Effect of small size of particles on thermal expansion and heat capacity of Ag2S silver sulfide

Thermochimica Acta 660 (2018) 1–10

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Effect of small size of particles on thermal expansion and heat capacity of Ag2S silver sulfide A.I. Gusev, S.I. Sadovnikov

T

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Institute of Solid State Chemistry, Ural Branch of the Russian Academy of Sciences, Ekaterinburg 620990, Russia

A R T I C L E I N F O

A B S T R A C T

Keywords: Silver sulfide Nanocrystalline and coarse-crystalline powders Thermal expansion Heat capacity Phase transformations

The thermal expansion and heat capacity of coarse-crystalline (bulk) and nanocrystalline silver sulfide Ag2S have been studied by dilatometry and differential scanning calorimetry methods in the temperature interval from 290 to 970 K. The thermal expansion coefficient and heat capacity of nanocrystalline silver sulfide in the examined temperature range are larger than the same properties of coarse-crystalline sulfide. It is shown that these differences are due to a small particle size which leads to the restriction of the phonon spectrum on the side of low and high frequencies. It is established that the acanthite α-Ag2S to argentite β-Ag2S and argentite β-Ag2S to γAg2S phase transformations are the first-order phase transitions, and the temperatures and enthalpies of these transformations have been determined.

1. Introduction The nanocrystalline silver sulfide Ag2S attracts recently much attention. This is due to the fact that the particle size reduction to nanosized scale can appreciably change the properties of solid-phase substances, especially semiconductors [1]. The semiconducting nanocrystals and nano-structured films of silver sulfide are used in optoelectronics, infrared equipment and energetic. The application of silver sulfide in Ag2S/Ag heteronanostructures, which can work as resistive switches and nonvolatile memory devices, holds much promise [2–7]. The operation of resistive switches based on Ag2S/Ag heteronanostructures is due to the reversible phase transformation of insulating semiconducting acanthite α-Ag2S into argentite β-Ag2S having superionic conduction [3,7–12]. For the application of nanocrystalline silver sulfide in infrared equipment, solar energy converters and resistive switches, it is necessary to have information about the variation in the thermal expansion coefficient of different Ag2S phases versus the temperature and about the effect of particle size on such lattice properties of silver sulfide as heat capacity and thermal expansion. Silver sulfide Ag2S has three basic polymorphic modifications [13]. Low-temperature monoclinic phase α-Ag2S (acanthite) exists at a temperature below ∼450 K. Cubic argentite β-Ag2S has body centered cubic (bcc) sublattice of S atoms and exists in the temperature range 452–859 K. High-temperature cubic phase γ-Ag2S with face centered cubic (fcc) sublattice of sulfur atoms is stable from ∼860 K up to melting temperature. For technical application, of most interest are

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low-temperature acanthite and argentite phases. According to Refs. [14,15], the coarse-crystalline silver sulfide with the average particle size of ∼500 nm and more has a monoclinic (space group No. 14 − P21/c (P121/c1)) structure of α-Ag2S acanthite type and is stoichiometric. The unit cell of acanthite α-Ag2S includes four formula units of Ag2S (z = 4). A thorough study of nanocrystalline silver sulfide revealed that it has the same monoclinic (space group P21/c) acanthite-type structure, but is nonstoichiometric and has the composition ∼Ag1.93S [16]. Argentite β-Ag2S phase with cubic (space group No 229 − Im3m (I 4/ m32/ m ) (Oh9 )) structure exists at temperatures above 443 K. The unit cell of argentite β-Ag2S includes two formula units of Ag2S (z = 2). According to high-temperature X-ray diffraction (XRD) data [17], four Ag atoms are statistically distributed in 54 positions 6(b) and 48(j) with the occupation probabilities ∼0.097 and ∼0.0715, respectively. The structure of argentite β-Ag2S is described in detail in study [18] and Crystallographic information file (CCDC reference number 1062400) attached thereto, and also in work [19] and Electronic supplementary information (ESI) for this paper [19]. According to Refs. [18,19], four silver atoms are statistically distributed in 6(b) and 48(j) positions with occupation degrees 0.0978(7) and 0.0711(0). At temperatures above 860 K silver sulfide contains cubic (space group Fm3m (F 4/ m32/ m ) (Oh5 )) γ-Ag2S phase. The unit cell of γ-Ag2S phase includes four formula units of Ag2S (z = 4). At a temperature of 923 K eight Ag atoms are statistically distributed in 88 positions 8(c), 32(f) and 48(i) with the occupation probabilities ∼0.088, ∼0.15, and ∼0.027, respectively [17].

Corresponding author. E-mail address: [email protected] (S.I. Sadovnikov).

https://doi.org/10.1016/j.tca.2017.12.013 Received 28 June 2017; Received in revised form 5 December 2017; Accepted 6 December 2017 Available online 07 December 2017 0040-6031/ © 2017 Elsevier B.V. All rights reserved.

Thermochimica Acta 660 (2018) 1–10

A.I. Gusev, S.I. Sadovnikov

Table 1 Composition of the reaction mixtures for silver sulfide synthesis, content of Ag, S and C in synthesized silver sulfide powders, specific surface area Ssp and average particle size D of Ag2S powders at temperature of 298 K and pressure of 1.01·105 Pa. No.

1 2 3 4

Concentration of reagents in the reaction mixture (mol l−1)a

Content of Ag, S, and C (wt.%)

AgNO3

Na2S

Na3C6H5O7

Ag

S

0.05 0.05 0.05 0.05

0.1 0.050 0.025 0.025

0.025 0.1 0.1 0.025

86.7 86.8 86.5 86.4

± ± ± ±

0.4 0.4 0.4 0.4

12.9 12.9 13.0 13.1

Ssp

b

(m2 g−1)

C ± ± ± ±

0.1 0.1 0.1 0.1

0.4 0.3 0.5 0.5

± ± ± ±

0.1 0.1 0.1 0.1

1.9 ± 0.1 1.8 ± 0.1 11.7 ± 0.2 12.4 ± 0.2

D (nm) BETc

XRDd

430 ± 10 460 ± 10 71 ± 5 67 ± 5

– – 66 ± 8 54 ± 6

Standard uncertainty in determination of concentrations not exceed 0.02 mol·l−1. Standard uncertainty in specific surface area measurements do not exceed 0.1 m2·g−1 for coarse-crystalline silver sulfide powders, and 0.2 m2·g−1 for nanocrystalline powders (0.68 level of confidence). c Standard uncertainty in estimation of average particle size from BET measurement do not exceed 10 nm for coarse-crystalline silver sulfide powders, and 5 nm for nanocrystalline powders (0.68 level of confidence). d Standard uncertainty in XRD measurement of average particle size do not exceed 8 nm (0.95 level of confidence). a

b

Under equilibrium conditions, a reversible phase transformation αAg2S − β-Ag2S takes place at a temperature about 448–453 K, and second reversible phase transformation β-Ag2S − γ-Ag2S occurs at a temperature about 845–860 K [13,20–22]. Until recently, few data on the α-Ag2S − β-Ag2S and β-Ag2S − γAg2S phase transformations were obtained only for coarse-grained (bulk) samples of silver sulfide Ag2S [20–23]. For example, “α − β” and “β − γ” phase transformations were studied in work [21] and heat capacity measurements of nanocrystalline silver sulfide were carried out in works [18,24]. Lately, dilatometric study of thermal expansion of coarse-crystalline (bulk) and nanocrystalline silver sulfide – monoclinic acanthite α-Ag2S and cubic argentite β-Ag2S – has been carried out [19,25]. The information about thermal expansion of silver sulfide is very limited, although such data are necessary for application of Ag2S at elevated temperatures. According to [26], the linear thermal expansion coefficient αac of acanthite is equal to ∼20×10−6 K−1. The temperature interval, to which this coefficient corresponds, and the method of its measurement in work [26] are not given. According to [27], the relative thermal expansions ΔL/L of coarse-grained (bulk) acanthite in the temperature range 293–450 K and bulk argentite at a temperature from ∼460 to 570 K linearly depend on temperature. This implies that the linear thermal expansion coefficients of acanthite and argentite in the mentioned temperature ranges are independent of temperature. Indeed, in the temperature range 293–450 K the linear thermal expansion coefficient αac of coarse-grained (bulk) acanthite is equal to 16.8×10−6 K−1, and the linear thermal expansion coefficient αarg of coarse-grained (bulk) argentite is equal to 45.8×10−6 K−1 at a temperature from ∼460 to 570 K [27]. According to high-temperature XRD data [19,25], the linear thermal expansion coefficient of coarse-crystalline (bulk) acanthite in the temperature region 300–433 K increases from ∼18.4 × 10−6 to ∼24.0×10−6 K−1. The linear thermal expansion coefficient of nanocrystalline acanthite in the same temperature region is almost 25 % more than the same coefficient of coarse-crystalline acanthite. From the measurement of the temperature dependence of the lattice constant of argentite follows that the thermal expansion coefficient of argentite decreases from ∼54 × 10−6 to ∼43 × 10−6 K−1 when the temperature rises from 443 to 623 K [19,25]. However, in the same temperature interval, the lattice constant a of argentite increases from 0.4856 to 0.4894 nm. According to the neutron diffraction data [28], the lattice constant aarg of argentite at 459, 473, 533, and 598 K is 0.4860, 0.4862, 0.4873, and 0.4889 nm, hence αarg ≈ 43 × 10−6 K−1. The data on the thermal expansion of cubic γ-Ag2S phase are not available in the literature. Direct dilatometric measurements of the thermal expansion of coarse-crystalline (bulk) and nanocrystalline silver sulfide at temperatures from 290 to 970 K in the region of existence of monoclinic acanthite α-Ag2S, cubic argentite β-Ag2S and cubic γ-Ag2S phase, as

well as the heat capacity measurement of these phases in nanocrystalline state in the temperature interval from 300 to 938 K have been carried out in this work for the first time. 2. Experiments Coarse-crystalline (bulk) and nanocrystalline silver sulfide powders with different particle size were synthesized by chemical deposition from aqueous solutions of silver nitrate AgNO3, sodium sulfide Na2S and sodium citrate Na3C6H5O7 ≡ Na3Cit. The particle size was controlled by varying the reagent concentrations and the duration of storage of deposits in the reaction mixtures. The synthesis technique is described earlier [10,15,16,29] (see also Supplementary Material for experimental details). Noted that the chemical deposition of sulfides from aqueous solutions is a particular case of one-pot synthesis [30] which makes it possible to synthesize the nanoparticles directly in the aqueous medium. Deposited Ag2S powders were washed with distilled water for removal of soluble impurity, filtered and dried in air at 323 K. Composition of the reaction mixtures and average particle size D of synthesized Ag2S powders 1, 2, 3 and 4 are given in Table 1. In situ high-temperature XRD (HT-XRD) experiments were performed using a X‘Pert PRO MPD (Panalytical) diffractometer equipped with a position-sensitive fast sector detector PIXCEL and an Anton Paar HTK-1200 Oven furnace. HT-XRD patterns were recorded in the angle interval 2θ = 20–67.5° with a step of Δ(2θ) = 0.026° and scanning time 200 s in each point. The diffraction measurements were performed at a temperature from 295 to 723 K with a step of ∼25–30 K (see also Supplementary Material). The synthesized silver sulfide samples were examined on a STADI-P (STOE) diffractometer in CuKα1 radiation. X-ray measurements were performed in the angle interval 2θ = 20–95° with a step of Δ(2θ) = 0.02° and large scanning time 10 s in each point. The resolution function FWHMR(2θ) = (utan2θ + vtan θ + w)1/2 of a STADI-P (STOE) diffractometer was determined in a special diffraction experiment using the cubic lanthanum hexaboride LaB6 (NIST Standart Reference Powder 660a) with lattice spacing a = 0.41569162 nm. The parameters of this resolution function FWHMR(2θ) are u = 0.00616, v = −0.00457, and w = 0.00778. The microstructure of the silver sulfide samples was studied by the scanning electron microscopy (SEM) method on a JEOL JSM 6390 LA microscope coupled with a JED 2300 Energy Dispersive X-ray Analyzer. Chemical composition of synthesized silver sulfide samples was estimated by EDX method. The contents of silver Ag and sulfur S in the synthesized coarse-crystalline (bulk) and nanocrystalline dried silver sulfide powders 1, 2, 3, and 4 are given in Table 1. The EDX spectra for all synthesized silver sulfide samples are shown in Fig. S1 (see Supplementary Material). The content of Ag and S in coarse-crystalline (bulk) powders 1 and 2 corresponds to stoichiometric sulfide Ag2S. The content of silver Ag and sulfur S in the silver sulfide nanopowders 3 and 2

Thermochimica Acta 660 (2018) 1–10

A.I. Gusev, S.I. Sadovnikov

Fig. 1. XRD patterns of studied silver sulfide Ag2S powders at 293 K. XRD numbers correspond to the sample numbers listed in Table 1. The vertical ticks correspond to diffraction reflections of monoclinic acanthite α-Ag2S. Insets show the dependences of reduced reflection broadening β*(2θ) on the scattering vector s = (2sin θ)/λ for nanocrystalline Ag2S powders.

3

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A.I. Gusev, S.I. Sadovnikov

from the sample and standard curves, respectively, and DSCbase(T) is value of DSC signal at temperature T from the baseline. The heat capacity of each silver sulfide samples was measured three times on samples with different masses. All measurements were carried out at heating and cooling. The measurements were carried out in Pt crucibles. According to the measuring procedure, the crucible was filled with a powder on 2/3 of the volume, so the mass of silver sulfide powders during loading was the same almost, and differs by several parts of milligram only. The mass of the samples was 25.52, 26.28, and 25.35 mg for nanocrystalline powders, and 25.91, 26.45, and 26.79 mg for coarse-crystalline powders. All measured data on thermal expansion and heat capacity of coarse-crystalline (bulk) and nanocrystalline silver sulfide are tabulated as supporting information in Tables S1-S4 (see Supplementary Material).

4 corresponds to sulfide which is close to stoichiometric composition ∼Ag2S but with small deficiency of silver. Besides silver and sulfur, the EDX spectra contain a small Kα line of carbon. The presence of Kα line of carbon in the EDX spectra is due to two reasons. Firstly, this line appears from the carbon planchet, onto which the examined powder was placed for EDX measurement. Secondly, the impurity of the initial reagent Na3Cit is the source of carbon C in the synthesized silver sulfide powders. The solution of sodium citrate is adsorbed by the surface of Ag2S powders during their deposition, and small amounts of Na3Cit is retained in the synthesized silver sulfide powders even after washing. No other impurity elements have been observed in the synthesized silver sulfide powders. The average particle size D (to be more precise, the average size of coherent scattering regions (CSR)) in the synthesized silver sulfide powders was estimated by XRD method from the diffraction reflection broadening β(2θ) using the dependence of reduced reflection broadening β*(2θ) = [β(2θ)cos θ]/λ on the scattering vector s = (2sin θ)/λ [31,32]. The value of broadening β(2θ) was determined by comparing the experimental width of each diffraction reflection, FWHMexp, with the instrumental resolution function FWHMR of the X-ray diffractometer as β(2θ) = [(FWHMexp)2–(FWHMR)2]1/2. Also the average particle size D was estimated from the specific surface Ssp value measured by the Brunauer-Emmett-Teller (BET) method, and from the SEM data (see Supplementary Material for experimental details). Thermal expansion and heat capacity were measured on silver sulfide powders annealed at 600–700 K in argon. The thermal expansion coefficient was measured on cylindrical samples 5 and 10 mm in diameter, which were pressed from synthesized Ag2S powders under a pressure of ∼260 MPa. The samples’ length was from 3.5 to 6.3 mm. Samples were pressed on a MP250D press (Maassen GmbH, Germany) with simultaneous pumping out of pressed powders to a residual pressure 13 Pa (0.1 mm Hg). Porosity of the pressed samples was 21–23%. The measurements of thermal expansion were made by means of a Linseis L75/1250 dilatometer under vacuum 0.0026 Pa (2·10−5 mm Hg) in the temperature range 293–985 K with a step 1 K and also on a NETZSCH DIL 402 C dilatometer in He atmosphere at the pressure 1.01·105 Pa at a temperature from 293 to 923 K with a step of 0.5 K. The heating rate was 4 K min−1. Heat capacity measurements of silver sulfide annealed powders were made by the differential scanning calorimetry method on an STA 449 F3 Jupiter (NETZSCH) thermal analyzer in Ar atmosphere at the standard pressure 1.01·105 Pa during heating from 300 up to 938 K with a step of 0.5 or 1.0 K. The STA 449 F3 Jupiter (NETZSCH) thermal analyzer belongs to the group of differential scanning calorimeters of variable temperature with continuous input of energy, which operate in adiabatic mode. The thermal analyzer was calibrated for five metals In, Sn, Bi, Zn, and Al. Sapphire was used as the standard. For measuring the heat capacity, the temperature interval was divided into parts 300–450 K, 470–840 K, and 870–970 K, corresponding to equilibrium α-Ag2S, β-Ag2S, and γ-Ag2S phases, and parts 450–470 K and 840–870 K in the vicinity the transformation temperatures Tα-β и Tβ-γ. At each heating or cooling part, a baseline with two empty crucibles was measured, then in the second stage the heat capacity of the sapphire (empty crucible and crucible with sapphire) was measured, and in the third stage the heat capacity of silver sulfide (crucible with Ag2S and crucible with sapphire) was measured. The initial measurement results were processed using a NETZSCH Proteus v.6.1.0 © NETZSCH-Geraetebau GmbH software. The heat capacity Cp was calculated from the formula

Cp (T ) = Cp-stand (T )

DSCsample (T ) − DSCbase (T ) mstand , × DSCstand (T ) − DSCbase (T ) msample

3. Thermal expansion of silver sulfide The XRD patterns for initial silver sulfide powders collected at a temperature of 293nK are shown in Fig. 1. The quantitative analysis of the XRD patterns and comparison with data [15,16] shown that all the powders are one-phase and contain a set of diffraction reflections of monoclinic (space group P21/c) α-Ag2S acanthite. The XRD patterns of nanocrystalline silver sulfides (Fig. 1, samples 3 and 4) contain the same set of broadened diffraction reflections. The average particle size D (to be more precise, the average size of CSRs) was estimated from the broadening of non-overlapping diffraction reflections (−1 0 2), (1 0), (−1 3), (−1 0 4), (0 3 1), and (0 1 4). According to these estimates, the average particle size D in the synthesized acanthite nanopowders (samples 3 and 4) is ∼66 and ∼54 nm. This agrees with the estimation of average particle size D from the value of specific surface area Ssp (Table 1). Change of XRD patterns of coarse- and nanocrystalline Ag2S powders at heating is shown in Fig. 2. It is seen that heating of silver sulfide up to temperature ≥ 453 K leads to transition of acanthite α-Ag2S into argentite β-Ag2S. Indeed, the XRD patterns of coarse- and nanocrystalline silver sulfide recorded at a temperature < 450 K (see Fig. 2) contain the diffraction reflections of monoclinic (space group P21/c) acanthite α-Ag2S. The XRD patterns recorded at > 450 K contain the diffraction reflections of cubic (space group Im3m ) argentite β-Ag2S. In situ high-temperature XRD measurements were performed in low vacuum 13 Pa (0.1 mm Hg) therefore oxidation started at heating of powders to temperature ≥ 623 K. The XRD patterns of Ag2S powders recorded at ≥ 623 K contain diffraction reflections of different oxide and oxide-sulfate phases of silver. The linear thermal expansion coefficient αaver is determined as the average expansion coefficient in the temperature interval between the initial temperature 293 K and measured temperature T

αaver (T ) =

ΔL L (T ) − L293K = , L293K ΔT L293K (T − 293)

(2)

where L(T) and L293K are the sample lengths measured at temperature T and at the initial temperature T0 = 293 K, respectively. The exact value of the thermal expansion coefficient α(T) at temperature T was found by numerical differentiation of the temperature dependence of elongation L(T) as

α (T ) =

1 dL (T ) . L293K dT

(3)

By determining the thermal expansion coefficient α(T) in the vicinity of the transformation temperatures Tα-β or Tβ-γ, it is possible to find out the order of the phase transition. The linear thermal expansion coefficient α(T) is related to heat capacity CV by the equation [33]:

(1)

where msample is mass of the sample, mstand is mass of the standard (sapphire), Cp-stand is specific heat of the standard (sapphire), DSCsample(T) and DSCstand(T) are values of DSC signal at temperature T 4

Thermochimica Acta 660 (2018) 1–10

A.I. Gusev, S.I. Sadovnikov

Fig. 2. XRD patterns of coarse- and nanocrystalline Ag2S powders at heating. The average particle size D of the coarse- and nanocrystalline Ag2S powders is equal to ∼400 and ∼60 nm, respectively. The “acanthite α-Ag2S − argentite β-Ag2S” phase transformation takes place at ∼450 K.

α (T ) =

γ CV (T ) , 3B vm

argentite β-Ag2S, the average coefficient αaver increases from ∼30.2 × 10−6 to ∼42.1 × 10−6 K−1. Further, after the β-Ag2S − γAg2S transformation, the coefficient αaver grows slightly from ∼45.3 × 10−6 to ∼45.9 × 10−6 K−1 as the temperature rises from ∼900 to ∼970 K. The temperature dependence of the average linear expansion coefficient αaver(T) exhibits smeared breaks in the vicinity of temperatures Tα-β or Tβ-γ (Fig. 3). According to [19,25], the average thermal expansion coefficient αaver of argentite during heating in the temperature range 443–623 K lowers from ∼55 × 10−6 to ∼42 × 10−6 K−1, whereas the results of the present study show that αaver of argentite in the temperature interval 500–800 K grows slowly from ∼30 × 10−6 to ∼42 × 10−6 K−1. This difference can be due to a large measurement step (25–30 K) of the expansion coefficient in works [19,25]; as a result, an overestimated value of αaver(443 K) is registered immediately in the region of the phase transition, i. e. in the region of the expansion coefficient discontinuity. Besides, observed small decreasing of thermal expansion coefficient of argentite [19,25] can be caused by its determination concerning the lattice constant for temperature 433 K instead of standard temperature 293 K. Explicit discontinuities are observed on the α(T) dependence in the temperature regions of α-Ag2S − β-Ag2S and β-Ag2S − γ-Ag2S transitions (see Fig. 3). This allowed us to determine rather accurately the transition temperatures Tα-β = 456 ± 5 K and Tβ-γ = 865 ± 10 K. The variation in the specific elongation ΔL/L293, the average thermal expansion coefficient αaver and the expansion coefficient α(T) during heating of silver sulfide sample 2 are close within the measurement error to those for sample 1. The average particle size in the initial powders 1 and 2 is almost the same and is equal to ∼430 and ∼460 nm.

(4)

where B is the bulk modulus, vm is the molar volume, and γ is the Grüneisen constant. In the case of first-order phase transition, at the transition temperature, the free energy exhibits a break, and jumps are observed on the temperature dependences of the entropy and the enthalpy. As follows from Eq. (4), the heat capacity and thermal expansion coefficient α(T) experience a discontinuity in the transition point. The temperature dependence of the average thermal expansion coefficient αaver(T) will have smeared jumps in the transition point. During second-order phase transitions, entropy changes continuously, and the heat (enthalpy) of transition is lacking. Therefore sudden changes, rather than discontinuities, are observed for the heat capacity and the thermal expansion coefficient α(T) at the transition point. The weak smeared breaks of the temperature dependence αaver (T) are observed in the second-order phase transition point. The effect of heating on the specific elongation ΔL/L293, the average linear thermal expansion coefficient αaver and the thermal expansion coefficient α(T) of coarse-crystalline silver sulfide sample 1 is shown in Fig. 3. All measured data on average linear thermal expansion coefficient of coarse-crystalline (bulk) silver sulfide without correction on porosity are given in Table S1 (see Supplementary Material). In the temperature region 293–455 K, where acanthite α-Ag2S exists, αaver increases from ∼14.3 × 10−6 to ∼18.9 × 10−6 K−1. These values agree with data [19,25] showing that the average linear expansion coefficient αaver of acanthite α-Ag2S in the temperature interval 300–433 K increases from ∼18.4 × 10−6 to ∼24.0 × 10−6 K−1, as well as with data [26] on αaver of acanthite equal to ∼20 × 10−6 K−1. At temperatures from ∼500 to ∼800 K, in the region of existence of 5

Thermochimica Acta 660 (2018) 1–10

A.I. Gusev, S.I. Sadovnikov

Fig. 3. Temperature dependences of specific elongation ΔL/L293, average coefficient of thermal expansion αaver, and thermal expansion coefficient α for coarse-crystalline Ag2S sample 1 (see Table 1). The discontinuities of α(T) dependence are shown by dashed lines. Temperatures Tα-β and Tβ-γ of α-Ag2S − β-Ag2S и β-Ag2S − γ-Ag2S phase transitions are marked by vertical dashed lines. Average particle size of the initial silver sulfide powder 1 is equal to 430 nm.

The temperature dependences of the specific elongation ΔL/L293 and the thermal expansion coefficients αaver and α of sample 3 produced from nanocrystalline silver sulfide powder 3 are displayed in Fig. 4. Measured data on average linear thermal expansion coefficient of nanocrystalline silver sulfide without correction on porosity are given in Table S2 (see Supplementary Material). The heating is accompanied by the growth of ΔL/L293 and αaver. The α(T) dependence has explicit discontinuities in the regions of transformations α-Ag2S − β-Ag2S and β-Ag2S − γ-Ag2S (see Fig. 4). The phase transition temperatures are equal to Tα-β = 450 ± 5 K and Tβ-γ = 863 ± 20 K and are close to those for sample 1. In whole, the character of αaver(T) and α(T) variation (Figs. 3 and 4) shows that the acanthite α-Ag2S to argentite β-Ag2S and argentite to γ-

Ag2S phase transformations take place as the first-order phase transitions. For direct comparison of the thermal expansion of coarse-crystalline and nanocrystalline silver sulfide, the average linear thermal expansion coefficients αaver can be used. The temperature dependences of αaver(T) for coarse- and nanocrystalline silver sulfide are demonstrated in Fig. 5. The largest coefficient αaver(T) in the examined temperature range 293–970 K belongs to nanocrystalline silver sulfide (sample 3) produced from the powder with an average particle size of ∼66 nm. The average linear thermal expansion coefficients αaver of coarsecrystalline (bulk) and nanocrystalline acanthite α-Ag2S and argentite βAg2S measured in works [19,25] by the high-temperature XRD method are shown in Fig. 5 for comparison. Considering the experimental errors Fig. 4. Effect of heating on the specific elongation ΔL/L293, average coefficient of thermal expansion αaver and thermal expansion coefficient α for nanocrystalline Ag2S sample 3 (see Table 1). The discontinuities of α(T) dependence are shown by dashed lines. Transition temperatures Tα-β and Tβ-γ are marked by vertical dashed lines. Average particle size of the initial silver sulfide powder 3 is equal about 66 nm.

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Fig. 6. Measured heat capacity of silver sulfide nanopowder 4 (see Table 1). The inset shows spasmodic change of Cp for Ag2S nanopowder in the regions of α-Ag2S − β-Ag2S and β-Ag2S − γ-Ag2S transformations.

Fig. 5. The average thermal expansion coefficients αaver of coarse- and nanocrystalline silver sulfide (samples 1 and 3, respectively) in the temperature range 293–970 K. Curve numbers correspond to the sample numbers listed in Table 1. The thermal expansion coefficient αaver of coarse-crystalline (●) and nanocrystalline ( ) silver sulfides measured in work [19,25] by the high-temperature XRD method are shown for comparison. The discontinuity regions of the coefficient α(T) are shown by a dotted line.

1) differs little from the heat capacity measured by other authors (see, for example, ref [21]). Measured data on the heat capacity of coarsecrystalline (bulk) Ag2S sample 1 are given in Table S3 (see Supplementary Material). The heat capacity of silver sulfide nanopowder 4 changes rather monotonically with rising temperature, except for the transition regions (Fig. 6). Measured data on heat capacity of silver sulfide nanopowder 4 are given in Table S4 (see Supplementary Material). In the temperature range 300–450 K, the heat capacity increases and then, near the transition temperature Tα-β, it experiences a discontinuity. In the region of existence of β-Ag2S phase, in the temperature interval from ∼470 to ∼840 K, the heat capacity first decreases slightly to ∼670 K and then grows slightly to the transition temperature Tβ-γ, where it discontinues. As the temperature increases further to ∼890 K, a small reduction of the heat capacity is observed, and at T > 890 K the heat capacity grows slightly. According to the heat capacity measurements, the transition temperatures Tα-β and Tβ-γ are equal to 451 and 858 K, which agrees with the data of thermal expansion measurements for silver sulfide. The experimental data on the heat capacity of nanocrystalline silver sulfide Ag2S in the temperature range 298–450 K, where the equilibrium phase is monoclinic acanthite α-Ag2S, are approximated by the following function

and different measurement methods, the results of this work and the data [19,25] on αaver obtained earlier satisfactorily agree with each other. The XRD patterns of silver sulfide samples obtained at 293 K after measurement of the thermal expansion coefficient revealed that the samples are still one-phase and contain only acanthite α-Ag2S (Fig. S2, Supplementary Material). Owing to a rather high heating rate, the measurement duration did not exceed 3 h, and the Ag2S particle size in the samples after measurements increased not more than by 20 nm.

4. Heat capacity of silver sulfide The differential scanning calorimetry of nanocrystalline powder 4 combined with thermo-gravimetric measurement revealed a mass loss of ∼1.0–1.5 % during heating of the nanopowder to ∼1000 K. The measurements showed that the temperature dependences of ionic current Iion exhibit distinct maxima for the mass numbers 64 and 48 corresponding to SO2 [34,35] in the temperature range 430–510 K (Fig. S2, Supplementary Material). There is a maximum at ∼570 K on the temperature dependence of ionic current for the mass number 44 corresponding to CO2 [34,35] (see Fig. S3, Supplementary Material). The liberation of SO2 and CO2 is caused by the oxidation of sulfur and carbon. The impurities of the initial reagents Na2S and Na3Cit are the sources of S and C in the syn-thesized silver sulfide powders. The solutions of sodium sulfide and citrate are adsorbed by the surface of Ag2S powders during their deposition, and small amounts of these impurities are retained in the synthesized silver sulfide powders even after washing. In order to avoid errors in heat capacity determination connected with isolation of S and C impurities from Ag2S powders, the powders were preliminarily annealed in argon during heating to 970 K, and then they were cooled. According to the XRD data, the annealed silver sulfide powders remain one-phase and contain only acanthite αAg2S; the BET and XRD data show that the particle size of the powders increases little if at all. The heat capacity of silver sulfide was measured on annealed coarse-crystalline (bulk) powder 1 and nanocrystalline powder 4. For measuring the heat capacity, the temperature interval was divided into parts (sections) 300–450 K, 470–840 K, and 870–970 K, corresponding to equilibrium α-Ag2S, β-Ag2S, and γ-Ag2S phases, and parts (sections) 450–470 K and 840–870 K in the vicinity the transformation temperatures Tα-β и Tβ-γ. The heat capacity of coarse-crystalline (bulk) silver sulfide (sample

Cp(α-Ag2S) = 161.8 − 0.15T + 0.14 × 10−3T2 − 15200/T (J mol−1 K−1) (298–450 K).

(5)

The heat capacity of nanocrystalline silver sulfide Ag2S in the region of existence of argentite β-Ag2S at temperatures from 470 to 840 K is described by the function Cp(β-Ag2S) = 318.8 − 0.42T + 0.25 × 10−3T2 − 39400/T (J mol−1 K−1) (470–840 K).

(6)

The heat capacity of nanocrystalline silver sulfide Ag2S in the region of existence of the cubic phase γ-Ag2S at temperatures from 890 to 930 K is described by the function Cp(γ-Ag2S) = 66.4 + 0.03T (J mol−1 K−1) (890–930 K).

(7)

The peaks of the heat capacity Cp of silver sulfide nanopowder in the αAg2S − β-Ag2S and β-Ag2S − γ-Ag2S transformation regions are not completely symmetric (Fig. 6, inset). However, the observed shape of the peaks is more characteristic of the first-order phase transitions. The heat capacity peaks are very narrow (the width of peak base is about 8 K) that also is typical for the first-order phase transitions. The enthalpies of phase transformations were determined by the integration of areas under Cp(T) dependence in regions of phase transitions. The enthalpies of the phase transformations α-Ag2S − β-Ag2S 7

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the temperature regions 300–440, 470–830, and > 880 K, respectively. The measured heat capacity Cp peaks of silver sulfide nanopowder are slightly broadened in temperature as compared with those for coarsecrystalline powder 1, and also for bulk Ag2S obtained in study [21].

Table 2 Temperatures T and enthalpies ΔH of phase transformations of silver sulfide Ag2S at pressure of 1.01·105 Pa. Acanthite α-Ag2S − argentite βAg2S

Argentite β-Ag2S- phase γAg2S

Tα-β/Ka

ΔHα-β/(kJ mol−1)

Tβ-γ/Ka

ΔHβ-γ/(kJ mol−1)

450 449.3–451.3 449 441–453 449–450 452 ± 1

3.98 ± 0.42 4.06 ± 0.03 3.93 ± 0.20 4.37 ± 0.02 3.7–3.9 4.2 ± 0.4b

– 865 859 – – 863 ± 1

– 0.78 ± 0.01 0.50 ± 0.54 – – 1.2 ± 0.3c

Refs. and methods

5. Additional contribution to the heat capacity and thermal expansion caused by small particle size d

20 21 e 22 f 23 f 18, 19, 24 g this study

The heat capacity of nanocrystalline silver sulfide has never been measured previously. Nevertheless, there are data that the low-temperature heat capacity of metals depends on the size of particles. So, in work [36] it was established that the specific heat capacity of nanocrystalline silver Ag measured at a temperature from 1 to 10 K is 3–4 times larger than that of bulk silver. According to data [37,38], in the temperature range 14–30 K, the heat capacity of palladium Pd with the average particle size of ∼7–8 nm is ∼30-40% larger than that of bulk palladium. In works [19,25,39,40] it was shown that a small particle size makes a positive contribution to the heat capacity and the average thermal expansion coefficient of silver and lead sulfides. According to [39,41,42], the main reason of variation of the lattice properties of nanocrystals in comparison with bulk substances is variation of the shape and boundaries of the phonon spectrum, i. e. change of the frequency distribution function of atomic vibrations. The main idea of the majority of models for modification of the phonon spectrum of small particles and/or nanostructured systems consists in the appearance of low-frequency modes in the phonon spectrum, which are absent in the spectrum of the bulk crystal [31,41]. Indeed, the lattice vibration spectrum in small particles differs drastically from the bulk, as demonstrated by authors of studies [43,44] and many others. According to [31,42,45], waves can occur in the nanoparticles, whose length does not exceed the doubled maximum size of the particle D, i. e. λ ≤ 2D; so, on the side of low-frequency vibrations the phonon spectrum is limited by a certain minimal frequency c ωmin ≥ 2π 2Dt , where ct is the velocity of propagation of transverse elastic vibrations (i. e. transverse velocity of sound). In bulk crystals, there is no limitation like this. Besides, the phonon spectrum is limited on the side of high frequencies. Taking into account the low- and high-frequency restrictions of the phonon spectrum, the molar heat capacity of a substance, whose molecule contains n atoms, is equal to

d

a Standard uncertainty in temperature measurements do not exceed 1 K (0.68 level of confidence). b Standard uncertainty in transition enthalpy do not exceed 0.4 kJ mol−1 (0.68 level of confidence). c Standard uncertainty in transition enthalpy do not exceed 0.3 kJ mol−1 (0.68 level of confidence). d Adiabatic calorimetry. e Drop calorimetry. f Differential thermal and thermogravimetric analysis. g Dilatometry and differential scanning calorimetry.

and β-Ag2S − γ-Ag2S were estimated to be ΔHα-β = 4.2 ± 0.4 and ΔHβ−1 , respectively. Within the measurement error, γ = 1.2 ± 0.3 kJ mol the obtained values of temperatures Tα-β and Tβ-γ and enthalpies ΔHα-β and ΔHβ-γ of these phase transformations are rather close to the literature data [18–24] (Table 2). The heat capacity of nanocrystalline and coarse-crystalline Ag2S samples with different mass is shown in Fig. S4 (see Supplementary Material). It is seen that the Cp does not depend on mass of the samples, but strongly differs for nanocrystalline and coarse-crystalline silver sulfide. For comparison, Fig. 7 presents our experimental data on the heat capacity Cp of nanocrystalline and coarse-crystalline (bulk) Ag2S and the most reliable literature experimental data on the Cp of coarsecrystalline (bulk) Ag2S [21] obtained with the use of adiabatic-shell calorimetry. Monoclinic silver sulfide was produced in study [21] by direct sintering of silver and sulfur powders at a temperature to 770 K for 11 days; the crystallite size in the synthesized powder was more than 1 μm. It is seen that the heat capacity of coarse-crystalline (bulk) silver sulfide (sample 1) measured in this work differs very little from the heat capacity measured in study [21]. The measured heat capacity of nanocrystalline silver sulfide is by ∼3-4 %, ∼5–7% and ∼11–12% larger than the heat capacity of coarse-crystalline (bulk) silver sulfide in

ωmax

CV (T ) = n

∫ ωmin

∂ε (ω, T ) g (ω) dω. ∂T

(8)

In studies [39,46], taking into consideration the restrictions of the phonon spectrum of the small particles and using the approach [47] for the upper boundary ωmax of the phonon spectrum of small particle, it was shown that molar heat capacity of a nanocrystalline substance with particles of a right-angled shape can be presented as a function not only of the temperature T, but also of the size D of the small particle:

CV (T , D) = CVbulk (T ) + n (k1 LΣ T + k2 SΣ T 2),

(9)

where the first summand represents the Debye heat capacity of bulk crystal, LΣ and SΣ are the total edge length and the total surface area of small particles. If we assume that small particles have a cubic shape with edges in length D, then the number of such particles in the volume vm is equal to np = vm/D3. In this case LΣ = 12npD = 12 vm/D2 and SΣ = 6npD2 = 6 vm/D. The values k1 = (kB2 c1−1/8π ℏ) I2 and k2 = (kB3 c2−1/2π ℏ2) I3 are positive ∞

constants, Im = (4m !/2m + 1)

Fig. 7. Measured heat capacity of coarse-crystalline (bulk) powder 1 (black solid line) and nanopowder 4 (blue dotted line) of silver sulfide (see Table 1). For comparison, heat capacity (circle) of coarse-crystalline (bulk) Ag2S [21] is shown. The unit cells of α-Ag2S acanthite, β-Ag2S argentite and γ-Ag2S phase are shown in the temperature regions of existence of these phases. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

∑

N−m ≡ (4m!/2m+1)ζ(m), ζ(m) is the

N =1

Riemann zeta function (I3 = 1.8031; I2 = π2/6), and c1−1 and c2−1 are effective propagation velocities of elastic vibrations determined through the velocities of longitudinal and transverse vibrations, cℓ and ct: c1−1 = cℓ−1 + 2ct−1 and c2−1 = 8

2ct4 − 3ct2 cℓ2 + 3cℓ4 ct2 cℓ2 (cℓ2 − ct2)

.

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With allowance for Eqs. (4), (9) and expressions for LΣ и SΣ, the thermal expansion coefficient of nanocrystalline substance can be presented as

α (T , D) =

Ag2S/Ag nanowire heterostructure, Nanotechnology 18 (2007) 5 (paper 485202). [3] Z. Xu, Y. Bando, W. Wang, X. Bai, D. Golberg, Real-time in situ HRTEM-resolved resistance switching of Ag2S nanoscale ionic conductor, ACS Nano 4 (2010) 2515–2522. [4] D. Wang, L. Liu, Y. Kim, Z. Huang, D. Pantel, D. Hesse, M. Alexe, Fabrication and characterization of extended arrays of Ag2S/Ag nanodot resistive switches, Appl. Phys. Lett. 98 (2011) 3 (paper 243109). [5] B. Liu, Z. Ma, Synthesis of Ag2S-Ag nanoprisms and their use as DNA hybridization probes, Small 7 (2011) 1587–1592. [6] A.N. Belov, O.V. Pyatilova, M.I. Vorobiev, Synthesis of Ag/Ag2S nanoclusters resistive switches for memory cells, Adv. Nanoparticles 3 (2014) 1–4. [7] S.I. Sadovnikov, A.I. Gusev, Facile synthesis, structure, and properties of Ag2S/Ag heteronanostructure, J. Nanopart. Res. 18 (2016) 277 (paper 277). [8] Yu.I. Kharkats, Electric-field induced transition to superionic conductive state, Fiz. Tverd. Tela 23 (1981) 2190–2192 (in Russian). [9] Yu.Ya. Gurevich, Yu.I. Kharkats, Features of the thermodynamics of superionic conductors, Sov. Phys. Usp. 25 (1982) 257–276. [10] S.I. Sadovnikov, A.I. Gusev, Universal approach to the synthesis of silver sulfide in the forms of nanopowders quantum dots, core-shell nanoparticles, and heteronanostructures, Eur. J. Inorg. Chem. 2016 (2016) 4944–4957. [11] S.I. Sadovnikov, A.I. Gusev, Structure and properties of Ag2S/Ag semiconductor/ metal heteronanostructure, Biointerface Res. Appl. Chem. 6 (2016) 1797–1804. [12] A.I. Gusev, S.I. Sadovnikov, Structure and properties of nanoscale Ag2S/Ag heterostructures, Mater. Lett. 188 (2017) 351–354. [13] R.S. Sharma, Y.A. Chang, The Ag-S (silver-sulfur) system, Bull. Alloy Phase Diagrams 7 (1986) 263–269. [14] R. Sadanaga, S. Sueno, X-ray study on the α-β transition of Ag2S, Mineral. J. Jpn. 5 (1967) 124–148. [15] S.I. Sadovnikov, A.I. Gusev, A.A. Rempel, Artificial silver sulfide Ag2S: crystal structure and particle size in deposited powders, Superlattices Microstruct. 83 (2015) 35–47. [16] S.I. Sadovnikov, A.I. Gusev, A.A. Rempel, Nonstoichiometry of nanocrystalline monoclinic silver sulfide, Phys. Chem. Chem. Phys. 17 (2015) 12466–12471. [17] T. Blanton, S. Misture, N. Dontula, S. Zdzieszynski, In situ high-temperature X-ray diffraction characterization of silver sulfide, Ag2S, Powder Diff. 26 (2011) 110–118. [18] S.I. Sadovnikov, A.I. Gusev, A.A. Rempel, An in situ high-temperature scanning electron microscopy study of acanthite −argentite phase transformation in nanocrystalline silver sulfide powder, Phys. Chem. Chem. Phys. 17 (2015) 20495–20501. [19] S.I. Sadovnikov, A.I. Gusev, Thermal expansion, heat capacity and phase transformations in nanocrystalline and coarse-crystalline silver sulfide at 290–970 K, J. Therm. Anal. Calor. (2018), http://dx.doi.org/10.1007/s10973-017-6691-8. [20] C.M. Perrott, N.H. Fletcher, Heat capacity of silver sulfide, J. Chem. Phys. 50 (1969) 2344–2350. [21] F. Grønvold, E.F. Westrum, Silver(I) sulfide: ag2S. Heat capacity from 5 to 1000 K, thermodynamic properties, and transitions, J. Chem. Therm. 18 (1986) 381–401. [22] W.T. Thompson, S.N. Flengas, Drop calorimetric measurements on some chlorides sulfides, and binary melts, Can. J. Chem. 49 (1971) 1550–1563. [23] K.P. Mamedov, M.F. Gadzhiev, Z.J. Suleimanov, Z.D. Nurieva, X-ray and thermographic study of phase transition α → β in Ag2S, Inorg. Mater. 16 (1980) 241–243. [24] S.I. Sadovnikov, A.I. Gusev, Thermal expansion and the heat capacity of nanocrystalline and coarse-crystalline silver sulfide Ag2S, Phys, Solid State 59 (2017) 1887–1894. [25] A.I. Gusev, S.I. Sadovnikov, A.V. Chukin, A.A. Rempel, Thermal expansion of nanocrystalline and coarse-crystalline silver sulfide Ag2S, Phys. Solid State 58 (2016) 251–257. [26] H. Okazaki, A. Takano, The specific heat of Ag2S in α-phase, Ztsch. Naturforsch. A 40 (1985) 986–988. [27] K. Honma, K. Iida, Specific heat of superionic conductor Ag2S, Ag2Se and Ag2Te in α-phase, J. Phys. Soc. Jpn. 56 (1987) 1828–1836. [28] R.J. Cava, F. Reidinger, B.J. Wuensch, Single-crystal neutron diffraction study of the fast-ion conductor β-Ag2S between 186 and 325 °C, J. Solid State Chem. 31 (1980) 69–80. [29] S.I. Sadovnikov, A.A. Rempel, Synthesis of nanocrystalline silver sulfide, Inorg. Mater. 51 (2015) 759–766. [30] N.S. Kozhevnikova, S.I. Sadovnikov, A.A. Rempel, One-pot synthesis of lead sulfide nanoparticles, Russ. J. Gen. Chem. 81 (2011) 2062–2066. [31] A.I. Gusev, A.A. Rempel, Nanocrystalline Materials, Cambridge Intern. Science Publ., Cambridge, 2004 (351 pp). [32] S.I. Sadovnikov, A.I. Gusev, Chemical deposition of nanocrystalline lead sulfide powders with controllable particle size, J. Alloys Compd. 586 (2014) 105–112. [33] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Cornell University, New York, Chicago, London, 1976 (826 pp). [34] P.J. Linstrom, W.G. Mallard (Eds.), NIST Standard Reference Database Number 69, NIST Chemistry WebBook, 2005. [35] http://webbook.nist.gov/chemistry/. [36] G. Goll, H. Lohneyen, Specific heat of nanocrystalline and colloidal noble metals at low temperatures, Nanostruct. Mater. 6 (5–8) (1995) 559–562. [37] G.H. Comsa, D. Heitkamp, H.S. Räde, Effect of size on the vibrational specific heat of ultrafine palladium particles, Solid State Commun. 24 (1977) 547–550. [38] Y.Y. Chen, Y.D. Yao, S.U. Jen, B.T. Lin, H.M. Lin, C.Y. Tung, S.S. Hsiao, Magnetic susceptibility and low temperature specific heat of palladium nanocrystals, Nanostruct. Mater. 6 (1995) 605–608. [39] S.I. Sadovnikov, A.I. Gusev, Effect of particle size on the thermal expansion of nanostructured lead sulfide films, J. Alloys Compd. 610 (2014) 196–202. [40] S.I. Sadovnikov, A.I. Gusev, A.A. Rempel, Nanostructured lead sulfide: synthesis,

γCV (bulk) (T ) γ (k1 LΣ T + k2 SΣ T 2) +n 3vm B 3vm B

= αbulk (T ) + n

γ ⎛ 12k1 T 6k T 2 + 2 ⎞. 2 D ⎠ 3B ⎝ D ⎜

⎟

(10)

The data on the propagation velocities of longitudinal and transverse elastic vibrations, cℓ and ct, for acanthite α-Ag2S, argentite β-Ag2S and γ-Ag2S phase are not available in the literature. Therefore the contribution of the particles with small size to the thermal expansion and heat capacity of silver sulfide phases cannot be estimated quantitatively. Nevertheless, from Eqs. (9) and (10) it is clear that the heat capacity and the thermal expansion coefficient of nanocrystalline substance contain an additional positive contribution as compared with the same properties of coarse-grained (bulk) substance. It is exactly this result that is observed in the present study for the heat capacity and the thermal expansion coefficient of nanocrystalline and coarse-grained (bulk) silver sulfide Ag2S. The appearance of a positive contribution is due to the restriction of the phonon spectrum on the side of low and high frequencies. Besides, the reduction of the particle size of silver sulfide should be accompanied by an increase in the anharmonicity of atomic vibrations [19,25]. This agrees with the conclusions [31,39] about a considerably more important role of the anharmonicity of thermal vibration for nanomaterials as compared with macrostructures. The data of this study and the previous results [19,25,39] show that decreasing the silver sulfide particle to the nanosized scale leads to enhanced values of heat capacity Cp and thermal expansion coefficient αaver at comparable temperatures. 6. Conclusions The average linear thermal expansion coefficient α aver of nanocrystalline silver sulfide Ag2S in the temperature interval ∼300-970 K is larger than α aver of coarse-crystalline silver sulfide. The heat capacity Cp of nanocrystalline silver sulfide in the same temperature range is by 3–9 % larger than Cp of coarse-crystalline (bulk) sulfide. The observed difference in the thermal expansion coefficient and heat capacity is due to a small size of Ag2S particles in nanocrystalline silver sulfide leading to the restriction of the phonon spectrum on the side of low and high frequencies. The presence of discontinuities on the temperature dependences of α (T) and Cp(T) in the vicinity of the α-Ag2S −β-Ag2S and β-Ag2S − γ-Ag2S phase transformation temperatures suggests these transformations should take place by the first-order phase transition mechanism. Acknowledgements This work is financially supported by the Russian Science Foundation (Grant 14-23-00025) through the Institute of Solid State Chemistry of the Ural Branch of the RAS. Authors are grateful to Dr. O.N. Leonidova and D.A. Yagodin for the help in thermal expansion and heat capacity measurements. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.tca.2017.12.013. References [1] A.I. Gusev, Effects of the nanocrystalline state in solids, Engl. Transl.: Physics − Uspekhi 41 (1998) 49–76 Uspekhi Fiz. Nauk 168 (1998) 55–83. (in Russian). [2] C.H. Liang, K. Terabe, T. Hasegawa, M. Aono, Resistance switching of an individual

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particles, Phys. Rev. B 1 (1970) 851–857. [44] P.M. Ajayan, L.D. Marks, Phase instabilities in small particles, Phase Trans. B 24–26 (1990) 229–258. [45] Yu.I. Petrov, Physics of Small Particles, Nauka, Moscow, 1982 360 pp. (in Russian). [46] S.I. Sadovnikov, A.I. Gusev, Thermal expansion of nanostructured PbS films and anharmonicity of atomic vibrations, Phys. Solid State 56 (2014) 2353–2358. [47] E.W. Montrol, Size effect in low temperature heat capacities, J. Chem. Phys. 18 (1950) 183–185.

structure, and properties, Russ. Chem. Rev 84 (2016) 731–758. [41] S.I. Sadovnikov, A.I. Gusev, Recent progress in nanostructured silver sulfide Ag2S: From synthesis and nonstoichiometry to properties, J. Mater. Chem. A 5 (2017) 17676–17704. [42] S.I. Sadovnikov, A.A. Rempel, A.I. Gusev, Nanostructured Lead, Cadmium and Silver Sulfides: Structure, Nonstoichiometry and Properties, Springer Int. Publ. AG, Cham – Heidelberg, 2017 (317 pp). [43] J.M. Dickey, A. Paskin, Size and surface effects on the phonon properties of small

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