# Dynamic Viscosity and Transverse Ultrasonic Attenuation of Engineering Materials
## 3 Results and Discussion
### Attenuation Behavior of Organic Materials  
Nine engineering polymers were evaluated for transverse attenuation and the results are summarized in Table 4 in the same format as before. Tests were conducted at 25 \(\pm\) 2 \({}^{\circ}\)C. TDM-2 was used for all the samples, and TDM-1 was also used for PMMA, PVC, and PC. Only the average values for PMMA from Table 1 are listed in Table 4. The transverse attenuation of eight other polymers was higher than that of PMMA. Their C\({}_{\rm dt}\) values ranged from 717 to 1830 dB/m/MHz, or about 3 to 7 times higher. The averaged C\({}_{\rm dt}\)/C\({}_{\rm d}\) value was 3.1. All the transverse attenuation spectra showed the linear frequency dependence and scattering effect was absent. The transverse damping factor, \(\eta_{\rm lt}\), did increase over the longitudinal damping factor, \(\eta_{\rm l}\), but \(\eta_{\rm l}\)/\(\eta_{\rm l}\) averaged to 1.53 or an increase of 53% since the starting average value of \(\eta\) was already high at 0.033. The average observed ratio of transverse and longitudinal damping factor of 1.62 agrees well with the same ratio of PMMA measured at lower frequencies (see Section 2). The transverse attenuation measurement used here can be used to extend low-frequency torsional methods for the study of transverse damping in polymers into the MHz range.  
Figure 8 shows the attenuation spectra of PMMA (green curve for a TDM-1 result and purple for TDM-2), PVC (blue), and PC (red). Each has the least-square fit dotted line in the same color and R\({}^{2}\) values were 0.994, 0.997, 0.996, and 0.998, indicating good overall linearity.  
When one attempts to correlate the C\({}_{\rm d}\) and C\({}_{\rm dt}\) values of polymers to the elastic moduli or sound velocities, general trends can be seen, except PMMA values always remain as outliers. The crystallinity, X\({}_{\rm c}\), is another possible microstructural feature [47; 48], but the attenuation values vary widely (by a factor of seven) among amorphous polymers of PMMA, PC, polypropylene (PP), and epoxy, for which X\({}_{\rm c}\) = 0. PMMA gives the lowest C\({}_{\rm d}\) or C\({}_{\rm dt}\) values and PC the highest. Semi-crystalline polymers of PVC (X\({}_{\rm c}\) = 10%), Nylon 6/6 (X\({}_{\rm c}\) = 40%), low-density polyethylene (LDPE; X\({}_{\rm c}\) = 40%), high-density polyethylene (HDPE; X\({}_{\rm c}\) = 63%), and UHMWPE (X\({}_{\rm c}\) = 70%) showed relatively unchanged attenuation values. Here, X\({}_{\rm c}\) values were from [49; 50; 51; 52]. Figure 9 shows the plots of C\({}_{\rm d}\) and C\({}_{\rm dt}\) vs. X\({}_{\rm c}\). Fitting curves are shown, but clearly these are of no value. The above attempts offered no clue in regard to the factor(s) controlling C\({}_{\rm d}\) or C\({}_{\rm dt}\) values. A possible source of varying attenuation among amorphous and semi-crystalline polymers is the internal stresses from processing. It was noted earlier (in Section 2) that the external stress affected the attenuation in PMMA, shown by [PERSON] et al. [38]. The cooling speed is a part of some polymer studies e.g., [47] and correlating it with attenuation and stressing may provide a needed key to understand the damping behavior of polymers. However, defining internal stresses carries its own problems since a reference state does not exist unlike its crystalline counterpart, as discussed by [PERSON] [53]. Even in the absence of a load, an amorphous solid always has internal stresses, which build up within the material as it solidifies. A new theory on amorphous solids was applied on sound attenuation of silica, but at THz frequencies [54; 55]. Its applicability to MHz attenuation is unknown. For our purpose, macroscopic residual stress measurement methods [56; 57] should work adequately.  
Figure 8: The transverse attenuation spectra of PMMA (green curve for TDM-1 and purple for TDM-2), PVC (blue), and PC (red). The least-square fits are shown in dotted lines of the same color.