Journal of Nanomaterials

Volume 2017 (2017), Article ID 2753934, 9 pages

https://doi.org/10.1155/2017/2753934

## Heat Dissipation of Resonant Absorption in Metal Nanoparticle-Polymer Films Described at Particle Separation Near Resonant Wavelength

^{1}Ralph E. Martin Department of Chemical Engineering, University of Arkansas, Fayetteville, AR 72701, USA^{2}Microelectronics-Photonics Graduate Program, University of Arkansas, Fayetteville, AR 72701, USA

Correspondence should be addressed to D. Keith Roper

Received 4 September 2016; Accepted 25 December 2016; Published 26 January 2017

Academic Editor: Ilaria Fratoddi

Copyright © 2017 Jeremy R. Dunklin and D. Keith Roper. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Polymer films containing plasmonic nanostructures are of increasing interest for development of responsive energy, sensing, and therapeutic systems. The present work evaluates heat dissipated from power absorbed by resonant gold (Au) nanoparticles (NP) with negligible Rayleigh scattering cross sections randomly dispersed in polydimethylsiloxane (PDMS) films. Finite element analysis (FEA) of heat transport was coordinated with characterization of resonant absorption by Mie theory and coupled dipole approximation (CDA). At AuNP particle separation greater than resonant wavelength, correspondence was observed between measured and CDA-predicted optical absorption and FEA-derived power dissipation. At AuNP particle separation less than resonant wavelength, measured extinction increased relative to predicted values, while FEA-derived power dissipation remained comparable to CDA-predicted power absorption before lagging observed extinguished power at higher AuNP content and resulting particle separation. Effects of isolated particles, for example, scattering, and particle-particle interactions, for example, multiple scattering, aggregation on observed optothermal activity were evaluated. These complementary approaches to distinguish contributions to resonant heat dissipation from isolated particle absorption and interparticle interactions support design and adaptive control of thermoplasmonic materials for a variety of implementations.

#### 1. Introduction

Thermal damping of resonant nanoparticles (NPs) dispersed in optically transparent polymer could affect potential implementations in biomedical therapeutics [1, 2], solar cells [3–6], optical interconnects [7–9], sensing [10–12], and chemical separation [13]. Optical damping by subwavelength Mie scatterers [14] in the Rayleigh regime [15] whose scattering cross sections are negligible [16] is due to absorption [17]. Resonant power absorption of isolated nanoparticles [16] and colloid suspensions [18] in homogeneous dielectric environments is describable using Beer-Lambert linearization [19, 20] of Mie theory. Interactions between isolated subwavelength scatterers and their effects on absorption require more extensive characterization, such as dipole approximation [21, 22], finite difference time domain, or T-matrix. Additionally, interference to forward scattering, for example, reflection, refraction, or diffraction, due to an obstacle, [23] dielectric interface [24], applied field [25, 26], or other heterogeneity, may also impact optical extinction [27–29], thermal dynamics [30], and temperature dispersion [31–33]. Evaluation of heat dissipated from power absorbed in nanocomposites to distinguish contributions from isolated scatterers, interacting scatterers, and heterogeneity is important to advance understanding and guide design of thermally responsive nanocomposite materials.

Mie’s solution to Maxwell’s equations elegantly determines scattering and absorption of an electromagnetic plane wave by an isolated, homogenous sphere [34]. The solution, which depends on particle geometry, composition, surrounding dielectric medium, and angle of incidence, allows calculation of efficiencies and cross sections of scattering and absorption as well as intensity distributions. The coupled dipole approximation (CDA) to Maxwell’s equations incorporates Mie particle polarizability and also accounts for interparticle interactions [21, 22]. Multiplying the extinction cross section from homogeneous nanocomposite media containing spherical particles determined via Mie or CDA by the number of particles yields the overall optical response [35]. This is valid when absorbance is linearly proportional to concentration, as in Beer-Lambert law [17]. Beer-Lambert’s law, , relates spectral absorbance, , to AuNP concentration, , and optical pathlength, , using a material- and wavelength-specific absorbance coefficient, [17, 36]. Deviations from Mie and Beer-Lambert Law arise due to particle interactions, for example, multiple scattering, or strongly absorbing media [37, 38]. For nanoparticles in the Rayleigh regime with appreciable absorption cross section, damping of resonant absorption yields a strong photothermal response influenced by incident intensity, NP morphology [39, 40], and host media [41–43].

Heating by absorptive nanoparticles has been examined widely, but accounting for incident and dissipated energy by all mechanisms, that is, calibrating heat dissipated per unit absorption, is rare. Pulsed or continuous resonant irradiation of isolated NP or their assemblies is sufficient to reshape NP [44] and melt (evaporate) surrounding solids (liquids) [42, 45, 46]. The first comprehensive microscale calorimetric description of resonant absorption by NP in colloidal suspension analyzed dissipation via convection, conduction, and radiation [36, 47]. It has since been applied to quantitate resonant absorptive heating of NP deposited randomly on ceramic [41] and polymer [31] substrates and in multiphase [46], phase-change, and open [42] systems. Compact analytic expressions are useful to intuit dissipation and characterize thermal dynamics from geometric and thermodynamic nanocomposite features for effectively one-dimensional dissipation [30]. However, computational analysis remains necessary for systems with multidimensional absorption and dissipation. Correlating resonant absorption due to isolated particles by Mie theory and to particle interaction by CDA with heat dissipation by finite element analysis (FEA) for NPs dispersed randomly in multidimensional systems would increase understanding of these systems to support their integration into opto- and bioelectronic devices [48–51].

The present work used FEA to account for heat dissipation from ca. 1 mm thick PDMS films in which 16 nm AuNP with negligible Rayleigh scattering cross sections were randomly dispersed. Absorbed resonant power was characterized by Mie and CDA solutions. Power dissipation corresponding to measured and FEA-computed temperature profiles at thermal equilibrium was compared with estimates of extinguished power from Mie and CDA. Heat dissipated per unit absorption in AuNP-PDMS was consistent with reported optoplasmonic efficiencies for 15 to 18 nm AuNPs [52]. The compact description developed herein could be useful to guide intuition, design, and development of responsive plasmonic energy materials, sensors, MEMS, and therapeutics for heat-sensitive applications.

#### 2. Materials and Methods

##### 2.1. AuNP-PDMS Thin Film Fabrication

Briefly [29], polydimethylsiloxane (PDMS) films containing randomly dispersed AuNP were fabricated by mixing 1 mg/mL isopropanol suspension of 16 nm ± 2.4 nm diameter AuNP into uncured PDMS at increasing volumes to achieve AuNP concentrations ranging from 0.234 to 3.52 × 10^{12} NP/cm^{3}. Resulting mixtures were degassed, poured in polystyrene sample boxes, covered, and cured in an oven at 60°C for 24 h.

##### 2.2. Optical Characterization

A light microscope (Eclipse LV100, Nikon Instruments, Melville, NY, USA) integrated with a spectrometer (Shamrock 303, Andor Technology, Belfast, UK) was used to measured AuNP-PDMS spectral response. Amplitudes at 532 nm, the excitation wavelength for subsequent thermal experiments, were determined as the difference between 532 nm extinction and extinction measured off-resonance (800 nm) for each sample. This accounted for contributions to extinction from the polymer matrix.

##### 2.3. Mie Solution

Results from Mie solutions for isolated particles were obtained using a publicly available resource, http://nanocomposix.com/pages/tools. Simulations used a 16 nm AuNP in a refractive index corresponding to PDMS (1.42). For these conditions, resonant absorption cross sections for 16 nm AuNP are ~99% of total extinction. Mie cross sections are reported in nm^{2} as a function of wavelength.

##### 2.4. Coupled Dipole Approximation

The CDA [21, 53] treats the nanoparticle in an ensemble [54, 55] as a single dipole whose polarization is directly proportional to the local electric field [56] where is a frequency-dependent polarizability and is the sum of incident irradiation () and fields scattered from other particles in the lattice (). Particle polarizability can be obtained analytically [55], computationally [56, 57], or by series approximations for higher order modes [58]. Effects of changing the angle of incident radiation can be evaluated [59]. The present work solved the CDA with a user-defined array by matrix inversion [51, 60]. Each dipole pair was calculated for a finite number of dipoles and superposed to determine the polarization vector, , at each dipole. Simulations in the present work were performed for 16 nm diameter spheres in a media refractive index for PDMS (1.42). A square 150 × 150 grid was used (90,601 dipoles) at a grating constant of double the Wigner-Seitz radius; , where is the particle radius, is the density of gold (19.3 g/cm^{3}), is gold mass per cubic centimeter of host media, is the media volume, and is the number of particles.

##### 2.5. Thermal Characterization

Resonant laser irradiation of AuNP-PDMS was performed with a 532 nm laser (MXL-FN-532, CNI, Changchun, CN) with an intensity of ca. 25 mW. An infrared camera (ICI 7320, P-Series, Beaumont, TX, USA) captured thermal images of the AuNP-PDMS at 5 Hz during a 10-second ambient capture, 2-minute-and-50-second heating (laser irradiation), and 3-minute cooling period. Tweezers were used to position AuNP-PDMS films such that the incident laser spot (~3 mm) was towards the top of the film, distal from the tweezer tip. Experimental apparatus was enclosed during data collection to minimize effects of forced convection. Each trial was performed in duplicate. Reported values were from a single representative trial.

##### 2.6. Finite Element Analysis (FEA)

Briefly [32], finite element analysis (FEA) used the Heat Transfer in Solids module in Version 5.2a of COMSOL Multiphysics (COMSOL, Stockholm, SE). Thermal equilibrium was simulated with an applied heat source that corresponded to laser irradiation for comparison with measured steady-state temperatures. Radiative, conductive, and convective cooling boundary conditions were used to estimate the rate at which heat was transferred to the surrounding environment (air). Thermal diffusivity controlled the developed temperature profile within the nanocomposite media. Measured two-dimensional AuNP-PDMS temperature profiles at the first instant of cooling were projected into the three-dimensional COMSOL geometry for comparison with steady-state simulated heating results.

The AuNP-PDMS films were assigned physical dimensions corresponding to measured height, length, and thickness. Each film had slightly different dimensions but was approximately 5 × 5 × 1 mm. Values of density and specific heat capacity for each AuNP-PDMS film were estimated as weighted averages of respective values for Au and PDMS based on mass fraction of Au in the film. The thermal conductivity of AuNP-PDMS was assumed to be that of Au-free PDMS, as low mass fraction metallic dispersions in polymer have a negligible effect on bulk thermal conductivity [61]. Density, specific heat capacity, and thermal conductivity values for PDMS used were 970 kg/m^{3}, 1460 kJ/kg K, and 0.16 W/m K, respectively [62]. For Au, values used were 19320 kg/m^{3}, 128 J/kg K, and 317.9 W/mK [63], respectively.

Plasmonic heating from laser irradiation was represented by a Gaussian volumetric heat source centered at the measured laser spot center with a standard deviation of , where was the measured laser spot radius of ca. 1.5 mm. The full width at half maximum (FWHM) of this Gaussian distribution was 1.8 mm. Based on a raw incident laser intensity of 25 mW and this Gaussian function, the laser power incident on the sample, , was estimated by centering the Gaussian function at the apparent laser spot center based on measured thermal results. This resulted in slightly suppressed powers ranging from 20 to 24 mW; that is, outer portions of the incident beam were off the sample. Spectral extinguished powers () were obtained from , where was spectral extinction in absorbance units. Thermally dissipated power value was the fitting parameter in FEA used to match model-derived equilibrium temperature distributions with temperature distributions measured by the infrared camera at steady-state. Its value was adjusted until resulting FEA equilibrium maximum and minimum temperatures within the laser spot were within 0.1°C of measured steady-state values.

#### 3. Results and Discussion

##### 3.1. Optical Response Increased as Particle Separation Decreased below Resonant Wavelength

Measured optical responses of AuNP-PDMS films increased with NP concentration consistent with Mie and CDA results at low concentrations. However, when interparticle separation exceeded resonant wavelength, measured responses increased above calculated extinction. Figure 1(a) compares measured (solid), Mie (dotted), and CDA (dashed) spectra from PDMS films containing 16 nm AuNP at 1.17 × 10^{12} NP/cm^{3} (0.005 mass-percent AuNP; blue) and 2.34 × 10^{12} NP/cm^{3} (0.01 mass-percent AuNP; red). Spectral units are absorbance coefficient, calculated as optical extinction in absorbance units (AU) at 532 nm divided by sample thickness (mm), relative to off-resonance extinction at 800 nm. At 1.17 × 10^{12} NP/cm^{3}, the maximum extinction per mm for measured, Mie, and CDA values was 0.121, 0.140, and 0.120, respectively. At 2.34 × 10^{12} NP/cm^{3}, the maximum measured extinction per thickness was 0.313, thirty percent on average above Mie and CDA estimates of 0.252 and 0.23, respectively. Whether the 2.6-fold increase in optical response from a 2-fold rise in AuNP concentration translated to increased heat dissipation motivated further study.