Appl. Phys. A 70, 573–580 (2000) / Digital Object Identifier (DOI) 10.1007/s003390000408

Applied Physics A Materials Science & Processing

Laser ultrasound characterization of chemically prepared nano-structured silver X.R. Zhang1 , W. Zhang2 , X.D. Wang1 , L.D. Zhang2 1 State Key Lab. of Modern Acoustics and Institute of Acoustics, Nanjing University, 2 Institute of solid Physics, Chinese Academy of Science, Hefei 230031, P.R. China

Nanjing 210093, P.R. China

Received: 23 November 1998/Accepted: 13 September 1999/Published online: 24 March 2000 –  Springer-Verlag 2000

Abstract. The ultrasonic properties of nano-structured silver (hereafter nm-Ag) are investigated by the laser ultrasonic technique. The nm-Ag superfine particles with a size of 20 to 27 nanometer (nm) are prepared by using a chemical method. Wafers of nm-Ag are fabricated under different pressing pressures and used as samples. The experimental results show that the ultrasonic velocity and attenuation of nm-Ag prepared by the chemical method depends on the pressing pressures and thus on the relative density. The elastic moduli values of nmAg prepared by the chemical method are deduced. The values are lower than those of polycrystalline and single-crystalline Ag. The detailed results, discussions, and comparisons with nm-Ag samples prepared by a physical method are presented. PACS: 62.65.+k; 43.35.Ud After Birringer et al. [1] first fabricated a nano-structured material (nm-material) by pressing fine polycrystalline particles with size less than 10 nm, nm-materials have attracted much attention in the past decades [2]. It is due to their manifestation of some novel or superior properties [2, 3], such as higher ductility in nm-ceramics [4], and higher strength (or hardness) in nm-metals [5], because of their small grain size (typical d < 100 nm) and large fraction of interfaces (grain boundaries). Obviously, such materials have great potential for application in industry. However, the sonic and mechanical properties, especially the elasticity and its relation to the microstructure of nm-materials, and to the fabrication conditions are not fully understood. The main reason is that few techniques are available for accurately determining the sonic and elastic constants of nm-materials with small weight, small size, and thin thickness. Since White [6] and Akar’yan [7] independently developed laser excitation of ultrasound in solid and liquid samples, the laser ultrasonic technique has become a useful technique for characterizing materials. The technique is particularly useful for application in industrial line inspection [8–11]. Previously, people measured the sound velocity, attenuation and elastic constants, etc., for general materials.

The accuracy of measurement is from 0.2% to 1% [12–16]. It is noteworthy to remember that Aussel et al. [17] investigated the diffractive correction of laser ultrasound and suggested that there are two limiting situations which can be used for accurately measuring the ultrasound velocity and attenuation of the material. One limiting case is by using a small-area excitation and a small-area detection, in other words, by using a point source for excitation and a point receiver for detection (hereafter point source/point receiver). The other limit is by using a large-area excitation and a large-area detection (hereafter large source/large receiver). One can use the approximation of a spherical wave for the former, whereas the approximation of a plane wave is used for the latter. Then they accurately determined the velocity of the ultrasonic wave excited by the laser for some materials [18]. In fact, this technique is suitable to characterize material with small size, thin thickness, or arbitrary shape, because it can provide a high resolution of time and space. Recently, this technique has been used to investigate the ultrasonic properties of nmAg [19–21], nm-Cu [22, 23] , nm-Zn [24–26] and nm-NiAl alloy [27, 28], which were prepared by physical methods and with thin thickness, The experimental results showed that the velocities and attenuation of nm-materials depend strongly on their fabrication conditions. However, nobody has investigated the sonic property of nm-Ag prepared by the chemical method. Recently, some researchers measured the Young’s modulus by using a tensile test on nm-metals (nm-Pb, nmMg, etc.) [5, 29], and by using indentation [30] and nanoindentation [31] on nm-TiO2. The methods they used are destructive and the results they obtained mainly involved the influence of the grain size on the Young’s modulus [3, 32]. Weller et al. [29] measured the shear modulus and internal friction of nm-Pb from the free decay of torsional oscillations in a torsion pendulum at 1 to 5 Hz. Now, we investigate for the first time the sonic and elastic properties of nm-Ag prepared by a chemical method (CMD), and focus our attention on the influence of relative density and pressing pressure on the ultrasonic velocity and elastic modulus. The experimental results and discussions including

574

the comparison of this work with the early work [19–21] are presented.

η is a constant related to the optical absorption. The symbol * is used to denote the convolution over t as follows: Zτ f(τ)g(t − τ) dτ .

f(t) ∗ g(t) =

1 Brief principle

(3)

0

Usually, three types of ultrasonic waves can be simultaneously excited and propagated in the solid when a pulsed laser beam irradiates the surface of solids. The shape of the waveform and which type of wave is detected depends on the detection method. By using a non-contact detector (laser probe) that can respond to both longitudinal (L-) and shear (S-) waves, an epicenter waveform is detected at the rear surface. In our experiment, we can obtain a good signal-tonoise ratio when adjusting the power of the laser beam keeping within the thermoelastic regime because the sample used in our experiment has a large optical absorption coefficient. Therefore, we can easily calculate the epicenter displacement U of the ultrasound generated by the pulsed laser in a plate with thickness h by the following equation: ZR U=

f(r)[ f(t)u z (0, h, t)] dr ,

(1)

0

when the ultrasound is excited by a pulsed laser beam with profiles in both optical intensity and time described by Gaussian functions f(r) and f(t), respectively. The functions f(r) = Imax exp(−r 2 /4R2 ) and f(t) = A exp(−t 2 /τ 2 ), where Imax , R, and τ are the center intensity, waist radius, and duration of the pulsed laser beam, respectively. u i is the displacement excited by the thermoelastic surface point source with optical intensity profile in time described by the Dirac delta function δ(t), given by Rose [33]. u z (0, h, t) =

Γδ(t) ∗ gzH (0, h, t)

,

(2)



 gzH (0, h, t) = 4Λb2h F˙L (t, h) + F˙T (t, h) ,

(2a)

FL (t, h) = −H(t − ah)(ς 2α2 βγ Re2 )ς=(t 2 /h 2 −a2 )1/2 ,

(2b)

FT (t, h) = H(t − bh)(ς 2αβ 2 γ Re2 )ς=(t 2 /h 2 −b2 )1/2 ,

(2c)

where 1

a = 1/cL , b = 1/cS , γ = (b2 + 2ζ 2 ) , α = (a2 + ζ 2 ) 2 , 1 κ β = (b2 + ζ 2 ) 2 , Re = (γ 2 − 4ζ 2αβ)−1 , Λ = , πµ c2 (1 − 2v) κ = 2S = . (2 − 2v) cL

αT , %Cp

C11 = %v12 , C44 = %vS2 ,  2   2 v1 v1 − 1 − 1 , λ = C12 = C11 − 2C4 , ν= 2vt2 vS2 2G µ = G = C44 , E = 2µ(1 + ν) , K = λ + . 3

(2d)

where αT , %, Cp are the linear thermal expansion coefficient (m/K), the mass density (kg/m3), and the specific heat (J/(kg K) of the solid, respectively. λ and µ are the Lame coefficients (N/m2 ). E 0 is the incident laser pulse energy (J).

(4) (5) (6)

By analyzing the waveforms received by the PVDF with fast Fourier transform (FFT), the phase velocity spectrum can obtained by using a phase spectrum method [35] . After obtaining the spectra of amplitude Ai (ω) and phase ϕi (ω), i = 0 or 1 for the first direct arrived pulse or the first echo, the spectra of phase velocity v(ω) and attenuation are respectively calculated by following equations: v(ω) =

F˙L (t, h) is the L-wave of direct arrival and its arrival time tL = ah. FT (t, h) is the S-wave arrival and its arrival time tS = bh. And Γ = ηE 0 (3λ + 2µ)

A waveform excited by a thermoelastic source with diameter R of 0.15 mm for sample nm-Ag fabricated under a pressing pressure of 1 GPa is calculated and shown in Fig. 2a (circle curve) as an example. We can see that there are two jumps appearing in the waveform. The first step denotes the L-wave arrival, and the second jump step denotes a S-wave arrival. Thus the time of flight tL and tS for L- and S-wave-packets may easily be obtained from this waveform. Then the velocity vl , and vS for L- and S-wave-packets of the sample can be calculated, respectively, using the time of flight and the thickness h of the sample, as well as the equations of vL = h/tL and vS = h/tS . Similarly, by using a PVDF transducer as a receiver, a series of pulses of L-wave are detected at the rear surface of the solid. From this waveform, we determine the time of flight for the L-wave to propagate back and forth (i.e. round trip) in the sample t2L . Then the group velocity of the L-wave for the sample can be calculated according to the pulsed-echo method using the equation vL = 2h/t2L . Once the values of the velocity in the solid are known, the elastic constants C11 , C44 , and Poisson’s ratio ν can be calculated from the values of the velocities and the density of sample. The Lame coefficients, Young’s modulus, shear modulus G, and bulk modulus K can be calculated from the elastic constants according to the elastic theory of solids using the following equations [34]:

ωh 1 A0 (ω) , α(ω) = . ϕ1 (ω) − ϕ0(ω) + 2mπ 2h A1 (ω)

(7)

Thus, we can obtain the sound and elastic property of materials. 2 Experiments 2.1 Sample The superfine nm-Ag particles with purity of 99.91% and size d of 20 to 27 nm are prepared by the sol-coagulate method [3]. Three samples are composed of these superfine

575 Table 1. Parameters of nm-Ag prepared by the chemical method Sample

h/ mm

d/ nm

P/ GPa

%/ g/cm3

P0 / %

rf# 2-0# 2-2# 2-3#

0.68 0.40 0.3 0.46

> 2E + 5 20–27 20–27 20–27

0.50 0.75 1.00

8.68 7.94 8.94 9.13

17.2 24.3 14.6 13.5

particles of nm-Ag and are pressed under different pressures P in wafers of 1 cm in diameter, with different thickness h < 460 µm, densities %, and porosity P0 . Table 1 shows parameters of thickness h, grain size of particles, pressing pressures, density, and porosity for the samples. A reference sample (noted rf#) composed of coarse particles is used for comparison. 2.2 Experiments The laser ultrasonic system used here has been reported previously [19, 20] and is shown in Fig. 1. A Nd:YAG Q-switched laser with duration 8 ns and adjustable pulse energy from 1 µJ to 10 mJ is used as excitation source. A laser heterodyne interferometer (SH-120) with frequency bandwidth of 18 MHz [20] and a polyvinylidene fluoride (PVDF) transducer with a frequency bandwidth of 125 MHz are used as receivers [19]. The ultrasonic signal received by the detector is fed to a digital oscilloscope (HP54510B), which has a sampling rate up to 1 Gbit/s. Then the signal is processed by a computer. First, the pulsed laser beam is focused on a spot with a diameter of less than 1 mm, and the probe beam is focused on a spot with diameter of 0.1 mm. We measure the

Fig. 1a,b. The schematic diagram of experimental system of laser ultrasonics: a using a laser-interferometer, and b using a PVDF transducer as detector

epicenter waveform repeatedly (ten times) for each sample, in a contactless way by using the experimental system shown in Fig. 1a. A detected epicenter waveform (solid curve) for sample 2-3# is shown in Fig. 2a as an example. Second, the pulsed laser beam is focused on a spot nearly 3 mm in diameter. We repeatedly measure the waveform of

Fig. 2a–d. The waveforms of a the comparison between calculated and measured by laser interferometer for sample 2-3#, b for sample rf#, c for sample 2-0#, and d for sample 2-3# received by PVDF transducer

576 Table 2. The values of ultrasonic velocities of nm-Ag prepared by the chemical method

Table 3. The values of effective engineering elastic moduli of nm-Ag prepared by the chemical method

Sample

tL / ns

tS / ns

vL / m/s

vS / m/s

C11 / GPa

C44 / GPa

Sample

λ/ GPa

µ/ GPa

E/ GPa

ν/ GPa

K/ GPa

rf# 2-0# 2-2# 2-3#

260 212 160 208

484 378 300 368

2615 1886 1875 2212

1404 1058 1000 1230

59.4 28.3 31.4 44.7

17.2 9.0 8.9 14.3

rf# 2-0# 2-2# 2-3#

25.0 10.3 13.6 16.1

17.2 9 8.9 14.3

44.5 22.9 23.3 36.1

0.30 0.27 0.30 0.275

36.6 16.3 19.5 25.6

the L-wave for each sample by using the experimental system shown in Fig. 1b. Since the size of the excitation spot is larger than the thickness of the sample, we can analyze the problem by using the plane wave approximation. Figures 2b–d show three waveforms for the samples detected by PVDF, as three examples. The reproducibility of the measurements is good and the precision of measurement is better than 1%. 3 Analysis and results

tL and tS . Then we calculate the elastic moduli according to (4)–(6). The calculated results are shown in Tables 2 and 3. We also obtain the spectra of phase velocity and attenuation by analysing the waveforms with fast Fourier transform (FFT). The phase velocity and attenuation spectra are obtained by using (7). Figures 3a,b show the obtained amplitude and phase spectra for the direct arrival, and echoes pulses for sample 2-0# (pressed by pressure 0.5 GPa) as two examples. Figures 4a,b show the comparisons of the velocity dispersions and attenuation spectra for three samples.

3.1 Analysis

3.2 Results

For obtaining the sonic property, we use the spherical wave approximation to analyse the waveforms detected by the laser interferometer, and use the plane wave approximation to analyse the waveforms detected by PVDF. Figure 2a shows that the experimental waveform is close to the calculated one. We calculate the group velocity for the each sample by referencing its ten waveforms and after this obtain an average

From Figs. 3 and 4, we know that the velocity dispersion is obvious at a frequency higher than 60 MHz. The velocities of nm-Ag are lower than that of the sample rfAg# composed of coarse particles, and the attenuation of nm-Ag is higher than that of rfAg#. The slopes of the linear fitting curves of attenuation for rfAg#, 2-0#, 2-2#, and 2-3# are 0.67, 1.55, 1.15, 0.33, respectively. After the analyses, the influences of the pressing pressure and the densification (relative

Fig. 3. a The amplitude spectra of direct pulse, first and second echo. b The phase spectra of direct pulse and the first echo for samples nmAg 2-0#

Fig. 4a,b. The velocities and attenuation of samples vs. frequency, a velocities for frequency up to 125 MHz, b attenuation for frequency up to 50 MHz

577

density) on the velocity, attenuation, and elastic moduli are obversed. 3.2.1 The influence of the pressing pressure on the velocity and attenuation. From Tables 2 and 3, we find that the values of the velocity increase with increasing pressing pressure. By analysing Fig. 4, the change of the group velocity and the phase velocity as well as the attenuation (at different frequency) with pressing pressure are obtained. Figure 5 shows the velocities and attenuation of samples versus the pressing pressure, respectively. 3.2.2 The influence of the pressing pressure on the elastic moduli. The elastic moduli increase with increasing pressure (cf. Fig. 6), but the change with the pressing pressure is nonlinear. It may relate to densification of the sample. Figure 7 shows the relation of Poisson’s ratio and the density with pressing pressure. We can see that the elastic moduli do not obviously change when the pressure is 0.5 GPa and 0.75 GPa, but the bulk modulus and Poisson’s ratio and density change noticeably. After comparing the values of the elastic moduli for samples pressed by pressures of 1.0 GPa and 0.75 GPa, one can see that the ratio of the moduli increases faster than that of the density, but Poisson’s ratio decreases with the increasing pressure. The reason for this is understandable (cf. text below).

Fig. 6. a The elastic constants C11 and C22 . b The Young’s and bulk moduli versus pressing pressure

3.2.3 The influence of the relative density on the elastic modulus. We let the relative density D = %/%0 , where %0 is the density of single-crystalline Ag. Figure 8 shows the elastic moduli are increasing with increasing D when D > 85%,

Fig. 7. The Poisson’s ratio and density of the samples versus pressing pressure

but only the bulk modulus still increases with increasing D when D < 85%.

Fig. 5a,b. The velocities and attenuation of samples vs. pressing pressure, a the velocities of L- and S-wave-packet, as well as the phase velocity of L-wave at frequency of 38 MHz, b the attenuation for frequency of 26, 33, and 50 MHz, respectively

Fig. 8. The elastic moduli of sample vs. the relative density

578

4 Discussion We try to compare the results obtained here for the samples prepared by the chemical method (CMD) with the results obtained before (cf. [20]) for samples prepared by a physical method (PMD). 4.1 The influence of the porosity on the elastic modulus Generally, the ultrasonic velocity and attenuation of solids are determined by their elasticity related to the microstructure properties, such as interatomic spacing. For nm-materials, there are a lot of factors, such as the grain size (causing a change in the fraction of the interface), the fraction of pores, the number and the properties of the interface, and the shape and the arrangement of pores etc., which may lead to the elastic soft-mode effect. However, the grain size and porosity are the two major influential factors. We consider only the porosity’s influence on the sample modulus and regard the nm Ag as a two-phase medium. One phase is the polycrystalline Ag (p-Ag) or single-crystalline Ag (s-Ag), the other phase is the pores. We then try to explain the relationship between Young’s modulus E e and the samples’ porosity P using the equation [36] E e = E 0 (1 − 1.9P + 0.9P 2) ,

(8)

where P = (%0 − %) × %−1 is the porosity of the sample, 0 E 0 and %0 are the Young’s modulus and the density of the p-Ag or s-Ag, respectively. E 0 are 112 GPa for singlecrystalline Ag and 77 GPa for polycrystalline Ag, respectively. The values of P are shown in Table 1 for the CMD sample and in Table 1 of [20], respectively. The calculated results for the samples of CMD and PMD are shown in Figs. 9a and 9b, respectively. It is obvious that the values of the calculated Young’s moduli for p-Ag and for s-Ag are higher than those measured for the nm-Ag, regardless of whether the sample is CMD or PMD. The relationship between the E and the porosity is linear for the calculated one, but is nonlinear for measured one. The theoretical results do not agree with the experimental results. This obvious discrepancy between calculated values and the experimental data is understandable (cf. text below). The values of the moduli for the CMD sample are lower than those for the PMD sample. It is because the porosity of the CMD sample (13.5% to 24.3%) is much more than that for the PMD sample (2.3% to 9.4%).

Fig. 9a,b. The calculated and the measured Young’s moduli versus porosity, a for the sample prepared by chemical method, and b for that prepared by physical method

where K m , G m = µm are the bulk and shear modulus (according to our assumption, they are equal to those of pAg or s-Ag, respectively) of the frame within the nm-Ag, K m and G m are equal to 103.5 GPa and 27 GPa for p-Ag,

4.2 The influence of the porosity on the bulk modulus for the CMD and PMD samples For comparison of the elasticity betweem the CMD nm-Ag and PMD nm-Ag, we try to explain the relationship between elasticity and densification (or porosity) of nm-Ag specimens based on a theoretical equation for an effective bulk modulus of spherical void-containing solids. We assume that the main difference in elasticity between nm-Ag and p-Ag or s-Ag is caused by porosity in the matrix of bulk p-Ag or s-Ag. Then, one can calculate the effective bulk modulus K e of nm-Ag by the following equation [37]: Ke = Km +

β(K i − K m )
, 1 + (1 − β)(K i − K m )/ K m + 43 G m

(9)

Fig. 10a,b. The comparison between the calculated and measured values of the bulk moduli, a for the sample prepared by chemical method, and b that prepared by physical method

579

or 62.5 GPa and 46.1 GPa for s-Ag. K i is the bulk modulus of the void (equal to the bulk modulus of air) , and β(= P) the porosity of the sample, is the same as that used above. The calculated results are shown in Fig. 10. It can be seen that the calculated effective bulk modulus is higher than the experimental values. Moreover, the calculated K e decreases linearly with increasing porosity; whereas the measured K e decreases nonlinearly with increasing porosity. This obvious discrepancy between calculated values and the experimental data is understandable, for (9) only considers the influence of porosity while neglecting the effects of the special microstructures of nm-Ag. In fact, the elastic property of nm-Ag depends certainly on its special microstructure details, which not only contain a lot of voids but also include various interfacial defects and varying atomic structures. The value of the bulk modulus for the CMD sample is lower than that for the PMD sample. It is due to the increased porosity of the CMD sample (13.5% to 24.3%) compared to the PMD sample (2.3% to 9.4%). 4.3 The relationship of the Young’s modulus to the relative density Figure 11 shows the relationship of the Young’s modulus to the relative density for CMD and PMD samples. We can clearly see that the relative density of CMD nm-Ag is much lower than that of PMD nm-Ag (cf. [20]), and the Young’s modulus of PMD nm-Ag decreases with decreasing relative density. It is noteworthy that the values of the moduli for all of the CMD nm-Ag are higher than those of the PMD nm-Ag with the relative density lower than 91%. Since the density of CMD nm-Ag used here is much lower than that of PMD nm-Ag, we can say that the velocity of the CMD sample is greater than that of the PMD sample. It is confirmed by comparison of the values of vL of the samples used here with sample A1 used in [20]. The sample A1 is composed of nm-Ag particles with a size of 27 nm prepared by a physical method. We find that the L-wave velocities in CMD samples with a size of 27 nm are higher than that in sample A1. The vL of A1 is 1750 m/s. However, the density of A1 is 9.74 g/cm3 and is higher than that of all the samples used here. So we say that the velocity of nm-Ag not only depends on its density or porosity, but also on the different microstructure of the boundary interfaces.

Fig. 11. The Young’s modulus versus relative density for CMD and PMD nm-Ag

5 Conclusions As mentioned above, some conclusions can be made: 1. The laser ultrasonic method has been successfully employed to investigate the ultrasonic and elastic properties of nm-Ag prepared by a chemical method, and it shows that the fabrication condition of the sample strongly influence the properties of nm-materials. 2. Both the velocities and attenuation of nm-Ag depend on the porosity (or the relative density) and pressing pressure. 3. The elastic moduli deduced for nm-Ag prepared by the chemical method are lower than that of the coarse particles of crystalline Ag, as well as nm-Ag prepared by a physical method when the relative density of the CMD sample is higher than 95%. 4. The velocity within nm-Ag prepared by the chemical method (used here) is faster than that prepared by the physical method when the relative density is lower than 95%. 5. The calculation of the influence of porosity on the elastic modulus shows that the velocity and elastic moduli of nm-Ag depend not only on its density or porosity, but also on the microstructure of the boundary interface which is different for the different samples. Further experiments are needed to fully understand the relationship between the ultrasonic properties and the interfacial structure of nm-Ag, and for building a suitable model. Acknowledgements. This work is supported by the Natural Science Foundation of China and State Key Lab. of Modern Acoustics, Nanjing University.

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