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J. Phys. Chem. B 2000, 104, 460-467
DNA-Linked Metal Nanosphere Materials: Structural Basis for the Optical Properties Anne A. Lazarides and George C. Schatz* Department of Chemistry, Northwestern UniVersity, EVanston, Illinois 60208-3113 ReceiVed: June 28, 1999; In Final Form: NoVember 3, 1999
The structural basis for the aggregation-induced optical properties of colloidal gold nanosphere aggregates is examined by means of electrodynamics calculations. Recently developed methods for calculating the electrodynamic response of aggregates composed of large numbers of small metal nanospheres in a dielectric medium are used to determine the optical changes associated with the formation of spherical aggregates. The calculations use accurate nanoparticle polarizabilities determined from Mie theory, an iterative conjugategradient solution algorithm, and fast-Fourier transform methods for efficient solution of the electrodynamic interacting nanoparticle equations. The UV extinction lowering and the shifting and broadening of the visible plasmon peak observed experimentally in solutions of DNA-linked gold nanospheres are explained as the collective electromagnetic response of thousands of nanoparticles.
I. Introduction Recent developments in the design of nanoparticle-based materials have yielded nanostructured materials with distinctive structural and optical properties. Among the new synthetic strategies is that of DNA-driven assembly of colloidal nanoparticle aggregates.1 In this process, oligonucleotide-functionalized gold spheres are exposed to free oligonucleotide, one end of which is complementary to the DNA on half of the nanoparticles, the other end of which is complementary to the DNA on the rest of the nanoparticles. DNA hybridization pulls the nanoparticles together and yields a reversibly aggregated material whose optical properties differ significantly from those of noninteracting, monodispersed particles. Whereas suspensions of dispersed gold colloid are red due to absorption of green light in the single particle plasmon region, aggregated colloid absorbs and scatters light over a broad band of longer wavelengths and appears purple. The extinction spectra2 of gold nanoparticle aggregates are characterized by reduced extinction in the UV, violet, and blue region and by a plasmon peak that is red shifted and broadened relative to that of the dispersed particles. Spectra for dispersed and DNA-linked 13 nm Au colloid are displayed in Figure 1. The electrodynamic modeling described in this paper was motivated by our interest in understanding the optical properties of the DNA-linked colloidal system. While much work has been done in the area of modeling metal inclusions in dielectric media,3,4 the DNA-linked system is, in some respects, unique. Firstly, the DNA coating prevents the nanoparticles from touching, and thus from coalescing. Secondly, DNA hybridization, while energetically favored, is reversible,1 and mild annealing produces aggregates with fairly uniform, high metal density, as observed by transmission electron microscopy.5 Lowdimension fractal structures, such as those formed from Au6 or Ag7 colloids that bind irreversibly due to dispersion forces and/ or metallic bonding, are not observed. Thirdly, the division of the nanospheres into two ‘species’ on the basis of their oligonucleotide coating can be expected to lead to structures that are topologically binary. Compact structures with elements of body-centered or simple cubic order are therefore favored
Figure 1. Extinction spectra for dispersed and DNA-linked Au nanoparticles, courtesy of the authors of ref 2. The 13 nm Au particles are capped with 3′- and 5′-(alkanethiol) 12-base oligonucleotides. The aggregated colloid is formed in the presence of a 24-base oligonucleotide each half of which is complementary to one of the two nanoparticle-capping oligonucleotides. Details of the preparative procedure are in ref 2.
over the face-centered cubic structures typically found under conditions of dense nonspecific aggregation.8,9 While it is well-established that the DNA-induced colloid color change is associated with DNA-driven assembly of otherwise dispersed particles,1,2,5,10 the structural basis for the plasmon shifting and broadening is not well understood. TEM imaging of deposited DNA-linked nanoparticle aggregates5 indicates that the suspended aggregates are large and threedimensional. However, the structure and density of the assembly unit responsible for the spectral changes is unknown. Our goals in this paper are to identify the extent of aggregation necessary to produce the observed color change and to determine the dependence of this result on particle density and organization within the aggregate. We will also clarify the role, if any, of small aggregates (dimers, trimers, etc.) in determining the observed optical properties of linked material. Prior to the development of the DNA-directed nanoparticle
10.1021/jp992179+ CCC: $19.00 © 2000 American Chemical Society Published on Web 12/31/1999
DNA-Linked Metal Nanosphere Materials assembly system, studies of the optical properties of the aggregated metal colloid focused upon the aggregate structures formed from underivatized colloid. Underivatized colloid is known to form chain-like clusters11 and fractal aggregates.6,7 Enhancements of the fields between clustered particles have been observed through surface-enhanced Raman spectroscopy, for example, and have been rationalized using dimer models rather than many-particle models of aggregated colloid.12 While the largest interparticle interactions occur only when particles are aligned with the polarization direction of incident polarized light, the relevant field enhancements that occur under these conditions are enormous, and the Raman signal of a population of molecules is controlled by a minority population that is situated in a statistically small number of hot sites. Modest effects, however, such as the shifts in nanoparticle plasmon extinction peaks that occur when dimers are aligned along an axis parallel to the polarization direction, may not be dominant enough to even be observable in colloidal suspensions where dimers are randomly oriented. Our initial efforts to understand the optical properties of materials formed from organic- or biomolecule-dressed nanoparticles focused upon the spectral changes associated with the formation of very small aggregates (clusters) such as dimers,13 trimers, tetramers,14 and linear arrays. These studies were based on high quality electrodynamics calculations, but we found that only the most extended of these aggregates bear any spectral similarity to DNA-linked colloid. We include in this paper the extinction spectra of the few particle aggregates that come closest to displaying the spectral changes observed in DNAlinked colloidal aggregates, but the conclusion of this part of our work is that the optical properties of the DNA-linked network materials cannot be explained using small aggregate models. In view of this, we have developed new methods, described in ref 15 and summarized here, for modeling aggregates containing hundreds to thousands of particles. The primary focus of this paper is therefore on the UV-vis extinction of large aggregates as calculated using these newly developed methods. We begin in section 2 with a description of an interacting dipole model for electrodynamically coupled spheres for which we have developed efficient methods of solution.15 In section 3 we present UV-vis extinction spectra for three-dimensional aggregates of 13 nm gold spheres and discuss the criteria for increased UV and short-wavelength visible transmission and for surface plasmon peak shifting and broadening. The dependence of spectral features on aggregate size, metal volume fraction, and microstructure is examined. In section 4 we consider very small aggregates, where it is possible to describe the electrodynamics accurately13 using finite element methods.16-18 We discuss the UV-vis extinction spectra of one- and two-dimensional arrays of nanoparticles, and conclude that three-dimensional aggregates are required to reproduce the observed spectral properties. In sections 5 and 6 we review and summarize the structural sensitivity of the aggregate spectra. Our fundamental conclusion is that the experimentally observed increased ultraviolet, violet, and blue transmission and plasmon shifting and broadening are theoretically reproducible only with aggregate models that include many hundreds of nanoparticles. II. Electrodynamic Theory for Nanosphere Aggregates A. Coupled Nanospheres. The behavior of light incident on a macroscopic target is governed by Maxwell’s equation for the electric and magnetic vector fields. The general framework for modeling the optical response of a collection of spheres
J. Phys. Chem. B, Vol. 104, No. 3, 2000 461 involves self-consistent solution of the response of each particle to the incident field and the scattered fields of the other particles. Spherical particles of nonmagnetic materials with sizes much smaller than the wavelength of light respond primarily to the electric dipole component of the local field unless the higher multipole components of the local field are very large. For isotropic materials, the frequency-dependent dipole polarizability R1 of the spheres is determined by the electric dipole scattering coefficient19,20 according to the expression21
3i R1 ) r3 a1 2(kr)3
(1)
where r is the sphere radius, k ) m0(2π/λ) is the magnitude of the wave vector in a dielectric medium with real refractive index m0, and a1 is a function of the size parameter kr and relative metal index of refraction m ) x/m0. For very small spheres, the bulk dielectric function must be corrected22 for scattering from the sphere surface (quantum size effect). While the dielectric modification of the solvent due to the presence of DNA in a layer surrounding the nanoparticles could be modeled, the effect on a1 is very small because of the low dielectric contrast between DNA and water.23 Effects of this type are reviewed elsewhere.24 The response of a dielectric aggregate of small nonmagnetic nanospheres to electromagnetic radiation can be determined by self-consistent solution of the electric dipole polarizations Pi of each sphere in the superposed field of the incident light and the dipole fields of the other particles. Thus, the coupled nanosphere problem is formulated analogously to the DDA approach referenced in section 4, though with one dipole per particle instead of many. For a given dipole at position ri,
Pi ) R‚Eloc(ri)
(2)
where Eloc(ri) is the sum of the incident plane wave, Einc,i ) E0 exp(ik‚ri - iωt) and the retarded fields
-Aij‚Pj )
{
1 - ikrij eikrij 2 k r × (r × P ) + × ij ij j r3ij r2ij [r2ijPj - 3rij(rij‚Pj)]
}
(3)
of the other N-1 dipoles, and R is a polarizability tensor with elements Rkl ) δklR1. Use of the retarded field expression eliminates the need for explicit modeling of the time dependence of the fields and polarizations. Thus,
(R-1)Pi +
Aij‚Pj ) Einc,i ∑ j*i
(4)
The N linear complex equations for 3-vectors Pi and Einc,i can be formulated as a single 3N-dimensional matrix equation,
AP ) Einc
(5)
where P and Einc are 3N vectors and A is a 3N × 3N symmetric matrix constructed from the 3×3 interparticle interaction matrices Aij, with additional terms R-1 along the diagonal. 1 Higher multipole couplings could also be included. However, for aggregates composed of fairly uniformly packed small spheres separated by distances that are substantial relative to the particle diameter, they are not needed. This has been verified in earlier work.15
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B. Conjugate Gradient Solution with Fast Fourier Transforms. Direct inversion of eq 5 is time consuming for targets with large numbers of spheres. An efficient alternative is to solve eq 5 by iteration, using a conjugate-gradient algorithm. The algorithm requires evaluation of matrix-vector products of the form AX and A†X but not the individual components of the interaction matrix. Because the elements of the interaction matrix depend only upon the interparticle vectors rij and not the individual particle positions ri, rj, the matrix A for a collection of particles on a lattice can be Fourier transformed and represented in terms of spatial frequencies.17 The fast-Fourier transform technique25 is used to calculate the Fourier space products. The application of the conjugate gradient and fastFourier transform techniques to the coupled sphere problem is described in ref 15. While use of Fourier methods is restricted to aggregates whose component spheres occupy sites on a cubic lattice, it does not impose any restrictions on the occupation of the lattice or the shape of the target. The methods are thus applicable to aggregates of any shape composed of nanospheres in simple cubic, body-centered cubic, or face-centered cubic arrays or disordered arrays of the lattice-gas variety. In ref 15 we have demonstrated the equivalence of the Fourier and direct solutions to eq 5 for these types of structures. C. Optical Cross Sections. In accordance with the optical theorem, extinction is determined by the forward scattering. In an extended system, each particle contributes to the forward scattering and hence, to the extinction cross section, according to the magnitude of its polarization Pj and the extent to which the polarization is out of phase with the incident field. The extinction cross section is calculated as16
Cext )
N
4πk |Einc|
/ Im{Einc,j ‚Pj} ∑ j)1
(6)
2
Absorption, for a dipole array, is calculated as
Cabs )
4πk
N
∑ 2 j)1
|Einc|
{
2 3 / / Im{Pj‚(R-1 j )*Pj } - k Pj‚Pj 3
}
(7)
and scattering as the difference Csca ) Cext - Cabs. We do not model extinction due to the DNA. While DNA absorbs in the UV region of interest, the magnitude of the absorption is insignificant relative to that of the metal (for linker lengths short enough to induce color-changing aggregation) unless unusually high concentrations of linker are present. III. Optical Properties of Space-Filling Aggregates Extinction, absorption, and scattering spectra were calculated for spherical aggregates of various sizes, microstructures, and nanoparticle densities. All of the calculations described here are for aqueous solutions of aggregates of 13 nm Au spheres. In the absence of detailed information about the shapes assumed by DNA-linked nanoparticle networks, we chose to model spherical aggregates, recognizing that a variety of shapes are undoubtably represented in the aggregate material. The calculations were performed using gold dielectric functions from the literature,26 corrected for increased damping due to electronsurface scattering.22 A primary goal was to determine the structural properties required to reproduce the aggregation-induced changes observed in DNA-linked nanosphere UV-visible extinction spectra.2,5 As indicated in Figure 1, the changes that occur when aqueous 13 nm gold spheres are linked with 24 base-pair lengths of
Figure 2. Extinction spectrum for a 300 nm spherical aggregate of 13 nm Au spheres in water with Au volume fraction 0.2. Thin solid lines (s) show contributions to the extinction from absorption and scattering. Spectrum of an isolated sphere in water (- -) is shown for comparison.
oligonucleotide are reduced extinction in the UV and shortwavelength visible range, a red shifting of the plasmon peak by approximately 50 nm, and a broadening of the plasmon peak by approximately a factor of 2. The optical properties of networks composed of spheres that are as small as 12 nm or as large as 16 nm have been observed to be similar. We focused on the 24 base-pair linked system because the plasmon peak location of this network is insensitive to annealing.5 We hypothesized that annealing increases the level of order and that a system whose spectrum was insensitive to annealing was more likely than other systems to be modeled successfully as an ordered array. However, as discussed below and in ref 15, the spectra of disordered arrays are similar to those of ordered arrrays unless the disorder involves particle spacings that are smaller than expected in a system with an oligonucleotide exclusion layer. UV-vis spectra were calculated for aggregates of 13 nm spheres separated by distances less than or equal to the lengths of DNA duplexes. Duplexes of 24 base-pairs are approximately 8 nm in length, so the nearest neighbor sphere separation in 24 base-pair linked aggregate is at most 8 nm. Spacings that are somewhat smaller than the length of coiled DNA duplex allow for the possiblility of multiple links between pairs of neighboring particles. Extinction spectra were calculated for a variety of aggregates composed of 13 nm spheres separated by 6.5 nm. Figure 2 presents a spectrum that is similar to the DNA-linked aggregate spectrum in Figure 1. The spectrum is for a 0.3 micron spherical aggregate composed of 2445 particles arranged on a body-centered cubic lattice with a gold volume fraction of 0.2. For comparison, the spectrum of a single sphere is shown (dashed line), calculated using Mie theory.19,20 The plasmon peak of the aggregate is red shifted (by 45 nm) and broadened relative to that of dispersed 13 nm Au spheres. While dispersed colloid extinction is due to absorption, aggregated colloid both scatters and absorbs visible light. The contributions to extinction from absorption and scattering for the model aggregate are included in the figure. The aggregate scatters light most efficiently on the low frequency (long wavelength) side of the shifted plasmon band, while absorbing light at higher frequencies. The absorption peak is nonetheless red shifted from the dispersed colloid location, which will allow blue light to be transmitted through the aggregate. Thus, both scattering and absorption contribute to the broadening and red-shifting of the extinction peak for this particular aggregate.
DNA-Linked Metal Nanosphere Materials
J. Phys. Chem. B, Vol. 104, No. 3, 2000 463
Figure 3. Extinction spectra for spherical aggregates of 13 nm Au spheres in water with Au volume fraction 0.2. The aggregate diameters are 153, 214, and 300 nm. The spectrum of an isolated sphere in water (- -) is shown for comparison.
Figure 4. Extinction spectra of spherical aggregates of 13 nm gold spheres in water for three colloid volume fractions: 0.068 (s), 0.123 (- -), and 0.256 (‚‚‚). All aggregates have 893 spheres arranged on a body-centered cubic lattice.
In the UV, the extinction in Figure 2 drops dramatically with aggregation, as has been observed (Figure 1) in spectral monitoring of DNA-directed aggregation in the laboratory.2,5 While aggregation induces scattering of UV light (though the scattering is weaker than the scattering of long wavelength visible light), the drop in absorption more than compensates for the initiation of scattering, and the transmission increases (extinction decreases). On the basis of the profile of induced polarization in the aggregate, we find that the decrease in absorption can be attributed to the screening of nanospheres embedded deeply within the aggregate interior. The 2445 nanosphere spectrum therefore displays the three features known from experiment to be the UV-vis extinction signature of DNA hybridization induced linking of small gold nanospheres, namely, substantial red shifting and broadening of the plasmon peak and enhanced UV transmission. A. Sensitivity of the Extinction Spectrum to Aggregate Size. Having established that aggregates composed of several thousand nanospheres have extinction spectra that exhibit the features associated with DNA hybridization-induced nanosphere aggregation, we then sought to understand the dependence of the aggregation-related spectral features upon structural properties of the aggregates. Figure 3 displays spectra for aggregates with the same volume fraction and structure, but differing sizes. Spectra for 153 and 214 nm aggregates are shown, along with the 300 nm aggregate spectrum illustrated in Figure 2. All three aggregates are 20% gold by volume, have component particles arranged on a bcc lattice with 6.5 nm between nearest neighbors, and are spherical in shape. The spectra of the smaller aggregates display decreased UV extinction (at wavelengths below 300 nm) and red shifting and broadening of the surface plasmon peak, though to a lesser extent than for the larger aggregate. While the amount of plasmon shifting observed in the laboratory varies with experimental conditions,5 DNA-induced aggregation consistently yields larger UV extinction changes and visible plasmon broadening than is observed in the spectra of the smaller (331 and 893 particle) aggregates displayed in the figure. These results indicate that aggregates that are 20% gold must contain more than a thousand 13 nm particles if their UV-vis spectra are to match the experimentally observed spectra. We are interested also in the structural basis for variations in the magnitude of plasmon extinction as colloid aggregates. From Figure 3, it is apparent that, if interparticle spacings remain
constant at 6.5 nm, the peak plasmon extinction increases as the first hundreds of particles link and then decreases as aggregates become larger. Similar size-dependent variations in the magnitude of plasmon extinction occur for more and less dense aggregates when aggregate size is varied while interparticle spacings are held fixed. B. Sensitivity of the Extinction Spectrum to Metal Volume Fraction. While DNA-linked nanoparticles appear to form dense materials with mild annealing,5 the actual metal volume fraction is not known. Because two different oligonucleotide sequences are used to coat the nanoparticles, and free oligonucleotide links a given nanoparticle only to particles with a coating composed of oligonucleotide of the other sequence, the densest possible DNA-linked material will have a binary structure. The upper bound on achievable metal volume fractions is, therefore, 0.68, the packing fraction of a body-centered cubic array of touching spheres. The DNA-dressed particles do not touch,1 however; rather, they are separated by an amount not exceeding the length of the coiled double-stranded DNA linker that may depend upon preparative conditions. It is important, therefore, to understand how aggregate spectra depend upon metal volume fraction. Extinction spectra were calculated for spherical aggregates of 893 spheres arranged on a body-centered cubic lattice with a range of metal volume fractions. Three such spectra are illustrated in Figure 4. The solid line is the spectrum for a 0.3 micron aggregate with 7% Au by volume. The long dashed line is the spectrum for a 0.25 micron aggregate with 12% Au by volume. The short dashed line is the spectrum for a 0.2 micron aggregate with 26% Au by volume. The volume fractions correspond to interparticle separations of 15, 10, and 5 nm respectively, for an ordered bcc array. These are plausible spacings for aggregates connected with 72, 48, and 24 basepair oligonucleotide linkers, respectively. The visible spectrum of the sparsest aggregate (15 nm separation, 6.8% Au) is very close to that of dispersed colloid; the plasmon peak is very slightly shifted and broadened. There is, however, a significant drop in the UV extinction relative to that of dispersed colloid. When the interparticle separation drops below a nanoparticle diameter, the UV extinction continues to drop and the plasmon peak shifting and broadening begins to become noticeable. Aggregates of this size (nearly a thousand particles) composed of spheres separated by less than a particle radius (e.g., with a 5 nm separation and 26% Au) have flattened extinction spectra in the UV, short-wavelength visible range
464 J. Phys. Chem. B, Vol. 104, No. 3, 2000
Figure 5. Extinction spectra of two 312 nm spherical aggregates of 13 nm gold spheres in water with 6.5% Au by volume. The solid line (s) corresponds to spheres arranged on a simple cubic lattice; the dashed line (- -) to a bcc array of spheres.
and a plasmon peak location that is at the long wavelength limit of experimentally observed peaks. There is, however, less extinction of red and near-IR light than observed experimentally in DNA-linked materials. Larger aggregates will have broader and, in the lower colloid-density materials (less than 25% Au by volume), redder plasmon extinction features. For small aggregates such as those whose spectra are shown in Figure 4, plasmon peak intensity increases with colloid volume fraction. C. Sensitivity of the Extinction Spectrum to Aggregate Structure. The sensitivity of extinction to the detailed arrangement of the colloidal particles was also explored. Figure 5 displays the spectra of two spherical aggregates of approximately the same size and colloid density, one with gold spheres arranged on a simple cubic lattice and the other with a bodycentered cubic arrangement of spheres. The two spectra are almost indistinguishable. The slight differences are caused by a small differences in the number of spheres in the two targets. Both aggregates are approximately 312 nm in diameter and 6.5% gold. Comparisons made for denser aggregates (up to 52% gold) yielded similarly matched spectra. These results correspond with theoretical results that indicate that the extinction of dipoles on a cubic lattice is insensitive to the basis when the volume fraction is held constant.3 Theoretical studies30 of randomly distributed metal nanoparticles have indicated that disorder can lead to red-shifting and broadening of the plasmon peak. The sensitivity of extinction to disorder was therefore also investigated. Arrays with substitutional disorder were generated by deleting randomly selected spheres from defect-free body-centered cubic arrays. The extinction spectra of the disordered arrays were compared with spectra for identically sized defect-free arrays with the same gold volume fraction. The results are illustrated in Figure 6 for a moderately dense spherical aggregate (Au volume fraction 0.2). The aggregates are 156 nm in size and contain 348 nanospheres arranged with or without vacancies on bodycentered cubic lattices of various lattice parameters. Spheres in the ordered array (solid line) are separated by a particle radius. The spectra of the disordered arrays display broadened, redshifted plasmon peaks, as anticipated. However, the effect is small if the nearest-neighbor spacings are confined to distances large enough to allow for the interparticle DNA layer. Thus, while a 30 nm shift with substantial broadening is calculated for a disordered aggregate in which nearest-neighbor sphere centers are allowed to touch (dashed line), only a 6 nm shift
Lazarides and Schatz
Figure 6. Extinction spectra of 156 nm spherical aggregates of 13 nm gold spheres in water with gold volume fraction 0.20. Two disordered arrays are compared with an ordered array (s). The dasheddotted line (- ‚) is for a disordered array with 4 nm minimum separation between spheres and a vacancy level of 35%. The dashed line spectrum (- -) is for a more highly disordered array (72% vacancy level) in which sphere centers may be as close as one diameter. All spectra are calculated at the interacting dipole level.
occurs when the disordered nanospheres are constrained to maintain at least a 4 nm separation from all neighbors. Interparticle interactions of higher order than the electric dipolar interaction can be expected to further red shift and broaden the spectra of disordered arrays in which the minimum particle spacing is less than several nanometers. If the neighboring nanospheres in the DNA-linked network penetrate deeply into the DNA layer, higher order interactions should be modeled. In this eventuality, modeling that neglects higher order interactions will underestimate the plasmon shifting and broadening. The extent of this error is examined quantitatively in ref 15. Previously, the convergence with respect to multipole order of polarization in disordered composite materials has been examined31 using electrostatic theory. Similar comparisons were made for aggregates with other metal volume fractions. Consistently, the disorder-induced plasmon peak shifts and broadening were observed to be modest except in cases where the nearest-neighbor spacings were unrealistically small for DNA-linked nanosphere network materials. IV. Optical Properties of Nanosphere Chains While DNA-linking appears to yield dense 3D assemblies,5 and the aggregate extinction properties are well reproduced using 3D models, we considered also the optical behavior of lower dimensional aggregates. In order to calculate accurately the optical properties of closely spaced particles, we used a method that is capable of representing high-order interactions between particles. Specifically, we used the discrete dipole approximation (DDA) of Purcell and Pennypacker32 as refined by Draine, Goodman, and Flatau,16-18 and modified for particles in dielectric media.33 Each nanoparticle is modeled as a cubic array of polarizable elements, and the coupled response to light of a given frequency of the array elements in all particles is calculated self-consistently. The extinction cross section is calculated from the dipole polarizations of the finite elements using the same expression (eq 6) that was used in section 3 to calculate large aggregate extinction from nanoparticle polarizations. In the DDA formulation, the polarizations, Pj, in eq 6 correspond to polarizations of finite elements, each of which
DNA-Linked Metal Nanosphere Materials
Figure 7. Extinction spectra (s) calculated for lines of 13 nm Au spheres in water separated by 6.5 nm. The chains are aligned along the polarization direction of the incident light. The spectrum of a single sphere (- -) is shown for comparison.
Figure 8. Extinction spectra calculated for linear arrays of 13 nm Au spheres composed of 2, 4, 6, 8, or 10 spheres separated by 1.1 nm in water. Solid lines (s) indicate spectra of chains of spheres aligned along the polarization direction of the incident light. Dotted lines (‚‚‚) are spectra of randomly oriented linear chains. The spectrum of a single sphere (- -) is shown for comparison.
represents the dipole moment of approximately a cubic nanometer of a nanoparticle. Although only the dipole response of each element is modeled, the collective response captures multipole interactions between particles. The method is described in detail in ref 13. Initially we consider the spectra of one-dimensional arrays. Figures 7 and 8 display spectra calculated for linear chains of Au nanospheres composed of varying numbers of particles and various separations. For the purpose of comparison with the three-dimensional aggregates whose spectra were discussed in section 3, we present in Figure 7 aqueous chains of 13 nm Au spheres with 6.5 nm interparticle spacings and include a 10sphere chain whose overall length (300 nm) is the same as the diameter of the aggregate shown in Figure 2. The extinction spectra for a 16-sphere chain is also shown, as is that of dispersed aqueous colloid. The spectra are normalized by the geometric cross section (nπr2) for consistency with the particle light-scattering literature.20 The chains are aligned along the polarization direction of the incident electric field, which is the orientation that maximizes the plasmon shift. However, while
J. Phys. Chem. B, Vol. 104, No. 3, 2000 465 the per particle extinction increases throughout the UV-vis region, only a small shift with negligible broadening is observed. As indicated by comparison of the 10- and 16-sphere spectra, lengthening the array beyond 10 spheres increases the per particle extinction cross section but does little to further shift or broaden the plasmon peak. Linear arrays aligned in the plane perpendicular to the electric field have spectra that are almost indistiguishable from the spectrum of dispersed spheres. The plasmon feature in the spectra (not shown) of randomly oriented linear chains, therefore, are broadened slightly relative to that of dispersed spheres but imperceptibly shifted. Chains of more closely spaced spheres have plasmon extinction peaks that are further broadened or split into two peaks,34 as chains aligned along the polarization direction interact more strongly and have plasmon frequencies that are shifted more than polarization-aligned chains of more widely spaced spheres. Figure 8 displays the spectra of lines of 13 nm spheres separated by 1.1 nm, i.e., one-sixth the separation of the arrays shown in Figure 7. While a 1.1 nm particle spacing is not realistic for DNA-linked nanoparticle networks, we pursue it here to clarify the spectroscopic properties of linear particle arrays in the strong interaction limit. Solid lines represent the spectra of lines of 2, 4, 6, and 10 spheres aligned along the polarization direction. For arrays aligned parallel to the polarization direction, the peak plasmon extinction red-shifts and broadens dramatically and the peak extinction per particle gets larger with increasing chain length. To model the spectra of randomly oriented chains, it is necessary to average spectra for all chain orientations relative to the polarization direction. Spectra for randomly-oriented chains are represented in the figure with dotted lines. While the plasmon peak of the aligned dimer is significantly red-shifted and broadened, and the dimer is more extinctive (per particle) than are dispersed nanospheres, the spectrum for randomlyoriented dimer (dotted line labeled 2) is very similar to that of dispersed nanospheres. For six-sphere chains, the orientationally averaged spectrum likewise displays little of the character displayed by spectra of chains aligned along the polarization direction. The splitting of the plasmon peak between the spectral region of the aligned six sphere orientation and the unshifted (unaligned) location can be described as broadening; however, because a minority of randomly oriented chains are aligned close to the polarization direction, the extinction maximum remains at the dispersed colloid value. The extinction spectra of randomly-oriented chains of other lengths also have doublet plasmon peaks, with both an unshifted peak and a weaker, chainlength dependent peak at longer wavelengths. The spectrum for a mixture of linear chains is shown in Figure 9. The distribution of chain lengths among two to ten sphere chains is such that equal numbers of nanospheres populate each represented chain length. The spectra have been renormalized for consistency with the extinction spectra for 3D aggregates illustrated in section 3. The parallel polarization peak survives the averaging and the plasmon maximum remains unshifted from the dispersed colloid value. The split plasmon peak is similar to that observed for fractal aggregates of underivatized Au nanoparticles.34 However, the chain spectra are distinctly different from the single-peaked spectra of DNA-linked nanoparticle aggregates. The red shifting observed in colloidal gold solutions upon DNA-directed aggregation is apparently not reproduced when the aggregates are modeled as mixtures of short, extended chains, even when particle spacings are reduced to implausibly short distances to increase the possibility of reproducing the experimental spectra. This is consistent with
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Figure 9. Extinction spectrum (s) for a mixture of linear chains of 13 nm Au spheres composed of 2, 4, 6, 8, or 10 spheres separated by 1.1 nm in water. Equal numbers of nanoparticles populate chains of each length. The spectrum is an average of spectra calculated for randomly oriented chains of a particular length, most of which are shown in Figure 8. The spectrum of a single sphere (- -) is shown for comparison. The cross sections are normalized by the metal volume.
experimental results which show that the DNA-linked structures are large 3D aggregates of nanoparticles rather than 1D chains. The insufficiency of chain models is even more easily deduced through examination of the UV and short-wavelength visible region of the extinction spectrum. Whereas DNA-directed nanoparticle assembly results in strongly enhanced transmission (reduced extinction) in the wavelength range from 250 to 500 nm,2 the extinction spectra of extended linear arrays display only slight transmission enhancement in this wavelength range. Two-dimensional subunit models were also explored. As with the linear arrays, the extinction spectra of planar colloidal arrays display noticeable deviations from the spectra of isolated colloid when the planes are aligned normal to the propagation direction (parallel to the polarization direction) of incident light. However, as with linear arrays of closely spaced nanoparticles, the extinction spectrum of an orientational average of circular disks displays peaks corresponding to both parallel and perpendicular orientations relative to the incident light, but does not display plasmon peak broadening or significantly enhanced UV transmission. V. Discussion The broadening and shifting of the plasmon peak in the extinction spectra of colloidal gold aggregates is an expression of the collective nature of the aggregate response to visible light. Close association of large numbers of nanospheres is required for significant broadening and red-shifting of the peak. The number of spheres required is a function of the density of the spheres within the aggregated material and the level of disorder. Spherical bcc arrays of 13 nm spheres with a separation between nearest neighbors equal to a radius (6.5 nm) must include several thousand spheres to achieve doubling of the plasmon peak width relative to that of dispersed spheres. Larger, sparser aggregates can have similarly shifted, broadened plasmon peaks, but require more particles. Denser aggregates can have similarly shifted, broadened plasmon peaks with slightly fewer particles. In disordered arrays, plasmon peak broadening is achievable with fewer particles and/or lower metal volume fraction. However,
significant reduction in the required particle number for a given plasmon shape generally requires interparticle spacings not likely to be achievable in DNA-linked networks. For example, the disordered 20% gold aggregate whose spectrum is illustrated in Figure 6 as a dashed line peaking at 575 nm contains one seventh as many particles as the ordered 20% gold array illustrated in Figure 2. However, the most closely spaced particles in the disordered array model allow no space for the DNA linkers. The extinction spectra of large aggregates differ significantly from that of dispersed colloid in the UV as well. Aggregation induces lowering of the UV extinction. This extinction lowering increases with aggregate size and colloid volume fraction and can be attributed to shadowing. Elsewhere,35 using effective medium methods, we have examined the size-dependence of the extinction for even larger aggregates and find that the extinction continues to drop in the visible, as well as in the UV, as aggregate size increases further. In contrast, small clusters and linear arrays of spheres have UV spectra that differ little from spectra of dispersed colloid and display negligible plasmon peak shifting. Linear aggregation can produce plasmon peak broadening, but the nanospheres must be spaced more closely than is plausible for DNA-linked particles. Substantial red shifting of the plasmon peak occurs only for strings that are aligned along the polarization direction of the incident light. However, colloidal aggregates are randomly oriented. The broadened red-shifted plasmon peak observed in aggregated gold colloid cannot, therefore, be reproduced using small aggregate models, even when aggregate size distributions are invoked. Clustering, which can explain broadened features in aggregates formed from underivatized metal particles,36 has less dramatic consequences in the DNA-linked material because of distance constraints imposed by the DNA. The physical basis for the many-particle requirement for aggregation-induced shifting and broadening is the same as that which controls the optical spectra of larger solid metal nanoparticles.19,20,37 While small gold nanospheres in aqueous solution have plasmon peaks near 525 nm, spheres larger than
DNA-Linked Metal Nanosphere Materials 80 nm have broadened red-shifted extinction peaks. The red shift can be explained in terms of dynamic depolarization38 and the broadening in terms of radiation damping,39 which shortens the plasmon lifetime. As the sphere size increases, broadening of the solid sphere plasmon extinction continues as the scattering of visible light becomes decreasingly dependent upon wavelength. VI. Conclusion By applying efficient CG/FFT methods for calculating coupled dipole optical response to the coupled nanosphere problem, we have been able to determine aspects of the structural basis for the UV-vis extinction properties of DNAlinked nanoparticle aggregates. The changes in the optical properties of oligonucleotide-coated colloidal gold in the presence of complementary oligonucleotide can be explained as a consequence of the DNA-directed assembly of many hundreds of nanoparticles. Aggregation-induced shifting and broadening of the plasmon peak is a manifestation of the development of a collective response involving the electrons of networked particles that assemble in regions with dimensions on the order of visible wavelengths. As the aggregate size increases beyond a critical number (which is dependent upon the particle volume fraction), the surface plasmon absorption peak shifts to longer wavelengths, and scattering becomes significant, first on the red edge of plasmon peak absorption region, and then, for larger aggregates, throughout the UV-vis range. UV absorption per particle decreases as screening prevents the interior aggregate particles from being fully illuminated. The optical properties of the aggregates have been shown to be sensitive to the size, density, and level of order of the spheres but, for cubic arrays, insensitive to the particular choice of lattice. Larger, denser aggregates have redder, broader surface plasmon features and lowered UV extinction. The signatures of increased metal density and increased aggregate size, while not identical, are similar. The calculations described here model intersphere interactions at the coupled dipole level. While higher order interaction models generally yield redder, broader plasmon features and increase the spectral sensitivity to microstructure, too little is known about the microstructure of the DNA-linked material to justify use of higher order models. Fortunately, at particle separations reasonable for DNA-linked colloid, higher order effects are small.15 In future work, however, we will extend our studies to aggregates of various shapes and packing properties. Acknowledgment. This research was supported by ARO Grant DAAG55-97-1-0133. We thank C. Mirkin and J. Storhoff for providing us with experimental results. We thank B. T. Draine and P. J. Flatau for use of their DDA codes which we used with minor modifications to calculate the extinction of small linear and planar arrays of nanospheres and which also provided the framework for our coupled nanosphere code. We particularly thank P. J. Flatau for advising our colleague, T. Jensen, on the modifications required to enable the DDSCAT program to calculate spectra for particles in a dielectric medium.
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