate its core concepts. In the process of doing so, we will revisit the Planck radiation law. Motivating the study of nanoscience is not difficult. “Nano” is arguably the most inspiring thing to happen in science and technology since the space race. It is hard not to get excited about what one might be able to accomplish by bringing the richness and atomic-precision of synthetic chemistry to manufacturing at the nanometer scale (1)! Applications already exist in stain- and wrinkle-resistant (NanoCare) fabric and wear-resistant coatings. Future applications in single-molecule detectors and drug delivery appear plausible. The next step is to introduce the “core concepts” of NSET (2)1: • As the size of a sample of material decreases, the fraction of atoms that are exposed on a surface increases. It is at the “nano” scale where the number of surface atoms becomes an appreciable percentage of the total number of atoms, and therefore surface properties become competitive with bulk properties. • As the size of a sample of material decreases below some threshold, the spatial confinement of the particle size starts to influence the spectral signature of the material. This threshold is in the “nano” regime.
One challenge in describing nanoscale systems is that they typically have properties that are determined by a large (but not effectively infinite) number of atoms, molecules, or subsystems. For this reason they are usually not amenable to theoretical treatment with simple analytical forms, such as those that arise in the treatment of few-body problems (i.e., the vibrations of a two-atom molecule may often be reason-
1516
Journal of Chemical Education
•
ably approximated by the dynamics of the simple harmonic oscillator, for which analytical expressions are well-known) or continuum systems (i.e., the vibrational free energy of a bulk material may often be reliably approximated by the Debye model, which is a simple analytic form that treats the material as a elastic continuum; ref 3 ). Computational modeling and numerical simulation therefore often play a critical role in the theoretical description of nanoscale systems and consequently it is useful to employ computational modeling and numerical simulation to introduce the core concepts of NSET. Modern computational tools such as spreadsheet software are a resource for numerical computation that is accessible to students even at the introductory level. Here we present two numerical implementations of theoretical calculations that are relatively simple to carry out in a standard spreadsheet and directly demonstrate the core concepts of NSET: I. The fraction of atoms on the surface of a simple solid is computed as a function of some characteristic length parameter. With decreasing particle size, the number of atoms on the surface becomes an appreciable fraction of the total number of atoms when the characteristic length scale is in the nanometer regime. II. The thermal emission spectrum of a solid is estimated for particles of various sizes. The thermal emission spectrum is “molecular” in nature for very small particles, but follows the Planck distribution law for macroscopic particles. The transition between these two regimes occurs when the characteristic length scale is in the nanometer regime.
The examples described here are each designed to highlight a single key NSET concept and to demonstrate the power of numerical computation in the theoretical description of nanoscale systems. Example I For the first concept, (suitable for discussion even at the first-year level) we use simple geometrical arguments to discover how the ratio (number of surface atoms)兾(total number of atoms) varies as a function of particle size. Estimating the fraction of atoms on the surface of a simple geometric solid as a function of some characteristic length parameter is not overly burdensome, once. Briefly, one divides the volume of the solid by the effective volume of an atom to determine the number of atoms in the structure. Next, one divides the surface area of the solid by the effective cross sectional area of an atom to determine the number of surface atoms and divides the latter by the former. Carrying out this procedure for a range of values of the characteristic length parameter to generate a plot becomes rapidly
Vol. 83 No. 10 October 2006
•
www.JCE.DivCHED.org
In the Classroom
tedious and would almost certainly lead to student mutiny, but when the derived expressions are entered into a spreadsheet program, production of such a curve becomes almost trivial. For the present case, our model will be a cube of closedpacked Cu atoms, the synthesis of which may be found in Jana et al. (4). We denote the cube edge length L. The metallic radius of Cu is r = 0.128 nm (5). The effective cross sectional area A´ subtended by an atom at the surface is
πr 2 pcircle
A ′ =
(1)
Here pcircle ≈ 0.9069, the packing efficiency of close-packed circles. Each atom effectively covers more area than its true cross section (A = πr 2) because it is impossible to pack circles with 100% efficiency (Figure 1). The number of surface atoms Ns is given by the total surface area (There are six faces each of area L2.) divided by the effective cross sectional area A´ of an atom: Ns =
6 L2 A ′
(2)
Figure 1. Packing efficiency of close-packed circles. The rectangle contains the equivalent of 4 circles. For simplicity we assume a radius of r = 1/2. The area covered by the four circles is π. Inspection of the triangle formed by the dashed line and use of the Pythagorean theorem shows that the width and height of the rectangle are 2 and √3, respectively. The area of the rectangle is 2 × √3 ≈ 3.464. The circle packing efficiency is therefore ≈ 0.9069.
The effective volume of an atom V´ is
V ′ =
4 πr 3 3 psphere
(3)
Here psphere ≈ 0.7405, the packing efficiency of close-packed spheres. The value of psphere may be arrived at by geometric arguments similar to those employed above to determine pcircle, as is presented in many general chemistry texts (6). (For students at an elementary level, one might simply supply the packing efficiencies.) The number of atoms N is given by the total volume divided by the effective volume V´ of an atom: N =
L3 V ′
(4)
The number of atoms within the bulk Nb is (5)
N b = N − Ns
The variation in the ratio Ns兾N of surface to bulk atoms as a function of particle size is 8 r pcircle Ns = L psphere N
(6)
and is shown in Figure 2. One special feature of the nanometer-size scale is highlighted in Figure 2: This is the size scale where the surface atoms represent a sufficiently large fraction of all atoms present that they can heavily influence the material properties. For example, surface atoms necessarily have different valence saturation (i.e., dangling bonds, extra free electron pairs) than their counterparts within the bulk, and therefore different chemical reactivity. Nanoparticles therefore exhibit different chemical reactivity from macroscopic samples of the same material. For example, while bulk gold is relatively unreactive, (as exhibited by its resistance to oxidation and corro-
www.JCE.DivCHED.org
•
Figure 2. Fraction of atoms on the surface as a function of the edge length for a cube of close-packed Cu atoms. Note that atoms exposed on the surface become an appreciable fraction of the total when the edge length drops into the 1–10 nm range.
sion) gold nanoparticles are highly reactive and consequently find use in catalysis (7). Once a spreadsheet is developed to generate a graph such as that shown in Figure 2, it is possible to “reverse engineer” the problem. The “goal seek” feature of a spreadsheet program can be used to find the characteristic length parameter that corresponds to a specific surface兾bulk ratio. (Effectively, this is solving a nonlinear equation!) Some students may even be inspired to pursue the concept further, say to investigate the role of special edge or corner atoms. A useful exercise is to repeat the calculation for particles of a different shape. For example, while Cu nanoparticles may be produced in cubic form (4), small clusters of atoms often form icosahedral particles and show greatly enhanced stability when the particle consists of an integer number of complete shells of atoms. A single close-packed shell of atoms surrounding one central
Vol. 83 No. 10 October 2006
•
Journal of Chemical Education
1517
In the Classroom
core atom yields an icosahedral particle with 13 atoms. Two complete shells yield a particle of 55 atoms. The sequence of these icosahedral “magic numbers” is 1(100%), 13 (92%), 55 (76%), 147 (63%), 309 (52%), 561 (45%), 923 (39%), … where the fraction of atoms on the surface of the particle is given in parentheses. Again we find that small particles have a very large fraction of their constituent atoms on the surface.
Hints for Spreadsheet Implementation First, a group of cells is reserved for various constants such as π, pcircle, psphere, A´, and V´ that are used in the following computations. A range of edge lengths is then tabulated in a spreadsheet column. A second column tabulates the number of surface atoms Ns based on eq 2. A third column tabulates the total number of atoms N based on eq 4. A final column tabulates the ratio Ns兾N. A scatter plot is generated, plotting the fourth column versus the first.
(8)
nc = 8
The number of edge atoms ne is given by counting the number of atoms along each edge, making sure to subtract 2 from each edge to account for the fact that the corner atoms have already been counted:
(
)
n e = 4 (nnx − 2) + 4 nny − 2 + 4 (nnz − 2) (9) In our case nnx = nny = nnz = nn since we are considering a cubic particle. Note that there are four edges parallel to the x-axis, four edges parallel to the y-axis, and four edges parallel to the z-axis. The number of face atoms nf is given by counting the number of atoms on each face, making sure to subtract the corner and edge atoms, that have already been counted:
(
nf = 2 nnx nny − 2nnx − 2 nny − 2
Example II For the second core concept we demonstrate the influence of particle size on the thermal emission spectrum. This example requires minimal familiarity with the eigenstates of a simple harmonic oscillator, or at least acceptance of the Planck quantization hypothesis. It could be used in the physical chemistry curriculum after introducing the Planck radiation law and the “ultraviolet catastrophe”, or after studying the quantum mechanics of the simple harmonic oscillator. A crude model is constructed as follows: We assume a cubic lattice of metal atoms with a bond length of d = 0.25 nm, a typical value for metals (8). For simplicity we consider a cubic particle. (The qualitative result reported here is essentially invariant with particle shape as long as the aspect ratio is ∼1.) Our task is to count the number of oscillators of each frequency present in a particle of a given size. The thermal emission spectrum is then found by multiplying the density of oscillators per unit wavelength by the average energy per oscillator, the latter also being a wavelength dependent quantity (9). The simplest oscillator is the local mode vibration of a single atom. Each atom possesses three such modes, nominally one for each spatial direction. One may think of the atoms as vibrating or of bond vibrations. In the language of theory, one may employ a basis of atoms or a basis of bonds. In the local mode picture, the two are equivalent and are related by a d兾2 translation of the lattice. We choose to think of bond vibrations, since this choice makes it simpler to keep track of edge effects as described below. Since each bond connects two atoms, we may count the number of atom–bond contacts and divide by two to find the number of bonds. In a cube of atoms arranged on a cubic lattice; each corner atom has 3 connecting bonds, each edge atom has 4 connecting bonds, each face atom has 5 connecting bonds, and each interior atom has 6 connecting bonds. Next we enumerate the number of atoms of each type. It is useful to first define the number of bond nodes nn along an edge: n n = int
1518
For a cube, the number of corner atoms nc is always eight:
L d
+1
Journal of Chemical Education
(7)
•
)
+ 2 nnx nnz − 2nnx − 2 (nnz − 2)
(10)
+ 2 nny nnz − 2nny − 2 (nnz − 2) Note that there are two faces parallel to the xy-plane, two faces parallel to the xz-plane, and two faces parallel to the yz-plane. The number of interior atoms ni is simply the total number of atoms, minus those counted as corner, edge, or face atoms, ni = nnx nny nnz − nc − ne − nf
(11)
Using this scheme, we can easily count the number of atoms of each type for a cube of edge length L. The total number of bonds nb is then nb =
(3nc
+ 4ne + 5nf + 6ni ) 2
(12)
This is the number of oscillators corresponding to the local mode vibration of an atom. The next simplest oscillation consists of two adjacent atoms vibrating in-phase. To count the number of such oscillators that are possible in the particle we find it simpler to count pairs of adjacent collinear bonds. Equations 7–12 can be reused, this time replacing d with 2d. (This time when we count “bonds,” we are actually defining a “bond” to be a pair of adjacent collinear bonds.) For oscillations of three adjacent atoms, the procedure is repeated with d in eqs 7–12 replaced by 3d. We continue this procedure until jd > L, in which case in-phase vibration of j adjacent collinear atoms is impossible because the particle is too small. Next we must assign frequencies to each type of oscillator. Each atom is assumed to have a local mode vibrational frequency of ν1 = 2000 cm᎑1, which is a typical value for an atomic local mode vibrational frequency in a bulk solid (10). By analogy to oscillations of a stretched string, if an adjacent pair of atoms vibrates together, the frequency will be ν2 = ν1兾2. (For a stretched string, frequency is inversely proportional to length between nodes.) If a trio of adjacent atoms vibrates together, the frequency will be ν3 = ν1兾3. A quartet
Vol. 83 No. 10 October 2006
•
www.JCE.DivCHED.org
In the Classroom
of adjacent atoms vibrating together has a frequency of ν4 = ν1兾4 and so forth, so that νj = ν1兾j. The lowest possible frequency is set by the size of the particle. As the size of the crystal increases, the lowest possible vibrational frequency decreases. The wavelength of radiation emitted from each oscillator is given by λj = c兾νj, where c is the speed of light. The four simplest local mode vibrations are shown schematically in two dimensions in Figure 3. Since n(λ) follows a power law, the number of oscillators per unit wavelength dn(λ)兾d λis proportional to n(λ)兾λ. The latter is used to approximate the density of oscillators per unit λ. Since each oscillator has an infinite series of eigenstates, the average energy of an oscillator 具U 典 is given by a sum of terms, each of which is a product of an oscillator eigenenergy times its Boltzmann population. The result for a harmonic oscillator may be found in many physical chemistry texts. (An older text by Castellan contains many details of the derivation; ref 9.)
U
hc λ = hc exp λ kBT
−1
Figure 3. Two-dimensional representation of local mode vibrations of a single atom (ν1) as well as several collective vibrations; pairs of atoms vibrating in-phase (ν2), trios of atoms vibrating in-phase (ν3), and quartets of atoms vibrating in-phase (ν4). Note that for lower frequencies, fewer oscillating centers “fit” within a lattice of fixed size. As shown here, along one dimension, n(λ1 = c/ν1) = 12, n(λ2 = c/ν2) = 6, n(λ3 = c/ν3) = 4, and n(λ4 = c/ν4) = 3.
(13)
Here h and kB are the Planck and Boltzmann constants, respectively, and T is the Kelvin temperature. Multiplying the density of oscillators per unit wavelength by the average energy per oscillator yields µ(λ), the thermal emission spectrum of the particle. Finally, we plot µ(λ) for various different edge lengths L. As shown in the series of four figures (Figures 4), the spectrum transitions from molecular in nature to that of a bulk material when the particle edge length is in the nanometer regime. For comparison, the solid curve on the plot for the largest nanoparticle shows the Planck distribution law of a perfect blackbody radiator, scaled so that the peak intensity matches the peak intensity of µ(λ)for the nanoparticle. Once a spreadsheet is developed to generate one of these figures, it is a trivial matter to vary T, ν1, or d to investigate the influence of these factors on the thermal emission spectrum. Numerous such opportunities exist for exploration and guided discovery.
Hints for Spreadsheet Implementation The second example may be implemented in two sheets. In the first sheet we determine the number of oscillators of each possible λ for various edge lengths L. It is useful to first reserve a group of cells for various constants such as ν0, d, k, T, and h that are used in the following computations. A range of edge lengths L is then tabulated in a spreadsheet column. Next, the six quantities nn, nc, ne, nf, ni, and nb are tabulated in six successive columns, and the group of columns tabulating the six quantities is repeated for each type of oscillator jd. Using a second sheet, we compute the thermal emission spectrum. The first column of sheet two lists all possible values of λ. A second column tabulates n(λ), the number of oscillators for each possible wavelength, as enumerated in the first sheet by the scheme discussed above. A third column www.JCE.DivCHED.org
•
Figure 4. Theoretical thermal emission spectra for four different sizes of cubic particles of a hypothetical metal with a cubic lattice and bond length of 0.25 nm. Note that for the smallest particle, only a small cluster of spectral lines is present, with the longer wavelengths generally having higher intensity, but for the largest particle the spectral distribution fits within an envelope described by the Planck radiation law, as indicated by the solid line in the figure for the 162-nm particle. The vertical axes represent intensity in unspecified units.
Vol. 83 No. 10 October 2006
•
Journal of Chemical Education
1519
In the Classroom
calculates n(λ)兾λ. A fourth column evaluates eq 13. Multiplying the density of oscillators per unit wavelength, (the third column) by the average energy per oscillator, (the fourth column) gives µ(λ), the thermal emission spectrum of the particle. This multiplication is carried out in a fifth column. Plotting the fifth column versus the first shows the predicted thermal emission spectrum. Concluding Comments The two examples presented here demonstrate the core concepts of nanoscience and nanotechnology. In example I, the fraction of atoms on the surface of a cubic nanoparticle is calculated as a function of the cube edge length to demonstrate the importance of surface atoms in nanometer-scale particles. In example II, the thermal emission spectrum of a cubic nanoparticle is computed for several different cube edge lengths to show the influence of particle size on spectral signature for nanometer-sized particles. In addition to introducing the special nature of the “nano” size scale, the examples presented here address another pedagogical goal that is beyond the core concepts of NSET, demonstrating the power of numerical computation. In example I, generating by hand a sufficiently large set of data to produce an informative plot of surface fraction versus particle size would be prohibitively tedious. In example II, manual enumeration of all the bonds is impractical for all but the tiniest cubes. Students are thereby introduced, albeit at a very primitive level, the power of numerical computation in the theoretical description of nanoscale systems. When students see that modern software tools can easily handle the burdensome computational aspect of problems of the type discussed here, they may be motivated to become adept in their use, a skill that they can employ not only in the description of nanosystems but across the spectrum of science and engineering problems. Acknowledgments This project was funded in part by grants NUE0304024, NER-0102848, and CAREER-0449595 from the
1520
Journal of Chemical Education
•
National Science Foundation. The author thanks professors S. Solomon and R. Schweitzer-Stenner for valuable comments on the manuscript. W
Supplemental Material
Sample spreadsheets for the two examples are available in this issue of JCE Online. Note 1. These core concepts are widely discussed, but rarely formally stated. The statements used here follow closely from a seminar given by Mostafa A. El-Sayed at Drexel University, Department of Chemistry, 22 May 2002.
Literature Cited 1. Small Wonders, Endless Frontiers: Review of the National Nanotechnology Initiative. http://nano.gov/ smallwonders_pdffiles.htm (accessed 2005). 2. Yarris, L. Clusters: A New State of Matter. http://www.lbl.gov/ LBL-Science-Articles/Archive/clusters.html (accessed Jun 2006). 3. Garbulsky, G. D.; Cedar, G. Phys. Rev. B 1996, 53, 8993– 9001. 4. Jana, N. R.; Wang, Z. L.; Sau, T. K.; Pal, T. Current Science 2000, 79, 1367–1370. 5. Porterfield, W. W. Inorganic Chemistry: A Unified Approach; Addison–Wesley: Reading, MA, 1984. 6. Petrucci, R. H. General Chemistry: Principles and Modern Applications; Macmillan: New York, 1989. 7. Valden, M.; Lai, X.; Goodman, D. W. Science 1998, 281, 1647–1650. 8. Greenwood, N. N.; Earnshaw, A. Chemistry of the Elements; Pergammon Press: Oxford, 1985. 9. Castellan, G. W. Physical Chemistry; Addison–Wesley: Reading, MA, 1983. 10. Cai, S.–H.; Rashkeev, S. N.; Pantelides, S. T.; Sohlberg, K. Phys. Rev. Lett. 2002, 89, 235–501.
Vol. 83 No. 10 October 2006
•
www.JCE.DivCHED.org
