A Set of Interaction Potentials for Molecular Dynamics Simulations of YPO₄ Crystal

Abstract

A set of empirical pair interaction potentials for molecular dynamics (MD) modeling of the YPO₄ crystal is proposed. The parameters of the potentials, as well as the effective charges of the ions, are recovered based on experimental data on the lattice constants of YPO₄, as well as on the structure of the PO₄ complex within this crystal. Using these potentials, an MD simulation of YPO₄ crystallites isolated in vacuum is performed in the temperature range from 300 K to 2200 K. A quantitative agreement between the thermal expansion coefficient of the crystal and experimental data is obtained. Simulation of the formation of Frenkel defects in the yttrium, oxygen, and phosphorus sublattices is carried out. The formation energy of the Frenkel defects in the oxygen and yttrium sublattices is in quantitative agreement with ab initio calculations.

1. Introduction

Yttrium orthophosphate, YPO₄, come into use as a luminophore due to its resistance to high temperatures, chemical stability, and high transparency in the visible and ultraviolet ranges. The YPO₄ crystal serves as a matrix for rare-earth metal ions (such as Eu³⁺, Tb³⁺, Dy³⁺, Ce³⁺) [1,2,3]. Impurity activators impart luminescent properties to the system by creating energy levels in the band gap of the matrix. Along with impurities, yttrium and oxygen vacancies have a significant effect on the light-emitting properties of YPO₄. Such vacancies affect the distribution of electron density, creating new energy levels in the band gap [4]. It is known, in particular, that oxygen vacancies can enhance or suppress luminescence, depending on their concentration [5,6,7]. The intrinsic defects of the host can reduce the symmetry of the local environment of activator ions, leading to an increased probability of radiative optical transitions and an enhancement of the quantum efficiency of luminescence [8]. In addition, intrinsic defects can act as electron and hole trap centers and play a crucial role in the mechanisms of persistent luminescence [9]. For instance, in YPO₄, oxygen vacancies serve as electron traps, while yttrium vacancies act as hole traps [4]. The thermal release of charge carriers from traps at room temperature underlies the phenomenon of persistent luminescence. In this context, the concentration of defects plays a decisive role in the resulting characteristics [10]. Insufficient or excessively high defect concentrations can lead to inefficient energy storage or significant energy losses, respectively, resulting in a decrease in both the intensity and duration of luminescence.

The energies of the formation of intrinsic point defects in YPO₄ were calculated from the first-principles by F. Gao et al. [11]. However, the capabilities of ab initio calculations are limited by the sizes of model systems (no more than several hundred particles). Meanwhile, there is an interest in the prediction defect structure of YPO₄ depending on the crystal growth methods, impurity composition, temperature, and substrate structure. It is relevant to study the composition and stability of clusters formed by defects in YPO₄ crystal. The mechanisms of formation, interaction, and transfer of defects can be clarified by classical molecular dynamics (MD) simulation, which requires suitable interaction potentials of crystal particles. Interaction potentials for classical modeling of YPO₄ were proposed in references [12,13,14]. The energies of defect formation calculated using these potentials differ significantly from quantum chemical calculations [11]. It is useful to propose an alternative set of potentials, which is the subject of this work.

2. Obtaining the Interaction Potentials

The YPO₄ crystal can be considered as a system formed by PO₄³⁻ ions, which are bound by Coulomb and dispersion attraction with Y³⁺ ions. The structure of the symmetric PO₄³⁻ ion is shown in Figure 1. The oxygen ions are placed on equal distances (R) from the center of the cube R = 1.54 Å [15], which remain almost unchanged when the PO₄³⁻ ion is incorporated into YPO₄ [16]. The angles α and β in an isolated PO₄³⁻ ion would be identical. In the YPO₄ crystal, these angles become different (α = 113°, β = 103°), since the cube shown in Figure 1 is compressed along the vertical axis.

The unit cell of the YPO₄ crystal is shown in Figure 2. For clarity, the sizes of the ions in Figure 2 are proportional to the atomic radii instead of the ionic radii. In this representation, the largest size is that of the yttrium particles, while the smallest size is that of the oxygen particles. The cell is a rectangular parallelepiped with sides a = b = 6.8947 Å, c = 6.0276 Å [17] (here and below, these values correspond to room temperature). The nearest environment of the yttrium ion is formed by four oxygen ions at a distance of R_Y-O = 2.3087 Å [17], and the other four oxygen ions are located at a close distance of 2.3813 Å [17].

The yttrium–oxygen distances in YPO₄ coincide with the distances between cations and anions in the ionic crystals of simpler structures. For example, in the Y₂O₃ crystal, the radius of the first coordination sphere of yttrium is 2.333 Å [19]. Thus, in the present work it was assumed that the yttrium–oxygen interaction could be described by a pair potential, depending only on the distance between the centers of the particles R, in the following form:

$$U_{Y–O}\left(R\right)=K_E{\displaystyle \frac{q_Y{q}_O}{R}}+A_{Y–O}·e^{-{R·B}_{Y–O}},$$

(1)

This takes into account the Coulomb attraction created by the effective charges q_y, q_o, (K_E = 14.39976 eV·Å/ē² is the Coulomb constant; ē is the elementary charge), as well as the valence repulsion of overlapping electron shells (the parameters A_Y–O, B_Y–O). Such potentials are successfully used to model ionic crystals [20]. The dispersion attraction was not included in the U_Y–O(R) potential, since its parameters are difficult to reconstruct from experimental data against the background of a stronger oxygen–oxygen dispersion attraction. The dispersion attraction of oxygen anions in YPO₄ turns out to be stronger than that of yttrium and phosphorus ions, since the polarizability of oxygen ions is significantly higher. In pairs of ions of the opposite sign Y–O and P–O, the dispersive forces are small compared to the Coulomb attraction. The approach in which non-zero dispersive attraction acted only between oxygen ions was successfully applied to restore pair potentials for modeling UO₂ in earlier work [21].

The PO₄³⁻ complex retains its structure not only in YPO₄, but also when incorporated into other crystals [22]. In particular, the phosphorus–oxygen distance within this complex remains approximately equal to 1.5 Å. In this work, it was assumed that the stability of the PO₄³⁻ complex corresponds to the existence of a chemical bond P–O. At the same time, the presence of non-zero effective charges on both oxygen (q_o) and phosphorus (q_p) ions was assumed. Thus, the phosphorus–oxygen interaction was described by the following potential:

$$U_{P–O}\left(R\right)=K_E{\displaystyle \frac{q_P{q}_O}{R}}+{\epsilon }_{P–O}·(e^{-2\mathsf{\beta }\left(R-R_m\right)}-2·e^{-\mathsf{\beta }\left(R-R_m\right)}),$$

(2)

which emulates chemical bonding using the Morse function “two exponents” [23]. Dispersive attraction was again not included, assuming it to be small compared to oxygen–oxygen attraction.

The interaction of a pair of oxygen ions was described by a potential that included Coulomb repulsion, valence repulsion of overlapping electron shells, and dispersion attraction.

$$U_{O–O}\left(R\right)=K_E{\displaystyle \frac{q_O{q}_O}{R}}+A_{Y–O}·e^{-{R·B}_{O–O}}-{\chi }_{O–O}({\displaystyle \frac{R}{d_a}})·{\displaystyle \frac{C_{O–O}}{R^6}},$$

(3)

The dispersion attraction was treated by taking into account its dumping at small distances corresponding to the overlap of the electron shells of the particles. The dumping function χ_O–O(d_a·R) proposed in [24,25] was used. The parameter d_a was chosen equal to 4.63 Å⁻¹, as suggested in [25]. For MD simulation, the function χ_O–O(d_a·R)/R⁶ at 1.8 Å < R < 3.7 Å was approximated in the following form:

$${\displaystyle \frac{{\chi }_{O–O}\left(R\right)}{R^6}}=1.4466·(0.3148·e^{-2.3623·R}+0.1415·e^{-1.8118·R}-0.2006·e^{-4.9845·R}).$$

(4)

The parameters of the interaction potentials Equations (1)–(3) were determined by achieving static equilibrium of the YPO₄ crystal lattice, which in dynamic modeling would correspond to zero absolute temperature. The values of the parameters were chosen to minimize the potential energy E of an ideal crystal, which was the sum of the potential energies of interaction of all pairs of particles.

$$E={\displaystyle \frac{1}{2}}·\sum _{i=1}^N\sum _{j=1, j\ne i}^N{U}_{ij}\left(R_{ij}\right).$$

(5)

where N is the number of particles in the model crystallite; and i, j are the particle indices, U_ij(R_ij) are the interaction potentials defined by Equations (1)–(3).

All positions of yttrium cations, as well as oxygen anions, in an ideal YPO₄ crystal are equivalent to each other. Accordingly, in order to minimize the energy E, it was sufficient to require zero values of the derivatives of this energy with respect to the coordinates of one arbitrary yttrium ion, as well as with respect to the coordinates of an arbitrary oxygen ion. Thus, the following derivatives should equal zero at static equilibrium when the crystal lattice parameters match the experimental values.

$${\displaystyle \frac{\partial E}{\partial x_Y}}=0,{\displaystyle \frac{\partial E}{\partial z_Y}}=0,{\displaystyle \frac{\partial E}{\partial x_O}}=0,{\displaystyle \frac{\partial E}{\partial z_O}}=0.$$

(6)

where E is energy from Equation (5), and x_y, z_y, x_o, z_o are the Cartesian coordinates of arbitrary yttrium and oxygen ions in the ideal YPO₄ lattice. The x axis is oriented along the longer edge of the unit cell (a = 6.8947 Å), while the z axis is oriented along the shorter edge c = 6.0276 Å.

The parameters of the interatomic potentials Equations (1)–(3), including effective charges q_p, q_y, and q_o, were determined through a combined approach involving minimization of the sum of squared residuals derived from the equilibrium conditions in Equation (6), and systematic identification of crystal lattice configurations exhibiting thermodynamic stability during molecular dynamics simulations. The effective charges of yttrium and phosphorus ions q_p and q_y were varied independently of each other, while the oxygen charge q_o was calculated using the following formula:

$$q_o=-\left(q_p+q_y\right)/8$$

(7)

which ensured the electroneutrality of the crystal.

The parameters of potentials Equations (1)–(3) are listed in

Table 1. Parameters of interaction potentials in Equations (1)–(3).

Parameter	P–O	Y–O	O–O
A, eV	–	2419.9	1691.1
B, Å−1	–	3.8170	3.3755
C, eV·Å−6	–	–	74.796
ε, eV	3.5295	–	–
β, Å−1	7.0	–	–
Rm, Å	1.5071	–	–

. The effective charges of intrinsic ions were obtained as follows, q_p = +1.323·ē, q_y = +1.8·ē, q_o = −0.78075·ē, where ē is the elementary charge.

3. Molecular Dynamics Simulation

The reconstructed interaction potentials were verified by molecular dynamics simulation of finite size YPO₄ crystallites that were isolated in vacuum. The boundaries of these crystallites were formed by crystallographic planes (100), (010), (001). The crystallite sizes varied from 8232 (343 unit cells) to 52,728 particles (2197 unit cells), and the temperatures were in a range from 300 K to 2200 K. The lower limit of the size range was determined by the need to exclude significant surface effects in the nanocrystal volume, while the upper bound was limited by the available computational performance. The temperature range boundaries were set between room temperature and the experimental melting point of YPO₄.

The equations of particle motion were integrated by the “leapfrog” method with a step of Δt = 3·10⁻¹⁵ s. The Berendsen thermostat [26] with the parameter τ = 9 ps was used to stabilize the temperature. As is known, the Berendsen thermostat is not ergodic. Nevertheless, the parameter value τ = 9 ps is sufficiently large for this thermostat to be comparable to the Nosé–Hoover [27,28] thermostat in terms of the quality of particle energy distribution [29]. The calculations were performed using the original (home-made) software (Ver. 1) [30] with parallelization of calculations on CUDA architecture graphics processors.

At the beginning of each computational experiment, the crystal particles were placed at the sites of an ideal lattice, with a Maxwellian velocity distribution. Subsequently, the surface and volume of the crystallite were allowed to relax freely. The initial and equilibrium states of the model systems are illustrated in Figure 3. It can be seen that the surface disordering was limited to the smoothing of the vertices and edges of the parallelepiped, while the bulk retained the structure of an ideal YPO₄ crystal.

In addition to the visual analysis, the correctness of the YPO₄ crystal simulation was checked by calculating the radial distribution functions of phosphorus–phosphorus, phosphorus–yttrium, phosphorus–oxygen, and by measuring the distances between the ionic planes (100), (010), and (001) in the bulk of the model crystallites. Figure 4 shows the characteristic time dependence of the specific volume v per YPO₄ molecule in the central region of the model nanocrystals of 52,728 particles during a computational experiment. It is evident that the equilibrium state was established within times of about 10 ns and did not change thereafter.

To investigate the possible influence of model crystallite size on the crystal lattice structure, this study obtained the dependencies of the lattice parameters a and c on size at temperatures ranging from 300 K to 1900 K. The linear crystallite size L varied from 7 to 13 unit cells, while the number of particles in the system changed from 8232 to 52,728. As an example, Figure 5 shows the a(L) and c(L) dependencies obtained at T = 1000 K. For comparison, experimental values of a and c for a macroscopic crystal at the same temperature are also plotted, calculated using the linear expansion coefficient from the work of Wang et al., 2010 [31]. The results show that the calculated lattice constants exhibit practically no dependence on L, and the difference between the calculated and experimental values of these constants remains unchanged as L varies. The same result was obtained at other temperatures. The simulation data discussed below correspond to the largest size L = 13 unit cells (52,728 particles). By selecting this size, we aimed to minimize the influence of surface effects on the bulk properties of the model.

As can be seen from Figure 5, the lattice parameter a in the present work is underestimated compared to the experimental data, while the parameter c is overestimated, to a lesser extent. At T = 1000 K, the calculated lattice parameters were a(1000 K) = 6.80 Å, c(1000 K) = 6.08 Å, while the experimental values, taking into account the linear expansion coefficient Wang-2010 [31], are a_exp(1000 K) = 6.911 Å, c_exp(1000 K) = 6.042 Å. Similarly, at room temperature, our parameters were a(300 K) = 6.765 Å, and c(1000 K) = 6.046 Å, in comparison with a_exp(300 K) = 6.8947 Å, c_exp(1000 K) = 6.0276 Å [17] (Δa = −1.9%, Δc = +0.3%).

The difference between both parameters and experimental data is a result of the displacement of yttrium ions relative to phosphorus ions, which we consider as the centers of PO₄ structural units. The pair of Y ions closest to the phosphorus ion has Cartesian coordinates (0, 0, 0.5 c) relative to it, where c is the lattice constant, and the axis z coincides with the vertical axis in Figure 2. In our model, the P–Y distance for these two ions is increased by 0.009 Å (0.3%) compared to experimental data, which explains the overestimation of the constant c.

The second coordination sphere of P–Y is formed by four yttrium ions lying in the same horizontal plane as the phosphorus ion in Figure 2, at an experimental distance of 3.762 Å. In our model, this distance is reduced to 3.70 Å (by 1.6%), leading to the error in the lattice constant a. Unfortunately, attempts to improve interaction potentials for simultaneous refinement of the constants a and c resulted in instability of the structure of the model nanocrystals during MD simulations.

The shape of the PO₄ complex in our model is somewhat distorted compared to the experiment: this complex is compressed by 0.07 Å (3.6%) along the vertical axis in Figure 2 and by 0.04 Å (1.6%) along the perpendicular horizontal axes. Our analysis did not reveal any direct relationship between the errors in the positions of yttrium and oxygen ions.

The examination of a range of temperatures from 300 K to 2200 K made it possible to observe the thermal expansion of the crystal lattice. Figure 6 shows the temperature dependence of the specific volume v per YPO₄ molecule in the central region of model nanocrystals consisting of 52,728 particles. For comparison, an experimental dependence obtained using the linear expansion coefficient from [31] is shown. At room temperature, the calculated value of v is underestimated by 3.5% relative to the experiment. The divergence decreases with increasing temperature.

Figure 7 illustrates the accuracy of the modeling of the YPO₄ unit cell shape. The temperature dependencies of the crystal lattice parameters a and c are shown in comparison with the experimental data from Wang et al., 2010 [31]. As discussed above, the constant a is underestimated compared to the experiment, while the constant c is overestimated, being significantly more accurate than the constant a. With increasing temperature, the error in the constant a decreases slightly, while the error in the constant c increases due to the overestimation of the coefficient of linear expansion.

The coefficient of thermal expansion (CTE) calculated in the present work is compared with the experimental data in Figure 8. The experimental data used for comparison were taken from [16,31]. In the relatively recent work by Wang et al., 2010 [31], the CTE is provided as a function of temperature. In the older work [16], the CTE values are given as constants, but these constants differ along the edges of the unit cell a and c: CTEa = 5.81·10⁻⁶ 1/K, CTEc = 7.16·10⁻⁶ 1/K [16]. In Figure 8, the values from these references are shown as horizontal dashed lines.

In the present study, no significant anisotropy of the CTE for model crystallites was detected. Therefore, the temperature dependence of the model CTE (Figure 8) was averaged over the a, b, and c directions. At room temperature, the thermal expansion coefficient obtained in this work is close to the experimental data. However, with increasing temperature, the CTE rises significantly faster than in the experiment.

The overestimation of the calculated coefficient of linear expansion at high temperatures may be due to the fact that only low-temperature data were used to reconstruct the interaction potentials. However, with increasing temperature, the amplitude of particle oscillations increases, which requires accurate reproduction of the shape of the short-range potentials in an increased range of distances.

The model crystal lattice of YPO₄ remained stable up to 2000 K. At 2200 K, the specific volume increased abruptly and the structure of the system became disordered. Further studies are needed to determine whether this process could be the melting point of the crystal. However, we note that the experimental melting temperature of YPO₄ is 2268 ± 20 K [32], which is close to the stability limit of the model in this work.

In order to provide a detailed comparison between the structure of model crystals and experimental data, radial distribution functions for oxygen, yttrium, and phosphorus ions relative to the phosphorus ion were calculated in this study. The average distances between the nearest neighbors, determined based on these functions, correspond to the crystal lattice structure of YPO₄ (

Table 2. Positions of first maximum of radial distribution functions of crystal ions relative to phosphorus.

Pair of Particles	Distance, 10−10 m
	300 K	1600 K	Experiment (293 K)
P–O	1.50	1.50	1.54
P–Y	3.025	3.066	3.014
P–P	3.69	3.75	3.76

). At the same time, the P–Y and P–P distances are underestimated relative to the experimental values to the same extent as the lattice constants.

On the other hand, the P–O distances in the present work were practically independent of temperature. This is due to the use of the Morse potential with a narrow and deep minimum ε = 3.5 eV (calculated in this work) at a distance R_P–O = R_m = 1.51 Å. The physical meaning of this model is that the PO₄ complexes act as distinct structural units of the crystal.

The accuracy of lattice constant calculations achieved in this work is comparable to the results obtained previously using classical potentials for modeling YPO₄ and structurally similar rare-earth phosphates. In work [11], dedicated to YPO₄, the potentials from [12,13,14] yielded lattice constants a = 6.760 Å and c = 6.260 Å. The trend to underestimate the constant a and overestimate the constant c aligns with our findings, though the overestimation of c in [11] was more pronounced.

In the study [33], two sets of interaction potentials were proposed for modeling LaPO₄ and one set for YbPO₄. The crystal LaPO₄ forms a monoclinic structure (space group P2₁/c) with three different lattice constants: a = 6.831 Å, b = 7.071 Å, and c = 6.503 Å [33]. The “Monazite 1” set allowed the authors of [33] to reproduce these constants with errors of Δa = +0.40%, Δb = +0.01%, and Δc = +2.03%, while the “Monazite 2” set resulted in Δa = −1.28%, Δb = −1.73%, and Δc = +3.31% [33]. Notably, in “Monazite 1,” which showed better agreement, the P–O bonding was described by a Morse potential Equation (2) with a minimum at R_m = 1.57 Å and well depth ε = 3.0 eV that are close to the parameters obtained in our work. In contrast, the set “Monazite 2” used a Buckingham potential with dispersive attraction for P–O bonding, which did not have a deep minimum. Strong P–O binding was achieved by increasing the effective charge of phosphorus from 1.784 ē to 3.2 ē (while the lanthanum charge was reduced from 3.0 ē to 1.6 ē, keeping the oxygen charge fixed at q_o = −1.196 ē).

YbPO₄ is a direct structural analog of YPO₄ with similar lattice constants. In [33], experimental values a = 6.809 Å and c = 5.964 Å were reproduced with errors Δa = −1.599% and Δc = +2.87% [33]. The interaction P–O was modeled using the same Buckingham potential as in the “Monazite 2” set for LaPO₄. This case again exhibited the trend of underestimating a and overestimating c.

4. Frenkel Disorder Energies

One of the applications of the interaction potentials proposed in this work might be modeling of the intrinsic disorder in the YPO₄ crystal. In this work, the energies of Frenkel pair formation in the yttrium, oxygen, and phosphorus sublattices were calculated using molecular dynamics. Interstitial ions were placed at positions with coordinates (0.25 a, 0.5 b, 0.5 c) relative to the lower left corner of the unit cell shown in Figure 2. In [11], this position is designated as Ta (Figure 2a,b). The vacancy was placed as far from the interstitial position as the model dimensions allowed.

In the presence of a free surface, the error in the calculation of Frenkel defect formation energies was found to be too high. Therefore, as a model system for calculating these energies, quasi-infinite crystallites consisting of 5184 particles (6 unit cells along each edge of the supercell) with periodic boundary conditions were used. Simulations were performed with dynamic variation in the supercell dimensions, ensuring zero mean values of the stress tensor components σ_xx, σ_yy, σ_zz. A barostat proposed by V.A. Ryabov in [34] was used. At a temperature of T = 25 K, the calculated lattice parameters of the ideal YPO₄ lattice were a = b = 6.7598 Å, c = 6.0374 Å, in agreement with the result of the simulation of the isolated nanocrystals.

To calculate the above-mentioned disordering energies, MD simulations of the quasi-infinite crystals were performed at a constant temperature of T = 25 K. The low temperature was chosen to avoid recombination of interstitial ions with the vacancy. At this temperature, the displaced ions remained in the interstitial site for at least 1 ns. Interstitial yttrium and phosphorus cations displaced their nearest neighbors of the same kind to an adjacent interstitial position, so that pair defects known as split interstitials were formed (Figure 9). These configurations remained stable during the computational experiments.

The defect formation energies were calculated as the differences in the total energies of the supercell with and without the defect. The dimensions of the supercell were fixed when calculating the energies of defect formation, since the system of defects located periodically could change the shape of the supercell in a non-physical way.

The calculated values of the formation energy of Frenkel defects are listed in

Table 3. Calculated energies of formation of Frenkel defects in YPO₄.

Disordered Sublattice	Energies of Formation of Frenkel Defects, eV
	This Work	Ab Initio [11]	Close Pairs [11]	Classical MD [11]
Yttrium	13.9 ± 0.1	16.2	7.37	13.7
Oxygen	4.7 ± 0.1	–	4.43	11.2
Phosphorus	6.9 ± 0.5	28.7	16.75	18.9

. For comparison, the results of first-principles calculations from reference [11] are given, as well as the classical MD modeling carried out in reference [11] with potentials [12,13,14]. The column “Ab initio” shows the energies of formation of the pairs of split interstitial cation (position Ta)–remote cation vacancy. We calculated these values based on the data from Table 2 and Table 3 of [11]. The formation energy of the pair interstitial oxygen (Ta)–remote vacancy could not be determined from the data of [11]. The column “Close pairs” contains the formation energies of close Frenkel pairs calculated in reference [11] from the first-principles. The column “Classical MD” lists the formation energies of close Frenkel pairs calculated in [11] using classical interaction potentials [12,13,14].

First-principles calculations [11] were performed using density functional theory (DFT) with the projector augmented wave (PAW) method [35] and the generalized gradient approximation (GGA) exchange-correlation functional [36]. The first-principles calculation of the lattice constants in [11] yielded values of a = 6.838 Å and c = 6.048 Å. These results are more accurate than those obtained in the present work, though they still exhibit the same trend of underestimating the constant a while overestimating the constant c.

The classical potentials [12,13,14] included the Coulomb interaction of the effective charges along with the valence interactions described by exponential repulsion and dispersive attraction (in the form of the Buckingham potential). The effective ionic charges were set as q_p = +3.0·ē, q_y = +1.8·ē, and q_o = −1.2·ē. In addition to pair potentials, the model accounted for bond bending interactions of O–P–O and P–O–P [14] in the form E(α) = 1.75·(α − 109.47°) eV, where α is the angle formed by the three particles.

As can be seen from Table 3, the energy of formation of the oxygen Frenkel pair obtained in the present work is closer to the first-principles calculation [11] than the value calculated with the alternative classical potentials [12,13,14]. The coincidence of this energy with the first-principles calculation [11] can be quantitative, if the formation energies of the close and remote Frenkel pairs are similar. The formation energy of the yttrium Frenkel pair is also close to the ab initio calculation.

On the other hand, the disordering energy of the phosphorus sublattice in this work is fundamentally underestimated relative to reference [11]. This is due to the use of the pair potential “two exponents” with a deep minimum for modeling the phosphorus–oxygen covalent bond. In the present work, the interaction potentials did not depend on the environment of the particles. As a consequence, the interstitial phosphorus ion formed strong bonds with the nearest oxygen ions that were already bound to another phosphorus ion. Thus, the potentials proposed in this work seem to be applicable only to those processes in which PO₄ complexes retain their integrity.

5. Conclusions

In this paper, empirical parameters of pair interaction potentials, as well as effective charges of yttrium, phosphorus, and oxygen ions, are obtained for classical molecular dynamic modeling of a YPO₄ crystal. The set of pair interaction potentials enabled stable molecular dynamics simulation of isolated YPO₄ crystals with free surface in the temperature range from 300 K to 2000 K. At a temperature of T = 2200, a process of disordering of the crystal lattice was registered, which may be associated with melting.

The restored lattice parameters are close to experimental values, and the thermal expansion coefficient at low temperatures quantitatively agrees with the experimental data. During the simulations, the PO₄ complexes behaved as structural units of the crystal, bonded to yttrium ions. In particular, thermal expansion led to an increase in the P–Y and P–P distances, whereas the P–O distance remained almost unchanged.

The calculated energy of the formation of the Frenkel defect in the oxygen and yttrium sublattices are close to the first-principles calculations [11]. The energy of Frenkel disordering of the oxygen sublattice obtained in this work is significantly closer to the first-principles calculation [11] than the value received in [11] using existing classical interaction potentials [12,13,14].

The proposed potentials can be applied to study phenomena related to thermal and impurity-induced disordering in YPO₄ crystals, vacancy transport in the yttrium and oxygen sublattices, interactions of the vacancies with activator ions, as well as vacancy–vacancy interactions.

Modeling the interaction of defects with each other, including the formation of their complexes, is of particular importance for YPO₄. For example, Y–O vacancy complexes provide hole traps with distributed (non-discrete) activation energy [4]. In turn, the hole traps in YPO₄ play a crucial role in determining the characteristics of long-lasting luminescence in the UV-C spectral range [37].

We believe that the interatomic potentials developed in this work can be directly applied to model YPO₄ crystals doped with trivalent impurities (Eu³⁺, Tb³⁺, Dy³⁺, Ce³⁺) without any modifications, provided that the effective charge of these dopants matches that of the Y³⁺ ion. In this case, no adjustments to the oxygen and phosphorus charges would be required.

For aliovalent doping (e.g., with Me²⁺ cations), charge compensation mechanisms must be considered. One possible mechanism involves the formation of anion vacancies, resulting in hypostoichiometric (Y,Me)PO_4−x crystals. Our potentials remain applicable for modeling such systems, as well as undoped nonstoichiometric crystals YPO_4±x. This requires maintaining the original effective charge of oxygen ions while assigning vacancies a charge equal in magnitude but opposite in sign to that of oxygen.

The pair potential reconstruction methodology developed here for YPO₄ can be extended to other rare-earth phosphates. An advantage of our approach lies in describing the strong (chemical) P–O bonding through a double-exponential Morse potential with a deep minimum at ~1.5 Å. This treatment considers the PO₄ complex as a structural unit whose size and geometry remain nearly invariant across different phosphate crystals, which is consistent with the experimental data [15,17,31]. We anticipate that the P–O and O–O potentials obtained here could be transferred to other rare-earth phosphates with minor adjustments. The main limiting factor would be their dependence on effective charges, which may vary in different crystal systems. Similarly, the Y–O potential could be adapted to other systems featuring comparable effective charges for yttrium and oxygen.

Another distinctive feature of this work is the implementation of a damping function [25] for the dispersive attraction at short distances. This modification enables the physically meaningful application of our O–O potential at small interatomic separations where unphysical growth of the C/r⁶ term would otherwise become problematic.