Finite element modeling of bulk ultrasonic waves generated by ring-shaped laser illumination in a diamond anvil cell

Wen Feng; Dexing Yang; Yuning Guo; Ying Chang

doi:10.1364/OE.20.006429

1. Introduction

Laser ultrasonic has shown significant advantages in nondestructive evaluation field (NDE) [1–4] as its characteristics of non-contact, wideband and simultaneously generating longitudinal, shear and surface waves on a range of materials. In recent years, the precision measurement of sound velocity, which can be applied to evaluate the elastic properties of materials, is an important issue in high pressure physics. Measuring sound velocity under high pressure by combining the diamond anvil cell (DAC) device with the technique of picosecond laser ultrasonic was proposed until 2008 by F. Decremps et al. [5]. In the same year, N. Chigarev et al. [6] developed laser ultrasonic in a diamond anvil cell (LU-DAC), they demonstrated that using this LU-DAC point-source-point-receiver technique made it possible to measure both longitudinal and shear velocities of a non-transparent iron film at pressures up to 23GPa. Later, the ultrasonic waves under high pressure were further analyzed by using LU-DAC technique in a point-source-point-receiver [7] or line-source-point-receiver [8,9] configurations. The main advantage of the LU-DAC technique over the technique proposed in Ref. 5 for non-transparent amorphous solids is that there is no need of measuring the thickness of the specimen independently of sound velocity [7]. In addition, it provides an opportunity to study specimen with certain thickness which is more than tens of micrometers, while picosecond ultrasonic technique requires the thickness of specimen to be thinner than a few micrometers due to the strong sound attenuation in the 10GHz-plus frequency range [8]. However, low intrinsic ultrasonic amplitude generated by a point-like laser source makes it difficult for the precision measurement of ultrasonic waves. Moreover, in those LU-DAC experiments, the probe point and excited source are arranged at the same side of sample. As a consequence, the wave system at the probe point becomes very complex due to the ultrasonic reflection at the aluminum/diamond interface and the wave amplitudes become lower due to the increase of soundpath. It also brings some difficulties when analyzing the signals. Furthermore, the obvious signals of longitudinal and shear waves cannot be detected simultaneously at the same point owing to their remarkable directivity [10]. In this article, we propose a LU-DAC ring-source-point-receiver technique to overcome the above drawbacks by taking advantage of constructive interference among waves generated by different parts of ring-shaped laser. The ring-shaped laser source can be treated as numerous point-like sources arranged as a circle, where each point-like source can excite the ultrasonic waves at the same time in one irradiation. In the free surface condition, some reports [11,12] have demonstrated that an annular illumination pattern, which can be easily formed by an axicon [13], can generate the strong ultrasonic signals along the central axis of the ring. Thus, not only the wave amplitudes can be highly enhanced even if not in the case of ablation but also strong signals of both the longitudinal and shear waves can be detected simultaneously at the same point by using LU-DAC ring-source-point-receiver technique. The sound velocities can also be measured without considering about the thickness of the specimen by modulating the ring radius of laser. However, both sides of the micron film are covered with the rigid diamond whose thickness is much greater than the film in a LU-DAC system. As a result, the excited source, waveforms and propagation laws of ultrasonic waves are different from that generated in the case of free surface. In this paper, our work is to develop numerical models to describe the generation and propagation of bulk ultrasonic waves generated by ring-shaped laser in a DAC system.

As a result of the complexity of the generation and propagation of ultrasonic waves in a film in a DAC, the finite element method (FEM), owing to its flexibility in modeling complicated geometry and its capability in obtaining full field numerical solutions [14], is adopted to deal with such complicated process in our model. In this article, the characteristics of ultrasonic displacement field generated by ring-shaped laser illumination have been simulated by the FEM based on the thermoelastic theory. The superposition effects of both the longitudinal and shear waves on the rear surface of aluminum film are analyzed in detail. By studying the displacement distributions along the central axis of the ring, the focal laws of waves are discussed. In addition, the full field numerical solutions at different times are presented, which makes it more intuitive to understand the evolution characteristics of ultrasonic waves within the specimen.

2. Thermoelastic theory model

The geometry model of ring-shaped laser irradiation on a DAC system which consists of two diamonds and an aluminum film is schematically shown in Fig. 1(a) . The contact between the diamond and the aluminum film is supposed to be perfect. A ring-shaped laser whose energy is below the sample damage threshold is directed on the top interface vertically because of the transparent diamond. The ring, which has two parameters: inner radius and thickness denoted by R and d, represents the laser illumination region. A cylindrical coordinate system is introduced in our model because the geometry of the problem has axial symmetry. Thus, the problem can be considered in the two-dimensional axial symmetric plane, which is shown in Fig. 1(b).

Fig. 1 (a) Schematic diagram of the system irradiated by ring-shaped laser. (b) Cross-section of the system.

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The distribution of transient temperature field is determined by thermal conduction equation, which can be described as [12, 15]

ρ_{i} c_{i} \frac{\partial T_{i} (r, z, t)}{\partial t} - \frac{1}{r} \frac{\partial}{\partial r} (r k_{i} \frac{\partial T_{i} (r, z, t)}{\partial r}) - \frac{\partial}{\partial z} (k_{i} \frac{\partial T_{i} (r, z, t)}{\partial z}) = Q (r, z, t)

where subscript i = 1 denotes the film while i = 2, 3 denote the diamond below and above the film respectively. T_i(r, z, t) is the temperature distribution; ρ_i, c_i and k_i are the density, thermal capacity and thermal conductive coefficient respectively; Q(r, z, t) is the heat source provided by ring-shaped laser, which can be expressed as

Q (r, z, t) = A I_{0} β \exp (- β z) f (r) g (t)

where A is the optical absorptivity of the specimen surface, I₀ is the peak intensity of the incident laser, β is the optical absorption coefficient, f(r) and g(t) are the spatial and temporal distributions of the pulsed laser, these two functions are

f (r) = \exp [- \frac{{(r - (R + d / 2))}^{2}}{{(d / 2)}^{2}}]

g (t) = \exp [- \frac{{(t - t_{0})}^{2}}{τ^{2}}]

where t₀ and τ are the arrival time of peak value and half pulse width of the pulsed laser.

The ultrasonic displacement induced by the transient temperature field satisfies the Navier-Stokes equation [14, 15]

(λ_{i} + 2 μ_{i}) \nabla (\nabla \cdot U_{i}) - μ_{i} \nabla \times \nabla \times U_{i} - ρ_{i} \frac{\partial^{2} U_{i}}{\partial t^{2}} = α_{i} (3 λ_{i} + 2 μ_{i}) \nabla T_{i} (r, z, t)

where U_i is the displacement vector, α_i is the thermoelastic expansion coefficient, λ_i and μ_i are the Lamé constants.

Through introducing the scalar potential function φ_i and vector potential function ψ_ie_θ, we can express U_i (r, z, t) as the functions of these two parameters [16]

U_{i} (r, z, t) = \nabla ϕ_{i} + \nabla \times (ψ_{i} e_{θ})

Then the following longitudinal and shear wave equations are obtained from Eq. (5).

\begin{array}{l} \frac{\partial^{2} ϕ_{i}}{\partial r^{2}} + \frac{1}{r} \frac{\partial ϕ_{i}}{\partial r} + \frac{\partial^{2} ϕ i}{\partial z^{2}} - \frac{1}{C_{L i}^{2}} \frac{\partial^{2} ϕ_{i}}{\partial t^{2}} = γ_{i} T_{i} \\ \frac{\partial^{2} ψ_{i}}{\partial r^{2}} + \frac{1}{r} \frac{\partial ψ_{i}}{\partial r} + \frac{\partial^{2} ψ_{i}}{\partial z^{2}} - \frac{ψ_{i}}{r^{2}} - \frac{1}{C_{S i}^{2}} \frac{\partial^{2} ψ_{i}}{\partial t^{2}} = 0 \end{array}

Here, C_Li and C_Si are the longitudinal and shear wave velocities respectively, γ_i = α_i(3λ_i + 2μ_i)/ (λ_i + 2μ_i).The decomposition of the displacement component normal to the interface can be written as

U_{z i} = \frac{\partial ϕ_{i}}{\partial z} + \frac{\partial ψ_{i}}{\partial r} + \frac{ψ_{i}}{r}

In order to solve the displacement field, it is necessary to determine the boundary and initial conditions in the model. As the contact between two materials is perfect, the temperature, heat flux, stress and displacement component are continual at interfaces. The boundary conditions at interfaces between materials are

\begin{array}{l} T_{1} = T_{i} \begin{matrix} , & - k_{1} \end{matrix} \frac{\partial T_{1}}{\partial z} = - k_{i} \frac{\partial T_{i}}{\partial z} \\ σ_{z z 1} = σ_{z z i} \begin{matrix} , & σ_{z r 1} \end{matrix} = σ_{z r i} (i = 2, 3) \\ U_{z 1} = U_{z i} \begin{matrix} , & U_{r 1} \end{matrix} = U_{r i} \end{array}

As the thickness of the diamond is much greater than that of the aluminum film, we can suppose that the front and back surfaces of the DAC system are thermally insulated and mechanically unconstrained.

Assuming the system has no initial heat source, stress and mechanical displacement, the initial conditions can be written as

\begin{array}{l} T_{i} |_{t = 0} = 300 K \\ U_{i} |_{t = 0} = 0 \begin{matrix} , & \frac{\partial U_{i}}{\partial t} |_{t = 0} = 0 \end{matrix} (i = 1, 2, 3) \end{array}

3. Numerical results and discussion

3.1 Lasers and materials parameters

The peak power intensity of laser is 3 × 10⁹ W/cm², and it reaches the top interface at 1ns. The pulse width and thickness of the ring-shaped laser are taken to be 0.5ns and 2μm. The inner radius of the laser ranges from 5μm to 60μm. The radius and thickness of the aluminum are 100μm and 30μm, respectively. In a LU-DAC experiment, the thickness of diamond window is usually within a millimeter magnitude. In order to not affect the accuracy of numerical results and calculate conveniently, the radius and the thickness of diamond windows are taken to be 100μm. The element size of grid is 0.1μm in the vicinity of the laser affected area, while the element size is 0.4μm outside the heat-affected zone. The properties of materials used in the calculation are listed in Table 1 , some parameters of aluminum film are from Refs. 14 and 17.

Table 1. Properties of aluminum and diamond used in calculation

View Table

3.2 Numerical results and discussion

Figures 2(a) -2(d) show the vertical displacement waveforms detected at the epicenter and the other positions of 4-μm-off, 8-μm-off, and 12-μm-off the epicenter on the rear surface of aluminum film respectively for a 30μm radius ring. In Fig. 2(a), a sharp rise at 7.91 ns represents the arrival of the longitudinal (L) wave, and an obvious peak at 14.3ns which represents the advent of the shear (S) wave comes next. The corresponding velocities of longitudinal and shear waves are 6243m/s and 3244m/s, respectively, which are very close to the theoretical value. Because the waves created by different portions of the ring source have no longer arrived at the receiver deviated from epicenter at the same time, the sharp longitudinal and shear peaks each splits into two separated peaks with much smaller amplitudes in Figs. 2(b)-2(d). Therefore, we can see that both the longitudinal and shear wave amplitudes fall off sharply and the waveforms are broadened as the probe point away from the epicenter. The longitudinal and shear wave amplitudes, as a function of radial distance from the epicenter, are shown in Figs. 3(a) and 3(b) for three varied ring radii. The amplitude is the first peak value of each waveform. It can be seen that the sharp focusing at the epicenter is evident, and wave amplitudes drop rapidly as the radial distance from the epicenter increases in these three cases. These phenomena discussed above indicate that the ring-shaped laser can simultaneously generate the amplified displacement signals of the bulk ultrasonic waves on the central axis of the ring-shaped laser.

Fig. 2 Vertical displacement of the bulk wave at different positions on the rear surface of specimen: (a) the epicenter; (b) 4μm; (c) 8μm and (d) 12μm from epicenter.

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Fig. 3 Amplitudes of (a) longitudinal wave and (b) shear wave at different positions along the radial direction on the rear surface of specimen.

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To further confirm the superposition effects of waves on the rear surface of sample, we have studied the frequency spectrums and superposition patterns by fast Fourier transform (FFT). Figures 4(a) and 4(b) show the spectrums of the longitudinal and shear waves detected at three different points on the rear surface of aluminum film, respectively. It can be seen that the frequency components are consistent in all cases, and the amplitudes of each frequency is larger at the epicenter than those at the other point. However, the spectral distributions are various for different probe points. The superposition patterns of the longitudinal and shear waves at 1.56GHz along the radial direction on the rear surface of aluminum film are presented in Figs. 5(a) and 5(b). It is obvious that the maxima and minima amplitudes of each wave appear alternately along the radial direction. The largest amplitudes appear at the epicenter as the result of constructive interference of multi-source with same intensity and phase. Meanwhile, the amplitudes at points deviated from epicenter are different as the superposition is created by multi-source with different intensity and phase.

Fig. 4 The spectrums of (a) longitudinal wave and (b) shear wave received at different points on the rear surface of specimen.

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Fig. 5 The interference patterns of (a) longitudinal wave (b) shear wave along the radial direction on the rear surface of specimen.

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In order to study the focal characteristics of bulk ultrasonic waves, we have calculated the displacement distributions of the longitudinal and shear waves along the central axis of the ring source, which are shown in Figs. 6(a) and 6(b). As the depth increases, we can see that the amplitudes of two waves increase and reach the maximum value at a focal depth, then decreased gradually. Meanwhile, the focal depths of each wave are various for different ring radii. The results in the figures suggest that the focal depths of two waves have the positive relationship with the radius of the ring, which means the bigger ring radius are able to produce the greater focal depth. Particularly in Fig. 6(b), the focal depths of the shear wave are 10μm, 15μm and 19μm, which correspond to the ring source with the radii of 10μm, 15μm and 20μm respectively. Thus the angle α ≈45° that denoted in the Fig. 1(b) can be calculated, which is very close to the directional angle of the shear wave induced by a point-like laser source in an aluminum film in a DAC [10]. For the longitudinal wave in Fig. 6(a), it is very hard to find a similar law because the majority of energy is concentrated near the point-like laser incident direction. However, we can find that the focal depth is larger than the corresponding ring radius. Therefore, the phenomena above suggest that the desired focal depths of two waves can be achieved by modulating the radius of the ring-shaped laser.

Fig. 6 Amplitudes of (a) longitudinal wave and (b) shear wave at different points along the central axis of ring source.

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From the above discussion, it can be seen that understanding the characteristics of two waves at the epicenter is very important for LU-DAC ring-source-point-receiver technique. Therefore, we changed the ring radius while the probe point was at the epicenter. Figures 7(a) and 7(b) plot longitudinal and shear wave amplitudes at the epicenter as a function of ring radius respectively. As the ring radius increases, the amplitudes of the longitudinal and shear waves reach their each maximum at ring radii of 40μm and 27μm respectively, and then decreased gradually. A ring angle of β = 42° [see Fig. 1(b)] is obtained, which is consistent with the directional angle of the shear wave. In addition, when the ring radius is small, the shear amplitude approaches zero because the illumination region is similar to a spot, which is consistent with our previous report [10]. This discussion suggests that we can obtain the strongest signals of two waves at the epicenter by modulating the ring-source radius for a given sample.

Fig. 7 Amplitudes of (a) longitudinal wave and (b) shear wave at the epicenter as a function of ring radius.

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As described in the introduction, the FEM has significant advantage in obtaining full field numerical solutions, which are more helpful for us to understand the evolution characteristics of ultrasonic waves within the sample. The displacement magnitude distributions generated by ring-shaped laser irradiation at different times are denoted by grayscale in Figs. 8(a) -8(d) for a 20μm radius ring. We can clearly observe the excited source in the aluminum film and the wavefronts of both the longitudinal and shear waves propagate away from the heated area. In Fig. 8(a), the displacement field is similar to that generated by a point-like source [10] and the superposition effects of waves are not obvious. However, it can be seen that the wavefronts are asymmetrical from Figs. 8(b)-8(d), the wave amplitudes in the area inside the ring radius are larger than that in the area outside the ring radius, which suggests that the enhancement effects of ultrasonic waves appear in the area inside the ring radius. From these figures, we can also see that the enhanced superposition effect is more apparent in the position closer to the axis, and particularly in Fig. 8(d), the shear wave displacement in the position Ι is far larger than that in the position ΙΙ. Consequently, these pictures further intuitively show the evolution characteristics of waves and the enhanced superposition effects of waves in the area near the central axis.

Fig. 8 Ultrasonic displacement fields at different times: (a) 3ns; (b) 6ns; (c) 8ns; (d) 11ns.

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Finally, in order to further demonstrate the enhancement effects of ultrasonic waves induced by the ring source, the wave amplitudes generated by a point-like source are analyzed for comparison. The peak power intensity and radius of the Gaussian laser pulse are taken to be 3 × 10⁹ W/cm² and 2μm, which are the same with the peak power and thickness of the ring-shaped laser. As a function of radial distance from the epicenter, the bulk ultrasonic wave amplitudes are displayed in Fig. 9 . By comparing Figs. 9(a) and 9(b) with Figs. 3(a) and 3(b), it shows that the amplitude distributions on the rear surface by the point-like source are different from those by the ring source. With the increasing of radial distance, the longitudinal wave amplitude gradually reduce, while the shear wave amplitude reach the maximum and then gradually decrease. That comparing Figs. 9(a) and 9(b) with Figs. 7(a) and 7(b) makes it clear that the maximums of wave amplitudes induced by the ring source are far greater than those induced by the point-like source. The amplitudes of longitudinal and shear waves generated by the ring source are almost 30 times and 40 times greater than those induced by the point-like source. This comparison further powerfully demonstrates that the ring-shaped laser has significant advantage in generating the strong signals of ultrasonic waves in a LU-DAC system.

Fig. 9 Amplitudes of (a) longitudinal wave and (b) shear wave as a function of radial distance from the epicenter for a Gaussian laser point-like source.

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4. Conclusions

In the study of high-pressure physics based on a LU-DAC technique, low intrinsic ultrasonic amplitudes make it difficult to detect the generated waves for the precision measurement of sound velocities. Therefore, it is very meaningful to generate the strong signals of both the longitudinal and shear waves by some feasible approaches. In this paper, based on the thermoelastic theory, finite element method is used to establish the model of bulk ultrasonic waves generated by ring-shaped laser in a DAC system. By analyzing the amplitude distributions in an aluminum film in detail, we have demonstrated that the strong enhancement effects of ultrasonic waves appear on the central axis of ring source owing to the constructive interference of the waves generated by different parts of the ring. The discussions of the displacement distributions along the central axis reveal that the directivity of shear wave plays a crucial role in determining its focal depth, whereas it is more complicated to determine the focal depth of longitudinal wave. By changing the ring radius, the strongest signals at the epicenter can be obtained for a given sample. The further comparative study indicates that the maximums of wave amplitudes induced by a ring source are far greater than those induced by a point-like source.

Acknowledgments

We acknowledge the project support from the China Academy of Engineering Physics. We also thank the technical support by the Center for High Performance Computing of Northwestern Polytechnical University.

References and links

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12. J. F. Guan, Z. H. Shen, X. W. Ni, J. Lu, J. J. Wang, and B. Q. Xu, “Numerical simulation of the ultrasonic waves genersted by ring-shaped laser illumination patterns,” Opt. Laser Technol. 39(6), 1281–1287 (2007). [CrossRef]

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Physical properties	Diamond	Aluminum
The optical absorptivity		0.08
Optical penetration depth (nm)		7
Thermal conductive coefficient K (W m⁻¹K⁻¹)	2000	249.45-0.085T, 200K<T ≤730K 198.47-0.014T, 730K<T ≤T_m
Density ρ (kg m⁻³)	3500	2769-0.22T, 300K≤T ≤T_m
Thermal capacity C (J kg⁻¹K⁻¹)	669.44-4.79 × 10⁻⁷T⁻² + 1.27T, 293K<T≤1100K	780.3 + 0.488T, 200K≤T ≤T_m
Thermal expansion coefficient	1.1 × 10⁻⁶	2.31 × 10⁻⁵
Young’s modulus (GPa)	1100	70.2
Poisson’s ratio	0.07	0.34

Finite element modeling of bulk ultrasonic waves generated by ring-shaped laser illumination in a diamond anvil cell

Abstract

1. Introduction

2. Thermoelastic theory model

3. Numerical results and discussion

3.1 Lasers and materials parameters

3.2 Numerical results and discussion

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (9)

Tables (1)

Equations (10)

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