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Abstract
Coated gold nanorods (GNRs) are attractive as chemical sensors because their plasmon resonance energy is strongly dependent on the value of the dielectric constant in the local environment. For thin coatings (<≈20 nm), the plasmon resonance is sensitive to both the coating and the surrounding medium, while for thicker coatings the plasmons are effectively screened from their surroundings. We use finite element modeling to develop a semi-empirical effective medium approximation for the dielectric constant surrounding GNRs 30-50 nm in length with coating thicknesses of 0.5-200 nm. We demonstrate that this approximation can be used to correctly interpret shifts in plasmon resonance energy when the dielectric constant of the surroundings changes with temperature. We compare plasmon resonances of gold nanorods embedded in an epoxy matrix when coated with polyethylene glycol or silica of various thicknesses during thermal cycling. The derived expression for the effective medium dielectric of a coated rod will help device engineers optimize the sensitivity and robustness of coated GNR plasmonic sensors.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years a wide variety of plasmonic nanostructures have been developed for optical sensing. Variations in substrate material and particle geometry have been explored, including many with sharp asymmetric features that provide a rich range of sensing modalities [1–8]. Among this smorgasbord of nanoparticle sensors, gold nanorods (GNRs) are widely used as a sensing platform for gas- and liquid-phase chemicals [9–11], including biomolecules [12–15]. GNRs are elongated plasmonic nanostructures whose surface plasmon resonances (SPRs) have large optical scattering and absorption cross-sections with high polarization anisotropy. The longitudinal surface plasmon resonance (LSPR) and transverse surface plasmon resonance (TSPR) energies and lineshapes are dependent on the nanorod geometry and on the dielectric properties of their immediate surroundings [16,17]. In particular, the TSPR is much more sensitive to local dielectric than the SPR of gold nanospheres of comparable size [9,18]. GNRs are easily synthesized via scalable wet chemistry techniques [19,20] and are available from several commercial sources. They can also be used as local temperature probes or photothermal heating sources [21–24]. In each of these application areas the shift of SPR energies, in response to in-operando changes in the local environment and/or changes in nanorod geometry, is key to sensor functionality.
In many cases it is advantageous to cover GNRs with a conformal coating to improve their processability and stability. For example, GNRs dispersed in aqueous or organic solvents are often coated with organic ligands to promote directed growth and improve solubility [20,25]. In nanocomposites and biological systems, polymers are directly grafted to the GNR surface to improve miscibility with the matrix [13,26,27]. Where temperature stability is required, GNRs are coated with carbon, silica, or other rigid coatings to minimize thermal reshaping [24,28–32]. However, because the operation of these sensors depends on the influence of the local dielectric environment on the surface plasmons, there is a trade-off between the sensitivity of the SPR to the surrounding environment and the advantages provided by the coating. Here, we define sensitivity broadly as the change in the LSPR peak energy $E$ per change in environment (ie, medium) dielectric constant $\epsilon _m$ ($s=\Delta E/\Delta \epsilon _m$). The design and calibration of sensing applications requires an understanding of the role that these coatings play in GNR sensitivity to the environment.
Two important experiments in recent decades have quantified the response of GNR sensors as a function of thickness of a specific analyte on the rod surface [9,33]. Marinakos, et al. demonstrated that a shift in SPR energy continues even for coatings larger than 40 nm, but that the greatest sensitivity is within the first 10 nm. Tian, et al. showed that the spatial extent of the SPR field depends primarily on the diameter of the GNR, and they measured decay lengths exceeding 30 nm in GNRs with diameters over 20 nm. However, two important questions remain with regards to the sensitivity of GNRs when they are coated with a thermal barrier such as silica. First, to what extent is the GNR sensitive to dielectric changes outside of a fixed coating of more than a few nanometers? Second, how will the SPR energies change if both the coating and the surrounding environment change?
Here, we investigate a material system comprising GNRs in a cross-linked polymer matrix with conformal coatings of either polyethylene glycol (PEG) or hydrothermally grown silica (SiO$_2$) of different thicknesses. We show through computation and experiment that the effective dielectric constant of these coated rods depends non-linearly on the thickness of the coating in a generalizable manner. We measure the changes in SPR energy as a function of temperature and demonstrate that resonance energy shifts are due to dielectric changes in both the coating and the surrounding matrix. We establish an analytical expression for the effective dielectric constant of a coated gold nanorod which accurately predicts anomalous energy shifts in plasmon resonances during heating and cooling of these materials. This expression can be used to calculate the sensitivity of a coated gold nanorod to changes in the dielectric constant on the outside of the coating, and thereby provides a critical design tool for plasmonic sensors.
2. Results and discussion
2.1 Computation of the thickness dependence
In the electrostatic approximation [17] (Mie-Gans theory), the frequency-dependent absorption cross-section for light polarized along the $j$ axis of a prolate gold ellipsoid with dielectric function $\tilde {\epsilon _g}=\epsilon _g'+i\epsilon _g''$ in a medium of (real) dielectric function $\epsilon _m$ can be calculated from the polarizability
Typical GNR sizes (>10 nm) and geometries (cylindrical rather than ellipsoidal) limit the applicability of the electrostatic approximation. As we show hereafter, Eqn. 3 does a poor job of predicting the absorption of realistic particles. Nevertheless, we find that Eqn. 1 can accurately predict the reversible temperature dependent shifts in SPR energies if a suitable approximation for the effective dielectric constant $\epsilon _m$ is used.
2.2 Effective dielectric constant of the medium
Since the electric field of the SPR extends well beyond the particle surface [9,33], the resonance energy is sensitive to changes in the dielectric responses of both the coating and the surrounding matrix. In order to use Eqn. 1 to predict shifts in SPR peak positions it is necessary to obtain a value for $\epsilon _m$ that reflects the dielectric response of both materials.
To investigate the influence that coating thickness has on the sensitivity of the SPR to the surrounding dielectric we used the finite element method (FEM) to calculate the response of conformally coated gold nanorods to plane electromagnetic waves. We modeled GNRs as hemispherically capped cylinders where the cylindrical and spherical radii are both $r$ and the distance from end to end at the rod surface is $l$ as illustrated in Fig. 1. The capped cylinders are a more accurate representation of the rod geometry than the equivalent ellipsoid (shown as a dashed line in Fig. 1(a)), as is apparent from the micrograph. The aspect ratio is defined as $a=l/2r$. The GNR, whose dielectric constant is $\varepsilon _g$, is coated uniformly to thickness $t$ with a material of dielectric constant $\varepsilon _c$ and surrounded by a matrix of dielectric constant $\varepsilon _m$. Figure 1 also shows the finite element model built in COMSOL Multiphysics to calculate the SPR energy. Solid lines in Fig. 1(a) represent region boundaries with individually assigned dielectric properties, and the heat map shows the energy density in the electromagnetic field near resonance conditions for an excitation field polarized along the rod axis. Figure 1(b) shows a transmission electron micrograph (TEM) of a typical population of GNRs coated with SiO$_2$. The coating is noticeably rough, but is of uniform average thickness; in this case, 10 nm as determined by TEM.
Fig. 1. COMSOL model for calculating the absorption due to LSPR in coated GNRs. (a) Geometric parameters referenced in the text illustrating the coating and emphasizing the distinction between the cylindrical and ellipsoidal approximations. (b) GNR with 10 nm SiO$_2$ coating (similar to Fig. 4(e), but before heating). (c) Measured and calculated absorption spectra for the GNR in (b).
We used the “Electromagnetic Waves, Frequency Domain” module in COMSOL v5.6 to solve the electromagnetic wave equations (Maxwell’s equations) for coated GNRs subject to a uniform external field of fixed frequency. This approach ignores retardation effects, which are thought to only be significant for larger rods than those considered in this study [35]. Regions associated with gold, coating, or dielectric medium were treated as having static dielectric functions represented by known material parameters. These were then bounded by a perfectly matched layer at a distance of 1 $\mu$m from the rod center to eliminate scattering at the model boundary. By computing the total energy flux at the boundaries of the model we calculate the absorption, scattering, and total extinction coefficients at discrete frequencies. The absorption (or scattering) spectra were then fit to a Gaussian lineshape to determine the peak SPR energy. Figure 1(c) shows the measured absorption spectrum of a nanocomposite consisting of silica coated GNRs of aspect ratio 4.2 in an epoxy matrix. The spectrum is plotted along with the FEM results and the analytical prediction for the same configuration using the coated ellipsoid expression Eqn. 3. Notably, the FEM model does a much better job of replicating the SPR peak energy and line width of the experimentally measured spectrum.
To probe the effect of surface coating on SPR resonance energy we calculated the spectral response for 30 different GNRs with aspect ratios of 2.5-5, radii of 5-10 nm, and having uniform coatings 0-200 nm in thickness. The dielectric constant of the coating and the surrounding dielectric were each varied between unity and six, representing a wide range of configurations. In each case we fit the computed absorption to a Gaussian line shape to determine the peak longitudinal SPR energy. Because we are fitting dozens of data points in each spectrum, the uncertainty in the Gaussian center is $<10^{-3}$ eV, which is smaller than the point size in Fig. 2. The influence of coating thickness on the homogenized dielectric environment is apparent when we compare the normalized relative change in SPR energy as a function of thickness $t$. We define this normalized change as
where $E(t)$ is the SPR peak energy for a rod with coating thickness $t$. $E_c$ and $E_m$ are the SPR energies for a bare rod encased in an infinite, homogeneous medium of dielectric $\varepsilon _c$ and $\varepsilon _m$, respectively. Figure 2 shows an exponential fit (black, solid line) to all of the model results for $\delta E(t)$. Different plot points correspond to different GNR geometries. The inset shows the peak energy as a function of coating thickness for twelve of the computed configurations spanning the parameter space. In each case, the SPR energy changes rapidly as the coating thickness increases from zero, but by the time the coating is $\approx 50$ nm thick it is indistinguishable from an infinite thickness coating (ie, a bare rod in a uniform matrix of dielectric $\varepsilon _c$).Fig. 2. Master plot of the relative change in SPR energy with coating thickness for 12 different combinations of geometry and coating materials (various open plot symbols). The solid black line corresponds to the exponential fit to all of the data points. The inset shows the calculated absolute LSPR peak energies for each case. The solid (red) points in the main plot correspond to Eqn. 3 for a rod with aspect ratio 5 and overall length of 80 nm.
It is clear from Fig. 2 that the normalized relative change in SPR energy predicted by FEM is remarkably consistent across all GNR sizes, aspect ratios, coating materials, and dielectric constants despite the fact that the absolute SPR energies vary widely ($>1$ eV). The same $\delta E(t)$ computed by the electrostatic model for a coated ellipsoid (red points) dramatically underestimates the thickness at which the GNR SPR continues to be influenced by the surrounding matrix, which may partially explain the difference between predicted absorption spectra in Fig. 1(c). While the electrostatic approximation suggests that coatings thicker than 5-10 nm completely screen the surrounding dielectric, the FEM results indicate that GNRs are affected by the dielectric constant of their environment out to much larger distances. This is consistent with previous experiments [9,33], and provides a foundation for predicting the sensitivity of the longitudinal SPR in coated GNRs to changes in the surrounding medium.
Since the SPR in an uncoated rod shifts almost linearly with the surrounding dielectric constant, our results suggest a universal form for the effective dielectric constant of a coated GNR in a particular embedding matrix. The associated empirical formula
serves as an accurate effective medium approximation (EMA) for the environment of a GNR with coating and medium having dielectrics $\epsilon _c$ and $\epsilon _m$, respectively. We find good agreement when $b=0.126\pm 0.004$ nm$^{-1}$, indicating an effective boundary between matrix-dominated and coating-dominated response at a thickness of around $1/b=8$ nm. We note that several other homogenization approaches for the effective medium of coated nanoparticles have been developed, but for rods of realistic sizes and morphology those approaches require modeling each geometric configuration. This empirical relationship is generally applicable and computationally cheap for designers of GNR based sensors to apply. The derived expression is in excellent agreement with previous reports of the thickness-dependence of GNR sensitivity (see Supplement 1).The sensitivity of a coated GNR plasmonic sensor to changes in the outer medium will be proportional to $d\epsilon _{EMA}/d\epsilon _m=e^{-bt}$. Thus, increasing the coating thickness will exponentially diminish the effectiveness of a coated GNR for sensing applications, regardless of the dielectric properties of the coating.
2.3 Thermal variations in the dielectric environment
We illustrate the accuracy and utility of Eqn. 5 by thermally modulating the dielectric properties of the coating and medium dielectrics. We dispersed GNRs in a transparent epoxy matrix and measured the plasmon resonance energies as a function of temperature via optical transmission in a manner previously reported [22].
Figure 3(a) shows the general spectral evolution of the extinction spectra (measured in transmission) for uncoated GNRs in an epoxy matrix during thermal cycling. At elevated temperatures, the GNRs change shape through surface diffusion of gold atoms towards more energetically favorable lower aspect ratios [36]. As the aspect ratio of the rods decreases the LSPR blue-shifts and the TSPR slightly red-shifts. However, between each period of heating, the spectra “recover” a fraction of the thermally induced shift as shown in Fig. 3(c). We attribute this reversal to the temperature dependence of the dielectric properties of the GNRs and the environment as illustrated in the cartoon in Fig. 3(b).
Fig. 3. Experimental extinction spectra and associated SPR energy shifts during two sets of heating/cooling (150 °C / 22 °C) cycles for uncoated GNRs embedded in epoxy resin. (a) and (c) correspond to the spectral evolution during an initial heating/cooling cycle while (b) and (d) correspond to the same process applied to the sample several days later. Red and blue curves in (c) and (d) correspond to LSPR and TSPR resonances, respectively.
As previously reported, a stiff embedding matrix limits the surface diffusion driven reshaping process of GNRs [29,36]. After an extended period of heating (usually minutes to hours) the GNRs stabilize at an equilibrium shape determined by the mechanical properties of the matrix. When the same sample is allowed to rest at room temperature for several days and then subjected to the same heating/cooling cycle, as shown in Fig. 3(b&d), the reversible temperature-dependent shift is significantly larger than any subsequent reshaping over the same time period. Thus, by thermally cycling pre-stabilized GNRs we can directly probe the temperature dependence of the surrounding dielectric constant independent from changes in geometry.
To show the influence of coating on GNR sensitivity, we measured the temperature dependence of the SPR resonances in three different nanocomposites comprising coated GNRs in an epoxy matrix. All of the GNRs in this study came from the same growth batch, having average lengths of 50.6 $\pm$ 3.5 and widths of 20 $\pm$ 1.7 before shape stabilization. They were then coated with $\sim$1-2 nm PEG followed by either 10 nm SiO$_2$ or 20 nm SiO$_2$. Figure 4 shows the SPR energies (a-c), TEM micrographs (d-f), and schematic representations (g-i) for GNRs subjected to repeated cycling at 140 $^\circ$C / 22 $^\circ$C after shape stabilization. There are a few key differences between the amplitudes and signs of the changes in SPR energy. For example, the total change in TSPR energy is more than twice as large for the SiO$_2$ coated rods as it is for the PEG coated rods. The change in LSPR, on the other hand, is much smaller for the SiO$_2$ coated rods. In particular we note the inversion of the sign of the shift in the LSPR for 20 nm versus 10 nm SiO$_2$ coatings. This difference is indicative of a change in the character of the local effective dielectric constant felt by the GNRs. Specifically, this shows a transition from a regime in which the SPR response is dominated by the coating (20 nm) to one in which the coating is dielectrically “transparent”(10 nm). We note that, as with the computational results in Fig. 2, the uncertainties associated with the fit peak centers in Fig. 3 and Fig. 4 are smaller than the point sizes in the graphs.
Fig. 4. Red and blue curves depict LSPR and TSPR energies respectively of gold nanorods embedded in an epoxy matrix during heating and cooling cycles between 140°C and room temperature. The reversible behavior of (a) PEG coated nanorods (b) 10 nm SiO$_2$ coated GNRs, (c) 20 nm SiO$_2$ coated GNRs. (d-f) Show the corresponding TEM micrographs of the coated GNRs. The energy ranges in each are shifted, but the scale is the same. (g-i) Show the corresponding illustrations of the PEG coated and SiO$_2$ coated nanorods, respectively.
2.4 Temperature dependence of the dielectric constants
To demonstrate the utility of the derived EMA we apply it to a quantitative analysis of the observed temperature dependent shifts in Fig. 4. These can be obtained from Eqns 1 and 5 if the coating, matrix, and GNR dielectric are known as a function of temperature. We used reported values or direct measurements of each constituent’s dielectric function to model the expected SPR resonance shifts. For each material we obtain the function $\epsilon (\omega, T)$ as described below.
PEG: Any direct contribution to the temperature induced shifts from the PEG coating is negligibly small because it is only $\sim$ 1 nm thick and its variation with temperature is small. Therefore, we consider only the contributions from SiO$_2$ and epoxy in the temperature variations of the SPRs.
SiO$_2$: Due to its importance in the semiconductor and thin film industries, the dielectric properties of SiO$_2$ are well studied. Reported values of $\epsilon (\omega, T)$ for SiO$_2$ vary with synthesis and processing history, but all have similar temperature dependencies that are not strongly frequency dependent. Thus, we have
where $\epsilon _{sio2}(\omega )$ is a numerical interpolation of the measured dielectric constant for optical grade silica at room temperature [37] and $\tau _{sio2}=2.6*10^{-5}$ $K^{-1}$ is the average measured temperature coefficient. $\Delta T$ is taken with respect to room temperature.Epoxy: The dielectric properties of epoxies vary significantly because of variations in stoichiometry and cross-link density between different formulations. Therefore, we used spectroscopic ellipsometry to directly measure $\epsilon (\omega, T)$ in neat epoxy films as a function of temperature. The data fit well to a Sellmeier equation
with a linear temperature coefficient $\tau _{epoxy}=-1.13*10^{-3}$ $K^{-1}$. Details and supporting data can be found in Supplement 1. We note that the temperature coefficient for epoxy is at least an order of magnitude larger than for silica, and of opposite sign. Thus, for thin silica coatings we expect qualitatively different SPR shifts than for thick silica coatings, consistent with the measurements shown in Fig. 4.Gold The complex dielectric function of gold can been accurately modeled using the Drude-Lorentz theory [38]. It is important to separate the contributions from free-carriers (fc) and interband transitions (ib) since the phenomena that govern the temperature dependencies are distinct. To that end we write the dielectric function of gold as
The temperature dependence of the Drude contribution is primarily in the plasma frequency $\omega _p$ and scattering term $\gamma _D$. For small changes in the dielectric relative to the room temperature ($T_0$) value we have
On the other hand, ultrafast laser heating experiments on gold nanorods [40,41] have shown that thermal fluctuations to the interband component of the dielectric function are dominated by a rigid band shift due to thermal expansion of the lattice. Therefore,
2.5 Combined response
Using Eqns 1 and 5 we calculated the absorption spectra for coated GNRs by substituting the appropriate temperature-dependent dielectric functions described above. Figure 5(a) shows the shifts (compared to room temperature) in the LSPR and TSPR energies for a range of temperatures determined by finding the local maxima of the calculated spectra. The calculated energy shifts largely agree with the experimental observations. The only case where there is an appreciable difference is the $\Delta$LSPR of the PEG-coated rods, which may be due to fluctuations in the local density of the polymer coating at elevated temperatures because of diffusion of the PEG into the epoxy matrix [28,29]. Because the outer medium (epoxy) is the same in each case, the slopes in Fig. 5 corresponding to the LSPR (red lines) are proportional to the sensitivity of the GNRs to changes outside of the coating in the usual chemical sensing modality.
Fig. 5. Measured and calculated GNR SPR energy shifts as a function of temperature. (a) Experimental data (markers) and computed values (solid lines) for the LSPR (red) and TSPR (blue) energies for three types of rods. (b) Room temperature (black) and 140 $^\circ C$ (orange) spectra calculated for 4 cases: GNR in an epoxy matrix without including matrix thermal effects, GNR in epoxy, GNR with 10 nm SiO$_2$ in epoxy, and GNR with 20 nm SiO$_2$ in epoxy.
Figure 5(b) shows representative calculated spectra at room temperature (black) and 140 $^\circ$C (orange) to illustrate the relative contributions to the effective medium from each constituent. When ignoring any temperature dependence of the matrix (top left of b) the intrinsic temperature dependence of the gold dielectric gives rise to a small blue shift in the LSPR and a moderate red shift in the TSPR. This is primarily due to the sign difference between the Drude and interband thermal coefficients as described above. On the other hand, the weakly dispersive epoxy dielectric uniformly pushes the SPRs to higher energies, resulting in a large blue shift in the LSPR and a much smaller red shift in the TSPR. Because the sign of the thermal coefficient of silica is opposite that of epoxy, a thin coating of SiO$_2$ pushes both peaks back to lower energies. The thicker coating continues that trend, resulting in a very slight red shift of the LSPR and a large red shift of the TSPR.
3. Conclusions
We have developed a simple analytical expression for the effective dielectric constant of a gold nanorod with a dielectric coating of arbitrary thickness surrounded by another dielectric. This EMA can be used to quantitatively predict the sensitivity of a coated gold nanorod to changes in the surrounding dielectric. As defined, the reported sensitivity ignores the ultimate detectability of a particular analyte, which will also depend on such parameters as the sharpness of the plasmonic absorption (ie, quality factor) or the total scattering or absorption cross-section, which are related to the modal volume, photon density of states, and number density of GNRs in the sensor system. For GNRs, these extrinsic parameters depend primarily on particle volume, and the results presented here are consistent across a wide range of commercially relevant particle sizes. We demonstrate the validity and utility of the EMA by modeling the temperature-dependent shifts in the SPR energies of coated GNRs using experimentally determined thermal coefficients of the dielectric constants for the gold, coating, and surrounding matrix. The electrostatic approximation for the polarizability of an uncoated ellipsoid yields resonance energies within 10% of experimental values when the dielectric constant of the medium is replaced by the effective value Eqn. 5. For intermediate coating thicknesses (5-20 nm) the predicted SPR energies using the EMA are much more accurate than those obtained from the electrostatic coated ellipsoid model. When the thermal coefficient of the coating and matrix have opposite signs the temperature-dependent SPR shifts in a manner that depends strongly on the thickness of the coating. It is therefore essential to take into account the thickness of the GNR coating when interpreting SPR shifts arising from changes in the dielectric environment, such as in GNR based chemical sensors. The EMA developed here can be used to predict the sensitivity of a plasmonic sensor with a coating of a fixed thickness if the dielectric properties of the coating and analyte are known. This knowledge can be combined with design rules based on nanorod geometry [12,42,43] to optimize GNR based nanosensors for specific applications. The sensitivity to the surrounding dielectric can be directly used to calculate a sensor figure of merit (FoM) if other relevant sensor system parameters are known [44].
Funding
Air Force Research Laboratory (FA8650-13-C-5800); Air Force Office of Scientific Research (16RXCOR324).
Acknowledgments
W. J. Kennedy thanks Rich Vaia and Kyoungweon Park for sharing their observations of heated gold nanorods in solution.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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