A model for two-dimensional graphene-based thermoacoustic membranes is investigated analytically and numerically validated using Bayesian statistics in this study. The temperature and the pressure variables are first analytically determined in one-dimension by noticing that the magnitude of the pressure time derivative is small in the heat transfer equations and by taking advantage of the large disparity between the length scales. The one-dimensional findings are then extended to three-dimensions, where pressure fluctuation produced by the surface temperature variation is determined using an acoustic piston model. Through the one and three-dimensional model analysis, the dependence of acoustic pressure as a function of frequency is studied. The acoustic response with respect to the frequency shows different characteristics when assuming Dirichlet (temperature) or Neumann (heat flux) boundary conditions. The thermoacoustic model is validated with a graphene-on-paper loudspeaker using Bayesian statistical methods and a Delayed Rejection Adaptive Metropolis algorithm to identify model parameters and their uncertainty. The findings provide insights into the heat transport mechanisms associated with sound generation from thermally cycling thin conductive membranes at high frequencies.

When a localized region of a solid wall surrounded by a compressible medium, such as air is subjected to a sudden temperature change, the medium near the wall experiences expansion/contraction in volume producing acoustic waves.1 These thermally produced waves are well known as thermoacoustic (TA) waves, based on the thermo-phone mechanism presented by Arnold and Crandall in 1917.2 TA waves generated by sudden heating (or cooling) in the localized region propagate at the local speed of sound of the medium and their amplitudes gradually decay due to thermal and viscous diffusion.

Several theoretical studies have been performed on TA wave generation from thin conductive isotropic, anisotropic, or thermoelastic metal layers.3–5 This phenomenon was experimentally examined by Shinoda et al.6 with a thermally induced high-intensity ultrasonic wave emitter using porous silicon (PS). They provided a formula for the acoustic pressure which showed a flat response over a wide frequency range. Boullosa and Santillan7 also derived a theoretical formula for the acoustic pressure due to any solid including porous silicon using a thermal piston model for ultrasonic radiation from a TA transducer. Both of these models are developed for the TA wave emission at high frequency. Hu et al.8 generalized the TA model to include the TA wave emission at low frequency from any solid by using thermal-mechanical coupling analysis. Vesterinen improved the efficiency of nanothermophones with thermoacoustically operated suspended arrays of nanowires, developed a Green's function formula for the TA pressure, and compared the findings with experimental and numerical results.9 Recently, Hu et al.8 extended their previous theoretical work to analyze the effect of thermal-mechanical coupling and thermal wave penetration depth to the TA wave emission in gas and revealed the existence of limits of both frequency and space of flat frequency response of TA ultrasound.10 Similar TA studies were conducted with a carbon nanotube thin film attached on the substrate11 and in suspension.12 

Recent studies have shown graphene to be an attractive material candidate for acoustic wave generation by Joule heating, particularly owing to its high thermal conductivity, low heat capacity, and its ability to form free-standing membranes.13–15 During Joule heating processes, the free electrons of graphene absorb electric energy applied through the electrode. This electron energy is transferred to the lattice vibrational modes increasing the lattice temperature of graphene and the surrounding media. This leads to sound generation in air as the volume of gas near the surface of graphene expands due to sinusoidal heating. Alternating electric current with angular frequency, ωs, produces a sinusoidal temperature variation with a double angular frequency, 2ωs, in the sample material (graphene) by Joule heating16 and leads to the generation of sinusoidal acoustic waves in the surrounding compressible medium.

We approximate heat generation in graphene and its effect on generating acoustic energy into the surrounding air using continuum thermomechanics. This approach may lead to model uncertainties due to the atomically thin characteristics of graphene. Bayesian statistics are introduced to quantify material parameter uncertainty and the propagation of uncertainty when inversely identifying graphene TA model parameters in light of sound pressure data. This technique has shown to provide new insights into complex field coupled mechanics of other functional materials and adaptive structures17,18 as well as a broader range of problems in weather and climate predictions, nuclear reactor design, and hydrology.19 A three parameter TA model is often implemented based on prior results summarized by Arnold and Crandall.2 We briefly review the physical motivation for this model using the first law of thermodynamics. It is shown that the three parameters within the original model quantify heat conduction, heat convection, and heat capacity. Using Bayesian analysis via the Delayed Rejection Adaptive Metropolis (DRAM) algorithm,20 we illustrate that the TA wave generation is predominantly driven by heat conduction and heat capacity in ambient air (no forced convection).

In what follows, an interface condition that describes temperature and pressure perturbations produced by the thin conductive membrane is developed and validated against past experimental reports on graphene-based speakers. The mathematical formulation of the TA generation from two-dimensional materials is first derived using the first law of thermodynamics. The boundary relations at the graphene interface are coupled with heat transport and wave equations of the surrounding media in contact with graphene. The analytic acoustic response with respect to the frequency in both one- and three-dimensions is carried out. The unknown heat transport parameters in the model are determined by using Bayesian analysis with the Markov Chain Monte Carlo (MCMC) method through comparisons with data.13 

The first law of thermodynamics states

(1)

where q is the heat flux, T is the Cauchy stress, D is the rate of deformation, and u is the internal energy per volume.21 These three terms must be balanced by heat generation g within the volume. For the TA problem, this heat generation is attributed to Joule heating produced by applying current to the membrane. This governing equation is applied over a volume denoted by Ω and subject to a set of boundary conditions (BCs) over the surface Γ.

Questions over the boundary conditions have focused attention on determining how input power in terms of voltage and current (V × I) or current and resistance (I2×R) is converted into heat. This heat subsequently creates temperature and heat flux variation on the surface of a material. Approximations of the heat transport become particularly challenging in nanoscale structures such as graphene. A general constitutive relation is presented as follows to gain insight on the appropriate interface conditions that satisfy Eq. (1).

This balance law can be re-expressed by making certain assumptions about the internal energy and deformation. We first assume that heat has negligible effects on the mechanical power such that T : D ≈ 0 in comparison to the divergence of the heat flux and the internal energy rate. We also assume that the internal energy only depends on entropy, u = u(s). The second assumption makes the claim that the entropy can be uniquely determined in terms of the absolute temperature T, i.e., u = u(s(T)). The rate of change of the internal energy per volume is then assumed to depend on the specific heat relation

(2)

where ρ denotes the density of the material and cp is the specific heat of the material at constant pressure. Note that this specific heat is a function of temperature, which has been documented in the literature for graphene.15 

Substituting Eq. (2) into Eq. (1) and integrating over the volume Ω, we have

(3)

where we applied the divergence theorem on the last term where n is the unit normal directed outwards on the material surface Γ.

For the case of graphene, we consider a two-dimensional material on the (y, z) plane with a small thickness in the x direction denoted by b. This gives

(4)

Assuming uniform properties in x and constant temperature through the thickness due to high thermal conductivity, Eq. (4) leads to

(5)

The heat flux is given as a function of the internal heat generation and rate of change of internal energy in terms of specific heat constitutive relations for any representative point on the surface. Typically, qini=qikni+qcni, where qik=kT/xi is the conduction term and qc=h0(TT) is the convection term. Here, k is taken to be the isotropic thermal conductivity coefficient, h0 is the heat transfer coefficient in air, and T is the surrounding temperature.

A plausible assumption is to allow gJ, where J is Joule heating that is proportional to the electric input power PE. This gives g = PESb, where S is the area of graphene and b is its thickness. The relation between applied electric power and temperature change can then be modeled using Eq. (5), which results in2,12

(6)

where Cb (=ρcp) is the heat capacity per unit volume, Tb is the temperature on the surface, and Q is the heat flow per unit area from the film defined by Q=kT/x|x=0. The coefficients aj (j = 1, 2, and 3) are uncertainty parameters associated with the amount of heat convection, heat conduction, and uncertainty in the product of the effective thickness and heat capacity, Cbb. All unknown parameters aj are unitless and contain uncertainties that are determined using Bayesian statistics in light of TA measurements given in the literature.13 

The input power considered, PE, is given by

  • Purely alternating current

(7)
  • (ii)

    Alternating current with a DC component

(8)

In each case, it can be observed that the electric input power has two components: one is a constant in time and the other is time varying. It is also recognized that electric power due to only an AC with angular frequency, ωs, produces the temperature response with a double angular frequency, 2ωs, but applying an AC with an DC component can lead to a sinusoidal temperature variation with the same frequency as the applied power2 if IdcIac.

The time-invariant power for each case can be determined by introducing constant time-averaged temperature and heat flux, Te and Qe from Eq. (6), which is given by

(9)

By defining ΔT = (TbTe) and ΔQ = (QQe), the interface condition due to the TA material is

(10)

where Um=Iac2R/2,ω=2ωs,ϕ=0 for case (i) and Um=2IdcIacR,ω=ωs,ϕ=π/2 for case (ii). Equation (10) states that only perturbation in the temperature is influenced by the AC source.

Figure 1 represents the schematic of the TA wave emission problem due to the temperature change in the thin membrane by Joule heating. The origin of coordinates is placed on the interface between the sample (graphene) and gas. The gas in the compressible medium is assumed to be ideal. The governing equations for the linear TA problem in isotropic materials are given by the heat conduction and wave equations describing the coupling of pressure and temperature fields

(11)
(12)
(13)

where the subscripts s and g denote values of the solid (substrate) and the gas, respectively, e represents the value in a quiescent equilibrium state, x = (x,y,z) is the spatial coordinates, t is the time, Ti is the temperature, p is the acoustic pressure, ω is the angular frequency, and α is the thermal diffusivity given by αi=ki/(ρicp,i) (i = s or g), where ki is the thermal conductivity, cp,i is the specific heat of the material, and ρi is the density of the medium (Cbi (=ρicp,i) is the heat capacity per unit volume). The corresponding interface condition to this system at x = 0 is given by Eq. (10). We note that Tg and Ts are bounded as x and x → −, respectively.

FIG. 1.

The schematic of the thermoacoustic wave emission problem.

FIG. 1.

The schematic of the thermoacoustic wave emission problem.

Close modal

The sinusoidal thermal diffusion process produces a sinusoidal temperature variation in the gas close to the wall. In addition, the temperature decays exponentially due to thermal and viscous diffusion. The thermal wavelength can be estimated as λth=2πxg=2π2αg/ω, which is much smaller than the acoustic wavelength (for example, λth/λacoustic[1.3×103,8.0×103] for f[1,40] kHz). In the present study, we consider the audible frequency range up to 30 kHz, for which the length of the thermal diffusion layer is negligible compared to the acoustic wavelength. Thus, the last term in the heat conduction equation of Eq. (13) is neglected due to the large disparity in length scales between the thermal and acoustic waves and 1kg1αg. (This was numerically tested by comparing the results with and without 1kgpt, however, is not shown in the paper.) Local temperature variation on the wall acts as an acoustic source term in the field. It is assumed that the solutions are dependent upon the sinusoidal boundary values with an angular frequency ω and the temperature and the pressure fields, T and p, are small perturbations relative to the temperature Te and the pressure pe in an equilibrium state. By letting Ti=T̃iejωt and substituting it into Eqs. (10), (11), and (13), the temperature variation resulting from the sinusoidal electric source is determined by the following ordinary differential equations:

(14)

with thermoacoustic interface conditions

(15)

where q̃=ksdT̃s/dxkgdT̃g/dx,σi=jω/αi=ω/(2αi)(1+j) for i = s or g. Since Tg and Ts are finite as x and x → −, respectively, the general solutions of Eq. (14) with interface conditions of Eq. (15) are given by

(16)

where xi=(2αi)/ω (i = s or g) is the thermal diffusion length. Equation (16) tells us that the temperature decays exponentially from the origin.

Next, let us consider the acoustic wave equation of Eq. (12). Local temperature variation in time on the right hand side of Eq. (12) acts as an acoustic source of a monopole type, which has already been determined in Eq. (16). By assuming p=p̃ejωt, Eq. (12) is given in the frequency domain by

(17)

where A(ω)=(ρeω2)/pe and

(18)

By applying the reflecting boundary condition (dp̃/dx=0) at the interface (x = 0), due to large disparity of impedance between two media, to Eq. (17), the analytic solution to this equation can be determined, which is given by

(19)

where

(20)

and D̃ is a constant determined by initial conditions. The pressure in one-dimension of Eq. (19) with isothermal assumption (Eq. (12)) is not a solution used for MCMC Bayesian analysis but only used for scaling analysis.

In Eq. (19), the first term is the homogeneous solution of the system and last two terms are the particular solutions produced by the external heat source: The first term is the near-field component decaying exponentially from the origin, and the last one is the far-field term with the wave number equal to A(ω) propagating to the far-field without decay.

We can further consider scaling analysis to arrive at the following relations:

(21)
(22)

since σω1/2,xgω1/2,A(ω)ω, and B(ω)ω2[ (a1h0+a2ksω1/2/αs)2+(a3Cbbω)2 ]1/2.

For example, consider two cases with simplified interface conditions:

  • For a sinusoidal heat flux (a1 = 0 and a3 = 0)

(23)
  • (ii)

    For a sinusoidal temperature (a2 = 0 and a3 = 0)

(24)

These results state that for a sinusoidal heat flux the exponentially decaying near-field pressure is a function of frequency, ω1∕2, but the far-field term has a flat response over the angular frequency. For the case with the sinusoidal boundary temperature, the far-field term is proportional to ω1∕2. These results can serve as references to check which boundary condition, i.e., a heat flux (Neumann BC), a temperature (Dirichlet BC), or a mixed BC, should be applied to the TA problem. If it is found from the experiment that the acoustic pressure produced by the TA actuator is proportional to ω1∕2, then the temperature boundary condition should be applied. However, if it is independent of the frequency, the heat flux boundary condition is appropriate.

Next, let us extend the analysis to consider three-dimensional effects in the acoustic wave equation

(25)

where c is sound speed in ambient condition, ψ is the acoustic source, and x is the coordinate from Fig. 1. Isentropic sound speed is used for the pressure in the far-field by assuming that temperature variation only exists near the wall (see Table II). By assuming that the magnitude of the sinusoidal pressure fluctuation in the thermal boundary layer is proportional to the amount of heat emanating from the solid, this layer of air is regarded as an acoustic piston generating sinusoidal pressure fluctuations propagating into the far-field. The source term is modeled as a monopole source on a rigid baffle, as shown in Fig. 2(a). First, let us consider acoustic waves generated by a monopole source in free space, as depicted in Fig. 2(b). In this case, ψp is the source strength surrounded by the surface Γs, which is determined by ψp(t)=Γsψ(x,t)ds and vn is the oscillation velocity of a sphere normal to the surface. When vn is uniform on the surface, the source can be easily determined by ψp(t)=4πa2vn(t), where a is the radius of the acoustic source which is the same as the acoustic piston, as seen in Fig. 2(a), by assuming that the source is compact. The pressure driven by this compact acoustic source is analytically given by p(r,θ,t)=(jkρecψp)/(4πr)ej(ωtkr) in the far-field. If this single point monopole source is located on a rigid baffle, as shown in Fig. 2(a), the source strength can be twice of that from the equivalent monopole in free space and the spatial dependency on the baffle. With additional manipulation, the sound pressure is analytically determined by22 

(26)

where J1 is the Bessel function of the first kind, k is the wavenumber, θ is the angle between the x axis and the observer from the origin, and the product ka is the Helmholtz number.

TABLE II.

Physical properties of the materials considered for the graphene based speaker.

Substrate (paper)AirGraphite
Density, kg/m3 800 1.2 2,250 
Thermal conductivity, W/m ⋅ K 0.05 0.025 … 
Specific heat, J/kg ⋅ K 1.336 ×106 1,006 800 
Ambient temperature, K … 300 … 
Speed of sound, m/s … 340 … 
Thermal expansion coefficient, 1/K … 0.00343 … 
Substrate (paper)AirGraphite
Density, kg/m3 800 1.2 2,250 
Thermal conductivity, W/m ⋅ K 0.05 0.025 … 
Specific heat, J/kg ⋅ K 1.336 ×106 1,006 800 
Ambient temperature, K … 300 … 
Speed of sound, m/s … 340 … 
Thermal expansion coefficient, 1/K … 0.00343 … 
FIG. 2.

Acoustic piston model. (a) Configuration and (b) monopole source model.

FIG. 2.

Acoustic piston model. (a) Configuration and (b) monopole source model.

Close modal

Temperature variation leads to sinusoidal volumetric change of the air near the material. The relative displacement, δ, is then given by

(27)

where β is the thermal expansion coefficient,2 Tg is the temperature given by Eq. (16), and T¯0 is the spatially averaged temperature of the gas within the thermal diffusion layer, which is expressed as T¯0=1/λth0λthex/xgcos(x/xg)dx=(1e2π)/4π. Accordingly, the velocity, vn, is directly given from Eq. (27) as

(28)

Finally, the sound pressure in the far-field is

(29)

The corresponding root mean squared pressure is

(30)

where

(31)

and

(32)

From Eq. (29), the magnitude of far-field pressure is

(33)

since σω1/2 and λthω1/2. Let us consider the following two cases with input of a sinusoidal heat flux (a1 = 0 and a3 = 0) and a sinusoidal temperature (a2 = 0 and a3 = 0), which are listed in Table I.

TABLE I.

Scaling analysis of acoustic waves generated in TA problem.

DimensionWave typeTemperature (Dirichlet BC)Heat flux (Neumann BC)
1D Near-Field |p| ∝ ω |p| ∝ ω1∕2 
 Far-Field |p| ∝ ω1∕2 |p| ∝ const 
3D Far-Field |p| ∝ ω3∕2r |p| ∝ ω∕r 
DimensionWave typeTemperature (Dirichlet BC)Heat flux (Neumann BC)
1D Near-Field |p| ∝ ω |p| ∝ ω1∕2 
 Far-Field |p| ∝ ω1∕2 |p| ∝ const 
3D Far-Field |p| ∝ ω3∕2r |p| ∝ ω∕r 

For both cases, we observe from Eq. (29) that the acoustic pressure is linearly proportional to the magnitude of the oscillation, Um, and inversely proportional to the distance from the origin, r, with frequency dependence associated with material properties and boundary conditions. For the sinusoidal heat flux, the pressure field represents a linear relation with the frequency. In contrast, the temperature boundary case results in the pressure that is proportional to ω3∕2. These results are also summarized in Table I. This table shows the strong dependence of acoustic waves on the boundary conditions. In the one-dimensional analysis, the acoustic pressure obtained by using a heat flux BC exhibits flat response over frequencies but that generated by a temperature BC is proportional to ω1∕2. However, this relationship changes for the three-dimensional problem. The strength of the acoustic waves is dependent on the frequency (ω) and the distance from the origin (r). These analyses provide significant insight on how the boundary condition affects the TA wave propagation. Through experiments, we can examine which boundary condition is more appropriate to the specific problem. It is also found that directivity only depends upon the Helmholtz number (ka) but the magnitude of acoustic pressure (or sound pressure level (SPL)) strongly depends on the boundary conditions as evident from Eq. (29).

Equation (31) shows that the directivity of sound is independent of the type of imposed thermal boundary condition. Figure 3 exhibits the directivity of sound from analytical and experimental results. The magnitude in each case is normalized by its maximum value. This figure represents that the analytical result is qualitatively in good agreement with experimental data. This means that the TA boundary model developed in this study predicts the acoustic wave propagation well.

FIG. 3.

Acoustic directivity of the graphene speaker in the far-field. Experimental data were from Tian et al.13 The model is compared with three graphene samples of different thickness.

FIG. 3.

Acoustic directivity of the graphene speaker in the far-field. Experimental data were from Tian et al.13 The model is compared with three graphene samples of different thickness.

Close modal

The three unknown model parameters in Eq. (29), {a1,a2,a3}, are determined by using Bayesian analysis with the MCMC method. Bayesian analysis is a statistical procedure used to estimate uncertain parameters by describing them in terms of an underlying probability distribution based upon model uncertainties and experimental uncertainties. The quantities of interest are the distribution of the parameters {a1,a2,a3} that can predict observed data.

In this study, the MCMC method is used together with the DRAM algorithm20,23 to obtain Bayesian statistical estimation of the model parameters. MCMC works by drawing a random sample of values for each parameter from its probability distribution.

We performed MCMC Bayesian analysis to determine the unknown parameters, {a1,a2,a3}, in Eq. (29). The measured data used for the current study were extracted from the reference of Tian et al.13 The experiment was performed on three different samples of a graphene-based sound source with a paper substrate. Graphene samples considered have thicknesses of 100 nm (sample 1), 60 nm (sample 2), and 20 nm (sample 3). This device is made up of a multilayered graphene (MLG) film of 1 cm × 1 cm in area. In the model, however, samples are assumed to have a radius a = 5 mm for the mathematical simplicity. The preliminary numerical test revealed that this assumption only leads to the difference in the SPL less than 1% over the frequency range considered in the study. Table II represents physical properties used in the study. The specific heat of graphene is approximately the value of graphite above 100 K.15 However, the specific heat and the density of a MLG film used in the experiment are not available from the literature. Therefore, the values of graphite were used instead and the multiplication of the parameter and the heat capacity of graphite, a3Cb, will become the heat capacity of a MLG film with uncertainties in the experiment and the model. We neglect any temperature dependence on material properties. The heat transfer coefficient in air is assumed to be h0 = 1 W/m2K. Samples have thicknesses of b = 100, 60, and 20 nm with the applied electric powers of Um = 0.19, 0.07, and 0.01 W, respectively. The sound pressure is computed at the distance r = 5 cm directly above the graphene sound source. The current study only considers inviscid wave equation to predict acoustics waves in the audible frequency range up to 30 kHz. In this range, the acoustic pressure increases with respect to the frequency (current study), but it will decrease due to large attenuation proportional to the frequency in the range of f ≫ 30 kHz and cancellation by phase differences in the waves.24–26 These will finally lead to a flat or decreasing response to the frequency as observed in the experiment.

We performed MCMC Bayesian analysis for all possible combinations of the three terms (i.e., T, Q, and Tt) to obtain a rigorous assessment of model parameters for the graphene-based TA actuator. Namely, we consider (i) only one parameter (T/Q/C), (ii) two parameters (TQ/TC/QC), and (iii) all three parameters (TQC), where T, Q, and C represent the inclusion of the temperature, the heat flux, and the heat capacity of the material, respectively, in the model. To ensure, we obtain accurate posterior probability densities, the number of iterations is set to be 5×106 for each simulation.

Figure 4 represents the results for all cases, (i)–(iii), for sample 1. Each figure plots the comparison between the experimental data (blue line) of sample 1 and the model prediction (black line). The gray area represents the confidence in the model prediction based on the uncertainty propagation of the model parameter distributions. In the figure, the light and the dark gray areas represent 95% prediction and credible intervals,19 respectively. The posterior density distributions of parameters in each case are inserted in each figure. The Bayesian routine samples the parameters using the DRAM algorithm20 and generates a probability distribution. In many of the cases, the distributions of the parameters have Gaussian distributions.

FIG. 4.

MCMC Bayesian analysis results for cases (i)-(iii) with sample 1.

FIG. 4.

MCMC Bayesian analysis results for cases (i)-(iii) with sample 1.

Close modal

Let us first discuss the results from the single parameter models. The temperature model (T model) does not appear to accurately model the sound generated by the current actuator made of the highly conductive material such as graphene as indicated by wide 95% prediction intervals. The literatures tell us that the heat flux boundary condition (Q model) is an appropriate TA model.6–8,10 However, present results in Figure 4 show that the heat flux model (Q model) is not the best model for the current actuator since larger uncertainty between the model and the measurement is observed to occur relative to the heat capacity model (C model). The model only considering the heat capacity (C model) exhibits the best match with the experimental data compared to the other two models, but this is not a physical model satisfying the first law of thermodynamics.

The second row of Fig. 4 shows the three possible models for case (ii). As expected from the C model in case (i), both the TC and the QC models containing the heat capacity term show better prediction of the SPL than the TQ model. In these two models, each parameter has an approximately Gaussian distribution and relatively small variance with respect to its mean value.

The last row of Fig. 4 illustrates the result of the model with all three parameters (TQC model). This model is seen to provide reasonable prediction of sound pressure level. However, the probability distributions of each parameter are not Gaussian but appear skewed. This figure also reveals that the heat convection most likely has minimal influence on the sound pressure level because the range for a1h0 is large relative to the typical value for free convection in air, h0free=525 W/m2K.

Table III summarizes the mean and the standard deviation values of the parameters obtained from the MCMC Bayesian analysis. From Table III, a3 is observed to have small variation in magnitude compared to other two parameters. The last column in the table denotes the error defined by 1N (SPLexpSPLmodel)2, where N is the number of data points. The models that contain heat capacity effects (C/TC/QC/TQC) possess the minimal errors relative to all possible TA models over the range of frequency considered.

TABLE III.

Parameters obtained from MCMC Bayesian analysis for sample 1. Np denotes the number of parameters and Std is the standard deviation.

NpCasea1a2a3Error
MeanStdMeanStdMeanStd
1570 526 … … … … 0.8124 
… … 0.0221 0.00278 … … 0.2178 
… … … … 0.104 0.0437 0.0032 
TQ 205 239 0.0201 0.00415 … … 0.2436 
TC 291 45.6 … … 0.0979 0.00272 0.0017 
QC … … 0.00535 0.00132 0.0829 0.0059 0.0028 
TQC 163 87.3 0.00285 0.00172 0.0894 0.00598 0.0016 
NpCasea1a2a3Error
MeanStdMeanStdMeanStd
1570 526 … … … … 0.8124 
… … 0.0221 0.00278 … … 0.2178 
… … … … 0.104 0.0437 0.0032 
TQ 205 239 0.0201 0.00415 … … 0.2436 
TC 291 45.6 … … 0.0979 0.00272 0.0017 
QC … … 0.00535 0.00132 0.0829 0.0059 0.0028 
TQC 163 87.3 0.00285 0.00172 0.0894 0.00598 0.0016 

Figure 5 represents the sensitivity of the SPL with respect to the frequency, Lp/aj(j=1,2,3), at the identified mean parameter values {a1,a2,a3}={163,0.00285,0.0894} of the TQC model for sample 1. This figure reveals that the sound pressure is strongly dependent upon the heat flux at low frequency (<7.6 kHz) and the heat capacity at high frequency (>7.6 kHz).

FIG. 5.

Sensitivity of SPL to each parameter aj as a function of the frequency for a sample 1. Lp represents the SPL.

FIG. 5.

Sensitivity of SPL to each parameter aj as a function of the frequency for a sample 1. Lp represents the SPL.

Close modal

From these analyses, we conclude that the model comprised of the heat flux and the heat capacity is the preferred choice for modeling TA behavior of graphene.

Equation (10) can be rearranged to assess parameter values relative to physical thermal transport properties using

(34)

In this form, the parameter, η, represents a measure of the amount of Joule heating converted to sound and ξ1 and ξ2 are uncertainty parameters associated with the amount of heat convection and uncertainty in the product of the heat capacity (Cb) and the effective thickness (b) of the TA actuator, respectively. According to Eq. (34), η=2.85×103,ξ1=5.72×104, and ξ2 = 31.4 for the mean values from the TQC model. Particularly, ξ1 is much greater than the commonly used free convection coefficient in air, h0free=525 W/m2K as well as the forced convection coefficient in air, h0forced=10200 W/m2K. This means that the parameter ξ1 in the TQC model is outside the typical physical range.

For the QC model (ξ1 ≡ 0), Table III gives η=5.35×103 and ξ2 = 15.5 from Eq. (34). This means that the product of the heat capacity (Cb) and the thickness (b) of the graphene is 15.5 times greater than that of the graphite parameters used in this study.15 

Figure 6 plots the comparison between the model and the experimental data for samples 2 and 3. For these samples, the QC and the TQC models are only presented based on the findings from sample 1. Errors in the sample 2 are 0.0542 (QC) and 0.0312 (TQC), and for sample 3, we have 3.6096×104 (QC) and 7.3398×104 (TQC). This figure reveals that the heat convection has minimal influence on the sound pressure level as observed for sample 1.

FIG. 6.

MCMC Bayesian analysis results for samples 2 and 3.

FIG. 6.

MCMC Bayesian analysis results for samples 2 and 3.

Close modal

Figure 7 illustrates the comparison between the model and the experimental data for three samples with respect to the frequency in log scale. This figure shows that the thinner membrane is efficient for sound generation and the slope varies from 1/2 to 1 depending upon its thickness. This phenomenon can be explained through the heat capacity of the material in the model. From Eq. (33), we can deduce that the sound pressure in the far-field |p|farfieldω1/2 for a comparatively thick membrane (a1h0+a2ksω/αsa3Cbbω). On the other hand, we find |p|farfieldω for a thin (two-dimensional) membrane (a1h0+a2ksω/αsa3Cbbω). These observations are in good agreement with the present TA model.

FIG. 7.

Comparison between the model and experimental data for samples 1, 2, and 3. The black solid lines are model predictions and the colored lines are measurements from Tian et al.13 Reference slopes are shown with broken lines.

FIG. 7.

Comparison between the model and experimental data for samples 1, 2, and 3. The black solid lines are model predictions and the colored lines are measurements from Tian et al.13 Reference slopes are shown with broken lines.

Close modal

A general thermoacoustic model for two dimensional membranes was examined to predict the emission of acoustic waves over a frequency range of 1–30 kHz. From the analysis, we found that the thermal wave length is very small compared to the acoustic wavelength and the thermal wave decays exponentially with the distance from the wall. We also extended the one-dimensional thermoacoustic model to a three-dimensional setting by using an acoustic piston model. The unknown parameters in the model were determined and analyzed using Bayesian statistics. This approach provides a general framework for assessing material property requirements for generating sound from two dimensional materials due to Joule heating phenomena. The present model was validated against experimental measurements from graphene-based loudspeakers with different membrane thicknesses available in the literature. From comparing data in the literature to our model, Bayesian statistics suggest that the heat flux and the heat capacity dominate thermoacoustics, while heat convection is negligible in ambient air condition. The model and analysis methodology presented in this paper can be extended to other membrane based speakers that rely on thermoacoustics.

We gratefully acknowledge the support from the US Army Research Office (Grant Nos. W911NF-13-1-0062 and W911NF-14-1-0224, program managers Dr. Frederick Ferguson and Dr. Bryan Glaz). We also acknowledge the insightful discussions with Dr. Matthew Munson at the US Army Research Laboratory (Aberdeen Proving Ground). William S. Oates also appreciates support from the NSF CDS&E Award (Grant No. 1306320). Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the funding sponsor.

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