Application to a tungsten-silver composite

To quantitatively test the reconstruction method, experimental data need to be available giving a picture of the material, properties for each individual phase, and overall composite properties. A well characterized system, suitable to test the reconstruction procedures, is provided by the tungsten-silver (W-Ag) composite of Umekawa et al. [45]. This composite was produced by infiltrating a porous tungsten solid with molten silver (volume fraction of silver p=20%). The Young's modulus of the composite was measured at a range of temperatures above and below the melting point of silver (960ºC). The elastic moduli of each phase were obtained by measuring the moduli of pure samples of tungsten and silver at each temperature. This data cannot be used directly because both phases of the composite actually contained tiny spherical pores. These will reduce the Young's moduli and Poisson's ratio of each phase. This effect can be accounted for by applying well known results for dilute spherical inclusions [12]. For porous materials (porosity 1) the formulae can be rewritten as

$$E_\phi E_m-\phi E_m\left( \frac{9-4\nu_m -5 \nu_m^2}{7-5\nu_m} \right)$$ = (19)

$$\nu_\phi \nu_m-\frac32 \phi \left(\frac{(5 \nu_m-1)(1-\nu_m^2)}{7-5\nu_m} \right)$$ = (20)

where E_m, _m denote matrix properties and E $E_\phi$, $\nu_\phi$ are the porosity modified values. The tungsten matrix had an internal porosity of 1% while the silver phase had a porosity of 10% at room temperature, decreasing linearly to 5% at the melting point. Table 3 shows the phase moduli used at different temperatures.

Table 3: The moduli of the silver and tungsten phases, as a function of temperature, after being corrected for the internal porosity of each phase.

	Silver		Tungsten
Temp(ºC)	E(GPa)	$\nu$	E(GPa)	$\nu$
25	71	0.36	400	0.28
200	69	0.36	392	0.28
400	63	0.36	383	0.28
600	54	0.36	373	0.28
800	45	0.37	363	0.28
860	42	0.37	361	0.28
910	39	0.37	359	0.28
950	37	0.37	357	0.28
960	37	0.37	356	0.28
960	0	0.50	356	0.28
1020	0	0.50	354	0.28

Figure 4: The original scanned image from Umekawa et al. [45] (236 x 204 µm at 759 x 657 pixels). The dark phase corresponds to tungsten and the lighter to silver.

To reconstruct the W-Ag composite we digitize a photograph of the sample (Fig. 4). All points below a selected threshold grey-level are set to black (the silver phase) while the remainder is set to white. The image was blurred and re-thresholded to remove the pores of the tungsten matrix (which appeared as silver). This had little effect on p_expt and p⁽²⁾_expt , but a significant effect on the measured silver chord distribution. The resulting image is shown in Fig. 5. The image actually has a silver content of only 13.5%, significantly lower than the nominal value of 20%. The statistical properties of the image are compared with those of 11 trial reconstructions in Table 4, while 2-D slices of the models themselves are shown, for purposes of illustration, in Fig. 6. Several of the models were unable to reproduce p⁽²⁾_expt and were considered no further. A comparison of the chord distributions indicated that model N (c=0) provided the best reconstruction. The auto-correlation function of this model is compared with experimental data in Fig. 7. The experimental and model chord-distributions are shown in Fig. 8. The silver distribution $\rho^{(1)}_{\rm {expt}}$ ⁽¹⁾_expt is well reproduced by the model at all lengths shown, while $\rho^{(2)}_{\rm {mod}}$ ⁽²⁾_mod performs less well at small chord lengths. Two and three dimensional images of the model are shown in Fig. 9 (shown at the same scale as Fig. 5) and in Fig. 10.

Figure 5: The cropped (640 x 640 pixels) binary image obtained from the scanned image. This is the sample used to calculate the statistics of the composite (side length 198.7 µm). The black phase corresponds to silver.

Table 4: A comparison of the statistical properties of 11 reconstructions with those of the experimental composite; p=13.5% and s_v=0.17 µm ^-1 (obtained from Fig. 5). Many of the models are able to reproduce the low order statistical properties of the composite (Ep⁽²⁾ <0.1). This shows that p⁽²⁾ (r) does not uniquely specify composite microstructure.

Mod.	c	rc	$\xi$	d	sv	Ep(2)	E $E\rho^{(1)}$ (1)	E $E\rho^{(2)}$ (2)
N	0	2.16	2.15	13.0	0.19	0.05	0.03	0.28
N	½	28.1	28.2	22.0	0.18	0.11
N	1	$\infty$	$\infty$	25.6	0.18	0.10
I	0	2.88	2.89	12.5	0.20	0.05	0.28	0.88
I	½	13.4	13.5	15.9	0.18	0.06	0.32	0.53
I	1	32.7	32.1	17.4	0.17	0.05	0.11	0.38
U	0	2.69	2.68	13.1	0.18	0.05	0.08	0.30
U	½	60.2	144	35.5	0.20	0.15
U	1		$\infty$	43.0	0.20	0.15
I10	0	4.75	4.76	12.6	0.21	0.06	0.33	0.58
OS		r0 =3.75			0.21	0.25	0.49	0.45

Figure 6: Cross-sections of a portion of the original image (a) and the eleven trial reconstructions (b-l) at p=13.5%. The length-scale parameters of the trials are chosen to match p⁽²⁾ _expt(r). The chord-distribution's (Table 4) indicate that model N (c = 0) [shown in (b)] provides the best reconstruction. Each image has side length 39.7 µm.

Figure 7: The auto-correlation function of the experimental image is well reproduced by the best reconstruction [model N (c=0)].

Figure 8: The good agreement between the model and experimental chord distributions [silver: $\rho ^{(1)}(r)$ ⁽¹⁾ (r); tungsten:
$\rho ^{(2)}(r)$ ⁽²⁾ (r)] indicates that model N (c=0) provides the best reconstruction.

Figure 9: The best reconstruction [model N (c=0)]. The region shown is 198.7 x 198.7 µm (cf. Fig. 5).

Figure 10: The silver phase (shown as solid) of the best reconstruction [model N (c=0)]. The side length
is 39.7 µm.

For the purposes of computing the elastic properties of the model we maintain the length scale parameters ($\xi$, r_c and d ) and alter the level cut parameters ($\alpha$ and $\beta$ ) of the model such that p_rec =20% (in accord with the experimental composite). The Young's modulus, computed using the finite element method, is compared with the experimental data of Umekawa et al. in Fig. 11. For the temperature region below the melting point of silver the maximum error is 4%, a very good result. Above the melting point of silver, when the silver phase is taken to have a zero bulk and shear modulus, the error is only 3%. The agreement may actually be better than that, however. Since the elastic measurements were dynamic measurements, the liquid silver can be considered as being trapped on the time scale of the experimental measurement, before any significant flow could take place, and so could still contribute to the effective moduli via its non-zero liquid bulk modulus. Just before melting, the silver had a bulk modulus of about 35GPa. If we take the bulk modulus to be somewhat lower, in analogy to the ice-water difference around 0ºC, then a bulk modulus of 23.1GPa causes the N (c=0) model to agree perfectly with experiment at temperature points above the melting point of silver.

Figure 11: The experimental Young's moduli compared with FEM data obtained from the best reconstruction [Fig. 10]. The Beran, Molyneux, Milton and Phan-Thien (BMMP) bounds for the reconstruction and Hashin and Shtrikman bounds are also shown.

The bounds are also shown in Fig. 11. For model N (c=0) the microstructure parameters are $\zeta_1=0.31$₁ = 0.31 and $\eta_1=0.27$₁ = 0.27. The results bound the experimental data and provide a reasonable prediction of the Young's modulus below the melting point of silver. Note that even if the silver phase is given a non-zero bulk modulus past its melting point, the zero shear modulus causes the lower bounds for shear modulus and therefore Young's modulus to be identically zero. Unfortunately, there was no reported Poisson's ratio results for the composite, so we cannot compare to the model results for this quantity.