Young’s modulus and Poisson’s ratio for TiO2-based Nanotubes and Nanowires: Modelling of Temperature dependence

In this work molecular mechanics simulations with the help of interatomic potentials were employed to predict the temperature dependence of the Young's modulus and Poisson's ratio of a number of TiO2-based four-facetted nanotubes and nanowires. The temperature dependence of the Young's modulus was obtained through the calculation of the Helmholtz free energy of the system under isothermal thermodynamic conditions. The Helmholtz free energy simulations were performed in the framework of quasi-harmonic approximation as a result of calculations of the potential energy and the harmonic phonon frequencies of the system under consideration. The Helmholtz free energy calculated for the set of fixed values of the nanoobject translational period allows obtaining the minimal Helmholtz free energy at specified temperatures in the range 0–1000 K. The present simulations demonstrate that the Young's modulus for the TiO2-based nanowires decreases with the increase of the nanowire diameters, and approaches the modulus for the rutile bulk crystal from above. Also, the temperature behavior of the Young's modulus, Poisson's ratio and period for the nanotubes and nanowires are considered. This study reveals that the Poisson's ratios of the nanotubes and nanowires depend on the surface atoms chosen to measure the transverse dimensions of the nanostructures.

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Introduction
Mechanical properties of TiO 2 -based nanostructures (nanotubes -NTs and nanowires -NWs) are important due to their practical applications.In particular, the strengthening of polystyrene films with small amounts of the TiO 2 -based nanotubes enables the increase of the Young's modulus of the nanohybrid films by 18% and the tensile strength by 30%. 1 In medicine, the treatment of the surface of metallic titanium implant by TiO 2 nanotubes improves osteointegration of the implant and prevents dislocation and premature loosening.
[4] Titania nanowires have exceptional photocatalytic and photovoltaic activity, outstanding optical properties. 5Mechanical characteristics of NWs is very important to understand the possibility of developing the reliable photovoltaic and photocalytic cells, optical and electrical devices based on titania nanowires. 6][8][9][10][11] Additional information on the behavior of TiO 2 nanostructures under mechanical loads at different temperatures can be obtained by theoretical modeling of the temperature dependence of the Young's modulus.
A number of computer-aided studies has been performed to investigate the temperature dependence of the mechanical properties of the single-walled, double-walled carbon nanotubes [12][13][14][15][16] and carbon-based nanomaterials such as the fullerene and nanotube derivatives. 17The main approaches used in these works are the classical molecular dynamics with empirical potentials and density-functional-based tight binding (DFTB) molecular dynamics. 12,13,168][19] Also, the temperature dependence of the stiffness of single-walled carbon nanotubes has been estimated by the molecular dynamics simulations of the intrinsic thermal vibrations of a single-walled carbon nanotube modelled as a clamped cantilever. 12The Young's modulus values were calculated from the equation relating the modulus and standard deviation of the vibrational amplitude of the tube's free tip displacement.
The results of Dereli and Süngü 14 indicate that the Young's modulus of (10, 10) nanotube decreases in the interval 0 K<T<1000 K but increases at temperature higher than 1000 K.The (10, 10) nanotube also has been studied by Zhang and Shen. 13The authors of mentioned papers employed different values of the nanotube's wall thickness δ.For comparing the results one can use an area-based Young's modulus Y S =Yδ. 20Our comparison of these studies' results reveals that the Y S values may differ by as much as 2.8 times.
Jin-Yuan Hsieh et al 12 investigated the carbon zig-zag nanotubes (n, 0) with n=7, 9, 12 and higher and a (5, 5) armchair nanotube.As follows from Figure 3 of this paper, 12 the Young's modulus of almost all considered nanotubes are invariable or demonstrate small growth in temperature interval from 100 K to 1000 K.
In the work of H. Jiang et al 21 a finite-temperature constitutive model not involving molecular dynamics simulation was established.The model using harmonic vibrational frequency and quasi-harmonic approximation 22 allows one to simulate the Helmholtz free energy and to compute the temperature dependence of the mechanical properties for the nanosystem.This model has been applied to the investigation of the Young's modulus and Poisson ratio of graphene.
In recent years the investigations of the mechanical properties of the inorganic nanotubes [23][24][25][26][27] and nanowires 11,[28][29][30][31][32] have been carried out.However, to the best of our knowledge, the only study on the temperature dependence of the mechanical properties of the inorganic nanoobjects is published in our recent article. 33ur previous paper 33 presents the results of the study of the Young's modulus temperature dependence of the TiO 2 -based cylindrical single-walled nanotubes with zig-zag and armchair chiralities and nanotubes with consolidated single wall (CSW NT).The latter are obtained from the cylindrical coaxial double-walled NTs by merging the single-walled constituents.The cylindrical single-walled constituents of the NTs were rolled up from the (111) 3-plane slab cut from a fluorite phase TiO 2 crystal, space group 3 ̅ .
The used 33 quasi-harmonic approximation and Helmholtz free energy simulation based on harmonic vibrational frequency is more simple method than the molecular dynamics modelling and it can produce the consistent results.
The present paper has five parts.In the section "Method" the theoretical background of the elastic moduli calculations and relevant formulae are described.The section "Nanostructure models and Computational Details" presents the nanoobject models used and describes the applied atom-atom force field and specific characteristics of the computational procedure.The analysis of the obtained data and the discussion of the results are arranged in the section "Results and Discussion".The summary of the present study conclusions is given in the last section "Conclusions".

Method
One of the main mechanical characteristics of a nanotube is the Young's modulus: where  =   ⁄ is the stress or uniaxial pressure due to the force F applied along the nanotube axis and S is the cross-sectional area.The strain along the nanotube axis  ║ is defined as follows: where L 0 is the initial length of the unit cell or period of the unstrained monoperiodic nanotube; L is the period of the nanotube under strain.The Eq. ( 1) is following from the consideration of the thermodynamics of the distortion of a solid.Also, as one of the results, this consideration leads to the important thermodynamics relation: where T is a temperature, E is the total internal energy of the nanosystem,  is the entropy, F Helmholtz free energy. 34ombining Eq. ( 1) and ( 2) one can get where  0 () is the volume of the unstrained nanosystem since the energies E and F must be related to the unit volume.
To obtain the Young's modulus from the results of computer simulation the change in the total internal energy of the nanosystem during an isothermal process is frequently applied as follows. 25,27() = However the more correct way to compute the Young's modulus () considering an isothermal process is to use the Eq.(4b).
The second quantity describing the mechanical properties of the nanostructures is the Poisson's ratio: The transverse strain   appears as a respond to exerting the longitudinal stress or uniaxial pressure  on the nanostructure.The strain   is defined as follows: where D 0 is the initial size of the nanostructure in the direction perpendicular to the nanotube or nanowire axis; D is this transvers size due to the longitudinal stress .Using Eqs. ( 2), ( 5), ( 6) and taking into account L=(L-L 0 ) and D=(D-D 0 ) as infinitesimal quantities the Poisson's ratio can be written in a differential form as (also see the Eq. ( 4) 26 ) The Young's modulus and the Poisson's ratio are independent elastic properties. 35e molecular mechanics (MM) modelling of the structural and mechanical properties of the nanosystems has been carried out by us with the help of GULP program package. 36Using an empirical force field the potential energy and the phonon harmonic frequencies of the considered nanotubes and nanowires were calculated.The calculated in harmonic approximation vibrational frequencies of the considered nanotubes or nanowires have been computed to obtain the temperature-dependent Helmholtz free energy.The temperaturedependent Helmholtz free energy is the sum of zero point vibrational energy  0 and the contribution of the temperature-dependent free harmonic phonon energy F(T).The zero point energy and the Helmholtz free energy are given by the well-known equation: where k B is the Boltzmann constant, h is the Plank constant, ν i (k) is ith mode frequency, cm -1 , n i is ith mode degeneracy, N k is the number of k-points in Brillouin Zone (BZ), N ν is the number of vibrational modes, E 0 is the zero point energy, U is the potential energy.In our calculations, the summation in Eq. ( 8) is made over 16 points uniformly distributed over the monoperiodic BZ of the nanotube.
The optimization of the atomic structure of the nanowire or nanotube is the first step of the elastic moduli simulation using quasi-harmonic approximation. 22Thus, the period of the optimized structure L * corresponds to the minimal potential energy of the system: the term U in Eq. 8. To obtain the dependence of the energy and transvers strain   on the axial strain  ║ seven contracted and nine elongated samples have been considered: L i =L * (1+i ║ ); i = -7, …, 0, 1, …, 9.The strain step  ║ =0.003 was taken the same as in molecular dynamics simulations of the Young's modulus for the carbon nanotubes. 19r each sample the period L i was fixed and potential energy minimization procedure executed to get the optimized atomic structure in local minimum of the potential energy U( ∥, * ).The Helmholtz free energy was calculated according to Eq. 8a for every optimized structure of the initial, contracted and elongated samples.The calculations were done at temperatures T j in the temperature interval from 0 K to 200 K with the step T=25 K and in the interval from 200 K to 1000 K with the step T=50 K.A number of data sets F( ∥, * , T j ) with  ∥, * =(L i -L * )/L * and T j from the abovementioned temperature interval have been obtained.
A five-degree polynomial was fitted to every Helmholtz free energy data set F( ∥, * , T j ) at fixed temperature T j .These polynomials allow one to calculate the second derivative of F( ∥ * , T j ) in respect to tensile uniaxial strain  ∥ * .
The minimum of U( ∥ * ) is located at  ∥ * =0 and independent of T, whereas a position of the F( ∥ * , T j ) minimum does not coincide with that of U( ∥ * )  ∥ * =0 and changes with T increase.As discussed above, the Young's modulus can be obtained using Eq.4b, where the Helmholtz energy is a function of strain  ∥ 0, =(L-L 0,T )/L 0,T and L 0,T is the period of the unstrained nanostructure with a the minimal Helmholtz free energy at temperature T. The relationship between the periods L * and L 0,T can easily be obtained: L 0,T =L * ( ∥ 0, +1).The relation between the second derivatives of F in respect to  ∥ * and in respect to  ∥ 0, is as follows: The usage of the unit cell parameters corresponding to the Helmholtz free energy minimum at fixed temperature determines the quasi-harmonic approximation.To find Helmholtz free energy minimum the vibrational frequencies are calculated for the certain range of the unit cell parameters.At the same time, the vibrational frequencies is obtained assuming that the atomic vibrations are harmonic.
Also, for each sample the transvers size of the sample D i and, consequently, the data set

Nanostructure models and Computational Details
As it was shown in our earlier study 37   A lot of atom-atom interaction potentials have been developed for simulating a variety of structures and diverse properties of the different polymorphs of TiO 2 .These force fields were employed in the Monte Carlo, 41 molecular mechanics and molecular dynamics studies [42][43][44][45][46][47][48][49] of the crystals.We have tested the ability of these different force fields to reproduce the values of the structural and mechanical parameters and phonon frequencies of four known phases of TiO 2 crystals: rutile, anatase, brookite, and columbite.In addition, we tested the ability of the force fields to reproduce the structure and phonon frequencies of the (111) 3-plane slab of a fluorite-like TiO 2 crystal and two cylindrical nanotubes (6, 6) and (12, 12) rolled up from such a slab.As far as we know, there are no the experimental data on the properties of the mentioned fluorite layer and two nanotubes.Therefore values of the structural parameters and phonon frequencies of the slab and nanotubes were obtained from our DFT calculations.The practice of employing the results of quantum chemical studies for parameterization of the force fields is widely used. 48,50,51r comparison of DFT and force field calculations results shows that the lattice parameters and mechanical properties of the four mentioned polymorphs of TiO 2 are reproduced well by the force field of Kim et al 46 and, slightly worse, by the one of Matsui and Akaogi (M&A). 45However, the phonon frequencies of these TiO 2 phases and nanotubes are reproduced more accurately by means of the M&A interaction potential.Thus, the force field used in the present work is a set of atom-atom interaction potentials developed by Matsui and Akaogi 45 with the use of the molecular dynamics modeling.Table 1 demonstrates the ability of the M&A force field to reproduce the lattice constants of rutile crystal.The optimization of atomic structure has been done through the minimization of potential energy of the system in the MM calculations with the GULP package 36 .The DFT calculation results are taken from Evarestov and Zhukovskii, 40 .the structure optimization is implemented as the potential energy minimization with the help of the conjugate gradient algorithm.As one can see, the discrepancy between DFT PBE results and experimental data is 1% and the corresponding discrepancy for the MM simulations with M&A force field is 2%.
Table 1 The lattice constants a and c of optimized TiO 2 rutile bulk.
Experimental data 52  The value of the nanostructure's volume per unit cell is the key parameter for the Young's modulus determination, see Eqs. 4a and 4b.The volume of an unstrained facetted nanowire at a temperature T can be calculated as a product of the period of unit cell, L 0,T , and the area of the cross-section of NW (Figures 2b and 3), S. The nanowire period is well defined quantity.However, the NW cross-section area cannot be defined unambiguously.The area of the largest FNW cross-section should be a product of the length of NW cross-section Moreover, the significantly smaller distances between the corner oxygen atoms (for example, in the smallest FNW, Figure 3a) become comparable with the ambiguous atomic diameter.
In the present study we applied the method of nanostructure volume calculation analogous to the one employed in the previous research. 37At first, the calculation of molar volume V M of the unstrained largest FNW at T=0 K was performed using the geometrical parameters of the nanowire and the effective anion radius.As in previous study, 37 the effective radius of the three-coordinated oxygen anion R=1.36 Å 53 was chosen for estimating the size of oxygen atom in the nanostructure.The area of the nanowire's cross-section was calculated as a product of ( Finally, the molar volume was estimated as a product of the cross-section area and the period of largest FNW L 0,T=0 divided by the number of TiO 2 formula units per unit cell.The obtained value of V M is 32.33 Å 3 .5][56][57][58][59] A three-degree polynomial was fitted to the 19 points data set as follows: where a=-3.7210-10 Å 3 K -3 ; b=9.1310 -7 Å 3 K -2 ; c=2.1110 -4 Å 3 K -1 ; d=31.69Å 3 .The volume of any nanostructure under consideration was calculated according to the equation where N FU is the number of formula units per unit cell of the nanowire or nanotube, and  The oxygen atoms of bulk crystal are three coordinated and titanium atoms are six coordinated.The surface corner (bridging) oxygen atoms of NWs are two coordinated while the titanium atoms are five coordinated (Figure 4).It is known [60][61][62] that the displacements upon the relaxation of the surface bridging oxygen and titanium atoms can differ by as much as 2 times according to the experimental data 60 and 25 times according to the results of ab initio calculations. 61,62.The similar effect is found here for the displacements of atoms in the ridges of the NW.Consequently, the change of the D OO due to uniaxial stress has a significant contribution of the surface oxygen atom shift besides of the total uniform transverse dilation or compression.To consider the influence of the different surface atoms on the Poisson's ratio we used two diagonals, D OO or D TiTi (see Figures 3b, 3c), as the size  in Eqs. ( 6), ( 7) and (10) for the computation of Poisson's ratio (ν OO or ν TiTi , respectively).Table 2 demonstrates the values of the distance between corner titanium atoms and between corner oxygen atoms or diagonal or diameter of the FNWs and FNTs under study (see Figures 2 and 3).As a whole, it follows from Table 2 that the differences between the results of the present simulations and data of DFT large scale study don't exceed 2-3%.

Results and Discussion
The results of our molecular-mechanics simulations demonstrate that CSW NT i. e.
distorted four facetted NT has a lower Young's modulus values than that for the four facetted NT.Simultaneously the temperature dependence of the Young's modulus for the distorted four facetted NT displays the more intensive decrease of the stiffness with the temperature increase, see Figure 5.These relations between the elastic characteristics of the two FNTs can easily be attributed to the disturbances which are brought in the structure of the distorted FNT by the fluorite-like elements, Figure 1b, region S. The presence of the fluorite-like elements in rutile structure provides higher coordination numbers for some titanium and oxygen atoms.
Also, Figure 6 indicates that the period of the distorted FNT is notably greater than the period of the intact (with regular rutile structure) FNT.This observation, probably, reveals one of the possible stages of the Young's modulus diminishing mechanism: the nanotube with mixed structure has additional number of degrees of freedom to relax under external loading and, consequently, to diminish stiffness.The nanotube pore of the intact facetted nanotube (Figure 2a) can also be treated as a sort of the FNW structure defect which should reduce the stiffness of the FNT relative to the FNW.As it was aforementioned, the FNT was obtained from largest FNW by cutting a pore.
Nevertheless, the Young's modulus for the four facetted NT with 72 formula units per UC exceeds the Young's modulus not only for the largest four facetted NW (81 FU), but also for large NW (64 FU) and is more close to the modulus of the middle FNW with 25 formula units per UC, Figure 5. Also, as it follows from Figure 6, in contrast to the previous case of the relations between two nanotubes, the FNT with greater stiffness has a longer unit cell than the largest FNW (81 FU per UC) and large FNW (64 FU per UC).The temperature dependence of the Young's modulus for the FNT is almost the same as one for the large FNW, see Figure 5 and Table 3.At the same time, the Young's modulus temperature dependence of the distorted FNT is the strongest one from all nanostructures under consideration.
The Young's modulus for the FNWs increases with the decrease of FNW's size.The smaller are the nanowire diameters the greater is the rate of modulus value growth.The difference between the Young's modulus values of nanowires with 81 (Y(0)=368 GPa) and 64 (Y(0)=371 GPa) FU per UC is small -1%.The difference is 6% for the small (16 FU) (Y(0)=404 GPa) and smallest (9 FU) (Y(0)=429 GPa) facetted nanowires.In other words, one needs to heat FNW (64 FU) up to ≈200 K to level the stiffness of this nanowire and the stiffness of the FNW (61 FU) at 0 K.However, in the case of middle FNW the temperature at ≈600 K is necessary.

RSC Advances Accepted Manuscript
crystal from above with the increase of nanowire size.At the same time, as it was shown in our study, 33 the Young's modulus values of the cylindrical nanotubes with the armchair chirality grow with the nanotube's diameter and approach to the crystal value from below.3.All Y versus T curves have a characteristic shape and can be split into two parts.In the temperature interval from 0 to ≈100 K, the Young's modulus decreases rather slowly.The rise of temperature above 150-200 K results in a sensitive increase in the slope of Y versus T.
As in the case of the cylindrical single-walled nanotubes, 33 the behavior of Young's modulus for the FNWs with temperature correlates with the behavior of the period of the nanowires.However, the length of unit cell of the nanowires increases with heating.
Nevertheless, the L 0,T versus T curves can also be split into two parts.In the temperature range 0 ≤ T ≤ 150 K the unit cell length is almost constant, Figure 6.The periods of all the FNWs and FNTs increase as temperature grows in the second range from ≈150 to 1000 K.

RSC Advances Accepted Manuscript
was obtained.A quadratic polynomial was fitted to the D i (L i ) data set, that allows one to calculate the transvers size D 0,T and the derivative (   ⁄ )  0, corresponding to L 0,T .The Poisson's ratio at temperature T was calculated according to the equation () = −  0

5 Figure 1 .
Figure1a.However it appears that modelling of the NTs elongation and compression allows us to perform a more detailed study of the NT potential energy surface and, in some cases, to find an optimized structure with lower potential energy.In that way in our present work a more stable optimized structure of the (8, 8)@(12, 12)@(16, 16) CSW NT with 72 TiO 2 formula units (FU) per NT unit cell (UC) was been obtained, see Figure1b.One can see from Figure1bthat this more stable structure is very close to the one of the rutile-based four

Figure 2 .
Figure 2. Cross-sectional images of the nanostructures of rutile morphology: a) FNT with 72 formula units per UC and b) FNW with 81 formula units per UC, largest.O 1 , O 2 , O 3 are corner oxygen atoms; O 4 , O 5 , O 6 and O 7 are edge oxygen atoms.

Figure 3 .
Figure 3. Cross-sectional images of the nanowires of rutile morphology: a) Ti atom-centered smallest; b) hollow site-centered small; c) Ti atom-centered middle and d) hollow sitecentered large with 9, 16, 25 and 64 formula units per UC, respectively.The distances d 1 , d 2 , d 3 , d 4 between edge oxygen atoms and diameter D OO are shown in b) and c) by twiceterminated arrows.The D OO and D TiTi diameters are shown in c) by twice-terminated arrows.O 1 , O 2 , O 3 , O 4 , O 7 , O 8 , O 9 , O 10 are edge oxygen atoms; O 5 and O 6 are corner oxygen atoms.Table1demonstrates the ability of the M&A force field to reproduce the lattice

(
110) facet side and the length of NW cross-section (11 ̅ 0) facet side.The length of NW crosssection (110) facet side can be estimated as the distance between nuclei of the two corner oxygen atoms O 1 and O 2 , d OO =27.52 Å (Figure 2b).In one's turn the length of the NW crosssection (11 ̅ 0) facet side is a distance between two corner oxygen atoms O 2 and O 3 , d OO =25.93 Å.On the other hand, the same quantities can also be estimated as the distances between O 4 -O 5 atoms and O 6 -O 7 ones which are 27.89Å and 27.88 Å, respectively, Figure 2b.To take into account the size of the oxygen atoms one should also add the quite ambiguous value of the oxygen atom diameter to the obtained values that introduces an extra uncertainty to the cross-section area.The calculation of volume of other FNWs is else less unequivocal process.The irregularities of the facets surface have noticeable contribution to the nanowire structure.
33 Å is the molar volume of the largest FNW.

Figure 4 .
Figure 4.The image of two cells of the smallest nanowire.The two-coordinated surface corner oxygen atom and five-coordinated surface corner titanium atom are marked.The cylindrical carbon nanotubes have a monoatomic wall and in the case of the Poisson's ratio calculation the only choice of the size  in Eqs.(5) and (6) is the diameter of the nanotube.The facetted nanotubes and nanowires have several structural parameters to be selected as the size , Figures 1-3.The aside facets (110) and (11 ̅ 0) of the hollow site-centered NWs have similar structures, see Figures3b and 3d.To measure a distortion perpendicular to the mechanical

Figure 5 .
Figure 5. Temperature dependence of the Young's modulus for the CSW NT, four facetted NTs and NWs.The nanotube pore of the intact facetted nanotube (Figure2a) can also be treated as a

Figure 6 .
Figure 6.Temperature dependence of the period of the four facetted NTs and NWs.The Young's moduli for the nanotubes and nanowires decrease with the temperature increase.The temperature dependence of the Young's modulus for the nanowires becomes more steeply with the decrease of nanowire size, Figure 5 and Table3.All Y versus T curves 40esent MM simulations DFT PBE calculations40

Table 2 .
The structural properties of optimized CSW NT, four facetted NT, Ti atom-centered and hollow site-centered [001]-oriented NWs.
63e MM simulation with the help of M&A force field gives Y(0)=366 GPa in the direction [001] and 154 GPa in the direction [100] for the rutile bulk crystal.The corresponding experimental values of the Young's modulus are 386 and 148 GPa.63The differences are -5% and +4% for the Y in the [001] and [100] directions, respectively, which can be treated as the success of M&A parametrization.According to MM simulation result, the minimal Young's modulus value of all [001]-oriented nanowires is Y(0)=368 GPa for the largest FNW which is only slightly greater than the corresponding Y for the rutile bulk crystal.Thus, the Young's modulus for the FNWs approaches to the Y for the rutile bulk