Letter pubs.acs.org/NanoLett
Fluctuation-Mediated Optical Screening of Nanoparticles Mamoru Tamura†,‡ and Takuya Iida*,†,§ †
Nanoscience and Nanotechnology Research Center, Osaka Prefecture University, 1-2, Gakuencho, Nakaku, Sakai, Osaka 599-8570, Japan ‡ Department of Physics and Electronics, Osaka Prefecture University, 1-1, Gakuen-cho, Nakaku, Sakai, Osaka 599-8531, Japan § PRESTO, Japan Science and Technology Agency, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan S Supporting Information *
ABSTRACT: Inspired by biological motors, we propose a guiding principle for selectively separating nanoparticles (NPs) by efficiently using the light-induced force (LIF) and thermal fluctuations. We demonstrate the possibility of transporting metallic NPs of different sizes with a size-selection accuracy of less than 10 nm even at room temperature by designing asymmetric spatiotemporal light fields. This technique will lead to unconventional nanoextraction processes based on light and fluctuations.
KEYWORDS: Nanoparticle, optical tweezers, fluctuations, chromatography, analytical chemistry
M
of the light irradiation is appropriately designed. Brownian motion due to the random collisions of molecules in the surrounding medium is usually considered as a significant disruption factor in the conventional optical manipulation. However, when the depth and time modulation of the optical potential well with a characteristic symmetry property are tuned to the Brownian motion, there is a possibility that we can efficiently harness the effects of thermal fluctuations. Changing our viewpoint to biological motors, there is a hypothesis that they utilize the fluctuations and the temporally modulated asymmetric interaction potential to effectively transport substances in the body,26,27 although the mechanism of realistic biological motors is complicated. In this Letter, inspired by this idea, we propose a new principle for separating NPs through a theoretical study that incorporates the effects of thermal fluctuations under designed spatiotemporal profiles of an electromagnetic light field. To evaluate the dynamics of NPs under light irradiation and thermal fluctuations, we employ the “Light-induced-force nano dynamics method (LNDM),”28 which we recently developed based on a general expression of the LIF derived from the Lorentz force29 and Langevin equation.30 If the velocity of the NP is assumed to decay and attain a constant value during the time step Δt, the Langevin equation can be written as
etallic nanomaterials show a strong optical response arising from the localized surface plasmon (LSP), even at room temperature, that is strongly dependent on the size, shape, and quality of the internal structure.1−4 This structuredependent property is utilized in various fields such as biosensors in analytical chemistry,5−7 single-molecule spectroscopy,8 and medical optics.9 In the chemical synthesis of metallic nanoparticles (NPs), a wide dispersion in the size, shape, and quality of the NPs is a crucial problem to be solved as many applications often demand a high homogeneity. A number of studies have consider specific size and shape selection of NPs based on chemical and physical methods such as gas expanding liquids,10 periodic gap structures,11 filtration,12,13 electrophoresis,14,15 depletion,16 centrifugation,17,18 and chromatography.18−22 In addition, an optical sorting method based on the quantum properties of semiconductor NPs have been theoretically proposed23,24 and its applicability has been experimentally demonstrated.25 This method is a certain type of optical chromatography (OC) using the radiation pressure from a laser whose wavelength is resonant with the NP quantum states. This provides noncontact and nondestructive sorting based on the light-induced force (LIF) and is a physical process independent of any chemical reaction. While the OC with quantum effect will be a useful method for extracting NPs with a nanoscale accuracy in size as discussed in ref 24, a special environment with small fluctuations at very low temperatures and a low viscosity like superfluid helium is required to effectively exploit the quantum effect. Previous reports have assumed that the target objects are manipulated by an LIF that exceeds the fluctuations, but we can make the most of fluctuations at room temperature if the spatiotemporal profile © XXXX American Chemical Society
r(t + Δt ) = r(t ) +
1 FLIFΔt + Δrran ξ
(1)
Received: July 23, 2012 Revised: August 24, 2012
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where r is the position of the NP at time t, ξ = 3πηd is the friction coefficient for a NP with mass m and diameter d in a medium with viscosity η, FLIF is the LIF exerted on the target NP, and Δrran is the random displacement for each time step due to the random force of the thermal fluctuations given by the standard deviation σ satisfying the relation σ2 = 2kBTΔt/ξ . Thus, when Δt ≫ tfric(= m/ξ) is satisfied and the displacement during Δt is sufficiently smaller than the size of the NP, eq 1 describes random motion from the collision of molecules in the supporting medium. In addition, the general expression of the LIF exerted on the NP is given by FLIF =
1 Re[∑ 2 ω
∫V
dr(∇E(0)(r, ω)*) ·P(r, ω)] (2)
where E is the electric field of the incident light, and P is the induced polarization. Equation 2 includes the dissipative force (scattering and absorbing forces) of the light momentum transfer and the gradient force (trapping force) depending on the inhomogeneity of the light intensity for a given electromagnetic field. For N NPs, the induced polarizations P can be calculated by solving the simultaneous equations (0)
N
Ei = E(0) i +
∑ Gimed , j · Pj Vj (3)
j=1
Pj = χj Ej
(4) 3
Gi,jmed
where Vj = (4π/3)(dj/2) is the volume of each NP, is the Green’s function in the homogeneous medium, and χj is the electric susceptibility in the light frequency domain. Metallic NPs are considered as targets, and χj is given by the Drude model as follows χj = (εb − εmed) −
Figure 1. Profiles of the electric fields and potential of the temporally modulated standing wave consisting of two Gaussian beams (wavelength: 780 nm, b = 0.20). The intensity distributions of the standing wave at (a) cos[Ωt] = 1 and (b) cos[Ωt] = −1 (both normalized by the maximum value in (a)). In (a), the NP is initially located at the origin in a region of high intensity. The arrows indicate counter-propagating waves from both ends (G1 and G2) with the same polarization. (c) The z-position dependence of the potential from the LIF on a NP (diameter: 40 nm) in the assumed standing wave.
(ℏΩ pl)2 ⎛ (ℏω)2 + iℏω⎜γ + ⎝
2Vf ⎞ ⎟ dj ⎠
(5)
where εb is the background dielectric constant, εmed is the dielectric constant of the surrounding medium, Ωpl is the bulk plasmon resonance frequency, γ is the bulk nonradiative decay rate, and Vf is the electron velocity at the Fermi level. Here, we assume gold NPs, where the parameters are given as εb = 12, εmed = 1.77 (in water), ℏΩpl = 8.958 [eV], γ = 72.3 [meV], and Vf = 0.922 [nm eV].31,32 Since we evaluate the dynamics of NPs in a dilute system by neglecting interparticle interactions, eq 3 reduces to E = E(0)+SP by integrating the Green’s function analytically for a single NP (i.e., i = j), where the self-term S is given by SI = −∫ Vi dr′Gmed(ri − r′). From this expression and eq 4, the induced polarization P can be described as P = χE(0)/ (1 − χS), where S gives the resonance frequency of LSP in a metallic NP as discussed in ref 9. To convert the isotropic energy of the thermal fluctuations into an anisotropic transport energy, we consider a superposition of light fields that breaks the spatiotemporal symmetry, as illustrated in Figure 1. The incident light is considered to be a standing wave consisting of two counter-propagating Gaussian beams. While beam G1 propagating in the +z direction has a fixed amplitude, beam G2 has a time-dependent amplitude. The two beams are described as EG1 = ŷE0ug exp[ikz − iωt ]
(6)
EG2 = ŷ E0ug*f (Ωt )exp[−ikz − iωt ]
(7)
where ŷ is the unit vector in y-direction, E0 = (2I/cn)1/2 is the electric field amplitude at the center of the spot (I[W/cm2] is the laser intensity obtained by dividing the laser power by the spot area πw20, c is the velocity of light, and n is the refractive index of the surrounding medium). The wavenumber in the medium is k = n(ω/c), ω is the angular frequency of the light, ug = −iw20Q(z)exp[iQ(z)(x2 + y2)] is the normalized electric field distribution of the Gaussian beam (Q(z) = k/(2z−ikw02)), and w0 is the spot radius (assumed as w0 = 1.0 μm in the numerical calculations). This assumed spot radius w0 = 1.0 μm is possible when the ratio of the lens focal distance to the lens diameter is 2.68 for the given laser wavelength 780 nm that can be generated, for example, by a Ti:Sapphire laser source. While the maximum power of G1 and G2 are the same, the amplitude of G2 is modulated proportional to cos[Ωt ] + 1 + b , (0 ≤ b ≤ 1) (8) 2 by assuming that an attenuating filter is inserted into the light path and rotates periodically with an angular frequency Ω. The function f(Ωt) varies periodically depending on cos[Ωt]; when cos[Ωt] is equal to 1, f(Ωt) is also equal to 1, and the amplitude of G2 takes its maximum value, giving a standing-wave intensity distribution as shown in Figure 1a. In contrast, when cos[Ωt] is equal to −1, then f(Ωt) is equal to b, and the maximum f (Ωt ) = (1 − b)
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stay near the initial potential well due to the relatively weak fluctuation effect. On the other hand, the gradient force on smaller NPs is weaker and the effect of fluctuations becomes relatively stronger. In other words, by controlling the optical trapping, optical transport, and thermal fluctuation balance, we can selectively separate out NPs with particular sizes under an optimized balance of these three factors. Also, it has been confirmed that the transport distance can be enhanced by the effect of fluctuations even when many interacting NPs with the interval of several hundred nanometers are in the laser spot as an initial condition. Moreover, the dissipative force can be also enhanced due to the redshift of LSP resonance arising from the electromagnetic interaction,28,36 which could be useful for the efficient transport. We now examine this possibility in detail. Results of the LNDM simulation based on eq 1 for gold NPs with diameters of 40, 50, and 60 nm in a standing wave defined by eq 2 for laser powers of 200, 300, and 400 mW and b = 0.20 are shown in Figure 2. We first consider NPs 40 nm in diameter as the
intensity of the standing wave decreases as shown in Figure 1b. In the optical manipulation of metallic NPs, the depth of the optical potential well for a laser of several hundred milliwatts is comparable to the energy of the thermal fluctuations at room temperature. For example, the considered laser intensity of each beam is 12.73 MW/cm2 for 400 mW input power and w0 = 1.0 μm, which is the same order as that used in refs 33 and 34. If the variation of the trapping potential created by the standing wave is tuned to the diffusion velocity of the thermal fluctuations, NPs can escape the optical potential wells with the help of thermal fluctuations. Here, we assume that the standing wave vibrates slowly with a time scale comparable to the hopping between several optical potential wells by diffusion (approximately 1.6 μm during 0.1 s), and we employ a time cycle of Ω = 8π[rad/s] for a vibration of 4 times per 1 s to utilize the fluctuations effectively. The laser wavelength is tuned to 780 nm giving optimum dissipative and gradient force magnitudes for gold NPs28 in a medium of water at room temperature (T = 298 K gives a thermal energy of kBT = 26 meV as the criteria for the optical potential, where kB is the Boltzmann constant). The balance of dissipative and gradient forces on a NP changes according to the intensity and spatial variation of the standing wave (Figure 1c). For the time that cos[Ωt] < 1 is satisfied, the negative slope of the potential well envelope function originates from the dissipative force since the light momentum from beam G1 exceeds that of beam G2 (green and blue lines). In particular, periodic potential wells generated by the standing wave show transiently an asymmetric spatial structure as the amplitude of G2 is decreased. The timedependent asymmetric potential is related to the thermal ratchet problem and controlling the periodic asymmetric potential well depth and modulation time are key factors for effective NP transport under thermal fluctuations from the viewpoint of stochastic resonance.35 To examine the effects of the fluctuations, we first consider the NP dynamics in the absence of thermal fluctuations. In the initial state, NPs are trapped by the gradient force at the origin where the optical field has a high intensity, and this gradient force gradually weakens with the decreasing electric field intensity of G2 and results in shallow potential wells. Conversely, the dissipative force on the NPs increases and pushes the particles in the propagation direction of G1 as the intensity of G2 is decreased. However, since the intensity of G2 always has a finite value, the NPs cannot escape the initial potential well without the assistance of thermal fluctuations. In reality, since the effect of the thermal fluctuations exists, the NPs exhibit Brownian motion arising from the random collisions of the molecules in the medium. When the amplitude of the standing wave is a maximum (for example, a potential well depth greater than 10kBT for a 400 mW laser), the gradient force dominates and random force from the thermal fluctuations does not disturb the trapping of NPs of a given size with a sufficiently high laser intensity. When the intensity of G2 decreases and the gradient force weakens, the effect of thermal fluctuations becomes non-negligible (as indicated by the blue line in Figure 1c). In this case, NPs are pushed into positive direction by the dissipative force arising from G1, and the thermal fluctuations accelerate the NP hopping between neighboring potential wells of depths less than 10kBT even for a laser power of 400 mW. Since the trapping strength under balanced gradient and dissipative forces changes depending on the size of the NP, larger NPs are trapped more strongly and
Figure 2. Separation of small NPs by changing the intensity of an asymmetric standing wave. The simulation time is 3 s (15 000 000 steps), and positions are recorded every 0.05 s (250 000 steps). The statistical average of the time dependence of the z-displacement is shown in (a), (c), and (e), where the simulations were repeated 2000 times for each laser power with b = 0.20. Each bar gives the root-meansquare of the z-displacement for that time interval. The number of NPs included in respective regions with an interval of 100 nm along zaxis at t = 3 s is shown in (b), (d), and (f) for the 2000 simulations.
separation targets. Simulations of the dynamics over 3 s (or 15 000 000 time steps) were repeated 2000 times, and statistical averages of the NP displacement are plotted as a function of time (Figure 2a,c,e) to sufficiently take the random effect of the thermal fluctuations into account. Figure 2b,d,f show the numbers of NPs at each z-displacement value after 3 s. Since the NPs exhibit hopping between neighboring potential wells C
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during the short times when the sum of the dissipative and random forces exceeds the gradient force as the electric field of G2 is decreased, the averages of the z-displacement also show steplike behavior. In Figure 2a, we can see that NPs with diameters of not only 40 nm but also 50 nm are transported because the gradient force for a low laser power of 200 mW is not strong enough to trap the 50 nm NPs. However, in Figure 2c,d,e,f with higher intensities (300 and 400 mW) only 40 nm NPs are transported, and the results provide a strong indication that size-sorting is feasible using the proposed principle. However, as shown in Figure 2b, the z-displacement distribution is somewhat broad since the diffusion effect is strong for a weak laser intensity. Examining Figure 2c,d in detail, with a 300 mW laser the depth of the potential well created by the gradient force is appropriate for the thermal fluctuations, and so we can conclude that the NP diffusion effect is smaller than in the case of a 200 mW laser and thus spatial dispersion is suppressed. Meanwhile, due to the synergetic effect of the modulating LIF and thermal fluctuations, the transport distance for a 300 mW laser is greater than in the case of the 400 mW laser, which provides stronger trapping conditions. These results indicate that we can extract small NPs by modulating the relationship between the standing-wave intensity and the thermal fluctuations. Figure 3 reveals the possibility of separating intermediatesized NPs (50 nm diameter). To control the separation, the value of b is varied from 0.15 to 0.25 for a fixed laser power of 150 mW. For a large b value, the degree of asymmetry decreases due to the modulation of the G2 amplitude and the
minimum gradient force on the NPs increases. However, for a small b value, the amplitude modulation of G2 is larger and the minimum gradient force is weaker. Therefore, under conditions shown in Figure 3a,b with b = 0.15, both 40 and 50 nm NPs display strong diffusion and a wide spatial dispersion. The situation changes dramatically in Figure 3c,d with b = 0.20 and a laser power that is 50 mW lower than that in Figure 2a,b. In contrast to Figure 2a,b, 50 nm NPs show a large z-displacement for a weaker laser power. Additionally, while the 40 nm NPs are widely dispersed and become extremely dilute due to the strong diffusion from the thermal fluctuations, the 50 nm NPs are transported unidirectionally and concentrated at a distance of around 4 μm from the origin. Furthermore, in the case of b = 0.25 (Figure 3e,f), the 50 nm NPs move a shorter distance than in the case of b = 0.20 and can be separated distinctly, while the 60 nm NPs remain trapped in the initial potential well. Since a difference in b is related to the spatiotemporal symmetry of the standing wave potential, these results indicate that we can extract particular NPs even of intermediate sizes by modulating the symmetry of the light field. While the averaged transport distance seems to be less than 15 μm during 3 s for the assumed condition, it can be controlled by changing w0 and laser irradiation time. The decay of the laser intensity in zdirection can be suppressed for large value of w0 = 2.0 μm, which leads to long transport distance. For example, the transport distance of 40 nm NP becomes about 30 μm for 1.0 W laser power with w0 = 2.0 μm during 10 s. This means that the separated NPs can be observed under the observation by optical microscope. In conclusion, we have demonstrated the possibility of sizesorting NPs by designing a multiple optical potential well structure whose symmetry is slowly modulated in time by the diffusion time scale arising from the thermal fluctuations in the surrounding medium. Efficient unidirectional transportation of the extracted NPs can be realized by adjusting the intensity and variation of the standing wave amplitude. While we have shown an example intended for extracting NPs with diameters on the order of 10 nm, we can realize selective manipulation with a higher accuracy by optimizing conditions such as the viscosity and temperature of the media, and the wavelength and polarization of the incident light. This fluctuation-mediated optical screening (FMOS) method will pave the way to a nextgeneration guiding principle for analytical nanoscience. In order to realize such a method, an experimental system with a prism to generate standing wave localized at its surface can be considered as an example, where the combination with microfluidic chip for the collection of extracted NPs would be convenient.
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ASSOCIATED CONTENT
S Supporting Information *
Video is provided. This material is available free of charge via the Internet at http://pubs.acs.org.
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Figure 3. Separation of intermediate-sized NP by changing the symmetry of the standing wave b. Simulation results are shown for the same conditions as those in Figure 2 with (a,b) b = 0.15, (c,d), b = 0.20, and (e,f) b = 0.25. The laser power is fixed at 150 mW. (Please see also the movie file in Supporting Information corresponding to Figure 3e,f.)
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest. D
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ACKNOWLEDGMENTS The authors thank Dr. S. Ito, Dr. S. Tokonami, and Dr. C. Kojima, for fruitful discussions about the experiments. They also thank Professor N. Fujimura and Professor H. Ishihara for their support and encouragement. A major part of this work was supported by PRESTO of the Japan Science and Technology Agency, the Special Coordination Funds for Promoting Science and Technology from MEXT (Improvement of Research Environment for Young Researchers (FY 2008-2012)), in part by Grant-in-Aids for Scientific Research (B) No. 23310079, and Exploratory Research No. 23655072 and No. 24654091 from JSPS.
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