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Determination of the Thickness of the Embedding Phase in 0D 2 Nanocomposites

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Determination of the Thickness of the Embedding Phase in 0D 2 Nanocomposites *Manuscript corrected with highlighted changes Click here to view linked References 1 Determination of the thickness of the embedding phase in 0D 2 nanocomposites 3 D. Martínez-Martínez1,*, J.C. Sánchez-López2 4 1Center of Physics of the University of Minho, Campus de Azurem, 4800-058 5 Guimaraes, Portugal. 6 2Instituto de Ciencia de Materiales de Sevilla, Américo Vespucio 49, 41092 7 Seville, Spain. 8 *corresponding author e-mail: [email protected] 9 10 Abstract 11 0D nanocomposites formed by small nanoparticles embedded in a second 12 phase are very interesting systems which may show properties that are beyond 13 the ones observed in the original constituents alone. One of the main 14 parameters to understand the behavior of such nanocomposites is the 15 determination of the separation between two adjacent nanoparticles, in other 16 words, the thickness of the embedding phase. However, its experimental 17 measurement is extremely complicated. Therefore, its evaluation is performed 18 by an indirect approach using geometrical models. The ones typically used 19 represent the nanoparticles by cubes or spheres. 20 In this paper the used geometrical models are revised, and additional 21 geometrical models based in other parallelohedra (hexagonal prism, rhombic 22 and elongated dodecahedron and truncated octahedron) are presented. 23 Additionally, a hybrid model that shows a transition between the spherical and 24 tessellated models is proposed. Finally, the different approaches are tested on 25 a set of titanium carbide/amorphous carbon (TiC/a-C) nanocomposite films to 26 estimate the thickness of the a-C phase and explain the observed hardness 27 properties. 28 29 Keywords: nanocomposite, tessellation, embedding, grain, crystal, grain 30 boundary thickness 31 Page 1 of 15 1 1. Introduction: 2 0D nanocomposites are formed by small particles with the three dimensions in 3 the nano range (contrary to e.g. 1D nanotubes or 2D nanolayers) embedded in 4 a secondary matrix [1]. The properties of these systems are very interesting, not 5 only because they combine properties of both mixing phases, but also because 6 the final properties may overcome the so-called rule of mixture, i.e. the 7 properties of the nanocomposite may exceed the properties of the constituents 8 alone [2]. The properties of the nanocomposite depend not only on the 9 composition and the particle size, but also on the distribution of these particles 10 within the embedding matrix, i.e. the interparticle distance. For instance, 11 superhard nanocomposites must be composed by hard small crystallites (ca. 5 12 nm) separated by an amorphous phase of 0.1-0.3 nm thickness [2]. A similar 13 concept has been developed to enhance the fracture toughness of 14 nanocomposites [3,4]. The electrical performance of nanocomposites is also 15 dependent on that parameter [5]. 16 However, the experimental measurement of inter-particle distance (or thickness 17 of the embedding phase) is difficult, even by high resolution electron 18 microscopy. Therefore, simple geometrical models have been generally used to 19 estimate that parameter. These models assume that all the particles are 20 identical and equally distributed. Two geometries have been used: spherical 21 and cubic [3–7]. The first one has the disadvantage of not filling the whole 22 space, while the second allows this possibility (so-called tessellation). However, 23 both of them are necessary to explain the observed distributions. Thus, Figure 1 24 shows two high resolution TEM images of titanium carbide/amorphous carbon 25 (TiC/a-C) nanocomposite coatings [8]. In the example of Figure 1b, only two 26 crystalline TiC particles with different orientations are observed (interplanar 27 distances of 0.22 and 0.25 nm), which are separated apart. Nevertheless, the 28 TiC particles are rounded, which would agree with a spherical model. In 29 contrast, in case of Figure 1a, the sample is practically fully occupied by 30 crystalline TiC grains (the Fourier transform of the central grain indicates that 31 the TiC is oriented along the [110] axis). Therefore, a spherical model would fail, 32 and a cube-like model would be more suitable. Page 2 of 15 1 The aim of this work is to analyze more geometrical models that tessellate the 2 space besides the cubic one, and also to present a unified hybrid model that 3 could combine the benefits of both approaches (spherical and tessellated). 4 2. Experimental details 5 TiC/a-C nanocomposite coatings were prepared by co-sputtering of Ti and 6 graphite targets under Ar pressure of 0.75 Pa. The power applied to each 7 magnetron was varied to get different chemical compositions and 8 microstructures. Further details about the film deposition are given elsewhere 9 [8–10]. Transmission electron microscopy (TEM) and electron energy loss 10 spectroscopy (EELS) were carried out in a Philips CM200 microscope operated 11 at 200 kV and equipped with a parallel detection EELS spectrometer from 12 Gatan (766–2K) [8–10]. X-ray diffraction (XRD) measurements were carried out 13 using Cu Kα radiation in a Siemens D5000 diffractometer at an incidence angle 14 of 1°. The Scherrer equation was used to estimate the grain size from the 15 broadening of the (220) peak. 16 17 3. Results and discussion 18 Figure 2 shows five polyhedra that tessellate the space (i.e. can fully occupy the 19 volume with face-to-face contact by translation) [11]. They are, in fact, the five 20 so-called parallelohedra, which are composed by 4-edge and/or 6-edge faces. 21 All the edges of the parallelohedra have the same size (L), which can be 22 considered as their characteristic length. For the sphere, the characteristic 23 length would be the radius. Table 1 shows the volume of these parallelohedra 24 depending on L, which can be represented in all the cases as the product of L3 25 times a constant K1 which depends on the parallelohedron. In this work, the 26 particle size will be considered as the distance (D) between opposed faces of 27 the parallelohedron (or the diameter in case of the sphere), which needs to be 28 calculated as a function of L for each system. Figure 3 shows appropriate views 29 of each prism to illustrate the different values of D in each case. In all cases, a 30 linear relationship between L and D can be found (D=K2 L), where K2 depends 31 on the geometrical system. Page 3 of 15 1 The first parallelohedron (Figs. 2a and 3a) is the cube or hexahedron, which is 2 composed by 6 square faces and 8 identical vertices. In this case, the edge size 3 L equals the interface distance D. The hexagonal prism (Figs. 2b and 3b) is 4 formed by 6 square faces and 2 regular hexagonal faces, with 12 identical 5 vertices. This body is also known as ‘elongated cube’, although this name may 6 induce confusion, because the hexagonal prism can be constructed from the 7 cube by adding two pairs of vertices in top (and bottom) of two opposite faces 8 (cf. Fig. 2a and 2b). As a consequence, the lateral faces are transformed from 9 squares to hexagons. Nevertheless, to tessellate the space, the hexagons need 10 to be regular, which obligates to increase the distance between the square 11 faces of the original cube. As a consequence, this body has two different inter- 12 face distances; between the hexagonal faces (Dhex), the distance is the same 13 than in the cube, while between two square faces (Dsq), the distance is larger 14 (cf. Fig. 3b and Table 1). The third body is the rhombic dodecahedron (Figs. 2c 15 and 3c), which is composed by 12 rhombic faces with a diagonal ratio equal 21/2 16 and two pairs of angles of ca. 70.5 and 109.5º. It has 14 vertices of two types, 6 17 of them connected to 4 faces (through the angle of 70.5º) and 8 vertices 18 connected to 3 faces (through the angle of 109.5º). The elongated 19 dodecahedron (Figs. 2d and 3d) is the result of inserting parallel additional 20 edges on 4 of the 6 tetra-coordinated vertices. As a consequence, four rhombs 21 are transformed in equiaxed hexagons, which conform the body together with 22 the unmodified 8 rhombic faces. It is worth mentioning that, in this case, the 23 hexagons are not regular, since not all the angles are the same (2 of 109.5º, as 24 before elongation and 4 of ca. 125.25º). In this case, there are also two different 25 inter-face distances, depending if we consider the rhombs or the hexagons (Drh 26 and Dhex in Table 1). The last parallelohedron is the truncated octahedron (Figs. 27 2e and 3e), which can be constructed by truncating an octahedron or a cube 28 with perpendicular planes to the 3 four-fold or to the 4 three-fold symmetry 29 axes, respectively. This body is formed by 6 squares and 8 regular hexagons, 30 and also two different inter-face distances can be found (Dsq and Dhex in Table 31 1). 32 Table 1 summarizes the values of K2 for all the geometrical bodies considered. 33 In all cases, K2≥1, which indicates that opposed faces are more separated than Page 4 of 15 1 the edge of the parallelohedron, as it can be expected for this type of bodies. It 2 can also be observed that the value of K2 increases for more complex 3 geometrical shapes (i.e. higher number of edges). The volume of the 4 parallelohedra can be expressed as a function of the inter-face distances 5 instead the edge size.
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