# Elastic Slow Dynamics in Polycrystalline Metal Alloys
## 4 Multi-Relaxation Theory and Results
### Theory
#### 4.1.2 Continuous Spectrum of Relaxation Times  
Distributions obtained in the \(log(\tau)\) space are continuous (see Figure 5) and suggest in most cases (see subplot (a)) a peaked distribution with asymmetry, i.e., a faster increase of the curve at lower values of \(\tau\) (the presence of semilog x-axes should not be misleading). Several choices of peaked and asymmetric functions are of course possible, but they are largely equivalent regarding the uniqueness of the solution, as we show in the Supplemental Material.  
In the following, we assume a Weibull distribution of the inverse of relaxation times, which has the expected asymmetry discussed above. This choice is plausibly suggested by the fact that Weibull distribution is typically observed as the most typical distribution of  
Figure 5: Multirelaxation process with decay times uniformly distributed in the log space. (**a**,**d**) Distribution of \(G_{i}\) in the \(log_{10}(\tau)\) space for different choices of the basis range \(B_{1}\) to \(B_{N}\) and two different metal alloy samples (two rows). (**b**,**c**,**e**,**f**) Results of the fitting procedure for one specific choice (\(K=1\)) of the selected bases are also reported.  
the size of the features present at the grain and crack scales [53; 54]. Thus, defining \(x=1/\tau\) we write:  
\[F(\tau)=\beta\frac{\gamma}{\alpha}(x/\alpha)^{\gamma-1}e^{-(x/\alpha)^{\gamma}} \tag{9}\]  
For the titanium sample, which has a two peaks distribution (see Figure 5d), the relaxation times distribution is the superposition of two Weibull distributions: \(F(\tau)=F_{1}(\tau)+F_{2}(\tau)\).  
The parameters of the distributions are then obtained fitting the curve \(\delta c/c(t)\) in the time domain, integrating in the full \(\tau\) space (from zero to infinity) using Equations (4) and (9). Results of the fit for a few cases are shown in Figure 6. The quality of the fitting function (continuous line) in describing the experimental behavior (dots) is evident, with good agreement also for the early stages of the evolution (see subplot (b) where a log time scale is used). In all cases, also the intermediate times logarithmic recovery is obtained. In the case of the Titanium alloy sample (black curve) the need to fit the data using the superposition of two Weibull distributions is particularly evident in subplot (b), where, after a first logarithmic in time phase and a bending at early times (arising from the \"saturation\" of the first distribution), at about 10 s velocity starts evolving rapidly again with a second logarithmic in time phase.  
The obtained best fitting distributions for the samples considered are shown in Figure 7. The distributions for the three metal alloys (Aluminum A and C samples and magnesium sample) reported in Figure 7 present a peak at \(\tau\approx 0.2\) s, which is remarkably similar for the three samples, corresponding to their similar grain size (\(\approx\)0.1 mm). The peak for mortar (\(\tau\approx 0.7\) s) is slightly higher, corresponding to a larger grain size (\(\approx\)0.8 mm). The two peaks in distribution for the Titanium alloy sample could be linked to the presence of grains with different sizes in this sample: Besides the very small structural grains (size \(\approx 0.05\) mm, corresponding to the peak at \(\tau\approx 0.08\) s), the columnar grain structure presents a larger grain size of \(\approx\)1 mm (resulting in a second peak at \(\tau\approx 0.6\) s, i.e., close to the peak of mortar which has a similar grain size).  
Figure 6: Fitting of experimental data using a continuous Weibull distribution of relaxation times for different samples. (**a**) Linear scale; (**b**) log time scale.  
Besides the link of the peak position to the grain size, intact metal alloys and mortar have a very similar behavior.