*4.3. Effect of Sintering*

Figure 5a,b portray the dependence of the thermal conductivity on the overlapping coefficient (*δ*) for nanoparticles that are organised into aggregates containing *N* = 9 and *N* = 50 particles, with volume fraction *f<sup>p</sup>* = 0.03 (Figure 5a) and *f<sup>p</sup>* = 0.1 (Figure 5b). The overlapping coefficient ranges from *δ* = 0, which indicates particles at single-point contact, to *δ* = 1, which corresponds to degeneration of the aggregate to a single particle. The corresponding thermal conductivity predictions of analytical models are also presented.

**Figure 5.** Effective thermal conductivity as a function of overlapping coefficient: (**a**) volume fraction *f<sup>p</sup>* = 0.03; (**b**) volume fraction *f<sup>p</sup>* = 0.1. Solid, black circles: *N* = 50, *d<sup>f</sup>* = 2.1. Open, red circles: (**a**) *N* = 9, *d<sup>f</sup>* = 2.1; (**b**) *N* = 50, *d<sup>f</sup>* = 2.5. Blue lines: two-step Maxwell model. Green lines: Maxwell model.

Upon the introduction of sintering, the effective conductivity is found to increase with the sintering level, up to a maximum value. Further sintering beyond that point has a negative effect on conduction until the value of Maxwell's model is obtained, for *δ* = 1. The value of the overlapping coefficient that offers the highest conductivity increase changes with the volume fraction and the number of particles per aggregate. A 15% maximum increase is shown for *f<sup>p</sup>* = 0.1, *N* = 42, and *δ* = 0.25 (Figure 5b). On the other hand, the conductivity calculated by the two-step Maxwell model decreases monotonically upon increase of the overlapping coefficient, due to the monotonic decrease of the radius of gyration (*Rg*).

The results showed that controlled aggregation and sintering can offer significantly improved thermal properties to nanofluids. From a physical point of view, higher values of the overlapping coefficient create an increased number of conduction pathways, which are also wider and longer, thus facilitating conduction along the macroscopic direction of heat transport. At the same time, the size of the aggregates decreases. For low sintering levels, the former factor prevails, whereas for high values of the overlapping coefficient, the aggregate tends to degenerate to an isolated body, thus reducing the thermal conductivity drastically.
