Modeling of Electrodynamic Interactions between Metal Nanoparticles

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Modeling of Electrodynamic Interactions between Metal Nanoparticles Aggregated by Electrostatic Interactions into Closely-Packed Clusters Anatoliy O. Pinchuk,†,‡ Alexander M. Kalsin,† Bartlomiej Kowalczyk,† George C. Schatz,‡ and Bartosz A. Grzybowski*,†,‡ Department of Chemical and Biological Engineering and Department of Chemistry, Northwestern UniVersity, 2145 Sheridan Road, EVanston, Illinois 60208 ReceiVed: May 3, 2007; In Final Form: June 14, 2007

This paper is a theoretical and experimental study of the optical properties of binary aqueous suspensions containing metal nanoparticles (NPs) coated with charged organics. If all nanoparticles bear charges of the same polarity, the NPs do not aggregate, and the solutions are stable. Under these circumstances, optical response of the mixture is a linear combination of the optical responses of the individual components. In contrast, when the NPs are oppositely charged, they aggregate into clusters, whose optical properties cannot be understood without taking into account electrodynamic coupling between the constituent NPs. To model such aggregates, we present two theoretical approaches: (i) an exact analytical model accounting for the granularity of the NPs and (ii) an approximation, in which the aggregates are represented as spherosymmetric core-and-shells. Both models reproduce optical spectra recorded experimentally and give physically reasonable estimates of the aggregates’ internal structure and composition.

Introduction Interest in nanoscopic objects bearing electric charges is motivated by their applications in nanostructured materials1 and ultrasensitive analytical probes2,6-10 and by the unique properties of their solutions.3 In the latter context, we have recently shown1,3 that noble-metal nanoparticles (e.g., gold and silver) stabilized with self-assembled monolayers (SAMs4) terminated in charged functionalities behave very differently from microscopic particles and in some ways analogously to molecular ions.5 In particular, when oppositely charged nanoparticles (NPs) are mixed, they remain stable in solution and precipitate only when their charges are compensated. We argued3,5 that this peculiar nanoscale effect is due to the formation of core-andshell aggregates, in which the “minority” NPs are surrounded by a shell of oppositely charged “majority” particles whose mutual repulsions are effectively screened by the ionic atmospheres surrounding the NPs. In subsequent work, we showed that the formation of core-and-shell structures accounts, at least qualitatively, for the experimentally observed optical properties of NP assemblies, including enhancement and extinguishing of surface plasmon resonance (SPR) bands in systems comprising nanoparticles of two types.3 The objective of the present work is to formalize these qualitative explanations and to provide a more rigorous theoretical framework, one based on the extended Mie theory,13,14 with which to model the electrodynamic response of mixtures of both like-charged (Figure 1a,b) and oppositely charged (Figure 1c,d) metallic NPs and their assemblies. The experiments accompanying this study indicate that, (i) when all NPs present in solution bear charges of the same polarity, they do not aggregate and the overall UV-vis spectrum of the mixture is a linear combination of the spectra of the component NPs, and that, (ii) in contrast, oppositely charged NPs assembling into core-andshell structures exhibit optical response that is strongly influenced by the electromagnetic interactions between nearby NPs. Guided by these observations, we show that while the behavior * Corresponding author. E-mail: [email protected]. † Department of Chemical and Biological Engineering. ‡ Department of Chemistry.

of like-charged NPs is accurately described by linear combinations of the absorption cross-sections of individual particles, modeling of oppositely charged interacting particles requires treatment of the retardation effects and multipole interactions within the core-and-shell assemblies. Specifically, for systems in which positively and negatively charged NPs are all made of the same material (e.g., Au+/Au- or Ag+/Ag-), interparticle coupling effects in their aggregates can be treated rigorously by approximating spatial arrangement of the shell NPs by regular (canonical) geometries. In this case, it is possible to solve numerically the exact equations based on the extended Mie’s theory and to reproduce the experimentally observed UV-vis spectra faithfully. For systems in which the NP cores are made of different materials (Au+/Ag- or Ag+/Au-), we adopt a simpler continuous-shell approximation, which matches reasonably well with the experimental spectra using only one fitting parameter, the effective thickness of the shell. It also provides more physical insight than the extended Mie calculations concerning the nonadditive absorption spectra. The ability of both types of models to rationalize the pronounced spectral changes accompanying electrostatic aggregation opens new avenues for the synthesis of NP aggregates exhibiting desired electrodynamic response and can have practical implications for detection schemes based on interacting nanoparticle probes. Experimental Section (i) General. We used gold and silver nanoparticles of average diameters 2R ) 5.2 nm (with standard deviation ca. 12-15%) and synthesized according to the procedure described in detail in ref 3. The NPs were coated with the SAMs of either positively charged HS(CH2)11NMe3+Cl- (henceforth, TMA) or negatively charged (at basic pHs) HS(CH2)10COOH (MUA) thiols. Both thiols were obtained from ProChimia Surfaces, Poland. Dodecyl amine (DDA) and decanoic acid (DA) used to stabilize the NPs prior to thiol functionalization were purchased from Aldrich. All reagents were used as received. (ii) Details of NP Functionalization: Ligand Exchange on Gold NPs. A toluene solution of DDA-capped gold particles (7 µmol/mL, 20 mL, 140 µmol) was quenched with 100 mL of

10.1021/jp073403v CCC: $37.00 © 2007 American Chemical Society Published on Web 07/20/2007

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Figure 1. Scheme of titration experiments of like-charged and oppositely charged nanoparticles. (a) Like charged NPs do not aggregate. This is evidenced by the DLS distribution of particle sizes (b) recorded during the titration of, in this example, AuMUA NPs with AgMUA particles (χAgMUA ) 0.14). (c) In contrast, oppositely charged NPs aggregate and, in the early stages of titration, form core-and-shell assemblies. The DLS distribution in (d) corresponds to such assemblies during titration of AgMUAs with AuTMAs (χAgMUA ) 0.14). In (c) and (d), 〈D〉 denotes the average NP/cluster hydrodynamic diameters measured by DLS.

methanol to give black precipitate. The supernatant solution with excess of capping agent and surfactant was decanted, and the precipitate was washed with methanol (50 mL) and dissolved in toluene (100 mL), to which a MUA solution (31 mg, 140 µmol) in 10 mL of CH2Cl2 was added upon stirring. The precipitate of AuMUA NPs was allowed to settle down, the mother-liqueur solution was decanted, and the solid was washed with CH2Cl2 (3 × 30 mL). The precipitate was then dissolved by sonication in 5 mL of methanol. The AuMUA NPs were deprotonated with 25% methanolic solution of NMe4OH (70 µL, 165 µmol), precipitated with acetone (30 mL), and washed with acetone (2 × 30 mL) and methanol (30 mL). Finally, AuMUA NPs were dried and dissolved in 13 mL of deionized (DI) water to obtain ∼10 mM (in terms of gold atoms) solutions of NPs. The pH of the solution was ∼7.5 and, when needed, was adjusted to a higher value by addition of NMe4OH. (iii) Ligand Exchange on Silver NPs. In a typical procedure, a toluene solution of DA-coated silver NPs (30 mg, ∼250 µmol) was filtered through a 0.2 µm syringe filter to remove the precipitate of large particles and was then quenched with an equal amount of methanol. Next, the precipitate was separated and washed with methanol (100 mL), and TMA (28 mg, 100 µmol) in methanol (10 mL) was added to give a yellow-brown solution. The solution was stirred for 1 h and precipitated with CH2Cl2 (50 mL); the precipitate was separated and then washed (3 × 30 mL) with CH2Cl2. The precipitate was redispersed in methanol (5 mL) and precipitated with 30 mL of acetone, washed with acetone, and then with CH2Cl2. The redispersion/ washing procedure was repeated twice. Finally, the product was dried and dissolved in 22 mL of DI water to obtain ∼10 mM (in terms of silver atoms) solutions of AgTMA.

(iv) NP Titrations, UV-vis Spectra, and DLS Measurements. Nanoparticle aggregation was studied by titrating a solution containing NPs of one type with small aliquots of a solution containing NPs of different type (cf. Figure 1). In a typical procedure (procedures were analogous for other thiols and NP metals), a 2 mM solution of AuMUA (400 µL, 0.8 µmol) was titrated with 0.05-0.1 equivalent aliquots (20-40 µL) of 2 mM solution of AuTMAs. Titration was carried out in a stirred vial, and after each addition the solution was allowed to equilibrate for 5-10 min and then transferred to a UV-vis cell. Progress of the titration was monitored by UV-vis spectra recorded on Cary Model 1 UV-vis spectrophotometer in the range 300-800 nm in an optical glass cell (1 mm of path length). In addition, the sizes of the NPs and NP aggregates were determined by DLS measurements performed on a Brookhaven BI-9000 instrument at a 90° scattering angle, 514 nm laser wavelength, and 25 °C temperature. Each sample was run two times; the correlation function was analyzed by the nonnegative least-squares (NNLS) method. Results and Discussion 1. Like-Charged Nanoparticles. (Figure 2). Irrespective of their material properties and their relative concentrations in solution, like-charged NPs do not aggregate or precipitate. This observation is supported by dynamic light scattering (DLS) measurements (cf. Figure 1b) and is reflected in the additivity of the UV-vis spectra characterizing the mixtures of NPs of different types (i.e., AuTMA/AgTMA or AuMUA/AgMUA; Figure 2a). To explain such additivity and to model the UV-vis spectra of mixtures of like-charged NPs, we first note that 2 mM

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Figure 2. (a) Experimental UV-vis spectra recorded during titration of AuMUA with AgMUA NPs for molar fractions χAgMUA ) 0-1; (b) modeled spectra corresponding to those in (a).

concentrations used here correspond to np ≈ 3 × 1014 particles in 1 mL of solution and translate into average interparticle ≈ 150 nm. Because electromagnetic coudistance ˜l ) n-1/3 p pling between NPs decays with separation rapidly and is negligible for interparticle distances larger than five times NP radii, ˜l > 5R,11 it can be neglected in describing the optical response of the solutions we study. Consequently, the optical density of a mixture of like-charged NPs should be a linear combination of the optical densities of each type of particles, D ) np dσAgχAg + σAu(1 - χAg), where σAg and σAu are absorption cross sections of respectively individual silver and gold NPs, and d is the length of the optical path of light through the solution. Furthermore, using simple electrostatic approximation,12 the values of the NP cross sections can be written as σ ) ω/cxm ImR. In this formula, ω is the angular frequency of the incident electromagnetic wave, c is the speed of light, m is the dielectric permittivity of the solution, and the polarizability of the particles is given by R ) 4πR3( - m)/( + 2m), where  is the complex dielectric function of the metal particles. This function can be expressed as  ) JC - Drude + Drude,A, in which JC is the experimental permittivity of bulk gold taken from Johnson and Christy,15 Drude is the free electron Drude dielectric function,13 and Drude,A is the size-corrected dielectric function of the nanoparticles. Finally, to account for the scattering of the conduction electrons inside NPs and consequent damping of the surface plasmon resonance (intrinsic size effect), a phenomenological A parameter is introduced16 such that γ ) γ∞ + A υF/R, where γ is the scattering rate of free electrons inside a nanoparticle, γ∞ is the bulk scattering rate, and υF is the Fermi velocity for the conduction electrons of the silver or gold. Using the values of A ) 0.35 for AgNPs and A ) 0.18 for AuNPs found in earlier experimental studies16 and confirmed by calculations,13,16 we find the UV-vis spectra can be modeled over the entire frequency domain and agrees with the experimental ones (Figure 2b). Since similar results are obtained for like-charged mixtures other than AuMUA/AgMUA, we conclude that in the concentration regime for which ˜l > 5R (i) the additivity of optical densities is a general property of NP solutions and (ii) the UV-vis spectra of NP mixtures can be expressed as linear combinations of the spectra of individual NPs taken with relative weights equal to the corresponding molar fractions, Abs(NP1, NP2) ) χNP1Abs(NP1) + χNP2Abs(NP2). 2. Oppositely Charged NPs. When solutions of NPs of one polarity are titrated with particles bearing opposite-charges, the NPs aggregate, interact electrodynamically, and give rise to “nonadditive” and sometimes counterintuitive UV-vis spectra. As we have shown previously, aggregating nanoparticles do not precipitate during the titration until NP charges are compensated. Before the precipitation point, the “minority” particles are

surrounded by those present in excess (Figure 1c) so that the formed aggregates are all like-charged and stabilized by mutual electrostatic repulsions.1,3 Importantly, for the fractions of added NPs, χminor, smaller than ∼0.14, DLS measurements indicate that the aggregates are disjoint core-and-shells, approximately two or three NPs in diameter (Figure 1d). For higher values of χminor, the added particles cause aggregation of these core-andshells into larger assemblies of unknown internal structures and have mostly the “majority” NPs on their surfaces.17 In the following, we limit our discussion to the isolated core-and-shells. 2.1. NPs with the Same Metal Core. (Figure 3). Figure 3e shows UV-vis spectra recorded during titration of AgTMA with AgMUA NPs. As the fraction of the added AgMUA increases, the SPR band of silver red shifts and its intensity decreases; this behavior is a typical signature of an aggregation process. To account for the changes in the optical properties of the system during NP aggregation, it is necessary to consider electrodynamic interactions between the nanoparticles within the clusters. Since the usual effective medium theories (e.g., Maxwell-Garnett or Bruggeman approximations13,14) are not applicable to systems of densely packed NPs, we follow the rigorous approach pioneered by Gerardy and Ausloos11,12 in which each NP experiences the field scattered by the particles in its vicinity. This treatment is an extension of the classical Mie theory for aggregates of spherical particles and accounts for both the retardations effects and the multipole interactions between the particles in the clusters as well as for the limitation of the mean free path of the electrons (i.e., an intrinsic size effect). Specifically, to calculate the extinction of light by NP clusters, the plane electromagnetic wave E0 incident at the NPs and the Ns scattered Es ) ∑i)1 Eis and internal E1 electromagnetic fields are first expanded into vector spherical harmonics9,10 ∞

E0 ) Ns

Es )

n

∑ ∑ (pnmN(1)nm (r,θ,φ) + qnmM(1)nm(r,θ,φ)) n)1 m)-n

∞

n

∑ ∑ ∑ (aimnN(3)mm(ri,θi,φi) + bimnM(3)nm(ri,θi,φi)) i)1 n)1 m)-n ∞

Ei1 )

(1)

(2)

n

i i i i (1) i i i (dimnN(1) ∑ ∑ mn(r ,θ ,φ ) + cmnMnm(r ,θ ,φ )) n)1 m)-n

(3)

In these equations, the vector spherical harmonics are given (j) (j) (j) by12 M(j) nm ) ∇ × r unm and Nnm ) 1/k ∇ × Mnm. The scalar (j) spherical harmonics unm are related to the spherical Bessel functions of the first, jn, and second kind, hn ) jn + iyn through

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Figure 3. Modeling of electrodynamic interactions between NPs arranged in canonical clusters: (a) tetrahedral, (b) octahedral, (c) cubic, and (d) icosahedral. Each plot shows spectra for different values of interparticle spacing, l ) 6-20 nm. (e) Experimental UV-vis spectra recorded during the titration of AgMUAs with AgTMAs. (f) Comparison of the experimental and modeled spectra for χAgTMA ) 0.14 indicates that the forming clusters are best approximated by an octahedral NP arrangement with l ) 8 nm. m imφ and u(3) ) h (r)Pm(cos θ) eimφ, where u(1) n nm ) jn(r)Pn (cos θ) e nm n m Pn (cos θ) denote the associated Legendre functions. The superscripts (1) and (3) in eqs 1-3 refer to the first and the third kind of the scalar spherical harmonics used in the expansion of the vector spherical harmonics. The choice of the corresponding scalar spherical harmonics is dictated by the requirement of regular behavior of the electric fields at infinity and the origin of the coordinate system (Figure 4). The unknown coefficients anm, bnm, cnm, dnm, pnm, and qnm of the expansions are found by applying standard boundary conditions at the surface of a chosen nanoparticle: (E0 + Es - E1) × eˆ r ) 0 and (H0 + Hs - H1) × eˆ r ) 0. This can be done by using the addition theorem for the vector spherical harmonics which transforms the expansion series of the vector spherical harmonics around each NP of the cluster to the chosen NP and, thus, allows one to match the

boundary conditions in the same, spherical coordinate system. After matching the standard boundary conditions, the coefficients of expansion anm and bnm can be found from a linear set of coupled equations: N

∆nmi ) -Rin(pinm +

∞

l

Aklnm (rij, θij, φij) ajkl + ∑ ∑ ∑ j)1,i*j l)1 k)-1 Bklnm (rij, θij, φij)bklj) N

Γinm ) -βin(qinm +

∞

l

∑ ∑ ∑ Aklnm (rij,θij,φij)bjkl + j)1,i*j l)1 k)-1 Bklnm (rij,θij,φij)aklj) (4)

where ∆inm and Γinm are the usual Mie coefficients

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∆in )

Γin )

Pinchuk et al.

miψ′n(xi)ψn(mixi) - ψn(xi)ψ′n(mixi) miξ′n(xi)ψn(mixi) - ξn(xi)ψ′n(mixi) ψ′n(xi)ψn(mixi) - miψn(xi)ψ′n(mixi) ξ′n(xi)ψn(mixi) - miξn(xi)ψ′n(mixi)

(5)

Here, mi is the refractive index of the metal NP relative to the refractive index of the host medium, mh, xi ) kRi is a dimensionless parameter, Ri is the radius of NPs, k ) 2π/λ where λ is the wavelength, and ψ, ψ′ and ξ, ξ′ are respectively the Riccati-Bessel functions and their derivatives. The extinction cross section of a system of nanoparticles arbitrarily arranged in space is then given by11,12

Qext ) 2π

N

∞

n

n(n + 1)(n + m)!

i *i Re[ainmp*i ∑ ∑ ∑ nm + bnmqnm] 2 i)1 n)1 m)-n (2n + 1)(n - m)!

(6)

k

where N is the number of the nanoparticles in a cluster. The expressions derived for the extinction cross section allow for the calculation of the optical densities of NP aggregates, D ) nlQext. We calibrated this approach against experimental results in the regime of isolated core-and-shell clusters (χminor < 0.14) and for varying spacing between the NPs (from 6 to 20 nm). Because the exact positions of the shell NPs in the clusters are not known, we considered their “average” regular arrangements for different numbers of particles, here, tetrahedral, octahedral, cubic, and icosahedral. Figure 3a-d shows calculated absorption spectra for these canonical geometries and for various distances l between the centers of the constituent NPs. When l is decreased and the electromagnetic coupling between the NPs becomes stronger, (i) a notable red shift of SPR wavelength is observed for octahedral (Figure 3b), cubic (Figure 3c), and icosahedral (Figure 3d) clusters but not for tetrahedral (Figure 3a) ones and (ii) the absorption spectra are broadened substantially for cubic, octahedral, and icosahedral clusters and only marginally for the tetrahedral arrangement. Comparison of the calculated and experimental (Figure 3f) spectra at the upper limit shows the isolated-cluster regime (χminor ∼ 0.14) gives the best agreement for octahedral (N ) 7 NPs) clusters with interparticle spacing l ) 8 nm (Figure 3f). These parameters are physically reasonable since (i) the ratio of shell-to-core NPs in octahedral clusters is 6:1, corresponding to χminor ∼ 0.14, and (ii) the calculated 8 nm interparticle spacing is close to the experimental value of ∼8.4 nm (two NP radii, 5.2 nm, plus two times the thickness of the coating SAMs,1 ∼3.2 nm). 2.2. NPs with Different Metal Cores. When the cores of oppositely charged metal NPs are made of different materials, the aggregating clusters have a nonadditive and sometimes a counter-intuitive optical response (Figure 5). A striking example is the spectra of clusters, in which the “minority” (i.e., added) silver NPs are surrounded by “majority” gold NPs.2 In this case, the addition of silver NPs does not give rise to the Ag SPR band at 424 nm but instead causes a pronounced increase of Au SPR at 520 nm (Figure 5c,d). Interestingly, when the silver NPs are in excess (i.e., they surround added gold NPs), the spectra exhibit both SPR bands (Figure 5 e,f). There is also a small red shift and broadening of the SPR band around 520 nm caused by NP aggregation (Figure 5c). Slight discrepancies between theory and experiment in the shift of the SPR band and in its broadening are due to the approximations made in the model. To model such systems, we first note that exact solutions of eqs 4-6 accounting for the granularity of individual NPs is

Figure 4. Coordinate system used to calculate optical response of the NP aggregates. Here, E0, E1, and Es stand for the incident, internal, and scattered electric fields, respectively; ri, θi, φi are the spherical coordinates, and mh is the refractive index of the host medium. The incident light is assumed to propagate along the z axis. The resulting spectra are averaged with respect to the polarization of the incident light and the φ and ϑ angles, i.e., the incoming light is assumed to be unpolarized.

complicated by nontrivially matching the boundary conditions of spherical harmonics describing NPs of different types. While, in principle, such matching should be possible, we have considered a simpler and more physically intuitive model that reproduces the experimental observations. Specifically, we approximated the majority NPs as forming a continuous shell of thickness, Rout - Rin (Figure 5a), determined by the condition of mass conservation, R3out - R3in ) (N - 1)R3, where N - 1 is the number of NPs in the shell, and R is the NP radius. This assumption appears reasonable because the diameters of the NPs and the distances between shell particles surrounding the core NP are much smaller than the wavelength of the illuminating light; therefore, light should not “feel” the discreteness of the shell. To test this approximation, we first calculated optical spectra of discrete and continuous shells. This procedure is illustrated in Figure 5a for the case of an icosahedral shell of discrete AgNPs and a corresponding, same-mass, continuous shell; in this and other examples, both types of calculations give similar results, which confirms our assumption that the incident electromagnetic wave is insensitive to the granularity of the cluster because of the small size of the NPs as compared with the incident wavelength. Next, we extended this approach to “full” NP clusters in which the shell NPs surround a core particle. Such structures can be modeled by the Mie theory, modified for the core-shell geometry.13 The absorption cross section for the core-shell structure is given by

Cabs )

2π

∞

∑(2n + 1)Re{an + bn} 2 n)1

(7)

k

where the coefficients age given by

an ) ψn(y)[ψ′n(n2y) - ∆nχ′n(n2y)] - n2ψ′n(y)[ψn(n2y) - ∆nχn(n2y)] ξn(y)[ψ′n(n2y) - ∆jχ′n(n2y)] - n2ξ′n(y)[ψn(n2y) - ∆nχn(n2y)] bn ) n2ψn(y)[ψ′n(n2y) - Γnχ′n(n2y)] - ψ′n(y)[ψn(n2y) - Γnχn(n2y)] n2ξn(y)[ψ′n(n2y) - Γnχ′n(n2y)] - ξ′n(y)[ψn(n2y) - Γnχn(n2y)] (8)

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Figure 5. Core-and-shell approximation and the optical properties of two-metal clusters. (a) Calculated absorption spectrum of an icosahedral cluster consisting of N ) 12 silver NPs (of radii R ) 2.5 nm) and the corresponding spectrum of a spherosymmetric silver shell of internal and external radii Rint ) 3 nm Rout ) 6 nm, respectively. (b) Experimental absorption spectrum of AuMUA titrated with AgTMA NPs at the molar fraction xAg ) 0.14 (red line) compared with the calculated absorption of a core-and-shell particle (R ) 2.5 nm Ag core, 2 nm thick Au shell, blue line). For comparison, spectrum of six isolated AuNPs is also given (green line). (c) Experimental and (d) calculated spectra for the titration of AuMUAs with AgTMAs. (e) Experimental and (f) calculated spectra for the titration of AgMUAs with AuTMAs. In (d) and (f), the calculated spectra take into account contributions both from the nanoparticles aggregated into clusters and the “unused”, free nanoparticles.

In these expressions, ∆n and Γn are the usual Mie coefficients given by eq 5, n1 and n2 are the refractive indices of respectively the core and shell relative to the surrounding medium, x ) kRc and y ) kRsh, Rc is the radius of the core, and Rsh is the radius of the shell. Using these equations, we first calculated the spectra of a core-and-shell structure at a given titration point. This is illustrated in Figure 5b, and it shows good agreement between the experimental spectrum for the AuMUA/AgTMA titration at xAgTMA ) 0.14 and a calculated spectrum of a core-and-shell particle, in which the Ag core is surrounded by a 2.0 nm thick shell of silver (i.e., a shell of mass equal to that of six AuMUA NPs.

Next, we applied the core-and-shell approximation to calculate the UV-vis spectra during different stages of titration of either AuMUAs with AgTMAs (Figure 5c,d) or AgMUAs with AuTMAs (Figure 5e,f). In the former case, we assumed that each cluster had a shell corresponding to six AuMUAs. The total absorption of the mixture was then expressed as a weighted average of the absorption of the core-and-shell structures and any unaggregated AuMUAs (such “unused” NPs were present up to xAg ) 0.14). The spectra calculated in this way agreed with the experimental ones and reproduced the extinguishing of the Ag SPR band and the enhancement of Au SPR. Interestingly, these peculiar effects can be explained qualitatively

11822 J. Phys. Chem. C, Vol. 111, No. 32, 2007 by the effects of the gold shell on the wavelength of light reaching the core. Because the high refractive index of the gold decreases the effective wavelength of light reaching the Ag core by the factor of λ/nc, resonant excitation of silver shifts from 424 nm to ∼520 nm and coincides with the SPR band of gold. Thus, instead of two resonances, the spectrum has only one, an enhanced peak at 520 nm. Finally, we considered the titration of AgMUA with AuTMA NPs (Figure 5e,f). Here, in contrast to the titration of AuMUA with AgTMA, the UV-vis spectra exhibit both gold and silver resonances suggesting the formation of less dense shells around the core NP (Figure 5f, insert). Calculated absorption spectra of gold-core/silver-shell aggregates with a fixed radius of the core, Rc ) 2.5 nm and different shell thicknesses, d ) 0.2 ÷ 3 nm, indicate that the silver SPR band appears if the thickness of the gold shell is smaller than 1 nm and that the best agreement between theory and experiment is obtained with d ) 0.6 nm. This effective thickness of the shell corresponds to approximately four silver nanoparticles surrounding the gold core. Conclusions In this work, we modeled the optical properties of solutions of charged metal nanoparticles and compared them with experimental spectra. While the optical response of mixtures of like-charged, nonaggregating NPs can be accurately described by linear combinations of the components’ individual spectra, mixtures of oppositely charged aggregating NPs have nonadditive spectra and require higher-level theoretical treatments. In particular, if the NPs in the mixture have the same metal cores, the aggregates can be approximated as canonical clusters, whose electrodynamics can be described analytically by extended Mie theory. With mixtures comprising NPs of different metal cores, the exact analytical modeling is complicated. In this case, it is possible to use a simpler model in which granularity of the nanoparticles is neglected and the aggregates are approximated as spherosymmetric, core-and-shells.

Pinchuk et al. We hope this work will help formalize theoretical approaches to the modeling of NP structures assembled by electrostatic interactions. From a practical perspective, the ability to model the optical response of such aggregates can guide the design of “supraparticles” of desired optical properties. Extension of the presented methods to particles of different sizes and materials remains an interesting challenge for future work. Acknowledgment. This work was supported by the NSF Award No. CHE-0503673. B.A.G. gratefully acknowledges financial support from the Alfred P. Sloan Fellows Program and the Camille Dreyfus Teacher-Scholar Awards Program. A.O.P. and G.C.S. were supported by the Northwestern MRSEC (NSF Grant DMR-0520513) and by DARPA. References and Notes (1) Kalsin, A. M.; Paszewski, M.; Smoukov, S. K.; Bishop, K. J. M.; Grzybowski, B. A. Science 2006, 312, 420-424. (2) McFarland, A. D.; Van Duyne, R. P. Nano Lett. 2003, 3, 10571062. (3) Kalsin, A. M.; Pinchuk, A. O.; Smoukov, S. K.; Paszewski, M.; Schatz, G. C.; Grzybowski, B. A. Nano Lett. 2006, 6, 1896-1903. (4) Witt, D.; Klajn, R.; Barski, P.; Grzybowski, B. A. Curr. Org. Chem. 2004, 18, 1763-1797. (5) Kalsin, A. M.; Kowalczyk, B.; Smoukov, S. K.; Klajn, R.; Grzybowski, B. A. J. Am. Chem. Soc. 2006, 128, 15046-15047. (6) Thaxton, C. S.; Mirkin, C. A. Nat. Biotechnol. 2005, 23, 681. (7) Stoeva, S. I.; Lee, J.-S.; Thaxton, C. S.; Mirkin, C. A. Angew. Chem., Int. Ed. 2006, 45, 3303-3306. (8) Han, M. S.; Lytton-Jean, A. K. R.; Oh, B.-K.; Heo, J.; Mirkin, C. A. Angew. Chem., Int. Ed. 2006, 45, 1807. (9) Rosi, N. L.; Mirkin, C. A. Chem. ReV. 2005, 105, 1547-1562. (10) Alivisatos, P. Nat. Biotechnol. 2004, 22, 47-52. (11) Gerardy, J. M.; Ausloos, M. Phys. ReV. B 1982, 25, 4204. (12) Gerardy, J. M.; Ausloos, M. Phys. ReV. B 1980, 22, 4950. (13) Kreibig, U.; Vollmer, M. Optical properties of Metal Clusters; Springer: Berlin, 1995, 275. (14) Bohren, C. F.; Huffman, D. R. Absorption and Scattering of Light by Small Particles; Wiley & Sons Inc.: New York, 1998, 625. (15) Johnson, P. B.; Christy, R. W. Phys. ReV. B 1972, 6, 4370-4379. (16) Pinchuk, A.; Kreibig, U. New J. Phys. 2003, 5, 151. (17) Kalsin, A. M.; Grzybowski, B. A. Nano Lett. 2007, 7, 1018-1021.