## Abstract

The diffusion of silver nanoparticles in water at 298K inside an optical vortex lattice is analyzed in detail by numerical simulations. At power densities of the order of those used to trap nanoparticles with optical tweezers, the dynamic response shows three different regimes depending on the light wavelength. In the first one particles get trapped inside the light vortices following almost closed trajectories. In the second one, around the plasmon resonance, the diffusion constant is dramatically enhanced with respect to the Brownian motion. In the third one, at longer wavelengths, nanoparticles are confined during a few seconds in quasi-one-dimensional optical traps.

© 2011 OSA

## 1. Introduction

Plasmon resonances are behind the unique properties of gold, silver and other metallic nanoparticles [1, 2]. An increasing number of applications involve the use of metallic nanoparticles as fluorescence and Raman-spectroscopy enhancers [3], tracers and nanoprobes for studying intra-cellular structure and biochemical pathways [4], and even as instruments of cell and molecular surgery. The diffusion of those particles in suspension is a preliminary and, in some cases, critical step in most of the relevant processes and applications.

Appropriate light fields can be used not only to benefit from the optical properties of plasmon resonances but also to manipulate and modify the Brownian dynamics of small metallic particles. Gold nanoparticles may be arrested and manipulated in three dimensions, demonstrating trapping down to 18nm in size [5]. The electric fields arising in the intersection region of two crossed optical standing waves provide an interesting laboratory to analyze nanoparticle diffusion on a two-dimensional spatially periodic force field landscape. By simply changing the phase of the two crossed beams, a transition from thermal activated to giant enhanced diffusion of gold nanoparticles has been recently predicted [6].

Due to the plasmon resonance, the forces induced by light on metallic nanoparticles strongly depend on both the field landscape and the wavelength. Although metallic nanoparticles like gold or silver present apparently similar plasmon resonances, the optical forces acting on them can be qualitatively different. Gold absorbs more than silver and its particle plasmon resonance is strongly damped. As a consequence, the real part of the polarizability of a gold nanosphere is always positive which leads to an optical gradient force towards the high field intensity regions. In contrast, silver nanospheres present a sharper plasmon resonance which makes them more suitable as fluorescence enhancers of specific fluorophores [7].

Interestingly, Ag nanospheres present a negative polarizability in a frequency window leading to repulsive gradient forces (in analogy to the so-called blue-detuned confinement of atoms within a low intensity region of a trapping laser). This suggests intriguing dynamic properties for silver nanoparticles. As we will see, for equivalent field landscapes, the dynamics of small silver nanoparticles is profoundly modified with respect to the reported behavior for gold nanoparticles.

Understanding the dynamics of particles moving through a periodic two-dimensional force field is important for both practical and conceptual reasons. Fascinating dynamics has been observed for a large variety of systems, including the motion of driven magnetic vortex lattices [8, 9], surface diffusion [10] colloidal particles or globular DNA in structured microfluidic devices [11–15] in optical [16–20] or magnetic [21, 22], lattices and in nanoscale friction [23, 24]. Manipulation of Brownian dynamics is even relevant in the analysis of financial systems or sociology studies [25]. In particular, manipulation of silver nanoparticles may be useful for several applications. Actually, they have been used in many bactericidal because of the strong toxicity for a wide range of micro-organisms [26–28]. However, the inability to control penetration of pharmacological agents into membrane, impeding the success of pharmacological treatment, is a major drawback. So, the study of nanoparticles transport and diffusion in aqueous media is fundamental to improve efficacy of drug delivering and to enable penetration of pharmacological agents.

In this work we analyze the complex dynamics of silver nanoparticles moving in a fluid inside the intersection region of two crossed optical standing waves. In particular we have computed the mean square displacement (MSD) by solving the Langevin molecular dynamics of the silver nanoparticles given by

**F**(r) is the optical force,

*γ*is the friction coefficient and

*ξ*represents an uncorrelated white noise that obeys the fluctuation-dissipation relation.

The simulation shows that the calculated diffusion paths followed by silver nanoparticles in the optical vortex lattice present remarkable features that differ substantially from those calculated for gold spheres. In particular, in the frequency region of negative polarizabilities, a silver nanoparticle can be trapped for long times inside an optical vortex following circular orbits. For very long times, the particle follows a diffusion path jumping from vortex to vortex in agreement with the expected thermally-activated Brownian diffusion. For longer wavelengths (red detuned), the nanoparticles are shown to be trapped in elongated regions with cross section much smaller than the wavelength.

## 2. Optical forces and silver polarizability

Let us consider a small (Rayleigh) silver nanosphere suspended in water. The water acts as a thermodynamic heat bath at temperature *T* = 298*K*. For simplicity, we consider a real relative permitivity *ε* for water in the visible spectral range of interest (*ε* = *n*
^{2} = 1.79, being n the refractive index). The optical properties of a small spherical particle, whose radius (*a*=50nm) is small compared with the incident wavelength, are characterized by the complex polarizability, *α* = *α*
^{′} + *iα*″ [29]

*α*

_{0}= (

*ε*(

_{g}*ω*) −

*ε*)/(

*ε*(

_{g}*ω*) + 2

*ε*) and

*ε*(

_{g}*ω*) is the macroscopic relative permitivity of silver [30],

*ω*the frequency and

*k*the wavenumber. Note that if particles’s radii are smaller than the skin depth (Rayleigh particles) entire volume is polarized and the gradient force scales with the volume of the particle.

In the presence of a light field, the particle moves under the influence of a time averaged light-induced force [31], which may be written as the sum of three terms [32]

*U*is the electric energy density,

**S**the Poynting vector,

*σ*=

*kα*

^{″}is the particle’s extinction cross section and $\u3008{\mathbf{\text{L}}}_{\mathbf{\text{S}}}\u3009\hspace{0.17em}=\hspace{0.17em}\frac{\epsilon {\epsilon}_{0}}{4\omega i}\hspace{0.17em}\{\mathbf{\text{E}}\hspace{0.17em}\times \hspace{0.17em}\text{E}*\}$ is the time averaged spin density of the electromagnetic field. Let us now consider a particular configuration consisting of two crossed one dimensional optical standings waves oriented along the X and the Y axis respectively, with a phase shift of

*π*/2 and both linearly polarized along the z-axis (see inset in Figure 1). The electric field in this configuration is simply given by

^{5}

*W*/

*cm*

^{2}) is similar to the one used in optical tweezers. Optical forces depend on the real and imaginary parts of the polarizability which show strong variations with wavelength. This is shown in Figure 1 where we plot the real and the imaginary parts of the polarizability versus wavelength for the silver spheres of radius

*a*= 50

*nm*. Specifically, at visible wavelength three different scenarios are possible: real part negative (Region I), real part positive and smaller than the imaginary part (Region II) and real part positive and larger than the imaginary part (Region III).

The real part of polarizability governs terms proportional to the potential energy landscape. If this term is positive, as it is the case for large wavelengths, particles become attracted towards the high field intensity regions. However, as the wavelength becomes smaller than the plasmon resonance (blue detuning), *α*
^{′} becomes negative and the particles would be attracted towards the low field intensity regions. On the other hand, the imaginary part, proportional to the extinction cross section, governs the scattering forces that, for this field configuration are non-conservative. The interference between the crossed beams leads to a scattering vortex force field around each node of the potential landscape. For *α*
^{′} positive, the combined effect will spin the particles out of the zero field nodes. However, for *α*
^{′} negative we will have the opposite: the force field (similar to the water flow in a sink) will spin the particles towards the zero field nodes! The interplay between the gradient forces and the non-conservative scattering forces suggests an intriguing behavior of the dynamics of silver particles under such field landscapes.

## 3. Wavelength dependence of silver nanoparticle dynamics.

The behavior of the mean square displacement (MSD) as a function of time gives interesting information about the dynamics of the nanoparticles. In Figure 2 we summarize our results for the time dependence of the MSD (projected on the XY plane) for different illuminating wavelengths. The functional dependence is usually written as *MSD* ∝ *t ^{β}* with

*β*= 2 at the short time ballistic regime, and

*β*= 1 at the diffusive regime. For the particular case of the diffusive regime

*Ln*(

*MSD*) =

*ln*(4

*D*) +

*Ln*(

*t*).

In the log-log plot, the intersection of each MSD curve with a vertical line at *t* = 1*s* in the diffusive regime gives directly the value of the corresponding diffusion constant multiplied by four. The zero field results, corresponding to the Brownian motion of the silver nanoparticles in water, are shown in Figure 2 together with a fit to a unit slope linear function. Motion is induced only by the thermal bath. From the numerical data we obtain (*D* ≃ 5 × 10^{−12}
*m*
^{2}/*s*) which reproduces the well known theoretical result *D* = *k _{B}T*/

*γ*with

*γ*= 6

*πaη*, being

*η*= 8.9 ×10

^{−4}

*kg*/

*ms*the water viscosity at

*T*= 298

*K*.

For *λ* = 350*nm*, i.e. in Region I where *α*
^{′} < 0 < *α*
^{″} (see Figure 1), the conservative gradient force contribution leads to an optical lattice where, due to the negative real part of the polarizability, particles are attracted to dark field regions. Bright intensity regions correspond to effective potential maxima, pushing nanoparticles towards the minimum of the potential where they get trapped, going along a spiral path induced by the non-conservative force (see Figure 3a ). Vortex from the non conservative term of optical forces match up with the potential minima (dark area in Figure 3). The length of the plateau in Figure 2 corresponds to the time expended by the nanoparticle to escape from the optical trap. The diffusion constant obtained from the figure has a value *D* ≃ 2 × 10^{−14}
*m*
^{2}/*s*, smaller than the value for the pure Brownian case. The trapping mechanism confining the nanoparticle in the dark regions, together with the remarkable spiral diffusion paths, differs from the trapping of metallic particles using an annular light field possessing a helical phasefront [33]. In the latter case, scattering forces confine the nanoparticle to the vortex core dark region.

As we enter in Region II, where 0 < *α*
^{′} < *α*
^{″} (see Figure 1), the diffusion constant dramatically increases with respect to the pure Brownian value. Now, the conservative field pattern changes and each minimum turns into a maximum. The diffusion behavior is similar to the one found for gold nanoparticles in a vortex lattice [6], where giant enhanced diffusion has been predicted. In this region, there are no stable equilibrium positions in the whole system. The particle is no longer trapped and moves very fast along vortex boundaries. The diffusion, is equivalent to the one expected for a medium with very low viscosity (see Figure 3b). For *λ* = 450*nm*, we find *D* ≃ 1 × 10^{−10}
*m*
^{2}/*s* (see Figure 2), almost two orders of magnitude larger than the value obtained without external illumination. In analogy with gold, near the silver plasmon resonance, the diffusion is strongly enhanced and vortex are localized around maximum of the potential (see Figure 3).

In Region III, for *λ* = 600*nm*, the conservative field pattern shows almost no difference when compared to the one at Region II however, the non-conservative part, has changed dramatically due to the smaller value of the imaginary part of the polarizability. Now there are no escape paths induced by the non-conservative forces and particles get trapped between maxima for times longer than *t* = 1*s* (see Figure 2). Interestingly, the particles are confined in elongated, cigar-shaped, regions with cross section much smaller than the wavelength (see Figure 3c).

Enhanced diffusion with a value for the diffusion close to *D* ≃ 4 × 10^{−11}
*m*
^{2}/*s* (smaller than the one reported for *λ* = 450*nm*) is also obtained in the borders of Region II. When ℜ (*α*) = 0 the conservative gradient force vanishes and the diffusion is enhanced due to the vortex fields. In this case the diffusion paths (not shown) are not strongly confined along the vortex boundaries. Enhanced diffusion is obtained in Region II even at the boundary when ℜ(*α*) = ℜ(*α*) since there are no stable equilibrium points in the whole system (even though the spatial average of the force is zero). As we enter in Region III, the MSD versus time plots (Figure 2) show a plateau whose length increases as ℜ(*α*) – ℜ(*α*) increases. Increasing the intensity of the laser will enhance the diffusion (in Region II) or trapping mechanisms (Region I and III) but will have no qualitative effect in swapping between the different regimens reported.

## 4. Ratchet effect

The long-time nanoparticles motion changes qualitatively if a second laser is added within a suitable design. It has been recently predicted [34] that by adding an additional independent standing wave, is it possible to build a deterministic ”ratchet” from purely stationary light forces. In the over-damped regime (where the particles follow the force lines), all particle paths were shown to converge to a discrete set of limit periodic trajectories following the same direction. As we will see, at room temperature it is possible to design a deterministic ratchet for silver nanoparticles although, due to the thermal fluctuations, limit trajectories can not be found. We consider the insertion of an independent laser which contributes with a force

*k*= 2

*π*/

*λ*and

*λ*= 450

*nm*.

The laser added produces a potential deformation since, for *λ* = 450*nm*, vortex centers do not coincide with the potential maxima any more. This results in the appearance of saddle points which are attractive in one direction and repulsive in the other one. The consequence is a ratchet anisotropic almost one dimensional motion of the nanoparticles independent of the initial conditions where no bias or noise is applied [34]. Figure 4 summarizes the different diffusion paths for the ratchet system and the three wavelengths regions previously considered.

## 5. Conclusion

To conclude, using Langevin molecular dynamics calculations we have studied the diffusion of silver nanoparticles in water under the electromagnetic interactions produced by two crossed optical standing waves. By tuning the wavelength, the diffusion constant can be changed from trapping to enhanced diffusion. When the real part of polarizability is negative, vortex match up with the minimum of the potential and particles get trapped. Diffusive movement between vortex has been reported. A quasi one dimensional trapping can be obtained when real part of polarizability is positive and larger than the imaginary part. In this region the diffusion constant is very small when compared with the pure Brownian value. Finally, there is a region of wavelengths where there is an enhanced diffusion similar to the one found for gold nanoparticles [6]. Moreover a one way ratchet motion has been reported adding a new laser requiring neither external noise nor driving.

## Acknowledgments

This work has been supported by the Spanish MICINN Consolider NanoLight (*CSD*2007 – 00046), *FIS*2009 – 13430 – *C*02 – 02 Projects and by the Comunidad de Madrid Microseres Program (*S*2009/*TIC –* 1476).

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