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УДК 539.32, 539.61
Numerical study of mechanical properties of nanoparticles of P-type Ti-Nb alloy under conditions identical to laser sintering. Multilevel approach
A.Yu. Nikonov12, A.M. Zharmukhambetova2, A.V. Ponomareva3, A.I. Dmitriev1,2
1 Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634055, Russia 2 National Research Tomsk State University, Tomsk, 634050, Russia 3 National University of Science and Technology MISiS, Moscow, Russia
A multilevel approach is used to numerically investigate physical and mechanical properties of titanium-based bcc alloys and their behavior under conditions identical to selective laser sintering. Elastic properties of P-Ti-Nb alloy are calculated within the first principles approach. An algorithm is proposed and tested to optimize the calculations and reduce their number by more than 5 times. A molecular dynamics method is employed to study structural changes of titanium and niobium powder particles during sintering and to calculate adhesion characteristics of nanoparticles of the produced alloy depending on the external action. The simulation results are in good agreement with the known experimental data and can be used as input data both for numerical models of a higher spatial scale and for the optimization of production parameters of titanium alloys by additive technologies.
Keywords: titanium alloys, laser sintering, first principles calculations, elastic properties, molecular dynamics, adhesion properties
Численное изучение изменения структуры и механических свойств наночастиц Р-сплава Ti-Nb в условиях идентичных лазерному спеканию. Многоуровневый подход
А.Ю. Никонов1,2, A.M. Жармухамбетова2, А.В. Пономарёва3, А.И. Дмитриев1,2
1 Институт физики прочности и материаловедения СО РАН, Томск, 634055, Россия 2 Национальный исследовательский Томский государственный университет, Томск, 634050, Россия 3 Национальный исследовательский технологический университет «МИСиС», Москва, Россия
В работе с использованием многоуровневого подхода проведено численное исследование физико-механических свойств ОЦК-сплавов на основе титана и изучено их поведение в условиях идентичных селективному лазерному спеканию. В рамках первоприн-ципного подхода вычислены упругие свойства Р-сплава Ti-Nb. Предложен и апробирован алгоритм оптимизации расчетов, позволивший сократить объем вычислений более чем в 5 раз. С использованием метода молекулярной динамики исследовано изменение структуры частиц порошка титана и ниобия в процессе плавления и рассчитаны адгезионные характеристики наночастиц образовавшегося сплава в зависимости от условий внешнего воздействия. Результаты моделирования хорошо коррелируют с известными экспериментальными данными и могут быть использованы в качестве входных данных для численных моделей большего пространственного масштаба, а также для оптимизации параметров процесса получения титановых сплавов с использованием аддитивных технологий.
Ключевые слова: титановые сплавы, лазерное спекание, первопринципные расчеты, упругие свойства, молекулярная динамика, адгезионные свойства
1. Introduction
With constantly increasing requirements for operation of materials and related products, a search for new materials becomes one of the most important directions of development for the modern materials science. This applies
equally to different economic branches, including medicine. Due to the unique combination of such properties as bioinertness, lightness, and strength, titanium and its al-
loys are currently the main materials for the production of bioimplants. It was found that P-titanium alloys, for example, Ti-Nb or Ti-Nb-Zr systems [1, 2], can be attractive in terms of biomechanical compatibility. Such alloys should also combine a high level of mechanical properties (yield strength, ultimate strength, fatigue limit and fatigue life, microhardness and hardness, ultimate plasticity) and a low modulus of elasticity close to elastic properties of bone tis-
© Nikonov A.Yu., Zharmukhambetova A.M., Ponomareva A.V., Dmitriev A.I., 2017
sue. Metallurgical production of such alloys is difficult on account of largely different densities and melting points of the components, their thermal conductivity and heat capacity values. In addition, to produce an alloy with a uniform structure and chemical composition requires multistage methods of thermomechanical processing, during which a specimen loses half its volume. Additive technologies recently developed allow us to effectively solve the listed problems as well as to meet the challenges of personalized medicine [3, 4]. The advantage of additive technologies and selective laser sintering lies in the possibility of forming not only the product volume, but also its internal structure by using powders with components mixed in the required ratio. At the same time, choosing of a technological mode of powder sintering calls for a large number of tests with varying control parameters of the process.
Not the least role in the fundamental problem of designing new materials is played by methods of the theoretical approach. At present, the problem of searching for new alloys with given physical and mechanical properties, in addition to experimental research, is solved in increasing frequency by computer simulation methods. In particular, first principles (ab initio) calculations represent quantum simulation at the fundamental level and allow for a prediction of properties of new alloys depending on the concentration of their components [5-7]. Such calculations make it possible to "relatively easily" verify many different combinations of substances and to choose the most suitable alloys for their further experimental study and testing. Thus, first principles calculations are used to find elastic constants and anisotropy values for ternary Ti^AlN alloy [6] and to investigate structural properties of binary Zr-Nb alloy [7]. The disadvantage of this approach is high requirements for computational resources, which not only makes it difficult to find exact concentration values, but also limits possible spatial dimensions of the systems under consideration. To study properties of the produced alloys on the nanoscale, we employ methods of a higher scale level. Among them is the molecular dynamics method, which is applicable to processes occurring in nanosecond time intervals and in systems measuring hundreds of nanometers [8-10]. It is on these scales that the main processes of selective laser sintering of metal powders are realized. Thus, numerical simulation methods open up new possibilities for the understanding of the physics of processes taking place under selective laser sintering, and, with the proper level of the model, allow for a qualitative determination of the required technological parameters of production.
In the present paper, the problem of designing a material with the given mechanical properties and of analyzing its behavior under conditions identical to the laser beam action is described using the multilevel numerical simulation approach. Beta-type Ti-Nb alloy is taken as the test material. The problem is solved on the two characteristic scales: quantum-mechanical calculations and molecular
dynamics investigations. The first principles calculations gave concentrations of the titanium-based alloy with the required elastic properties. To reduce the number of necessary ab initio calculations, an original algorithm for optimizing the search for alloys with specified characteristics is developed and tested. The results obtained are compared with the known experimental and theoretical data. On the basis of this information within the molecular dynamics method, a model is developed to investigate processes that occur when a laser beam is applied to individual particles of metal powders. Adhesion properties of particles of the produced Ti-Nb alloy are evaluated depending on the duration of high-energy laser action.
2. Procedure of first-principles calculation of thermodynamic and mechanical characteristics of crystalline materials
Ab initio calculations stem from the solution of the quantum mechanical problem within the density functional theory. In this paper, chemical chaos is described using the coherent potential approximation when a three-dimensional periodic phase of the solid solution A1-j-Bj is reconstructed by depicting it in the ordered crystal lattice of "effective" atoms [11]. Effective atoms are placed in nodes of the crystal lattice of the original system and their properties are determined self-consistently providing that electron scattering of alloy components embedded into the medium as impurities is on average zero. In this paper the method of coherent potential approximation is applied within the exact muffin-tin orbital theory [12]. The effectiveness of the coherent potential approximation within the exact muffin-tin orbital theory using the total charge density method [12] was proved by a successful simulation of physical and mechanical properties for a wide range of technologically important systems [13, 14].
In the present study the basis set of wave functions of the exact muffin-tin orbital theory includes s-, p-, d- and f-orbitals. To calculate elastic characteristics of an alloy, integration over the irreducible Brillouin zone is performed using the grid of 31x31x31 k-points in the reciprocal space for cubic lattices. When integrating over energy in the complex plane, 24 points are taken on the semispherical contour. Convergence of energy with respect to calculation parameters comprises 10-8 Ry. Exchange-correlation effects in electron gas are given consideration within the generalized gradient approximation [15].
The equation of state of the simulated alloy is determined from the calculation of the total energy of the material E(Vat) at the corresponding values of atomic volume Vat. In this paper, E(Vat) is approximated by a modified Morse function [16]. Adiabatic elastic constants for single crystals are found by calculating the total energies of the alloy, which are obtained for a number of small volume-preserving (V= = const) deformations in the region where Hooke's law is
valid. Since the alloy energy depends more strongly on the volume than on the stress, the condition of volume preservation under deformation allows us to ignore the contribution of the volume variation to the energy change.
For cubic crystals there exist three independent elements of the tensor of elastic constants Cn, C12, and C44. The constant C44 is calculated using the monoclinic distortion. Its value is found through the coefficient of the corresponding energy change versus the squared strain:
AE/V = 2C44S2 + O(84).
(1)
Thus, we first calclulate the total energy AE for six different strains 8 in the range 0.00-0.05. Then we determine the elastic constant C44 using the linear approximation of AE as a function of the squared strain. Constants C11 and C12 are found from the ratios
C11 = (3B + 4C 0/3, (2)
C12 = (3B - 2C 0/3, (3)
where B is the bulk modulus and C' is the tetragonal shear modulus.
By analogy with the calculation of C44, the elastic constant C' can be found by estimating the change in the total energy under orthorhombic deformation:
AE/V = 2C '82 + O(84). (4)
Elastic constants C11 and C12 are calculated using bulk moduli determined from the equation of state and tetragonal shear moduli found from Eqs. (2) and (3).
The criterion for stability of the crystal lattice is obtained provided that the energy density can be represented by a positive definite quadratic form in such a way that the energy increases at any small deformation. Therefore, all diagonal components of the elastic constant tensor must have positive values. Consequently, for cubic crystals the stability criterion is determined by the expressions
C44 > 0, C11 >|C12|, C11 + 2C12 > 0. (5)
Elastic characteristics for polycrystals, which can be considered as quasi-isotropic materials, are described using the bulk and shear moduli G. Knowing them, you can calculate Young's modulus E and Poisson ratio v:
E = 9BG/ (3B + G), (6)
v = (9B - 2G)/(2(3B + G)). (7)
In this paper, elastic characteristics of polycrystals are obtained using the Hill averaging (H) of Reuss (R) and Voigt (V) bulk and shear moduli, which represent the upper and lower boundaries of variation of the corresponding moduli:
BH = ( Bv + BR )/2, (8)
Gh = (Gv + CR)/2, (9)
Bv = BR = (Cn + 2^/3, (10)
GV = (C11 - C12 + 3C44)/5,
Gr =-
5C44(C11 C12)
4C44 + 3(C11 C12)
(11) (12)
3. Algorithm of optimization of first principles calculations
First principles calculations enable a prediction of mechanical properties of metals and alloys of any composition. However, despite the use of high-performance computing systems, these calculations are time consuming. Our task is to find an alloy with the required characteristics rather than to determine alloy characteristics in the entire range of component concentrations. To reduce the number of calculations, an algorithm of optimization of calculations was developed.
The main idea of the optimization algorithm of calculations will be illustrated with the example of searching for a specific configuration of a ternary titanium-based alloy whose Young's modulus E would be equal or close to 90 GPa. At first, a grid is constructed on the ternary graph. In this case, the concentration step dl should be the same for all alloy components, so that the sum of all concentrations equals 1 (100%) at the point. In our calculations, the grid step dl is 0.1 (10%). At the next algorithm step, we choose an arbitrary grid triangle and calculate the desired variable for all its vertices (Fig. 1, a). By comparing the three values corresponding to the vertices of the chosen triangle, we find the highest/lowest value of them. The highest value is chosen if all the three values are greater than the sought quantity, and conversely the lowest value is chosen if all the three values are less than the sought quantity. Another triangle that we consider in the algorithm of searching for the required configuration will share a common side with the first triangle. This side is opposite to the previously chosen point. Thus, in order to determine the way forward, the sought parameter should be calculated at the only vertex since the other two values are already known from the previous triangle. The sequence of steps is repeated until we find the triangle where the sought parameter is as close to the specified value as possible. The variant when the sought and calculated values at the triangle vertices differ in sign means that the graph point corresponding to the concentration of the alloy components with the required characteristics is located inside this triangle.
A reference point can be chosen in different ways. The most obvious choice is a triangle located in the center of all considered configurations of the ternary alloy. The advantage of this method is the equidistance from possible limiting concentrations of the alloy components, which reduces the number of necessary calculations to the limiting case when the search for concentrations leads to pure substance. So, at the grid step dl = 0.1 the maximum number of calculated configurations of the alloy will be 15 instead of 66 with an exhaustive search for all possible alloy configurations.
Another way to choose the reference point is to start with a triangle near the sought concentration. To do this, we first calculate the sought parameter values for pure substances. Then, on the assumption that the calculated char-
vAxBv-dlCz+dl
AxByCz j
Ax+dlBy-dlCz
Fig. 1. A separate element of the ternary graph (a) and the scheme of the optimization algorithm of calculations where the reference point is a triangle with identical component concentrations and Young's modulus close to 90 GPa (b)
acteristics depends almost linearly on the concentration of alloy components, we calculate a stoichiometry of the alloy with the parameter value close to the sought one. The reference point will be one of the triangles into which the found alloys fall. Figure 1, b shows the corresponding two search trajectories of the given bulk modulus starting with triangles designated as 1. The digits (2 etc.) indicate successive triangles whose vertices correspond to the stoichiometry of the alloy with the calculated parameter values. According to the results obtained, for the variant with the central reference triangle Young's modulus has to be calculated for 6 alloy configurations; while for the second variant, only for 4. This advantage is obtained, in part, due to the fact that the sought parameter in the considered example is achieved for several different variants of the alloy stoichiometry.
The above method with the central reference triangle can be formulated as an algorithm:
1. Calculate variable values at the reference point. In our case, it is a triangle with vertices 334, 343, and 433 (hereinafter each digit denotes a tenth of the alloy component concentration, for example, 136 corresponds to Ti10-Nb30-Zr60 all°y)-
Table 1
Element concentration of ternary alloy TixNbyZrz and Young's moduli calculated for the components at the triangle vertices depicted in Fig. 1
Ti Nb Zr E, GPa Ti Nb Zr E, GPa
1 0
0.3 0.3 0.4 108.51 0.0 0.0 1.0 76.75
0.3 0.4 0.3 115.62 0.0 1.0 0.0 134.22
0.4 0.3 0.3 110.37 1.0 0.0 0.0 55.67
2 1
0.3 0.3 0.4 108.51 0.0 0.2 0.8 97.17
0.4 0.3 0.3 110.37 0.0 0.3 0.7 102.83
0.4 0.2 0.4 101.16 0.1 0.2 0.7 98.06
3 2
0.3 0.3 0.4 108.51 0.0 0.2 0.8 97.17
0.4 0.2 0.4 101.16 0.1 0.2 0.7 98.06
0.3 0.2 0.5 100.14 0.1 0.1 0.8 89.69
4
0.4 0.2 0.4 101.16
0.3 0.2 0.5 100.14
0.4 0.1 0.5 89.20
2. Check signs of the difference between the calculated and sought values. If the signs are the same, then among the three points we choose the most different value from the sought one to eliminate this point. If the signs are different, then this is a required triangle.
3. Calculate a new point that is found by the following data:
- From the three coordinates we choose the coordinate that does not have a pair with the other two points. For example, among three points 343, 442, and 352 the unpaired coordinate for point 352 will be the second one that values 5.
- Compare values of this coordinate at the points (they are 5, 4, and 4 in our example).
- If the coordinate of the eliminated point is greater than that of the others, then its value decreases by 2, and the other two coordinates increase by 1. (The new point for 352 is 433.) In the opposite case, the coordinate value increases by 2, and the other two coordinates decrease by 1.
4. Pass to step 1 with a new set of three points.
How this algorithm can be realized for the two different reference points is presented in Table 1 where digits 1, 2, 3, 4 denote sequential triangles. Three table lines for each triangle contain information about the current concentration (in fractions) of each component and the calculated value of the sought parameter at every triangle vertex. A configuration of the ternary alloy excluded from fur-
ther consideration is marked in bold face (the highest of the three values in our example).
To find an alloy with a more precisely calculated parameter of the multicomponent alloy, consider the finite triangle determined by the optimization algorithm. At the triangle vertices, the calculated parameter values will be higher or lower than the sought ones. A further search for the required alloy configuration can be realized by one of the triangle sides. This will change the concentration of only two of the three components. A precise concentration value can be also found by the linear interpolation of values inside the finite triangle.
The proposed algorithm can be similarly applied to alloys with four or more components. The application of this approach to a binary alloy degenerates to a motion along a straight line, which is a particular case or final phase of the considered example.
4. Results of first principles calculations
The above-described algorithm is intended to find an optimal composition of a ternary alloy. For a binary alloy, the problem of searching for the optimal stoichiometry is substantially simplified and reduced to a variation of the only variable X in the AX-B1-X-type alloy. Since the total number of calculations of the sought parameter for various configurations of the binary alloy at the grid step dl = 0.1 does not exceed 11, the bulk moduli and elastic constants are calculated for all configurations of Ti-Nb bcc alloy. Young's moduli for polycrystalline material are determined using Hill, Reuss and Voigt averaging. The calculation results are plotted in Fig. 2. To verify the mathematical model, we give the available experimental data and values calculated elsewhere.
The accuracy of elastic constants calculated from first principles is about 10%. Thus, for Nb the agreement between the theoretical and available experimental data is in the given range. The estimation error of Ti bcc properties is explained by the lack of sufficient experimental data as well as by the dynamic instability of the Ti lattice at low temperatures. According to Fig. 2, different theoretical
Fig. 2. Elastic moduli for beta Ti-Nb alloy obtained from first-principles calculations for a polycrystal. The results of experiments and calculations are taken elsewhere: 1 and 5 [18], 2 [21], 3 [22], 4 [19], and 6 [20]
methods also give a spread in calculated values. Nevertheless, the moduli change in a similar way in the two ab initio calculation methods. It was earlier noted [17, 18] that alloys with a low content of niobium most likely contain significant amounts of a- and ^-phases, i.e. the experimentally obtained rigidity values give an incomplete picture of pure bcc properties.
Based on the available numerical simulation results, including [18-20], the stoichiometry of the P-Ti-Nb alloy comprises 40 wt % for the biomechanical compatibility with bone tissue, which agrees with the known experimental data [18, 21, 22].
5. Results of molecular dynamics simulation
5.1. Special features of crystallization of Ti-Nb alloy
Special features of crystallization of Ti-Nb alloy are studied on the scale of individual atoms using the molecular dynamics method implemented via the LAMMPS software package [23]. The Ovito program is used to visualize and analyze the crystalline structure [24]. A system for simulation consists of two spherical crystallites. One crystallite represented a titanium nanoparticle, and the other consisted of niobium atoms (Fig. 3, a). In both crystallites, the crystal lattice orientation is set in such a way that the [100], [010] and [001] directions correspond to the X, Y and Z axes of the laboratory coordinate system. Nanoparticles are 10 nm in diameter. The interaction between particles is described by the interatomic potential developed within the embedded atom method [25, 26]. This many-body potential proves effective in solving various problems of description of deformation of metal clusters and crystalline structures [27, 28]. Numerical integration of classical equations of motion of atoms is performed using the Verlet velocity algorithm. The integration step is 0.001 ps. The temperature of the system under consideration is controlled using the atomic velocity scaling algorithm on the assumption of the equality of energies
3/2 NkT = 2 mV2/2,
i
where N is the number of atoms in the system, i is the atom number, and Vi is the absolute magnitude of the velocity of the ith atom. The system is heated by the following algorithm. First, we calculate the starting temperature Tstart of the system and set its end temperature Tend, the number of steps M, and frequency of scaling of atom velocities K. Then, we determine the system temperature Tcurrent every K time steps of the atomic system evolution and calculate the required temperature by the formula
T = Tend - Tstart)/M •
If the current and required temperatures differ, absolute magnitudes of atom velocities are multiplied by the corresponding coefficient an. The scaling coefficient at stage n is evaluated by the kinetic temperature ratio
a n = VTI Tcurrent •
a b c d e
Fig. 3. Structure of the simulated specimen at the initial time instant (a) and at subsequent stages of the melting process: 30 (b), 70 (c), 120 (d), and 220 ps (e). Hereinafter titanium atoms are marked in dark color, niobium atoms are in light color
To study effects of various laser pulse durations, we simulate high-rate heating of both crystallites that are in contact to the temperature 6000 K. Then, we simulate relaxation of the atom system for 300 ps under the achieved temperature. Figures 3, b-e demonstrates the atomic structure of both crystallites at different time instants corresponding to the melting process. After both metals are uniformly molten, we simulate a crystallization process, during which the system temperature decreases by the linear law to 300 K.
The following cooling rates of the system are simulated in the paper: 9.5, 5.7, and 3.1 deg/ps. Solid lines in Fig. 4, a show temperature-time dependences of the system. The specimen structure derived after cooling is analyzed using the algorithm of searching for local topology of atomic bonds, which enables a determination of bcc, fcc and hcp structures. We also study how the cooling rate affects the proportion of atoms with local bcc topology of local bonds of the crystal lattice of the derived alloy.
Final structures for the considered cooling rates are shown in Figs. 4b-4d. The proportion of atoms with bcc lattice for the resulting state depends on the cooling rate and comprises 11.2, 56.2, and 87.7%, respectively. A further decrease in the cooling rate of the system hardly affects the proportion of atoms with bcc lattice in the resulting state. Thus, by varying cooling modes, we find that the cooling rate 3.1 deg/ps or less is sufficient for the two-component system under consideration to form the resulting polycrystalline structure of P-Ti-Nb alloy. The remaining 12% correspond to surface atoms as well as to atoms located near the formed grain boundaries.
A qualitative agreement is seen between the simulation results and experimental data on the effect of preliminary heating of the powder mixture and the use of the substrate in the production of P-Ti-Nb alloy under laser sintering conditions. According to the obtained results, mechanical and physical characteristics of the formed Ti-Nb alloy particles can be used as parameters necessary for describing the particle interaction within mesomodels.
5.2. Analysis of adhesion properties of Ti-Nb particles
At the next stage of our investigation, we analyze the influence of the heating time of the Ti-Nb particle system
and its cooling rate on the resulting adhesive force. In so doing, two particles are simulated in the form of hemispheres with an internal structure corresponding to the structure of P-Ti-Nb alloy derived after simulation of the crystallization process. Testing is carried out by the following scheme. The hemispheres approach each other to a distance of =1 nm between their apexes. The resulting con-
3000
* MkÉÈ ^
mjgm
Ésatik
p. ,»jr*: .
Fig. 4. Simulation results of the crystallization process of the Ti-Nb system. Temperature dependence of the system (a), final structure for different cooling rates: 9.5 (1), 5.7 (2), 3.1 K/ps (3). Shown are only atoms with bcc structure
Fig. 5. Structure of particles of P-Ti-Nb alloy at various stages of testing: before heating (a), after heating for 30 (b) and 50 ps (c)
figuration of the simulated system is shown in Fig. 5, a. The crystallographic orientation of the particles is chosen in such a way that an intergranular boundary is formed in the region where the particles approach each other.
Heating in the region where the particle apexes come close is simulated by adding kinetic energy to surface atoms with regard to the ratio 1 eV/ps for the entire system. This procedure results in melting of a part of hemispheres with their further sintering. The sintering area varies with the heating time. The resulting structure corresponding to the formed contact area 30 nm2 is shown in Fig. 5, b. This configuration is obtained on heating of the particle system in the contact region to -3000 K for 30 ps with subsequent cooling. Heating for 50 ps gives a structure shown in Fig. 5, c where the sintering area is 83 nm2.
To evaluate adhesive forces, we simulate configurations not only with different contact areas but also with different structure of the contact. A structure in the alloying zone is changed by varying the cooling rate of the molten region. At low cooling rates the molten region crystallizes in the contact region with the formation of a grain boundary. At high cooling rates an amorphous layer forms in the contact region. The final stage of evaluation of adhesive forces includes a simulation of rupture of the formed alloying zone. To do this, atoms located at the bases of the hemispheres are assigned constant velocities but in opposite directions
Fig. 6. Dependence of the adhesive force on the alloying area and structure of the contact region of two particles of P-Ti-Nb alloy
shown in Figs. 5, b and 5, c by arrows. The thickness of the loaded layers is 1 nm. Figure 6 plots the dependence of the ultimate resistance to rupture on the initial alloying area, which is seen to remain linear for the studied crystalline structures in the contact region. In so doing, structural features of the alloying area hardly affect the obtained dependence.
Thus, the main factor determining the adhesive force value is the contact area of two particles, which in turn depends on the heating time. According to the numerical simulation results, the cooling rate and the related structure of the formed contact region exert no effect on the resulting adhesive properties in the investigated spatial and time intervals.
6. Conclusion
The computer simulation results on the melting process with subsequent crystallization of a mixture of isolated titanium and niobium particles at different temperature gradients make possible a determination of production conditions for a system with the maximum content of a crystalline structure of P-Ti-Nb alloy. It is found that the cooling rate 3.1 deg/ps or less is sufficient for the two-component system to form the resulting polycrystalline structure of the P-alloy with the maximum bcc lattice content. In this case, a variation in temperature gradients is governed by the initial temperature of the system and by different heat dissipation rate, which depends on thermal and physical properties of the substrate. The obtained results are in qualitative agreement with the experimental data on the effect of preliminary heating of the powder mixture and the use of the substrate in the production of P-Ti-Nb alloy under selective laser sintering [14]. According to the simulation results, the main factor determining the adhesive force between two particles is their contact area. The cooling rate and the related structure of the formed contact region hardly affect the resulting adhesive properties. This is apparently due to the instability of the amorphous state of the resulting "bridge" since under subsequent tension its disordered structure begins to be reconstructed into a crystal lattice with grain boundaries.
The investigation results will be used to specify mechanical and physical characteristics of formed Ti-Nb alloy particles when describing the interparticle interaction within the mesoscopic model.
Acknowledgments
The work was financially supported by Russian Science Foundation grant No. 15-19-00191. The ab initio calculations were performed by A.M. Zharmukhambetova and A.V. Ponomareva with support of the TSU Competitiveness Improvement Program.
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Поступила в редакцию 22.04.2017 г.
Сведения об авторах
Anton Yu. Nikonov, Cand. Sci. (Phys.-Math.), Researcher, ISPMS SB RAS, Researcher, TSU, anickonoff@ispms.tsc.ru Albina M. Zharmukhambetova, Post-Graduate, TSU, zharmukhambetova@gmail.com Alena V. Ponomareva, Cand. Sci. (Phys.-Math.), Leading Researcher, MISiS, alenaponomareva@yahoo.com Andrei I. Dmitriev, Dr. Sci. (Phys.-Math.), Leading Researcher, ISPMS SB RAS, Prof., TSU, dmitr@ispms.tsc.ru
