Optodynamic phenomena in aggregates of polydisperse plasmonic nanoparticles

Abstract

We propose an optodynamical model of interaction of pulsed laser radiation with aggregates of spherical metallic nanoparticles embedded into host media. The model takes into account polydispersity of particles, pair interactions between the particles, dissipation of absorbed energy, heating and melting of the metallic core of particles and of their polymer adsorption layers, and heat exchange between electron and ion components of the particle material as well as heat exchange with the interparticle medium. Temperature dependence of the electron relaxation constant of the particle material and the effect of this dependence on interaction of nanoparticles with laser radiation are first taken into consideration. We study in detail light-induced processes in the simplest resonant domains of multiparticle aggregates consisting of two particles of an arbitrary size in aqueous medium. Optical interparticle forces are realized due to the light-induced dipole interaction. The dipole moment of each particle is calculated by the coupled dipole method (with correction for the effect of higher multipoles). We determined the role of various interrelated factors leading to photomodification of resonant domains and found an essential difference in the photomodification mechanisms between polydisperse and monodisperse nanostructures.

Appendix: The heating rate of particles in a polydisperse dimer

Consider a polydisperse dimer (R ₂ < R ₁, R ₁, R ₂ are the radii of dimer particles) in the external field E ₀ . The heating rate is determined by the absorbed power density

$${\text{d}}T/{\text{d}}t \sim W/R^{3} ,\quad W \sim |{\mathbf{d}}|^{2} \text{Im} \left( {\frac{1}{{\alpha^{*} }}} \right)$$

(37)

where W is the power of radiation absorbed by the particle. Compare the heating rates of the particles. The interaction of particles will be considered in dipole approximation. We assume that

$$(\omega_{\text{pl}} )_{1,2} \gg \Upgamma_{2} > \Upgamma_{1} \quad u\quad (\omega_{\text{pl}} )_{1} \approx (\omega_{\text{pl}} )_{2} = (\omega_{\text{pl}} )$$

(38)

that is, the surface plasmon resonances of both particles differ very little. Hence, for the dipole moments of the particles, we can write down:

$$|{\mathbf{d}}_{1} | = \varepsilon_{m} |\alpha_{1} | \cdot |{\mathbf{E}}_{0} + {\mathbf{E}}_{2} |,\quad |{\mathbf{d}}_{2} | = \varepsilon_{m} |\alpha_{2} | \cdot |{\mathbf{E}}_{0} + {\mathbf{E}}_{1} |,\quad {\mathbf{E}}_{1} = \frac{{2{\mathbf{d}}_{1} }}{{\varepsilon_{m} r_{12}^{3} }},\quad {\mathbf{E}}_{2} = \frac{{2{\mathbf{d}}_{2} }}{{\varepsilon_{m} r_{12}^{3} }},$$

(39)

where E ₁ and E ₂ are the fields produced by the first and the second particle at the location of the neighboring one, r ₁₂ = R ₁ + R ₂ + h _ij , h _ij is the interparticle gap. Substituting (39) into (37) yields

$$\frac{{W_{1} }}{{R_{1}^{3} }}\sim \varepsilon_{m}^{2} \frac{{\left| {\alpha_{1} } \right|^{2} }}{{R_{1}^{3} }}|{\mathbf{E}}_{0} |^{2} \left| {1 + \frac{{2\alpha_{2} }}{{r_{12}^{3} }}} \right|^{2} \cdot \text{Im} \left( {\frac{1}{{\alpha_{1}^{*} }}} \right),\quad \frac{{W_{2} }}{{R_{2}^{3} }}\sim \varepsilon_{m}^{2} \frac{{\left| {\alpha_{2} } \right|^{2} }}{{R_{2}^{3} }}|{\mathbf{E}}_{0} |^{2} \left| {1 + \frac{{2\alpha_{1} }}{{r_{12}^{3} }}} \right|^{2} \cdot \text{Im} \left( {\frac{1}{{\alpha_{2}^{*} }}} \right).$$

(40)

Polarizability of the particles $\alpha_{1,2} = \chi_{1,2}^{\prime } - i\chi_{1,2}^{\prime \prime }$ at ω ≈ ω _pl can be represented as [4]:

$$\alpha_{1,2} = - i\chi_{1,2}^{\prime \prime } ,\quad \chi_{1,2}^{\prime \prime } = r_{1,2}^{3} \frac{{\omega_{\text{pl}} }}{{\Upgamma_{1,2} }}.$$

(41)

The specific absorbed power is found by substituting (41) into (40):

$$\frac{{W_{1} }}{{R_{1}^{3} }}\sim \varepsilon_{m}^{2} \left( {\frac{{\omega_{\text{pl}} }}{{\Upgamma_{1} }}} \right)|{\mathbf{E}}_{0} |^{2} \left| {1 - i\frac{{2\omega_{\text{pl}} }}{{\Upgamma_{2} }} \cdot \frac{{r_{2}^{3} }}{{r_{ij}^{3} }}} \right|^{2} ,\quad \frac{{W_{2} }}{{R_{2}^{3} }}\sim \varepsilon_{m}^{2} \left( {\frac{{\omega_{\text{pl}} }}{{\Upgamma_{2} }}} \right)|{\mathbf{E}}_{0} |^{2} \left| {1 - i\frac{{2\omega_{\text{pl}} }}{{\Upgamma_{1} }} \cdot \frac{{r_{1}^{3} }}{{r_{ij}^{3} }}} \right|^{2} .$$

(42)

Hence, the heating rates will relate as:

$$\frac{{W_{2} /R_{2}^{3} }}{{W_{1} /R_{1}^{3} }} \approx \frac{{\Upgamma_{2} }}{{\Upgamma_{1} }} \cdot \frac{{\Upgamma_{1}^{2} + 4\omega_{\text{pl}}^{2} (R_{1} /r_{12} )^{6} }}{{\Upgamma_{2}^{2} + 4\omega_{\text{pl}}^{2} (R_{2} /r_{12} )^{6} }}.$$

(43)

Using (43) for our case, we obtain: R ₁ = 8 nm, R ₂ = 2 nm, r ₁₂ = 11 nm, ω _pl = 4.7 × 10¹⁵ s⁻¹, $\Upgamma_{1,2} = \Upgamma_{\infty } + \frac{{v_{\rm F} }}{{r_{1,2} }},\,\Upgamma_{\infty } = 1.5 \times 10^{14} \,s^{ - 1} ,\,\Upgamma_{1} = 3.25 \times 10^{14} \,s^{ - 1} ,\,\Upgamma_{2} = 8.5 \times 10^{14} \,s^{ - 1} .$ As the result (W ₂/R ³₂ )/(W ₁/R ³₁ ) ≈ 47. That is, the initial heating rate of the small particle is much higher than that of the large particle.