Merging Past Century Theories into a Singular Scali - American

Article pubs.acs.org/JPCC

Cumulative Effect of Solvent and Ligand Dielectric around the Nanoparticles: Merging Past Century Theories into a Singular Scaling Equation Hirak Chatterjee and Sujit Kumar Ghosh* Department of Chemistry, Assam University, Silchar-788011, India ABSTRACT: The brilliant colors of dispersions of metallic colloids have been fascinating since antiquity, long before our understanding of light−matter interactions. The ability of noble-metal colloids to manipulate light at the nanoscale has opened up the emerging research area called plasmonics. Metals are considered to be either conductors in electronics or reflectors in optics. The Drude model of electron conduction, which is an application of kinetic theory to electrons in a solid, was proposed in 1900 by Paul Drude to explain the transport properties of electrons in metals. On the other hand, in 1908, Gustav Mie published his seminal work on the simulation of the color effects connected with colloidal gold particles using the classical Maxwell equations, which is popularly known as Mie scattering theory. The physical origin of light absorption by metal nanoparticles is the coherent oscillation of the conduction-band electrons, coined as localized surface plasmon resonance (LSPR). The resonance frequency of this LSPR is strongly dependent on the size, shape, interparticle interactions, dielectric properties, and local environment of the nanoparticles. In this article, we aim to elucidate the epicenter of the sensitivity of the localized surface plasmon resonance to the local dielectric environments around the ligand-stabilized gold nanoparticles that merges the Drude electron conduction model and Mie scattering theory proposed from two different perspectives in the historical achievements of scientific discoveries.

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INTRODUCTION The colors in nature have fascinated humans from the ancient days of civilization.1 From the artistic representations in the Paleolithic cave paintings of Altamira to modern impressionism, the colors of objects play a pivotal role in human creativity.2 The manifestation of unusual colors by colloidal particles has been exploited in ancient paintings and crafts; as colorants in glasses, ceramics, china, and pottery; and in curative and aesthetic applications.3 However, the historical landmark of scientific investigations of the significant variation of colors displayed by colloidal metal particles can be traced to the revolutionary insight of Michael Faraday in 1857 calling a solution “a beautiful ruby fluid” and noting that “a mere variation in the size of particles gave rise to a variety of resultant colors”.4 The physical origin of light absorption by metal nanoparticles is the coherent oscillation of the conduction-band electrons (just as water ripples travel along the surface of a pond after a stone is thrown into it and, therefore, have been explored as an intriguing surface− wave phenomenon) induced by the interacting electromagnetic field.5 Such strong absorption induces strong coupling of the nanoparticles to the electromagnetic radiation of light and called the localized surface plasmon resonance (LSPR). Surface plasmons (SPs) are coherent oscillations of conduction electrons on a metal surface excited by electromagnetic radiation at a metal−dielectric interface. The transfer of information by surface plasmons, that is, the merging of photonics and © XXXX American Chemical Society

electronics, has given birth to the concept of new physics for materials at nanoscale dimensions, referred to as plasmonics.6,7 Nanoplasmonics deals with the study of light−matter interactions that is based on the ability of small metallic particles to interact strongly with light of wavelengths significantly larger than their size.8 The spectral position and magnitude of the LSPR is strongly dependent on the size, shape, interparticle interactions, dielectric properties, and local environment of the nanoparticles.9−11 As the LSPR originates from the contributions of many electrons, the absorption and scattering cross sections of metal nanoparticles become very large, resulting in a high intensity of the plasmon resonance12 and its sensitivity to the local dielectric environment of the particles.13 The oscillation frequency is, critically, determined by four factors: the density of electrons, the effective electron mass, and the shape and size of the charge distribution. Real-time monitoring of the optical properties of metallic nanoparticles demands that the following parameters be taken into consideration: the shape and size of the particles, the presence of a supporting substrate or stabilizing ligand shell, the solvent dielectric continuum around the particles, and the electromagnetic interactions among particles that are close enough in the ensemble to influence the optical Received: May 30, 2017 Revised: September 11, 2017

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DOI: 10.1021/acs.jpcc.7b05243 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C properties.14 Therefore, for a sufficiently dilute (such that interparticle interactions can be disregarded) colloidal dispersion (devoid of any substrate), the optical properties of size- and shape-selective metallic particles are governed by the stabilizing ligand shell and the solvent medium in which the nanoparticles are dispersed. Under such constraints, the change in color of the nanoparticles is due to changes in the local dielectric environment arising from the specificity of interaction of the dispersed ligand-stabilized particles with the electromagnetic radiation.15 Recent advances in synthetic strategies, characterization techniques, the fundamental understanding of relevant theories, and methodologies for simulating plasmonic nanostructures have enabled researchers to study their optical properties under a variety of microenvironmental conditions and in a diverse range of niche applications that merge physics, chemistry, biology, materials science, and engineering on the nanoscale.16−18 A plethora of theoretical investigations, computational studies, and seminal experiments by eminent research groups throughout the world have illuminated the research arena of the dielectric sensitivity of the localized surface plasmon resonances of a variety of noble-metal nanostructures in colloidal dispersions. Underwood and Mulvaney19 described the synthesis and transfer of gold hydrosol using a polymeric comb stabilizer into various mixtures of organic solvents with different refractive indices. They investigated the change in color of the sol from red to purple depending on the refractive indices of the solvents and found the corresponding optical characteristics to be in excellent agreement with the predictions of Mie theory. Murray and co-workers20 reported that the surface plasmon absorption of alkanethiolate-protected gold clusters remains almost unchanged and concluded that dielectric environment of the particles is more influenced by the optical dielectric of the organic ligand shell rather than that of the bulk solvent. Murray, Mulvaney, and colleagues21 described the effects of the solvent refractive index on the surface plasmon absorbance of alkanethiolate-monolayer-protected gold clusters. A notable shift in the surface plasmon band position (∼8 nm) was seen as the solvent refractive index was varied from 1.33 to 1.55, in agreement with predictions of Mie theory. Kamat and coworkers22 studied the surface interactions of gold nanoparticles (Au NPs) with solvents following the shifts in the surface plasmon band position. They found that the plasmon absorption of the particles is influenced by the refractive index of the medium in the case of solvents with no active functional groups, whereas polar solvents with nonbonding electrons undergo direct complexation with the particle surface. Scrimin and colleagues23 studied the surface plasmon band of triethylene glycol-functionalized gold nanoparticles and reported that the presence of a protecting monolayer does not affect the optical properties of the gold core and that the shifts of the SPR band in different alcohols are in accordance with Mie theory. Ghosh et al.24 studied the effects of changing the dielectric properties of solvents and stabilizing ligand shell on cetylpyridinium chloride-stabilized gold nanoparticles dispersed in toluene. They found that the surface plasmon absorption maximum of Au NPs varies between 520 and 550 nm, depending on the refractive index and chemical nature of the solvent medium and the stabilizing ligand shell. Miller and Lazarides25 performed electrodynamic simulations of the spectra of Au NPs of various shapes and found that the sensitivity of the dipole resonance positions depends on the dielectric properties of the metal and the medium. Wang and

co-workers26 investigated the size- and shape-dependent refractive index sensitivity of gold nanoparticles dispersed in water−glycol mixtures of varying volume ratios and observed that the refractive index sensitivity increases as the gold nanoparticles become elongated and their apexes become sharper. Wang and co-workers27 also studied the effects of multipolar plasmon resonances on the refractive index sensitivity and figure of merit of noble-metal nanocrystals and noted that the dipolar resonances have the highest refractive index sensitivity of 336 nm RIU−1 (refractive index unit, which corresponds to the per unit change in the refractive index), whereas the quadrupolar resonances exhibit the largest figure of merit of 4.55. El-Sayed and co-workers28 investigated the effects of the dielectric constant of the surrounding medium on the surface plasmon resonance spectrum and the sensitivity factors of highly symmetric silver nanocubes. They determined that, when the dielectric medium is changed anisotropically, either by placing the particle on a substrate or by coating it asymmetrically with a solvent, the plasmon field is distorted and the plasmonic absorption and scattering spectra can shift depending on the conditions. Therefore, the rich literature enhanced by several research groups provides a compelling story about the health and prospects of refractive-index-mediated changes in the plasmonic sensitivity of noble-metal nanocrystals. Based on these perspectives, we have elucidated the epicenter of exquisite sensitivity of the localized surface plasmon resonance to local changes in the refractive index around the nanoparticles on the basis of the fundamentals of plasmonics that merges the Drude electron conduction model and Mie scattering theory proposed to account for two different perspectives of the metals. The present investigation thus offers a critical and analytical condensation to correlate the underlying physics behind the rationalization of the perspectives and designing a landscape toward the objectives in developing real-time plasmonic sensors.

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THEORETICAL PERSPECTIVES The intriguing optical properties of metals at the nanometer length scale have been known since antiquity, quite long before our scientific understanding of light−matter interactions. As in many disciplines of fundamental physics, chemistry, and biology, the past two centuries have witnessed significant contributions to the investigation of optical phenomena at the nanometer scale pioneered by specific theoretical approaches to solve Maxwell’s equations, together with powerful simulation techniques that are able to anticipate experimental observations. Theoretical predictions of the optical properties of metal nanoparticles have been obtained using traditional Drude free electron model29 and Mie scattering theory30 to explicate the appearance of the surface plasmon band position under different microenvironmental conditions. Drude Free Electron Model. According to the Lorentz model,31 in a dielectric material, the electrons are bound to the nuclei by quasi-elastic forces. Upon interaction with an electromagnetic wave, the bound electrons oscillate and generate a periodic dipole moment. Therefore, the movement of the bound electrons under the influence of an external electric field of a linearly polarized wave, E, can be described by the harmonic oscillator model as ⎞ ⎛ d2r dr me⎜ 2 + Γ + ω0 2r ⎟ = −eE dt ⎠ ⎝ dt B

(1) DOI: 10.1021/acs.jpcc.7b05243 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C where e is the charge and me is the effective mass of the electron, k = meω02k is the spring constant that describes the restoring forces due to electrostatic attraction, where ω0 is called the resonant (angular) frequency of the oscillator; and E = E0 exp(−iωt)E is the external electric field, where ω is the angular frequency of the time-varying electric field driving the oscillator. Because the dielectric dipole moment of the system is independent of the distance, r, between the nuclei and the vibrating electrons, an electron relaxation rate, Γ, was introduced into the equation of motion to incorporate losses due to scattering (emission of electromagnetic waves that carry energy away by the oscillating electrons) and other dissipation mechanisms (e.g., collisions between atoms). However, in the presence of a perfectly conducting medium, the electrons are free to move between the molecules and are said to be free electrons. Metals can be considered as free electron metals, so that their optical and electronic properties originate from their conduction electrons. The classical free electron model, proposed by Paul Drude in 1900, assumes the existence of an electron gas moving between the positive ions that form the crystal lattice of the metal.29 The electron gas can be treated as a classical ideal gas that can freely move with a common average relaxation time, τ. The drift motion of free electrons (meω02 = 0) in the presence of an external electric field can, then, be simplified to me

d2r dr + me Γ = −eE 2 dt dt

and gold when the frequency approaches the plasma frequency. This model describes the optical response of metals in the limit of zero losses when there is no damping of the free electron oscillations inside the metal due to the presence of and external electromagnetic field. The main advantage of using the Drude model is that it allows for the derivation of various simple analytical expressions for electrodynamic wave propagation in metals and at the metal−dielectric interface. Mie Scattering Theory. To understand the origin of the creation of localized surface plasmon resonances (LSPRs) in metallic particles at the nanoscale, one must turn to scattering theory. In 1908, German physicist Gustav Mie published his seminal work on the simulation of the color effects of colloidal gold particles through an exact analytical solution to the classical Maxwell equations that describes the scattering and absorption of light by spherical particles and is popularly known as Mie scattering theory.30 Mie theory is a mathematical-physical description of the scattering of a plane monochromatic wave by a homogeneous sphere immersed in a homogeneous medium for any particle radius and any material. It deals with the problem of the continuity of the tangential component of the total electromagnetic fields satisfying Maxwell’s equations outside and inside the sphere. Mie theory requires the wavelength-dependent dielectric constants of the particle and the surrounding medium as input parameters that can be described by their bulk optical dielectric functions. The dielectric constants of metals are strongly frequency-dependent and contain both real and imaginary parts, and their optical properties are best described by a complex, wavelengthdependent dielectric constant, ε(λ) = ε1(λ) + iε2(λ), where ε = m2 and m = n + ip is the complex refractive index given as a function of the refractive index, n, and the absorption coefficient, p. In the limit in which the particle diameter, d, is much smaller than the wavelength of light (d ≪ λ), which is often called the quasistatic approximation, only the dipolar term should be taken into consideration, and under these conditions, the extinction efficiency (σext) produced by a plane wave incident on a homogeneous conducting sphere is given by

(2)

where Γ is a phenomenological damping coefficient (Γ = 1/τ) that quantifies the dissipation of energy due to electron− phonon interactions, collisions of the electron gas with the lattice ions, lattice defects, and impurities. Now, when an electric current is passing through the conductor, a linear relationship exists between the current density (J) and the electric field (E), such that J = σE, where the electrical conductivity (σ) is given by

σ=

n 0 e 2τ me

(3)

σext =

where n0 = N/V is the number density of the electrons. When the electrons in the plasma are displaced from a uniform background of positive ions, an electric field will be established in such a direction as to restore the neutrality of the plasma by pulling the electrons back to the original positions. Because of their moment of inertia, the electrons will oscillate around their equilibrium positions with a characteristic frequency, known as the plasma frequency (ωp), which can be defined as

ωp2 =

n 0e 2 ε0me

(4)

ωp2 ω(ω + iΓ)

(6)

where V is the volume of the particle and εm is the dielectric function of the medium. From this expression, it is evident that the size of the effect is proportional to the volume of the particle and becomes stronger in higher-dielectric environments. It logically follows that the resonance condition for σext lies between the dichotomy and can be achieved when the function reaches the extremum condition. The extinction cross section will be maximized when the denominator in eq 6 is minimized, and the plasmon resonance occurs when ε1 = −2εm. This explains the dependence of the resonance properties on the dielectric constant of the metal particles and the surrounding environment because they polarize each other and the charge density has to adjust not only to the incident fields but also to the fields caused by polarization. The marked dependence of the extinction peak on the refractive index of the surrounding medium is the basis of sensing applications. The sensitivity to εm originates from the slope of the real part of the dielectric function in the observed wavelength range. Roughly, the real part of the dielectric function determines the position of the LSPR, and the imaginary part leads to the dephasing, that is, resonance peak broadening, that is observed. Now, assuming a

where ε0 is the permittivity of a vacuum. This equation thus offers an expression for the plasma frequency defining the conduction electron concentration of the bulk metal, where the bulk refers to materials that are sufficiently large compared to the wavelength of light in all three dimensions. The dielectric constant of the metal is given by ε(ω) = 1 −

24πεm 3/2V ε2(λ) λ [ε1(λ) + 2εm]2 + [ε2(λ)]2

(5)

Thus, it is important to emphasize that the Drude model is a crude approximation for noble metals, such as copper, silver, C

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with stirring for 30 min following the protocol of DiScipio,38 and each of the hydrosols was transferred to toluene with five individual long-chain amines of the same homologous series, namely, decylamine (DA), dodecylamine (DDA), tetradecylamine (TDA), hexadecylamine (HDA), and octadecylamine (ODA), according to the Brust method.39 The size distribution of the gold particles was found to remain unchanged upon transfer of the hydrosol to the organic medium. The gold particles prepared by this method were employed to study the effect of refractive index of the solvent medium and stabilizing ligand shell around the gold nanoparticles.

similar dielectric function for bulk ideal free electron metal (the Drude dielectric function) and for a metal nanoparticle, ωp2

ε(ω) = 1 − ω2 + iω Γ , one can use the real frequency-dependent coefficients, ε1(ω) from the Drude model of the electronic structure of metals ε1(ω) = ε∞ −

ωp2 ω2 + Γ 2

(7)

where ε∞ is the dielectric constant at infinite (sufficiently high) frequency when all of the polarization mechanisms die out, thus giving rise to its contribution from core-to-interband transitions. For visible and near-infrared frequencies, Γ ≪ ωp, eq 7 can be simplified to ε1(ω) = ε∞ −

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SIMULATION TECHNIQUES During the past few decades, numerical simulation techniques have been developed to solve Maxwell’s equations in understanding and manipulation of the optical response of small metallic particles. Different simulation tools, namely, MATLAB software package, discrete dipole approximation scattering (DDSCAT) code,32−34 N-MIE online tool35 and COMSOL finite-element method (FEM) solver36 have been used to perform the computations. The optical constants of bulk gold provided by Johnson and Christy37 have been used in all the calculations.

ωp2 ω2

(8)

The advantage of the Drude model is that it allows changes in the absorption spectrum of the metallic particles at the nanoscale dimension to be interpreted directly in terms of the material properties of the metal and the solvent medium. The power of this method is that the problem can be separated into two independent components: the electromagnetic one that can be fully solved and the material one that requires the determination of the dielectric constant. This approach enables the study of all important cluster effects through this phenomenological constant that can, indeed, be evaluated both by a theoretical approach and by experimental measurements.

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SOLVENT EFFECT Change in color of the nanoparticles in different solvents as the dispersion media has been amazing because of the specificity of the solvent-particle interaction with incident electromagnetic radiation.5,12 Ten different solvents were, judiciously, selected in such a way that there is a wide variation of refractive index and functionality to study the optical properties of three sizespecific gold organosols by measuring the LSPR spectrum of the metallic dispersion. Therefore, to establish a correlation, it is desirable to use nanoparticles with tight size distribution and stabilized by capping agents with the same type of functional groups. The gold hydrosols containing particles of various sizes (6 ± 0.5, 8 ± 0.7 and 10 ± 1.0 nm, designated as Au NPs@1, Au NPs@2 and Au NPs@3, respectively) were prepared by the reduction of choloroauric acid (0.25 mM) with different concentrations of sodium borohydride (ca. 0.25, 0.35, and 0.47 mM) by following the protocol of DiScipio38 and each of them was, separately, transferred to toluene with five individual long-chain amines, namely, decylamine (DA), dodecylamine (DDA), tetradecylamine (TDA), hexadecylamine (HDA) and octadecylamine (ODA) (5.0 mM) of the same homologous series according to the Brust method.39 The size distribution of the gold particles was found to remain unchanged upon transfer of the hydrosol to the organic medium. The gold particles prepared by this method were used to study the effects of the refractive index of the solvent medium and the stabilizing ligand shell around the gold nanoparticles. To gain clear insight into the dependence of the position of the surface plasmon band of the gold organosols, a highly concentrated toluenic dispersion of gold was suspended in a series of polar and nonpolar solvents (50 μM) in a well-stoppered quartz cuvette. All of the solvents were dried according to the literature method40 prior to the suspension of the gold nanoparticles. Then, the normalized LSPR spectra of the dispersions were measured by UV−vis spectroscopy. Because of the poor solubility of decylaminecapped gold nanoparticles in nonpolar solvents, these particles were not employed for the solvent effect study.

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EXPERIMENTAL SECTION Reagents and Instruments. All of the reagents used were of analytical reagent grade. Choloroauric acid (HAuCl4·3H2O); sodium borohydride (NaBH4); and five ong-chain amines of the same homologous series, namely, decylamine (DA), dodecylamine (DDA), tetradecylamine (TDA), hexadecylamine (HDA), and octadecylamine (ODA), were purchased from Sigma-Aldrich and used without further purification. Spectroscopic-grade solvents, namely, cyclohexane, toluene, o-xylene, n-heptane, dichloromethane, chloroform, dimethylformamide (DMF), methanol, acetone, and tetrahydrofuran (THF) were obtained from Sigma-Aldrich and were used for the spectroscopic measurements after purification. Solvents were dried according to published methods and freshly distilled before use, except for methanol, dichloromethane, and DMF, which were dried using a Puresolv solvent purification system.40 Doubly distilled water was used throughout the course of the investigation. Absorption spectra were recorded on a Shimadzu UV-1601 spectrophotometer (Shimadzu, Kyoto, Japan) in a 1-cm wellstoppered quartz cuvette, and solvent background was subtracted each time. Transmission electron microscopy (TEM) of the metal colloids was performed on a Hitachi H-9000NAR transmission electron microscope operating at 200 kV. The samples were prepared by mounting a drop of the solution on a carbon-coated copper grid and allowing them to dry in air. Synthesis of Size-Selective Gold Nanoparticles. Gold hydrosols containing particles of various sizes (6 ± 0.5, 8 ± 0.7, and 10 ± 1.0 nm, designated as Au NPs@1, Au NPs@2, and Au NPs@3, respectively) were prepared by the reduction of choloroauric acid (0.25 mM) with different concentrations of sodium borohydride (ca. 0.25, 0.35, and 0.47 mM, respectively) D

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results in Table 1 shows that the interaction of the solvents with the gold particles in terms of solvent polarity follows two different trends.19,21,22,24 It can, primarily, be summarized that solvents such as n-heptane, cyclohexane, toluene, o-xylene, and chloroform render a red shift in the LSPR maximum. In contrast, solvents such as tetrahydrofuran, dichloromethane, dimethylformamide, and acetone impart a blue shift, whereas methanol leads to an unprecedentedly high red shift of the LSPR maximum. A critical analysis of these observations is provided in Figure 1. Numerical simulations based on DDA predict a gradual red shift of the extinction spectra (Figure 1a) of a 6-nm bare gold nanoparticle immersed in solvent media with varying refractive indices. A schematic representation of a bare nanoparticle dispersed in the solvent medium conceived for DDA simulations and the Drude fit of the maxima of the simulated extinction spectra are shown in the left and right insets, respectively. Therefore, the linearity of the Drude fit cannot account for the irregularities in the observed LSPR spectra. The variation of the maxima of the LSPRs of gold nanoparticles immersed in different solvent media (Figure 1b) shows that the position of the plasmon absorption maximum depends on the refractive index of the surrounding medium. A critical analysis shows that the variation of the maxima of the LSPRs with the refractive indices of the nonpolar solvents can be treated within the framework of the Drude model.29 The linear variation of the plot of λ2 versus 2εm for gold nanoparticles immersed in different nonpolar solvent media (Figure 1c) shows that the position of the plasmon absorption

The effect of surrounding solvent medium on the plasmon resonance wavelength of the metallic nanoparticles can be explained within the framework of the Drude model.29 The bulk metal plasmon wavelength (λp) is defined by λp =

2πc ωp

(9)

where c is the speed of light in a vacuum. Upon substitution of the frequency terms in eq 8, the real part of the dielectric function can be expressed in the form ε1 = ε∞ −

λ2 λp2

(10)

Therefore, according to the Drude model of electronic structure of metal and invoking the resonance condition, ε1 = −2εm, the surface plasmon peak position, λ, is related to the dielectric constant of the surrounding medium (εm) by the expression λ 2 = λp2(ε∞ + 2εm)

(11)

Although eq 11 represents only a qualitative theory, it does reveal that the plasmon wavelength varies roughly linearly with the index of refraction of the surrounding solvent. The positions of the surface plasmon band maxima (λ) of three different sizes of HDA-capped gold nanoparticles with variations in three plausible physical parameters, including the refractive index (n) and conductivity (S/m), as taken from the literature,41 are presented in Table 1. A close inspection of the

Table 1. Summary of the Shifts of the LSPR Maximum with Solvent Parametersa for Three Size-Specific Gold Nanoparticles Capped with Hexadecylamine λ (nm) of HDA-capped Au NPs

a

solvent

refractive index (n)

methanol acetone n-heptane tetrahydrofuran dichloromethane cyclohexane dimethylformamide chloroform toluene o-xylene

1.3246 1.3570 1.3850 1.4040 1.4211 1.4240 1.4270 1.4459 1.4940 1.5054

conductivity (S/m) 1.5 5.0 1.0 4.5 4.3 9.0 6.0 1.0 8.0 8.0

× × × × × × × × × ×

10−7 10−7 10−14 10−3 10−9 10−16 10−6 10−8 10−14 10−14

set 1

set 2

set 3

556.0 515.5 516.0 517.5 514.0 520.0 520.5 524.5 521.5 528.5

566.0 517.0 517.0 520.5 516.5 517.0 521.0 526.0 523.0 528.5

568.0 518.0 519.0 522.5 518.0 520.5 522.5 527.5 525.0 530.0

Solvent parameters were taken from ref 41.

Figure 1. (a) Simulated extinction spectra (normalized) based on DDA for 8-nm gold particles dispersed in different solvents, (b) experimental surface plasmon band (normalized) of Au-HDA NPs@2 dispersed in different solvents, and (c) plot of λ2 as a function of 2εm of three size-specific HDA-capped gold nanoparticles dispersed in nonpolar solvents. Insets in panel a show (left) a schematic of a bare nanoparticle dispersed in the solvent medium conceived for DDA simulation and (right) the Drude fit of the maxima of the simulated extinction spectra. E

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on the electromagnetic perspective of environmental dielectric response, we solved Maxwell’s equations with an FEM solver to obtain eigenstates for dipolar modes at the experimentally observed LSPR frequencies, employing the bare gold nanoparticle surface as the boundary element. Figure 2 shows the electric field distribution patterns of an 8-nm gold particle dispersed in 10 different solvents in ascending order of electrical conductivity (Figure 2a), the dipolar modes in ascending order of out-of-plane propagation constant [Re(k)] using the optical conductivity of gold in the terahertz frequency range (Figure 2b), the electric field as a function of the refractive index of the solvents (Figure 2c), and the effect of solvent permittivity (εs) on the quality factor (Q), which is defined as Im(k)/[2Re(k)] at the LSPR frequency (Figure 2d). The general pattern of the electric field distribution (as shown in the inset in Figure 2d) around a nanoparticle (using the dielectric field of toluene) obtained by FEM simulation offers an estimation of the local electric field strength in the vicinity of gold particles due to a change in the refractive index of the solvents. It can be seen that the values of the Q-factor obtained from the FEM analysis decrease almost linearly with the permittivity of the ambient medium. The lowest values of both Re(k) and Im(k) correspond to the exceptional decrease in Q-factor for methanol, accounting for the broad and red-shifted band observed experimentally. The change in the surrounding solvent medium also influences the plasmon bandwidth, as shown in the experimental spectra in Figure 1b. It is noted that, for polar

maximum varies according to the Drude model. It is obvious from the expression of the Mie extinction cross section (eq 6) that changing the solvent to a medium with a markedly different refractive index surrounding the NPs strongly alters the plasmon behavior of the particles and plays a predominant role in determining both the position and intensity of the plasmon peak.30 On the basis of these results, solvent−particle interactions can be grouped into two general categories: (i) solvents that alter the refractive index surrounding gold nanoparticles and (ii) solvents that can coordinate with the gold surface. Therefore, it can be inferred that the manifestation of the LSPR spectra due to solvent−particle interactions becomes a myriad combination of refractive indices and alterations of the electron density due to surface coordination with the gold particles.21,22,24,42 From the plot of λ2 versus 2εm for three different particle sizes in nonpolar solvents, the bulk plasma wavelengths (λp) are ca. 129, 127, and 125 nm, and the high-frequency dielectric constants (ε∞) are estimated to be 12.4, 12.7, and 13.3 for Au NPs@1, Au NPs@2, and Au NPs@3, respectively, which are not far from other estimates for bulk gold (λp = 131 nm, ε∞ = 12.2).29,43 Bare nanoparticles in a solvent cage become polarized in incident electric field. The position of this resonance red shifts with an increase in the magnitude of the dielectric constant of the medium surrounding the NP as a result of the buildup of polarization charges on the dielectric side of the interface, which is responsible for weakening the restoring force. Based

Figure 2. Finite-element simulation of 8-nm gold particles dispersed in different solvents showing (a) the electric field distribution pattern in ascending order of electrical conductivity; (b) the dipolar modes in ascending order of out-of-plane propagation constant, Re(k), using the optical conductivity of gold in the terahertz frequency range; (c) the electric field as a function of solvent refractive index, and (d) the effect of solvent permittivity on the Q-factor (Q). Inset in panel d presents the general pattern of electric field distribution. F

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(τ).44,45 Because methanol is highly polar, it gives rise to a strong dipolar field at the gold nanoparticles, and thereby, using plasmon ruler equation,46 Δλ/λ0 = ae−x/τ, it is quite certain that this solvent will exhibit the highest plasmon shift. Interband charge transfer plays an important role in determining the dielectric contribution to the optical width of the observed plasmon peak position.47 Contributions of electron−electron, electron−phonon, electron−surface, and electron−defect coupling lead to higher relaxation times in nanostrucutures observed at the femtosecond scale. 47 The surface electron contribution of the nanospheres is noticeable in the case of polar solvents, and the effect on the Drude forbidden plasmon shift can be elucidated in this context. The Drude effect is well pronounced in nonpolar solvents but not obeyed in polar solvents. Through FEM calculations, we found a linear decrease in the k-vector with increasing field strength for polar solvents only, which was correlated with the damping constant to explain the experimentally observed enhanced broadening in LSPR peaks with polar solvents. The hot electrons are cooled efficiently by polar solvents, which can provide extra stability by taking up the excess surface charge. Thus, it is convenient to elucidate the effect of the solvent dielectric constant on the plasmonic nature of noble-metal nanoparticles, simply by verifying the Drude relationship through the alteration of the solvent dielectric constants of the respective dispersion media. However, it was found that the model actually predicts the overall trend with a nonlinear shift of the plasmon peaks that increases with the solvent dielectric constant. The contribution from the solvent dielectric to Mie scattering calculated by DDA simulations also fails to explain the spectral pattern observed experimentally. Therefore, from the viewpoint of scattering phenomena, an increase in the refractive index of the solvent contributes to the red shift of the LSPR maximum. On the contrary, the conductance of the solvents determines the local electric field intensity around the solvent−particle interface that also contributes to the optical absorption and determines the overall plasmonic shift of bare metal nanoparticles as a result of the collective effects of the two parameters. It is, therefore, apparent that both the solvent refractive index and the conductance should be taken into consideration as variables in explaining the plasmonic features of colloidal dispersions of metallic nanoparticles.

solvents, namely, methanol, DMF, and dichloromethane, a regular shift in the plasmon width is seen, whereas in the case of nonpolar solvents, namely, chloroform, cyclohexane, toluene, o-xylene, and n-heptane, a regular shift in the plasmon peak is observed, as discussed earlier. The breadth of the LSPR spectra depends on the damping due to surface electron scattering. The damping constant (Γ) is given by

Γ=

2ε2bulk δε1bulk

2 +

ωp2 ω3

δε1bulk

δω

AvF leff

δω

(12)

which can be approximated as Γ ≈ Γbulk +

AvF leff

(13)

The constant A comprises both bulk and interface effects as A = Abulk + Aeff

(14)

The dependence of Aeff can be assigned by the charge flow through the density of states around the Fermi level of nanocluster (ρa) as Γ

ρa =

1 2 π (ϵ − ϵ ) + a

2

( Γ2 )

(15)

where ϵ and ϵa denote the energy of the particle and an adsorbate, respectively. The group delay time (Yg) of the surface electron scattering due to the presence of the stabilizing ligand shell around the nanoparticles is related to the propagation constant (k) as Yg = −

λ 2 dk 2πc dλ

(16)

which shows that, with a decrease in the k value, Yg increases, and the breadth of the LSPR bandwidth decreases. The FEM analysis clearly shows that, for the strongly polar solvents, the propagation constant (k) decreases exponentially with field strength. Because the damping constant (Γ) also increases with the propagation constant, the bandwidth decreases with the wavelength, thus explaining the appearance of the broad LSPR band observed for the methanolic dispersion of gold nanoparticles. Anomalies in the LSPR profile for gold nanoparticles in methanol arise in two respects, namely, an abnormal shift and an unexpected breadth of the LSPR peak. To account for the unexpectedly high LSPR peak shift, we explicitly modeled a bare nanoparticle surrounded by solvents with the finiteelement method to elucidate the field strength and propagation constant (k), which clearly indicates that the higher the value of the electrical conductance of the solvent, the smaller the value of the propagation constant; consequently, the lowest value of the propagation constant was obtained for the methanol suspension. The high conductivity (σ) of methanol leads to a greater attenuation (α) at near field, as α = ωμσ , where ω is the angular frequency of the time-varying electric field driving the oscillator and μ is the dipole moment, and this causes greater electromagnetic loss from the gold−methanol interface. The experimental results also show a broad LSPR line width for the methanol suspension, which can account for the chemical interface effect in plasmon dephasing giving rise to a high group delay time (Yg), which, in turn, increases the relaxation time

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LIGAND EFFECT The stability of metallic nanoparticles in the solution phase is an important issue, given that bare metal nanoparticles are unstable in organic solvents, and therefore, some form of stabilizer is necessary to prevent particle aggregation.20−22,24,48 The encapsulation of the particle core with an appropriate shell material offers a means of protection from the surrounding environment. When the nanoparticle is small, the dimensions of the ligand layer are comparable to the nanoparticle size, which causes the ligand layer to be highly disordered, which is, in turn, reflected in the field distribution around the nanoparticle.49 However, for larger nanoparticles, one could envision that the field would be less perturbed by the presence of the ligand layer because of the larger relative dimensions of the nanoparticle and could be described as a homogeneous layer with a small real refractive index.49 Therefore, it was assumed that the ligand layer can be described by a macroscopic homogeneous dielectric constant for the computation of their optical properties using classical electromagnetism. These types of nanoparticles are commonly referred to as “core−shell” particles, G

DOI: 10.1021/acs.jpcc.7b05243 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C consisting of a metal core with a dielectric shell. However, a dense shell of stabilizing ligand, such as surfactant, polymer, or alkanethiol, will alter the refractive index at the particle surface and, eventually, form a thick protective skin from the incoming radiation to perturb the surface electrons.21 The presence of the monolayer shell is also polarized by light and sets up a dipole that augments the core polarization charge and, as a consequence, adds a new level of complexity to the electromagnetic modeling of the LSPR band due to the asymmetric environment. Moreover, good stabilizers, such as alkanethiolates, form strong covalent bonds with the coordinatively unsaturated gold atoms at the particle surface through back π-bonding from the sulfur, making the surface plasmon band more damped than simply predicted by Mie theory. A more accurate description of the extinction cross section, σext, of the monolayer-protected gold clusters considered as dipole oscillators can be expressed on the basis of electrostatics as12,50,51

Assuming the monolayer shells to be thin relative to the metallic core, g ≪ 1, the condition of resonance becomes εc = −2εm −

⎡ 2g (εs − εm) ⎤ λ 2 = λp2⎢(ε∞ + 2εm) + ⎥ ⎣ ⎦ 3

where εc is the optical dielectric function of the metallic core and εm is that of the surrounding medium. The medium is assumed to be nonabsorbing so that εs is the dispersionless optical dielectric function of the shell layer. The volume fraction of the shell layer, g, is defined by21 [(R core + R shell)3 − R core 3] (18)

and the size parameter, x, can be expressed as

x=

2πRεm1/2 λ

(19)

where R is the radius (sum of the radii of the core and the monolayer chain) of the ligand-stabilized particle. The condition of plasmon resonance for such ligand-stabilized particles arises when the denominator in eq 16 becomes zero, that is εc = −2εs

[εsg + εm(3 − g )] [εs(3 − 2g ) + 2εmg ]

(22)

A closer analysis of eq 22 reveals two important consequences: First, as the ligand chain length increases, the volume fraction of the shell layer (g) increases and shifts the plasmon band position, λ, to higher wavelengths. Second, as the dielectric constant of the ligand shell layer (εs = ns2) increases, the plasmon band position, λ, is also shifted to higher wavelengths. Moreover, when the solvent and ligand shell have very similar refractive indices (εs = εm), the contribution of the additional term becomes zero, and the particles can be treated as if they were in a homogeneous solvent. Under other conditions (εs ≠ εm), the distribution of the plasmon field around the particle is no longer homogeneous and is, thus, sensitive to asymmetric dielectric perturbations.52 It is, therefore, important to understand the effect of the ligand chain length by considering the contribution of the dielectric of the organic shell and the complexation interaction with the particle surface to the LSPR spectrum of the metal colloids. To study the effects of a stabilizing ligand shell on the LSPR of gold nanoparticles, we selected five homologous long-chain amines with subsequent variations in a pair of methylene (−CH2) units, namely, decylamine, dodecylamine, tetradecylamine, hexadecylamine, and octadecylamine, to detect the LSPR changes induced by binding of these analytes to the gold colloids. In this case, the LSPR spectrum of each solution was measured upon binding of these analytes to Au NPs@2 in toluene. Figure 3 shows the normalized extinction spectra simulated with the N-MIE online tool (Figure 3a) and the experimental normalized surface plasmon bands of toluenic dispersions of Au NPs@2 capped with ligands of varying chain length (Figure 3b). The inset in Figure 3a shows a schematic representation of a ligand-stabilized nanoparticle dispersed in a

(17)

(R core + R shell)3

(21)

Upon invoking this resonance condition, the surface plasmon peak position (λ) for monolayer-protected metallic clusters is modified by

⎡ (ε − εm)(εc + 2εs) + (1 − g )(εc − εs)(εm + 2εs) ⎤ σext = 4 Im⎢ s ⎥ ⎣ (εs + 2εm)(εc + 2εs) + (1 − g )(εc − εs)(2εs − 2εm) ⎦

g=

2g (εs − εm) 3

(20)

Figure 3. (a) Extinction spectra (normalized) simulated with the N-MIE online tool and (b) experimental surface plasmon band (normalized) of the toluenic dispersion of Au NPs@2 capped with ligands of varying chain lengths. Inset in panel a shows a schematic representation of a ligandstabilized nanoparticle dispersed in the solvent medium conceived for N-MIE simulation. H

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Figure 4. (a) Change in Murray volume fraction parameter (g) with the concomitant variation in core radius and shell thickness of three size-specific Au NPs capped with different ligands and (b) comparison of the observed LSPR maximum with the Murray-modified Drude model of Au NPs@2 capped with five different ligands. The scale difference between the calculated and observed data corroborates that the change in the LSPR shifts over two different wavelength regimes.

solvent medium conceived for N-MIE simulations. It can be seen that the maximum of the LSPR spectrum is gradually shifted to the red with increasing ligand chain length whereas an unprecedentedly high red shift is noted for tetradecylamine as the ligand in the absorption spectral features of the metallic particles. We tried to correlate the effects of ligand binding through theoretical and simulation results and experimental observations (Figure 4). Figure 4a shows the change in Murray volume fraction parameter (g) with the concomitant changes in core radius and shell thickness. For a common set of core radii, if the refractive indices of the ligands remain almost constant with increasing ligand chain length, the volume fraction of the shell layer (g) decreases almost exponentially. However, when the core radius increases for a constant ligand chain length, a steep increase in the g value is noted. The increase in g is more prominent for smaller nanoparticle cores and decreases with increasing core radius. Therefore, the volume fraction of the shell layer (g) is a rapidly changing function for smaller nanoparticles (R ≤ 10 nm), but a limiting condition appears in the case of larger core radii where the volume fraction parameter becomes almost constant and tends to converge toward unity. Figure 4b shows a comparison of the LSPR maxima of Au NPs@2 between experimental observations and theoretically calculated values from the Murray modification of the Drude model as a function of ligand chain length. It can be observed that both theory and experiment show similar patterns; the differences between the observed results and the calculated values are smaller for the smaller ligands, and the experimental and theoretical results converge toward each other when the ligand size further decreases. The range of values of LSPR shifts obtained from experiments and the range of values predicted by the Drude− Murray model are quite different. The deviation of the Drude− Murray-predicted LSPR wavelengths from the experimentally observed values occurs, presumably, because of the presence of inhomogeneous adsorption of ligand layers instead of a singular monolayer considered in the model and the modification of the nanosphere surface states as a result of the chemisorption of ligands. Although there is a considerable scale difference between the calculated and observed data, it was found that they both exhibit a nonlinear trend of the LSPR profile with the ligand chain length. Thus, we aimed to corroborate these

changes over two different wavelength regimes, eventually leading to a singular framework of scaling equation. It is evident that the dense shell of stabilizing ligands provides a dielectric coating on the particle surface, amounting to a change in the dielectric constant of the medium and causing a red shift of the λmax value of the LSPR. Because of the presence of coordinatively unsaturated orbitals of the surface atoms at the metal cluster, amine molecules are bound to the gold surface through the headgroups and are connected to the outer layer through hydrophobic interactions.53,54 It is wellknown that the refractive index of the ligands increases almost linearly with increasing chain length; therefore, it is expected that, as the chain length of the amine molecules increases, the band position should gradually shift toward higher wavelengths.48 However, the salient feature of the physical significance is that the TDA-stabilized particles exhibit an irregular trend in the LSPR maximum. This is due to the fact that hydrocarbon monolayer with the C14 chain is partially compressed and, hence, provides an effective shield for the metal core from interactions with the solvent molecules. For the alltrans zigzag configuration, Bain et al.55 showed that the length of the alkyl chain (l) can be represented by the equation l (nm) = 0.25 + 0.127q

(23)

where q is the number of methylene units in the alkyl chain. However, a significant deviation from this linearity occurs for q = 14; for alkylamines with q = 8, 10, and 12, the ligand chain lengths are 1.27, 1.52, and 1.77 nm, respectively, whereas for q = 14, because of the high chain end gauche interaction, a pseudorotational twist occurs in the stable zigzag conformation and the ligand chain length becomes compressed to ∼1.8 nm, which increases its density as well as its refractive index in comparison with those of its homologues. This effect of ligand coiling is also responsible for the unexpected decrease of the interparticle distance (∼1.2 nm),56 which accounts for the more compact ligand packing around the nanoparticles. Assuming that free electrons act as a classical ideal gas, the Drude macroscopic free electron model can account for the optical properties of bare metal nanoparticles dispersed in a solvent medium.29 However, because bare metal nanoparticles are unstable in organic solvents and some form of stabilizer is necessary to prevent the particle aggregation, toward a more I

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where μ′ = μ/(1 − αf), where μ is the dipole moment, α is the mean isotropic polarizability of the solute, and f is called the reaction field factor and is given by

rational approach, Murray generalized the Drude model for “coated” metal nanoparticles dispersed in a solvent medium.21 However, further modification of this model is necessary to illustrate the dielectric sensitivity around nanospheres stabilized by a ligand shell, whose thickness and organization pattern depend on the stereochemical properties of the chain. However, the polarization of the stabilizing ligand shells by the solvent environment can arise in two ways, either through solvent dielectric field or through electromagnetic field of incident radiation, which should also be taken into consideration. The polarization induced at the metal core was elucidated by the investigation of Silsbee,57 who calculated the frequency-dependent dielectric function of a damped oscillator placed at the center of a spherical cavity in a homogeneous dielectric medium. According to this treatment, the dipole moment (μ) and electric field (E) of a single point dipole inside the center of the spherical cavity of volume V and dielectric constant ε can be given by ⎛ 1−ε ⎞ ⎟VE μ = ⎜3ε ⎝ 2ε + 1 ⎠ ext

E in =

3ε Eext 2ε + 1

f=

(27)

where ε0 is the permittivity of the vacuum. Now, the electric polarization (P) on a dielectric substance placed in an electromagnetic field can be written as, P = n0′ αε0ELoc, where n0′ is the number density of the molecules and ELoc is the local electric field experienced by an individual molecule.60 Considering Onsager−Böttcher theory of nonpolar solutes, ELoc depends on the average volume of the cavity in which the nanoparticles are placed and the dielectric constant of the medium, which remain constant under the experimental conditions. Considering a constant ELoc value for a particular size of the particles encapsulated with different ligand shells, electrical polarization (P) becomes a linear function of the mean molecular polarizability (α). Following these conditions, the value of the effective ligand polarizability (α) for each ligand can be calculated from the Lorentz−Lorenz equation as31,61

(24)

αρ n2 − 1 = 2 3 n +2

(25)

where Eext is the external electric field and Eint is the induced electric field inside the particles. Now, the cumulative effect of the solvent dielectric environment and incident electromagnetic field can be explained within the framework of the Onsager−Böttcher dielectric continuum model, which is based on the concept of a macroscopic reaction field.58,59 The success of various dielectric continuum models is embodied in the constraints imposed on the “local electric field” experienced by the solute due to the dielectric continuum. Traditionally, Onsager−Böttcher theory has most frequently been employed to interpret experiments, as the reaction field is reasonably well-predicted by this theory, provided that the cavity radius is adjusted to reproduce the exact value at small polarizability. According to Onsager− Böttcher theory, the reaction field (R) acting on a solute molecule in a spherical cavity of radius a in a dielectric continuum with permittivity ε can be expressed by the formula R = fμ′

1 2(ε − 1) 1 4πε0 2ε + 1 a3

(28)

where n and ρ are the refractive index and density, respectively, of the ligand molecules. Figure 5 shows the consequence of effective ligand polarizability on the LSPR maximum of the ligand-stabilized gold nanoparticles. Figure 5a shows that the effective ligand polarizability (α) varies linearly with effective refractive index of the ligand molecules. The effect of ligand interaction on the LSPR maximum has been explored through the plot of LSPR maximum (λLSPR) as a function of ligand polarizability (α) for three different sizes of gold nanoparticles as shown in Figure 5b. The inset in Figure 5b shows an enlarged view of the profile of λLSPR as a function of ligand polarizability for Au NPs@2 capped with different ligands excluding tetradecylamine. Halas and colleagues62 introduced the idea to core−shell nanoparticles from molecular orbital perspectives, where electronic transitions take place between bonding and antibonding energy levels upon interaction of the electron-donating ligand molecules to the coordinatively unsaturated orbitals on

(26)

Figure 5. (a) Effective ligand polarizability as a function of effective refractive index of five different ligands and (b) observed LSPR maxima (λLSPR) of three size-specific gold nanoparticles as a function of ligand polarizability of five different ligands. Inset in panel b shows an enlarged view of the profile of λLSPR as a function of ligand polarizability for Au NPs@2 capped with different ligands excluding tetradecylamine. J

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CUMULATIVE EFFECT OF SOLVENT AND LIGAND In the previous sections, we discussed the explicit contributions of the solvent and ligand dielectric around the nanoparticles. However, in practice, while ligand-stabilized nanoparticles are dispersed in a solvent medium, the overall contribution is not the simple sum of their individual contributions. To circumvent this insufficiency, we studied the cumulative effect of the solvent and ligand dielectric to offer a complete description of the influence of the local dielectric environment around the nanoparticles. In this regard, we performed computations using the concentric Mie scatterer model in which a metal core is wrapped with another spherical shell that makes a continuum of the ligands surrounding the nanoparticles and immersed in toluene as the solvent medium. The LSPR extinction efficiencies of particles dispersed in toluene and capped with homologous amine ligand series were calculated with the N-MIE online tool. A comparative study among the experimentally observed, theoretically calculated, and simulated LSPR maximum of the Au NPs@2 immersed in toluene as a function of ligand chain length and refractive index is summarized in Table 3. The salient feature of physical significance is that the LSPR maxima calculated from the N-MIE simulations is contrary to the theoretical calculations based on the Drude−Murray equation but shows a pattern similar to that of the experimental results with a different scale factor. However, neither Drude− Murray calculations nor the N-MIE tool predicts the results observed, and the latter merely shows a similar trend. Therefore, taking into account the scale effect, an empirical formula could be designed for the LSPR shift for a concomitant change in the solvent refractive index and the chain length of the homologous amine ligand series as

the surface of the metal nanoparticles. This concept also rationalizes the relative lowering of the energy due to the dielectric continuum of the ligand shell around the nanoparticles. It is found that the increase in λLSPR with ligand polarizability can be fit to a generalized logistic Gompertz function, where growth is slowest at the start and end of a time period.63 The fitting equation is given by ln|λLSPR | = ln a − e−k(α − αc)

(29)

The constants a and k represent the upper convergence limit and rate of “growth” in the LSPR wavelength (λLSPR), respectively; αc is the critical ligand polarizability, which represents lowest possible threshold value of ligand polarizability, where growth of λLSPR is the slowest. A curve with a similar nature was seen for three size-specific gold nanoparticles. With the increase in the value of αc and with the condition that α > αc, the overall spread increases. To fit a broad range of data, the asymptotic function becomes slower with concomitant decrease in the values of constants so that the function approaches linearity. The flattening of the curves increases with the increase of nanoparticles radii, showing a high sensitivity of smaller nanoparticles toward the dielectric environment. Moreover, the values of k decrease with increasing particle size; the fitting parameters for different sizes of gold nanoparticles are listed in Table 2. Table 2. Fitting Parameters for λLSPR versus Ligand Polarizability sample

a

k (×1024)

αc (×10−23 cm3)

set 1 set 2 set 3

530.50 532.50 533.92

1.105 1.460 1.510

2.788 3.115 2.978

Article

obsesrved NMIE Drude Δλ ligand − ligand = k ligand |Δλ ligand − ligand | + 40Δλ ligand − ligand j

i

i

j

i

j

i

(30)

Therefore, the incorporation of the contribution of the dielectric of the stabilizing ligand shell could be considered a pioneering landscape for elucidating ligand binding events on the LSPR band of the metallic nanoparticles. However, the significant deviation in the experimental observations points to the consideration of some additional ligand characteristics, namely, ligand polarizability and the refractive index of ligands. For the model of a point dipole inside a polarizable ligand core, the Gompertz plot has been employed to model the interaction energy of a ligand shell with a nanoparticle core and the distribution of interaction energy over a critical polarizability. Penetration of incident electromagnetic waves polarizes the ligand shell, and to account for the field, the Onsager−Böttcher theory of point dipole trapped in a dielectric environment is taken into consideration. It can be seen that the upper convergence limit of the polarizability increases with increasing particle radius and the formation of larger polar shell around nanoparticles by longer ligand chains, and the resultant model satisfies experimental observations at a point-to-point level of accuracy.

where the value of kligand,i depends on the value of Δni, where Δni = nligand − nsolvent, and an excellent fit for kligand,i versus Δni was obtained by setting the function as kligandi = −1.287 + 0.036e−Δni/0.01142, as shown in Figure 6. The inset shows comparative profiles of experimentally observed, theoretically calculated, and numerically simulated LSPR maxima as functions of the ligand chain length. The Drude−Murray calculations are dependent on the classical kinetic theory of electrons, where the plasma frequency is considered to be distributed among the core-to-shell volume fraction, and the N-Mie calculations depend on the thickness of the coating, size, and dielectric properties of the core and shell, respectively. According to Drude theory, the plasma frequency is given by the equation ωp =

Ne 2 ε0me

, where N is the bound surface free

electron density responsible for intraband charge transfer, e is the electronic charge, ε0 is the dielectric constant of free space, and me is the effective mass of an electron, and the surface

Table 3. Experimentally Observed, Theoretically Calculated, and Simulated LSPR Maxima for Au NPs@2 ligand

chain length (nm)

refractive index

λLSPR (nm) (observed)

λLSPR (nm) (Drude−Murray)

λLSPR (nm) (N-MIE)

DA DDA TDA HDA ODA

1.80 2.08 2.31 2.57 2.83

1.4369 1.4377 1.5055 1.4522 1.4414

522.0 522.5 532.0 523.0 523.5

532.9 33.0 535.1 533.5 533.2

529.0 529.5 531.0 528.5 529.5

K

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velocity overestimates of the value of λp, which might result in the shift of the resonance condition toward the lower-energy end of the spectrum as a constant error of 10 nm. To account for these irregularities, finite-element simulations of the ligand-stabilized nanoparticles dispersed in an ambient solvent medium with defined relevant physical parameters, namely, conductance, refractive index, and relative permeability, were performed, as shown in Figure 7. A pictorial depiction of the model showing a ligand-stabilized nanoparticle dispersed in the solvent medium is presented in Figure 7a. Now, the modified Mie scatterer model developed by Fleury65 was used to illustrate the localized electromagnetic field strength. Considering the ligand shell as a perfect electric conductor, it was found that the electric field at the ligand−solvent interface increases linearly with solvent permeability, with a regular increase in slope with increasing core size, as shown in Figure 7b. Next, we studied the relative peak shift as a function of refractive indices of the ligands attached to three different sizes of gold nanoparticles; the slopes of the curves give the refractive index sensitivity (mi) of each size of gold particles. Through a plot of these values as a function of ligand chain length for each size of nanoparticles (Figure 8), the optical sensory response (R) for each size of nanoparticles was obtained. It was found that the value of the optical sensory response decreases almost exponentially (as shown in the inset) with increasing size of the nanoparticles. Campbell and co-workers66 established that, for a flat metal film, the decay of the penetrating electric field at a height z above the metal surface is an exponential function of (−2z/ld), and the corresponding change in effective refractive index (neff) is measured by the depth integral as

Figure 6. Modeled exponential fit for kLigand,i vs Δni for Au NPs@ 2 dispersed in toluene. Inset shows a comparative study of the results of experimental observations, theoretical calculations based on the Murray-modified Drude model, and simulations based on the N-MIE tool for the LSPR maximum as a function of ligand chain length.

plasmon resonance condition is given by ωsp =

ωp2 1 + 2εm

− Γ2 .

Upon substitution of ω = 2πc/λ, this equation becomes, λmax = λp 2εm + 1 , where εm is the dielectric constant of the medium and Γ is the bulk collision frequency. This equation could be represented based on sensor response as λmax =

3ε∞ Γλ p Γ 2 − ε∞λ p2

, where ε∞ is the high-frequency dielectric neff =

constant, to obtain the relative dielectric coefficient as a measure of plasmon sensitivity.64 Because the electronic equation of motion as predicted by Drude theory is mostly used to define the frequency-dependent dielectric constants of the materials considering their intraband charge transfer, the interband charge-transfer mechanism and the relaxation time dependence of surface states due to chemisorption aare neglected throughout. Another limitation of Drude theory is the underestimation of the mean square velocity, which leads to inaccuracy in the calculation of the angular frequency of surface free electrons by an order of two. An underestimation in the

2 ld

∫0

∞

n(z) exp( −2z /ld) dz

(31)

where ld is the electromagnetic field decay length of the nanoparticles. The proposition of this concept was successfully smployed by the Van Duyne group67 for an array of silver nanoparticles. In the present experiments, the evanescent electromagnetic field decays exponentially into this medium with a characteristic decay length, ld, of ∼25−50% of the wavelength of the light. The wavelength is typically ∼500−900 nm at the SPR minimum. The intensity of light is the field strength squared, so it decays with height z above the metal surface as

Figure 7. (a) Finite-element modeling of ligand-capped nanoparticles dispersed in a solvent medium and (b) variations of the localized electric field strength with relative permeability of the dispersion medium for three size-specific DDA-capped gold nanoparticles calculated by FEM simulation. L

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the particles increases, the slope increases as 0.053, 0.100, and 0.347, respectively, indicating different electric field decay lengths for different sizes of particles. In addition, the intercept of the line also increases as the size of the particles increases, which demonstrates that the penetration depth depends not only on the ligand thickness but also on the core size of the nanoparticles. Therefore, the overall mechanism of electromagnetic interaction with ligand-stabilized nanoparticles involves penetration of the ligand shell in the first step, which, in addition, is dependent on the refractive indices of the ligand. The higher the refractive index, the higher the velocity and kinetic energy of photons, which corresponds to low electromagnetic loss. It follows that the absolute value of the electromagnetic decay length becomes higher with increasing refractive index of the ligand shell and decreases with the relative thickness of shell, that is, the Murray volume fraction parameter, g. Figure 9b presents a plot of ld as a function of g, which shows a linear variation for each of the ligands; the slopes of the curves decrease linearly with increasing refractive index. The linear dependence of ld/g with the refractive index of the ligand (nligand) is shown in the inset. Upon invoking the proportionality constant (x) and replacing the value of the Murray volume fraction parameter (g) in the modified Drude equation (eq 22), the LSPR maximum can be expressed as

Figure 8. Sensitivity of the refractive index as a function of ligand chain length of three size-specific gold nanoparticles. Inset shows the relative increase in slope, which is called the optical sensory response, taking the size of the nanoparticles as a key parameter for the sensing of the medium refractive index.

[exp(−z/ld)]2. Thus, the proper weighting factor in calculating this average refractive index should be simply exp(−2z/ld), and the effective index of refraction is calculated by averaging the index of refraction over the depth of the whole bilayer structure, always weighting the local index with this factor. This average is, therefore, calculated with the depth integral where n(z) is the index of refraction at height z. This equation is not restricted to the bilayer structure and should generally be useful even for complex multilayer structures. From the above relationship, the optical sensory response (R) can be obtained as a function of the refractive index difference (Δni) as R = miΔni(1 − e−2di / ld)

⎡ ⎤ 2ld (εs − εm)⎥ λ 2 = λp2⎢(ε∞ + 2εm) + 3nligand ⎢⎣ ⎦⎥

Thus, an empirical equation can be formulated that fits the data set through good agreement among the theoretical, experimental, and numerical results and, in essence, provides a scaling law to best optimize the plasmonic sensitivity68−70 by mimicking the finer experimental details of the dielectric environment around the nanoparticles. The ultimate success of the basic philosophy of causality is to propose a model to mimic natural phenomena. Like a wand in the hand of a wizard, a suitable theoretical model reflects true understanding and helps to endow predictability toward material properties, which are difficult to achieve because of nonlinear, temporal, and phenomenological effects. The quantitative shift of the LSPR wavelength of gold nanoparticles as a function of the local dielectric field is one of the basic problems that has remained unanswered through the past five decades and forms the keystone of some marvelous applications, such as nanosensing, metamaterial science, and radiative decay engineering. Despite being addressed by numerous eminent scientists with effective theories (namely, the Drude model, Mie scattering, electrostatic approximation, Murray modification), none of the theories scales the shift with quantitative accuracy. The equation bridges between the classical Drude model of the free electron contribution of metals to the optical properties and Mie scattering theory of electromagnetism of metal plasmonics through the traditional Onsager−Böttcher cavity model providing the connection between macroscopic properties, such as the dielectric constant, to microscopic properties, such as polarizability, as depicted in Scheme 1. This stepwise development from three different perspectives, namely, experimental observations, theoretical modeling, and numerical simulations, ends up with a singular scaling equation, which, on one hand, matches experimental results perfectly and, on the other hand, describes the relative sharing of electrostatic and scattering approximated values of the observed LSPR shift. The proposition of this

(32)

where di the chain length of the ligands. This equation shows that the sensing range can be determined by the decrease of the average induced electric field from the surface of the nanoparticles. The negative exponent indicates that, when the di values are increased, ld should have more and more negative values. To interpret the decay length in a physical manner, we considered the absolute value of ld throughout the calculations; then, setting the value for the chain length, the values for ld, in each cases, could be obtained. Plotting the values of ld as a function of di, values for ld/di can be obtained as listed in Table 4. Table 4. Values of Electromagnetic Decay Lengths for Different Sets of Gold Nanoparticles electromagnetic decay length (nm) ligand

chain length (nm)

set 1

set 2

set 3

DDA TDA HDA ODA

2.08 2.31 2.57 2.83

1.6890 1.1146 1.7712 1.7140

1.7720 1.1542 1.8821 1.8308

2.3940 1.4416 2.6757 2.6119

(33)

Figure 9 shows the cumulative effect of the stabilizing ligand shell and size of the nanoparticles on the electromagnetic decay length. Although it is expected that, in each case, the ld/di values would be equal, experimentally, three different slopes are obtained for the three sets of particles (Figure 9a); as the size of M

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Figure 9. Cumulative effect of the stabilizing ligand shell and the size of the nanoparticles on the electromagnetic field decay length: (a) variation of the electromagnetic field decay length with the chain length of the ligands and (b) variation of the electromagnetic field decay length with the Murray volume fraction parameter (g). Inset in profile b shows the linear relationship ld/di as a function of ligand refractive index.

dependence of the localized surface plasmon band of the gold sols on the dielectric constant and local environment around the nanoparticles, and therefore, the sensitivity is strongly correlated with the LSP electric field as the charge density has to adjust not only to the incident fields but also to the fields arising from polarization. The significance of Murray’s modification of the Drude model, based on the volume fraction parameter (g) has been studied explicitly to provide a complete description of the ligand binding event on the experimentally observed LSPR spectra. The penetration of incident electromagnetic waves polarizes the ligand shell, and to account for the field, the Onsager−Böttcher theory of a point dipole trapped in a dielectric environment has been taken into consideration, and the resultant model satisfies experimental observations at a point-to-point level of accuracy. The integration of the experimental observations and fundamentals of the theories underlying the relevant area could give rise to an accelerating pace for the design and fabrication of ultrasensitive plasmonic chemical and biological sensors with exquisite sensitivity. Finally, the rich literature summarized in this article develops a compelling story about both the health and prospects of the leading current advances to design a landscape of the local environments around nanoparticles for their numerous possible applications from theoretical perspectives.

Scheme 1. Schematic Depiction Showing Plasmonics in Accounting for the Cumulative Effect of Solvent and Ligand Dielectric Environments as a Triple-Point Crossover of the Drude Free-Electron Model and Mie Scattering Theory through the Onsager−Böttcher Cavity Model

■

AUTHOR INFORMATION

Corresponding Author

analytical model could be extended to ligand-stabilized metallic nanostructures of arbitrary size and shape dispersed in ambient solvent media that could provide ubiquitous relationships of plasmonic sensitivity toward the local dielectric environment.

*E-mail: [email protected]. Telephone: +91-3842240848.

CONCLUSIONS In conclusion, the cumulative effect of the solvent and ligand contributions has been studied to provide clear and rigorous insight into the total dielectric environment around size-specific gold nanoparticles and to bridge the correspondence between the observed LSPR peak shift and that calculated by the N-MIE and Drude models, an empirical equation was formulated that fits throughout the data set. We have reviewed fundamental concepts, recent advances, and applications concerning the

Notes

ORCID

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Sujit Kumar Ghosh: 0000-0001-6657-7396 The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

We gratefully acknowledge financial support from DBT, New Delhi, India (Project BT/277/NE/TBP/2012). H.C. is thankful to UGC, New Delhi, India, for a Senior Research Fellowship. N

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