A. Surface energy

The surface energy is the criterion to evaluate the surface stability, which is defined as the excess energy at the surface of a material compared to the bulk ^{Reference Gale and Rohl25}

(1) $${E_{{\rm{surf}}}} = {1 \over A}\left( {{E_{{\rm{slab}}}} - {E_{{\rm{bulk}}}}} \right)\quad ,$$

where A is the surface area, E _slab is the total energy of the surface slab, and E _bulk is the energy of the bulk material with the same component. The accurate E _surf can only be calculated when the surface model has the same stoichiometry as the bulk material. So the two-region modeling method is involved. ^{Reference Gale and Rohl25} As shown in Fig. 1, each model employs a two-dimensional slab that contains layers of Ti and B atoms and is separated by a 15 Å vacuum gap, and the slab has been divided into two regions. Region 1 contains the surface and several atomic layers below it, while region 2 is assumed as the inside part of the bulk material. The initial model is built according to the bulk crystal structure, and then its cell is locked both in size and shape. The atoms in region 1 are free to relax during the geometry optimization; in the meantime, those in region 2 are totally fixed. In other words, region 1 represents the real surface slab, and there is only one surface which needs to be considered in this configuration. In this way, the number of atomic layers for each model is more freely to choose to construct a stoichiometric surface slab. In addition, it is necessary to insure that region 1 and region 2 are thick enough because the properties of the innermost layer of region 1 must converge to those of the bulk. Then a convergence test on the relaxation of the interlayer distance, Δ _i–j = d _ij /d _bulk − 1, of the surface models with different layer numbers is arranged. As shown in Table I, when the layer numbers in region 1 and region 2 are set as 6 and 6 for (0001)-Ti, 8 and 6 for (0001)-B, 6 and 5 for $\left( {11\bar 20} \right)$ , 9 and 9 for $\left( {10\bar 10} \right)$ -Ti, and 12 and 9 for $\left( {10\bar 10} \right)$ -B(Ti) and $\left( {10\bar 10} \right)$ -B(B), geometry optimizations show that the relaxations of the innermost interlayer distance in all of these models are less than 0.16%, indicating a well converged lattice structure.

FIG. 1.

Side view of the two-region models of TiB₂ (a) (0001)-Ti, (b) (0001)-B, (c) $\left( {11\bar 20} \right)$ , (d) $\left( {10\bar 10} \right)$ -Ti, (e) $\left( {10\bar 10} \right)$ -B(Ti), and (f) $\left( {10\bar 10} \right)$ -B(B) surfaces. The Ti atoms are shown as big gray balls, the B atoms are shown as small pink balls, and the dark blue balls represent the fixed atoms during the geometry optimization.

TABLE I.

Interlayer distance relaxations, Δ _i–j = d _ij /d _bulk − 1, for TiB₂ surfaces.

The total energy of this two-region model can be divided into three parts,

(2) $${E_{{\rm{total}}}} = {E_{11}} + {E_{22}} + {E_{12}}\quad ,$$

where E ₁₁ and E ₂₂ represent the interaction energies within each region, and E ₁₂ corresponds to the interaction energy between two regions. Since E ₁₁ and E ₂₂ can be calculated directly with the partial model by removing the atoms in the other region from the surface model, E ₁₂ is obtained by deducting the former ones from the total energy

(3) $${E_{12}} = {E_{{\rm{total}}}} - {E_{11}} - {E_{22}}\quad .$$

Since region 2 is assumed to be the bulk part, the energy in region 2 should not be included by the surface energy. But the energy change of region 2 induced by the atoms in region 1 still contributes to the total energy. Therefore, the surface slab energy only contains the interaction energy within region 1 and half of the interaction energy between the two regions

(4) $${E_{{\rm{slab}}}} = {E_{11}} + {1 \over 2}{E_{12}} = {1 \over 2}\left( {{E_{{\rm{total}}}} + {E_{11}} - {E_{22}}} \right)\quad .$$

The reliability and accuracy of the two-region models have already been demonstrated in the investigation of ZrB₂ surfaces. ^{Reference Sun, Liu, Xiang and Zhou26} In the same way, the surface energies of six kinds of TiB₂ surfaces are calculated, as listed in Table II. The surface energy of (0001)-B is 4.207 J/m², while that of (0001)-Ti is only 0.026 J/m² higher. Compared with the two (0001) surfaces, $\left( {10\bar 10} \right)$ -B(B) has a lower surface energy of 4.192 J/m², but the other $\left( {10\bar 10} \right)$ surfaces hold much higher energies. The $\left( {11\bar 20} \right)$ surface keeps the lowest surface energy of 4.112 J/m².

TABLE II.

Calculated surface energies of TiB₂. M represents the metal element in diborides.

^a Ref. Reference Sun, Liu, Xiang and Zhou26.

It is worth noting that the surface energies of TiB₂ are much higher than those of ZrB₂ with the similar structure (as listed in Table II). The surface energies of TiB₂ arise for about 0.3 J/m² on the (0001) surfaces, and the surface energy increases are even up to 0.5–0.8 J/m² on $\left( {10\bar 10} \right)$ and $\left( {11\bar 20} \right)$ surfaces. This is because the chemical bonds in TiB₂ are much stronger than those in ZrB₂, which can be deduced from the higher modulus and shorter bond lengths of TiB₂. ^{Reference Zhou, Xiang, Feng and Li5} In other words, a higher cleaving energy is required for breaking the stronger chemical bonds during the formation of surface, which finally leads to the higher surface energy of TiB₂. Since a higher surface energy indicates a less stable surface, to reduce the surface area, the TiB₂ grain will grow much bigger than that of ZrB₂ if they were in the same condition. Thus, the grain coarsening may occur easily during the sintering of TiB₂. It has been found that the TiB₂ grain size increases quickly with the increasing temperature and dwell time in the course of hot pressing sintering of TiB₂. ^{Reference Wang, Fu, Wang and Yuan27} Ferber et al. also reported that sintering of TiB₂ at high temperatures resulted in grains above a critical size of 15 μm, which caused spontaneous microcracking of the material upon cooling. ^{Reference Ferber, Becher and Finch28} So the high surface energy of TiB₂ is one of the essential factors for the difficulty of sintering.

B. Anisotropic surface stability

Besides the high surface energy, it is also noticeable that the surface stability of TiB₂ is anisotropic. Since a lower surface energy always leads to an energetically stable surface, the surface stability sequence of TiB₂ is $\left( {11\bar 20} \right)$ > $\left( {10\bar 10} \right)$ -B(B) > (0001)-B > (0001)-Ti > $\left( {10\bar 10} \right)$ -B(Ti) > $\left( {10\bar 10} \right)$ -Ti. It is also notable that although this sequence resembles that of ZrB₂ surfaces, the surface energy differences in TiB₂ are much different from those in ZrB₂, which will lead to the different anisotropy in surface stability. This is a result of the anisotropic bonding feature on surfaces and will be discussed in the next section.

The anisotropy in surface stability is helpful in understanding the growth morphology of TiB₂ grains. In general, crystal growth is a combination of mass transport, surface attachment, and crystallization. Since TiB₂ is usually synthesized by a solid-state reaction, the crystal growth of TiB₂ during synthesis can be regarded as following. TiB₂ initially nucleates at the boundary between boron source and titanium source particles. The two source substances are separated by the grown TiB₂ grain, and the subsequent reaction relies on the atomic diffusion across the TiB₂ grain. ^{Reference Fan, Guo and Cantor29} Then the generated TiB₂ units crystallize on the grain surfaces which results in the grain growth. Therefore, atomic diffusion and crystallization on surfaces are the main procedures dominating the crystal growth. Higher surface energy generally corresponds to strong attachment and quick crystallization. Hence, the crystallization rate on $\left( {11\bar 20} \right)$ , $\left( {10\bar 10} \right)$ -B(B), (0001)-B, and (0001)-Ti surfaces may be similar, which are slower than those on $\left( {10\bar 10} \right)$ -B(Ti) and $\left( {10\bar 10} \right)$ -Ti surfaces. Since the three $\left( {10\bar 10} \right)$ surfaces alternate during the growth, the crystallization rate along the direction normal to $\left( {10\bar 10} \right)$ surface can be faster than those along 〈0001〉 and the direction normal to $\left( {11\bar 20} \right)$ surface. Besides, the atomic diffusion within the (0001) plane requires lower activation energy than that along the normal direction, owing to the compact arrangement in the (0001) plane. ^{Reference Fan, Guo and Cantor29} In other words, diffusion rates along directions normal to $\left( {10\bar 10} \right)$ and $\left( {11\bar 20} \right)$ surfaces are faster than that along 〈0001〉. Combining both the fast diffusion rate and crystallization rate, the growth rate along the direction normal to $\left( {10\bar 10} \right)$ is faster than that along the direction normal to $\left( {11\bar 20} \right)$ , and much faster than that along 〈0001〉. This anisotropic growth rate suggests a hexagonal plate-like grain morphology of TiB₂ with the plate perpendicular to 〈0001〉 directions, and the final grain is supposed to be covered by the more stable surfaces, i.e., $\left( {11\bar 20} \right)$ and (0001)-B. This hexagonal shape coincides with the observations in as-synthesized TiB₂ powders. ^{Reference Kang and Kim15–Reference Shahbahrami, Fard and Sedghi17}

The grain coarsening procedure of TiB₂ during high temperature sintering is schematically shown in Fig. 2. The initial TiB₂ powders are thin plate like, then the grains become large sized and close packed during the sintering. Due to the smaller anisotropy in the grain growth at high temperatures, the aspect ratio of grains may be increased mildly in the course of growth. However, the anisotropy in grain morphology leads to the irregular grain distribution during packing [Fig. 2(b)], which also presents an opportunity of abnormal grain growth. As can be seen from Fig. 2(c), owing to the anisotropy in thermal expansion, ^{Reference Xiang, Feng, Li and Zhou9} the mismatch of crystal orientations between neighboring grains can cause residual stress at the grain boundary during cooling. Then, microcracking occurs easily with the high residual stress, as the enlarged grain size will aggravate the mismatch of thermal expansion. Therefore, microcracking in sintered TiB₂ cannot be avoided through this grain growth procedure.

FIG. 2.

Illustration of the grain coarsening procedure of TiB₂: (a) The plate-like grains in TiB₂ powders, (b) packed grown grains, and (c) densified TiB₂. The anisotropic arrows illustrate the anisotropy in thermal expansion. The mismatch of thermal expansion can cause residual stress at the grain boundary (shown in red) and result in microcracking during cooling.

C. Bonding features on surfaces

The anisotropic surface stability of TiB₂ is resulted from the anisotropic bonding features on the surfaces. The electronic structure and chemical bonding of bulk TiB₂ had been discussed in our previous work. ^{Reference Zhou, Xiang, Feng and Li5} There are both σ-type and π-type covalent bonds between the neighboring B atoms in the graphite-like B layer on the (0002) plane, and the interaction between Ti and B atoms is ionic-covalent, while only the metallic bond appears in the Ti layer. To further understand the anisotropic surface stability of TiB₂, the electronic structure and bonding nature on these surfaces are investigated by analyzing the charge distribution and DOS of surface atoms.

Due to the different bonding environments, the DOS of the surface atoms is much different from those in the bulk, as can be seen from Fig. 3. The surface Ti atom contains much more electrons than that in the bulk. On (0001)-Ti and $\left( {10\bar 10} \right)$ -Ti, the surface Ti atom is only surrounded by 6 B atoms (versus 12 B atoms in the bulk). The excess electrons mainly fill in the high energy orbitals and cause a huge peak at the Fermi level [Figs. 3(b) and 3(e)]. On the $\left( {11\bar 20} \right)$ surface, although the Ti atom also appears in the first layer, it has a bigger coordination number of 8, so the DOS at the Fermi level is much smaller in Fig. 3(d). In contrast, the surface B atom is electron deficient owing to the lack of Ti–B and B–B interactions. To compensate for the bonding deficiency of surface atoms, the surrounding chemical bonds are automatically enhanced during the relaxation of surface structures, which can be deduced from the contracted bond lengths and the overlapping of DOS peaks. So the surface structure is stabilized through relaxation, and the surface energy is relatively lower than the cleaving energy that required for breaking the chemical bonds.

FIG. 3.

The projected DOS of the Ti and B atoms (a) in the bulk, (b) on (0001)-Ti, (c) (0001)-B, (d) $\left( {11\bar 20} \right)$ , (e) $\left( {10\bar 10} \right)$ -Ti, (f) $\left( {10\bar 10} \right)$ -B(Ti), and (g) $\left( {10\bar 10} \right)$ -B(B) surfaces of TiB₂. The layer in which the atom locates is marked for each atom. The vertical dotted line at 0 eV marks the Fermi level.

On (0001)-B, as shown in Fig. 3(c), multiple electron states of the B atom overlap with those of the Ti atom from −2.5 eV to the Fermi level, indicating the new Ti–B interaction on (0001)-B, the additional charge accumulating between surface B atoms and the Ti atom in the second layer could also confirm the extra electron transfer from Ti to the surface B atom [Fig. 4(a)]. However, as can be seen from Fig. 3(b), there are no such new bonding states in the DOS on (0001)-Ti and the DOS of the B atom in the second layer of (0001)-Ti is very similar to that of the B atom in the bulk. As shown in Fig. 4(b), only a small part of the excess electrons of surface Ti atoms is involved by the B–B π bonds; the other parts still distribute in the (0001) Ti plane. In other words, the relaxations of the remaining bonds on the two (0001) surfaces are quite different. This is because the Ti atom in the second layer of (0001)-B still keeps several d electrons in the Ti–Ti metallic bonds, which can be deprived by the surface B atom to form new Ti–B bonds, but the orbitals of B atom in the second layer of (0001)-Ti are almost saturated. Thus, the Ti–B bond on (0001)-B becomes much stronger and the bond length is shorter than that on (0001)-Ti. This shorter bond length contributes to the bigger interlayer distance relaxation of (0001)-B (Table I), which coincides with the results of Han et al. ^{Reference Han, Dai, Shu, Wang and Sun13} Since the required cleaving energies for the two (0001) surfaces are the same, the difference in the surface relaxation will lead (0001)-B to be more stable.

FIG. 4.

The differential charge density of (a) (0001)-B and (b) (0001)-Ti surfaces on the $\left( {11\bar 20} \right)$ plane. The positive charges are shown as blue, and negative ones are shown as red.

Besides the surface Ti atoms, B atoms also exist on the $\left( {11\bar 20} \right)$ surface. Different from that on (0001)-B, the B atom on the $\left( {11\bar 20} \right)$ surface is lack of one B–B covalent bond. The unbonded electrons of surface B atoms can participate in the π bonding in the B plane; as shown in Fig. 5(a), the surface B–B bond is enhanced and contracts from 1.749 to 1.684 Å in length. Meanwhile, as can be seen from Fig. 5(b), there is an electron accumulation between surface B and surface Ti atoms, which indicates that the covalent part of Ti–B interaction increases and it becomes stronger than those in the bulk. The surface B atom still keeps the sp ² hybridization, and the unbonded sp ² orbital combines with a part of the p _z orbital and turns to a spindle shape perpendicular to the B plane. Then this large shape orbital can overlap with the 3d(t _2g) orbitals of two surface Ti atoms simultaneously. The overlapping of DOS near the Fermi level also indicates the strengthened interaction between surface Ti and B atoms [Fig. 3(d)]. In this way, the bonding statuses for both surface Ti and B atoms are ameliorated. Combined with the Ti–B bonds between first two layers, the interactions around the surface atoms make more contributions to the relaxation of the $\left( {11\bar 20} \right)$ surface, which results in a lower surface energy.

FIG. 5.

The differential charge density of the $\left( {11\bar 20} \right)$ surface on the (a) (0001) B-plane and (b) $\left( {11\bar 20} \right)$ plane across the surface B atoms. The positive charges are shown as blue, and negative ones are shown as red.

Because of the bonding deficiency of the atoms in the first two layers, the remaining bonds around the atoms on $\left( {10\bar 10} \right)$ surfaces are also obviously enhanced, which result in the more evident relaxations of the first interlayer distance (in Table I). Similar to the $\left( {11\bar 20} \right)$ surface, there is an enhanced Ti–B interaction on $\left( {10\bar 10} \right)$ -Ti due to the unfilled orbitals of the B atom in the second layer, which can be deduced from the overlapping of DOS at −2.5 eV [Fig. 3(e)]. Unfortunately, the B–B bonds between the second and third layers are not strengthened but extend to 1.757 Å in length, as can be seen from Fig. 6(a). So the contribution of the surface relaxation to the surface energy is limited on $\left( {10\bar 10} \right)$ -Ti. The B atom on $\left( {10\bar 10} \right)$ -B(Ti) can only form one B–B covalent bond with the B atom in the third layer. As shown in Fig. 6(b), this B–B interaction involves part of the unbonded electrons and contracts to 1.684 Å in length, but the other electrons still distribute in the unbonded p orbitals of the surface B atom and participate in the Ti–B interaction, which leads to the increased DOS at −2.1 eV [Fig. 3(f)]. However, on $\left( {10\bar 10} \right)$ -B(B), the unbonded electron of the surface B atom mainly fills in the B–B bond within the first two layers, which contract to 1.613 Å in length and become quite strong, as shown in Fig. 6(c). This significant change of chemical bonding can strongly make up the electron deficiency of B atoms in the first two layers simultaneously. Furthermore, there are also enhanced Ti–B interactions [the overlapped peaks at −0.5 eV in Fig. 3(g)] around the surface B atoms. Thus, despite the same cleaving energies, $\left( {10\bar 10} \right)$ -B(B) is much more stable than $\left( {10\bar 10} \right)$ -Ti owing to the more evident surface relaxation.

FIG. 6.

The differential charge density of (a) $\left( {10\bar 10} \right)$ -Ti, (b) $\left( {10\bar 10} \right)$ -B(Ti), and (c) $\left( {10\bar 10} \right)$ -B(B) surfaces on the (0001) B-plane. The positive charges are shown as blue, and negative ones are shown as red.

On the cleaving aspect, only Ti–B bonds are broken on (0001) surfaces, while B–B interactions are also involved on $\left( {10\bar 10} \right)$ and $\left( {11\bar 20} \right)$ surfaces. Since the strength difference between these two types of bonds in TiB₂ is enlarged from that in ZrB₂, the surface energies of $\left( {10\bar 10} \right)$ and $\left( {11\bar 20} \right)$ surfaces increase more evidently than that of (0001) surfaces in TiB₂. And on the aspect of surface relaxation, the graphite-like conjugation structure in the B plane can gather electrons and contribute to the enhanced B–B and Ti–B interactions around the surface B atom. The contribution of enhanced interactions and the absence of huge peak at the Fermi level make the B-terminated surfaces to be more stable than the Ti-terminated ones. It is also obvious that the enhancement of B–B covalent bonding makes a greater contribution to compensate for the bonding deficiency of the surface atom than that of the Ti–B interaction. Thus, the relaxation aspect also plays an important role in surface energy, which even conquer the barrier of cleaving energy and lead to the better stability of the $\left( {10\bar 10} \right)$ -B(B) surface.