# Evolution Mechanism of Metallic Dioxide MO2 (M = Mn, Ti) from Nanorods to Bulk Crystal: First-Principles Research

Authors: Pengsen Zhao; Guifa Li; Bingtian Li; Haizhong Zheng; Shiqiang Lu; Ping Peng
Journal: Journal of Nanomaterials
 (2018)

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## Abstract

Using first-principle calculations, the surface energy, cohesive energy, and electronic properties ofα-MnO2 and rutile TiO2 nanorods and microfacets were investigated and clarified to, in the first instance, determine the evolution mechanism. The results show that the surface energies of α-MnO2 nanorods and microfacets conform to function 1.0401 Jm−2 +N× 0.608 Jm−2, while the surface energies of the rutile TiO2 nanorods and microfacets are governed by a 1.0102 × 1.1997 rule. Their electronic properties, such as the Mulliken population and Mulliken charge, can only be normalized by their surface areas to attain a linear function. Meanwhile, the surface energy of α-MnO2 with the nanostructure closely conforms to the function for normalized Mulliken population and Mulliken charge as f(x)=102.9×x+0.101 with an R2 value of 0.995. Thus, our research into the evolution mechanism affecting the surface effect of nanometer materials will be useful for investigating the intrinsic mechanism of the nanometer effect and doping process of metallic dioxide catalysts.


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## Body

## 1. Introduction

TiO2, which is a vital inorganic functional nanomaterial, has been widely used in down-flop pigments, ultraviolet screening, photoelectric conversion, photocatalysis, and so on [1]. MnO2 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 TiO2 and MnO2 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 (AsIII) is oxidized to AsV for more than 0.4 hours by manganese dioxide, and the Mn-As bond length is from 0.271 nm to 0.34 nm [6]. For the rutile TiO2, however, it is found that the Ti-As bond length is from 0.283 nm to 0.405 nm, and the adsorption energy of AsV on TiO2(110) is greater than that on MnO2 [7]. Regarding the decomposition of CO, Chen et al. [8] indicated that the CO adsorbed onto the anatase TiO2 resulted in a moderate adsorption energy (about 0.3 eV) and a positive shift of the C-O stretching frequency (about +44 cm−1) whereas the CO could no longer be adsorbed onto the MnO2 [9]. Considering the adsorption of O2 onto MnO2, the oxygen reduction reaction can occur either in solution [10] or in air [11]. Meanwhile, Petrik and Kimmel [12] stated that O2 could be adsorbed onto rutile TiO2 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 TiO2 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 TiO2 only composed by 110 Miller index [15]. Nevertheless, Deringer and Csányi [16] and Tompsett et al. [17] discovered that nanorods of rutile TiO2 and α-MnO2 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 TiO2 were dependent with edges and corners of nanocrystal at particle size ≤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 TiO2 and α-MnO2 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 α-MnO2 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 MO2 (M = Mn, Ti) metallic dioxide.


## 2. Simulation Models and Method

To elucidate the nanometer effect of metallic dioxideMO2(M=Mn,Ti) with a nanostructure, several models of MO2(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 TiO2 model construction, only two prominent and stable Miller index planes, such as {100} and {110}, are considered. In the present study, the α-MnO2 and rutile TiO2 nanorod models were constructed based on the experimental results obtained by Barnard et al. [13–15] and Deringer and Csányi [16]. All the simulation models are shown in Figure 2. For the rutile TiO2 (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 TiO2 [(100 × 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 Ti32O64 supercell, as shown in Figure 2(e). The second rutile TiO2 nanorod (NR(2)) contains two units each of {100} and {110} Miller index slabs to construct a Ti52O104 supercell (Figure 2(f)). The largest rutile TiO2 nanorod contains triple {110} and double {100} Miller index slab units named NR(3), which form a Ti88O176 supercell (Figure 2(g)). Similar way of constructed configuration is forced to α-MnO2 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 bulk surface models were calculated assuming slabs with a minimum thickness of 14 Å. In all the bulk surface and nanorod models, a separating vacuum distance of at least 12 Å 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 MO2 (M = Mn, Ti) will be studied in our further research. To distinguish the difference between α-MnO2 and rutile TiO2 in similar configurations, every simulated model was labeled “M” or “T” to represent the α-MnO2 and rutile TiO2 series, respectively. For example, the (100) bulk surface for α-MnO2 was labeled M(100), while T(100) represents the (100) bulk surface for rutile TiO2. This convention is used for every model shown in Figures 1 and 2.Figure 1
Simulated several morphologies of Hollandite MnO2 models, where (a) α-MnO2 crystal (Mn8O16), (b) (100) bulk surface (Mn32O64, M(100)), (c) (110) bulk surface (Mn16O32, M(110)), (d) [(100 × 110)] microfacet (Mn32O64, M[(100 × 110)]), (e) nanorod(I) (Mn28O56, MNR(I)), (f) nanorod(II) (Mn68O136, MNR(II)), and (g) nanorod(III) (Mn112O224, MNR(III)).
(a) (b) (c) (d) (e) (f) (g)Figure 2
Simulated several morphologies of rutile TiO2 models, where (a) TiO2 crystal supercell (Ti16O32), (b) (100) bulk surface (Ti27O54, T(100)), (c) (110) bulk surface (Ti20O40, T(110)), (d) [(100×110)] microfacet (Ti34O68, T[(100×110)]), (e) nanorod(1) (Ti32O64, TNR(1)), (f) nanorod(2) (Ti52O104, TNR(2)), and (g) nanorod(3) (Ti88O176, TNR(3)).
(a) (b) (c) (d) (e) (f) (g)Based on the calculated sets of Deringer and Csányi [16], all the above α-MnO2 and rutile TiO2 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 α-MnO2 [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 TiO2, however, Deringer and Csányi [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 TiO2 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 × 1 × 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/Å, a stress of 0.05 GPa, and a displacement of 0.001 Å). The convergence criteria for the self-consistent field (SCF) and energy tolerances were set to 1.0 × 10−6 and 5.0 × 10−5 eV/atom, respectively.


## 3. Results and Discussion

### 3.1. Test of Potential

The value of theU 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 α-MnO2. Table 1 lists the calculated lattice parameters for α-MnO2 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 TiO2 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 MO2(M=Mn,Ti).Table 1
Predicted PBE +U, experimental and theoretical lattice parameters for α-MnO2.
TiO2 a (Å) b (Å) c (Å) This work 9.922 9.922 2.904 Ref. [17] 9.907 9.907 2.927 Exp. [20] 9.750 9.750 2.861Table 2
Predicted DFT + TS, experimental and theoretical lattice parameters for rutile TiO2.
TiO2 a (Å) b (Å) c (Å) This work 4.62 4.62 2.95 Ref. [16] 4.61 4.61 2.97 Exp. [21] 4.58 4.58 2.95


### 3.2. Evolution Character of Surface Energy

In Tables3 and 4, the surface energy Esurface for α-MnO2 and rutile TiO2 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 α-MnO2 or rutile TiO2 formula units in the bulk [19]:(1)Esurface=Etotal-nEbS,where Etotal is the energy of a surface or nanorod model containing n formula crystal units, Eb 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 Esurface values of the α-MnO2(100) and (110) bulk surfaces are equal to 0.6503 Jm−2 and 0.6794 Jm−2, respectively, which are similar to the results reported by Tompsett et al. (0.64 Jm−2 and 0.75 Jm−2, resp.) [17]. Regarding the rutile TiO2, the Esurface value for the T(100) is equal to 1.2492 Jm−2, which is a little larger than that (1.00 Jm−2) obtained by Deringer and Csányi [16]. Furthermore, the Esurface value of the T(110) is equal to 1.1503 Jm−2, which is similar to that (1.10 Jm−2) obtained by Ramamoorthy et al. [26], but a little larger than the 0.81 Jm−2 obtained by Lindan et al. [27] and the 0.80 Jm−2 obtained by Deringer and Csányi [16]. However, none of them influence the trend and the internal law governing the surface energy in this manuscript.Table 3
Surface energy (Esurface) and cohesive energy (Ecohesive) of α-MnO2 crystal, bulk surface, and nanorod models.
Models E t o t a l (eV) A (Å) B (Å) S (Å2) E s u r f a c e (eV) E s u r f a c e (Jm−2) E c o h e s i v e (eV) Crystal Mn8O16 - 11734.8960 — — — — — - 4.7223  Bulk surface M(100) - 46934.8997 2.9040 19.8440 57.6265 4.6843 0.6503 - 4.6735 M(110) - 23466.2580 2.9040 14.3292 41.6116 3.5340 0.6794 - 4.6487  Microfacet M[(100 × 110)] - 11734.8960 2.9040 24.2512 70.4255 44.4003 5.0437 - 4.2598  Nanorod MNR(I) - 41029.8368 2.9040 68.3480 198.4811 42.2991 1.7040 - 4.2188 MNR(II) - 99651.0223 2.9040 108.0360 313.7342 95.5937 2.4376 - 4.2537 MNR(III) - 164137.7267 2.9040 137.0160 397.8914 150.8172 3.0323 - 4.2735Table 4
Surface energy (Esurface) and cohesive energy (Ecohesive) of rutile TiO2 crystal, bulk surface, and nanorod models.
Models E t o t a l (eV) a (Å) B (Å) S (Å2) E s u r f a c e (eV) E s u r f a c e (Jm−2) E c o h e s i v e (eV) Crystal Ti2O4 - 2481.8319 — — — — — - 7.8669  Bulk surface T(100) - 67003.1105 2.9528 13.7748 81.3473 6.3510 1.2492 - 7.7885 T(110) - 49631.1245 2.9528 12.9870 76.6949 5.5137 1.1503 - 7.7750  Microfacet T[(100 × 110)] - 84372.5298 2.9528 44.6000 131.6929 9.7551 1.1852 - 7.7712  Nanorod TNR(1) - 79405.0467 2.9528 60.6920 179.2087 13.5744 1.2119 - 7.7255 TNR(2) - 129037.4570 2.9528 77.9360 230.1260 17.8022 1.2377 - 7.7528 TNR(3) - 218377.7220 2.9528 102.5320 302.7520 23.4859 1.2412 - 7.7779Figure 3
Surface energy ofα-MnO2 and rutile TiO2 in different morphologies.
(a) (b)The results are shown in Figures3 and 4. For α-MnO2, it is found that the Esurface value increases according to a trend of EsurfaceM(100) < EsurfaceM(110) < EsurfaceMNR(I) < EsurfaceMNR(II) < EsurfaceMNR(III) < EsurfaceM[(100 × 110)], as shown in Figure 3(a) (labeled by ①). 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 × 110)] is 9, which is greater than that of MNR(III) by 3 points, as shown in Figure 3(a) (labeled by ②). It is found that the surface energies of α-MnO2 of different morphologies fall in line with the relationship among them, implying the existence of an internal correlation. Furthermore, the sequence of M[(100 × 110)] does not correspond to nanorod(III), because other nanorods may exist. Upon closely analyzing their geometric structure, it is found that all the α-MnO2 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 EsurfaceM(100) and EsurfaceM(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 Jm−2, labeled A, and another constant parameter of the surface energy, 0.6648 Jm−2 labeled B, which is equal to the average of EsurfaceM(100) and EsurfaceM(110), that is, (0.6593 Jm−2 + 0.6974 Jm−2)/2 = 0.6648 Jm−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×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 (R2) value is equal to 0.999, as shown in Figure 4. We can thus derive a function for the surface energy: EsurfaceMNR(I) = 1.0401 Jm−2 + 1 × 0.6648 Jm−2, EsurfaceMNR(II) = 1.0401 Jm−2 + 2.1021 × 0.6648 Jm−2, EsurfaceMNR(III) = 1.0401 Jm−2 + 2.9967 × 0.6648 Jm−2, and EsurfaceM[(100 × 110)] = 1.0401 Jm−2 + 6.0226 × 0.608 Jm−2. Thus, it is clear why a previous adjustment of the abscissa in the surface energy of α-MnO2 was a line correlation (Figure 3(a) labeled by ②). This reveals the evolution character of the surface energy for α-MnO2 nanorods and microfacets.Figure 4
Formulized treatment on surface energy ofα-MnO2 and rutile TiO2 in different morphologies.It is well known that the geometrical and chemical performances of rutile TiO2 are similar to those of α-MnO2. From the results of analysis, the surface energy of α-MnO2 nanorods and microfacets has a quantization character. Therefore, it is needed to determine whether there is the same for rutile TiO2. To do so, the surface energies of the rutile TiO2 are shown in Figure 3(b) and in Table 4. The trend in the surface energy Esurface for rutile TiO2 from bulk surface → nanorod → microfacet was found to be different from that of α-MnO2. 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 Esurface is for T(100), which is equal to 1.2492 Jm−2. The smallest is for T(110), which is equal to 1.1503 Jm−2. Their difference is only 0.0989 Jm−2. Furthermore, the differences between the microfacet and nanorods are obviously very small. Therefore, the trend in the surface energy for the rutile TiO2 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 EsurfaceT(100) (1.2492 Jm−2) and EsurfaceT(110) (1.1503 Jm−2) as one constant B′, where B′ is equal to (1.2492 Jm−2 + 1.1503 Jm−2)/2 = 1.1997 Jm−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 EsurfaceTNR(1) = 1.0102 × 1.1997 Jm−2,  EsurfaceTNR(2) = 1.0317 × 1.1997 Jm−2,  EsurfaceTNR(3) = 1.0347 × 1.1997 Jm−2, and EsurfaceT[(100 × 110)] = 0.9879 × 1.1997 Jm−2. This evolution character of the surface energy of rutile TiO2 is different from that for α-MnO2. The quantization character for α-MnO2 nanorods is a positive integer, while that for rutile TiO2 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, theEcohesive value for several α-MnO2 or rutile TiO2 models has been calculated from the following equation [19]:(2)Ecohesive=1l+mEtotalMlOm-lEgasM-mEgasO,where l and m represent the number of M (M = Ti or Mn) and O atoms in the respective morphologies of rutile TiO2 or α-MnO2, EtotalMlOm denotes the total energy of the MlOm models, and EgasM and EgasO are the energies of the gaseous M (M = Ti or Mn) and O atoms, respectively. Before optimizing the gaseous atoms, a 10 × 10 × 10 (Å3) 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 EgasMn=-588.1855 eV and EgasO=-432.2548 eV, as given in Table 3. For rutile TiO2, the results are EgasTi=-1594.3577 eV and EgasO=-431.9368 eV, wherein the difference of energy for oxygen in α-MnO2 and rutile TiO2 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 α-MnO2 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 Ea-cohesive for M(110) and M(100) to establish a basic standard, which is equal to -4.6611 eV/atom (where EcohesiveM(110) = -4.6487 eV/atom and EcohesiveM(100) = -4.6735 eV/atom) in Figure 5. Considering the ratio n′ of the cohesive energy for microfacets and nanorods to Ea-cohesive, it is found that every instance of n is nearly equal to 1.09, where n′1=Ea-cohesive/Ecohesive MNR(I) = 1.1048, n′2=Ea-cohesive/Ecohesive MNR(II) = 1.0958, n′3=Ea-cohesive/Ecohesive MNR(III) = 1.0907, and n′4=Ea-cohesive/Ecohesive M[(100 × 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 TiO2, the difference is found to be very small, unlike the case of α-MnO2. For example, the most stable structure is the rutile TiO2 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 × 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 TiO2, 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 EcohesiveT(110) = -7.7750 eV/atom and EcohesiveT(100) = −7.7885 eV/atom). Considering the ratio n′′ of the cohesive energy for the nanorods/microfacets to Ea-cohesive, it is found that every instance of n is nearly equal to 1.00, where n′′1=Ea-cohesive/EcohesiveTNR(1) = 1.0073, n′′2=Ea-cohesive/EcohesiveTNR(2) = 1.0037, n′′3=Ea-cohesive/Ecohesive MNR(3) = 1.0049, and n′′4=Ea-cohesive/EcohesiveT[(100 × 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 α-MnO2 and rutile TiO2 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 Ea-cohesive is found to be nearly equal to 1.Figure 5
Cohesive energy ofα-MnO2 and rutile TiO2 in different morphologies.Figure 6
Formulized treatment on cohesive energy ofα-MnO2 and rutile TiO2 in different morphologies.


### 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 ofα-MnO2 and rutile TiO2 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 QA-B between atoms A and B and the Mulliken charge QA are defined as follows [21]:(3)QAB=∑kwk∑μA∑vA2PμvkSvμk,QA=∑kwk∑μA∑vAPμvkSvμk,where Pμv and Svμ 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 QA-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 dA-B, Mulliken population QA-B, and Mulliken charge QA of the crystal as the base values. These base values are equal to the average value of the bond length d-A-B, the Mulliken population Q-A-B, and the Mulliken charge Q¯A in α-MnO2 or rutile TiO2 crystal, respectively. Then, the bond length variance (Δd)2 is set equal to (dA-B-d-A-B)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 (ΔQA-B)2 is equal to (QA-B-Q-A-B)2. However, the Mulliken charge for α-MnO2 or rutile TiO2 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 (ΔQA)2 is equal to the sum of (ΔQM)2 and (ΔQO)2. In this case, d-A-B, Q-A-B, Q¯Mn, and Q¯O for α-MnO2 are equal to 1.9255, 0.3925, −0.59, and 1.05, respectively. For rutile TiO2, d-A-B, Q-A-B, Q¯Mn, and Q¯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 ∑(Δd)2, ∑(ΔQA-B)2, and ∑(ΔQA)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.Table 5
Normalizing variance of bond length ∑(Δd)2/s, Mulliken population ∑(ΔQA-B)2/s, and Mulliken charge ∑(ΔQA)2/s of α-MnO2 bulk surface, nanorod, and microfacet models.
Models S (Å2) ∑(Δd)2 ∑(Δd)2/s ∑(ΔQA-B)2 ∑(ΔQA-B)2/s ∑(ΔQA)2 ∑(ΔQA)2/s M(100) 57.6265 0.114424 0.001986 3.138325 0.05446 0.2984 0.005178 M(110) 41.6116 0.033487 0.000805 1.508575 0.036254 0.1639 0.003939 MNR(I) 198.4811 0.114594 0.000577 2.611850 0.013159 0.2932 0.001477 MNR(II) 313.7342 0.248664 0.000793 6.574800 0.020957 0.6644 0.002118 MNR(III) 397.8914 0.332759 0.000836 10.75690 0.027035 1.0484 0.002635 M[(100 × 110)] 44.4003 0.095139 0.001351 3.009660 0.042735 0.3248 0.004612Table 6
Normalizing variance of bond length ∑(Δd)2/s, Mulliken population ∑(ΔQA-B)2/s, and Mulliken charge ∑(ΔQA)2/s of rutile TiO2 bulk surface, nanorod, and microfacet models.
Models S (Å2) ∑(Δd)2 ∑(Δd)2/s ∑(ΔQA-B)2 ∑(ΔQA-B)2/s ∑(ΔQA)2 ∑(ΔQA)2/s T(100) 81.3473 0.423012 0.0052 4.611500 0.056689 0.0252 0.00031 T(110) 76.6949 0.507518 0.006617 5.946150 0.07753 0.0582 0.000759 TNR(1) 179.2087 1.646453 0.012502 6.671700 0.050661 0.1028 0.000574 TNR(2) 230.1260 2.358380 0.010248 11.433600 0.049684 0.1240 0.000539 TNR(3) 302.7520 2.796938 0.009238 20.220300 0.066788 0.1796 0.000593 T[(100×110)] 131.6929 0.899027 0.006827 7.728600 0.058687 0.0837 0.000636The final results are shown in Figure7. Figure 7 shows that the normalized parameters ∑(Δd)2/s, ∑(ΔQA-B)2/s, and ∑(ΔQA)2/s for the bulk surface, nanorod, and microfacet models for α-MnO2 and rutile TiO2 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 α-MnO2 or rutile TiO2, respectively, as shown in Figures 7(a) and 7(d), whose the surface energy is the smallest for each model. In the case of α-MnO2, 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 ∑(ΔQA-B)2/s and ∑(ΔQA)2/s for MNR(I)-MNR(III) and M[(100 × 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 R2 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 α-MnO2 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 (Esurface) is a function of the normalized Mulliken population (∑ΔQA-B2/s) and normalized Mulliken charge (∑ΔQA2/s). Their summation (∑ΔQA-B2/s+∑ΔQA2/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 Esurface versus (∑(ΔQA-B)2/s +∑(ΔQA)2/s) is f(x)=102.9×x+0.101 with an R2 value of 0.995. This indicates that the surface effect in nanomaterials differs from that in bulk materials. Regarding rutile TiO2, 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 ∑(ΔQA-B)2/s and ∑(ΔQA)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 ∑(ΔQA-B)2/s and ∑(ΔQA)2/s in M(100) and M(110) to be KM(QA-B)=0.04535 and KM(QA)=0.004559, respectively. Second, we abstracted the quantization number by the quotient of the average value of K divided by the corresponding value of ∑(ΔQA-B)2/s and ∑(ΔQA)2/sfor the nanorod and microfacet models. The same treatment was applied to rutile TiO2, wherein the average value K′ of ∑(ΔQA-B)2/s and ∑(ΔQA)2/s in T(100) and T(110) is found to be K′M(QA-B) = 0.06711 and K′M(QA) = 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 α-MnO2, regardless of the Mulliken population and Mulliken charge. After linear fitting, they obtain a function f(l)=-0.5921∗l+4.011 for ∑(ΔQA-B)2/s and f(m)=-0.42∗m+3.461 for ∑(ΔQA)2/s, respectively, where the quantization number l 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 TiO2, regardless of the Mulliken population and Mulliken charge, there is another evolution law which is different from that for α-MnO2. After linear fitting, a near-horizontal line for ∑(ΔQA-B)2/s and ∑(ΔQA)2/s in the evolution from nanorod to microfacet for rutile TiO2 is obtained, while the quantization number l 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.Figure 7
Normalizing bond length, Mulliken population, and Mulliken charge ofα-MnO2 and rutile TiO2 in different morphologies.Figure 8
Schematic curve of Normalizing Mulliken population and Mulliken charge ofα-MnO2 in different morphologies with nanostructure.Figure 9
Schematic curve of surface energy versus Normalizing Mulliken population or Mulliken charge ofα-MnO2 in different morphologies with nanostructure.
(a) (b) (c)Figure 10
Formulized treatment on Mulliken population and Mulliken charge ofα-MnO2 and rutile TiO2 in different morphologies.


## 3.1. Test of Potential

The value of theU 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 α-MnO2. Table 1 lists the calculated lattice parameters for α-MnO2 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 TiO2 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 MO2(M=Mn,Ti).Table 1
Predicted PBE +U, experimental and theoretical lattice parameters for α-MnO2.
TiO2 a (Å) b (Å) c (Å) This work 9.922 9.922 2.904 Ref. [17] 9.907 9.907 2.927 Exp. [20] 9.750 9.750 2.861Table 2
Predicted DFT + TS, experimental and theoretical lattice parameters for rutile TiO2.
TiO2 a (Å) b (Å) c (Å) This work 4.62 4.62 2.95 Ref. [16] 4.61 4.61 2.97 Exp. [21] 4.58 4.58 2.95


## 3.2. Evolution Character of Surface Energy

In Tables3 and 4, the surface energy Esurface for α-MnO2 and rutile TiO2 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 α-MnO2 or rutile TiO2 formula units in the bulk [19]:(1)Esurface=Etotal-nEbS,where Etotal is the energy of a surface or nanorod model containing n formula crystal units, Eb 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 Esurface values of the α-MnO2(100) and (110) bulk surfaces are equal to 0.6503 Jm−2 and 0.6794 Jm−2, respectively, which are similar to the results reported by Tompsett et al. (0.64 Jm−2 and 0.75 Jm−2, resp.) [17]. Regarding the rutile TiO2, the Esurface value for the T(100) is equal to 1.2492 Jm−2, which is a little larger than that (1.00 Jm−2) obtained by Deringer and Csányi [16]. Furthermore, the Esurface value of the T(110) is equal to 1.1503 Jm−2, which is similar to that (1.10 Jm−2) obtained by Ramamoorthy et al. [26], but a little larger than the 0.81 Jm−2 obtained by Lindan et al. [27] and the 0.80 Jm−2 obtained by Deringer and Csányi [16]. However, none of them influence the trend and the internal law governing the surface energy in this manuscript.Table 3
Surface energy (Esurface) and cohesive energy (Ecohesive) of α-MnO2 crystal, bulk surface, and nanorod models.
Models E t o t a l (eV) A (Å) B (Å) S (Å2) E s u r f a c e (eV) E s u r f a c e (Jm−2) E c o h e s i v e (eV) Crystal Mn8O16 - 11734.8960 — — — — — - 4.7223  Bulk surface M(100) - 46934.8997 2.9040 19.8440 57.6265 4.6843 0.6503 - 4.6735 M(110) - 23466.2580 2.9040 14.3292 41.6116 3.5340 0.6794 - 4.6487  Microfacet M[(100 × 110)] - 11734.8960 2.9040 24.2512 70.4255 44.4003 5.0437 - 4.2598  Nanorod MNR(I) - 41029.8368 2.9040 68.3480 198.4811 42.2991 1.7040 - 4.2188 MNR(II) - 99651.0223 2.9040 108.0360 313.7342 95.5937 2.4376 - 4.2537 MNR(III) - 164137.7267 2.9040 137.0160 397.8914 150.8172 3.0323 - 4.2735Table 4
Surface energy (Esurface) and cohesive energy (Ecohesive) of rutile TiO2 crystal, bulk surface, and nanorod models.
Models E t o t a l (eV) a (Å) B (Å) S (Å2) E s u r f a c e (eV) E s u r f a c e (Jm−2) E c o h e s i v e (eV) Crystal Ti2O4 - 2481.8319 — — — — — - 7.8669  Bulk surface T(100) - 67003.1105 2.9528 13.7748 81.3473 6.3510 1.2492 - 7.7885 T(110) - 49631.1245 2.9528 12.9870 76.6949 5.5137 1.1503 - 7.7750  Microfacet T[(100 × 110)] - 84372.5298 2.9528 44.6000 131.6929 9.7551 1.1852 - 7.7712  Nanorod TNR(1) - 79405.0467 2.9528 60.6920 179.2087 13.5744 1.2119 - 7.7255 TNR(2) - 129037.4570 2.9528 77.9360 230.1260 17.8022 1.2377 - 7.7528 TNR(3) - 218377.7220 2.9528 102.5320 302.7520 23.4859 1.2412 - 7.7779Figure 3
Surface energy ofα-MnO2 and rutile TiO2 in different morphologies.
(a) (b)The results are shown in Figures3 and 4. For α-MnO2, it is found that the Esurface value increases according to a trend of EsurfaceM(100) < EsurfaceM(110) < EsurfaceMNR(I) < EsurfaceMNR(II) < EsurfaceMNR(III) < EsurfaceM[(100 × 110)], as shown in Figure 3(a) (labeled by ①). 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 × 110)] is 9, which is greater than that of MNR(III) by 3 points, as shown in Figure 3(a) (labeled by ②). It is found that the surface energies of α-MnO2 of different morphologies fall in line with the relationship among them, implying the existence of an internal correlation. Furthermore, the sequence of M[(100 × 110)] does not correspond to nanorod(III), because other nanorods may exist. Upon closely analyzing their geometric structure, it is found that all the α-MnO2 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 EsurfaceM(100) and EsurfaceM(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 Jm−2, labeled A, and another constant parameter of the surface energy, 0.6648 Jm−2 labeled B, which is equal to the average of EsurfaceM(100) and EsurfaceM(110), that is, (0.6593 Jm−2 + 0.6974 Jm−2)/2 = 0.6648 Jm−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×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 (R2) value is equal to 0.999, as shown in Figure 4. We can thus derive a function for the surface energy: EsurfaceMNR(I) = 1.0401 Jm−2 + 1 × 0.6648 Jm−2, EsurfaceMNR(II) = 1.0401 Jm−2 + 2.1021 × 0.6648 Jm−2, EsurfaceMNR(III) = 1.0401 Jm−2 + 2.9967 × 0.6648 Jm−2, and EsurfaceM[(100 × 110)] = 1.0401 Jm−2 + 6.0226 × 0.608 Jm−2. Thus, it is clear why a previous adjustment of the abscissa in the surface energy of α-MnO2 was a line correlation (Figure 3(a) labeled by ②). This reveals the evolution character of the surface energy for α-MnO2 nanorods and microfacets.Figure 4
Formulized treatment on surface energy ofα-MnO2 and rutile TiO2 in different morphologies.It is well known that the geometrical and chemical performances of rutile TiO2 are similar to those of α-MnO2. From the results of analysis, the surface energy of α-MnO2 nanorods and microfacets has a quantization character. Therefore, it is needed to determine whether there is the same for rutile TiO2. To do so, the surface energies of the rutile TiO2 are shown in Figure 3(b) and in Table 4. The trend in the surface energy Esurface for rutile TiO2 from bulk surface → nanorod → microfacet was found to be different from that of α-MnO2. 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 Esurface is for T(100), which is equal to 1.2492 Jm−2. The smallest is for T(110), which is equal to 1.1503 Jm−2. Their difference is only 0.0989 Jm−2. Furthermore, the differences between the microfacet and nanorods are obviously very small. Therefore, the trend in the surface energy for the rutile TiO2 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 EsurfaceT(100) (1.2492 Jm−2) and EsurfaceT(110) (1.1503 Jm−2) as one constant B′, where B′ is equal to (1.2492 Jm−2 + 1.1503 Jm−2)/2 = 1.1997 Jm−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 EsurfaceTNR(1) = 1.0102 × 1.1997 Jm−2,  EsurfaceTNR(2) = 1.0317 × 1.1997 Jm−2,  EsurfaceTNR(3) = 1.0347 × 1.1997 Jm−2, and EsurfaceT[(100 × 110)] = 0.9879 × 1.1997 Jm−2. This evolution character of the surface energy of rutile TiO2 is different from that for α-MnO2. The quantization character for α-MnO2 nanorods is a positive integer, while that for rutile TiO2 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, theEcohesive value for several α-MnO2 or rutile TiO2 models has been calculated from the following equation [19]:(2)Ecohesive=1l+mEtotalMlOm-lEgasM-mEgasO,where l and m represent the number of M (M = Ti or Mn) and O atoms in the respective morphologies of rutile TiO2 or α-MnO2, EtotalMlOm denotes the total energy of the MlOm models, and EgasM and EgasO are the energies of the gaseous M (M = Ti or Mn) and O atoms, respectively. Before optimizing the gaseous atoms, a 10 × 10 × 10 (Å3) 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 EgasMn=-588.1855 eV and EgasO=-432.2548 eV, as given in Table 3. For rutile TiO2, the results are EgasTi=-1594.3577 eV and EgasO=-431.9368 eV, wherein the difference of energy for oxygen in α-MnO2 and rutile TiO2 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 α-MnO2 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 Ea-cohesive for M(110) and M(100) to establish a basic standard, which is equal to -4.6611 eV/atom (where EcohesiveM(110) = -4.6487 eV/atom and EcohesiveM(100) = -4.6735 eV/atom) in Figure 5. Considering the ratio n′ of the cohesive energy for microfacets and nanorods to Ea-cohesive, it is found that every instance of n is nearly equal to 1.09, where n′1=Ea-cohesive/Ecohesive MNR(I) = 1.1048, n′2=Ea-cohesive/Ecohesive MNR(II) = 1.0958, n′3=Ea-cohesive/Ecohesive MNR(III) = 1.0907, and n′4=Ea-cohesive/Ecohesive M[(100 × 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 TiO2, the difference is found to be very small, unlike the case of α-MnO2. For example, the most stable structure is the rutile TiO2 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 × 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 TiO2, 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 EcohesiveT(110) = -7.7750 eV/atom and EcohesiveT(100) = −7.7885 eV/atom). Considering the ratio n′′ of the cohesive energy for the nanorods/microfacets to Ea-cohesive, it is found that every instance of n is nearly equal to 1.00, where n′′1=Ea-cohesive/EcohesiveTNR(1) = 1.0073, n′′2=Ea-cohesive/EcohesiveTNR(2) = 1.0037, n′′3=Ea-cohesive/Ecohesive MNR(3) = 1.0049, and n′′4=Ea-cohesive/EcohesiveT[(100 × 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 α-MnO2 and rutile TiO2 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 Ea-cohesive is found to be nearly equal to 1.Figure 5
Cohesive energy ofα-MnO2 and rutile TiO2 in different morphologies.Figure 6
Formulized treatment on cohesive energy ofα-MnO2 and rutile TiO2 in different morphologies.


## 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 ofα-MnO2 and rutile TiO2 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 QA-B between atoms A and B and the Mulliken charge QA are defined as follows [21]:(3)QAB=∑kwk∑μA∑vA2PμvkSvμk,QA=∑kwk∑μA∑vAPμvkSvμk,where Pμv and Svμ 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 QA-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 dA-B, Mulliken population QA-B, and Mulliken charge QA of the crystal as the base values. These base values are equal to the average value of the bond length d-A-B, the Mulliken population Q-A-B, and the Mulliken charge Q¯A in α-MnO2 or rutile TiO2 crystal, respectively. Then, the bond length variance (Δd)2 is set equal to (dA-B-d-A-B)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 (ΔQA-B)2 is equal to (QA-B-Q-A-B)2. However, the Mulliken charge for α-MnO2 or rutile TiO2 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 (ΔQA)2 is equal to the sum of (ΔQM)2 and (ΔQO)2. In this case, d-A-B, Q-A-B, Q¯Mn, and Q¯O for α-MnO2 are equal to 1.9255, 0.3925, −0.59, and 1.05, respectively. For rutile TiO2, d-A-B, Q-A-B, Q¯Mn, and Q¯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 ∑(Δd)2, ∑(ΔQA-B)2, and ∑(ΔQA)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.Table 5
Normalizing variance of bond length ∑(Δd)2/s, Mulliken population ∑(ΔQA-B)2/s, and Mulliken charge ∑(ΔQA)2/s of α-MnO2 bulk surface, nanorod, and microfacet models.
Models S (Å2) ∑(Δd)2 ∑(Δd)2/s ∑(ΔQA-B)2 ∑(ΔQA-B)2/s ∑(ΔQA)2 ∑(ΔQA)2/s M(100) 57.6265 0.114424 0.001986 3.138325 0.05446 0.2984 0.005178 M(110) 41.6116 0.033487 0.000805 1.508575 0.036254 0.1639 0.003939 MNR(I) 198.4811 0.114594 0.000577 2.611850 0.013159 0.2932 0.001477 MNR(II) 313.7342 0.248664 0.000793 6.574800 0.020957 0.6644 0.002118 MNR(III) 397.8914 0.332759 0.000836 10.75690 0.027035 1.0484 0.002635 M[(100 × 110)] 44.4003 0.095139 0.001351 3.009660 0.042735 0.3248 0.004612Table 6
Normalizing variance of bond length ∑(Δd)2/s, Mulliken population ∑(ΔQA-B)2/s, and Mulliken charge ∑(ΔQA)2/s of rutile TiO2 bulk surface, nanorod, and microfacet models.
Models S (Å2) ∑(Δd)2 ∑(Δd)2/s ∑(ΔQA-B)2 ∑(ΔQA-B)2/s ∑(ΔQA)2 ∑(ΔQA)2/s T(100) 81.3473 0.423012 0.0052 4.611500 0.056689 0.0252 0.00031 T(110) 76.6949 0.507518 0.006617 5.946150 0.07753 0.0582 0.000759 TNR(1) 179.2087 1.646453 0.012502 6.671700 0.050661 0.1028 0.000574 TNR(2) 230.1260 2.358380 0.010248 11.433600 0.049684 0.1240 0.000539 TNR(3) 302.7520 2.796938 0.009238 20.220300 0.066788 0.1796 0.000593 T[(100×110)] 131.6929 0.899027 0.006827 7.728600 0.058687 0.0837 0.000636The final results are shown in Figure7. Figure 7 shows that the normalized parameters ∑(Δd)2/s, ∑(ΔQA-B)2/s, and ∑(ΔQA)2/s for the bulk surface, nanorod, and microfacet models for α-MnO2 and rutile TiO2 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 α-MnO2 or rutile TiO2, respectively, as shown in Figures 7(a) and 7(d), whose the surface energy is the smallest for each model. In the case of α-MnO2, 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 ∑(ΔQA-B)2/s and ∑(ΔQA)2/s for MNR(I)-MNR(III) and M[(100 × 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 R2 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 α-MnO2 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 (Esurface) is a function of the normalized Mulliken population (∑ΔQA-B2/s) and normalized Mulliken charge (∑ΔQA2/s). Their summation (∑ΔQA-B2/s+∑ΔQA2/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 Esurface versus (∑(ΔQA-B)2/s +∑(ΔQA)2/s) is f(x)=102.9×x+0.101 with an R2 value of 0.995. This indicates that the surface effect in nanomaterials differs from that in bulk materials. Regarding rutile TiO2, 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 ∑(ΔQA-B)2/s and ∑(ΔQA)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 ∑(ΔQA-B)2/s and ∑(ΔQA)2/s in M(100) and M(110) to be KM(QA-B)=0.04535 and KM(QA)=0.004559, respectively. Second, we abstracted the quantization number by the quotient of the average value of K divided by the corresponding value of ∑(ΔQA-B)2/s and ∑(ΔQA)2/sfor the nanorod and microfacet models. The same treatment was applied to rutile TiO2, wherein the average value K′ of ∑(ΔQA-B)2/s and ∑(ΔQA)2/s in T(100) and T(110) is found to be K′M(QA-B) = 0.06711 and K′M(QA) = 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 α-MnO2, regardless of the Mulliken population and Mulliken charge. After linear fitting, they obtain a function f(l)=-0.5921∗l+4.011 for ∑(ΔQA-B)2/s and f(m)=-0.42∗m+3.461 for ∑(ΔQA)2/s, respectively, where the quantization number l 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 TiO2, regardless of the Mulliken population and Mulliken charge, there is another evolution law which is different from that for α-MnO2. After linear fitting, a near-horizontal line for ∑(ΔQA-B)2/s and ∑(ΔQA)2/s in the evolution from nanorod to microfacet for rutile TiO2 is obtained, while the quantization number l 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.Figure 7
Normalizing bond length, Mulliken population, and Mulliken charge ofα-MnO2 and rutile TiO2 in different morphologies.Figure 8
Schematic curve of Normalizing Mulliken population and Mulliken charge ofα-MnO2 in different morphologies with nanostructure.Figure 9
Schematic curve of surface energy versus Normalizing Mulliken population or Mulliken charge ofα-MnO2 in different morphologies with nanostructure.
(a) (b) (c)Figure 10
Formulized treatment on Mulliken population and Mulliken charge ofα-MnO2 and rutile TiO2 in different morphologies.


## 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 areα-MnO2 and rutile TiO2. 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 Csányi [16] and Tompsett et al. [17] determined the geometric configuration of α-MnO2 and rutile TiO2 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 TiO2 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 α-MnO2 nanorods exhibit a quantization phenomenon. Following the growth of the nanorods, the surface energies of the nanorods are defined by the function Esurface = 1.0401 Jm−2 + N×0.608 Jm−2, where N is a positive integer and is a maximum value of 6. Then, considering only the surface energies of the α-MnO2 nanorods, the optimal structure is identified. Considering their stability, they also obey the following law: Ea-cohesive/Ecohesive MNR ≈ 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 α-MnO2 nanorods differs from that of rutile TiO2. It is found that the surface energies of rutile TiO2 nanorod and microfacet conform to another function: EsurfaceTNR ≈ 1.0102 × 1.1997 Jm−2, which means that the surface energy will remain nearly constant during the growth of rutile TiO2 nanomaterials. This phenomenon indicates that the surface effect would have a similar impact on a catalyst consisting of rutile TiO2 nanorods, ignoring their morphologies. Regarding their structural stability, it is found that their cohesive energy conforms to the following rule: Ea-cohesive/EcohesiveTNR ≈ 1.00. This phenomenon indicates that the rutile TiO2 nanorods will exhibit a better structural stability during the manufacturing process, relative to α-MnO2 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 α-MnO2 nanorods and microfacet are increased straightly with the summation of ∑(ΔQA-B)2/s and ∑(ΔQA)2/s, but they keep constant for rutile TiO2. 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 α-MnO2 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 MnO2 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 TiO2 nanorods and microfacet, their surface energies are fluctuating smoothly with the summation of ∑(ΔQA-B)2/s and ∑(ΔQA)2/s. So if we dope the rutile TiO2 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 TiO2 is sensitive to its change of energy gap [36]. Then the doping processes for rutile TiO2 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 TiO2 nanomaterials, the optimized way to enhance the catalytic performance of rutile TiO2 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 forα-MnO2 or rutile TiO2 nanorods, such as {112}, {211}, and {111} for the α-MnO2 nanorods and {101} for the rutile TiO2 nanorods, as mentioned by Deringer and Csányi [16] and Tompsett et al. [17], respectively, although their proportions are smaller than those of the (100) and (110) Miller indexes for α-MnO2 or rutile TiO2 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 MO2 (M = Mn, Ti) nanorods and microfacet, which have a correlation with their bulk surfaces and structures. The overall evolution character of metallic oxidation MO2 (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 MO2 (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 α-MnO2 and rutile TiO2 nanorods and microfacets have a quantization phenomenon. For α-MnO2, it is found that the surface energy conforms to the function: Y=A+N×B, where A is equal to 1.0401 Jm−2, B is equal to 0.6648 Jm−2, and N is equal to a positive integer of no more than 6. For rutile TiO2, the surface energy conforms to another function: EsurfaceTNR ≈ 1.0102 × 1.1997 Jm−2, which remains constant regardless of the geometric structure of the rutile TiO2 nanorods.( 2 ) The cohesive energies of the α-MnO2 and rutile TiO2 microfacets and nanorods also have a quantization phenomenon. For α-MnO2, it is found that the cohesive energy conforms to the function Ea-cohesive/Ecohesive MNR ≈ 1.09, where Ea-cohesive is equal to the average of EcohesiveM(110) and EcohesiveM(100). For rutile TiO2, the cohesive energy conforms to Ea-cohesive/Ecohesive TNR ≈ 1.00, where Ea-cohesive is equal to the average value of EcohesiveT(110) and EcohesiveT(100).( 3 ) The electronic properties of α-MnO2 and rutile TiO2 nanorods and microfacet also exhibit a quantization phenomenon. After being normalized by their surface area, the Mulliken population and Mulliken charge variance of α-MnO2 exhibit a linear function as f(n)=0.5921∗l+4.011 for ∑(ΔQA-B)2/s and f(n)=0.42∗m+3.461. However, the Mulliken population and Mulliken charge variance of rutile TiO2 exhibit a nearly horizontal line in the evolution from nanorod to microfacet.



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