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Evolution Mechanism of Metallic Dioxide M[O.sub.2] (M = Mn, Ti) from Nanorods to Bulk Crystal: First-Principles Research.

1. Introduction

Ti[O.sub.2], which is a vital inorganic functional nanomaterial, has been widely used in down-flop pigments, ultraviolet screening, photoelectric conversion, photocatalysis, and so on [1]. Mn[O.sub.2] is a popular and cost-effective material for the removal of pollutants in air, water, and industry [2]. Both have been widely investigated and improved to enhance their catalytic performance, such as by doping with metallic elements [1, 3], incorporation into carbon nanotubes [4], and manufacturing with a nanometer structure [5]. Especially, in the nanocrystallization process, the Ti[O.sub.2] and Mn[O.sub.2] nanometer materials exhibit additional surface and nanometer effects although they have the same components and skeleton units as the bulk morphology. Both have been successfully applied to catalytic redox for some pollutants. But they exhibit different catalytic capabilities for the same reactant in somewhere. To remove arsenite oxidation, the arsenite ([As.sup.III]) is oxidized to [As.sup.V] for more than 0.4 hours by manganese dioxide, and the Mn-As bond length is from 0.271 nm to 0.34nm [6]. For the rutile Ti[O.sub.2], however, it is found that the Ti-As bond length is from 0.283 nm to 0.405 nm, and the adsorption energy of [As.sup.V] on Ti[O.sub.2](110) is greater than that on Mn[O.sub.2] [7]. Regarding the decomposition of CO, Chen et al. [8] indicated that the CO adsorbed onto the anatase Ti[O.sub.2] resulted in a moderate adsorption energy (about 0.3 eV) and a positive shift of the C-O stretching frequency (about +44 [cm.sup.-1]) whereas the CO could no longer be adsorbed onto the Mn[O.sub.2] [9]. Considering the adsorption of [O.sub.2] onto Mn[O.sub.2], the oxygen reduction reaction can occur either in solution [10] or in air [11]. Meanwhile, Petrik and Kimmel [12] stated that [O.sub.2] could be adsorbed onto rutile Ti[O.sub.2] only at very low temperatures. Their different catalytic activities have attracted the attention of many researchers. Barnard et al. [13] had modeled the electronic properties of Ti[O.sub.2] nanoparticles and pointed out that the free energy of surface would keep constant after the sizes of nanoparticles were larger than 100 nm [14]. After studying a series of low stoichiometric surfaces, they found the effects of edges and corners were omitted when the nanoparticles were larger than ~2 nm, and constructed the morphology of rutile Ti[O.sub.2] only composed by {110} Miller index [15]. Nevertheless, Deringer and Csanyi [16] and Tompsett et al. [17] discovered that nanorods of rutile Ti[O.sub.2] and [alpha]-Mn[O.sub.2] have the same equilibrium geometric morphologies, with a structure consisting mainly of {100} and {110} Miller indexes. Furthermore Hummer et al. [18] pointed out that the surface energies of Ti[O.sub.2] were dependent with edges and corners of nanocrystal at particle size [less than or equal to]3 nm. At present, former researches do not identify the intrinsic mechanism between bulk and nanorods although they have studied the nanoscale morphology of rutile Ti[O.sub.2] and [alpha]-Mn[O.sub.2] for a long time. But such an intrinsic mechanism plays a vital role in the design and optimization of metallic oxide nanomaterials. In a previous paper [19], it have been stated that there may be an optimal [alpha]-Mn[O.sub.2] nanorod, which has a surface energy suitable for promoting enhanced surface activity, together with an appropriate degree of cohesive energy for maintaining structural stability. As an extension of previous work, the present study further sets out to investigate the evolution mechanism of bulk and nanorods of M[O.sub.2] (M = Mn, Ti) metallic dioxide.

2. Simulation Models and Method

To elucidate the nanometer effect of metallic dioxide M[O.sub.2] (M = Mn, Ti) with a nanostructure, several models of M[O.sub.2] (M = Mn, Ti) in crystal, bulk surface, nanorod, and microfacet topological configurations were constructed and studied systematically according to their stoichiometric proportions. Their corresponding simulated models are shown in Figures 1 and 2. Regarding the rutile Ti[O.sub.2] model construction, only two prominent and stable Miller index planes, such as {100} and {110}, are considered. In the present study, the [alpha]-Mn[O.sub.2] and rutile Ti[O.sub.2] nanorod models were constructed based on the experimental results obtained by Barnard et al. [13-15] and Deringer and Csanyi [16]. All the simulation models are shown in Figure 2. For the rutile Ti[O.sub.2] (100) bulk surface, there are triple units of {100} Miller index slabs, while, for the {110} bulk surfaces, there are double {110} Miller index slabs, as shown in Figures 2(b) and 2(c), respectively. A microfacet rutile Ti[O.sub.2] [(100 x 110)] supercell structure containing only double units of {100} and {110} Miller index slabs, which had been proven to exhibit a similar catalysis performance as a nanostructure [19,22], was built as a bulk surface with nanometer morphologies only, as shown in Figure 2(d). Regarding the nanorod (NR) models, all of them were combined with only {110} and {100} Miller index slabs, as shown in Figures 2(e)-2(g). (In our future work, further Miller index slabs will be considered to represent a more complicated situation.) The smallest nanorod addressed in the present study, that is, (NR(1)), consisting of two {110} and one {100} Miller index slab unit, was built as a [Ti.sub.32][O.sub.64] supercell, as shown in Figure 2(e). The second rutile Ti[O.sub.2] nanorod (NR(2)) contains two units each of {100} and {110} Miller index slabs to construct a [Ti.sub.52][O.sub.104] supercell (Figure 2(f)). The largest rutile Ti[O.sub.2] nanorod contains triple {110} and double {100} Miller index slab units named NR(3), which form a [Ti.sub.88][O.sub.176] supercell (Figure 2(g)). Similar way of constructed configuration is forced to [alpha]-Mn[O.sub.2] nanorods. The latter is the largest that can be handled in the calculation limits of our computer cluster. All these primitive nanorods can be regarded as being free nanomaterials in morphologies by using their periodic boundary conditions and transitional symmetry. The purpose of such constructions is to investigate the surface effect of different Miller indices over several models. All the bulksurface models were calculated assuming slabs with a minimum thickness of 14 [Angstrom]. In all the bulk surface and nanorod models, a separating vacuum distance of at least 12 [Angstrom] was used to distance the slabs from their periodic image. For the first step, all of the models were not terminated by hydrogenation as followed by Barnard et al.'s report [13]. The complicated surface models of metallic dioxide M[O.sub.2] (M = Mn, Ti) will be studied in our further research. To distinguish the difference between [alpha]-Mn[O.sub.2] and rutile Ti[O.sub.2] in similar configurations, every simulated model was labeled "M" or "T" to represent the [alpha]-Mn[O.sub.2] and rutile Ti[O.sub.2] series, respectively. For example, the (100) bulksurface for [alpha]-Mn[O.sub.2] was labeled M(100), while T(100) represents the (100) bulk surface for rutile Ti[O.sub.2]. This convention is used for every model shown in Figures 1 and 2.

Based on the calculated sets of Deringer and Csanyi [16], all the above [alpha]-Mn[O.sub.2] and rutile Ti[O.sub.2] simulation models were relaxed by applying the following process: a first-principles pseudopotential plane-wave method, based on density functional theory, was implemented in the Cambridge Sequential Total Energy Package (CASTEP) code [23]. The electronic structure was calculated using the Generalized Gradient Approximation (GGA) devised by Perdew et al., with a Tkatchenko-Scheffler approach (TS) being used for the dispersion corrections [16]. The PBE + U exchange-correlation function has been demonstrated to give a good description of the defect properties in [alpha]-Mn[O.sub.2] [17]. All calculations were performed in a ferromagnetic spin polarized configuration, while effects of more complex magnetic orders were left for future work due to their low energy scale. For the rutile Ti[O.sub.2], however, Deringer and Csanyi [16] pointed out that adding U terms caused the results to steadily be worsened, in much the same way as in [22,24]. In the present study, therefore, U correction was not applied to any of the rutile Ti[O.sub.2] models. All the subsequent calculations were performed based on the equilibrium lattice constants obtained without cell relaxation using a cutoff of 500 eV, which was more precise than previous papers [13-15,18]. This included the recalculation of the energy for the bulk unit cell so that all the comparative energies could be obtained. A minimum of 8 x 1 x 1 k-points were used in the Brillouin zone of the conventional cell and scaled appropriately for supercells. All the atomic positions in these primitive cells were relaxed in spin polarized situation according to the total energy and force using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm scheme [25], based on the cell optimization criteria (a root mean square (RMS) force of 0.03 eV/[Angstrom], a stress of 0.05 GPa, and a displacement of 0.001 [Angstrom]). The convergence criteria for the self-consistent field (SCF) and energy tolerances were set to 1.0 x [10.sup.-6] and 5.0 x [10.sup.-5] eV/atom, respectively.

3. Results and Discussion

3.1. Test of Potential. The value of the U parameter for our PBE + U calculations is determined by ab initio calculations. Previous study [10] has demonstrated that a good description of the structural stability, band gaps, and magnetic interactions can be obtained when PBE + U is applied with the fully localized limit, which is therefore also used in the present study. U = 2.0 eV is employed for [alpha]-Mn[O.sub.2]. Table 1 lists the calculated lattice parameters for [alpha]-Mn[O.sub.2] obtained from PBE + U. These results are within 1.8% of the theoretical [17] and experimental [20] parameters, but the common tendency for PBE + U to overestimate the unit cell volume is evident. Regarding the values listed for rutile Ti[O.sub.2] in Table 2, the results are also similar to those of theoretical [16] and experimental [21] reports on DFT + TS. Therefore the calculated sets are appropriate for investigating the surface effects of M[O.sub.2] (M = Mn, Ti).

3.2. Evolution Character of Surface Energy. In Tables 3 and 4, the surface energy [E.sub.surface] for [alpha]-Mn[O.sub.2] and rutile Ti[O.sub.2] obtained via PBE + U and DFT + TS calculations, respectively, is shown. The surface energy is calculated by taking the difference between the energy of a constructed slab and the same number of [alpha]-Mn[O.sub.2] or rutile Ti[O.sub.2] formula units in the bulk [19]:

[E.sub.surface] = [E.sub.total] - n[E.sub.b]/S, (1)

where [E.sub.total] is the energy of a surface or nanorod model containing n formula crystal units, [E.sub.b] is the total energy of the crystal, and S is the surface area of the simulated models, where the bulk surface and microfacet contain two surfaces by their periodic boundary condition. The results are shown in Figure 3 (as well as in Tables 3 and 4). In [19], it is shown that the [E.sub.surface] values of the [alpha]-Mn[O.sub.2](100) and (110) bulk surfaces are equal to 0.6503 J[m.sup.-2] and 0.6794 J[m.sup.-2], respectively, which are similar to the results reported by Tompsett et al. (0.64 J[m.sup.-2] and 0.75 J[m.sup.-2], resp.) [17]. Regarding the rutile Ti[O.sub.2], the Esurface value for the T(100) is equal to 1.2492 J[m.sup.-2], which is a little larger than that (1.00 J[m.sup.-2]) obtained by Deringer and Csanyi [16]. Furthermore, the [E.sub.surface] value of the T(110) is equal to 1.1503 J[m.sup.-2], which is similar to that (1.10 J[m.sup.-2]) obtained by Ramamoorthy et al. [26], but a little larger than the 0.81 J[m.sup.-2] obtained by Lindan et al. [27] and the 0.80 J[m.sup.-2] obtained by Deringer and Csanyi [16]. However, none of them influence the trend and the internal law governing the surface energy in this manuscript.

The results are shown in Figures 3 and 4. For [alpha]-Mn[O.sub.2], it is found that the [E.sub.surface] value increases according to a trend of [E.sub.surface]M(100) < [E.sub.surface]M(110) < [E.sub.surface]MNR(I) < [E.sub.surface] MNR(II) < [E.sub.surface] MNR(III) < [E.sub.surface]M[(100 X 110)], as shown in Figure 3(a) (labeled by [C]). Given that the surface energies of M(100) and M(110) are nearly equivalent, the abscissa has no physical significance. Therefore, it assumes that the abscissa value of M(100) and M(110) is 2.5 and that of M[(100 x 110)] is 9, which is greater than that of MNR(III) by 3 points, as shown in Figure 3(a) (labeled by [C]). It is found that the surface energies of [alpha]-Mn[O.sub.2] of different morphologies fall in line with the relationship among them, implying the existence of an internal correlation. Furthermore, the sequence of M[(100 x 110)] does not correspond to nanorod(III), because other nanorods may exist. Upon closely analyzing their geometric structure, it is found that all the [alpha]-Mn[O.sub.2] nanorods and microfacets are composed of two Miller indexes, for example, the (110) and (100) microsurfaces. They differ only in the numbers of the (110) and (100) microsurfaces. If it is hypothesized that the average of the values of [E.sub.surface]M(100) and [E.sub.surface]M(110) is one component element of the surface energy for the nanorods and microfacets, their surface energies exhibit some linear relationship. This has not been observed previously. Their formulized relationship can be explained as follows: one constant parameter of the surface energy, 1.0401 J[m.sup.-2], labeled A, and another constant parameter of the surface energy, 0.6648 J[m.sup.-2] labeled B, which is equal to the average of [E.sub.surface]M(100) and [E.sub.surface]M(110), that is, (0.6593 J[m.sup.-2] + 0.6974 J[m.sup.-2])/2 = 0.6648 J[m.sup.-2]. The correlation function for the surface energy of the nanorods and microfacets to the average value of the surface energy for M(100) and M(110) is given as Y = A + N x B, where N is the quantization parameter for different nanorods and microfacet models. Quantization parameter N plays two roles in this paper: firstly it implies that their surface energies have some relationship between nanorods and corresponding bulk surface; secondly it restricts the maximum value of surface energy for nanorods. And N is defined as the additional number of {110} or {100} Miller index slab units from minimum nanorod MR(I) or NR(I), wherein the quantization parameter N for MNR(I) or TNR(I) is equal to 1, respectively. Their quantization parameter N is shown Figure 4. Their N values closely follow a linear relationship. After fitting by linear regression, the adjusted R square ([R.sup.2]) value is equal to 0.999, as shown in Figure 4. We can thus derive a function for the surface energy: [E.sub.surface]MNR(I) = 1.0401 J[m.sup.-2] + 1 x 0.6648 J[m.sup.-2], [E.sub.surface]MNR(II) = 1.0401 J[m.sup.-2] + 2.1021 x 0.6648 J[m.sup.-2], [E.sub.surface]MNR(III) = 1.0401 J[m.sup.-2] + 2.9967 x 0.6648 J[m.sup.-2], and [E.sub.surface]M[(100 x 110)] = 1.0401 J[m.sup.-2] + 6.0226 x 0.608 J[m.sup.-2]. Thus, it is clear why a previous adjustment of the abscissa in the surface energy of [alpha]-Mn[O.sub.2] was a line correlation (Figure 3(a) labeled by [C]). This reveals the evolution character of the surface energy for [alpha]-Mn[O.sub.2] nanorods and microfacets.

It is well known that the geometrical and chemical performances of rutile Ti[O.sub.2] are similar to those of [alpha]-Mn[O.sub.2]. From the results of analysis, the surface energy of [alpha]-Mn[O.sub.2] nanorods and microfacets has a quantization character. Therefore, it is needed to determine whether there is the same for rutile Ti[O.sub.2]. To do so, the surface energies of the rutile Ti[O.sub.2] are shown in Figure 3(b) and in Table 4. The trend in the surface energy [E.sub.surface] for rutile Ti[O.sub.2] from bulk surface [right arrow] nanorod [right arrow] microfacet was found to be different from that of [alpha]-Mn[O.sub.2]. Their surface energies were found to be similar to each other. The difference between them is very small, as shown in Figure 3(b). For example, the largest [E.sub.surface] is for T(100), which is equal to 1.2492 J[m.sup.-2]. The smallest is for T(110), which is equal to 1.1503 J[m.sup.-2]. Their difference is only 0.0989 J[m.sup.-2]. Furthermore, the differences between the microfacet and nanorods are obviously very small. Therefore, the trend in the surface energy for the rutile Ti[O.sub.2] in a bulk surface and nanorods assumes a horizontal line, as shown in Figure 3(b). In deep analysis, the microfacet and nanorod models are also composed of two Miller indexes, for example, (110) and (100) microsurfaces. And it takes the average of [E.sub.surface]T(100) (1.2492 J[m.sup.-2]) and [E.sub.surface]T(110) (1.1503 J[m.sup.-2]) as one constant B1, where B' is equal to (1.2492 J[m.sup.-2] + 1.1503 J[m.sup.-2])/2 = 1.1997 J[m.sup.-2]. It is found that the relationship between the surface energy of microfacets/nanorods and constant B' can also be fitted by linear regression, as shown in Figure 4. The surface energy function is given by SurfaceTNR(1) = 1.0102 x 1.1997 J[m.sup.-2], Surface TNR(2) = 1.0317 x 1.1997 J[m.sup.-2], [E.sub.surface]TNR(3) = 1.0347 x 1.1997 J[m.sup.-2], and [E.sub.surface]T[(100 x 110)] = 0.9879 x 1.1997J[m.sup.-2]. This evolution character of the surface energy of rutile Ti[O.sub.2] is different from that for [alpha]-Mn[O.sub.2]. The quantization character for [alpha]-Mn[O.sub.2] nanorods is a positive integer, while that for rutile Ti[O.sub.2] is equal to 1. However, they all have a quantization phenomenon in their surface energies.

3.3. Evolution Character of Cohesive Energy. The cohesive energy represents the work that is required for a crystal to be decomposed into atoms, which in turn denotes the stability of the respective simulation model. Here, the [E.sub.cohesive] value for several [alpha]-Mn[O.sub.2] or rutile Ti[O.sub.2] models has been calculated from the following equation [19]:

[mathematical expression not reproducible], (2)

where I and m represent the number of M (M = Ti or Mn) and O atoms in the respective morphologies of rutile [mathematical expression not reproducible] denotes the total energy of the [M.sub.l][O.sub.m] models, and [E.sup.M.sub.gas] and [E.sp.O.sub.gas] are the energies of the gaseous M (M = Ti or Mn) and O atoms, respectively. Before optimizing the gaseous atoms, a 10 x 10 ([[Angstrom].sup.3]) vacuum box is constructed and a single atom is placed, such as Ti, Mn, or O, in the center of the box to be relaxed and thus to obtain its global minimum energy. The results are given as [E.sup.Mn.sub.gas] = -588.1855 eV and [E.sup.O.sub.gas] = -432.2548 eV, as given in Table 3. For rutile Ti[O.sub.2], the results are [E.sup.ti].sub.gas] = -1594.3577 eV and [E.sup.O.sub.gas] = -431.9368 eV, wherein the difference of energy for oxygen in [alpha]-Mn[O.sub.2] and rutile Ti[O.sub.2] was originated from their different calculated sets in previous part of Simulation Models and Method. The results are given in Table 4. A previous study [19] has discussed the evolution of cohesive energy for [alpha]-Mn[O.sub.2] and found that the structural stability of the nanorods and microfacet is lower than that of the crystal and bulk surfaces. The present study will examine the relationship between the bulk surface, nanorods, and microfacet. If only the absolute values of the cohesive energy are considered, it can only determine that the structural stability of nanorods and microfacet is lower than that of bulk surfaces and crystals, as is already known. Analyzing their geometric morphologies, it is found that they are also composed by two Miller indexes, for example, the (110) and (100) microsurfaces. This paper applies the same treatment to the average value of cohesive energy [E.sub.a-cohesive] for M(110) and M(100) to establish a basic standard, which is equal to -4.6611 eV/atom (where [E.sub.collesive] M(110) = -4.6487 eV/atom and [E.sub.cohesive]M(100) = -4.6735 eV/atom) in Figure 5. Considering the ratio n of the cohesive energy for microfacets and nanorods to -[E.sub.a-cohesive], it is found that every instance of n is nearly equal to 1.09, where [n'.sub.1] = [E.sub.a-cohesive]/[E.sub.cohesive] MNR(I) = L1048, [n'.sub.2] = [E.sub.a-cohesive]/[E.sub.cohesive] MNR(II) = 1.0958, [n'.sub.3] = [E.sub.a-cohesive]/[E.sub.cohesive] MNR(III) = 1.0907, and [n'.sub.4] = [E.sub.a-cohesive]/[E.sub.cohesive] M[(100 x 110)] = 1.0942. It is found that all the correlation constants are equal to 1.09, as shown in Figure 6. This trend, which presents as a horizontal line, is different from the trend in the surface energy. Obviously, a quantization phenomenon can be seen. Regarding the cohesive energy of rutile Ti[O.sub.2], the difference is found to be very small, unlike the case of [alpha]-Mn[O.sub.2]. For example, the most stable structure is the rutile Ti[O.sub.2] crystal, for which the cohesive energy is equal to -7.8669 eV/atom. The least stable structure is TNR(1), for which the cohesive energy is equal to -7.7255 eV/atom. The difference between them is only equal to 0.1414 eV/atom. The second stable structure is the bulk surface T(100), for which the cohesive energy is equal to -7.7885 eV/atom. The cohesive energies of the bulk surface T(110), microfacets [(100 x 110)], and TNR(3) are very similar, being -7.7750 eV/atom, -7.7712 eV/atom, and -7.7779 eV/atom, respectively. The difference in their cohesive energies is only 0.0067 eV/atom, which may be regarded as being the calculation error, so that they can all be regarded as having the same structural stability. Regarding the trend in their cohesive energies, shown in Figure 5, they closely approximate to each other. Applying the same treatment to the cohesive energy of rutile Ti[O.sub.2], it can also take the average value of cohesive energy -Ea-cohesive for T(110) and T(100) to be a basic standard, which is equal to -7.7817 eV/atom (where [E.sub.cohesive]T(110) = -7.7750 eV/atom and [E.sub.cohesive]T(100) = -7.7885 eV/atom). Considering the ratio n" of the cohesive energy for the nanorods/microfacets to [E.sub.a-cohesive], it is found that every instance of n is nearly equal to 1.00, where [n".sub.1] = [E.sub.a-cohesive]/[E.sub.cohesive]TNR(1) = 10073, [n".sub.2] = [E.sub.a-cohesive]/[E.sub.cohesive]TNR(2) = 1.0037, = [E.sub.a-cohesive]/[E.sub.cohesive] T[100 x MNR(3) = 1.0049, and [n".sub.4] = [E.sub.a-cohesive]/[E.sub.cohesive]T[(100 X 110)] = 1.0013. All the normalized parameters are nearly equal to 1.00, as shown in Figure 6. Then, the evolution character of the cohesive energy for [alpha]-Mn[O.sub.2] and rutile Ti[O.sub.2] is abstracted absolutely. In line with the evolution of the geometric morphologies in the growth of nanorods, the ratio of their cohesive energies divided by [E.sub.a-cohesive] is found to be nearly equal to 1.

3.4. Evolution Character of Electronic Structure. From the above analysis, it is found that, from the bulk surface to nanorods, and even to microfacet with a nanometer structure, the surface and cohesive energies of [alpha]-Mn[O.sub.2] and rutile Ti[O.sub.2] exhibit some quantization phenomena. It is well known that the surface and cohesive energies are derived from the geometric or electronic structures. Therefore, in the next section of this paper, we study the evolution of the geometric and electronic structures by applying Mulliken analysis to reveal whether there is any quantization phenomenon corresponding to those energies. The Mulliken population [Q.sub.A-B] between atoms A and B and the Mulliken charge [Q.sub.A] are defined as follows [21]:

[mathematical expression not reproducible], (3)

where [P.sub.[mu]v] and [S.sub.v[mu]] are the density and overlap matrices, respectively, and wk is the weight associated with the calculated k-points in the Brillouin zone. Usually, the magnitude and sign of Q(A) characterize the ionicity of atom A in the supercell, while [Q.sub.A-B] can be used to approximate the average covalent bonding strength between atoms A and B. It is known that all the bulk surface, nanorod, and microfacet models originate from their crystal. As a result, the area of the crystal can be regarded as being infinite. To determine the intrinsic mechanism of the evolution character on the surface and cohesive energies, it sets a bond length [d.sub.A-B], Mulliken population [Q.sub.A-B], and Mulliken charge [Q.sub.A] of the crystal as the base values. These base values are equal to the average value of the bond length [[bar.[d.sub.A-B], the Mulliken population [Q.sub.A-B], and the Mulliken charge [Q.sub.A] in [alpha]-Mn[O.sub.2] or rutile Ti[O.sub.2] crystal, respectively. Then, the bond length variance [([DELTA]d).sup.2] is set equal to [([d.sub.A-B] - [[bar.d].sub.A-B]).sup.2] to elucidate the influence of the other geometric morphologies, such as the bulk surface, nanorods, and microfacet by their growth. The Mulliken population variance [([DELTA][Q.sub.A-B]).sup.2] is equal to ([Q.sub.A-B] [Q.sub.A-B])2. However, the Mulliken charge for [alpha]-Mn[O.sub.2] or rutile Ti[O.sub.2] consists of two parts, namely, the lost charge of metallic elements Mn or Ti and the reception charge of the oxygen elements. Therefore, the Mulliken charge variance [([DELTA][Q.sub.A]).sup.2] is equal to the sum of [([DELTA][Q.sub.M]).sup.2] and [([DELTA][Q.sub.O]).sup.2]. In this case, [bar.[d.sub.A-B]], [bar.[Q.sub.A-B]], [bar.[Q.sub.Mn]], and [bar.[Q.sub.O]] for [alpha]-Mn[O.sub.2] are equal to 1.9255, 0.3925, -0.59, and 1.05, respectively. For rutile Ti[O.sub.2], [bar.[d.sub.A-B]], [Q.sub.A-B], [bar.[Q.sub.Mn]], and [bar.[Q.sub.O]] are equal to 1.9714, 0.5050, -0.66, and 1.31, respectively. To identify the quantization phenomenon of the surface effect, the summation of [summation][([DELTA]d).sup.2], [summation] [([DELTA][Q.sub.A-B]).sup.2], and [summation][([DELTA][Q.sub.A]).sup.2] by their surface area S is normalized. Although the size and shape of nanocrystal have been set as a function of its free energy [15], the surface area S is the vital factor to affect the chemical performance of nanomaterials. Then the surface area S is chosen to be a normalized parameter in this paper. The detailed data is exhibited in Tables 5 and 6.

The final results are shown in Figure 7. Figure 7 shows that the normalized parameters [summation][([DELTA]d).sup.2]/s, [summation][([DELTA][Q.sub.A-B]).sup.2]/s, and [summation][([DELTA][Q.sub.A]).sup.2]/s for the bulk surface, nanorod, and microfacet models for [alpha]-Mn[O.sub.2] and rutile Ti[O.sub.2] fluctuate considerably and exhibit irregularities. Regarding the electronic properties of the bond strength and atomic charge, the largest is for M(100) or T(110) to [alpha]-Mn[O.sub.2] or rutile Ti[O.sub.2], respectively, as shown in Figures 7(a) and 7(d), whose the surface energy is the smallest for each model. In the case of [alpha]-Mn[O.sub.2], there is an intrinsic law conforming to the configuration evolution, such as the linear correlation between MNR(I) and MNR(III), shown in Figures 7(b) and 7(c). Upon plotting [summation][([DELTA][Q.sub.A-B]).sup.2]/s and [summation][([DELTA][Q.sub.A]).sup.2]/s for MNR(I)-MNR(III) and M[(100 x 110)], along with the quantization number N of the abscissa value of 1, 2, 3, and 6, as shown in Figure 8, it is found that they exhibit a linear correlation with [R.sup.2] of 0.989 and 0.992, respectively. It is well known that the value of the surface energy sometimes reflects the catalytic performance of materials, in which case the evolution in the normalized Mulliken population and the Mulliken charge of [alpha]-Mn[O.sub.2] nanorods and microfacet are very similar to those of the surface energy, as shown in Figures 3 and 4, thus confirming how the surface catalytic performance of nanomaterials is mainly controlled by the electronic structure. Furthermore, we can assume that the surface energy ([E.sub.surface]) is a function of the normalized Mulliken population ([summation][([DELTA][Q.sub.A-B]).sup.2]/s) and normalized Mulliken charge ([summation][([DELTA][Q.sub.A]).sup.2]/s). Their summation ([summation][([DELTA][Q.sub.A-B]).sup.2]/s + I[([DELTA][Q.sub.A]).sup.2]/s) is shown in Figure 9. It noted an absolute linear relationship, as shown in Figures 9(a), 9(b), and 9(c), where the function of [E.sub.surface] versus ([summation][([DELTA][Q.sub.A-B]).sup.2]/s +I[([DELTA][Q.sub.A]).sup.2]/s) is f(x) = 102.9 x x + 0.101 with an [R.sup.2] value of 0.995. This indicates that the surface effect in nanomaterials differs from that in bulk materials. Regarding rutile Ti[O.sub.2], a near-horizontal correlation between TNR(1) and TNR(3) is shown in Figures 7(e) and 7(f). Because all the nanorods and microfacets are composed of {100} and {110} Miller indexes, it can be assumed that their electronic structure originates from that of the (100) and (110) bulk surfaces. These were treated in the same way as their surface and cohesive energies. If we hypothesize that the average value of [summation][([DELTA][Q.sub.A-B]).sup.2]/s and [summation][([DELTA][Q.sub.A]).sup.2]/s for M(100) and M(110) is one component element contributing to the evolution character of nanorods and microfacet, their electronic structures can be explored and some linear function can be found. First, we abstracted the average value K of [summation][([DELTA][Q.sub.A-B]).sup.2]/s and [summation][([DELTA][Q.sub.A]).sup.2]/s in M(100) and M(110) to be [K.sub.M]([Q.sub.A-B]) = 0.04535 and [K.sub.M]([Q.sub.A]) = 0.004559, respectively. Second, we abstracted the quantization number by the quotient of the average value of K divided by the corresponding value of [summation][([DELTA][Q.sub.A-B]).sup.2]/s and [summation][([DELTA][Q.sub.A]).sup.2]/s for the nanorod and microfacet models. The same treatment was applied to rutile Ti[O.sub.2], wherein the average value of [summation][([DELTA][Q.sub.A-B]).sup.2]/s and [summation][([DELTA][Q.sub.A]).sup.2]/s in T(100) and T(110) is found to be [K'.sub.M]([Q.sub.A-B]) = 0.06711 and [K'.sub.m]([Q.sub.A]) = 0.000545, respectively. The results are shown in Figure 10 and in Tables 5 and 6. Figure 10 shows that there is an obviously linear correlation in the evolution from nanorod to microfacet for [alpha]-Mn[O.sub.2], regardless of the Mulliken population and Mulliken charge. After linear fitting, they obtain a function f(l) = -0.5921 * l + 4.011 for [summation][([DELTA][Q.sub.A-B]).sup.2]/s and /(m) = -0.42 * m+3.461 for [summation][([DELTA][Q.sub.A]).sup.2]/s, respectively, where the quantization number I or m is a positive integer. This evolution law is the same as that of the surface energy shown in Figure 3. For the rutile Ti[O.sub.2], regardless of the Mulliken population and Mulliken charge, there is another evolution law which is different from that for [alpha]-Mn[O.sub.2]. After linear fitting, a near-horizontal line for [summation][([DELTA][Q.sub.A-B]).sup.2]/s and [summation][([DELTA][Q.sub.A]).sup.2]/s in the evolution from nanorod to microfacet for rutile Ti[O.sub.2] is obtained, while the quantization number I or m is close to 1, which is the same as the surface energy shown in Figure 3. This abstracts the quantization phenomenon in an electronic structure and its relationship with the surface energy.

4. Discussion

It is well known that metallic oxides offer great potential for application to catalysts, not only for clean energy applications but also for pollution mitigation. Typically applied dioxides are [alpha]-Mn[O.sub.2] and rutile Ti[O.sub.2]. Their nanometer structure is ideal for attaining the greatest catalytic action. However, there are only a few valid theories that can be used to guide their design. The determination of the intrinsic mechanism of the surface effects and the correlation with the bulk surface or crystal has attracted the attention of many researchers. Deringer and Csanyi [16] and Tompsett et al. [17] determined the geometric configuration of [alpha]-Mn[O.sub.2] and rutile Ti[O.sub.2] nanorods by applying the Wulff construction method, the results of which were corroborated by experiment. Hummer et al. [18] compared the discrepancy between the total calculated nanoparticles surface energies and the summed energies of the constituent faces for rutile Ti[O.sub.2] and inferred that they uncorrelated with each other as the discrepancy was large. However, they are not able to identify the contributors to the surface effect. In the present study, the surface energies of the [alpha]-Mn[O.sub.2] nanorods exhibit a quantization phenomenon. Following the growth of the nanorods, the surface energies of the nanorods are defined by the function [E.sub.surface] = 1.0401 J[m.sup.-2] + Nx 0.608 J[m.sup.-2],where N is a positive integer and is a maximum value of 6. Then, considering only the surface energies of the [alpha]-Mn[O.sub.2] nanorods, the optimal structure is identified. Considering their stability, they also obey the following law: -[E.sub.a-cohesive]/[E.sub.cohesive] MNR [approximately equal to] 1.09. The interaction between the surface energy and cohesive energy in the quantization phenomenon also conforms to the following commonly held view: the growth of the surface energy of nanometer materials will adversely affect their structural stability. This evolution character of the [alpha]-Mn[O.sub.2] nanorods differs from that of rutile Ti[O.sub.2]. It is found that the surface energies of rutile Ti[O.sub.2] nanorod and microfacet conform to another function: [E.sub.surface]TNR = 1.0102 x 1.1997 J[m.sup.-2], which means that the surface energy will remain nearly constant during the growth of rutile Ti[O.sub.2] nanomaterials. This phenomenon indicates that the surface effect would have a similar impact on a catalyst consisting of rutile Ti[O.sub.2] nanorods, ignoring their morphologies. Regarding their structural stability, it is found that their cohesive energy conforms to the following rule: [E.sub.a-cohesive]/[E.sub.cohesive]TNR = 1.00. This phenomenon indicates that the rutile Ti[O.sub.2] nanorods will exhibit a better structural stability during the manufacturing process, relative to [alpha]-Mn[O.sub.2] nanorods. With further analysis, their quantization phenomenon originates from the evolution character of the electronic structure in terms of the difference in the bond strength and the atomic charge, rather than the geometric configuration. From the previous analysis, the surface energies of [alpha]-Mn[O.sub.2] nanorods and microfacet are increased straightly with the summation of [summation][([DELTA][Q.sub.A-B]).sup.2]/s and [summation][([DELTA][Q.sub.A]).sup.2]/s, but they keep constant for rutile Ti[O.sub.2]. Mulliken population and charge are originated from the valence electrons of component elements from their formulas [28]. In other words, if we enhance the valence electrons of [alpha]-Mn[O.sub.2] catalysts by doping process, their surface activity would be improved because of the increased surface energy. So it is not hard to understand why their doping elements for Mn[O.sub.2] catalysts are Pt [29], Pd [29], Ag [30], Nb [31], Fe [32], and so on, which are translation metals with abundance of valence electrons instead of metalloid elements. But for rutile Ti[O.sub.2] nanorods and microfacet, their surface energies are fluctuating smoothly with the summation of [summation](A[Q.sub.A-B]) /s and [summation][([DELTA][Q.sub.A]).sup.2]/s. So ifwe dope the rutile Ti[O.sub.2] catalysts with translation metals, such as Fe, V, and Cr [33-35], their improved effect would be very limited because the catalytic performance of rutile Ti[O.sub.2] is sensitive to its change of energy gap [36]. Then the doping processes for rutile Ti[O.sub.2] catalysts are used for the metalloid elements, such as N [37], Sn [38], and S [39], which affects the internal bonding orbitals. Conclusively limited by their surface energies of rutile Ti[O.sub.2] nanomaterials, the optimized way to enhance the catalytic performance of rutile Ti[O.sub.2] is that doping technology appending nanofabrication instead of single nanofabrication method. Then our investigation has a vital significance to understand and help the optimized processed of metallic oxides catalysts.

Space limitations mean that, within the scope of this paper, we have not been able to address other Miller indexes for [alpha]-Mn[O.sub.2] or rutile Ti[O.sub.2] nanorods, such as {112}, {211},and {111} for the [alpha]-Mn[O.sub.2] nanorods and {101} for the rutile Ti[O.sub.2] nanorods, as mentioned by Deringer and Csanyi [16] and Tompsett et al. [17], respectively, although their proportions are smaller than those of the (100) and (110) Miller indexes for [alpha]-Mn[O.sub.2] or rutile Ti[O.sub.2] nanorods. However, we do not think that this flaw influences the significance of this paper, given that we began by identifying the evolution mechanism of the metallic oxidation of M[O.sub.2] (M = Mn, Ti) nanorods and microfacet, which have a correlation with their bulk surfaces and structures. The overall evolution character of metallic oxidation M[O.sub.2] (M = Mn, Ti) nanorods and their other nanometer structures will be revealed and addressed in our future research. Furthermore, the evolution mechanism between a nanometer structure and bulk surface will be useful for investigating the intrinsic mechanisms of nanoeffects.

5. Conclusion

The evolution mechanism of metallic dioxide M[O.sub.2] (M = Mn, Ti) from nanorods to bulk crystal has been investigated by first-principles calculation. The results of the investigation show the following:

(1) The surface energies of [alpha]-Mn[O.sub.2] and rutile Ti[O.sub.2] nanorods and microfacets have a quantization phenomenon. For [alpha]-Mn[O.sub.2], it is found that the surface energy conforms to the function: Y = A + N x B, where A is equal to 1.0401 J[m.sup.-2], B is equal to 0.6648 J[m.sup.-2], and N is equal to a positive integer of no more than 6. For rutile Ti[O.sub.2], the surface energy conforms to another function: [E.sub.surface]TNR = 1.0102 x 1.1997 J[m.sup.-2], which remains constant regardless of the geometric structure of the rutile Ti[O.sub.2] nanorods.

(2) The cohesive energies of the [alpha]-Mn[O.sub.2] and rutile Ti[O.sub.2] microfacets and nanorods also have a quantization phenomenon. For [alpha]-Mn[O.sub.2], it is found that the cohesive energy conforms to the function [E.sub.cohesive]/Cohesive MNR [approximately equal to] 1.09, where [E.sub.a-cohesive] is equal to the average of [E.sub.cohesive]M(110) and [E.sub.cohesive]M(100). For rutile Ti[O.sub.2], the cohesive energy conforms to [E.sub.cohesive]/[E.sub.cohesive] TNR [approximately equal to] 1.00, where [E.sub.cohesive] is equal to the average value of [E.sub.cohesive]T(110) and [E.sub.cohesive]T(100).

(3) The electronic properties of [alpha]-Mn[O.sub.2] and rutile Ti[O.sub.2] nanorods and microfacet also exhibit a quantization phenomenon. After being normalized by their surface area, the Mulliken population and Mulliken charge variance of [alpha]-Mn[O.sub.2] exhibit a linear function as f(n) = 0.5921 * l + 4.011 for [summation][([DELTA][Q.sub.A-B]).sup.2]/s and f(n) = 0.42 * m + 3.461. However, the Mulliken population and Mulliken charge variance of rutile Ti[O.sub.2] exhibit a nearly horizontal line in the evolution from nanorod to microfacet.

https://doi.org/10.1155/2018/9890785

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the projects of National Natural Science Foundation of China (Grant no. 51361026), Foundation of Jiangxi Educational Committee (GJJ160684), and Key Laboratory of Jiangxi Province for Persistent Pollutants Control and Resources Recycle (Nanchang Hangkong University) (ST201522014).

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Pengsen Zhao, (1) Guifa Li [ID], (1) Bingtian Li, (1) Haizhong Zheng (1) Shiqiang Lu, (1) and Ping Peng (2)

(1) Key Laboratory of Jiangxi Province for Persistent Pollutants Control and Resources Recycle, Nanchang Hangkong University, Jiangxi 330063, China

(2) School of Material Science and Engineering, Hunan University, Hunan 410082, China

Correspondence should be addressed to Guifa Li; lgf_918@126.com

Received 23 November 2017; Accepted 4 March 2018; Published 4 April 2018

Academic Editor: Nageh K. Allam

Caption: Figure 1: Simulated several morphologies of Hollandite Mn[O.sub.2] models, where (a) [alpha]-Mn[O.sub.2] crystal ([Mn.sub.8][O.sub.16]), (b) (100) bulk surface ([Mn.sub.32][O.sub.64], M(100)), (c) (110) bulk surface ([Mn.sub.16][O.sub.32], M(110)), (d) [(100 x 110)] microfacet ([Mn.sub.32][O.sub.64], M[(100 x 110)]), (e) nanorod(I) ([Mn.sub.28][O.sub.56], MNR(I)), (f) nanorod(II) ([Mn.sub.68][O.sub.136], MNR(II)), and (g) nanorod(III) ([Mn.sub.112][O.sub.224], MNR(III)).

Caption: Figure 2: Simulated several morphologies of rutile Ti[O.sub.2] models, where (a) Ti[O.sub.2] crystal supercell ([Ti.sub.16][O.sub.32]), (b) (100) bulk surface ([Ti.sub.27][O.sub.54], T(100)), (c) (110) bulk surface ([Ti.sub.20][O.sub.40], T(110)), (d) [(100 x 110)] microfacet ([Ti.sub.34][O.sub.68], T[(100 x 110)]), (e) nanorod(l) ([Ti.sub.32][O.sub.64], TNR(1)), (f) nanorod(2) ([Ti.sub.52][O.sub.104], TNR(2)), and (g) nanorod(3) ([Ti.sub.88][O.sub.176], TNR(3)).

Caption: Figure 3: Surface energy of [alpha]-Mn[O.sub.2] and rutile Ti[O.sub.2] in different morphologies.

Caption: Figure 4: Formulized treatment on surface energy of [alpha]-Mn[O.sub.2] and rutile Ti[O.sub.2] in different morphologies.

Caption: Figure 5: Cohesive energy of [alpha]-Mn[O.sub.2] and rutile Ti[O.sub.2] in different morphologies.

Caption: Figure 6: Formulized treatment on cohesive energy of [alpha]-Mn[O.sub.2] and rutile Ti[O.sub.2] in different morphologies.

Caption: Figure 7: Normalizing bond length, Mulliken population, and Mulliken charge of [alpha]-Mn[O.sub.2] and rutile Ti[O.sub.2] in different morphologies.

Caption: Figure 8: Schematic curve of Normalizing Mulliken population and Mulliken charge of [alpha]-Mn[O.sub.2] in different morphologies with nanostructure.

Caption: Figure 9: Schematic curve of surface energy versus Normalizing Mulliken population or Mulliken charge of [alpha]-Mn[O.sub.2] in different morphologies with nanostructure.

Caption: Figure 10: Formulized treatment on Mulliken population and Mulliken charge of [alpha]-Mn[O.sub.2] and rutile Ti[O.sub.2] in different morphologies.
Table 1: Predicted PBE + U, experimental and theoretical lattice
parameters for [alpha]-Mn[O.sub.2].

Ti[O.sub.2]   a ([Angstrom])   b ([Angstrom])   C ([Angstrom])

This work         9.922            9.922            2.904
Ref. [17]         9.907            9.907            2.927
Exp. [20]         9.750            9.750            2.861

Table 2: Predicted DFT + TS, experimental and theoretical lattice
parameters for rutile Ti[O.sub.2].

Ti[O.sub.2]   a ([Angstrom])   b ([Angstrom])   c ([Angstrom])

This work          4.62             4.62             2.95
Ref. [16]          4.61             4.61             2.97
Exp. [21]          4.58             4.58             2.95

Table 3: Surface energy ([E.sub.surface]) and cohesive energy
([E.sub.cohesive]) of [alpha]-Mn[O.sub.2] crystal, bulk surface,
and nanorod models.

                   Models       [E.sub.total]       A
                                    (eV)        ([Angstrom])

Crystal            Mn8016        -11734.8960        --
Bulk surface       M(100)        -46934.8997      2.9040
                   M(110)        -23466.2580      2.9040
Microfacet     M[(100 x 110)]    -11734.8960      2.9040
                   MNR(I)        -41029.8368      2.9040
Nanorod           MNR(II)        -99651.0223      2.9040
                  MNR(III)      -164137.7267      2.9040

                   Models           B             S
                                ([Angstrom])  ([[Angstrom].sup.2])

Crystal            Mn8016           --            --
Bulk surface       M(100)         19.8440       57.6265
                   M(110)         14.3292       41.6116
Microfacet     M[(100 x 110)]     24.2512       70.4255
                   MNR(I)         68.3480      198.4811
Nanorod           MNR(II)        108.0360      313.7342
                  MNR(III)       137.0160      397.8914

                   Models       [E.sub.surface]   [E.sub.surface]
                                     (eV)          (J[m.sup.-2])

Crystal            Mn8016             --                --
Bulk surface       M(100)           4.6843            0.6503
                   M(110)           3.5340            0.6794
Microfacet     M[(100 x 110)]       44.4003           5.0437
                   MNR(I)           42.2991           1.7040
Nanorod           MNR(II)           95.5937           2.4376
                  MNR(III)         150.8172           3.0323

                   Models       p cohesive
                                   (eV)

Crystal            Mn8016        -4.7223
Bulk surface       M(100)        -4.6735
                   M(110)        -4.6487
Microfacet     M[(100 x 110)]    -4.2598
                   MNR(I)        -4.2188
Nanorod           MNR(II)        -4.2537
                  MNR(III)       -4.2735

Table 4: Surface energy ([E.sub.surface]) and cohesive energy
([E.sub.cohesive]) of rutile Ti[O.sub.2] crystal, bulk surface,
and nanorod models.

                     Models          [E.sub.total]        a
                                         (eV)        ([Angstrom])

Crystal        [Ti.sub.2][O.sub.4]    -2481.8319          --
Bulk surface         T(100)           -67003.1105       2.9528
                     T(110)           -49631.1245       2.9528
Microfacet       T[(100 x 110)]       -84372.5298       2.9528
                     TNR(l)           -79405.0467       2.9528
Nanorod              TNR(2)          -129037.4570       2.9528
                     TNR(3)          -218377.7220       2.9528

                     Models               B
                                     ([Angstrom])

Crystal        [Ti.sub.2][O.sub.4]        --
Bulk surface         T(100)            13.7748
                     T(110)            12.9870
Microfacet       T[(100 x 110)]        44.6000
                     TNR(l)            60.6920
Nanorod              TNR(2)            77.9360
                     TNR(3)            102.5320

                     Models                   S
                                     ([[Angstrom].sup.2])

Crystal        [Ti.sub.2][O.sub.4]            --
Bulk surface         T(100)                81.3473
                     T(110)                76.6949
Microfacet       T[(100 x 110)]            131.6929
                     TNR(l)                179.2087
Nanorod              TNR(2)                230.1260
                     TNR(3)                302.7520

                     Models          [E.sub.surface]   [E.sub.surface]
                                          (eV)          (J[m.sup.-2])

Crystal        [Ti.sub.2][O.sub.4]         --                --
Bulk surface         T(100)              6.3510            1.2492
                     T(110)              5.5137            1.1503
Microfacet       T[(100 x 110)]          9.7551            1.1852
                     TNR(l)              13.5744           1.2119
Nanorod              TNR(2)              17.8022           1.2377
                     TNR(3)              23.4859           1.2412

                     Models          [E.sub.cohesive]
                                           (eV)

Crystal        [Ti.sub.2][O.sub.4]       -7.8669
Bulk surface         T(100)              -7.7885
                     T(110)              -7.7750
Microfacet       T[(100 x 110)]          -7.7712
                     TNR(l)              -7.7255
Nanorod              TNR(2)              -7.7528
                     TNR(3)              -7.7779

Table 5: Normalizing variance of bond length
[summation][([DELTA]d).sup.2]/s, Mulliken population
[summation][([DELTA][Q.sub.A-B]).sup.2]/s, and Mulliken charge
[summation][([DELTA][Q.sub.A]).sup.2]/s of [alpha]-Mn[O.sub.2] bulk
surface, nanorod, and microfacet models.

Models                    S                [summation]
                 ([[Angstrom].sup.2])   [([DELTA]d).sup.2]

M(100)                 57.6265               0.114424
M(110)                 41.6116               0.033487
MNR(I)                 198.4811              0.114594
MNR(II)                313.7342              0.248664
MNR(III)               397.8914              0.332759
M[(100 x 110)]         44.4003               0.095139

Models               [summation]             [summation]
                 [([DELTA]d).sup.2]/s        [([DELTA]
                                         [Q.sub.A-B]).sup.2]

M(100)                 0.001986               3.138325
M(110)                 0.000805               1.508575
MNR(I)                 0.000577               2.611850
MNR(II)                0.000793               6.574800
MNR(III)               0.000836               10.75690
M[(100 x 110)]         0.001351               3.009660

Models                [summation]            [summation]
                       [([DELTA]              [([DELTA]
                  [Q.sub.A-B]).sup.2]/s    [Q.sub.A]).sup.2]

M(100)                  0.05446                0.2984
M(110)                 0.036254                0.1639
MNR(I)                 0.013159                0.2932
MNR(II)                0.020957                0.6644
MNR(III)               0.027035                1.0484
M[(100 x 110)]         0.042735                0.3248

Models               [summation]
                      [([DELTA]
                  [Q.sub.A]).sup.2]/s

M(100)                0.005178
M(110)                0.003939
MNR(I)                0.001477
MNR(II)               0.002118
MNR(III)              0.002635
M[(100 x 110)]        0.004612

Table 6: Normalizing variance of bond length
[summation][([DELTA]d).sup.2]/s, Mulliken population
[summation][([DELTA][Q.sub.A-B]).sup.2]/s, and Mulliken charge
[summation][([DELTA][Q.sub.A]).sup.2]/s of rutile Ti[O.sub.2] bulk
surface, nanorod, and microfacet models.

Models                   S                 [summation]
                 ([[Angstrom].sup.2])  [([DELTA]d).sup.2]

T(100)                 81.3473              0.423012
T(110)                 76.6949              0.507518
TNR(1)                179.2087              1.646453
TNR(2)                230.1260              2.358380
TNR(3)                302.7520              2.796938
T[(100 x 110)]        131.6929              0.899027

Models               [summation]            [summation]
                 [([DELTA]d).sup.2]/s        [([DELTA]
                                        [Q.sub.A-B]).sup.2]

T(100)                  0.0052                4.611500
T(110)                 0.006617               5.946150
TNR(1)                 0.012502               6.671700
TNR(2)                 0.010248              11.433600
TNR(3)                 0.009238              20.220300
T[(100 x 110)]         0.006827               7.728600

Models               [summation]              [summation]
                     [([DELTA]                 [([DELTA]
                  [Q.sub.A-B]).sup.2]/s    [Q.sub.A]).sup.2]

T(100)                 0.056689               0.0252
T(110)                 0.07753                0.0582
TNR(1)                 0.050661               0.1028
TNR(2)                 0.049684               0.1240
TNR(3)                 0.066788               0.1796
T[(100 x 110)]         0.058687               0.0837

Models               [summation]
                 [([DELTA][Q.sub.A]).sup.2]/s

T(100)                 0.00031
T(110)                0.000759
TNR(1)                0.000574
TNR(2)                0.000539
TNR(3)                0.000593
T[(100 x 110)]        0.000636
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Title Annotation:Research Article
Author:Zhao, Pengsen; Li, Guifa; Li, Bingtian; Lu, Haizhong Zheng Shiqiang; Peng, Ping
Publication:Journal of Nanomaterials
Date:Jan 1, 2018
Words:10208
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