## 3 Results and Discussion
Plots based on the equation were drawn with 4 sin\(\theta\) along the x-axis and \(\beta_{hkl}\)cos\(\theta\) along the y-axis, as shown in Figure 5. These graphs make it possible to determine microstrain and crystallite size by the slope of the straight line and its intersection with the vertical axis, respectively. The microstructure parameters calculated for the aluminum matrix are shown in Table 2.  
Figure 4: XRD patterns of as-milled and hot-pressed composites with (**a**) 2% stearic acid and (**b**) 5% Cu. Figure 5: The Williamson-Hall plots for (**a**) Al-5 TiO\({}_{2}\)-2 PCA and (**b**) Al-5 TiO\({}_{2}\)-5 Cu composites. The following formula was used for the calculation of the dislocation density (\(\rho\)) [73]:  
Figure 3.0  
where \(b\) is the Burgers vector (0.286 nm). The slope of the straight lines (Figure 5) for the materials, both after milling and after hot pressing, suggests that the dislocation density (Table 2) varies slightly depending on the composition and synthesis mode. The main difference in the graphs is that for the material with the addition of copper, the distance between the straight lines after milling and after hot pressing increases noticeably, as compared with that for the material with the addition of stearic acid. This fact is associated with a significant increase in the size of grains during hot pressing. Thus, the increment in the crystallite size of the Al-5 TiO\({}_{2}\)-5 Cu material is 32 nm, while for Al-5 TiO\({}_{2}\)-2 PCA it is only 23 nm (Table 2). In any case, the aluminum matrix remains nanocrystalline after hot pressing, which contributes to the preservation of the strength parameters. The grain boundaries are known to be effective barriers for moving dislocations that are blocked there, forming a pile-up [74]. Elastic stress fields arise around the pile-ups, which act on the boundaries and adjacent areas of neighboring grains in addition to external stresses. Under these conditions, new dislocation sources may form at the borders and in border areas. Accordingly, the finer the grains, the more boundaries/barriers are encountered in the path of gliding dislocations, and the greater the stresses required to continue plastic deformation even at its initial stages. As a result, more effective strengthening of grain boundaries is achieved. It is interesting to note the change in the microhardness of the studied materials. Table 2 shows that for the material with copper addition, hot pressing leads to a decrease in the microhardness from 291 to 233 HV due to the precipitation of a stable Al\({}_{2}\)Cu phase from a supersaturated solid solution and grain growth during thermal impact. In this case, the decomposition of the solid solution is accompanied by an increase in the Al lattice parameter. However, for the material with a PCA additive, the microhardness increases from 162 to 187 HV after hot pressing despite a slight increase in the grain size. This effect may be due to the formation of dispersed particles of aluminum carbide, as noted above. A slight increase in the lattice parameter of this material after hot pressing is probably caused by the precipitation of impurity elements from the aluminum matrix. The hardness of materials depends on their microstructure and the distribution of hardening particles. The microstructure of hot-pressed samples of the studied materials is shown in Figure 3.1 It can be seen that the microstructure of the materials in both compositions is homogeneous. The microstructure is a dark aluminum matrix with evenly distributed light particles not larger than 1 \(\upmu\)m. However, in the material with the addition of copper, the amount of these particles is noticeably higher than in the material with the addition of PCA. This is because, whereas only TiO\({}_{2}\) particles are visible in the Al-5 TiO\({}_{2}\)-2 PCA material, the Al-5 TiO\({}_{2}\)-5 Cu material, in addition to submicron TiO\({}_{2}\) particles, also contains dispersed Al\({}_{2}\)Cu particles with a size of about 100 nm that precipitated from the