**2. Methods**

### *2.1. Experimental Procedure*

As initial materials, we used nanopowders of ScF3 (~80 nm), obtained via the sol–gel method [38], and alumina (~80 nm), obtained via the method of electric explosion of a conductor (EEW) [39], aluminum micropowder (<20 μm) [39], and A356 aluminum alloy. Figure 1 shows TEM images of the initial powders.

**Figure 1.** Transmission electron microscopy (TEM) images of initial powders: ScF3 (**a**); Al2O3 (**b**); Al (**c**).

As initial components for the synthesis of ScF3 nanoparticles, scandium oxide, hydrofluoric acid, and hydrochloric acid were used, which makes the process safer, without the use of poisonous fluorine, which is used in the direct synthesis of ScF3 from metallic scandium.

For the introduction of nanoparticles, a powder mixture of Al–5 wt.% nanoparticles was prepared. For deagglomeration and distribution of nanoparticles in the powder mixture, stearic acid was used as a surfactant. First, 200 mL of petroleum ether and 1.5 wt.% stearic acid were added to the powder mixture. Then, for 20 min, the powder mixture was mechanically mixed. After mechanical mixing, the aluminum–nanoparticle powder mixture was air-dried (90 ◦C) and sieved. The resulting powder mixture was introduced into the melt using ultrasonic treatment, which allows degassing and refining the melt, preventing deagglomeration, and evenly distributing nanoparticles in the volume [40].

Aluminum alloy A356 was melted in a graphite crucible with a total melt volume of 500 g. Ultrasonic treatment was carried out using a water-cooled magnetostrictive transducer (power—5 kW, frequency—17.5 kHz, RELTECH, Saint-Petersburg, Russia). The waveguide of the ultrasonic equipment was made in the form of a niobium alloy (VN2AE) cone, the working amplitude of which was 30 μm. At a melt temperature of 730 ◦C, ultrasonic degassing was carried out for 1 min. After degassing, an aluminum–nanoparticle master alloy (5 wt.%) was introduced into the melt with ultrasonic treatment. After dissolution of the master alloy in the melt, ultrasonic treatment lasted for 2 min (temperature, 730 ◦C). The melt obtained at a temperature of 710 ◦C was poured into a steel chill mold. The amount of ScF3 and Al2O3 nanoparticles in the A356 aluminum alloy ranged from 0 to 1 wt.%. The A356 alloy without particles was obtained with similar melt processing parameters. Mechanical tests were performed on a universal testing machine (Instron Europian Headquarters, High Wycombe, UK), Instron 3369, at a speed of 0.2 mm/min. The samples were tensioned at room temperature (25 ◦C), and the number of samples for each composition was at least four pieces. The structures of the obtained materials were investigated through optical microscopy, Olympus GX71 (Olympus Scientific Solutions Americas, Waltham, MA, USA). Samples were subjected to preliminary mechanical polishing, electrolytical etching, and anodization. The electrochemical oxidation of the metallographic specimen surface in a 5% solution of hydrofluoric acid (HBF4) at a voltage of 20 V and a current of 2 A was carried out to identify grain boundaries.

### *2.2. Strengthening Mechanism*

It is generally accepted that nanocomposite hardening occurs due to the load transfer, Orowan mechanism, and CTE mismatch mechanism [41]. The total strengthening can be calculated as a superposition of the individual strengthening mechanisms.

$$
\sigma\_{\rm Y} = \sigma\_{\rm Ym} + \sigma\_{\rm NP} + \sigma\_{\rm HP} + \sigma\_{\rm Cr} + \sigma\_{\rm CTE} \tag{1}
$$

where *σ*Ym is the yield strength of the matrix, *σ*NP is the stress caused by the transfer of the load from the matrix to the particles, *σ*Or is the Orowan stress, *σ*CTE is the thermal stress arising from the difference in the coefficients of thermal expansion, elastic modulus, and shear modulus of the matrix and the particle.

The load transfer mechanism is the most accepted strengthening mechanism. The transfer of the load from the soft matrix to the hard particles when an external load is applied helps to harden the material, as proposed by Nardon and Prewo [42].

$$
\sigma\_{\rm NP} = 0.5 f\_{\rm p} \sigma\_{\rm YM} \tag{2}
$$

where *f*p is the volume fraction of particles.

The grain size has an influence on the strength of the alloy, since the grain boundaries can impede the movement of dislocations. This is due to the different orientations of adjacent grains and to the high lattice disorder characteristic of these regions, which prevents the glide of dislocations in a continuous slip plane. The Hall–Petch equation relates stress to average grain size. There are empirical models available for predicting the yield strength change due to the change in the grain size in metal matrix composites by extending the Hall–Petch [43,44] relationship as follows:

$$
\sigma\_{\rm HP} = k\_{\rm Y} \left( \frac{1}{\sqrt{d\_{\rm gr}}} - \frac{1}{\sqrt{d\_{\rm gr0}}} \right) \tag{3}
$$

where *σ*HP is the change in tensile strength due to the contribution of the Hall–Petch law, *k*Y is the hardening coefficient (constant for each material), and *<sup>d</sup>*gr and *<sup>d</sup>*gr<sup>0</sup> are the average grain sizes of the obtained alloys. This equation assumes that the Hall–Petch parameters *k*Y ≈ 68 MPa·(μm)1/2 for Al alloys [45].

Particles perform a fundamental role in the final grain size of the matrix of composites as they can interact with grain boundaries and act as nuclei of crystallization, slowing or stopping grain growth. An increase in the volume fraction *f*p and a decrease in the particle diameter *δ*p lead to a finer-grained structure, which is theoretically modeled by the Zener–Smith equation [46].

$$d\_{\rm gr} = k\_z \frac{\delta}{f\_{\rm P}},\tag{4}$$

where *kz* coefficient values are in the range 0.1 < *kz* < 1. Equation (5) is theoretically derived to describe the deceleration of migrating grain boundaries by particles.

The Orowan mechanism is based on the interaction of nanoparticles with dislocations. Solid "noncutting" particles impede the movement of dislocations, which leads to bending of the dislocation line around the particles (Orowan loops) under the action of an external load [47]. The Orowan mechanism is very important in metal matrix composites, hardening by fine particles when the interparticle distance is not large. An increase in the composite yield strength by Orowan stress may be expressed as follows [47]:

$$
\sigma\_{\rm Or} = G\_{\rm m} \frac{b}{\Lambda\_{\rm P} - 2R\_{\rm P}},
\tag{5}
$$

where *G* m is the matrix shear modulus, *b* is the Burgers vector of the matrix (*b* = 0.202 nm), and <sup>Λ</sup>p is the distance between particles.

When a composite is subjected to heating or cooling, the difference in the coefficients of thermal expansion, elastic modulus, and shear modulus between the matrix and hardening particles produce internal stress state changes. The improvement in yield strength from thermal mismatch can be calculated using the equation suggested in [48–50].

$$
\sigma\_{\rm CTE} = 6G\_{\rm m} \sqrt{\left(a\_{\rm m} - a\_{\rm p}\right) \Delta T \frac{b}{\delta\_{\rm P}} \frac{f\_{\rm P}}{1 - f\_{\rm P}}} \,. \tag{6}
$$

Equation (6) was obtained on the basis of the simplest geometric estimates under the assumption that the volume mismatch between the matrix and the reinforcing particles, arising from the difference in thermal expansion coefficients, leads to the appearance of geometrically necessary dislocations around the reinforcing particles. The disadvantages of the above approach include the fact that the stresses in Equation (6) do not depend on the elastic properties of the strengthening particles.

A more rigorous assessment of the thermal stresses arising from the difference in the coefficients of thermal expansion, elastic modulus, and shear modulus of the matrix and particle can be carried out using the methods of solid mechanics.

#### *2.3. Mathematical Model of the Stresses Caused by the Thermal Expansion Mismatch between the Matrix and Strengthening Particles*

Let us consider the stress–strain state that arises as a result of a change in the temperature of a spherical particle with a radius *<sup>R</sup>*p surrounded by a matrix. The coefficients of linear thermal expansion of materials are assumed to be different.

The equation of equilibrium of an elastic medium in spherical coordinates under the assumption of spherical symmetry can be written in the following form:

$$\frac{\partial \sigma\_{rr}}{\partial r} + \frac{2\sigma\_{rr} - \sigma\_{\varphi\varphi} - \sigma\_{\theta\theta}}{r} = 0 \tag{7}$$

The relationship between stresses *<sup>σ</sup>ij* and strains *<sup>ε</sup>ij*, expressing the generalized Hooke's law, under nonisothermal conditions, according to the von Neumann hypothesis, has the following form:

$$
\sigma\_{ij} = \lambda \varepsilon\_{kk} \delta\_{ij} + 2G\varepsilon\_{ij} - 3\text{Ka}\Delta T \delta\_{ij\prime} \tag{8}
$$

where *λ* is the the Lame coefficient, *G* is the shear modulus, *K* is the bulk strain modulus, *α* is the coefficient of linear thermal expansion, Δ *T* = *T*test − *T*room is the temperature difference between the testing temperature *T*test and room temperature *T*room, and *δij* is the Kronecker symbol.

The components of the strain tensor under spherical symmetry are

$$
\varepsilon\_{rr} = \frac{\partial u}{\partial r}, \; \varepsilon\_{\varphi\varphi} = \frac{u}{r}, \; \varepsilon\_{\theta\theta} = \frac{u}{r}, \; \varepsilon\_{r\theta} = 0, \; \varepsilon\_{r\theta} = 0, \; \varepsilon\_{r\theta} = 0. \tag{9}
$$

Substituting into the equilibrium equation stresses expressed in terms of displacements, we obtain

$$\frac{\partial}{\partial r}\left[\frac{1}{r^2}\frac{\partial}{\partial r}(\mu r^2)\right] = 0.\tag{10}$$

First, we integrate Equation (11) for a spherical particle for the following boundary conditions:

$$r = 0: \quad \mu = 0; \quad r = R\_{\mathbb{P}}: \quad \mu = \mathcal{U} \text{ .} \tag{11}$$

As a result of simple calculations, we obtain the displacement field in the particle.

$$
\mu = \mathcal{U} \frac{r}{R\_{\mathcal{P}}}.\tag{12}
$$

The stress field in a particle using the generalized Hooke's law has the following form:

$$
\sigma\_{rr} = \left(3\lambda\_{\text{P}} + 2G\_{\text{P}}\right) \frac{\text{II}}{R\_{\text{P}}} - 3K\_{\text{P}}\alpha\_{\text{P}}\Delta T\_{\text{A}} \tag{13}
$$

$$
\sigma\_{\mathfrak{P}\mathfrak{P}} = \left(3\lambda\_{\mathbb{P}} + 2G\_{\mathbb{P}}\right) \frac{\mathcal{U}}{R\_{\mathbb{P}}} - 3K\_{\mathbb{P}}a\_{\mathbb{P}}\Delta T\_{\prime} \tag{14}
$$

$$
\sigma\_{\theta\theta} = \left(3\lambda\_{\text{P}} + 2G\_{\text{P}}\right) \frac{\text{II}}{R\_{\text{P}}} - 3K\_{\text{P}}a\_{\text{P}}\Delta T. \tag{15}
$$

The displacement of the matrix material can be defined as

$$
\mu = \alpha\_{\rm m} \Delta T r + \frac{\mathcal{U} R\_{\rm p}^2 - \alpha\_{\rm m} \Delta T R\_{\rm p}^3}{r^2}. \tag{16}
$$

The parameter *U* is determined from the condition of continuity of the radial stresses at the boundary between the particle and the matrix: <sup>σ</sup>*rr*|− = <sup>σ</sup>*rr*|<sup>+</sup> . As a result, we obtain

$$
\Delta U = \frac{3K\_{\text{P}}\alpha\_{\text{P}} + 4G\_{\text{m}}\alpha\_{\text{m}}}{3K\_{\text{P}} + 4G\_{\text{m}}} \Delta T R\_{\text{P}}.\tag{17}
$$

Using the generalized Hooke's law, one can determine the stress field in the matrix.

$$\sigma\_{\rm II} = -12 \frac{K\_{\rm p} G\_{\rm m}}{3 K\_{\rm p} + 4 G\_{\rm m}} (\alpha\_{\rm P} - \alpha\_{\rm m}) \Delta T \frac{R\_{\rm P}^3}{r^3} \,\mathrm{}^{\prime} \tag{18}$$

$$
\sigma\_{\varphi\varphi} = 6 \frac{K\_{\rm P} G\_{\rm m}}{3 K\_{\rm P} + 4 G\_{\rm m}} \left( \alpha\_{\rm P} - \alpha\_{\rm m} \right) \Delta T \frac{R\_{\rm P}^3}{r^3} \,\mathrm{}\,\mathrm{}\tag{19}
$$

$$
\sigma\_{\theta\theta} = 6 \frac{K\_{\text{P}} G\_{\text{m}}}{3 K\_{\text{P}} + 4 G\_{\text{m}}} \left( \alpha\_{\text{P}} - \alpha\_{\text{m}} \right) \Delta T \frac{R\_{\text{P}}^3}{r^3} . \tag{20}
$$

We now turn from considering the stresses created by a single particle to the stresses caused by an ensemble of particles. The stress intensity characterizing the stress state of the material is determined by the following equation:

$$
\sigma\_{\rm int} = \sqrt{\frac{1}{2} \left( \sigma\_{rr}^2 + \sigma\_{\rho\rho}^2 + \sigma\_{\theta\theta}^2 - 3 \left( \sigma\_{rr} + \sigma\_{\theta\theta} + \sigma\_{\theta\theta} \right)^2 \right)}.
\tag{21}
$$

Taking into account Equations (13)–(15) and (18)–(20), the intensity of stresses caused by the difference in the coefficients of thermal expansion of the strengthening particles and the matrix is equal to

$$
\sigma\_{\rm int} = 6\sqrt{3} \frac{K\_{\rm P} G\_{\rm m}}{3K\_{\rm P} + 4G\_{\rm m}} \left| \alpha\_{\rm P} - \alpha\_{\rm m} \right| \Delta T \min \left[ 1, \frac{R\_p^3}{r^3} \right]. \tag{22}
$$

The average value of the thermal stress arising from the difference in the coefficients of thermal expansion, elastic modulus, and shear modulus of the matrix and the particle is determined using the traditional procedure of averaging over a spherical volume with a radius <sup>Λ</sup>p/2.

$$
\sigma\_{\rm CTE} = \frac{24}{\Lambda\_{\rm P}^3} \int\_0^{\Lambda\_{\rm P}/2} \sigma\_{\rm int} r^2 dr. \tag{23}
$$

Thus, the average value of thermal stresses can be estimated as

$$
\sigma\_{\rm CTE} = 6\sqrt{3} \frac{K\_{\rm P} G\_{\rm m}}{3K\_{\rm P} + 4G\_{\rm m}} f\_{\rm P} |u\_{\rm P} - u\_{\rm m}| \Delta T \left(1 - \ln f\_{\rm P} \right). \tag{24}
$$

#### **3. Results and Discussion**
