Assessing the degree of heating present when a metal nanoparticle is trapped in an optical tweezers is critical for its appropriate use in biological applications as a nanoscale force sensor. Heating is necessarily present for trapped plasmonic particles because of the non-negligible extinction which contributes to an enhanced polarisability. We present a robust method for characterising the degree of heating of trapped metallic nanoparticles, using the intrinsic temperature dependence of the localised surface plasmon resonance (LSPR) to infer the temperature of the surrounding fluid at different incident laser powers. These particle specific measurements can be used to infer the rate of heating and local temperature of trapped nanoparticles. Our measurements suggest a considerable amount of a variability in the degree of heating, on the range of 414–673 K/W, for different 100 nm diameter Au nanoparticles, and we associated this with variations in the axial trapping position.
© 2015 Optical Society of America
Svoboda and Bloch  first studied stable optical trapping of Au nanoparticles and demonstrated superior trap stiffnesses k compared to similar-sized dielectric particles . The enhanced trapping properties were attributed to the larger polarisability of Au at the trapping laser wavelength, which is an important factor in determining the strength of the optical gradient force in the Rayleigh regime . The size range over which stable three-dimensional trapping of Au nanoparticles can be achieved has been experimentally and theoretically considered by a number of groups [4–7]. It is found that for non-resonance trapping: (i) effective particle diameters larger than 150 nm experience a dominant axial radiation pressure that results in an unstable trapping configuration; and (ii) particles with smaller sizes experience a gradient force that scales with the volume of the particle and may be ultimately overcome by associated thermal forces. In considering these limits Hajizadeh et al.  reported stable trapping of larger Au nanospheres as small as 9.5 nm in diameter.
Whilst trapping of Au nanoparticles can be achieved with as little as a few tens of milliwatts - large trapping forces (30 pN) are anticipated at very higher laser power (∼ 4.5 W). Such high powers, however, are implicitly linked to significant localised heating due to the plasmonic enhancement of optical absorption of Au. Indeed even at moderate trapping powers the level of localised heating is not negligible and has significant implications for accurate determination of trapping parameters such as trap stiffness, damping coefficients and stochastic forces [8, 9]. Based on a comparison of three-dimensional trapping metrics Seol et al. found trapped 100 nm Au nanoparticles exhibited heating of 266°C/W. Bendix et al. studied the temperature surrounding single Au nanoparticles optically trapped on a lipid bilayer, and used temperature dependent transitions of lipid bilayers to infer the degree of heating experienced in the trap . They experimentally found the heating at the surface of 100 nm diameter Au nanospheres to be of the order of 452 K/W. Whilst this study was performed using a two-dimensional trapping geometry, similar fluorescence based studies in a three-dimensional trapping geometry revealed a markedly different response in the local temperature increase for different sized particles . These results imply that the degree of heating is very specific for the particular trapping geometry under investigation, and as such, in situ characterisation techniques are required for accurate determination of local thermal effects.
In this paper we investigate the local temperature of a laser irradiated single nanoparticle by combining optical tweezers, dark-field illumination and localised surface plasmon resonance (LSPR) spectroscopy . Our method makes use of the temperature dependent refractive index of the surrounding water to correlate the change in the scattering intensity over a fixed spectral window (550–750 nm) with the heating produced at different trapping powers. Experimentally we find that Au nanoparticles of 100 nm in diameter experience heating, on average, of 554 K/W by the 1064 nm trapping laser. However, some variability exists between individual particles (between 414 ± 20 and 673 ± 19 K/W) that is not accounted for by the size dispersion of the particles. We associated the spread in the degree of heating of the nanoparticles with variations in their axial trapping position. This method is generally applicable for determination of the temperature of metallic nanoparticles with well described plasmonic resonances and can be used as an intrinsic method for nanoscale thermometry in trapping experiments.
The temperature of Au nanoparticles in aqueous solution can be estimated through the localised surface plasmon resonance (LSPR) peak shift . The LSPR peak is dependent on refractive index changes in the medium surrounding the particle, and is thus a sensitive probe of the temperature. Temperature dependence of the LSPR has been utilised previously for non-trapping experiments by Honda et al. . They monitored the localised heating effect on a 40 nm in diameter Au nanoparticle coated with a thermo-responsive polymer through the plasmon peak shift. Their results indicate that the particle experiences a temperature rise of around 10°C when irradiated with a tightly focused beam of ∼ 1 mW at 532 nm.
The quasi-static or dipole approximation of Mie theory (2R ≪ λ, for Au 2R < 25nm ) can be used for a qualitative analysis of the temperature dependence of the nanoparticles’ LSPR peak. The extinction cross section σext of the nanoparticles then reduces to the following relationship,Eqn. 1 we can see that a decrease in the dielectric constant of the medium surrounding the particle due to a temperature increase will result in a decrease of the extinction cross section σext and a blue shift of the resonant frequency.
However, full Mie theory  has been used in this work to quantitatively study the temperature dependence of the nanoparticles’ LSPR peak, as they are too large to satisfy the dipole approximation. Au nanospheres of 100 nm in diameter were used and white light scattering calculations were performed. The temperature of the surrounding aqueous medium was varied from 20°C to 250°C. The wavelength and temperature dependent refractive index of water was calculated with the Cauchy formula and the temperature dependent coefficients were obtained from Bashkatov and Genina . Since the change in the refractive index of Au with temperature is minimal, due to the thermal conductivity of Au remaining almost constant over the temperature range of interest , it was not included in the calculations. For an absence of trapping laser the ambient temperature was T0 = 20°C, hence the local temperature of the trapped nanoparticle and its surrounding medium depend on the heating rate H of the nanoparticle and the trapping laser power P as follows 
A schematic diagram of the experimental set-up used for this study is presented in Fig. 1. The optical tweezers comprise of a linearly polarised 5W, CW Nd:YAG laser (Laser Quantum, 1064 Ventus, λ = 1064 nm, TEM00), which is focused using an oil-immersion microscope objective (Olympus, E-plan 100×, Numerical Aperture (NA) = 1.25), via relay optics, in order to create a diffraction limited spot in the front focal plane. A two-axis acousto-optic deflector (2D-AOD) (Gooch and Housego, 45035 AOBD) is used to control the power of the beam and a half-wave plate is used to control the polarisation in the focal plane. A spatial light modulator (SLM) (Hamamatsu LCOS-SLM x10468-03) corrects the aberrations in the beam profile to optimise the trap shape. We note that the trapping power could be also controlled by the SLM directly, leading to a simplified experimental arrangement. The absolute power at the focus is measured using a calibrated photodetector taking into account the transmission losses of the optical system and microscope objective.
Dark field microscopy can be combined with optical tweezers to measure LSPR spectra on single isolated nanoparticles without compromising the strength of the optical trap [12, 19]. In our setup, we achieve dark field microscopy by using a backscattered light to image the sample. An axicon lens was used to form a ring of light, which was imaged onto the back focal plane of the objective. The reflected light was split into two paths for simultaneous dark-field imaging and spectroscopy. Dark-field images of the nanoparticles were obtained using a charge coupled device (CCD) (AVT Stingray), labelled as CCD1 in Fig 1. Backscattered spectra of the nanoparticles were recorded using a spectrometer (Acton SP2300, Princeton Instruments) and a scientific thermoelectrically cooled CCD optimised for spectroscopic applications (PIXIS 256, Princeton Instruments), labelled as CCD2 in Fig 1. The focal plane for the spectroscopy was selected by placing a 100 μm pinhole as the spectrometer aperture. This ensured that our spectroscopy measurements were performed only on the trapped particle and the signal from other particles in the sample was eliminated.
We studied Au nanoparticles of 100 nm in diameter supplied by Sigma-Aldrich (product number 753688). The colloids have a size dispersion of between 98–115 nm and polydispersity index (PDI) ≤ 0.2. Samples were diluted with milli-Q water from stock solution to ensure that only one particle enters the tweezers during the course of the measurements. A 10 μL volume of the nanoparticle suspension was pipetted into a sealed microfluidic chamber with a height of approximately 100 μm. Trapping measurements were performed on individual nanoparticles at a height of approximately 40 μm from the cover slip. Trapping power at focus ranged from 30 mW to 180 mW. Dark-field spectra of each nanoparticle were collected with the spectrometer CCD2 (1 sec exposure time) for different trapping powers in order to observe the spectral changes that were induced by the heating. Power dependent trapping measurements of non-absorbing particles, polystyrene beads of 1 μm in diameter, were used to verify that the spectral changes were not due to systematic effects.
In Fig. 2 we present a comparison of scattering spectra of a single 100 nm Au nanoparticle calculated using full Mie theory, IMie(λ, T) (a), and typical scattering spectra of a trapped nanoparticle recorded experimentally in our setup, IExp(λ, P) (b). The power dependent response behaves as expected; for increasing trapping powers the local temperature increases, hence the refractive index of water decreases, resulting in a decrease in the LSPR peak’s intensity and a blue shift of the resonance wavelength.
The rate of heating experienced by the Au nanoparticle due to the trapping laser can be obtained by relating the integrated scattering intensity measured for different trapping powers with the integrated scattering intensity calculated for different temperatures of the nanoparticle and its surrounding medium. We define the integrated intensity, IInt,Exp(P) and IInt,Mie(T), as the sum of the wavelength-dependent scattering intensity, IExp(λ, P) and IMie(λ, T), between λi = 550 nm and λf = 750 nm, being a function of the trapping power for the experimental data (Eqn. 3a) and of the temperature for the Mie theory calculations (Eqn. 3b).
We note that the dominant changes in the nanoparticle spectra for increasing temperature is the decrease in the peak intensity; the resonance wavelength being only marginally blue shifted by a few nanometers compared with a peak full width half maximum of 100 nm, and barely resolvable in our system. The method we adopted to assess the degree of heating is then to look at the changes in the integrated scattering intensity from 550 nm to 750 nm. This measurement reports on the most salient changes to the spectra upon heating with the required signal to noise ratio. Our approach is also particularly amenable to fluorescence based microscopy where band pass filters are almost universally employed for spectral selectivity.
The heating of the nanoparticles due to the trapping laser is a reversible process, allowing the repeated heating and cooling of the same particle by the trapping laser in a single experiment, as shown in Fig. 3, where the integrated intensity and trapping power are presented as a function of time. This observations excludes the possibility of permanent modification of the nanoparticle. Our system response time is of the order of 2 seconds, limited by the exposure time set for the CCD2. Since the characteristic time of transient heating of nanoparticles is of the order of 100 ns  we are not able to measure it in our experiment.
By non-linear least squares fitting of the power-dependent integrated intensity measured for each nanoparticle, IInt,Exp(P), to the temperature-dependent integrated intensity calculated using Mie theory, IInt,Mie(T), taking into account the temperature’s power dependence (Eqn. 2) the heating rate is obtained from the fitting parameters, as shown in the following relationship:
Heating measurements of a number of individual nanoparticles are presented in Fig. 4(a), showing the measured integrated scattering intensity vs. power of the trapping laser, and the calculated curve of best fit obtained from Eqn. 4, where the integrated intensity of each nanoparticle has been normalised to be 100% for 0 mW of laser power. The obtained heating of the nanoparticles due to the trapping laser are 414 ± 20 K/W (▵), 480 ± 24 K/W (○), 558 ± 14 K/W (▿), and 673 ± 19 K/W (□). Figure 4(b) shows the local temperature immediately adjacent to the nanoparticle for different trapping laser powers calculated using Eqn. 2, where T0 = 20°C.
It is important to note that the LSPR is sensitive to changes only within the plasmon penetration depth, estimated to be of the order of the particle diameter [21, 22], and so the effect is sensitive only to local changes in temperature.
The trapped 100 nm Au nanoparticles experience heating of, on average, 554 K/W, which considerably larger than that predicted for direct heat of the surrounding water . For a typical trapping experiment that utilises a minimum of 30 mW of trapping power, the local temperature of the nanoparticle is on average 37 ± 3°C, but can be as large as 146 ± 4°C for higher trapping powers. The water surrounding the particle reaches temperatures above 100°C without any explosive boiling visible during the whole experiment duration. Laser heating experiments in the literature have already reported super heating of water to such high temperatures .
From Fig. 4 there is clear evidence of particle specific variability in the level of heating observed by the nanoparticles, ranging from 414 ± 20 K/W to 673 ± 19 K/W. There are a number of factors that should influence the specific heating observed for individual measurements. Firstly, the size dispersion (∼ 17 nm) within the sample solution will yield slight changes in the scattering cross-section of the nanoparticles. However, when fitting the measured integrated intensity vs. trapping power data points to simulated curves with a range of different particle diameters, which reflect the sample size dispersion, the small change in the resulting heating cannot account for the observed variability in the experimental measurements.
Another possible source of variation is in the detailed structure of the nanoparticle and its deviation from a perfect spherical shape , which may present local hot spots that modify the LSPR at the trapping wavelength (1064 nm). Whilst we do not measure the scattering spectrum at the trapping wavelength we see no evidence of significant changes to the LSPR spectrum in the visible and near-infrared and so shape is likely to have only a perturbative influence in our measurements.
Finally, we may consider the influence of variations in the particle dimensions on the stable equilibrium axial trapping position and how this may dictate the degree of heating observed in the trap . The power absorbed by the optically trapped nanoparticle that will dissipate as heat may be estimated using the absorption cross-section Cabs and the intensity Itrap of the field at the trapping position, using Pabs = Cabs · Itrap. In this case, Itrap refers to the local intensity at the equilibrium trapping position, which is typically displaced from the focal plane along the axial direction. We may estimate the axial displacement by calculating the relevant optical trapping forces acting on Au spherical nanoparticles using vector diffraction integration  and then decomposing the total force into the scattering Fscat, absorption Fabs and gradient forces Fgrad [1, 5]. In Fig. 5 we present the stable axial trapping positions for different sized Au nanoparticles for our given experimental configurations (trapping power = 10 mW). These calculations used Mie theory to obtain the scattering and absorption cross-sections, which are then used to derive the scattering and absorption forces. The gradient force was calculated using the method described by Chaumet and Nieto-Vesperinas . We note that this method has deficiencies at larger particle sizes and polarisabilitiy and we use it here only as a semi-quantitative tool in understanding the origin of experimentally observed variability in the rate of heating. What we observe is that the larger particles experience a greater extinction force (Fscat + Fabs) relative to the axial gradient force, which will displace their equilibrium trapping position away from the focus. Over the size range of interest (diameter ≃ 100–120 nm) we note that the local intensity can vary by a factor of two, which would certainly account for the observed variability in the degree of heating within the sample dispersion. We note that the stable axial trapping position may also be susceptible to the detailed geometry of the particles , which will in turn affect the degree of heating experienced by the particle in the trap.
We note that whilst our results are comparable with previously published values of heating of Au nanoparticles, there is a discrepancy that is unaccounted for in the particle to particle variations. We suggested that the origin of the variability might be expected when considering differences in the experimental apparatus used in the study. The Perkins Group [8,28] used a 1.4 NA oil-immersion objective with no aberration correction, the two works by the Oddershede Group [10, 11] used a 1.4 NA oil-immersion objective for two-dimensional trapping and a 1.2 NA watter-immersion objective for three-dimensional trapping, whilst in our study we employ a 1.25 NA oil-immersion objective that has been corrected for aberration using an SLM. These differences alone will invariably lead to variations in the axial equilibrium trapping position and hence local intensity seen by the nanoparticles. The sensitivity of the heating to axial trap stiffness is again suggestive of the need of an intrinsic calibration method.
In conclusion, we have characterised the degree of heating of Au nanoparticles of 100 nm in diameter in an optical tweezers using intrinsic properties of the nanoparticles. We have observed particle specific variability in the degree of heating of the trapped nanoparticles, larger than expected from the variations in the scattering cross section. The technique can be extended to other nanoparticle sizes and shapes. It is a powerful tool that can be used not only to characterise the localised heating of nanoscale particles, which is of great importance when using them as nanoprobes or other biological sensors, but also to correct for the influence of temperature in trapping Brownian motion dynamics.
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