Ultimate Photo-Thermo-Acoustic Efficiency of Graphene Aerogels

The ability to generate, amplify, mix, and modulate sound with no harmonic distortion in a passive opto-acoustic device would revolutionize the field of acoustics. The photo-thermo-acoustic (PTA) effect allows to transduce light into sound without any bulk electro-mechanically moving parts and electrical connections, as for conventional loudspeakers. Also, PTA devices can be integrated with standard silicon complementary metal-oxide semiconductor (CMOS) fabrication techniques. Here, we demonstrate that the ultimate PTA efficiency of graphene aerogels, depending on their particular thermal and optical properties, can be experimentally achieved by reducing their mass density. Furthermore, we illustrate that the aerogels behave as an omnidirectional pointsource throughout the audible range with no harmonic distortion. This research represents a breakthrough for audio-visual consumer technologies and it could pave the way to novel opto-acoustic sensing devices.


PHOTO-THERMO-ACOUSTIC MODEL
A general analytical solution for the root-mean-square sound pressure amplitude in the photo-thermo-acoustic (PTA) model can be derived by the thermo-acoustic model 1,2 as follows with R 0 /r 0 the Rayleigh correction for far-field regime, being r 0 the distance from the sound source and R 0 = S/λ g with λ g = v g /f the gas acoustic wavelength, S the sample illuminated surface area, and v g and f the gas speed and frequency of sound, respectively; e i = k i ρ i C p,i the thermal effusivity of the sample (i = s) and the gas (i = g), with k i the thermal conductivity, ρ i the mass density, and C p,i the specific heat capacity; γ = C p,g /C v,g the gas adiabatic index, q 0 (λ) = Q 0 (λ)/S the power density of the light of wavelength λ absorbed by the sample with power amplitude , with a(λ) = 1 − R(λ) the sample absorptivity and R(λ) the sample reflectivity, σ the Stephen-Boltzmann constant, T 1,2 the light source and sample temperature, respectively, and h the convection coefficient; ] the far-field directivity, with κ = 2π/λ g the sound wavevector, L x,y the lateral dimensions of the illuminated spot (i.e., the PTA diaphragm), and θ and φ the azimuthal and polar angle, respectively. For a self-standing sample with no backing material 2 M (f ) = k s σ s tanh (σ s L s ), where L s is the sample thickness and σ s = i2πf /α s , with α s the thermal diffusivity of the sample. Equation 1 practically reduces to with T g the gas temperature. Eq. 2 holds at low frequency when the sample thickness L s < 2πµ, being µ = α s /πf the thermal diffusion length. On the other hand, at high frequency when L s > 2πµ, M (f ) = 1 and Eq. 2 reads This is our case, as for graphene aerogels µ = 10 µm−0.1 mm and d = 2 − 20 mm. From the above equation is clear that e s > −e g , therefore ρ s ≥ ρ 0 ≡ e 2 g /k s C p,s . The ultimate limit of the PTA model is for D(θ, φ) = 1 and e s → e g or ρ → ρ 0 , which reads pressure is only frequency limited. Since the corresponding sound intensity is I = p 2 rms /ρ g v g , the acoustic power in half space P ac = Ir 2 2π 0 π/2 0 D 2 (θ, φ) sin θdθdφ has a maximum at distance r = r 0 for a point-source, which reads and for a directive source being f = v g /πL x the high-frequency cut-off of the acoustic power. Therefore, the ultimate and beyond the audible range, where the source becomes directive reads

MASS DENSITY CHARACTERIZATION
The mass density of the graphene aerogels can be estimated at first approximation by ρ = m/V . However, this is the total mass density, as aerogels are matter with a mix phase of gas and solid. Therefore, a better estimation of the solid mass density can be provided by the effective medium approximation ρ = ρ c (1 − Φ air ) + ρ air Φ air , being Φ c + Φ air = 1, where ρ c and Φ c = V c /V are respectively the density and the volume fraction of the carbon phase, while ρ air = 1.225 kg/m 3 and Φ air = V air /V are the density and the volume fraction of air, respectively. Hence, the effective mass density of the sample is defined as In order to obtain the carbon mass density ρ c , we estimated the air volume fraction by carrying out contact angle measurements, being the aerogels hydrophobic in the Cassie-Baxter regime 3 ( Figure S1). Images of sessile water drops cast on over 30 graphene aerogels were acquired by Dataphysics OCA instrument and analyzed by its software. In order to estimate the average, maximum, and minimum contact angle on the heterogeneous and porous aerogel surface (contact angle hysteresis ≈ 15 • ), static, advanced, and receding contact angles 3 were measured, respectively, by increasing and decreasing the volume of the water drop by 1 µL step. The deionized water (18 MΩcm) drop volume used to achieve the static contact angles of the samples was V = 15 µL. Moreover, every contact angle was measured 15 s after drop casting, to ensure that the droplet reached its equilibrium position.
The Cassie-Baxter law in the hydrophobic regime reads 3 where Φ is the volume fraction of air pockets underneath the water droplet, Θ * is the measured apparent contact angle, and Θ Y = 87 • the corresponding Young contact angle measured on a highly oriented pyrolytic graphite (HOPG) substrate. Since the air pockets occur due to the aerogel porous surface, we estimated an aerogel air volume fraction Φ air = 0.43 − 0.92 by considering the air pocket volume fraction underneath the water droplets cast on several spots of the aerogel surface. Since the aerogel is a three-dimensional fractal solid with a self-similar structure, we assumed that the air volume fraction on the aerogel surface is the same in the aerogel volume.

THERMAL CHARACTERIZATION
The graphene aerogel thermal properties were investigated by a Fluke Ti20 thermal camera. The sample base surface were illuminated by a panchromatic LED light source of power Q 1 = 0.5 W with a Gaussian spot profile ( Figure S3a, b). Heat is exchanged in time between the LED and the sample by radiation and convection, which reads with S ≈ 2 cm 2 and a(λ) ≡ 1 − R(λ) ≈ 0.99 the sample illuminated surface area and absorptivity, respectively, R(λ) < 0.01 the sample reflectivity, σ the Stephen-Boltzmann constant, T 1,2 the light source and sample temperature, respectively, and h = 1 − 10 W/m 2 K the experimental air convection coefficient, which is compatible with the value reported for air laminar flow 4 . In our experiments, we observed that the heat convection term is negligible and all the light power is absorbed by the sample (Q 0 = Q 1 ).
The heat-up phase was recorded in order to obtain the heat capacity C s ≈ 10 −2 ± 10 −3 J/K of the samples ( Figure S3c). Since the Debye temperature of carbon allotropes is Θ D > 1000 K 5 , the heat capacity is not stationary in the range of temperature considered.
Therefore, since the steady state is the one that contributes to the sound generation, we took into account the maximum value of heat capacity where t 0 is the initial time and T M AX,M IN the sample maximum and minimum temperature, respectively.
Thermal conductivity k s ≈ 10 −2 ± 10 −3 W/mK of the samples was derived by a thermal profile image of the illuminated sample base surface at the steady state ( Figure S3d), by fitting data with the Fourier law of heat conduction in cylindrical coordinates where r 1,2 are the coordinates of the radial thermal profile of temperature T and d ≈ 1.5 cm is the sample thickness.
Thermal diffusivity α s of the samples was studied by illuminating the samples on the base surface and recording the heat-up phase at the orthogonal lateral surface ( Figure S3e).
Data was fit by the law 6 in order to achieve the value of α s ≈ 10 −6 ± 10 −7 m 2 /s.