## Abstract

Metallic nanoparticles are of significant interest due to their particular optical and biological applications. Gold nanoparticles are proven to be excellent candidate for *in vivo* micro-manipulation using Optical Tweezers. This manuscript reports on stable 3-D trapping of 9.5–254nm gold nanospheres using substantially decreased laser power. The lower limit is ∼2 times smaller than previous record. 5.4nm gold nanospheres were trapped for only 2–3 seconds. For the first time, our experimental data verify the volume corrected Rayleigh model for particles smaller than 100nm in diameter. Measuring the maximum applicable force for gold nanoparticles, we have shown that a few tens of milli-Watts of laser power can produce pico-Newton range forces.

©2010 Optical Society of America

## 1. Introduction

The unique size-dependent properties of Gold Nanoparticles (GNPs) make them superior and indispensable in many areas from biology to electronics. In cancer research, nontoxic GNPs can be used to target tumors providing *in vivo* tumor spectroscopic detection based on the use of Surface-Enhanced Raman Scattering (SERS) [1]. The surface properties of metallic nanoparticles can be changed, using intermediate molecules, to make them specifically attach to predefined structures, e.g., cancer cells [1, ]. Gold nanoparticles are also used as drug carriers in Drug Delivery Project(DDP) [2, ]. In electronics, GNPs have proved very useful, e.g., they provide organic memory devices [3, ]. Nanogold-coated bacteria have been used for electronic wiring [4, ]. Widely growing applications imply that the micromanipulation using GNPs would be of great interest in a variety of scientific communities.

The single beam Optical Tweezer (OT) was introduced by Ashkin *et al*. [5, , 6, ]. It refers to a particular geometry where a laser beam is focused through a high numerical aperture objective lens to generate a three-dimensional optical trap by exerting a radiation force to small objects. Present days optical traps have become calibrated tools to immobilize, orient and transport submicron particles. The nanometer positioning ability along with sub-picoNewton force resolution have turned OT into a valuable tool in biology and physical sciences [7, 8]. It is shown that the trapping efficiency of a Rayleigh (diameter *d* ≪ wavelength *λ*) gold particle is 7 fold better than a similar-sized latex particle [9]. This property makes the metallic nanoparticles a very good candidate for OT-based micromanipulation inside living cells. However, increase in temperature due to the light absorbed by a trapped metallic particle and the probable damage in the cell is the main concern in this picture [10]. Therefore, finding a method to overcome this problem will be very valuable.

Historically, three-dimensional trapping of 36nm GNPs was reported for the first time by Svoboda *et al*. [9]. Trapping of Mie gold particles (diameter *d* ≃ wavelength *λ*) with diameters of 0.5–3*μ*
*m* has been restricted to two-dimensions [11] because of the high extinction of the particles. Later, the trapping range of the GNPs was expanded to 18–254nmby Hansen *et al*. [12]. However, due to the spherical aberration present in their optical system, a very intense laser beam was used for trapping smaller particles, e.g., 900mW of laser power at the sample was used to trap 18nm GNPs. While trapping 18nm gold [12] and 20nm silver nanospheres [13] are reported, to our knowledge, the gold nanorods with a volume of 2080*nm*
^{3} [14] are the thinnest particles that have been trapped by a single beam optical tweezers so far.

In the present letter, we report on stable 3-dimensional trapping of gold nanospheres ranging from 9.5 to 254nm in diameter using a nearly aberration-free single beam optical trap. To our knowledge, this is the first stable trapping report of a single gold nanosphere with a diameter down to 9.5nm (volume≃500 nm^{3}). This substantially decreases the lower size limit of the trapped GNPs. The range of the laser power (at the sample) used in this research was 30∓415mW which is significantly lower than the previous works [12], e.g., a 20nm gold particle was trapped using 300mW of laser power which is lower by factor ∼3. The results reported in this letter, for the first time, quantitatively support the theoretical predictions for GNPs smaller than 100 nm [15]. 4.9nm gold spheres were trapped only for 2–3 seconds using 415mW of laser power at the sample (this was the highest power which our laser source could provide). Our results could be of interest for *in vivo* manipulation using optical tweezers because of the substantial decrease in the temperature of the trapped metallic nanoparticles.

## 2. Theory

A focussed Gaussian intensity profile provides a 3-D Hookean restoring force for a particle near the focus. In one dimension, the restoring force can be written as *F* = -*κ*(*x* - *x*
_{0}), with *x*, *x*
_{0} and *κ* being the position, equilibrium position, and the stiffness of the trap, respectively. The stochastic force along with the frictional force of the surrounding medium, and the restoring force, altogether result in the Langevin equation, *γx*̇ + *κ*(*x* - *x*
_{0}) = *η*(*t*) in which *γ* and *η*are drag coefficient and stochastic force, respectively. Fourier transform of the Langevin equation results in a Lorentzian power spectrum,

where ${f}_{c}=\frac{\kappa}{2\mathrm{\pi \gamma}}$
*D _{SI}* are corner frequency and diffusion constant, respectively [16].

The optical forces exerting on a metallic Rayleigh particle in an optical trap are absorption force, ${F}_{\mathrm{abs}}=\frac{{n}_{m}}{c}<S>{C}_{\mathrm{abs}},$ scattering force, ${F}_{\mathrm{scat}}=\frac{{n}_{m}}{c}<S>{C}_{\mathrm{scat}},$ and gradient force, ${F}_{\mathrm{grad}}=\frac{\mid \alpha \mid}{2}\nabla <{E}^{2}>\left[9\right],$ where *S* is the Poynting vector of the electromagnetic wave. *C _{scat}* =

*k*

^{4}|

*α*|

^{2}/4π and

*C*=

_{abs}*k*Im[

*α*] are scattering and absorption cross sections, with $k=\frac{2\pi {n}_{m}}{\lambda},{n}_{m}$ and

*α*being the wave number, refractive index of the surrounding medium and polarizability, respectively.

For very small particles the scattering and absorption forces can be neglected in comparison to the gradient force [9]. Therefore the total force acting on a trapped particle can be estimated by the gradient force. Assuming that the amplitude of the illuminating light is constant over the extent of the trapped particle, *α* can be given by Claussius-Mossotti relation $\alpha =3V\frac{\hat{\epsilon}-{\epsilon}_{m}}{\hat{\epsilon}+2{\epsilon}_{m}}$ in which *ε*̂(*λ*) = *ε*
_{1}(*λ*) + *iε*
_{2}(*λ*) and *ε _{m}* =

*n*

_{m}^{2}are the dielectric constants of the particle and the surrounding medium (water in our case), respectively [9].

*V*(∝

*r*

^{3}) is the fraction of the particle’s volume which is polarized. It is known that the electromagnetic wave attenuates inside a metallic medium. the skin depth can be written as

*δ*=

*λ*/2

*πn*

_{2}, with

*n*

_{2}being the imaginary part of the refractive index of the medium. At

*λ*= 1064

*nm*one can estimate

*δ*= ≅ 23

*nm*and

*ε*̂ ≅ -54 + 5.9

*i*[12]. For the cases where

*r*≃

*δ*, the decrease of the gradient force due to the attenuation of the electric field inside the particle can be accounted by replacing

*V*with the corrected volume

*V'*= 4

*π*∫

_{0}

^{a}

*r*

^{2}exp[(

*r*-

*a*)/

*δ*]

*dr*[9]. Therefore, the gradient force can be written as:

where *δ*, *a* and *A* are skin depth, radius of the particle and a constant, respectively.

For particles bigger than 100nm, the extinction increases dramatically [15] and it can not be neglected, anymore. In such cases all three forces should be considered for calculation of the trapping force. To Our knowledge, due to cumbersome calculations and some complicated processes, e.g., bubble formation [15] and shape transformation [17], there is no accurate formulation for trapping force at these sizes yet.

## 3. Experiment

Our optical tweezer set-up is based on a custom-designed inverted microscope with a CW laser source (Nd:YAG, Compass 1064, Coherent). The laser is tightly focused using an objective lens (Olympus, UPlan FLN, 100x, NA=1.3) mounted on a piezo equipped objective holder (Physik Instrumente, P-723.10) which can displace the objective with a nanometer resolution in the vertical direction. A quadrant Back Focal Plane (BFP) detection scheme [18] was used for 3-dimensional detection of the trapped particle. More instrumental details can be found in the reference 19. It is shown that the tube length [19] and the refractive index of the immersion medium [20] can change the stable trapping depth range. In our setup the tube length was tuned so that the most efficient trap (minimal aberration) occurred at the height of 3*μ*m from the inner surface of the coverglass using normal immersion oil (Zeiss, n=1.518) and the trap was stable up to depth of 20*μm* (depending on particle size and input power of the laser). Furthermore, the trapping range could be shifted inside the chamber either by altering the tubelength [19] or the refractive index of the immersion medium [20]. All trapping experiments were conducted at the height of 3*μ*m where the proximity effect of the glass wall can be neglected for all GNPs used in this research [21, 22]. The GNPs with mean diameters of 5.4, 9.5, 15.6, 20.5, 31.4, 40.9, 50.4, 60.7, 78, 97, 149.4, 194.4 and 254nm were purchased from British Biocell (BBI). The standard deviation of the particle sizes was at most 10%, based on the company’s report. The sample chamber was assembled from a coverslip, a microscopy slide and two strips of a double stick tape as spacers. The GNPs solution were diluted in double-distilled water. The optimized concentration for each size was chosen so that there was a minimum waiting time of 5 minutes for a bead to diffuse into the trap. This was to reduce the risk of having multiple beads in the trap at same time. To break possible agglomerates between the particles, the solution was ultrasonicated for 15 minutes before it was introduced into the sample chamber. Based on the company’s report, the GNPs were already stabilized to prevent aggregation, therefore, there was no need to coat them with extra polymers [12]. At each experiment the positional data were recorded as time series at sampling rate of 22kHz, using a custom-made LabView program. The corner frequency, *f _{c}*, and the spring constant,

*κ*, of each experiment resulted from power spectrum analysis (fit to equation 1) of the recorded data using a Matlab program [23]. Figure 1(a) shows typical power spectra, as well as the positional histograms, of a trapped 97, 194.4 and 254nm GNPs. The width of the histogram, or equivalently, the corner frequency of the power spectrum, can be used to determine the spring constant of the trap. The higher the corner frequency (equivalently, the narrower the histogram), the stronger the trap. Since the GNPs used in this research can not be visualized, it is crucial to have a consistent method to make sure that there is

*only*one particle in the trap during the measurement. We used power spectrum (online) and standard deviation of positional signal (off-line) as criteria for this reason. Figure 1(b) shows typical power spectra of a single and double 149.4nm GNPs in the trap. It anticipates that when the second particle diffuses into the trap the corner frequency, the power spectral value and the standard deviation of the signal (inset) increase accordingly (the increment can vary from one bead size to another). Considering this fact, on-line power spectrum and off-line standard deviation monitoring doing repeatability of the data helped us to make sure that the sampled data belong to a single particle.

Different sizes of the GNPs were trapped and the trapping strength for each case was calculated from power spectrum analysis of the positional data [23]. The resulted stiffness in the lateral directions for different bead sizes are shown in the Fig. 2 in log-log format. Each data point in the graph is an average of at least 25 measurements (5 particles with 5 measurements for each). It is important to note that a particle could be trapped using a wide range of the laser powers. However, as the heat induced by the absorption can increase the error of the power spectrum analysis [10], the particles were trapped using the minimum possible laser power. This minimum power were size-dependant and it was chosen so that the particle could stay in the trap for at least 3 minutes. In order to compare the results of different sizes, the resulted spring constants were normalized to the laser power at the sample.

Figure 2 illustrates that: 1)The GNPs with diameters of 9.5–254nm were stably trapped in 3-dimensions. To our knowledge, the 9.5nm gold spheres (volume≃500 nm^{3}) are the smallest metallic volume which has ever been trapped using a single beam optical tweezer. This volume is at least 4 times smaller than the previous record [14]. 2)From inset-a it is clear that the trap was weaker in the polarization direction (X) for all examined sizes, in accordance with the theoretical prediction for the Rayleigh particles [24]. 3) In both lateral directions two different regimes can be recognized (diameter of 100nm as turning point), in accordance with the theoretical prediction described earlier. For first regime (particles below 100nm in diameter) data were fitted to *a ^{α}* (red dashed lines) to be able to compare our results to previous reports[12] with results of

*α*=2.68±0.08 and

*α*=2.68±0.07 for

*X*and

*Y*directions, respectively. It is clear that the resulted values are not same as suggested value of

*α*=3[12]. The reason is that, not for all particle sizes in this regime the whole body of the particle is polarized, therefore, the

*corrected-volume*concept must be considered. To account for this correction, the data were fitted to equation 2 (inset-b) considering

*δ*and

*A*as fitting parameters with results of

*δ*=22.4±9.6nm and $A=\left(2.0\pm 0.7\right)\times {10}^{-5}\frac{{p}^{N}}{n{m}^{3}.W}$ (balck solid line) for

*X*and

*δ*=20.2±0.9nm and $A=\left(4.1\pm 0.1\right)\times {10}^{-5}\frac{{p}^{N}}{n{m}^{3}.W}$ (red dashed line) for

*Y*directions, respectively. It should be mentioned that resulted values for skin depth (

*δ*) are in very good agreement with the theoretical prediction of 23nm [9]. The larger error bar in

*X*direction could be because of the lower trapping strength in that direction. 4)Although, there is not a concrete theory for second regime (because of high extinction), a footprint of a power-law behavior can still be seen on the graph. Fitting to

*a*at this regime (blue dotted lines) resulted in

^{α}*α*=1.92±0.07 (

*α*=2.13±0.11) for

*X*(

*Y*) direction. 5) It is worth mentioning that the GNPs with diameters of 5.4nm were stayed in trap for only 2–3 seconds using 415mW of the laser power at the sample (data are not shown in Fig. 2). This was the highest power which our laser source could provide.

To map the optical potential well, one can use the positional distribution, *P*(*x*), of the trapped particle. Since *P*(*x*) obeys Boltzmann statistics, the optical potential, *U*, can be estimated as *U* ∼ ln[*P*(*x*)]. It should be mentioned that, since it is based on the Brownian excursion of the particle in the trap, this method fails to estimate the full depth of the potential well, unless the particle escapes the trap. For a stably trapped particle, this method can only monitor a window of 7–10*k _{B}T* from the potential well. Figure 3 shows typical potential wells for 9.5, 15.6 and 31.4nm GNPs. Symbols represent the measured values while the full lines show harmonic fit to the experimental data. It can be seen that the produced optical potentials are fairly harmonic. The depth of 8–9

*k*implies that the particles were stably trapped. We checked this point experimentally by monitoring for longer periods which showed no change in the results after couple of minutes. The potential depth for 5.4nm particles were ∼3

_{B}T*k*(data are not shown) which explains why they did not stay in the trap for longer than 2–3 seconds.

_{B}TIt is crucial to know the ultimate forces which can be applied by trapped GNPs. One simple way to measure this force is to push the trapped particle to borders of the potential well using drag force [25]. Figure 4 shows the results of a set of such experiments for different GNPs. At each experiment once the particle was trapped (at depth of 3*μm*), first the positional signal from photodiode was recorded and a Matlab program was used to fit a Lorentzian function to power spectrum of the recorded data, giving rise to spring constant, *κ*, and diffusion constant, *D _{v}* [23]. It is worth mentioning that since the recorded positional signals are in Volt scale, the diffusion constant given by Matlab fitting (

*D*) would be in scale of

_{V}*Volt*

^{2}/

*S*. The conversion factor,

*β*, which results

*D*to

_{v}*D*(diffusion constant in scale of

_{SI}*m*

^{2}/

*s*) can be calculated by $\beta \left(m/\mathrm{Volt}\right)=\sqrt{\frac{{D}_{\mathrm{SI}}({m}^{2}/S)}{{D}_{V}({\mathrm{Volt}}^{2}/S)}}$ (Figure 4(a)).Then the piezo stage was moved in triangular manner applying constant forces in the lateral directions while recording photodiode signal. Figure 4(b) shows the histogram of a typical positional signal. Note that the positions of the maximums in Fig. 4(b) represents the displacement of the particle (in Volt scale) in the trap in response to applied drag force. The real displacement can be resulted by multiplying

*β*to the positional signals. Figures 4(c) and 4(d) illustrates the results of such drag-force experiments using 194nm and 254nm GNPs. It can be seen that the optical potential is fairly harmonic. For each size the stiffness resulted from harmonic fit to data points was in very good agreement with the one from power spectrum analysis. It should be mentioned that using this method one can map both shape and depth of the potential well which enables one to measure the maximum applicable force in each case. For 194nm and 254nm GNPs the maximum force was measured to be 1.9pN and 2.38pN using 50mW and 35mW of laser at the sample, respectively. One can see that a few tens of milli-Watts of laser power can produce pico-Newton range forces. Comparison of our data to previous reports [12] reveals that our improved trapping method can produce a trap which is ∼4 times more efficient for nanoparticles.

## 4. Conclusion

We have stably trapped the gold nanospheres ranging from 9.8 to 254nm in diameter using a nearly-aberration-free single beam optical trap. To our knowledge, the 9.5nm nanogold is the smallest metallic volume ever been trapped by a single beam optical tweezer. The 5.4nm gold beads were trapped only for 2–3 seconds. Our experimental results quantitatively verifies the volume corrected Rayleigh model for particles below 100nm in diameter. Although there is not a concrete theory for particles bigger than 100nm, our experimental data show a power-law behavior of *a ^{α}* with

*α*≃2 for normalized stiffness of the trap. Measuring the maximum applicable fore using gold nanoparticles, we have shown that our trap is ∼4 times more efficient for nanoparticle trapping compared to previous works.

## Acknowledgments

We acknowledge stimulating discussion with P. M. Bendix, L. B. Oddershede and A. Richardson as well as their valuable comments.

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