Polymorphic Phase Transformations in Nanocrystalline Ag2S Silver Sulfide in a Wide Temperature Interval and Influence of Nanostructured Ag2S on the Interface Formation in Ag2S/ZnS Heteronanostructure

Phase transformations that take place in nanocrystalline Ag2S silver sulfide have been systematically studied at temperatures from 298 to 893 K. The crystal structures of the polymorphic modifications α-Ag2S, β-Ag2S, and γ-Ag2S of nanocrystalline Ag2S have been found. It is established that the interstitial spacings between ions of silver in the superionic phases β-Ag2S and γ-Ag2S are noticeably smaller than diameter of the Ag+ ion. As a result of which, the probabilities of filling the sites of the metal sublattices of these phases with Ag atoms are very small. It was found that the “α-Ag2S—β-Ag2S” and “β-Ag2S—γ-Ag2S” transitions between polymorphic modifications of silver sulfide occur as phase transformations of the first order at temperatures of ~440–442 K and ~850–860 K. The structure of interface forming by nanostructured Ag2S and ZnS is considered, taking into account the anisotropy of elastic properties of these sulfides. It is established that a large amount of cubic zinc sulfide stabilizes the cubic structure of β-Ag2S argentite at 300 K during the co-deposition of Ag2S/ZnS heteronanostructures from colloid solutions. It is found that placing Ag atoms at four crystallographic positions located in one plane of the unit cell of cubic β-Ag2S argentite is most favorable for the appearance of Ag2S/ZnS heterostructures. The smallest strain distortions at the interface are observed at the minimum difference of shear moduli of the components forming heteronanostructure. The distributions of elastic characteristics, including the shear moduli of monocrystalline particles of cubic β-Ag2S argentite and ZnS sphalerite from the [hkl] direction, are found. The formation of Ag2S/ZnS heteronanostructures, in which the interface is formed by the (hk0) ≡ (110) plane of ZnS sphalerite and the (hk 0.4123) ≡ (1 1 0.4123) plane of β-Ag2S argentite, is the most energetically favorable.


Introduction
Among binary semiconductor compounds, such compounds of metals with sulfur as sulfides CuS, Cu 2 S, Ag 2 S, ZnS, CdS, HgS, SnS, and PbS occupy an important place. These sulfides are used in various fields of modern technology.
Semiconductor sulfide nanostructures represent the promising group of nanocrystalline materials for various fields of application [1][2][3]. Silver sulfide Ag 2 S attracts a lot of attention [4][5][6]. Silver sulfide is a semiconductor at temperature <~420-450 K, and this sulfide possesses a superionic conductivity at temperature more than 452 K. Nanocrystalline silver sulfide is the versatile semiconductor for applications in different optoelectronic devices such as photocells, photoconductors, and infrared detectors [6][7][8].
The use of nanocrystalline silver sulfide is promising for creating Ag 2 S/Ag heteronanostructures intended for operation in memory devices and resistance switches. Their action is based on the transformation of α-Ag 2 S acanthite into β-Ag 2 S argentite and the formation of a conducting channel from silver Ag and superionic β-Ag 2 S argentite [9][10][11][12].
Ag 2 S and ZnS are among the most sought-after semiconductor sulfides [5,13]. Recently, much attention has been attracted by heteronanostructures based on semiconductor Ag 2 S and ZnS sulfides [14][15][16][17][18][19][20][21]. Composite heteronanostructures can exhibit enhanced nonlinear phenomena in optical properties due to the quantum size effect of nanoparticles. The light emission properties of such heteronanostructures are important for their applications in optical imaging, sensors, or lasers. Composite heteronanostructures represent an interesting class of materials because their applications are of multidisciplinary importance. The main hindrance for zinc sulfide widespread use is the low emission properties of pure ZnS nanoparticles. Formation of the heteronanostructures is one way that has been used to improve the emission properties of pure ZnS in recent years. For example, in study [17], it was reported that the light emission properties of double-shell ZnS-Ag 2 S hollow nanoparticles are maximum compared to those of pure ZnS nanoparticles, and these properties highly depend on the external ZnS layer thickness. Ag 2 S/ZnS type heteronanostructures make it possible to control the band gap and are considered promising nanomaterials for solid-state ultraviolet lasers and fast-acting resistive switches as well as for catalysis.
In Ag 2 S/ZnS heteronanostructures, an important role belongs to the interface between silver and zinc sulfides. Strain distortions at the interface should be minimal.
Information on the elastic characteristics of ZnS and Ag 2 S is needed to assess the strain (deformation) distortions at the interface between silver and zinc sulfides. The elastic properties of cubic ZnS are well known [22][23][24][25][26][27][28][29][30][31][32]. The theoretical data on the elastic properties of semiconductor monocrystalline particles of silver sulfide Ag 2 S with monoclinic (space group No. 11-P2 1 /m) and orthorhombic (space group No. 63-Cmcm) structures are available in the form of databases on the websites [33,34]. According to [33], the maximal value of the Young's modulus of modeling monoclinic (space group P2 1 /m) silver sulfide is equal to~37.9 GPa. According to data [34], the maximal Young's modulus E of orthorhombic (space group Cmcm) Ag 2 S is~48.1 GPa. The elastic constants are calculated by the ab initio method, described in general terms in the study [35].
In electronics, the potentially most applicable superionic is β-Ag 2 S argentite, which has a body-centered cubic (bcc) (space group Im3m) lattice. Recently, the thermal expansion and heat capacity of coarse-crystalline and nanocrystalline silver sulfide were measured in the temperature range 300-930 K [36][37][38] including the region of existence of argentite. The elastic properties of cubic (space group Im3m) superionic β-Ag 2 S argentite were determined for a wide temperature range in studies [39,40].
Unfortunately, intensive studies of the synthesis and properties of nanocrystalline sulfides are not accompanied, as a rule, by a detailed study of the crystal structure. Many data on the crystal structure of coarse-crystalline sulfides were obtained back in the 1950s-1970s of the last century and even earlier on samples of natural minerals. The structure of artificial (synthetic) silver sulfides, especially nanocrystalline ones, can differ markedly from that of natural minerals in terms of unit cell parameters and even in the arrangement of atoms at other crystallographic positions. For example, the X-ray diffraction (XRD) reflections not related to the structure of monoclinic acanthite were detected in the XRD pattern of Ag 2 S nanoparticles [41].
Ag 2 S is one of a few sulfides that have three polymorphous modifications: Semiconductor monoclinic α-Ag 2 S acanthite, superionic argentite β-Ag 2 S with body centered cubic (bcc) structure, and superionic face centered cubic (fcc) γ-Ag 2 S phase. Application of silver sulfide in microelectronic devices is based on the reversible transition "α-Ag 2 S β-Ag 2 S". Therefore, an accurate determination and refinement of the structure of different phases is especially important in the study of phase transformations in silver sulfide, for the use of silver sulfide in devices operating in a wide temperature range.
Studies of bulk silver sulfide, carried out from the 1930s of the last century to the beginning of the XXI century, are described in detail in the monograph [6]. Blanton et al. [42] made the first modern determination of the crystal structure of all three phases for coarsecrystalline synthetic powders of silver sulfide. Prepared silver sulfide Ag 2 S powder was studied by XRD at temperatures of 293, 523, and 923 K.
A kinetic investigation of phase transitions in bulk silver sulfide Ag 2 S by the differential thermal analysis was carried out by Živković et al. [43]. According to [43], the temperatures of phase transitions α-Ag 2 S-β-Ag 2 S и β-Ag 2 S-γ-Ag 2 S are equal to 458 and 867 K, respectively. However, no study of the structure of Ag 2 S different phases was performed in work [43].
Il'inskii et al. [44] studied the phase transition from the monoclinic semiconducting to the bcc superionic state in thin films of silver sulfide by dielectric spectroscopy in the temperature region 273-473 K. According to [44], the temperature dependences of the dielectric loss tangent of the Ag 2 S film at 425 K exhibit a hysteresis corresponding to the taking place reversible "semiconductor-superionic phase" transition. In work [44], experimental data related to the structure of the studied films of silver sulfide are absent.
Recently, the authors of work [45] studied the phase transformations in silver sulfide with sheet-like morphology under high pressure up to 30 GPa. Ran Liu et al. have found two different structural phase transitions. The first transition from monoclinic (space group P2 1 /n) acanthite α-Ag 2 S to orthorhombic (space group P2 1 2 1 2 1 ) phase II is observed under the pressure of 8.9 GPa. The second transformation of orthorhombic phase II to monoclinic phase III, isosymmetric to a phase I, takes place under the pressure of 12.4 GPa. Unfortunately, the structure of the Ag 2 S monoclinic phases is described in the outdated space group P2 1 /n, therefore the convergence of the experiment and calculation is low: the Rietveld factors R p and ωR p are equal to 14-20% and 30-31%, respectively [45].
Thus, studies of phase transformations in silver sulfide have been carried out in few works, moreover, on coarse-crystalline samples. In a number of works, the crystal structure of the phases of Ag 2 S was not studied at all or was not exactly determined.
In works [14,[16][17][18][19]21] devoted to the synthesis and study of heteronanostructures based on semiconductor Ag 2 S and ZnS sulfides, the main attention is paid to the producing conditions and optoelectronic properties of synthesized Ag 2 S/ZnS heteronanostructures, whereas the structure of the interfaces in general was not being discussed. However, the properties of heteronanostructures strongly depend on the structure of the interfaces [46,47], and on the elastic properties of nanoparticles [48]. The structure of Ag 2 S/ZnS heteronanostructures was analyzed, taking into account the morphology of Ag 2 S and ZnS sulfide monocrystalline particles in work [49] using the outdated Hartman-Perdok theory [50]. Additionally, the structure of Ag 2 S/ZnS heteronanostructures was considered in work [51] without analysis of possible mutual positions of planes of β-Ag 2 S argentite and ZnS sphalerite on their interface.
In this regard, in present study, an in situ systematic investigation of polymorphic phase transitions in nanocrystalline silver sulfide is performed in a wide temperature interval from 298 to 893 K, where semiconductor monoclinic α-Ag 2 S acanthite, superionic bcc β-Ag 2 S argentite, and superionic fcc γ-Ag 2 S phase exist. X-ray diffraction over a wide temperature range and high-resolution transmission electron microscopy (HRTEM) are used as the main experimental methods.
The other aims of this work are the following: (1) to determine possible combinations of crystallographic planes of cubic β-Ag 2 S argentite and ZnS sphalerite, which can physically form the interface of the Ag 2 S/ZnS heteronanostructure, and (2) to study the structure of the interface between zinc and silver sulfides, taking into account the structural features and elastic characteristics of the monocrystalline cubic β-Ag 2 S argentite and ZnS sphalerite.

Samples
Silver sulfide Ag 2 S, in a form of colloid solution and powder, was prepared by hydrochemical bath deposition, i.e., the chemical deposition from aqueous solutions. It is a conversant universal technique for preparing colloid solutions, coarse-crystalline, and nanocrystalline powders of silver sulfide [52].
Colloid solutions and powders of Ag 2 S were produced by deposition from aqueous solutions of AgNO 3 (ACS reagent, ≥99.0%, Sigma-Aldrich, St. Louis, MO, USA), Na 2 S (ACS reagent, ≥98.0%, Sigma-Aldrich, St. Louis, MO, USA), and sodium citrate Na 3 C 6 H 5 O 7 (Na 3 Cit) (ACS reagent, Merck KGaA, Darmstadt, Germany) using the same chemical reactants by altering their concentration in a solution. The synthesis of silver sulfide was carried out at pH ≈ 7 in the dark by the following reaction scheme: with an excess 0.01 ≥ δ ≥ 0.5 of sodium sulfide Na 2 S. An excess of Na 2 S ensures producing silver sulfide without Ag impurity. The concentrations of main reactants AgNO 3 , Na 2 S, and Na 3 Cit were 0.625, 0.313, and 1.25 mmol·L −1 for the preparation of a stable colloid Ag 2 S solution, and 50, 50, and 100 mmol·L −1 for deposition of Ag 2 S powder. The average size of colloid nanoparticles of silver sulfide was~4 ± 2 nm, the average particle size for powder was~90 ± 10 nm. According to XRD data, the synthesized colloid nanoparticles and silver sulfide powder at room temperature had a monoclinic (space group P2 1 /c) crystal structure of α-Ag 2 S acanthite.
In our work, Ag 2 S powders with particles smaller than 50 nm are believed to be nanocrystalline, and powders with particle of about 100 and 200 nm are called finedispersed and coarse-crystalline powders, respectively. The procedure for the synthesis of Ag 2 S powders and colloid solutions is described in detail in [52][53][54][55].
Heteronanostructures Ag 2 S/ZnS have been produced by two-stage synthesis in detail, described in work [19]. First, silver sulfide Ag 2 S was synthesized by chemical deposition from aqueous solutions of AgNO 3 and Na 2 S in the presence of sodium citrate Na 3 Cit. Next, Trilon B (ACS reagent, 99.0-101.0%, Sigma-Aldrich, St. Louis, MO, USA) was added to an aqueous Zn(NO 3 ) 2 (reagent grade, 98%, Sigma-Aldrich, St. Louis, MO, USA) solution with a sodium sulfide solution. The use of Trilon B promoted the appearance of nanostructured zinc sulfide on a surface of silver sulfide nanoparticles. Thus, Ag 2 S/ZnS heteronanostructures of core-shell type were formed.

Experimental Techniques
Primary certification of synthesized Ag 2 S colloid solution and deposited silver and zinc sulfides powders was carried out by X-ray diffraction (XRD) at 298 K in CuKα 1radiation in the interval of angles 2θ from 20 to 95 • with a step 0.02 • . The determining parameters of lattice and refinement of structure of the prepared colloid nanoparticles and silver sulfide powder were made using the X'Pert HighScore Plus program (version 2.2e Phase transformations in silver sulfide in a temperature region 298-893 K were studied by in situ high-temperature XRD in filtered CuKα radiation using a high-resolution "Empyrean" diffractometer (X'Pert PRO MRD, PANAlytical, The Netherlands) supplied with an Anton Paar HTK-1200 Oven furnace (Graz, Austria). The diffraction measurements were made in high vacuum 0.01 Pa (7.7 × 10 −5 mm Hg) in the interval of angles 2θ from 20 to 60 • with a step 0.026 • . The cylindrical sample in 10.0 mm diameter and 1.5 mm thickness, produced by uniaxial pressing of the synthesized silver sulfide powder in a steel compression mold, was used for measurements. Pressure of pressing was equal to 20 MPa. The surface of the sample was polished before XRD measurements.
A scanning JEOL-JSM LA 6390 electron microscope (JEOL Ltd., Tokyo, Japan) equipped with a JED 2300 Energy Dispersive X-ray Analyzer (JEOL Ltd., Tokyo, Japan) has been used for direct observation of the changes at heating and cooling of silver sulfide pressed sample.
The microstructure, particle size, and element chemical composition of sulfide powders were studied by the high-resolution transmission electron microscopy (HRTEM) on a JEOL JEM-2010 microscope (JEOL Ltd., Tokyo, Japan) with 0.14 nm lattice resolution. The elemental chemical composition of heteronanoparticles was studied on the same microscope with the use of an Phoenix (EDAX) Energy Dispersive Spectrometer (AMETEK Inc. Berwyn, PA, USA), with a Si(Li) detector having an energy resolution of 130 eV.
The method of HRTEM has been used for investigation of transformations in colloid silver sulfide nanoparticles. The HRTEM images were registered by a Tecnai G2 30 Twin microscope (FEI company, Hillsboro, OR, USA) with 0.14 nm lattice resolution. For study, colloid solution of silver sulfide was placed on the perforated carbon substrates which are fixed on copper grids. After deposition, the substrates with nanoparticles were dried and placed in a microscope on a heated holder.
In situ XRD and HRTEM measurements were performed in the "Test Centre of Nanotechnologies and Perspective Materials" of the Institute of Metal Physics (Ural Branch of Russian Academy of Sciences).

Results and Discussion
The XRD patterns of prepared fine-dispersed powder of Ag 2 S silver sulfide were recorded at temperatures of 298, 353, 423, 443, 453, 463, 523, 573, 623, 643, 723, 773, 843, 863, and 893 K. A preliminary analysis has shown that silver sulfide is monoclinic α-Ag 2 S at a temperature of 298-423 K, it has bcc structure of β-Ag 2 S argentite at a temperature from 443 to 843 K, and at a temperature of 863-893 K it has fcc crystal structure of γ-Ag 2 S phase.

α-Ag 2 S Acanthite
The XRD patterns of pressed sample at 298 and 423 K are presented in Figure 1. Quantitative refinement of structure of synthesized powder has shown that silver sulfide has monoclinic (space group P2 1 /c) symmetry of acanthite α-Ag 2 S in the temperature region 298-423 K.
Experimental, calculated, and difference (I exp -I calc ) XRD patterns of synthesized fine-dispersed silver sulfide powder are presented in Figure 1. The refinement of structure at 298 K made it possible to achieve a high convergence of experiment and calculation: the Rietveld factors R p , ωR p , R I (R B ), and R expect are equal to 3.218%, 4.147%, 2.949%, and 4.374%, respectively. The parameters of the monoclinic unit cell of fine-dispersed Ag 2 S powder and the atomic coordinates in this unit cell at 298 K are given in Table 1. These magnitudes are close to those found in work [53,57] for bulk Ag 2 S.  Some difference in the experimental and calculated intensities of diffraction tions ( Figure 1) is caused by the presence of texture in pressed sample caused by un pressure during powder pressing. In XRD patterns, in addition to the main mono phase, there is only the reflection of the impurity phase at 2θ ~26.75°, correspond the (002) reflection of orthorhombic cementite Fe3C [58,59]. The appearance of the rity is associated with the pressing of Ag2S powder in a steel compression mold amount of the impurity phase is noticeably less than 1 wt. %.
The stability of α-Ag2S acanthite is due to the mutual arrangement of Ag ato distances exceeding the covalent diameter of silver atom, equal to ~0.2883 nm [60 temperature of 298 K, the interatomic spacing Ag1-Ag1 in monoclinic acanthite α-A Figure 1. Experimental (×), calculated (-), and difference (I exp -I calc ) XRD patterns of Ag 2 S pressed pill at temperatures of 298 and 423 K. According to refinement, Ag 2 S at 298 and 423 K is monoclinic (space group P2 1 /c) α-Ag 2 S acanthite. Some difference in the experimental and calculated intensities of diffraction reflections ( Figure 1) is caused by the presence of texture in pressed sample caused by uniaxial pressure during powder pressing. In XRD patterns, in addition to the main monoclinic phase, there is only the reflection of the impurity phase at 2θ~26.75 • , corresponding to the (002) reflection of orthorhombic cementite Fe 3 C [58,59]. The appearance of the impurity is associated with the pressing of Ag 2 S powder in a steel compression mold. The amount of the impurity phase is noticeably less than 1 wt. %. The stability of α-Ag 2 S acanthite is due to the mutual arrangement of Ag atoms at distances exceeding the covalent diameter of silver atom, equal to~0.2883 nm [60]. At a temperature of 298 K, the interatomic spacing Ag1-Ag1 in monoclinic acanthite α-Ag 2 S is 0.33464 nm, the spacing Ag1-Ag2 changes from 0.30758 to 0.31850 nm, and the interatomic distance Ag2-Ag2 is 0.31144 nm.
Minimal interatomic spacings S-Ag1 in monoclinic α-Ag 2 S at 423 K are in the range from 0.24801 to 0.25231 nm, and spacings S-Ag2 change from 0.25201 to 0.27126 nm. The interatomic spacing Ag1-Ag1 in α-Ag 2 S at 423 K is equal to 0.33781 nm, and interatomic spacings Ag1-Ag2 change from 0.30959 to 0.32010 nm. Thus, atoms of Ag in monoclinic acanthite at a temperature of 298-423 K are at rather large distances from each other (greater than the atomic diameter of silver). Therefore, Ag and S atoms fill their crystallographic positions in monoclinic acanthite with a probability equal to 1.
Temperature increasing from 298 to 423 K leads to a slight growth of unit cell volume of monoclinic acanthite α-Ag 2 S from~0.2274 to~0.2288 nm 3 .

β-Ag 2 S Argentite
Diffraction reflections of the body-centered cubic (space group Im3m) β-Ag 2 S phase with an argentite structure are observed in XRD patterns at higher temperatures of 443-843 K ( Figure 2). Thus, the monoclinic α-Ag 2 S acanthite transforms into argentite at a temperature of~440-442 K. According to the quantitative full-profile refinement of XRD pattern of silver sulfide registered at 443 K (Figure 2), the experiment and calculation converge satisfactory: Rietveld factors are R p = 22.34%, ωR p = 29.42%, R I (R B ) = 1.28%, and R expect = 15.69%. The parameters of bcc (space group Im3m) unit cell of fine-dispersed powder of silver sulfide and the atomic coordinates in this unit cell at 443 K are given in Table 2. h )) fine-dispersed powder of silver sulfide with β-Ag 2 S argentite structure at 443 K: Z = 2, a = b = c = 0.4859(3) nm.

Atom Position and Multiplicity
Coordinates of Atoms Occupancy B iso × 10 −4 (pm 2 ) x y z The unit cell of β-Ag 2 S involves two formula units Ag 2 S. S atoms fill the sites of a bcc sublattice with a probability equal to 1. Four Ag atoms are randomly placed in 54 sites, 6(b) and 48(j), with small filling probabilities~0.0974 and~0.0716 (Table 2). Within measurement errors, these filling probabilities coincide with the occupation probabilities 0.0978 and 0.0711 of 6(b) and 48(j) argentite sites by Ag atoms, which were found in study [61]. As a result, Ag atoms are in permanent motion over these 54 possible crystallographic sites of metallic sublattice of argentite, providing superionic conductivity of argentite. According to [11,62,63], the conductivity of supercooled argentite β-Ag 2 S at room temperature is (1.3-1.6) × 10 3 Ohm −1 ·cm −1 , which is about 6 × 10 5 times larger than the conductivity 2.5 × 10 −3 Ohm −1 ·cm −1 of monoclinic phase.
In cubic superionic phases, ions Ag + are surrounded by six sites of nonmetal sublattice. The ionic diameter of Ag + ion for coordination number 6 is~0.252 nm [60]. The interstitial distances between silver ions in bcc β-Ag 2 S argentite at 443 K are significantly less than the interatomic distances in monoclinic α-Ag 2 S acanthite at a close temperature of 423 K. In β-Ag 2 S at 443 K, the distances Ag1-Ag1 between silver ions are 0.24297 nm, the distance between the nearest Ag1 and Ag2 ions is 0.0943 nm, and spacings Ag2-Ag2 between corresponding nearest silver ions are in the range from 0.6803 to 0.07628 nm. These spacings are noticeably less than the ionic diameter of Ag + ion, equal to~0.252 nm. Therefore, silver ions in argentite cannot occupy the nearest neighboring sites of the crystal lattice. Changes in XRD patterns and parameters of unit cell for β-Ag 2 S argentite upon heating from 443 to 843 K are depicted in Figures 3 and 4. The XRD spectra taken during heating from 443 to 843 K include diffraction reflections of only cubic β-Ag 2 S argentite ( Figure 3). Rise in temperature leads to a continuous shift of the position of the argentite's reflections to the field of smaller angles 2θ (for example, a shift of the (200) diffraction reflection is shown in Figure 3 (inset)). This corresponds to a growth in the lattice parameter a arg of argentite upon heating.
Diffraction reflections of the body-centered cubic (space grou with an argentite structure are observed in XRD patterns at h 443-843 K ( Figure 2). Thus, the monoclinic α-Ag2S acanthite trans temperature of ~440-442 K. According to the quantitative full-pro pattern of silver sulfide registered at 443 K (Figure 2          The temperature dependence a arg (T) of the lattice period for β-Ag 2 S in comparison with the literature data [42,[64][65][66] is shown in Figure 4. Within the accuracy of experimental measurements, the dependence of the lattice period a arg (T) in the range 443-853 K is described by a linear function: where a(443) = 0.48632 nm and b = 1.95684 × 10 −5 nm·K −1 . For this linear dependence a arg (T), the coefficient of thermal expansion α arg is a constant equal to~4.024 × 10 −5 K −1 .

γ-Ag 2 S Phase
The XRD patterns of silver sulfide at 863 and 893 K exhibit reflections of the fcc (space group Fm3m) γ-Ag 2 S phase and broad diffuse bands ( Figure 5). The Rietveld convergence factors for refining the structure of Ag 2 S powder at 863 K are R p = 20.67%, ωR p = 25.52%, R I (R B ) = 7.973%, and R expect = 17.41%. The atomic coordinates and the parameters of fcc unit cell of fine-dispersed powder of silver sulfide at 863 K are presented in Table 3.

γ-Ag2S Phase
The XRD patterns of silver sulfide at 863 and 893 K exhibit reflections of (space group m m F 3 ) γ-Ag2S phase and broad diffuse bands ( Figure 5). The R convergence factors for refining the structure of Ag2S powder at 863 K are Rp = ωRp = 25.52%, RI (RB) = 7.973%, and Rexpect = 17.41%. The atomic coordinates and rameters of fcc unit cell of fine-dispersed powder of silver sulfide at 863 K are pre in Table 3.    [37,68,69], it follows that the transformation of β-Ag 2 S argentite intoγ-Ag 2 S phase occurs in the temperature range 843-863 K, i.e., at a temperature of~850-860 K.
Four Ag 2 S formula units enter into the unit cell of γ-Ag 2 S phase ( Figure 6). S atoms are in 4(a) crystallographic positions. Ag atoms are randomly situated on 88 positions 8(c), 32(j), and 48(i) with the probabilities of filling~0.088,~0.150, and~0.027 (Table 3). Figure 6 presents the exact positions of S atoms and the sites of three types on which the Ag atoms can be located with the mentioned probabilities. According to the refined structure, a noticeable silver deficit corresponding to the nonstoichiometric sulfide Ag 1.7 S is observed in the γ-phase. The γ-Ag 2 S phase has superionic conductivity, like β-Ag 2 S argentite, due to disordered distribution of Ag atoms, low probabilities of filling the lattice sites by Ag atoms, their continuous motion, and silver deficiency. The lattice period of cubic γ-Ag 2 S at 863 and 893 K is 0.62706 and 0.62747 nm, respectively.   (6) [37,68,69], it follows that the transformation of β-Ag2S argentite intoγ-Ag2S phase occurs in the temperature range 843-863 K, i.e., at a temperature of ~850-860 K.
Four Ag2S formula units enter into the unit cell of γ-Ag2S phase ( Figure 6). S atoms are in 4(a) crystallographic positions. Ag atoms are randomly situated on 88 positions 8(c), 32(j), and 48(i) with the probabilities of filling ~0.088, ~0.150, and ~0.027 (Table 3). Figure 6 presents the exact positions of S atoms and the sites of three types on which the Ag atoms can be located with the mentioned probabilities. According to the refined structure, a noticeable silver deficit corresponding to the nonstoichiometric sulfide Ag1.7S is observed in the γ-phase. The γ-Ag2S phase has superionic conductivity, like β-Ag2S argentite, due to disordered distribution of Ag atoms, low probabilities of filling the lattice sites by Ag atoms, their continuous motion, and silver deficiency. The lattice period of cubic γ-Ag2S at 863 and 893 K is 0.62706 and 0.62747 nm, respectively.  Table 3), as shown.
The calculated interatomic (interstitial) spacings for different coordination spheres (CS) of structure of the fcc (space group m m F 3 ) γ-Ag2S phase are given in Table 4.   Table 3), as shown.
The calculated interatomic (interstitial) spacings for different coordination spheres (CS) of structure of the fcc (space group Fm3m) γ-Ag 2 S phase are given in Table 4.
Interstitial spacings d were calculated using a simple formula for describing structures with a cubic unit cell: For the calculation, we used the coordinates (x, y, z) of atoms in γ-Ag 2 S phase (Table 3) and the value of lattice period a of γ-Ag 2 S phase at 863 K. The occupation probabilities of the sites of metallic sublattice by Ag atoms are small, and therefore it is more correct to speak about the spacings between sites at which the disposition of Ag atoms is possible.
The distances (spacings) between Ag + ions in the γ-Ag2S phase at a temperature of~860 K were compared with the diameter of Ag + ion at the same temperature. The diameter of Ag + ion for coordination number 6 is~0.252 nm at 298 K [60]. The linear coefficient of thermal expansion α Ag for silver in the temperature region 300-1000 K varies from 19 × 10 −6 to 25 × 10 −6 K −1 [70], the average coefficient of expansion of silver for this temperature range is α Ag-aver ≈ 22 × 10 −6 K −1 . Taking this into account, the ionic diameter of silver at~860-900 K is~0.255 nm. At 863 K in the lattice of cubic γ-Ag 2 S phase, the possible smallest interionic spacings Ag1-Ag1 are 0.3135 nm, and the distances Ag1-Ag2 between corresponding silver ions are in the region from 0.0909 to 0.2719 nm. Interionic distances Ag2-Ag2 are in the region of 0.1481 to 0.2953 nm, distances between ions Ag2 and Ag3 are in the interval of 0.1076 to 0.2516 nm, and spacing Ag3-Ag3 between corresponding ions lie in the range from 0.0940 to 0.3026 nm. The sites of silver sublattice, especially 48(i) sites, are located very close to each other. Therefore, the disposition of the Ag + ion in any site excludes the filling of adjacent site by another silver ion Ag1 or Ag2. Indeed, the diameter of Ag + ion (≥0.255 nm) is more the distances Ag1-Ag3 (0.1637 nm) and Ag2-Ag3 (0.1076-0.2516 nm). The interstitial spacings in γ-Ag 2 S phase are such that, in the appearance of Ag + ion at site 8(c), another Ag + ion can fill one of the sites 32(j), which is only in the 2nd or 3rd CS at a spacing of 0.2712-0.2719 nm from site 8(c) ( Table 4). The presence of Ag + ion at site 8(c) excludes the filling of any site 48(i) with Ag3 atom, and vice versa. Similarly, if Ag + ion fills one of the sites 32(j), then another Ag + ion can be disposed at site 32(j), located at a spacing of leastwise 0.255 nm from the first site, i.e., in the 7th or more distant CS. When the Ag + ion is placed at the site 48(i), another Ag + ion can fill one of the 48 sites (i) situated in the 5th or more distant CS relative to the first site (see Table 4).
Thus, in cubic γ-Ag 2 S phase, the possible spacings between silver atoms (ions) are too small for positions (c), (j), and (i) to be filled by silver ions with a probability equal to 1. Therefore, the degree of occupying positions (c), (j), and (i) by Ag atoms are very small and amount to 0.088, 0.15, and 0.027 (see Table 3).
This means that in the lattice of the fcc γ-Ag 2 S phase, as in the lattice of cubic argentite, Ag atoms are in permanent motion along 88 crystallographic positions which are possible for them. Moreover, the very close arrangement of the sites of the silver sublattice in γ-Ag 2 S phase leads to a noticeable deviation of its composition from stoichiometry (Ag 1.7 S) with a silver deficiency. Continuous motion of Ag atoms and their deficiency ensures the stability of the fcc γ-Ag 2 S phase and its superionic conductivity.

Diffuse Scattering in XRD Patterns of the Superionic β-Ag 2 S and γ-Ag 2 S Phases
Regions of strong diffuse scattering are visible in XRD patterns of the superionic β-Ag 2 S and γ-Ag 2 S phases. Figure 7 shows the XRD patterns of β-Ag 2 S argentite at 453 and 623 K and γ-Ag 2 S phase at 893 K with regions of strong diffuse scattering.
Nanomaterials 2022, 12, x FOR PEER REVIEW 13 of 34 possible for them. Moreover, the very close arrangement of the sites of the silver sublattice in γ-Ag2S phase leads to a noticeable deviation of its composition from stoichiometry (Ag1.7S) with a silver deficiency. Continuous motion of Ag atoms and their deficiency ensures the stability of the fcc γ-Ag2S phase and its superionic conductivity.

Diffuse Scattering in XRD Patterns of the Superionic β-Ag2S and γ-Ag2S Phases
Regions of strong diffuse scattering are visible in XRD patterns of the superionic β-Ag2S and γ-Ag2S phases. Figure 7 shows the XRD patterns of β-Ag2S argentite at 453 and 623 K and γ-Ag2S phase at 893 K with regions of strong diffuse scattering. In XRD patterns of argentite at temperature 453 and 623 K, along with diffraction reflections of cubic β-Ag2S, diffuse reflection in the region 2θ ≈ 30.0-31.0° and a wide diffuse halo in the region 2θ ≈ 34-40° are observed (see Figure 7). The intensity of argentite reflections decreases with increasing temperature, while the intensity of the diffuse halo increases. In the XRD pattern of γ-Ag2S phase at 893 K, there is a wide diffuse halo in In XRD patterns of argentite at temperature 453 and 623 K, along with diffraction reflections of cubic β-Ag 2 S, diffuse reflection in the region 2θ ≈ 30.0-31.0 • and a wide diffuse halo in the region 2θ ≈ 34-40 • are observed (see Figure 7). The intensity of argentite reflections decreases with increasing temperature, while the intensity of the diffuse halo increases. In the XRD pattern of γ-Ag 2 S phase at 893 K, there is a wide diffuse halo in the region 2θ ≈ 30-39 • between the diffraction reflections (200) and (220) of the fcc γ-phase.
The appearance of diffuse scattering is due to the very high mobility of Ag atoms in the superionic phases. Earlier, Blanton et al. [42] observed wide diffuse halos in the XRD pattern of argentite at 523 K and in the XRD pattern of γ-Ag 2 S phase at 923 K in the same interval of 2θ angles. They noted that the weak diffraction of the β-Ag 2 S and γ-Ag 2 S phases, and the observed peaks of diffuse scattering are due to the large disordering and high mobility of silver atoms.
After heating the sample to 893 K, the XRD pattern (see Figure 7) contains only weak reflections of the fcc γ-Ag 2 S phase and a wide diffuse halo. Subsequent cooling of the sample to room temperature restores the monoclinic structure. However, in the XRD pattern, in addition to the reflections of the main monoclinic phase, there is a reflection (111) of cubic (space group Fm3m) metallic silver. This is clearly seen in Figure 8. Apparently, silver sulfide decomposes at temperatures above 800 K with isolation of metallic silver. Indeed, the composition of the γ-phase corresponds to the nonstoichiometric sulfide Ag 1.7 S (see Table 3) due to the precipitation of metallic Ag. The appearance of Ag whiskers on the surface of a pressed pill of silver sulfide heated in air to 570 K was detected in work [71]. The authors of work [42] observed the spontaneous appearance of silver whiskers as a result of heating of Ag 2 S to 800-900 K.
The appearance of diffuse scattering is due to the very high mobility of Ag at the superionic phases. Earlier, Blanton et al. [42] observed wide diffuse halos in th pattern of argentite at 523 K and in the XRD pattern of γ-Ag2S phase at 923 K in th interval of 2θ angles. They noted that the weak diffraction of the β-Ag2S and phases, and the observed peaks of diffuse scattering are due to the large disorderi high mobility of silver atoms.
After heating the sample to 893 K, the XRD pattern (see Figure 7) contains onl reflections of the fcc γ-Ag2S phase and a wide diffuse halo. Subsequent cooling sample to room temperature restores the monoclinic structure. However, in th pattern, in addition to the reflections of the main monoclinic phase, there is a re (111) of cubic (space group m m F 3 ) metallic silver. This is clearly seen in Figure parently, silver sulfide decomposes at temperatures above 800 K with isolation of lic silver. Indeed, the composition of the γ-phase corresponds to the nonstoichio sulfide Ag1.7S (see Table 3) due to the precipitation of metallic Ag. The appearanc whiskers on the surface of a pressed pill of silver sulfide heated in air to 570 K w tected in work [71]. The authors of work [42] observed the spontaneous appear silver whiskers as a result of heating of Ag2S to 800-900 K. Scanning electron microscopy of a sample pressed from a synthesized fine-di Ag2S powder heated to 893 K and cooled to room temperature ( Figure 9) confir change in the phase composition due to the appearance of Ag whiskers. Numero whiskers of metallic silver with a diameter of 1-2 µm appeared on the surface of sample (Figure 9a). In some places of the sintered sample, the whiskers are combin splices up to 40 µm thick (Figure 9b). Scanning electron microscopy of a sample pressed from a synthesized fine-dispersed Ag 2 S powder heated to 893 K and cooled to room temperature ( Figure 9) confirms the change in the phase composition due to the appearance of Ag whiskers. Numerous long whiskers of metallic silver with a diameter of 1-2 µm appeared on the surface of cooled sample (Figure 9a). In some places of the sintered sample, the whiskers are combined into splices up to 40 µm thick (Figure 9b).

Transitions "α-Ag2S-β-Ag2S" and "β-Ag2S-γ-Ag2S" as the First-Order Transformations
The α-Ag2S, β-Ag2S, and γ-Ag2S phases, observed at a temperature of 298-893 K, have a different crystal structure. Therefore, to compare the changes occurring in silver sulfide at heating from 298 to 893 K, we used reduced volumes of the unit cells of these phases. The reduced unit cell volume is the volume of unit cell, Vun.cell, divided by the quantity of Ag2S formula units, z, in the cell, i.e., Vun.cell/z (z = 4 for acanthite α-Ag2S and fcc γ-Ag2S phase, and z = 2 for argentite β-Ag2S). With an increase in temperature from 298 to 893 K, the reduced volume increases and at temperatures of ~420-422 K and ~840-850 K, its abrupt increase is observed. The abrupt changes in the reduced volume are associated with the transformations of acanthite into argentite and argentite into the γ-phase of Ag2S ( Figure 10). Figure 10. Temperature dependence of reduced volume Vun.cell/z of silver sulfide in the region 298-893 K. At ~440-442 K and ~850-860 K, jumps in the reduced volume are observed, which correspond to the first order phase transformations "α-Ag2S-β-Ag2S" and "β-Ag2S-γ-Ag2S", respectively. 3.5. Transitions "α-Ag 2 S-β-Ag 2 S" and "β-Ag 2 S-γ-Ag 2 S" as the First-Order Transformations The α-Ag 2 S, β-Ag 2 S, and γ-Ag 2 S phases, observed at a temperature of 298-893 K, have a different crystal structure. Therefore, to compare the changes occurring in silver sulfide at heating from 298 to 893 K, we used reduced volumes of the unit cells of these phases. The reduced unit cell volume is the volume of unit cell, V un.cell , divided by the quantity of Ag 2 S formula units, z, in the cell, i.e., V un.cell /z (z = 4 for acanthite α-Ag 2 S and fcc γ-Ag 2 S phase, and z = 2 for argentite β-Ag 2 S). With an increase in temperature from 298 to 893 K, the reduced volume increases and at temperatures of~420-422 K and~840-850 K, its abrupt increase is observed. The abrupt changes in the reduced volume are associated with the transformations of acanthite into argentite and argentite into the γ-phase of Ag 2 S ( Figure 10). Nanomaterials 2022, 12, x FOR PEER REVIEW Figure 9. The surface of sample pressed from synthesized fine-dispersed Ag2S powder, hea 893 K and cooled to 298 K: (a) long whiskers of metallic silver; (b) splices (aggregates) of m silver whiskers.

Transitions "α-Ag2S-β-Ag2S" and "β-Ag2S-γ-Ag2S" as the First-Order Transform
The α-Ag2S, β-Ag2S, and γ-Ag2S phases, observed at a temperature of 298have a different crystal structure. Therefore, to compare the changes occurring in sulfide at heating from 298 to 893 K, we used reduced volumes of the unit cells o phases. The reduced unit cell volume is the volume of unit cell, Vun.cell, divided quantity of Ag2S formula units, z, in the cell, i.e., Vun.cell/z (z = 4 for acanthite α-Ag fcc γ-Ag2S phase, and z = 2 for argentite β-Ag2S). With an increase in temperatur 298 to 893 K, the reduced volume increases and at temperatures of ~420-422 ~840-850 K, its abrupt increase is observed. The abrupt changes in the reduced v are associated with the transformations of acanthite into argentite and argentite i γ-phase of Ag2S ( Figure 10). The change in the reduced volume upon the "α-Ag2S acanthite-β-Ag2S arg transition is noticeably less than that during the "β-Ag2S argentite-γ-Ag2S phase sition. The small value of the jump in the reduced volume at the transition " Figure 10. Temperature dependence of reduced volume V un.cell /z of silver sulfide in the region 298-893 K. At~440-442 K and~850-860 K, jumps in the reduced volume are observed, which correspond to the first order phase transformations "α-Ag 2 S-β-Ag 2 S" and "β-Ag 2 S-γ-Ag 2 S", respectively.
The change in the reduced volume upon the "α-Ag 2 S acanthite-β-Ag 2 S argentite" transition is noticeably less than that during the "β-Ag 2 S argentite-γ-Ag 2 S phase" transition. The small value of the jump in the reduced volume at the transition "α-Ag 2 S acanthite-β-Ag 2 S argentite" is due to the following. According to [57], the structure of α-Ag 2 S acanthite is a result of small displacements of S and Ag atoms from positions of bcc lattice of β-Ag 2 S argentite. The "acanthite-argentite" transformation occurs due to the jump of Ag + ion from (e) acanthite site to (j) argentite site [72]. The distance on which Ag + ion is displaced during such jumping is small and amounts to only 0.0425 nm. The energy barrier for the jump of the Ag + ion from the acanthite sites to the argentite sites is from 89 to 410 meV [12,72,73]. During the transition "β-Ag 2 S argentite-γ-Ag 2 S phase", the displacements of atoms, especially sulfur atoms S, are larger than at the transition "acanthite-argentite". As a result, the transition "β-Ag 2 S argentite-γ-Ag 2 S phase" is accompanied by a larger jump in the reduced volume.
The discontinuous change of the reduced volume during the "acanthite α-Ag 2 Sargentite β-Ag 2 S" transition corresponds to the transformation of first-order [61,64,69,74]. Measuring the thermal expansion coefficient α and the heat capacity C p of silver sulfide [37,38] at temperatures from 290 to 970 K also confirms that the "β-Ag 2 S-γ-Ag 2 S" transition is a first-order phase transformation.
An additional theoretical confirmation of the first order of phase transformation "α-Ag 2 S acanthite-β-Ag 2 S argentite" is the symmetry analysis of this transformation. The transformation of argentite into acanthite, which occurs with a decrease in temperature, can be discussed as a transition from random distribution of Ag atoms in cubic argentite to their ordered arrangement in monoclinic acanthite with a simultaneous change in the positions of sulfur atoms. The location of the monoclinic unit cell in basic bcc nonmetallic sublattice of argentite is shown in Figure 11a, with allowance for the ratio of the axes of acanthite and argentite. When the temperature drops below the transition temperature T trans , the S atoms, which equally fill the sites of bcc nonmetallic argentite sublattice, concentrate on the 4 sites of monoclinic nonmetallic sublattice (Figure 11b). anomaterials 2022, 12, x FOR PEER REVIEW the jump of Ag + ion from (e) acanthite site to (j) argentite site [72]. The distance Ag + ion is displaced during such jumping is small and amounts to only 0.0425 energy barrier for the jump of the Ag + ion from the acanthite sites to the argent from 89 to 410 meV [12,72,73]. During the transition "β-Ag2S argentite-γ-Ag the displacements of atoms, especially sulfur atoms S, are larger than at the "acanthite-argentite". As a result, the transition "β-Ag2S argentite-γ-Ag2S ph companied by a larger jump in the reduced volume.
An additional theoretical confirmation of the first order of phase trans "α-Ag2S acanthite-β-Ag2S argentite" is the symmetry analysis of this transf The transformation of argentite into acanthite, which occurs with a decrease i ature, can be discussed as a transition from random distribution of Ag atom argentite to their ordered arrangement in monoclinic acanthite with a sim change in the positions of sulfur atoms. The location of the monoclinic unit ce bcc nonmetallic sublattice of argentite is shown in Figure 11a, with allowance fo of the axes of acanthite and argentite. When the temperature drops below the temperature Ttrans, the S atoms, which equally fill the sites of bcc nonmetallic sublattice, concentrate on the 4 sites of monoclinic nonmetallic sublattice (Figur This makes it possible to discuss the formation of a monoclinic cell as an o S atoms in the bcc sublattice. Four Ag atoms, randomly situated on 54 crysta positions of argentite and being in continuous motion due to small distances these positions, during transition to acanthite find themselves in the sites 4 probability equal to 1. Thus, ordering sulfur atoms in argentite is accompani 4 , k (7) 4 , and k (8) 4 , are the phase transition channels associated with the formation of the simulated monoclinic (space group P2/c) α-Ag 2 S ordered phase ( Figure 12). are given according to [75,76]; b1 = (0 1 0 1), and b3 = (1 1 0) are the structure vectors for reciprocal lattice of the basic bcc 2π/a units (a is the lattice period for the bcc lattice). These six superstructura the simulated monoclinic ordered phase α-Ag2S in the first Brillouin zone of the bcc lattic Thus, the "argentite-acanthite" transformation can be discussed as a disord transition in both (nonmetallic and metallic) sublattices of cubic argentite, compl atomic static displacements. This transformation occurs with a distortion of cu metry of argentite for the Lifshitz star {k9} and the non-Lifshitz star {k4}. The pr the rays of a non-Lifshitz star in the phase transition channel means that the mation "β-Ag2S-α-Ag2S" does not fulfill the Landau group-theoretical crit second-order phase transformations. Therefore, the transformation "β-Ag2Soccurs on the first-order transition mechanism.

High-Resolution Transmission Electron Microscopy
The TEM image of prepared colloid Ag2S nanoparticles is presented in Figu Along with separate Ag2S nanoparticles of almost the same size 5-6 nm, agglomerates −20 nm in size. Agglomerated colloid nanoparticle of silver sulfide 4 , k (7) 4 and k (8) 4 of the simulated monoclinic ordered phase α-Ag 2 S in the first Brillouin zone of the bcc lattice.
Thus, the "argentite-acanthite" transformation can be discussed as a disorder-order transition in both (nonmetallic and metallic) sublattices of cubic argentite, complicated by atomic static displacements. This transformation occurs with a distortion of cubic symmetry of argentite for the Lifshitz star {k 9 } and the non-Lifshitz star {k 4 }. The presence of the rays of a non-Lifshitz star in the phase transition channel means that the transformation "β-Ag 2 S-α-Ag 2 S" does not fulfill the Landau group-theoretical criterion for second-order phase transformations. Therefore, the transformation "β-Ag 2 S-α-Ag 2 S" occurs on the first-order transition mechanism.

High-Resolution Transmission Electron Microscopy
The TEM image of prepared colloid Ag 2 S nanoparticles is presented in Figure 13.  Colloid nanoparticle of α-Ag2S acanthite is shown in Figure 15a. The increased area of the HRTEM image, in which atomic planes are visible, is highlighted with a white square (Figure 15b). The electron diffraction pattern calculated by the Fast Fourier Transform (FFT) of highlighted area of HRTEM image is shown in Figure 15c. To determine the interplanar distance, a mask is created from the FFT image (Figure 15d), which is used to obtain an inverse FFT image of atomic planes and a linear scale of interplanar distances (Figure 15e). Interplanar distances were determined using the DigitalMicrograph software known as the Gatan Microscopy Suite [77]. A detailed description of the determination of interplanar distances using this program [77] is given on the website [78].
The observed distance between ten atomic planes is ~4.504 nm (Figure 15e). The distance between adjacent atomic planes is ~0.450 nm and corresponds to the local ordering of atoms, which leads to the appearance of a superperiodic lattice. A period of a superperiodic lattice is equal to doubled interplanar spacing (−114) of monoclinic acanthite, i.e., 2 × d-114 ≈ 2 × 0.226 nm. Thus, the  Along with separate Ag 2 S nanoparticles of almost the same size 5-6 nm, there are agglomerates −20 nm in size. Agglomerated colloid nanoparticle of silver sulfide at 298 K is presented in Figure 14a. It can be seen that a nanoparticle with a size of~20-50 nm is agglomerated and combines five or six separate nanoparticles. The electron diffraction of the agglomerated nanoparticle confirms the formation of acanthite. The electron diffraction pattern (Figure 14b  Colloid nanoparticle of α-Ag2S acanthite is shown in Figure 15a. The increased area of the HRTEM image, in which atomic planes are visible, is highlighted with a white square (Figure 15b). The electron diffraction pattern calculated by the Fast Fourier Transform (FFT) of highlighted area of HRTEM image is shown in Figure 15c. To determine the interplanar distance, a mask is created from the FFT image (Figure 15d), which Colloid nanoparticle of α-Ag 2 S acanthite is shown in Figure 15a. The increased area of the HRTEM image, in which atomic planes are visible, is highlighted with a white square ( Figure 15b). The electron diffraction pattern calculated by the Fast Fourier Transform (FFT) of highlighted area of HRTEM image is shown in Figure 15c. To determine the interplanar distance, a mask is created from the FFT image (Figure 15d), which is used to obtain an inverse FFT image of atomic planes and a linear scale of interplanar distances (Figure 15e). Interplanar distances were determined using the DigitalMicrograph software known as the Gatan Microscopy Suite [77]. A detailed description of the determination of interplanar distances using this program [77] is given on the website [78].  The observed distance between ten atomic planes is~4.504 nm (Figure 15e). The distance between adjacent atomic planes is~0.450 nm and corresponds to the local ordering of atoms, which leads to the appearance of a superperiodic lattice. A period of a superperiodic lattice is equal to doubled interplanar spacing (-114) of monoclinic acanthite, i.e., 2 × d -114 ≈ 2 × 0.226 nm. Thus, the (-114) P2 1 /c atomic planes of monoclinic α-Ag 2 S are observed in the HRTEM image (Figure 15a).
HRTEM image of a large colloid nanoparticle of α-Ag 2 S acanthite in size~60 nm, combining several individual nanoparticles, and the observed multiple electron diffraction are shown in Figure 16a To separate sets of diffraction reflections corresponding to individual nanoparticles, the experimental interplanar distances dhkl and the angles ϕrefl between (h1k1l1)mon and (h2k2l2)mon reflections were compared with the calculated values of interplanar distances and angles. The interplanar distances and the angles between reflections have been calculated using formulae [6,76]: where i = 1 or 2; a, b, c, β are the parameters of unit cell of fine-dispersed monoclinic silver sulfide (see Table 1).
Among the observed diffraction reflections, one can distinguish a set of reflections

Phase Transformations in Silver Sulfide at Temperature from 298 to 893 K
According to the authors of study [79], structural polymorphism and related phase transformations in sulfide nanocrystals are not yet well understood. The research of silver sulfide undertaken fills the noted gap in relation to nanocrystalline Ag2S.

Phase Transformations in Silver Sulfide at Temperature from 298 to 893 K
According to the authors of study [79], structural polymorphism and related phase transformations in sulfide nanocrystals are not yet well understood. The research of silver sulfide undertaken fills the noted gap in relation to nanocrystalline Ag 2 S.
The results of high-temperature XRD showed that the reversible phase transformations of acanthite to argentite and of argentite to γ-Ag 2 S phase take place in the process of heating Ag 2 S silver sulfide. The first transformation occurs at a temperature of~440-442 K, the second transformation is observed at~850-860 K ( Figure 17). The crystal structures of polymorphic modifications of silver sulfide, especially superionic modifications β-Ag 2 S and γ-Ag 2 S, are rather complex. In the β-Ag2S and γ-Ag2S phases, Ag atoms are randomly located on several positions of different types with occupancy probabilities less than 1. For β-Ag2S argentite, there are 54 positions of (b) and (j) types whose degrees of filling are equal ~0.0974 and ~0.0716. In the γ-Ag2S, Ag atoms are located at 88 positions of (c), (j), and (i) types with probabilities of filing ~0.088, ~0.150, and ~0.027, respectively. Therefore, the exact positions for S atoms and sites of two and three types are shown for the β-Ag2S and γ-Ag2S phases, respectively ( Figure 17). Ag atoms can be located at these sites with the indicated probabilities.
The abrupt (discontinuous) change in the reduced volumes of unit cells during the transitions from acanthite to argentite and from argentite to γ-Ag2S phase indicates that these transitions occur as phase transformations of the first order.
When silver sulfide is heated to a temperature of more than 850 K, the transformation of β-Ag2S argentite into the γ-Ag2S phase is accompanied by the isolation of metallic silver. Therefore, the metallic sublattice of the γ-phase contains a large number of vacant sites, and γ-phase has a nonstoichiometric composition Ag1.7S.

Simulated Arrangements of Ag Atoms in β-Ag2S Argentite
The unit cells of cubic zinc and silver sulfides α-ZnS and β-Ag2S are shown in Figure  S1 (see Supplementary Material).
The low-temperature modification α-ZnS has a cubic (space group m F 3 4 ) structure of zinc blende or sphalerite (type B3) ( Figure S1a) (see Supplementary Material) [80]. The Zn and S atoms occupy the sites of ZnS crystal structure with the probability equaling 1. In contrast to ZnS sphalerite with a fixed arrangement of Zn and S atoms in the unit cell, body-centered cubic (space group m m I 3 ) β-Ag2S argentite is characterized by a statistical distribution of Ag atoms [61].
In β-Ag2S argentite, four Ag atoms are statistically distributed in 54 positions of 6(b) and 48(j) types ( Figure S1b) with degrees of filling ~0.0978 and ~0.0711, respectively [61]. The degrees of filling [61] almost coincide with the found degrees of occupancy (see Table 2). Degrees of filling of 54 positions of argentite silver sublattice are less than 0.1, Figure 17. Change of crystal structure of silver sulfide during reversible phase transformations. The unit cells of monoclinic (space group P2 1 /c) α-Ag 2 S acanthite, bcc (space group Im3m) β-Ag 2 S argentite, and fcc (space group Fm3m ) γ-Ag 2 S phase are presented. The exact positions of S atoms and sites of different types, on which Ag atoms can be located with different probabilities (see Tables 2 and 3), are shown for the β-Ag 2 S argentite and γ-Ag 2 S phase.
In the β-Ag 2 S and γ-Ag 2 S phases, Ag atoms are randomly located on several positions of different types with occupancy probabilities less than 1. For β-Ag 2 S argentite, there are 54 positions of (b) and (j) types whose degrees of filling are equal~0.0974 and~0.0716. In the γ-Ag 2 S, Ag atoms are located at 88 positions of (c), (j), and (i) types with probabilities of filing~0.088,~0.150, and~0.027, respectively. Therefore, the exact positions for S atoms and sites of two and three types are shown for the β-Ag 2 S and γ-Ag 2 S phases, respectively ( Figure 17). Ag atoms can be located at these sites with the indicated probabilities.
The abrupt (discontinuous) change in the reduced volumes of unit cells during the transitions from acanthite to argentite and from argentite to γ-Ag 2 S phase indicates that these transitions occur as phase transformations of the first order.
When silver sulfide is heated to a temperature of more than 850 K, the transformation of β-Ag 2 S argentite into the γ-Ag 2 S phase is accompanied by the isolation of metallic silver. Therefore, the metallic sublattice of the γ-phase contains a large number of vacant sites, and γ-phase has a nonstoichiometric composition Ag 1.7 S.

Simulated Arrangements of Ag Atoms in β-Ag 2 S Argentite
The unit cells of cubic zinc and silver sulfides α-ZnS and β-Ag 2 S are shown in Figure S1 (see Supplementary Material).
The low-temperature modification α-ZnS has a cubic (space group F43m) structure of zinc blende or sphalerite (type B3) ( Figure S1a) (see Supplementary Material) [80]. The Zn and S atoms occupy the sites of ZnS crystal structure with the probability equaling 1. In contrast to ZnS sphalerite with a fixed arrangement of Zn and S atoms in the unit cell, body-centered cubic (space group Im3m) β-Ag 2 S argentite is characterized by a statistical distribution of Ag atoms [61].
In β-Ag 2 S argentite, four Ag atoms are statistically distributed in 54 positions of 6(b) and 48(j) types ( Figure S1b) with degrees of filling~0.0978 and~0.0711, respectively [61]. The degrees of filling [61] almost coincide with the found degrees of occupancy (see Table 2). Degrees of filling of 54 positions of argentite silver sublattice are less than 0.1, therefore only sets of 6(b) and 48(j) positions on which silver ions can be located, are shown on Figure  S1b (Supplementary Material). The sites of silver sublattice are so close to each other that the presence of Ag + ion in one of these sites makes the filling of the nearest site by other silver ions impossible because the diameter of Ag + ion is larger than the distance between these sites [81].
To simulate the interface between zinc and silver sulfides, the following should be taken into account. According to [40,53,57,61,72,82], the structure of monoclinic α-Ag 2 S acanthite can be considered as result of distortion of body-centered cubic (bcc) sublattice of sulfur atoms S in the structure of cubic β-Ag 2 S argentite. At a temperature below 450 K, cubic β-Ag 2 S argentite transforms in monoclinic α-Ag 2 S acanthite. This transformation is accompanied by a distortion of the bcc sublattice of S atoms to the monoclinic sublattice. Ag atoms that statistically distributed over the sites 6(b) and 48(j) of the bcc structure of argentite concentrate at the sites of the monoclinic structure of acanthite and occupy them with probability close to one. Thus, at the first stage of simulating the interface, one can consider the interface between cubic zinc sulfide and cubic silver sulfide.
In reality, the distances between Ag atoms in cubic argentite should be larger than the doubled atomic radius of silver, i.e., than~2.52 Å (0.252 nm). In other words, four Ag atoms in cubic silver sulfide can be located at a distance of at least 2.52 Å from each other. Taking this into account and 48 + 6 = 54 crystallographic positions possible for 4 Ag atoms, there are C 54 4 = 316,251 options for the arrangement of silver atoms in a cubic lattice. In fact, the number of such arrangements is even larger, since it is also necessary to take into account positions in which any coordinate equal to 0 changes to 1. There are 48 + 24 = 72 positions of (j) type and 6 + 12 = 18 positions of (b) type, i.e., 90 positions for which the number of variants of the arrangement of Ag atoms at arbitrary four positions is C 90 4 = 2,551,190. The coordinates of all positions for the arrangement of Ag and S atoms in the cubic (space group Im3m) unit cell of β-Ag 2 S argentite are given in Table S1 (see Supplementary Material file). Among these arrangements, of interest are those in which the minimum distance between silver atoms exceeds 2.52 Å. The number of such arrangements is 596,898. Most of these options are equivalent and can be obtained from one another by rotating. The number of nonequivalent variants of the arrangement of 4 Ag atoms is much less and is equal to 13,116. Within the boundaries of the cubic unit cell of argentite, four Ag atoms can be located at four or six positions, depending on the coordinates of these positions. In most cases, silver atoms will not be in the same plane, but there are also such variants of arrangement when all or part of Ag atoms will be in the same plane, or all Ag atoms are in two planes with a certain angle between them.
Such  (Figure 18f). The arrangement of some Ag atoms in the same plane (z = 0.5) at three (b) positions and one (j) position is impossible, since in this case the distance between a pair of silver atoms in this plane is less than~2.52 Å (0.252 nm). Placing Ag atoms in one plane (z = 0.5) only at (b) positions is also impossible, since in this case the rest of Ag atoms are located at positions (b) and/or (j), and the distances between some pairs of silver atoms turn out to be less than~2.52 Å (0.252 nm). As an example, a few simulated possible arrangements of Ag atoms at different crystallographic positions of the crystal lattice of cubic β-Ag 2 S argentite are shown in Figure 18.  Figure 19. The XRD pattern of the (Ag2S)0.25(ZnS) heteronanostructure with the highest content of silver sulfide contains diffraction reflections of monoclinic silver sulfide α-Ag2S (see Section 3.1) [54,57] and cubic sphalerite ZnS [83]. Quantitative analysis of XRD patterns of (Ag2S)0.1(ZnS) and (Ag2S)0.025(ZnS) heteronanostructures with a high content of zinc sulfide and their comparison with the data of Section 3.2 and [42,61,66,83] showed the presence of strong diffraction reflections of cubic sphalerite ZnS and very weak reflections of cubic argentite β-Ag2S (Figure 19). The intensity of diffraction reflections of silver sulfide is lower than the intensity of reflections of zinc sulfide due to the low relative content of Ag2S in these heteronanostructures. The diffraction reflections of Ag2S and, especially, ZnS in heteronanostructures (Ag2S)0.1(ZnS) and (Ag2S)0.025(ZnS) are strongly broadened. Broadening of all the reflections of these heteronanostructures is caused by the small size of particles (~9 and ~10 nm for (Ag2S)0.1(ZnS) and (Ag2S)0.025(ZnS) heteronanostructures), and the presence of microstrains of crystal lattice owing to its deformation distortions.
Earlier, the formation of Ag2S/ZnS heteronanostructures was confirmed in study [84]. According to [84], the central portion of heteronanoparticles ~9-10 nm in size is formed by silver sulfide, and the surface of heteronanoparticles is coated with smaller nanoparticles of cubic zinc sulfide ZnS.  Figure 19. The XRD pattern of the (Ag 2 S) 0.25 (ZnS) heteronanostructure with the highest content of silver sulfide contains diffraction reflections of monoclinic silver sulfide α-Ag 2 S (see Section 3.1) [54,57] and cubic sphalerite ZnS [83]. Quantitative analysis of XRD patterns of (Ag 2 S) 0.1( ZnS) and (Ag 2 S) 0.025 (ZnS) heteronanostructures with a high content of zinc sulfide and their comparison with the data of Section 3.2 and [42,61,66,83] showed the presence of strong diffraction reflections of cubic sphalerite ZnS and very weak reflections of cubic argentite β-Ag 2 S (Figure 19). The intensity of diffraction reflections of silver sulfide is lower than the intensity of reflections of zinc sulfide due to the low relative content of Ag 2 S in these heteronanostructures.  Figure 20 shows the electron diffraction patterns obtained by FFT of areas (1)-(4). The set of diffraction reflections and interplanar distances ~0.311 nm (see Figure 20) for areas (1)-(3) correspond to the interplanar distance between the atomic planes (111) of cubic (space group m F 3 4 ) ZnS. The interplanar distance ~0.246 nm of area (4) coincides with the distance between the atomic planes (200) of silver sulfide with a cubic β-Ag2S argentite structure. Thus, the surface of the core from cubic silver sulfide is covered with a layer of nanoparticles of cubic zinc sulfide ZnS. EDS (EDX) data confirm the presence of Zn, Ag and S in this heteronanostructure (see Figure S7, Supplementary Material). The size of the heteronanoparticle is close to the size estimated from diffraction data. HRTEM image of (Ag2S)0.025(ZnS) heteronanostructure and its EDS (EDX) data are shown in Figure S8 and S9, respectively (see Supplementary Material). The formation of an interface in the heteronanostructure leads to a change in the band gap Eg compared to the band gaps Eg of individual sulfides. Such a change in Eg from 3.56 eV for (Ag2S)0.025(ZnS) to 2.06 eV for (Ag2S)0.1(ZnS) was found in synthesized Ag2S/ZnS heteronanostructures with nanoparticles of ~9-10 nm in size [19]. The observed change in the band gap Eg additionally confirms the formation in these heteronanostructures of interfaces with minimal strain distortion.
With allowance for the experimental diffraction and TEM data, it can be assumed that a large amount of cubic zinc sulfide (more than 72-73 wt.% ZnS or more than 87 Figure 19. XRD patterns of core-shell Ag 2 S@ZnS (Ag 2 S/ZnS) heteronanostructures: short black, middle red, and long blue ticks indicate reflections from monoclinic (space group P2 1 /c) α-Ag2S acanthite, cubic (space group Im3m) β-Ag 2 S argentite, and cubic (space group F43m ) ZnS sphalerite, respectively.
Earlier, the formation of Ag 2 S/ZnS heteronanostructures was confirmed in study [84]. According to [84], the central portion of heteronanoparticles~9-10 nm in size is formed by silver sulfide, and the surface of heteronanoparticles is coated with smaller nanoparticles of cubic zinc sulfide ZnS.
HRTEM confirms the formation of Ag 2 S/ZnS heteronanostructures. HRTEM image (Figure 20) confirms the formation of (Ag 2 S) 0.1 (ZnS) heteronanostructure. The dark central part of the heteronanoparticle (selected area (4)) is surrounded by smaller nanoparticles; some of them are selected in areas (1), (2), and (3). The bottom row of Figure 20 shows the electron diffraction patterns obtained by FFT of areas (1)-(4). The set of diffraction reflections and interplanar distances~0.311 nm (see Figure 20) for areas (1)-(3) correspond to the interplanar distance between the atomic planes (111) of cubic (space group F43m) ZnS. The interplanar distance~0.246 nm of area (4) coincides with the distance between the atomic planes (200) of silver sulfide with a cubic β-Ag 2 S argentite structure. Thus, the surface of the core from cubic silver sulfide is covered with a layer of nanoparticles of cubic zinc sulfide ZnS. EDS (EDX) data confirm the presence of Zn, Ag and S in this heteronanostructure (see Figure S7, Supplementary Material). The size of the heteronanoparticle is close to the size estimated from diffraction data. HRTEM image of (Ag 2 S) 0.025 (ZnS) heteronanostructure and its EDS (EDX) data are shown in Figures S8  and S9, respectively (see Supplementary Material). The formation of an interface in the heteronanostructure leads to a change in the band gap E g compared to the band gaps E g of individual sulfides. Such a change in E g from 3.56 eV for (Ag 2 S) 0.025 (ZnS) to 2.06 eV for (Ag 2 S) 0.1 (ZnS) was found in synthesized Ag 2 S/ZnS heteronanostructures with nanoparticles of~9-10 nm in size [19]. The observed change in the band gap E g additionally confirms the formation in these heteronanostructures of interfaces with minimal strain distortion.  β-Ag2S silver sulfide.

Elastic Properties of Cubic Silver and Zinc Sulfides
The elastic stiffness constants c11, c12, and c44 of cubic β-Ag2S argentite at 470 K and the temperature dependences dc11/dT, dc12/dT, and dc44/dT of these stiffness constants were found in study [40]. According to approximating calculation with allowance for these data, the elastic constants c11, c12, and c44 of cubic β-Ag2S at 300 K are equal to 99.4, 7.6, and 19.0 GPa, respectively.
For cubic (space group m F 3 4 ) ZnS with a B3 type structure, the data [23,27] are the closest to the elastic properties of cubic Ag2S. According to [27], the elastic stiffness constants of cubic ZnS sphalerite at T = 0 K are c11 = 96.9, c12 = 48.3, and c44 = 55.8 GPa.
The values of elastic stiffness constants cij of zinc sulfide ZnS with cubic (space group m F 3 4 ) sphalerite structure at a temperature of 300 K were estimated using the temperature dependences of the isothermal bulk modulus B presented in studies [31,32]. According to [31], the slope dB/dT is equal to −0.0109 GPa K −1 . As a first approximation, we will assume that the relative decrease of elastic stiffness constant c11 with an increase in temperature from 0 to 300 K is the same as the decrease in the bulk modulus B. The elastic stiffness constants cij of ZnS at a temperature of 300 K can be represented as cij(T) = cij(0) + Tdcij/dT. According to estimation performed for ZnS, the values of dc11/dT, dc12/dT, and dc44/dT are equal to −0.0109, −0.0057, and −0.022 GPa K −1 , respectively. The elastic stiffness constants of cubic ZnS sphalerite at a temperature of 300 K are c11 = 93.6, c12 = 46.6, and c44 = 49.2 GPa. The composition of (Ag 2 S) y (ZnS) heteronanostructures can be represented as (Ag 2 S) 1-x (ZnS) x , where x = 1/(1 + y). The obtained experimental diffraction data and TEM results show that (Ag 2 S) y (ZnS) heteronanostructures with y > 0.15 (or (Ag 2 S) 1-x (ZnS) x with x < 0.87), consisting of less than 72 wt. % of cubic zinc sulfide, contain monoclinic silver sulfide. Heteronanostructures (Ag 2 S) y (ZnS) with y < 0.15 (or (Ag 2 S) 1-x (ZnS) x with x > 0.87), consist of more than 72 wt. % of cubic zinc sulfide, and contain cubic silver sulfide.
With allowance for the experimental diffraction and TEM data, it can be assumed that a large amount of cubic zinc sulfide (more than 72-73 wt.% ZnS or more than 87 mol.% ZnS) stabilizes the cubic structure of β-Ag 2 S argentite at 300 K during the deposition of Ag 2 S/ZnS heteronanostructures from colloid solutions.

Elastic Properties of Cubic Silver and Zinc Sulfides
The elastic stiffness constants c 11 , c 12 , and c 44 of cubic β-Ag 2 S argentite at 470 K and the temperature dependences dc 11 /dT, dc 12 /dT, and dc 44 /dT of these stiffness constants were found in study [40]. According to approximating calculation with allowance for these data, the elastic constants c 11 , c 12 , and c 44 of cubic β-Ag 2 S at 300 K are equal to 99.4, 7.6, and 19.0 GPa, respectively.
For cubic (space group F43m) ZnS with a B3 type structure, the data [23,27] are the closest to the elastic properties of cubic Ag 2 S. According to [27], the elastic stiffness constants of cubic ZnS sphalerite at T = 0 K are c 11 = 96.9, c 12 = 48.3, and c 44 = 55.8 GPa.
The values of elastic stiffness constants c ij of zinc sulfide ZnS with cubic (space group F43m) sphalerite structure at a temperature of 300 K were estimated using the temperature dependences of the isothermal bulk modulus B presented in studies [31,32].
According to [31], the slope dB/dT is equal to −0.0109 GPa K −1 . As a first approximation, we will assume that the relative decrease of elastic stiffness constant c 11 with an increase in temperature from 0 to 300 K is the same as the decrease in the bulk modulus B. The elastic stiffness constants c ij of ZnS at a temperature of 300 K can be represented as c ij (T) = c ij (0) + Tdc ij /dT. According to estimation performed for ZnS, the values of dc 11 /dT, dc 12 /dT, and dc 44 /dT are equal to −0.0109, −0.0057, and −0.022 GPa K −1 , respectively. The elastic stiffness constants of cubic ZnS sphalerite at a temperature of 300 K are c 11 = 93.6, c 12 = 46.6, and c 44 = 49.2 GPa.

Anisotropy of Elastic Properties
According to [40,48,85,86], the values of the Young's modulus E, the shear modulus G, and the Poisson's ratio µ of cubic monocrystals depend a direction [hkl] through parameter Γ = (h 2 k 2 + h 2 l 2 + k 2 l 2 )/(h 2 + k 2 + l 2 ) as The bulk modulus B of cubic crystals equals B = 1/[3(s 11 + 2s 12 )]. The distributions of elastic characteristics of monocrystalline particles of cubic ZnS sphalerite and cubic β-Ag 2 S argentite from the [hkl] direction were calculated using the obtained data on s 11 , s 12 , and s 44 of these sulfides. The dependences of the elastic properties of cubic ZnS and cubic β-Ag 2 S from the [hkl] direction are shown in Figure S10 (Supplementary Material).

Anisotropy of Elastic Properties
According to [40,48,85,86], the values of the Young's modulus E, the shear modulus G, and the Poisson's ratio μ of cubic monocrystals depend a direction [hkl] through pa- 2 ) 2 ( 1 12 11 The bulk modulus B of cubic crystals equals B = 1/[3(s11 + 2s12)]. The distributions of elastic characteristics of monocrystalline particles of cubic ZnS sphalerite and cubic β-Ag2S argentite from the [hkl] direction were calculated using the obtained data on s11, s12, and s44 of these sulfides. The dependences of the elastic properties of cubic ZnS and cubic β-Ag2S from the [hkl] direction are shown in Figure S10 (Supplementary Material).
The spatial distributions of the Young's moduli Ehkl and shear modulus Ghkl for cubic ZnS sphalerite and cubic β-Ag2S argentite at 300 K are presented in Figures 21 and 22.

Interface between Cubic ZnS and Ag2S with Simulation of Ag Atoms Arrangement
According to the Hartman-Perdok theory [50], for the considered cubic zinc and silver sulfides, the (111) planes should coincide at the interface. The average shear moduli Ghkl of sphalerite ZnS and argentite β-Ag2S in the (111) planes are ~41.6 and ~21.2 GPa, respectively. In the case of the formation of an interface along the (111) planes of sphalerite and argentite, the difference in their shear moduli Ghkl is very large, and the elastic strain energy Estr of the (Ag2S)0.09(ZnS)0.91 heteronanostructure is large and amounts to 9680 J⋅mol −1 [49]. Therefore, the morphological approach [50] used in study [49], according to which the (111) planes of sphalerite and argentite can be combined at the interface in the Ag2S/ZnS heteronanostructure, gives an incorrect result.
In Section 4 and Supplementary Material file, it is shown that among the nonequivalent variants of the arrangement of Ag atoms with distances between them not less than ~2.52 Å (0.252 nm), there are the arrangements of silver atoms at four crystallographic positions located in one plane of the unit cell of cubic β-Ag2S. The crystallographic positions of these arrangements have the same coordinate z ≡ l, and the z value (taking into The spatial distribution of the Young's modulus E hkl of cubic ZnS sphalerite at 300 K is shown in Figure 21. According to the E hkl surface visualization, the Young's modulus E hkl of cubic ZnS has the maximal value~116. 6 (Figure 22).

Interface between Cubic ZnS and Ag 2 S with Simulation of Ag Atoms Arrangement
According to the Hartman-Perdok theory [50], for the considered cubic zinc and silver sulfides, the (111) planes should coincide at the interface. The average shear moduli G hkl of sphalerite ZnS and argentite β-Ag 2 S in the (111) planes are~41.6 and~21.2 GPa, respectively. In the case of the formation of an interface along the (111) planes of sphalerite and argentite, the difference in their shear moduli G hkl is very large, and the elastic strain energy E str of the (A g 2S) 0.09 (ZnS) 0.91 heteronanostructure is large and amounts to 9680 J·mol −1 [49]. Therefore, the morphological approach [50] used in study [49], according to which the (111) planes of sphalerite and argentite can be combined at the interface in the Ag 2 S/ZnS heteronanostructure, gives an incorrect result.
In Section 4 and Supplementary Material file, it is shown that among the nonequivalent variants of the arrangement of Ag atoms with distances between them not less than~2.52 Å (0.252 nm), there are the arrangements of silver atoms at four crystallographic positions located in one plane of the unit cell of cubic β-Ag 2 S. The crystallographic positions of these arrangements have the same coordinate z ≡ l, and the z value (taking into account coordinates of Ag atoms (see Table S1, Supplementary Material)) for such arrangements is 0.1694, 0.3306, 0.4123, 0.5, 0.6694, or 0.8306. The distances between the nearest neighboring Zn atoms in the (110) plane of ZnS sphalerite are 0.383 nm. The distances between the nearest neighboring Ag atoms in the (11l) planes with l = const of β-Ag 2 S argentite depend on the value of l and are equal to 0.363 nm at l = 0.4213. At l = 0.5, interatomic Ag-Ag distances are 0.364 and 0.406 nm. At l = 0.6694, interatomic Ag-Ag distances are equal to 0.349 nm. Thus, the distances between the nearest neighboring Zn atoms in the (110) plane of ZnS sphalerite, on the one hand, and between the nearest neighboring Ag atoms in the (11l) planes with l = const of β-Ag 2 S argentite, on the other hand, differ by no more than~5.2% at l = 0.4213, by~4.7-6.0% at l = 0.5, and by~8.9% at l = 0.6694. Since the faces (110) of sphalerite and (11l) with l = constant of argentite are characterized by rather close distances between metal atoms, the formation by these faces of the interfaces in Ag 2 S/ZnS heteronanostructures is quite possible. An even more important parameter determining the possibility of forming an interface in Ag 2 S/ZnS heteronanostructure is the smallest strain distortion at the interface between two sulfides.
The distributions of the shear modulus G hkl of argentite in the (hkl) planes with l = 0.1694, 0.3306, 0.4123, 0.5, 0.6694, or 0.8306 calculated in study [84] are shown in Figure 23. The maximum shear modulus G hkl in the (hkl) planes with l = 0.1694, 0.3306, 0.4123, 0.5, 0.6694, or 0.8306 takes in the [±10 l] and [0±1 l] directions, and the minimum shear modulus G hkl of β-Ag 2 S argentite is observed in four equivalent directions [±1±1 l]. As the value of l increases, the shear modulus of β-Ag 2 S generally decreases. For comparison, a calculated distribution of the shear modulus G hk0 of cubic ZnS sphalerite is shown in Figure 23.  Table S1, Supplementary Material)) for such arrangements is 0.1694, 0.3306, 0.4123, 0.5, 0.6694, or 0.8306. The distances between the nearest neighboring Zn atoms in the (110) plane of ZnS sphalerite are 0.383 nm. The distances between the nearest neighboring Ag atoms in the (11l) planes with l = const of β-Ag2S argentite depend on the value of l and are equal to 0.363 nm at l = 0.4213. At l = 0.5, interatomic Ag-Ag distances are 0.364 and 0.406 nm. At l = 0.6694, interatomic Ag-Ag distances are equal to 0.349 nm. Thus, the distances between the nearest neighboring Zn atoms in the (110) plane of ZnS sphalerite, on the one hand, and between the nearest neighboring Ag atoms in the (11l) planes with l = const of β-Ag2S argentite, on the other hand, differ by no more than ~5.2% at l = 0.4213, by ~4.7-6.0% at l = 0.5, and by ~8.9% at l = 0.6694. Since the faces (110) of sphalerite and (11l) with l = constant of argentite are characterized by rather close distances between metal atoms, the formation by these faces of the interfaces in Ag2S/ZnS heteronanostructures is quite possible. An even more important parameter determining the possibility of forming an interface in Ag2S/ZnS heteronanostructure is the smallest strain distortion at the interface between two sulfides. The distributions of the shear modulus Ghkl of argentite in the (hkl) planes with l = 0.1694, 0.3306, 0.4123, 0.5, 0.6694, or 0.8306 calculated in study [84] are shown in Figure  23   The smallest strain distortions at the interface between silver and zinc sulfides in Ag 2 S/ZnS heterostructures will be observed with a minimum difference in shear moduli of β-Ag 2 S and ZnS sulfides. In cubic β-Ag 2 S argentite, there are 13,116 nonequivalent variants of the arrangement of 4 Ag atoms (see Section 4). For the formation of Ag 2 S/ZnS heterostructures, such arrangements of Ag atoms in β-Ag 2 S argentite are most favorable when all four Ag atoms are in one plane. For the binary or pseudobinary systems, the lowest elastic strain energy is observed when the difference in shear moduli of the system's components is minimal [87]. Thus, the energetically most favorable will be the Ag 2 S/ZnS heterostructures, with the minimum difference in the shear moduli of the β-Ag 2 S and ZnS sulfides.
In accordance with study [87], the elastic strain energy E str as the function of the (Ag 2 S) 1-x (ZnS) x composition can be represented as where G Ag 2 S and G ZnS are the shear moduli of Ag 2 S and ZnS sulfides; V Ag 2 S = 3.43 × 10 −5 m 3 ·mol −1 and V ZnS = 2.39 × 10 −5 m 3 ·mol −1 are the molar volumes of Ag 2 S and ZnS sulfides; x is the relative content of zinc sulfide ZnS in the (Ag 2 S) 1-x (ZnS) x heteronanostructure. For (Ag 2 S) 0.1 (ZnS) and (Ag 2 S) 0.025 (ZnS) heteronanostructures, the x value is~0.91 and~0.95, respectively. The shear moduli that correspond to (hk 0.4123) and (hk 0.5) planes of argentite β-Ag 2 S is closest to the shear modulus G hk0 of sphalerite ZnS (see Figure 22). The average shear modulus G hk0 of ZnS sphalerite is~30.5 GPa, and the average shear modulus in the (hk 0.4123) and (hk 0.5) planes of β-Ag 2 S argentite is~24.9 and~23.4 GPa, respectively. Thus, the Ag 2 S/ZnS heterostructures, in which the interface is formed by the (hk0) ≡ (110) plane of ZnS sphalerite and the (hk 0.4123) ≡ (1 1 0.4123) plane of β-Ag 2 S argentite, are the most energetically favorable.
In the first approximation, the interfacial energy γ can be estimated as γ = E str /S, where S is the surface area of the interface. The surface area of the interface can be found as the average value S = (V ZnS /D ZnS + V Ag 2 S /D Ag 2 S )/2 where D ZnS and D Ag 2 S is the size of ZnS and Ag 2 S nanoparticles. According to TEM data, the size of nanoparticles of zinc and silver sulfides is the same approximately, and equal to~10 nm (see Figure 20 and Figure S9), therefore S = 2910 m 2 ·mol −1 . The energies γ of the Ag 2 S(1 1 0.4123)/ZnS(110) interface for the (Ag 2 S) 0.1 (ZnS) and (Ag 2 S) 0.025 (ZnS) heteronanostructures at this S value are~2.5 and 1.5 J·m −2 , respectively. Knowledge of the elastic stiffness constants of a monocrystal allows one to find the elastic characteristics of a polycrystal using the Voigt-Reuss-Hill method [88].
The bulk B and shear G moduli are related to the elastic stiffness constants c ij and the elastic compliance constants s ij (see Section S2, Supplementary Material). For any crystal system, the bulk and shear moduli are given by the known equations of Voigt [89] and Reuss [90] where B V and B R are the upper and lower limits of bulk modulus B, and G V and G R are the upper and lower limits of shear modulus G, respectively.
The Voigt-Reuss-Hill averages of the bulk and shear moduli, as well as estimated isotropic for the Young's modulus E and the Poisson's ratio µ for cubic β-Ag 2 S and ZnS sulfides, are given in Table 5. Universal criterion A U = (5G V /G R +B V /B R -6) is used to quantify the anisotropy of elastic properties of crystals [91]. The value A U = 0 shows the isotropy of crystal. The higher the value of A U is, the more intense the elastic anisotropy is. We used the criterion A U for cubic β-Ag 2 S argentite and ZnS sphalerite to evaluate their elastic anisotropy. For the temperature 300 K, the anisotropy criterions A U are 1.008 and 0.687 for cubic β-Ag 2 S argentite and ZnS sphalerite (Table 5). Thus, polycrystalline β-Ag 2 S argentite and ZnS sphalerite possess the anisotropy of elastic properties. Table 5. Elastic stiffness constants c ij (GPa), elastic compliance constants s ij (10 −12 Pa −1 ), bulk B, shear G, and the Young's E moduli (GPa), the Poisson's ratios µ and the universal anisotropy criterion A U for cubic β-Ag 2 S (space group Im3m) and ZnS (space group F43m ) sulfides at a temperature of 300 K.

Conclusions
Polymorphism, being the ability of certain substances to exist in states with different crystal structures, is quite common among sulfides. Silver sulfide Ag 2 S has three polymorphic modifications. Literature data on polymorphic modifications of Ag 2 S refer mainly to bulk (coarse-crystalline) silver sulfide.
In this work, we have systematically studied the structure of polymorphic modifications of nanocrystalline silver sulfide with a particle 90 nm in size and less. The structure of the superionic β-Ag 2 S and γ-Ag 2 S phases has been studied in particular detail. It is shown that the interstitial distances between silver ions in the β-Ag 2 S and γ-Ag 2 S phases are noticeably smaller than the diameter of the Ag + ion, as a result of which the probabilities of filling the sites of the metallic sublattices of these phases with Ag atoms are very small. As a result, silver atoms are in permanent motion over all the crystallographic positions, which is possible for them. Permanent motion of silver atoms creates the superionic conductivity of the β-Ag 2 S and γ-Ag 2 S phases, and offers the stability of these phases.
In this work, we succeeded in obtaining a core/shell heteronanostructure in which the core of cubic superionic β-Ag 2 S argentite is covered with a layer of cubic zinc sulfide ZnS nanoparticles. Ag 2 S/ZnS heteronanostructures were prepared via a facile two-stage hydrochemical co-deposition of Ag 2 S and ZnS from aqueous solutions. Deposition of ZnS on a surface of Ag 2 S nanoparticles has led to the formation of Ag 2 S/ZnS heteronanostructures.
Experimental diffraction and TEM data allow supposing that a large amount of cubic zinc sulfide (more than 87 mol.% ZnS) stabilizes the cubic structure of β-Ag 2 S argentite at 300 K during the deposition of Ag 2 S/ZnS heteronanostructures from colloid solutions.
The physically possible variants of the arrangement of silver atoms with a minimum interatomic Ag-Ag distance of more than 2.52 Å on fixed crystallographic positions of cubic argentite are determined.
The elastic stiffness constants c 11 , c 12 , and c 44 of cubic β-Ag 2 S argentite and ZnS sphalerite at a temperature of 300 K are estimated.
It is shown that the formation of Ag 2 S/ZnS heteronanostructures, in which the interface is formed by the (hk0) plane of ZnS sphalerite and the (hk 0.4123) plane of β-Ag 2 S argentite, is the most energetically favorable. The smallest strain distortions at this interface are observed. The interfacial energy of the Ag 2 S/ZnS heteronanostructures is estimated.
Calculation of universal criterion of the anisotropy of elastic properties of cubic β-Ag 2 S argentite and ZnS sphalerite shows that the polycrystalline cubic silver and zinc sulfides are elastically anisotropic.

Supplementary Materials:
The following are available online at https://www.mdpi.com/article/ 10.3390/nano12101668/s1, Figure S1: (a) The cubic (space group F43m) unit cell of ZnS sphalerite: (•) metal sites 4(a) occupied by Zn atoms, (•) nonmetal sites 4(c) occupied by S atoms; (b) the cubic (space group Im3m) unit cell of β-Ag 2 S argentite: (•) nonmetal sites 2(a) occupied by S atoms, (*) sites 6(b) occupied by Ag atoms with probability~0.0978, and (X) sites 48(j) filled by Ag atoms with occupation degree~0.0711. Table S1: Coordinates of all sites for Ag and S atoms in the unit cell of cubic (space group No. 229-Im3m) silver sulfide with β-Ag 2 S argentite-type structure. Figure  S2: Arrangement of four Ag atoms in one plane (z = 0.5) at one (b) type position and three (j) type positions of the unit cell of a cubic (space group Im3m) silver sulfide with argentite β-Ag 2 S structure. Figure S3: Two variants of arrangement of four Ag atoms in one plane (z = 0.8306 or 0.1694) at four (j) type positions of the unit cell of a cubic (space group Im3m) silver sulfide with argentite β-Ag 2 S structure. Figure S4: Arrangement of four Ag atoms in one plane (z = 0.3306) at (j) type positions of the unit cell of a cubic (space group Im3m) silver sulfide with argentite β-Ag 2 S structure. Figure  S5: Arrangement of four Ag atoms in one plane (z = 0.6694) at (j) type positions of the unit cell of a cubic (space group Im3m) silver sulfide with argentite β-Ag 2 S structure. Figure S6: Arrangement of four Ag atoms in one plane (z = 0.4123) at (j) type positions of the unit cell of a cubic (space group Im3m) silver sulfide with argentite β-Ag 2 S structure. Figure S7: The EDX spectrum of (Ag 2 S) 0.1 (ZnS) heteronanostructure. The contents of Zn, Ag and S f are equal to 43.5 ± 0.2, 8.7 ± 0.1 and 47.8 ± 0.2 at.% (or 53.4 ± 0.3, 17.6 ± 0.2 and 28.7 ± 0.2 wt. %), respectively. Figure S8: HRTEM image of (Ag 2 S) 0.025 (ZnS) heteronanostructure. Selected areas (1)-(4) correspond to cubic ZnS sphalerite, and selected area (5) corresponds to cubic silver sulfide with β-Ag 2 S argentite structure. Figure S9: The EDX spectrum of (Ag 2 S) 0.025 (ZnS) heteronanostructure. The contents of Zn, Ag and S are equal to 48.2 ± 0.3, 2.4 ± 0.1 and 49.4 ± 0.3 at.% (or 63.1 ± 0.3, 5.2 ± 0.1 and 31.7 ± 0.2 wt. %), respectively. Figure S10: The elastic characteristics of monocrystalline ZnS and β-Ag 2 S particles as a functions of the crystallographic direction at 300 K: (a) the Young modulus E 0kl , (b) the shear modulus G 0kl , (c) the bulk modulus B 0kl , and (d) the Poisson's ratio µ 0kl in (100) plane of ZnS sphalerite; (e) the Young modulus E 0kl , (f) shear modulus G 0kl , (g) bulk modulus B 0kl , and (h) the Poisson's ratio µ 0kl in (100) plane of β-Ag 2 S argentite. Distributions of E hkl , G hkl , B hkl , and µ hkl in (010) and (001) planes have the same shape. Section S2: The bulk B and shear G moduli as functions of the elastic stiffness constants c ij and the elastic compliance constants s ij .
Author Contributions: A.A.V. proposed the main idea and methodology of research and supervised work; S.I.S. carried out the synthesis of silver sulfide and Ag 2 S/ZnS heteronanostructures, preparation of investigated samples, and the data analysis; and A.I.G. wrote the manuscript with contribution from all other co-authors. All authors have read and agreed to the published version of the manuscript.
Funding: The research of Ag 2 S phase transformations has been supported by the Russian Science Foundation (grant 19-73-20012), and the investigation of interface in Ag 2 S/ZnS heteronanostructure has been supported by the Russian Science Foundation (grant 19-79-10101), respectively, through the Institute of Solid State Chemistry of the Ural Branch of the RAS.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.