1. Introduction

Light interaction with metal nanoparticle aggregates results in interparticle forces and torques [1–4]. These forces can move particles and change the geometry of the aggregates. These changes will affect all optical responses of the aggregates because of their strong dependence on the interparticle distances and the overall geometry in the general case. The electron resonances of plasmon-coupled metal nanoparticles enable many known applications. In addition to surface-enhanced phenomena including Raman scattering, fluorescence, absorption and optical nonlinearities, new applications such as plasmonic molecular rulers [5] and cancer phototherapy [6–8] reinforce interest to the optical properties of metal particles aggregates. The dynamics of nanoparticles in an optical electromagnetic field can be an important factor for nanostructures without fixed positions. This is the case for many realizations of plasmonic nanomaterials in solutions or nanoparticles assembled by organic or bio-molecules.

The optics of nano-aggregates has been under comprehensive theoretical study for decades [9–17]. All previous theories including the latest extensions [18–20] involve the assumption of fixed positions for the nanoparticles in an aggregate. However, the concept of the particles at fixed spatial positions cannot be fully consistent because particle interaction implies forces [21–24] that act to move the irradiated particles toward some new equilibrium state. As we pointed out previously [25], this leads to an optical nonlinearity that is specific to the aggregates of nanoparticles, such as colloidal metal clusters, and which does not require any optical nonlinearity from the isolated nanoparticles.

In this work we focus on the effect of light-induced particle motions on the basic properties of metal aggregates, including absorption and scattering of light, and scattering of the probe wave at another frequency. The coupled dipole equations along with Newton’s differential equations are numerically solved for the aggregate of five metal particles, with each particle having purely linear optical polarizability.

We leave beyond the scope of this paper the radiation forces which are exerted on an aggregate of particles as whole [1–4] and play an important role in astrophysics, optical levitation, and optical tweezers. Note that the static problem of the radiation forces was considered in Refs 1–4. We consider the dynamic problem of the optical response of the particles in an aggregate where light-induced forces change only the interparticle geometry.

2. Numerical simulation

An electromagnetic wave expressed as

induces the dipole moment d _i = ε_hα_i E _i for the i-th spherical particle with a complex polarizabil-ity α_i, where E _i is the local electric field at the particle’s position r _i, and ε_h is the permittivity of the host dielectric medium. In the dipole approximation each of the N particles in an aggregate can be replaced by its dipole moment located at the particle center. A self-consistent system of linear equations describes the interaction of the dipoles with the incident field and each other:

where k is the modulus of the wave vector. Also, α_i = α ³ _i(ε_m - ε_h)/(ε_m + 2ε_h), where the metal permittivity ε_m = ε′_m + iε′′ _m is taken at the corresponding optical frequency. The Drude formula for ε_m is given by ε_m = ε_b - ω ² _p/ω(ω + iΓ), where ω_p and Γ are the frequency and decay rate of the plasma oscillation, respectively. The term ε_b, (instead of unity) is introduced to account for the interband electron transitions. The decay rate also includes a size dependent term, Γ = Γ_∞ + Av_F/a , where v_F is the Fermi speed of electrons and a is the particle radius [10]. In the case of Ag particles and Al₂O₃ host, experiments [26] give A = 0.6 for the particles on a substrate and A = 1.6 for fully imbedded particles in a matrix in contrast to A = 0.25 for spherical particles in vacuum. The specific values for silver are the following: ω_p ≈ 9.1 eV Γ ≈ 0.02 eV, ε_b ≈ 5.3, v_F ≈ 1.4 × 10⁸ cm/s [27, 28].

All optical properties of the aggregate within the dipole approximation can be expressed in terms of the coupled dipole equation (CDE) solution, d _i(r ₁,…, r _N). To account for particle displacement, the CDE should be accompanied by Newton’s equations for the particle coordinates:

where

The term U_EM represents the potential energy of the nanoparticles interaction caused by the electromagnetic field. To derive (4), we use an expression for the potential of a point-like dipole D in the electric field E: U = -(E ∙ D), assuming a time average over the fast oscillations [29]. The second term in the right-hand side of (4) is, in fact, a total potential energy of N isolated particles in the field; by subtracting this term we leave in U_EM only the energy of the mutual inter-particle forces.

It is known that the dipole approximation is inaccurate for touching spheres and for their absorption in the infrared spectral range. This issue was discussed in the literature (Ref. 17 and references there). However, it is recognized that the dipole approximation captures the main features of the light interaction with metal aggregates. With the goal of this paper to evaluate distinct optical features of the nanoparticle aggregate introduced by the particle motions, we consider the dipole approximation to be a reasonable approach for the aggregate geometry without touching spheres and the spectral range considered here. Indeed, the exact solution for the polarizability of a particle dimer [30] is within 10% to the accuracy of the dipole approximation in our case.

The binding forces represented in (3) by the potential U ₀ can be of different origins. The origins can be volume forces, van der Waals force, and surface forces such as electrostatic forces and steric forces of polymer-coated particles. The aggregate geometry results from the equilibrium of the attractive and repulsive forces present in the aggregate. The particular potential U ₀ depends on many factors, including the specific system chosen, and its study is the subject of many papers. We will model the potential energy as mainly determined by the surface-to-surface distance between particles, treated as solid balls. In other words, the interaction potential of the i-th and j-th particles U^ij ₀ is a function of Δ_ij ≡ ∣r _i - r _j∣ -a_i -a_j. For instance, the van der Waals interaction between two spheres in the Derjaguin approximation corresponds to the following potential [31]: U_vdw ≃ -a_ia_jA_H/6Δi_j(a_i + a_j), where A_H is the Hamaker constant.

Specifically, the potential is set to the following form:

where ⋃ scales the binding potential. The corresponding potential shown in Fig.1 describes, for example, coalescent aggregates in colloids where the nanoparticles are separated by finite distances with ⋃ ≈ 0.1 e V and the depth of the potential well is about 0.5 eV. In this particular case, the equilibrium distance Δ_eq/a ₁ = 1/3 is taken as a typical value for the aggregated silver colloids in accord with electron microscopy data [32]. The chosen ⋃ corresponds to the van der Waals attractive term with the Hamaker constant A_H = 2 eV, which is close to both the calculated and measured values for metal nanoparticles [31].

The frictional forces in viscous fluid can be evaluated with the Stokes law provided that the typical sizes of the nanoparticles are at least one order of magnitude greater than that of the solvent molecules. This leads the coefficient of the friction forces, which are proportional to the particle radius, to be γ ∼ 6πηa, where η is the fluid viscosity.

The characteristic relaxation time following from (3) is τ_ri = m_i/γ_i. Below we use a designation τ_r for τ _r1 = τ _r3 = τ _r4 = τ _r5. A Gaussian pulse will be considered here for the incident field intensity:

with τ_p = 100τ_r.

 

Fig. 1. The model potential describing the aggregation forces for a pair of nanoparticles.

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By solving the coupled dipole and Newton’s equations one can obtain time dependent particle coordinates and their dipole moments d _i(t). We will be interested in the time/intensity dependent absorption, scattering, and probe field scattering calculated via dipole moments as follows:

where k is the wave vector, k = ω/c, and the scattering amplitude is:

The probe wave scattering cross section with pump at ω_pump can be expressed as

where the corresponding scattering amplitude is:

We consider penta-particle aggregates with a central particle radius a ₂ = 2a ₁ of about 10 nm as shown in Fig. 2. The 4-fold symmetry allows an analytical solution for the eigenvalues of the dipolar interaction operator studied in Ref. 19. The equilibrium distance Δ_eq = 0.326a ₁ between the nearest particles in this aggregate without incident light is slightly different than that for two particles due to the distant particles’ interactions. The calculated absorption spectrum for such an aggregate with Ag in a liquid with ε_h = 1.78 (Fig. 2) has three peaks shifted relative to the single particle surface plasmon resonance (the latter is close to 3 eV). It is known that, in case of the particles of equal size, some resonances are very weak in the quasistatic approximation (when the retardation effect is negligible). For instance, this is the case for the higher energy resonance of a dimer oriented along the field vector [18]. However these resonant modes have comparable amplitude with the “main” resonances in the case of particles with different size. This effect was discussed in more detail for the considered penta-particles aggregate [19]. In our simulations both the retardation effect and the effect of different sizes are taken into account.

 

Fig. 2. The structure (left) and the linear absorption spectrum (right) of the Ag nano-aggregate. The light wave is polarized along the y axis. The dashed lines on the spectrum mark three frequencies, 2.83 eV, 2.865 eV, and 2.89 eV, which are specifically considered in the text.

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Particle motions depend on the light frequency relative to the peak position. A corresponding change in the absorption power normalized per absorption power at low intensity, P_NL(t) = [P(t) - P ₀]/P ₀ is obviously frequency and intensity dependent (Fig. 3). As we see, the overall spectral dependence is rather complicated. There are two different modes of the particles’ light-induced motion: a symmetric one, where all particles move to (or from) the cluster’s center, and an asymmetric mode with two particles moving to (from) the center and other two moving from (to) it. The sign of the contribution from a particle to P_NL of the whole aggregate depends on whether this particle is approaching the center or moving away from it since the dipolar resonance condition is mainly governed by the distance between the nearest particles. From these circumstances, it is possible to fully compensate the contribution of one pair of particles with that of another pair at a corresponding light frequency. At such a frequency the nonlinear part of the aggregate’s absorption equals zero. That is the case for ω ≈ 2.837 eV in Fig. 3(a). The qualitative explanation of the spectral features including the very sharp peak near ω = 2.89 eV is not as straightforward and requires the complete analysis of the temporal-frequency dependence of P_NL, which will be discussed elsewhere.

The following general feature of the studied dynamics is worth mentioning. According to the simulations, the dipole-dipole forces between the particles aligned along the y-axis can be much stronger (1–2 order of magnitude for I_p ∼ 20 MW/cm²) compared to those of the particles aligned along the x-axis. The dynamics of the particles aligned along the two axes is very different too. However, there is no such a difference in the displacements of the corresponding particles. This is because the value of the particle displacement is, basically, determined by the location of the light-induced potential well rather than by its depth.

The pulse-average nonlinear absorption P_NL is less than peak value as shown in Fig. 3(b). For an aggregate concentration providing a linear absorption coefficient of δ ₀ = 0.7 cm^-1 (about 50% in transmittance for a 1-cm sample), at 2.87 eV one can estimate the nonlinear absorption coefficient β. Assuming the absorption coefficient form to be δ = δ ₀ + βI and with a given nonlinear to linear absorption ratio and linear absorption spectrum, β ≈ 2 × 10^-9 cm/W at 2.8 eV and β ≈ -16 × 10^-9 cm/W at 2.87 eV. We note also that β becomes intensity dependent at higher intensities.

Similar behavior is obtained for the nonlinear light scattering in the forward direction (Fig. 4). Specifically, the scattering grows at ω = 2.83 eV and decreases at ω = 2.865 eV in comparison with the linear (low intensity) scattering, as does the nonlinear absorption. A very sharp non-monotonic intensity dependence characterizes the scattering changes near the frequency of 2.89 eV. This correlates with the peculiarity in the nonlinear absorption spectrum, which is also near 2.89 eV (see Fig. 3).

 

Fig. 3. The normalized nonlinear absorption P_NL as a function of light frequency: (a) the peak values P_NL(t ₀) (t ₀ corresponds to the maximum of the incident light pulse); (b) the pulse-average values. The peak intensity I_p of the incoming light amounts to 1.75 MW/cm² for square data points; 17.5 MW/cm² for the open circles, and 57.8 MW/cm² for the solid circles.

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Fig. 4. Relative cross section of the nonlinear forward scattering: F_NL(t ₀) = [F(t ₀)-F ₀]/F ₀ vs. the peak intensity of the excitation pulse. F(t ₀) is given by (8), and F ₀ is the linear (low intensity) scattering cross section. Circles, triangles, and diamonds correspond to different frequencies of the incident light at 2.83 eV, 2.865 eV, and 2.89 eV, respectively.

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We also investigated aggregate particle motion in the case of pump-probe scheme. The pump-probe technique is widely used in optical experiments, and an understanding of the expected motion in such experiments may prove useful. In our case the particle motion is induced by the pump field and the scattering signal is calculated for the probe wave at various frequencies. Figure 5 illustrates the spectrum of the normalized scattering cross section of the probe wave, S_NL(ω) = [F_probe(t, ω) - F ⁰ _probe(ω)]/F ⁰ _probe(ω) at a pump intensity of I_p = 17.5 MW/cm² and three pump frequencies, ω_pump = 2.83 eV, 2.865 eV, and 2.89 eV Here, F ⁰ _probe(ω) is the linear (low pump intensity) cross section of the scattering of the probe wave with frequency ω. The scattering signal with the particle motion is strongly increased on the blue side of the pump frequency by a factor of about 4.6.

 

Fig. 5. Relative scattering cross section of the probe beam in the forward direction vs. the probe frequency ω at pump intensity I_p = 17.5 MW/cm² for (a) pulse-peak values and (b) pulse-average values. The three curves correspond to different frequencies of the pump radiation (the values indicated are in eV), and are the same frequencies as for Fig. 4

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Let us now estimate the characteristic values for the light-induced forces and times. For silver particles with radius a = 7 nm and embedded in a solvent of viscosity η = 10^-3 Pa∙s, the relaxation time is about 0.1 ns. An estimate for the light-induced force gives the following value for a pair of particles (a dimer) with E ₀ directed along the dimer axis:

where ξ = (a ₁ a ₂)^3/2 R ^-3, R[nm] = (a ₁ + a ₂ + Δ) [nm] is the distance between the particles, and κ = α_i/a ³ _i. The derivative in the first term of (12), for Δ = a ₁/3, is about unity in the range ω = 2.8 ÷ 3 eV while the derivative in the second term reaches 3 × 10³ in this spectral region. Thus, we obtain F_EM ≈ 10 pN at I ₀ = 1 × 10⁶ W/cm², a ₁ = 5 nm, a ₂ = 10 nm.

The characteristic time of the particle displacement D without friction can be estimated as τ_D ≈ (2Dμ/F)^1/2, where μ = m ₁ m ₂/(m ₁ + m ₂) = 4.9 × 10^-21 kg. For D = 0.2 nm the displacement time τ_D ≈ 0.5 ns. If τ_D exceeds the relaxation time τ_r = m/γ, the friction is important and increases the characteristic time τ_fD = Dμ/F_EM τ_r = τ ² _D/4τ_r. Indeed, the latter follows from the estimate for the displacement of two equal particles: $D=\frac{2}{\gamma }{\int }_0^{{\tau }_{fD}}{F}_{\mathrm{EM}}\mathrm{dt}=2F_{\mathrm{EM}}{\tau }_{\mathrm{fD}}/\gamma =F_{\mathrm{EM}}{\tau }_{\mathrm{fD}}{\tau }_r/\mu .$

Large changes in the absorption and scattering (nonlinear response) caused by light-induced motion in aggregates enables an interesting way to probe the interparticle potential. Indeed, ∂U ₀(Δ_eq)/∂Δ = 0 at equilibrium at low intensity and ∂U ₀(Δ^I _eq)/∂Δ = -∂U_EM(Δ^I _eq)/∂Δ at a new equilibrium distance Δ^I _eq at high intensity. By varying the intensity one can obtain force-distance dependence for interparticle potential. The typical force value is comparable with the tension of biomolecules and polymers. For instance, the tension for DNA at relative extension 0.2 is about 0.02 ÷ 0.3 pN [33]. If the metal aggregate is formed by biomolecule or polymer conjugation, light-induced motion will probe the tension of the conjugated molecules. In particular, the pump-probe scattering technique gives a variety of options to study induced motion at one frequency/polarization and detect scattering at different frequencies and polarizations of the probe wave. Time-resolved measurements introduce a complimentary option to extract information about the interparticle potential and friction force, employing intensity and pulse energy dependence of rise time and relaxation time τf_D(I) = Dμ/(F ₀+F_EM))τ_r forrectangu-lar pulses as is illustrated in Fig. 6. In addition, the specific frequency dependence of the effects of light-induced motion considered herein enables a method to distinguish these effects from thermal effects which are maximal at plasmon resonances.

 

Fig. 6. Temporal dependence of the relative displacement of particle #3 (dashed line, right axis) and #5 (solid line, left axis). The excitation is a 20-ns rectangular pulse with an intensity I_p = 17.5 MW/cm².

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3. Conclusion

To summarize, the dynamics of metal nanoparticle aggregates irradiated by light affect all of the optical properties of the aggregates. This enables control of interparticle distances through use of nanoscale light-induced forces. We have studied light absorption, scattering and scattering of a probe beam using Newton’s equations and the coupled dipole equations for an artificial medium composed of penta-particle aggregates. All studied optical responses change in time at pulse excitation and depend on light intensity. The sign of the nonlinear coefficients in absorption and scattering depends on the light frequency relative to the plasmon resonance. The relative changes in optical response are large and relatively fast with nanosecond characteristic times. Scattering intensity can be increased by factor 4.6 and absorption by factor 1.4 (specifically, for I_p = 17.5 MW/cm²). That is, the nonlinear contribution caused by the particle motion is comparable with the linear contribution for the absorption and can be quite prevailing for the scattering. Time and intensity dependencies are shown to be sensitive to the initial potential of aggregation forces. Also, the specific frequency dependence provides a method of distinguishing changes due to light-induced motion from those of thermal effects. The typical interparticle force value is comparable with the tension of biomolecules so that this mechanism can be employed to probe interparticle potential including molecular forces.