**1. Introduction**

Numerous contemporary applications are based on the incorporation of nanoparticles into convectional fluids [1,2]. The resulting nanofluids may have drastically increased thermal properties and reduced sedimentation. Nanofluids and nanoparticles are increasingly used in a variety of fields. Heating and cooling systems using nanofluids show significant improvement in energy consumption, while new medical techniques are developing using nanoparticles in bioliquids. [3,4]. A large number of recent publications study the potential use of nanofluids in multidisciplinary fields [5,6]. Although much effort has been placed on the correlation of the heat transfer properties of nanofluids with the underlying phenomena, it appears that there is no widely accepted explanation of their behaviour or a reliable way to predict their heat conduction properties [7,8].

Many models have been developed to predict the effective conductivity of nanoparticles. The effect of Brownian motion, interfacial resistance, the existence of nanolayers, and the aggregation mechanism have been discussed in detail in the literature [9–13]. A large increase in effective thermal conductivity has been detected experimentally when nanoparticles are organized in small aggregates [14,15]. The increased contact of the particles within the aggregate was found to facilitate heat transfer compared to fully dispersed particles. On the contrary, larger mass aggregates have a negative effect on the stability of the nanofluid and, therefore, on heat transport properties [16]. Lotfizadeh et al. [11], Prasher et al. [17], Evans et al. [18], and Liao et al. [19], among others, developed models to

**Citation:** Karagiannakis, N.P.; Skouras, E.D.; Burganos, V.N. Modelling Thermal Conduction in Polydispersed and Sintered Nanoparticle Aggregates. *Nanomaterials* **2022**, *12*, 25. https://doi.org/10.3390/ nano12010025

Academic Editor: S. M. Sohel Murshed

Received: 17 November 2021 Accepted: 18 December 2021 Published: 22 December 2021

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predict the thermal conductivity of nanofluids based on the morphology of the aggregates. These works showed that the configuration of the nanoparticles and the morphological parameters of the aggregates can alter the effective conductivity of the nanofluids noticeably. A typical assumption made in several studies is that the nanoparticles are of the same size and the neighbouring monomers are mainly at single point contact.

In real conditions, samples always come with a certain distribution in particle size [20–22], which is held responsible for altering the thermal properties of nanofluids [21]. It has been noticed that polydispersity may occur during dispersion of the nanoparticles in the base fluid [23,24]. Chon et al. [21] used commercial, uniform nanoparticles for the preparation of nanofluids. They measured the size distribution after dispersion and found a significant deviation. In fact, it is technically challenging to synthesize and disperse a large quantity of highly monodispersed nanoparticles [25]. Zhiting et al. [26] studied the effect of polydispersity, among other parameters, on the heat transfer coefficient of nanocomposites with molecular dynamics simulations. They concluded that polydispersity negatively affects the effective conductivity. However, the nature of the used method does not allow the simulation of large systems, such as that of aggregated nanoparticles.

Strong electrostatic forces between particles, collision of particles during the formation of the aggregates, and high-temperature environments are some of the factors that contribute to a certain degree of overlapping between particles [27,28]. Two methods have been widely used for the preparation of nanofluids. The one-step nanoparticle production and blending method produces nanofluids with increased stability and offers elevated thermal conductivity, but has a relatively high production cost and is not yet suitable for large-scale production [29]. The two-step method is one that proceeds in two sequential steps, namely, the step of separate production of nanoparticles followed by suspension in a base fluid [29]. Commercial nanoparticles are usually found in powder form. The production of nanoparticles commences with the creation of a nanoparticle suspension from their precursors. This suspension is dried using various methods and, eventually, a powdered form of nanoparticles is obtained [23,30]. Thermal decomposition of organic precursors is a well-established process for the fabrication of solid nanoparticles [23]. It is temperature dependent and has been reported to result in sintered and/or polydispersed particles [23,24,30]. Other methods include nanoparticle production in the gas phase [31,32]. During the creation of these nanoparticles, formation processes, including surface reactions, condensation, coagulation and sintering, are some of the key mechanisms that take place [32–35]. The kinetics of each process determines the final structure morphology, which can vary among spherical particles, agglomerates, or compact aggregates [34]. The resulting system usually includes partially coalesced particles with sintering necks [32,35].

Attempts to model the aforementioned process usually start with the formation of an aggregate using a stochastic method. Eggersdorfer et al. [36] modelled the sintering process in aggregates, formed by Diffusion Limited Aggregation (DLA), Diffusion Limited Cluster–Cluster Aggregation (DLCCA), and Ballistic Aggregation (BA). The driving force for sintering was the minimization of free energy. They noted that, during sintering, primary particles approach each other. Sander et al. [31] presented an analytical description of the underlying phenomena during the production of nanoparticles, such as coagulation and sintering. The primary particles were modelled having a spherical shape and a polydispersed size, with each particle described by its sintering level and radius. The final structure has been compared with experimental data and transmission electron microscopy (TEM) images. An overlapping algorithm, developed by Brasil et al. [27], has studied the effect of sintering on the morphological properties of the aggregates, such as the fractal dimension and the radius of gyration. Schmid et al. [33,37] have developed a model for aggregates subjected to coagulation and sintering. The sintered aggregates were presented as the result of successive overlapping of spherical, primary particles.

The previous discussion underlines two major open issues in the study of the thermal conductivity of aggregated nanoparticles. Even though the effect of aggregation has been extensively studied, there is a dearth of research dealing with the effect of polydispersity of the nanoparticles within the aggregate. Moreover, sintering and the concomitant partial coalescence are most likely to occur in nanoparticle systems, yet the study of their effects on heat conduction remains a challenging, open field. These configurations complicate the determination of heat transfer properties and, therefore, reliable simulations of transport phenomena are required.

The present work examines the effects of polydispersity and sintering of particles on the effective thermal conductivity of nanofluids that contain particle aggregates. To this end, the method developed by the authors [38] for reconstructing particle aggregates is extended to include particle overlapping due to sintering as well as non-uniform particle size as a realistic outcome of nanofluid preparation. Among the merits of this method is the algorithmically rapid reconstruction of agglomerated systems with predetermined properties, namely, the fractal dimension and the average number of particles in the aggregates. As a case study, the particle size here follows the normal distribution and the standard deviation is expressed as a fraction of the mean size. Moreover, a technique has been developed to simulate sintered aggregates. The sintering process is expected to change the particle position and size while, naturally, preserving the mass of the working sample. An overlap parameter and the morphology of the primary aggregate determine the final morphology. The effective thermal conductivity is calculated through the temperature distribution obtained from the solution of the heat transfer equation. The Meshless Local Petrov–Galerkin (MLPG) method [39–41] is used here as it was shown to provide stable and fast solutions to particulate systems even with point contact. The Discretisation-Corrected Particle Strength Exchange (DC PSE) method [42,43] is used to approach the field function and its derivatives, while the meshless nature of the method allows local increase of the domain discretisation at the interface between the base fluid and solid particles.

The effect of the overlap parameter and the polydispersity level of particle aggregates on the thermal conductivity is studied by changing the number of particles in the aggregate, the fractal dimension of the aggregates, and the volume fraction of the particles. The effective conductivity of the polydispersed nanoaggregates, as predicted by the present method, is compared to the effective conductivity of the corresponding systems, as these result from the Diffusion Limited Aggregation (DLA) method. Moreover, the effective conductivity of aggregates consisting of polydispersed particles is compared with that of aggregates of monodispersed particles, keeping all other morphological parameters constant. Notable deviations between monodispersed and polydispersed cases are observed and discussed. In addition, the effect of sintering is examined by varying the overlap parameter. The results are compared with predictions of analytic expressions from the literature.
