Using theory and computation to model nanoscale properties

Abstract

This article provides an overview of the use of theory and computation to describe the structural, thermodynamic, mechanical, and optical properties of nanoscale materials. Nanoscience provides important opportunities for theory and computation to lead in the discovery process because the experimental tools often provide an incomplete picture of the structure and/or function of nanomaterials, and theory can often fill in missing features crucial to understanding what is being measured. However, there are important challenges to using theory as well, as the systems of interest are usually too large, and the time scales too long, for a purely atomistic level theory to be useful. At the same time, continuum theories that are appropriate for describing larger-scale (micrometer) phenomena are often not accurate for describing the nanoscale. Despite these challenges, there has been important progress in a number of areas, and there are exciting opportunities that we can look forward to as the capabilities of computational facilities continue to expand. Some specific applications that are discussed in this paper include: self-assembly of supramolecular structures, the thermal properties of nanoscale molecular systems (DNA melting and nanoscale water meniscus formation), the mechanical properties of carbon nanotubes and diamond crystals, and the optical properties of silver and gold nanoparticles.

Nanoscience deals with the behavior of matter on length scales where a large number of atoms play a role, but where the system is still small enough that the material does not behave like bulk matter. For example, a 5-nm gold particle, which contains on the order of 10⁵ atoms, absorbs light strongly at 520 nm, whereas bulk gold is reflective at this wavelength and small clusters of gold atoms have absorption at shorter wavelengths. The special properties associated with nanoscale systems like this have provided both challenges and opportunities for the use of theory and computation to play a role in the discovery process. The challenges arise from the fact that most theories that describe the properties of matter by using first-principles approaches in which all atoms (and even all electrons in these atoms) are explicitly described are close to or (more often) beyond their capability to describe such systems, even using the largest computer available. At the same time, continuum theories, which play such a useful role for micrometer-scale systems, are sometimes incapable of describing nanoscale properties due to incomplete incorporation of the underlying physics in the size-dependent materials parameters. However, the opportunities for theory to play a role are significant, as nanoscience often provides the smallest systems that are amenable to study using methods that involve macroscopic manipulation of nanoscale structures. Thus, it is possible to measure the structure and thermal properties of a single supramolecular assembly, observe the thermal properties of a single nanodroplet, or study the mechanical properties of a single carbon nanotube or light scattering from a single metal nanoparticle. Nanoscale structures often have properties that are familiar in much larger systems, such as defects and thermal instability; however, nanoscale structures can sometimes be synthesized with molecular perfection, and when this is the case, the resulting properties of these structures are often remarkable, including carbon nanotubes that are 10 times stronger than steel, nanoscale particles that can enhance the spectroscopic properties of nearby molecules by many orders of magnitude, nanoscale wires in which electrons move ballistically, etc.

Experiments on nanoscale objects are often fraught with uncertainty due to the difficulty of fabricating and manipulating these objects at length scales below ≈10 nm. In many experiments, the device that holds or measures the nanoscale structure can also produce irreversible degradation. Thus, electron microscopy can produce defects in the structure being measured, optical measurements on the smallest metal nanoparticles are subject to serious heating effects that can melt or destroy the particle, and atomic force microscopy studies of soft materials often lead to structural reorganization. In addition, the highest resolution measurements of structure, such as electron microscopy measurements, require that the nanostructure be removed from its natural environment, such as from solution, and placed on a grid under circumstances where aggregation and restructuring can occur. In situ measurements, for example, of the structure of a self-assembled monolayer on a colloidal particle in solution or defects in diamond crystals under high stress, are either not possible or are of limited resolution. Thus, there is a huge role for theory in “filling in the gaps” in our understanding of phenomena at the nanoscale, and to predict new properties and phenomena.

This paper is designed to provide an overview of the use of theoretical methods to describe nanoscience problems. A number of examples are provided where theory has been used for nanoscale problems, and I hope to use these examples to illustrate some of the possibilities for getting theory to work, and also some of the existing challenges. This is a field where no one type of theory can be used in all cases, and where the marriage of theories associated with different length scales is still somewhat rocky. Thus, some of the problems will be addressed by pushing traditional atomistic theories, such as electronic structure theory, to systems that are much larger than they have been traditionally calibrated for, and for which serious approximations need to be introduced to calculate useful numbers. In other areas, one tries to push continuum theories down to smaller length scales than they were originally developed for, and again one needs to introduce approximations to make these produce useful results. Another approach involves multiscale theories, in which the atomistic and continuum scales are matched together. Alternatively, one can use coarse-grained models, in which one attempts to describe the effective properties of small groups of atoms that are contained within a larger nanoscale structure.

One reason why this diversity of theoretical approaches is needed is that nanoscale structures and nanoscale phenomena do not always vary with length in the same way. Thus, “quantum dot” behavior in which the colors of semiconductor nanoparticles vary with size due to electron confinement effects is associated with sizes on the few-nanometer-length scale, whereas the variation of the color of metal nanoparticles with size due to electrodynamic effects is associated with particles in the 10- to 200-nm regime. The formation of covalent bonds is often highly localized, so it is possible to use small clusters to determine adsorbate binding energies that are useful for much larger particles; however, electron delocalization effects in aromatic systems can sometimes lead to properties, such as bond length alternation in carbon nanotubes, which require structures with >10³ atoms to converge.

The studies to be presented in this paper are taken from three broad areas of research that I have worked on during the last 10 years. I first discuss some examples from the soft materials field where structure and thermal properties are of primary interest. These include studies of self-assembly of peptide amphiphiles (PAs) to give cylindrical micelles, where theories at many length scales can be explored. I also discuss the thermal melting properties of DNA, both at the atomistic and coarse-grained level, and I describe studies of a water meniscus that has nanoscale dimensions, and where coarse-grained and continuum theories can be applied.

A second area of research involves the mechanical properties (modulus and fracture) of carbon nanotubes and diamond crystallites. Here we will see that atomistic electronic structure theories can play a useful role with suitable calibration, and we will also examine empirical potential models and continuum theory.

The third area of research is concerned with the optical properties of metal nanostructures, with emphasis on anisotropic silver nanoparticles, and on small clusters of these nanoparticles. Here we will see that continuum theory in the form of Maxwell's equations plays a very useful role, but at the same time there are problems in the treatment of some important properties, such as electric fields that are very close to the particle surface. We will also describe the use of electronic structure approaches which provide an accurate description of the near-field properties, but only for particle structures that are much smaller than are typically amenable to experimental studies.

This paper will not go into the details of any of the theories that underlie each method. Instead, we hope that, by providing physically motivated examples of the application of these theories, the readers will see how these work and how theory can play a useful role despite the present limitations. The original literature can be consulted for further details.

Structural and Thermal Properties

Self-Assembly of Peptide Amphiphiles.

The fabrication of complex structures at the nanoscale is a process that biology does with exquisite precision, but that nonbiological synthesis is only slowly learning to do. In particular, it is a major challenge to predict what structures will result from the aggregation of a large number of identical molecules, particularly amphiphilic molecules that have both hydrophilic and hydrophobic interactions. Such interactions often lead to structures in which the hydrophilic parts interact with solvent, whereas the hydrophobic parts interact with each other. Even with these very simple rules for how the molecules interact, there are many possible assemblies of these molecules that can result when the molecules aggregate. Often only one structure is found, but the factors that determine which one will dominate is an important subject of study.

An example of this is the work of Stupp and coworkers (1–4), in which amphiphilic molecules composed of peptides and alkane chains self-assemble into cylindrical micelles as schematically illustrated in Fig. 1a under appropriate conditions of pH and salt concentration, and an appropriate choice of amino acid residues. Fig. 1b shows the PA that makes up this structure, and Fig. 1c shows a stylized structure, in which the amphiphile has a roughly cone-shaped appearance (5). The hydrophobic tail forms the core of the cylindrical micelle, but what is less clear was what is the driving force for creating a cylindrical structure. It is not hard to show that simple packing of cones favors spherical micelles provided that the only interactions between the cones involve short ranged van der Waals interactions (6). However, the actual peptide amphiphiles that lead to cylindrical micelle formation have charged head groups due to the presence of acidic and basic peptide residues. Also, the central part of the peptide amphiphile involves residues that are efficient beta-sheet formers.

Fig. 1.

Models used to describe the aggregation of peptide amphiphiles (PA) to form cylindrical micelle structures. (a) Atomistic cartoon of a cylindrical micelle that is fabricated by using peptide amphiphiles. (b) Simplified atom-level picture of monomer PA. (c) Coarse-grained PA structure. (d) Atomistic minimum energy structure of PA dimer which shows the dipoles associated with the head groups. (e) Coarse-grained representation of the 16 PA system, showing voids in the tail region. (f) Bead-model representation structure of minimum energy cylindrical micelle formed from many PAs. a is adapated from ref. 4, b–e are adapted from ref. 5, and f is adapted from ref. 8.

Although atomistic theory (molecular dynamics in which all atoms are included, often with empirical force fields such as Amber or MM3) can be used to describe a single peptide amphiphile (5), it is not feasible to simulate even a short section of a complete cylindrical micelle to simulate micelle formation. However, some guidance is provided by using molecular dynamics to simulate small clusters of PAs (7). Fig. 1d shows the minimized structure of a dimer of PAs. If 16 monomers are organized into a 4 × 4 array, and the energy is minimized subject to a planar overall alignment of the PAs, this leads to a structure whose stylized presentation is presented in Fig. 1e. This structure has the ionic head groups with dipoles located head-to-tail as indicated by the arrays in Fig. 1e. Beta-sheet interactions reinforce the head-to-tail structure. The presence of voids in the hydrophobic tail region (see label in Fig. 1e) drives the formation of a cylindrical patch on the micelle when the planar constraint is released. Fig. 1f shows the cylindrical structure that results, but to generate this we have used a coarse-grained model (8) rather than an atomistic representation. The coarse-grained model describes each PA in terms of a linear string of five beads, with bead sizes chosen to produce an overall conical shape. In addition, the head bead in the PA is assumed to have a dipole embedded into it that allows for formation of the same head-to-tail pattern as in Fig. 1e. If the dipole is removed, the cylinder transforms into a sphere. Thus, we see that the cylindrical structure results from the choice of residues, with both the acid/base head group and the beta-sheet formers leading to the same result.

Thermal Melting of DNA Oligonucleotides.

DNA is an important example of a macromolecule which spans length scales from nanoscale dimensions (for short oligonucleotides) to macroscopic. In the last 10 years, there has been growing interest in the use of DNA to make nanoscale structures. One version of this has involved using DNA to link together nanoparticles, viruses, and polymeric materials (9–14). These materials have found use in sensing applications, in part because they exhibit much narrower melting transitions (thermal dehybridization) than occur for the same duplexes in solution (15). This behavior, which also contrasts with the normal melting curves that are observed when DNA links microparticles, has been the subject of several theoretical studies (15–22), and some general principles that can produce this behavior have been established. A limitation in modeling this process is that the simulation of DNA melting at the atomic scale is not feasible, even for a 10-bp duplex such as that illustrated in Fig. 2a, as the time scale (approximately microsecond) for melting is much longer than the few-nanosecond time scale that is possible for molecular dynamics simulations. As a result, the detailed mechanism of the melting processes, including the kinetics of dissociation and recombination, can only be described by using simple kinetics models. Unfortunately, these models require rate parameters that are unknown for DNA linked to gold particles. Simulating the melting of very short oligonucleotides, such as 2-bp duplexes, is feasible with atomistic simulation methods, so very recent experiments in which the melting of hairpin or dumbbell structures has been observed spectroscopically (23–25) provide promise that, at least in this limit, the transition can be understood in detail.

Fig. 2.

DNA structural models. (a) Atomistic model. (b) Coarse-grained model. (c) Use of the coarse-grained model to simulate melting. These figures are adapted from ref. 38.

An alternative method to describe DNA melting is to use coarse-grained dynamics. Over the years, there have been a large number of coarse-grained models developed for DNA (refs. 26–38 are representative), mostly designed to learn about the structural and mechanical properties of DNA. Only a few of these models have been developed in such a way that melting could occur (35, 37, 38). Fig. 2b shows a model that was developed by Drukker, Wu, and Schatz (37, 38) in which each base is represented by a sphere, and each sugar–phosphate moiety is another sphere. Force field parameters for this model were designed to give DNA a stable double helix structure and to provide reasonable melting behavior. Fig. 2c shows an example of a trajectory in which a 10-bp duplex dehybridizes over a 100-ns time scale. Indeed, with this model, it is possible to follow trajectories for a microsecond or more for oligonucleotides up to 30 bp. In addition, the dependence of melting temperature on DNA length and base pair composition is at least qualitatively correct. Although this model is still limited in its ability to describe subtle features of the base pair dependence of the melting transition, it provides a start to the development of models that can be used to describe a variety of complex processes in which DNA is dehybridized and rehybridized, and it can be used to describe the structural properties of DNA-linked nanoparticle structures.

Thermodynamics of Nanoscale Water Droplets.

Crucial tools for nanoscale characterization are the scanning probe tips that are used in atomic force microscopy (AFM), scanning tunneling microscopy, and many other experiments. When such tips are near to any surface under ambient humidity, a nanoscale water droplet spontaneously appears between the tip and the surface. Such droplets contribute to the force between the tip and the surface, and they are important to a new kind of lithography known as Dip Pen Nanolithography (DPN) (39, 40) in which molecules diffuse down the tip and are delivered to the surface. The precise function of the droplet in facilitating molecular transport in this case is not known, but the overall resolution of DPN is sometimes limited by its size and shape.

The structural and thermodynamic properties of droplets which form a meniscus between an AFM tip and a surface are not well known. One might imagine that the drop would look something like that in Fig. 3a, in which one uses continuum mechanics, such as is embodied in the Kelvin Equation, to describe the droplet in terms of contact angles and radii of curvature. Indeed, this approach has often been used to describe meniscus properties at larger length scales, and corrections that enable the description of nanoscale menisci have been proposed (41–43). However, the applicability of classical theory for a structure that could contain <100 molecules is uncertain, and as a result the variation of droplet width with relative humidity, temperature, and tip structure is not known.

Fig. 3.

Nanodroplet at the end of an AFM tip. (a) Modeled with classical continuum theory. (b) Modeled with a lattice model in which the AFM tip is a portion of an ellipsoid, and the water distribution is determined from grand canonical Monte Carlo calculations. (c) Minimum droplet radius as a function of AFM tip radius of curvature. b and c are adapted from ref. 46.

Recently, Jang, Ratner, and I have developed grand canonical Monte Carlo (GCMC) methods in combination with a lattice-gas model of water to establish the drop properties (44–48). In these calculations, the molecules are assumed to occupy sites on a cubic lattice, as this greatly simplifies the process of sampling possible configurations, and it also makes the relationship between the detailed atomic structure of the drop and its macroscopic properties especially simple to determine. In the GCMC calculations, water molecules are randomly placed at lattice sites, and then the configuration is either accepted or rejected depending on whether the energy is consistent with a Boltzman distribution. Fig. 3b shows a picture of a droplet that we have obtained based on these calculations. This picture exhibits the typical shape of a macroscopic meniscus; however, the structural and thermodynamic properties which it has are different. Continuum theory allows a meniscus to exist at any separation between the tip and the surface; however, in the MC calculations, we find that the meniscus is unstable when it becomes narrower than ≈10 molecules in width. This result is apparent in Fig. 3c, where we show the meniscus width, as a function of the diameter of the end of the tip with all length units given in terms of the diameter of water, 0.37 nm. Here we see that, when the tip is smaller than 5 nm, the meniscus width remains at 5 nm. Anything that is done to make the meniscus smaller, such as pulling the tip away from the surface, leads to total loss of the meniscus. The minimum width limits the resolution of patterns that are made by using tip-based lithography tools such as Dip Pen Nanolithography.

Mechanical Properties

Fracture of Carbon Nanotubes.

Nanoparticles, nanotubes, and nanowires are the smallest structures for which conventional mechanical properties such as Young's modulus and fracture stress can be determined, while at the same time they can be studied by using theory approaches that are at least in part atomistic in nature. Thus, we have a natural meeting point between top-down measurements and bottom-up theory, providing the ability to test fundamental limits on the strength and stiffness of materials. In recent years, there have been a large number of studies aimed at making this connection, particularly for carbon nanotubes, as reviewed in refs. 49 and 50. Carbon nanotubes provide a structure in which tube thickness can easily be a nanometer, whereas tube length can be microns such that it is possible to attach the tubes to AFM tips for mechanical property measurements. In addition, carbon nanotubes are extremely strong structures, nominally 10 times the strength of steel, and they are potentially of interest in making polymer composites for applications where stiffness and strength can be combined with other nanotube properties such as electrical conductivity. However, experimental measurements of carbon nanotube mechanical properties have yielded a wide range of results, including fracture stress values that range from <10 to >100 GPa (51–56). This dramatic variation of results points to the importance of defects. A number of theory studies have been performed to help sort this out (49, 55, 57–62), but developing a quantitative theory of the effect of defects on nanotube fracture has been challenging.

Fig. 4a shows the types of theory that we have used (61, 63) to study nanotube mechanical properties. Here we show a vacancy defect in what is otherwise a perfect nanotube. To describe the fracture, it is necessary to use a quantum mechanical (QM) method for the region where bonds are being broken, and in the present case this is the region around the defect. However, QM calculations of fracture properties are expensive, as the minimum energy structures of nanotubes under stress need to be evaluated many times to determine stress-strain behavior. At the same time, the smallest nanotube structures that are useful for theoretical studies often involve >100 atoms, as the mechanical properties of very small nanotubes show strong size dependence such that these calculations are not useful for predicting the properties of the larger tubes that are typically accessible to experimental study.

Fig. 4.

Multiscale modeling of carbon nanotube mechanical properties. (a) QM/MM/CM hybrid method. (b) Results from QM calculations for a two atom vacancy [5,5] SWCNT, including comparisons of several levels of theory and experimental results. Figures are adapted from ref. 60.

To circumvent the limitations of calculations that use QM methods for all atoms, we have implemented QM/MM methods in which QM calculations are done for a patch of the nanotube close to the defect, and then a molecular mechanics (MM) force field is used to represent the region around the patch (63). In addition, we have also studied MM/CM methods in which the molecular mechanics calculations (which are still atomistic) are interfaced with continuum mechanics (CM) to extend the nanotube stress field to large distances from the initial defect (61). The QM/MM and MM/CM interfaces both require careful definition, and indeed both are the subject of ongoing research. We have used a QM/MM interface in which QM atoms that are on the edge of the QM region are terminated with H atoms so that there are no dangling bonds. The total energy is then written as E(QM/MM) = E(QM) + E(MM: all atoms) − E(MM: QM patch), where the second and third terms on the right represent the difference between MM calculations for all atoms, and MM calculations for just the QM patch. Also, it is important to scale the MM force field so that the MM and QM force fields are matched as well as possible for strains close to the fracture point (63). The MM/CM interface is done somewhat differently by using an overlay swath between MM and CM regions. Here, both MM and CM calculations are used to represent all atoms in the swath, and the MM and CM regions are linked together by applying Lagrange constraints in the equations of motion of each region. Here, truncation errors are less important than in the QM calculations due to the importance of delocalized electrons in the latter.

Fig. 4b shows typical results comparing theory and experiment, in this case for a [5,5] single-wall carbon nanotube with a two-atom vacancy (60). The top curve in Fig. 4b shows the stress-strain behavior from a semiempirical QM method known as PM3, whereas the next highest curve shows results from density functional theory (a variant of DFT that includes the PBE gradient corrected density functional), which is the highest level of QM theory that can be used for this application. The DFT curve shows a linear stress-strain relationship at low strains, with a slope (Young's modulus) of ≈1,000 GPa (compared with 210 GPa for steel). Also, the fracture stress is ≈100 GPa (compared with 1.4 GPa for steel). The dashed curve shows what is obtained by using a reactive bond-order MM potential function known as MTB-G2 for all atoms. QM/MM calculations (combining DFT with MTB-G2) lead to results (63) (not shown) that are comparable to the DFT results, but with smaller computational effort.

The three theory curves (PM3, DFT, MTB-G2) have fracture stresses that differ by 15%, which provides an indication of the uncertainties in the theoretical models. However, these uncertainties are small compared with the differences between the theory results and experimental results that are also plotted in the figure. The theory results refer to CNTs with a vacancy defect, and we have demonstrated that other kinds of isolated defects lead to similar results (60). Furthermore, the fracture stresses are only a few percent lower than those obtained for undefected nanotubes. Very large defects, such as a slit that covers a substantial fraction of the circumference of the tube, are needed to reduce the fracture stress to measured values. Recent measurements for less-defected tubes have produced fracture stress values that are closer to the calculations in Fig. 4b (50, 53), but the uncertainties in these measurements are large, so it is not yet certain how significant is the comparison of theory and experiment.

Modulus and Fracture of Nanoscale Diamond Crystals.

An analogous comparison of theory and experiment arises in studies in the fracture of nanoscale diamond crystals. Interest in this material arises because of the development of ultrananocrystalline diamond (UNCD) films as potentially important coatings in the manufacture of microelectrical mechanical systems (MEMS) devices (64–68). UNCD consists of few-nanometer diamond crystallites that meet at atom-wide grain boundaries (69). UNCD films have fracture stress values in the few GPa range (67, 68), so a useful question for theory is to understand what determines these stress values. To study this, we have considered a model for a grain boundary structure that is presented in Fig. 5 (70, 71). This model consists of perfect diamond crystals (with periodic boundary conditions) that meet at a grain boundary structure known as the Σ13 twist in which one crystal is twisted relative to the other by 67.4° (69). The structure in Fig. 5 involves two grain boundaries with ≈200 carbon atoms total. We have performed DFT (PBE) studies of this system to determine modulus E, fracture stress σ, and strain ε_f (70). The results are presented in Table 1, and we also include values for a perfect diamond crystal for comparison. The results show modulus values that are about the same for perfect and grain-boundary diamond; in fact, these are very close to what we presented above for carbon nanotubes. The fracture stress of the perfect diamond crystal, estimated to be 233 GPa (70), is one of the largest values known for any material, whereas the grain boundary structure result is substantially smaller. However, even the grain boundary result, 100 GPa, is significantly higher than is found in the mechanical property measurements. This finding suggests that fracture of UNCD involves features other than the grain boundaries, and indeed we find that a large crack (several hundred nanometers in length) is required to reproduce the observed results. This result is analogous to what we found for carbon nanotubes, where it is only the largest defects that limit fracture behavior. Defects of the size needed to explain the measured values have been observed for UNCD (71).

Fig. 5.

Diamond fracture modeling using QM methods showing a double grain boundary structure that was used to model fracture in UNCD. This figure is adapted from ref. 71.

Table 1.

A brief summary of results for undefected single crystal diamond and for the grain boundary structure of Fig. 5, showing Young's modulus (E), fracture strain σ_f, and stress ε_f

Diamond structure	E(TPa)	εf	σf(GPa)
Single-crystal diamond	1.09	0.35	233
UNCD grain boundary	1.05	0.13	100

Results are adapted from refs. 70 and 71.

Optical Properties

Electrodynamics of Silver Nanoparticles.

Interest in the optical properties of silver and gold nanoparticles goes back to work by Faraday in 1856. At the same time, these properties have been the subject of intense interest in the last 10 years with the emergence of biological sensing applications based on these particles (10, 72). Key to this interest is the intense absorption and scattering that these particles exhibit as a result of plasmon excitation. This is a collective excitation of the conduction electrons, and the excitation wavelength depends on the shape and size of the nanoparticles. In many applications it is desirable to vary this shape and size to optimize sensitivity, so the ability to predict plasmon wavelengths is important to the design of experiments.

One of the triumphs of classical physics is that the intensity and wavelength of the plasmon excitation in nanoparticles is explained with high precision by classical electromagnetic theory, i.e., solving Maxwell's equations for light scattering from the appropriate particle structure, with the only materials parameters needed being the frequency dependent dielectric constants of the metal and surrounding material. Mie (73) presented a detailed solution for light scattering from a sphere that is very commonly used, even for nonspherical particles. Indeed, in most colloidal systems the dispersion of particle shapes is sufficiently ill-defined that Mie theory is the only practical theory for describing extinction (absorption and scattering) spectra. However, one of the important developments in nanoscience has been methods for making metal nanoparticles with tailorable shapes and sizes using a variety of lithography methods, and even wet chemistry, and this has provided significant motivation for implementing computational methods that can describe light scattering from particles of arbitrary shape. Fortunately, the current generation of numerical methods is capable of doing this for structures as large as several micrometers in size, so this application of theory is finding many important uses.

Fig. 6a shows examples of the results of applications that my group has done (74, 75) using a method known as the discrete dipole approximation (DDA) (76–78). The results include extinction spectra for a silver sphere, cylinder, cube, trianglar prism, and triangular pyramid (75). In all cases, the volume of the particle is constrained to be the same (equivalent to a 50-nm sphere) so that the change in spectrum that is being displayed is due to shape changes. What we see is that the sphere shows the bluest peak. The other particles show red-shifted plasmon resonance peaks, with the amount of the red-shifting being roughly determined by the sharpness of the features on the particles. We also note that, in addition to the most intense peak for each particle shape, which is typically associated with dipole excitation of the electrons in the particle, there are often additional peaks at shorter wavelengths. These are higher multipoles, and in some cases the multipoles have properties that make them useful for sensing applications. The results in Fig. 6a are representative of spectra that have been used to interpret a number of experiments, and the agreement between theory and experiment is of sufficient quality that deviations between theory and experiment are now used to identify when the particle structures are not being correctly identified.

Fig. 6.

Optical properties of silver nanoparticles from DDA calculations. (a) Extinction spectra of silver particles of different shapes (sphere, cylinder, cube, triangular prism, and tetrahedron), all having the same volume as that of an R = 50-nm sphere. Contours of the local field (|E|²) for sphere (b), cube (c), and tetrahedron (d). These results are adapted from ref. 75.

Another feature of nanoparticle plasmon excitation is the behavior of the electromagnetic field around the particles, the so-called near-field, as this field is important in sensing molecules which are either on the particle surfaces or nearby. Fig. 6 b–d shows examples of this field that have been obtained by using the DDA method (75). For a sphere, the behavior of the field is largely dictated by dipole excitation. However, for the cube and triangular prism, the sharp tips on the particle produce hot spots in which the near-field can be enhanced by many orders of magnitude. If molecules are adsorbed on the particle surfaces, then the hot spots are responsible for most of the dielectric shift that leads to red-shifted plasmon wavelengths. In addition, these hot spots are a key mechanism responsible for surface-enhanced Raman spectroscopy (SERS) (72). Classical electromagnetic theory makes important errors in determining the near-field behavior very close to the surfaces of nanoparticles as will be discussed below, so the ability of calculations such as those presented in Fig. 6 to quantitatively explain SERS and other measurements is still uncertain. However, the qualitative predictions have been verified in a recent study of methylene blue adsorbed onto pairs of gold disks, where the effects of disk spacing and thickness were examined, and good correspondence between theory and experiment was found (79).

Electronic Structure Modeling of Extinction and Surface Enhanced Raman Spectra.

Classical electrodynamics is a continuum approximation that replaces the response of the atoms in a solid to an applied electromagnetic field by the response of a continuous object that is characterized by a dielectric function. This approximation can break down whenever the underlying atomic structure is significant. This can happen at the interface between two materials where the dielectric function changes abruptly, and this is important in the interpretation of SERS measurements. In addition, it can be important for particles that are sufficiently small that there can be size dependence to the dielectric function (as in quantum dots) or where the discreteness of the electronic structure leads to deviations from the continuum assumption. The development of theory that describes optical phenomena under these circumstances is an important challenge.

Recently, my group, in collaboration with Jochen Autschbach, has developed an electronic structure method that provides new capabilities for describing the optical properties of silver and gold metal clusters (80). The algorithm is based on time-dependent density functional theory (TDDFT), which is a useful (although often criticized) tool for describing the excited states of molecules and materials. Our version of this algorithm enables us to use TDDFT to calculate frequency-dependent polarizabilities and polarizability derivatives for wavelengths where resonant excitations in the molecule or cluster can occur. Often, such resonant excitations must be treated quite distinctly from nonresonant excitation, but in the case where excited state dephasing is rapid compared with the time scale of molecular vibrations, it is possible to implement a short time approximation to the Kramers–Heisenberg formula in such a way that it is possible to implement this approach with the same formalism for both resonant and nonresonant processes. We have used this method to describe a number of problems in molecular spectroscopy (81, 82), and we have also studied the optical properties of metal clusters (83–85).

Fig. 7a shows the extinction spectrum of a silver cluster, Ag₂₀, that we have studied with this approach (83). This spectrum is the result of a large number of electronically excited states, but one in particular dominates to give the peak at 3.4 eV. The structure assumed for this calculation is a perfect tetrahedron, which is a low-energy structure for Ag₂₀, although not necessarily the lowest in energy. However, in the present case, it is only important that we have a representative structure, as we can use it to study the analog of plasmon excitation for a small (1 nm) particle. The 3.4 eV feature in Fig. 7a is not a collective excitation, nor should it be, as the density of electronic states is too small to provide the serious configuration mixing that is needed to produce a collective excitation. However, the spectrum in Fig. 7a does match what has been found in matrix isolation spectra (86) (Fig. 7b), and it furthermore can be used to understand SERS intensities, as is demonstrated in Fig. 7 c and d. Fig. 7c shows the Raman spectrum of pyridine that we have calculated at a wavelength that matches the resonance wavelength (the analog of the peak at 3.4 eV) for the pyridine/Ag₂₀ system. Fig. 7d shows the normal Raman spectrum of pyridine that we have obtained from the same code. The two spectra differ in intensity for some of the vibrational modes, but a more important difference is 10⁵ to 10⁶ enhancement in the intensity in Fig. 7c compared with Fig. 7d. This is the small-particle analog of the surface enhanced Raman effect, and the magnitude of the enhancement factor that we find is surprisingly close to enhancement factors that have often been estimated for SERS on much larger nanoparticles. Indeed, although the excited state of the metal cluster does not involve plasmon excitation, it still leads to dipole excitation, and therefore electromagnetic enhancement. The close correspondence of experimental and estimated enhancement factors is somewhat of an accident, as the enhancement factor is not converged with respect to cluster size for a 20-atom cluster. However, the calculations can be used to provide insight concerning a number of others issues, such as the relative importance of electromagnetic and chemical contributions to the SERS enhancement factor, the influence of adsorption geometry, and coverage on the SERS spectrum, and even the role of junction structures in producing the extra large enhancement factors that are postulated to exist in single-molecule SERS measurements.

Fig. 7.

Electric structure studies of silver cluster optical properties. (a) Calculated time-dependent density functional theory absorption spectrum of Ag₂₀. (b) Corresponding experimental results (86) for clusters varying from Ag₁ to Ag₂₅. (c) SERS spectrum of pyridine/Ag₂₀. (d) Corresponding calculated static Raman spectrum. These results are adapted from ref. 83.

The electronic structure methods we have developed provide a powerful way to describe the optical properties of particles with up to a few hundred atoms, but ultimately it will be essential to combine quantum mechanics and electrodynamics (QM/ED methods) to describe many optical properties in nanoscale systems. In particular, the near-field behavior associated with the nanoparticles in Fig. 6 is imperfectly described by the clusters in Fig. 7, and until this is done, the description of SERS will be incomplete.

Conclusion

Table 2 presents a summary of the applications we have presented in this manuscript. This includes the limiting case atomistic and continuum descriptions available for the properties we have considered, as well as hybrid and mesoscale methods that help bridge the gap between length and time scales. The examples we have considered show that purely atomistic theory or purely continuum theory usually have important limitations for modeling nanoscale properties. Instead, it is hybrid and mesoscale theories that can often play a leading role. In most of the cases we considered, there has been important progress in the development of these theories in the past few years. However, this is still just the beginning of what will be needed to describe real materials.

Table 2.

Hierarchies of theoretical models

Property	Atomistic level theory	Hybrid/mesoscale theory	Continuum theory
Structural, thermal	QM, empirical potentials	Coarse-grained MD, GCMC on lattices	Static structure models, thermodynamics
Mechanical	QM, empirical potentials	Coarse-grained models, QM/MM/CM	Elasticity theory
Optical	QM	Multipole coupling of particles, QM/ED	Continuum electrodynamics

The applications we have described show how theory can sometimes play a critical role in the progress of nanoscience research. For example, we have seen that the optical properties of silver and gold nanoparticles are well predicted by continuum electrodynamics, and this makes it possible to use theory in a predictive mode to determine optimal nanoparticle shape for a given application. In other applications, such as the fracture of nanomaterials, the gap between theory and experiment is large enough that the primary role of theory has been to suggest the origin of the gap. Still another role of theory shows up in the treatment of self-assembly processes, and in the description of DNA melting in DNA-linked aggregates, where it can be used in a qualitative way to rationalize why structures of a given type result from specified monomer structures.

Abbreviations

PA

peptide amphiphile

AFM

atomic force microscopy

QM

quantum mechanics

MM

molecular mechanics

CM

continuum mechanics

UNCD

ultrananocrystalline diamond

SERS

surface-enhanced Raman spectroscopy.