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Thermodynamic Modeling of Oxidation of Tin Nanoparticles

Abstract

A thorough thermodynamic analysis of oxidation of tin nanoparticles was performed. Solid tin oxides SnO2, Sn3O4 and SnO were considered according to the bulk phase diagram and a number of experimental results on tin nanostructures oxidation were taken into account in the assessment. Two equilibrium models with different spatial configuration, namely two single-component particles and core–shell model were explored. The surface energies for solid SnO and Sn3O4 were obtained on the basis of DFT calculations while the interfacial energies at SnO2(s)/Sn(l) and Sn3O4(s)/Sn(l) interfaces were assessed using a broken bond approximation. The opposite influence of nanosizing on stability of SnO2 and SnO/Sn3O4 oxides is demonstrated. It is due to the surface contribution which is higher for SnO2(s) than Sn(l) while lower for SnO(s) and Sn3O4(s) compared to Sn(l). This situation can explain some experimental findings during oxidation of Sn nanoparticles, namely an increased stability of SnO(s) and Sn3O4(s) with respect to both liquid tin and solid tin dioxide.

Introduction

Oxidation of metal nanostructures is one of size/shape dependent phenomena which are intensively studied now. A huge number of experimental works as well as theoretical studies on this subject have been published recently. Equilibrium thermodynamics provides very useful tools for studying various size/shape effects in the top-down approach.[1,2,3] Whereas the effect of particle size and surface energy on the melting point has been studied both by experimental methods and theoretical modeling for many systems, the phase transitions in partly open systems involving exchange of one or more components with surrounding atmosphere have been less explored. Within the family of metal–oxide systems, Navrotsky et al.[4,5,6] have calculated phase equilibria in the systems Co-CoOx-O2(g), Fe-FeOx-O2(g) and Mn-MnOx-O2(g) for bulk and nanoparticles forms and have shown substantial influence of nanosizing on stability of various oxide phases. The aim of the present paper is to carry out a similar analysis for the system Sn-SnOx-O2(g).

Thermodynamic Description of the bulk Sn-O System

A thorough thermodynamic analysis of the Sn-O system including an assessment of thermodynamic data for tin oxides has been performed by Cahen et al.[7] It should be noted, however, that substantial differences between the optimized and experimental values of entropy at 298.15 K exist for SnO2 and Sn3O4 oxides. The assessed value Sm,298(SnO2) = 73.23 J K−1 mol−1 is 1.4-times higher than the experimental value 51.82 J K−1 mol−1 obtained from low-temperature heat capacity measurements.[8] A possible reason for this discrepancy may lie in a variable oxygen stoichiometry of SnO2 (oxygen vacancy formation yielding a sub-stoichiometric SnO2−δ) that may stabilize this phase towards higher temperatures and/or lower partial pressures of oxygen. In fact, the Sn3O4 phase can be considered as oxygen deficient SnO2−δ with rutile structure and one-third of oxygen vacancies cumulated in (011) planes in an ordered pattern.[9] Based on the assessed data[7] the Sn-O phase diagram was calculated using the FactSage software[10] as shown in Fig. 1. Three invariant points can be identified in the temperature range 300-800 K under the conditions summarized in Table 1.

Fig. 1
figure 1

Phase diagram of the system Sn-O calculated with data from Ref 7

Table 1 Invariant points in the Sn-O system calculated with data from[7]

Tin sesquioxide Sn2O3 has been identified as a product of bulk SnO disproportionation reaction[11, 12] as well as a product of nano-Sn oxidation.[13] Being an intermediate oxide between SnO and SnO2, its stability has been examined on the basis of first-principle calculations.[14, 15] Nevertheless, Sn2O3 was not included into the present calculations due to absence of complete thermodynamic data.

Oxidation of Nano-Sn

Oxidation of nano-Sn as well as the stability of nano-SnO2 have been subjects of many studies. Although some intermediate oxides, namely SnO, Sn3O4 and Sn2O3, have been observed during oxidation, the most stable oxide, SnO2, has been obtained in most cases and the resulting particles frequently adopted a core(Sn)–shell(SnO2) structure.

Using XRD and HRTEM Huh et al.[16] have observed oxide formation on a surface of almost spherical tin nanoparticles with an average diameter 28 nm. Oxidic shells were composed of SnO2 which was amorphous at low temperatures while it crystallized in tetragonal rutile structure after heating above 500 K. A metastable (high-pressure) orthorhombic form of SnO2 has been also observed. In their subsequent study[13] they have observed the tin particles oxidation (6-40 nm in diameter) in air which resulted in an oxide shell composed of SnO and Sn2O3 oxides. Using a simultaneous TG/DSC technique combined with XRD characterization of product oxides Song and Wen[17] have investigated the oxidation of tin (nano)particles with average diameter 110 nm in air. Their results reveal a two-stage oxidation process. During a continuous heating (2-20 K min−1) SnO shell is created first at temperatures 473-523 K. The second step in the temperature range 673-973 K results in a formation of SnO2 coexisting with SnO. At temperatures above 1073 K, only SnO2 is present in the oxidic shell of particles. These results have been also confirmed under isothermal heating at various temperatures in the range 473-1173 K for 4 h. A similar sequence of oxidation products (Sn → SnO + SnO2 → SnO2) has been also observed in the case of tin nanowires.[18] Sn3O4 as a product of liquid tin nanodroplets oxidation was identified by Mima et al.[19] who employed TEM for an in situ analysis of Sn(l)/tin oxide boundary during oxidation. Sutter et al.[20] have studied the size-dependent room temperature oxidation of tin nanoparticles in air. SnO has been identified as an oxidation product and using TEM images of core/shell particles with various sizes (3-100 nm in diameter) the dependence of SnO shell thickness on particle diameter has been found out. It should be noted that the oxidation of tin nanoparticles is influenced by a lowering of tin melting temperature which has been observed experimentally.[21, 22] This decrease in temperature can be also calculated using the Gibbs–Thomson equation or the Pawlow equation.[23] Using the data from Table 2 one can calculate a decrease as low as 60 K for nanoparticles with a diameter of 5 nm.

Table 2 Basic physicochemical data for tin

Thermodynamic Modeling of Nano-Sn Oxidation

Equilibrium relations in nanosystems are fundamentally dependent on topology of the system under consideration, e.g. on spatial configuration of coexisting phases. In the following analysis we consider two different configurations, namely (1) two single-component particles (analogy to the Pawlow equation for the melting point decrease) and (2) core–shell geometry, which are both applied on the Sn(l)-SnO2(s) and Sn(l)-Sn3O4(s) equilibrium. According to the proposed classification scheme[29] the respective configurations represent models of the first and the second generation. Although core–shell model is physically more realistic, simpler two particles model is often used for the thermodynamic modeling of equilibria in oxide systems as well as for evaluation of structural stability of polymorphic oxide nanoparticles.[4,5,6, 30,31,32]

Single-Component Particles

The spatial configuration of such a system is shown in Fig. 2. The equilibrium condition for the reaction Sn(l) + O2(g) = SnO2(s) is in detail derived in “Appendix” (Eq 23, 26 and 27) yielding Eq 1 with the respective bulk and surface contribution given in Eq 2 and 3:

Fig. 2
figure 2

Spatial configuration of the two single-component particles system Sn-O ((α) = Sn(s,l), (β) = SnOx(s))

$$\Delta_{\text{r}} G^{\text{np}} \left( {T,p_{{ ( {\text{O}}_{ 2} ) , {\text{np}}}} } \right) = \Delta_{\text{r}} G^{\text{bulk}} \left( {T,p_{{ ( {\text{O}}_{ 2} ) , {\text{np}}}} } \right) + \Delta_{\text{r}} G^{\text{surf}} \left( {T,p_{{ ( {\text{O}}_{ 2} ) , {\text{np}}}} } \right) = 0$$
(1)
$$\Delta_{\text{r}} G^{\text{bulk}} \left( {T,p_{{ ( {\text{O}}_{ 2} ) , {\text{np}}}} } \right) = \Delta_{\text{r}} G^{\text{o,bulk}} \left( {T,p^{\text{o}} } \right) - RT\ln p_{{({\text{O}}_{ 2} ),{\text{np}}}} = RT\ln p_{{({\text{O}}_{ 2} ),{\text{bulk}}}} - RT\ln p_{{({\text{O}}_{ 2} ),{\text{np}}}}$$
(2)
$$\Delta_{\text{r}} G^{\text{surf}} \left( {T,p_{{ ( {\text{g)}}}} } \right) = \frac{{2\,V_{{{\text{m(SnO}}_{ 2} )}} \gamma_{{ ( {\text{SnO}}_{ 2} )}} }}{{r_{{ ( {\text{SnO}}_{ 2} )}} }} - \frac{{2\,\,V_{\text{m(Sn)}} \gamma_{{ ( {\text{Sn)}}}} }}{{r_{{ ( {\text{Sn)}}}} }}$$
(3)

If the stoichiometric amounts of Sn and SnO2 are considered, then

$$\frac{{n_{{ ( {\text{SnO}}_{ 2} )}} }}{{n_{{ ( {\text{Sn)}}}} }} = \frac{{r_{{ ( {\text{SnO}}_{ 2} )}}^{3} }}{{V_{{{\text{m(SnO}}_{ 2} )}} }}\frac{{V_{\text{m(Sn)}} }}{{r_{\text{Sn}}^{3} }}\quad \Rightarrow \quad r_{{ ( {\text{SnO}}_{ 2} )}} = r_{{ ( {\text{Sn)}}}} \left( {\frac{{{\kern 1pt} V_{{{\text{m(SnO}}_{ 2} )}} }}{{V_{\text{m(Sn)}} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}}}$$
(4)

Equation 4 can be used for substituting r(SnO2) into the equilibrium relation resulting from Eq 1 to 3

$$RT\ln \frac{{p_{{({\text{O}}_{ 2} ),{\text{np}}}} }}{{p_{{({\text{O}}_{ 2} ),{\text{bulk}}}} }} = \frac{{2\,V_{{{\text{m(SnO}}_{ 2} )}} \gamma_{{ ( {\text{SnO}}_{ 2} )}} }}{{r_{{ ( {\text{SnO}}_{ 2} )}} }} - \frac{{2\,\,V_{\text{m(Sn)}} \gamma_{{ ( {\text{Sn)}}}} }}{{r_{{ ( {\text{Sn)}}}} }}$$
(5)

which represents an implicit function of temperature, oxygen partial pressure and tin nanoparticle size, F(T, p(O2),np, r(Sn)) = 0. It is thus possible to calculate the dependence of p(O2) on r(Sn) at given temperature or the dependence of T on r(Sn) at given oxygen pressure. Similar calculations can be performed for Sn-SnO and Sn-Sn3O4 equilibria.

Using this topology it is convenient to introduce the standard chemical potential of a single-component nanoparticle as

$$\mu_{i}^{\text{o,np}} \left( {T,p^{\text{o}} } \right) = \mu_{i}^{\text{o,bulk}} \left( {T,p^{\text{o}} } \right) + \mu_{i}^{\text{surf}} \left( {T,p^{\text{o}} } \right) = G_{{{\text{m}},i}}^{\text{o,bulk}} \left( {T,p^{\text{o}} } \right) + \frac{{2\,V_{{{\text{m}},i}} \gamma_{i} }}{{r_{i} }}$$
(6)

At the temperature of tin fusion, 505.1 K, the surface contributions for tin are: 21.787/r(nm) (kJ/mol) for solid and 19.841/r(nm) (kJ/mol) for liquid Sn. Temperature dependences of the molar volumes and the surface energies were neglected in the following calculations.

The molar volumes, surface energies and surface contribution for tin oxides are summarized in Table 3. Surface energy is the crucial quantity for the surface contribution to the Gibbs energy of nanoparticles. In the case of SnO2, the value γ(SnO2,sg) = 1200 mJ m−2[33] was chosen. This value based on adsorption calorimetry measurements is markedly lower than 1720 mJ m−2 as previously obtained at the same laboratory using a combination of high-temperature oxide melt solution calorimetry and water adsorption calorimetry.[34] On the other hand ab initio calculated values for SnO2/vacuum interface for various (hkl) surface planes give mean values 1261 mJ m−2,[35] 1438 mJ m−2,[36] 1407 mJ m−2.[37] These mean values were calculated according to the expression[38]

$$\bar{\gamma } = \frac{n}{{\sum\nolimits_{(hkl)} {\frac{1}{{\gamma_{(hkl)} }}} }}$$
(7)

where n means the number of (hkl) surface planes for which γ(hkl) is calculated. Only one calculated value γ(SnO,sg) = 392 mJ m−2[37] is available in literature for SnO surface. As in the case of SnO2 this mean value was obtained using ab initio γ(hkl) values according to Eq 7. Due to the lack or the complete absence of surface energy data for SnO and Sn3O4 we performed DFT calculations for slabs consisting of six and four unit cells stacked along [001] and [010] directions, respectively, with a 15 Å thick vacuum region inserted between the slabs. The calculations were carried out using all electron full potential method as implemented in WIEN2k program (APW + lo basis set, GGA-PBE96 functional, RMT * Kmax in the range 7.5-8, typical k-mesh sampling density 0.8 nm3).[39] As expected, the resulting surface energy values (see Table 3) are much lower compared to rutile structure of SnO2 due to the layered character imposing a relatively weak cohesion between the layers in both SnO and Sn3O4.

Table 3 Basic physicochemical data for solid tin oxides

Using Eq 5 in the form

$$\log \frac{{p_{{({\text{O}}_{ 2} ),{\text{np}}}} }}{{p_{{({\text{O}}_{ 2} ),{\text{bulk}}}} }} = \frac{1}{RT\ln 10}\left( {\frac{{2\,V_{{{\text{m(SnO}}_{ 2} )}} \gamma_{{ ( {\text{SnO}}_{ 2} )}} }}{{r_{{ ( {\text{SnO}}_{ 2} )}} }} - \frac{{2\,\,V_{\text{m(Sn,l)}} \gamma_{{ ( {\text{Sn,l)}}}} }}{{r_{{ ( {\text{Sn,l)}}}} }}} \right)$$
(8)

the ratio of O2 equilibrium pressure in Sn(l)-SnO2(s)-O2(g) nano-system and bulk-system as a function of temperature for tin nanoparticles of radius 2.5, 5, 10 and 20 nm was calculated. As the surface term for solid SnO2 is greater than that for liquid tin, the decrease of nanoparticle size brings about an increase of this ratio (Fig. 3).

Fig. 3
figure 3

The ratio of O2 equilibrium pressure in Sn(l)-SnO2(s)-O2(g) nano-system and bulk-system as a function of temperature (r(Sn) = 2.5, 5, 10 and 20 nm)

Similar calculations were performed for reaction 1.5 Sn(l) + O2(g) = 0.5 Sn3O4(s) using Eq 9

$$\log \frac{{p_{{({\text{O}}_{ 2} ),{\text{np}}}} }}{{p_{{({\text{O}}_{ 2} ),{\text{bulk}}}} }} = \frac{1}{RT\ln 10}\left( {\frac{{\,V_{{{\text{m(Sn}}_{ 3} {\text{O}}_{ 4} )}} \gamma_{{ ( {\text{Sn}}_{ 3} {\text{O}}_{ 4} )}} }}{{r_{{ ( {\text{Sn}}_{ 3} {\text{O}}_{ 4} )}} }} - \frac{{3\,\,V_{\text{m(Sn,l)}} \gamma_{{ ( {\text{Sn,l)}}}} }}{{r_{{ ( {\text{Sn,l)}}}} }}} \right)$$
(9)

Since the surface contributions for solid Sn3O4 are smaller than that for liquid tin, we observe an opposite trend in this case, namely a decrease of the oxygen pressure ratio with a decrease of nanoparticle size (Fig. 4).

Fig. 4
figure 4

The ratio of O2 equilibrium pressure in Sn(l)-Sn3O4(s)-O2(g) nano-system and bulk-system as a function of temperature (r(Sn) = 2.5, 5, 10 and 20 nm)

Using the surface terms defined by 2Vmγsg values given in Table 3 we can calculate the Sn-O phase diagram for a selected nanoparticle size. The calculated phase diagram presented in Fig. 5 for tin oxides nanoparticles whose sizes are deduced from solid tin nanoparticles of radius 5 nm exhibits notable differences compared to the bulk version. Indeed, the SnO and Sn3O4 stability is substantially enhanced at the expense of SnO2 due to the size effect (significantly larger surface energy of SnO2). Furthermore, there is only one invariant point in the temperature range 300-800 K, namely Sn(s)-SnO(s)-Sn3O4(s) at T = 460 K and p(O2)/p° = 4.6 × 10−58.

Fig. 5
figure 5

Phase diagram of the system Sn-O at nanoscale (sizes of nanoparticles included into the calculation are derived from r(Sn,s) = 5 nm considering the differences in molar volumes: r(Sn,l) = 5.04 nm, r(SnO) = 5.42 nm, r(Sn3O4) = 5.86 nm, r(SnO2) = 5.47 nm)

Core–Shell Topology

According to the most experimental findings, the oxidation of Sn nanoparticles leads to a core–shell geometry. This situation is depicted in Fig. 6. The equilibrium condition for the reaction Sn(l) + O2(g) = SnO2(s) is derived in “Appendix” (Eq 23, 27, 29, 30 and 31). Using a material balance (r0 = r(Sn,l,np), r = r(Sn,l,core)):

$$n_{{{\text{SnO}}_{ 2} }} = \frac{4\pi }{{3\,V_{{{\text{m(SnO}}_{ 2} )}} }}\left[ {\left( {r + \delta } \right)^{3} - r^{3} } \right] = n_{\text{Sn}}^{\text{o}} - n_{\text{Sn}} = \frac{4\pi }{{3\,V_{\text{m(Sn)}} }}\left( {r_{0}^{3} - r^{3} } \right)\quad$$
(10)

we obtain for the SnO2 shell thickness

Fig. 6
figure 6

Spatial configuration of core–shell system Sn-O ((α) = Sn(s,l), (β) = SnOx(s))

$$\delta = \left[ {\frac{{V_{{{\text{m(SnO}}_{ 2} )}} }}{{V_{\text{m(Sn)}} }}r_{0}^{3} + \left( {1 - \frac{{V_{{{\text{m(SnO}}_{ 2} )}} }}{{V_{\text{m(Sn)}} }}} \right)r^{3} } \right]^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}}} - r$$
(11)

Similarly to Eq 1, ΔrGnp(T,p(O2)np) = 0 at equilibrium and

$$RT\ln \frac{{p_{{({\text{O}}_{ 2} ),{\text{np}}}} }}{{p_{{({\text{O}}_{ 2} ),{\text{bulk}}}} }} = \left( {1 + x} \right)\left( {\frac{{2\,\gamma_{{({\text{SnO}}_{ 2} , {\text{s}})}} V_{{{\text{m}}({\text{SnO}}_{ 2} , {\text{s}})}} }}{r + \delta } - \frac{{2\,\gamma_{{({\text{Sn,l}})}} V_{{{\text{m}}({\text{Sn,l}})}} }}{{r_{0} }}} \right) + x\frac{{2\,\gamma_{{({\text{Sn/SnO}}_{ 2} )}} V_{{{\text{m}}({\text{Sn,l}})}} }}{r}$$
(12)

Equation 12 represents a relation F(T, p(O2),np, r0, δ) = 0 and it is possible to calculate the dependence of δ on temperature or oxygen pressure at a given value o r0. The x parameter stands for the amount of unoxidized molten tin when 1 mol of SnO2 is formed as the shell of nanoparticles: x = n(Sn,l,core)/(n(Sn,l,np) − n(Sn,l,core)), which can be expressed as

$$x = \frac{1}{{\left( {{{r_{0} } \mathord{\left/ {\vphantom {{r_{0} } r}} \right. \kern-0pt} r}} \right)^{3} - 1}}$$
(13)

It should be noted that for x = 0 (r → 0), Eq 12 is identical with the previously introduced Eq 5 with r + δ = r(SnO2,s) and r0 = r(sn).

One more parameter, namely the interfacial energy γ(Sn/SnO2) at the liquid tin/solid SnO2 interface, is included in Eq 12. It can be assessed from contact angle of Sn(l) sessile drop on SnO2(s) substrate measurement, but such a measurement has not been performed yet. Alternatively, it can be calculated from the work of adhesion Wad for Sn(l)/SnO2(s) interface and the surface energies of Sn(l) and SnO2(s) as

$$\gamma_{{ ( {\text{Sn,l/SnO}}_{ 2} )}} = \gamma_{{ ( {\text{SnO}}_{ 2} )}} + \gamma_{{ ( {\text{Sn,l)}}}} - W_{\text{ad}}$$
(14)

Some empirical estimates for the work of adhesion at liquid metal/solid oxide interfaces or just for interfacial energy at liquid metal/solid oxide interfaces have been proposed in literature.[42,43,44,45,46] Here we use a model based on broken bond approximation on the SnO2 interface. While the Sn-O bond energy in bulk SnO2 is 229.5 kJ mol−1 (obtained from the cohesive energy), the value Eb(Sn-O) = 40.5 kJ mol−1 assessed from the surface energy 1200 mJ m−2 (Table 3) and the number of broken Sn-O bonds on (001) surface represents only to 18 per cent of the bulk value. This is likely due to relaxation of both the valence electron density and the atomic positions close to the surface. Considering that oxygen atoms on the surface are lacking this portion of energy to saturate their bonding state and there are enough Sn atoms available for bonding on the Sn(l) counterpart (1.22 Sn per O), we can evaluate the adhesion work as Wad = = 600 mJ m−2. Let us note that this result is in a very good agreement with the values obtained from models[42] (Wad = = 688 mJ m−2) based on partial molar enthalpy of oxygen dissolution in liquid tin, \(\Delta \bar{H}_{{ [ {\text{O]Sn(l)}}}}^{\infty }\) = − 182 kJ mol−1.[47] Substituting our value of Wad into Eq 14 we obtain the corresponding interfacial energy γ(Sn/SnO2) = 1190 mJ m−2.

Equation 12 was used for the calculation of the surface oxide layer thickness δ (expressed as δ/(r + δ)) as a function of relative equilibrium O2 pressure (at T = 800 K) and temperature (at p(O2,np)/p(O2,bulk) = 5) for a given initial tin particle radius r0 (Fig. 7 and 8). Due to the large interfacial energy γ(Sn/SnO2) the core–shell structure is not a stable configuration with respect to the complete SnO2 particle and thus the oxidation of tin particles proceeds spontaneously under these T and p(O2) conditions. This is imposed by a descending character of the δ/(r + δ) versus log (p(O2,np/p(O2,bulk)) curve, where for a given oxygen pressure the core–shell structure corresponding to a point on the calculated curve has always higher Gibbs energy compared to a single phase SnO2 nanoparticle with δ/(r + δ) = 1. The experimentally observed core–shell structures formed on Sn(l) oxidation are thus the result of kinetic effects (diffusion of oxygen).

Fig. 7
figure 7

The dependence of SnO2 surface layer thickness δ (expressed as δ/(r + δ) as a function of relative equilibrium O2 pressure in Sn(l)-SnO2(s)-O2(g) nano-system (T = 800 K, r0 = 2.5, 5, 10 and 20 nm)

Fig. 8
figure 8

The dependence of SnO2 surface layer thickness δ (expressed as δ/(r + δ) as a function of temperature in Sn(l)-SnO2(s)-O2(g) nano-system (p(O2,np)/p(O2,bulk) = 5, r0 = 2.5, 5, 10 and 20 nm)

Similar calculations were performed for the shell composed of Sn3O4 using Eq 15

$$\frac{2}{3}RT\ln \frac{{p_{{({\text{O}}_{ 2} ),{\text{np}}}} }}{{p_{{({\text{O}}_{ 2} ),{\text{bulk}}}} }} = \left( {\frac{1}{3} + x} \right)\frac{{2\,\gamma_{{({\text{Sn}}_{ 3} {\text{O}}_{ 4} , {\text{s}})}} V_{{{\text{m}}({\text{Sn}}_{ 3} {\text{O}}_{ 4} , {\text{s}})}} }}{r + \delta } - \left( {1 + x} \right)\frac{{2\,\gamma_{{({\text{Sn,l}})}} V_{{{\text{m}}({\text{Sn,l}})}} }}{{r_{0} }} + x\frac{{2\,\gamma_{{({\text{Sn/Sn}}_{ 3} {\text{O}}_{ 4} )}} V_{{{\text{m}}({\text{Sn,l}})}} }}{r}$$
(15)

which was derived in the similar way as the relations for SnO2 shell (see “Appendix”). In this case it holds for the surface layer thickness

$$\delta = \left[ {\frac{{V_{{{\text{m(Sn}}_{ 3} {\text{O}}_{ 4} )}} }}{{3\,V_{\text{m(Sn)}} }}r_{0}^{3} + \left( {1 - \frac{{V_{{{\text{m(SnO}}_{ 2} )}} }}{{3\,V_{\text{m(Sn)}} }}} \right)r^{3} } \right]^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}}} - r$$
(16)

The contribution of work of adhesion to the interfacial energy is much smaller for the Sn(l)/Sn3O4(s) interface due to very low value of γ(Sn3O4) (Table 3). As mentioned above, Sn3O4 exhibits a layered structure which can be derived from rutile-SnO2 by periodically inserting 1/3 of oxygen vacancies arranged in parallel layers. Hence, only weak Sn-Sn bonds are broken when an interface is formed at this vacancy-rich plane, and these interactions are assumed to be re-established on the interface with liquid Sn. The respective interfacial energy can be thus put equal to that of liquid tin, γ(Sn/Sn3O4) = γ(Sn,lg) = 588 mJ mol−2 (Table 2). Using Eq 15 the surface oxide layer thickness δ (expressed as δ/(r + δ)) as a function of relative equilibrium O2 pressure (at T = 800 K) and temperature (at p(O2,np)/p(O2,bulk) = 5) for a given initial tin particle radius r0 were calculated (Fig. 9 and 10). The same conclusion regarding the instability of core–shell structure with respect to a single phase Sn3O4 nanoparticle can be drawn as in the case of Sn/SnO2 particles but with the difference that Sn/Sn3O4 core–shell structures are more stable (closer to the equilibrium state) than the Sn/SnO2 ones.

Fig. 9
figure 9

The dependence of Sn3O4 surface layer thickness δ (expressed as δ/(r + δ) as a function of relative equilibrium O2 pressure in Sn(l)-Sn3O4(s)-O2(g) nano-system (T = 800 K, r0 = 2.5, 5, 10 and 20 nm)

Fig. 10
figure 10

The dependence of Sn3O4 surface layer thickness δ (expressed as δ/(r + δ) as a function of temperature in Sn(l)-Sn3O4(s)-O2(g) nano-system (p(O2,np)/p(O2,bulk) = 5, r0 = 2.5, 5, 10 and 20 nm)

Discussion

Thermodynamic calculations whose results are presented here show a substantial influence of the considered system topology. In the case of single-component nanoparticles the system formed by two independent components (tin and oxygen) has three degree of freedom (T, p(O2), r(Sn)) and it is possible to formulate three distinct dependences: T = f(p(O2)) at [r(Sn)], T = f(r(Sn)) at [p(O2)] and p(O2) = f(r(Sn)) at [T]. Thus, for the constant value of r(Sn), the oxygen pressure p(O2) is unambiguously determined by temperature (Fig. 3 and 4). On the other hand, core–shell topology brings four degrees of freedom (T, p(O2), r0, δ). For a given size of tin nanoparticles r0, the thickness of surface oxide layer δ is determined by the oxygen pressure (Fig. 7 and 9) at constant temperature or by the temperature (Fig. 8 and 10) at constant oxygen pressure. Our results, which show the instability of core–shell structure, are perhaps surprising. Using the core–shell model Vegh and Kaptay[29] have shown that in the case of lead melting, a core solid and a liquid shell is found in a finite temperature range below the macroscopic melting point. It is a consequence of the simple relation γ(Pb,sg) > γ(Pb,sl) + γ(Pb,lg). In analogy, if the relation γ(Me) > γ(Me/MeOx) + γ(MeOx) was satisfied, a core-shell structure might be stable. But it is not the case for many of the liquid metal/solid oxide interfaces including our system under study. By contrast, in an opposite process of SnO2 reduction into liquid Sn, the SnO2 core—Sn(l) shell structure would be stable for a certain range of conditions. Another aspect that should be mentioned is the sensitivity of the calculated results on the input parameters. While thermodynamic data for pure substances are rather reliable, the values of surface/interface energies are inconsistent. Considering two single-component particles Sn(l), r(sn) = 2.5 nm, and SnO2(s) as an example, the calculated oxygen pressures for γ(SnO2) = 1720 mJ m−2[31] are 2-28 times higher than those for γ(SnO2) = 1200 mJ m−2[30] in the temperature range 1300-300 K. The same applies for the interfacial energy γ(Sn/SnO2) within the core–shell topology. Also other approximations considered in the calculations, e.g. spherical shape of nanoparticles and temperature-independent molar volumes and surface/interface energy, bring some uncertainty in the results. Therefore, the results of these calculations must be understood as semi-quantitative estimate rather than the exact values.

The last issue we would like to mention here is the opposite influence of nanosizing on the stability of SnO2 and SnO/Sn3O4 oxides within the two single-component nanoparticles model. It is due to the surface contribution which is higher for SnO2(s) than Sn(l) while lower for SnO(s) and Sn3O4(s) compared to Sn(l). This situation can explain some experimental findings during oxidation of Sn nanoparticles, namely an increased stability of SnO(s) and Sn3O4(s) with respect to both liquid tin and solid tin dioxide.

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Acknowledgments

This work was supported by Czech Science Foundation, Grant No. 17-13161S.

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Correspondence to Jindřich Leitner.

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This invited article is part of a special issue of the Journal of Phase Equilibria and Diffusion in honor of Prof. Jan Vrestal’s 80th birthday. This special issue was organized by Prof. Andrew Watson, Coventry University, and Dr. Ales Kroupa, Institute of Physics of Materials.

Appendices

Appendix 1

Let us consider a closed heterogeneous system at constant T and V. It follows for the equilibrium from basic thermodynamics relations

$${\text{d}}F_{\text{syst}} = \sum\nolimits_{\text{bulk phases}} {{\text{d}}F_{(\varphi )} } + \sum\nolimits_{\text{interfaces}} {{\text{d}}F_{(\sigma )} } = 0,\quad \left[ {n,T,V^{\text{syst}} } \right]$$
(17)

F is the Helmholtz energy and superscripts φ and σ stand for coexisting bulk phases and interfaces/surfaces, respectively. Let us now describe the equilibrium in the system in Fig. 2 where the reaction Sn(l) + O2(g) = SnO2(s) takes place ((α) = Sn(l) and (β) = SnO2(s)) in terms of the condition (17). The equilibrium corresponds to coexistence of two single-component particles Sn(l) and SnO2(s) and gaseous oxygen at temperature T and oxygen pressure p(O2,g):

$${\text{d}}F_{\text{syst}} = {\text{d}}F_{{({\text{Sn,l}})}}^{\text{bulk}} + {\text{d}}F_{{({\text{SnO}}_{ 2} , {\text{s}})}}^{\text{bulk}} + {\text{d}}F_{{({\text{O}}_{ 2} , {\text{g}})}}^{\text{bulk}} + {\text{d}}F_{{({\text{Sn,l}})}}^{\text{surf}} + {\text{d}}F_{{({\text{SnO}}_{ 2} , {\text{s}})}}^{\text{surf}}$$
(18)
$${\text{d}}F_{{({\text{Sn,l}})}}^{\text{bulk}} = - p_{{ ( {\text{Sn,l)}}}} {\text{d}}V_{{ ( {\text{Sn,l)}}}} + \mu_{{ ( {\text{Sn,l)}}}} (p_{{ ( {\text{Sn,l)}}}} ){\text{d}}n_{{ ( {\text{Sn,l)}}}}$$
(19)
$${\text{d}}F_{{({\text{SnO}}_{ 2} , {\text{s}})}}^{\text{bulk}} = - p_{{ ( {\text{SnO}}_{ 2} , {\text{s)}}}} {\text{d}}V_{{ ( {\text{SnO}}_{ 2} , {\text{s)}}}} + \mu_{{ ( {\text{SnO}}_{ 2} , {\text{s)}}}} (p_{{ ( {\text{SnO}}_{ 2} , {\text{s)}}}} ){\text{d}}n_{{ ( {\text{SnO}}_{ 2} , {\text{s)}}}}$$
(20)
$${\text{d}}F_{{({\text{O}}_{ 2} , {\text{g}})}}^{\text{bulk}} = - p_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} {\text{d}}V_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} + \mu_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} (p_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} ){\text{d}}n_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} = p_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} \left( {{\text{d}}V_{{ ( {\text{Sn,l)}}}} + {\text{d}}V_{{ ( {\text{SnO}}_{ 2} , {\text{s)}}}} } \right) + \mu_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} (p_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} ){\text{d}}n_{{ ( {\text{O}}_{ 2} , {\text{g)}}}}$$
(21)

Summation of these bulk contributions yields

$$\begin{aligned} & - \left( {p_{{ ( {\text{Sn,l)}}}} - p_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} } \right){\text{d}}V_{{ ( {\text{Sn,l)}}}} + \mu_{{ ( {\text{Sn,l)}}}} (p_{{ ( {\text{Sn,l)}}}} ){\text{d}}n_{{ ( {\text{Sn,l)}}}} \\ & \quad - \left( {p_{{ ( {\text{SnO}}_{ 2} , {\text{s)}}}} - p_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} } \right){\text{d}}V_{{ ( {\text{SnO}}_{ 2} , {\text{s)}}}} + \mu_{{ ( {\text{SnO}}_{ 2} , {\text{s)}}}} (p_{{ ( {\text{SnO}}_{ 2} , {\text{s)}}}} ){\text{d}}n_{{ ( {\text{SnO}}_{ 2} , {\text{s)}}}} + \mu_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} (p_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} ){\text{d}}n_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} \\ \end{aligned}$$
(22)

Supposing incompressibility of liquid tin and solid oxide (molar volume does not depend on pressure) and applying material balance conditions in the stoichiometric form \(n_{i} = n_{i}^{o} + \nu_{i} \xi ,\;(\nu_{{({\text{Sn}},{\text{l}})}} = - \,1,\;\nu_{{ ( {\text{O2,g)}}}} = - \,1,\;\nu_{{ ( {\text{SnO2,s)}}}} = 1)\) one can arrange Eq 22 as

$$\begin{aligned} & \left[ {\mu_{{ ( {\text{Sn,l)}}}} (p_{{ ( {\text{Sn,l)}}}} ) - \left( {p_{{ ( {\text{Sn,l)}}}} - p_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} } \right)V_{\text{m(Sn,l)}} } \right]{\text{d}}n_{{ ( {\text{Sn,l)}}}} \\ & \quad + \left[ {\mu_{{ ( {\text{SnO}}_{ 2} , {\text{s)}}}} (p_{{ ( {\text{SnO}}_{ 2} , {\text{s)}}}} ) - \left( {p_{{ ( {\text{SnO}}_{ 2} , {\text{s)}}}} - p_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} } \right)V_{{{\text{m(SnO}}_{ 2} , {\text{s)}}}} } \right]{\text{d}}n_{{ ( {\text{SnO}}_{ 2} , {\text{s)}}}} + \mu_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} (p_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} ){\text{d}}n_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} \\ & \quad = \left[ {\nu_{{ ( {\text{Sn,l)}}}} \mu_{{ ( {\text{Sn,l)}}}} (p_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} ) + \nu_{{ ( {\text{SnO}}_{ 2} , {\text{s)}}}} \mu_{{ ( {\text{SnO}}_{ 2} , {\text{s)}}}} (p_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} ) +\, \nu_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} \mu_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} (p_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} )} \right]{\text{d}}\xi = \Delta_{\text{r}} G^{\text{bulk}} (p_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} ){\text{d}}\xi \\ \end{aligned}$$
(23)

The surface terms in Eq 18 can be expressed as

$${\text{d}}F_{{({\text{Sn,l}})}}^{\text{surf}} = \gamma_{{({\text{Sn,l}})}} {\text{d}}A_{{({\text{Sn,l}})}} = \gamma_{{({\text{Sn,l}})}} \frac{{{\text{d}}A_{{({\text{Sn,l}})}} }}{{{\text{d}}V_{{({\text{Sn,l}})}} }}\frac{{{\text{d}}V_{{({\text{Sn,l}})}} }}{{{\text{d}}n_{{({\text{Sn,l}})}} }}{\text{d}}n_{{({\text{Sn,l}})}} = \gamma_{{({\text{Sn,l}})}} \frac{{2\,V_{{{\text{m}}({\text{Sn,l}})}} }}{{r_{{({\text{Sn,l}})}} }}\nu_{{({\text{Sn,l}})}} {\text{d}}\xi$$
(24)
$${\text{d}}F_{{({\text{SnO}}_{ 2} , {\text{s}})}}^{\text{surf}} = \gamma_{{({\text{SnO}}_{ 2} , {\text{s}})}} {\text{d}}A_{{({\text{SnO}}_{ 2} , {\text{s}})}} = \gamma_{{({\text{SnO}}_{ 2} , {\text{s}})}} \frac{{{\text{d}}A_{{({\text{SnO}}_{ 2} , {\text{s}})}} }}{{{\text{d}}V_{{({\text{SnO}}_{ 2} , {\text{s}})}} }}\frac{{{\text{d}}V_{{({\text{SnO}}_{ 2} , {\text{s}})}} }}{{{\text{d}}n_{{({\text{SnO}}_{ 2} , {\text{s}})}} }}{\text{d}}n_{{({\text{SnO}}_{ 2} , {\text{s}})}} = \gamma_{{({\text{SnO}}_{ 2} , {\text{s}})}} \frac{{2\,V_{{{\text{m}}({\text{SnO}}_{ 2} , {\text{s}})}} }}{{r_{{({\text{SnO}}_{ 2} , {\text{s}})}} }}\nu_{{({\text{SnO}}_{ 2} , {\text{s}})}} {\text{d}}\xi$$
(25)

and the summation gives

$$\left[ {\nu_{{({\text{Sn,l}})}} \frac{{2\,\gamma_{{({\text{Sn,l}})}} V_{{{\text{m}}({\text{Sn,l}})}} }}{{r_{{({\text{Sn,l}})}} }} + \nu_{{({\text{SnO}}_{ 2} , {\text{s}})}} \frac{{2\,\gamma_{{({\text{SnO}}_{ 2} , {\text{s}})}} V_{{{\text{m}}({\text{SnO}}_{ 2} , {\text{s}})}} }}{{r_{{({\text{SnO}}_{ 2} , {\text{s}})}} }}} \right]{\text{d}}\xi = \Delta_{\text{r}} G^{\text{surf}} {\text{d}}\xi$$
(26)

dFsyst = 0 at equilibrium thus

$$\Delta_{\text{r}} G^{\text{bulk}} (p_{{ ( {\text{O}}_{ 2} , {\text{g)}}}} ) + \Delta_{\text{r}} G^{\text{surf}} = 0$$
(27)

Somewhat more complicated is the derivation of equilibrium condition in the system in Fig. 6 where the same reaction Sn(l) + O2(g) = SnO2(s) takes place ((α) = Sn(l) and (β) = SnO2(s)). The equilibrium corresponds to a coexistence of unoxidized Sn(l) core and SnO2(s) shell in gaseous oxygen at temperature T and oxygen pressure p(O2,g). Whereas Eq 19-23 for bulk contribution hold as in the previous case, the surface contribution is qualitatively different. Due to different interfaces of Sn(l,np) nanoparticle and unoxidized Sn(l,core) the oxidation reaction should be written as (1 + x) Sn(l,np) + O2(g) = SnO2(s) + x Sn(l,core) however, the bulk properties of Sn(l,np) and x Sn(l,core) are the same. Thus the surface contribution should read

$${\text{d}}F^{\text{surf}} = {\text{d}}F_{{({\text{Sn,l,np}})}}^{\text{surf}} + {\text{d}}F_{{({\text{Sn,l,core}})}}^{\text{surf}} + {\text{d}}F_{{({\text{SnO}}_{ 2} , {\text{s}})}}^{\text{surf}}$$
(28)
$${\text{d}}F_{{({\text{Sn,l,np}})}}^{\text{surf}} = \gamma_{{({\text{Sn,l}})}} {\text{d}}A_{{({\text{Sn,l,np}})}} = \gamma_{{({\text{Sn,l}})}} \frac{{{\text{d}}A_{{({\text{Sn,l,np}})}} }}{{{\text{d}}V_{{({\text{Sn,l,np}})}} }}\frac{{{\text{d}}V_{{({\text{Sn,l,np}})}} }}{{{\text{d}}n_{{({\text{Sn,l,np}})}} }}{\text{d}}n_{{({\text{Sn,l,np}})}} = \gamma_{{({\text{Sn,l}})}} \frac{{2\,V_{{{\text{m}}({\text{Sn,l}})}} }}{{r_{{({\text{Sn,l,np}})}} }}\nu_{{({\text{Sn,l,np}})}} {\text{d}}\xi$$
(29)
$${\text{d}}F_{{({\text{Sn,l,core}})}}^{\text{surf}} = \gamma_{{({\text{Sn/SnO}}_{ 2} )}} {\text{d}}A_{{({\text{Sn,l,core}})}} = \gamma_{{({\text{Sn/SnO}}_{ 2} )}} \frac{{{\text{d}}A_{{({\text{Sn,l,core}})}} }}{{{\text{d}}V_{{({\text{Sn,l,core}})}} }}\frac{{{\text{d}}V_{{({\text{Sn,l,core}})}} }}{{{\text{d}}n_{{({\text{Sn,l,core}})}} }}{\text{d}}n_{{({\text{Sn,l,core}})}} = \gamma_{{({\text{Sn/SnO}}_{ 2} )}} \frac{{2\,V_{{{\text{m}}({\text{Sn,l}})}} }}{{r_{{({\text{Sn,l,core}})}} }}\nu_{{({\text{Sn,l,core}})}} {\text{d}}\xi$$
(30)
$$\begin{aligned} {\text{d}}F_{{({\text{SnO}}_{ 2} , {\text{s}})}}^{\text{surf}} & = \gamma_{{({\text{SnO}}_{ 2} , {\text{s}})}} {\text{d}}A_{{({\text{SnO}}_{ 2} , {\text{s}})}} = \gamma_{{({\text{SnO}}_{ 2} , {\text{s}})}} \frac{{{\text{d}}A_{{({\text{SnO}}_{ 2} , {\text{s}})}} }}{{{\text{d}}V_{{({\text{core/shell}})}} }}\frac{{{\text{d}}V_{{({\text{core/shell}})}} }}{{{\text{d}}n_{{({\text{core/shell}})}} }}{\text{d}}n_{{({\text{core/shell}})}} \\ \,& \doteq\, \gamma_{{({\text{SnO}}_{ 2} , {\text{s}})}} \frac{{2\,V_{{{\text{m}}({\text{SnO}}_{ 2} , {\text{s}})}} }}{{r_{{({\text{Sn,l,core}})}} + \delta }}\left( {\nu_{{({\text{SnO}}_{ 2} , {\text{s}})}} + \nu_{{({\text{Sn,l,core}})}} } \right){\text{d}}\xi \\ \end{aligned}$$
(31)

To simplify Eq 31 the difference between molar volumes of Sn(l) and SnO2(s) was neglected in the derivative dV(core/shell)/dn(core/shell). Since (ν(Sn,l,np) = − (1 + x), ν(SnO2,s) = 1, ν(Sn,l,core) = x) the surface contribution has got a form

$$\begin{aligned} & \left[ { - \left( {1 + x} \right)\frac{{2\,\gamma_{{({\text{Sn,l}})}} V_{{{\text{m}}({\text{Sn,l}})}} }}{{r_{{({\text{Sn,l,np}})}} }} + x\frac{{2\,\gamma_{{({\text{Sn/SnO}}_{ 2} )}} V_{{{\text{m}}({\text{Sn,l}})}} }}{{r_{{({\text{Sn,l,core}})}} }} +\, \left( {1 + x} \right)\frac{{2\,\gamma_{{({\text{SnO}}_{ 2} , {\text{s}})}} V_{{{\text{m}}({\text{SnO}}_{ 2} , {\text{s}})}} }}{{r_{{({\text{Sn,l,core}})}} + \delta }}} \right]{\text{d}}\xi \\ & \quad = \left[ {\left( {1 + x} \right)\left( {\frac{{2\,\gamma_{{({\text{SnO}}_{ 2} , {\text{s}})}} V_{{{\text{m}}({\text{SnO}}_{ 2} , {\text{s}})}} }}{{r_{{({\text{Sn,l,core}})}} + \delta }} - \frac{{2\,\gamma_{{({\text{Sn,l}})}} V_{{{\text{m}}({\text{Sn,l}})}} }}{{r_{{({\text{Sn,l,np}})}} }}} \right) +\, x\frac{{2\,\gamma_{{({\text{Sn/SnO}}_{ 2} )}} V_{{{\text{m}}({\text{Sn,l}})}} }}{{r_{{({\text{Sn,l,core}})}} }}} \right]{\text{d}}\xi = \Delta_{\text{r}} G^{\text{surf}} {\text{d}}\xi \\ \end{aligned}$$
(32)

Equation 27 is valid and applicable for both topologies, two single-component particles and core–shell model, with the difference in surface term being described by Eq 26 and 32, respectively.

Appendix 2

List of symbols

A :

Surface area

E b :

Bond energy

F :

Helmholtz energy

G :

Gibbs energy

ΔrG:

The change of Gibbs energy due to reaction (reaction Gibbs energy)

ΔrGo:

The change of standard Gibbs energy due to reaction (standard reaction Gibbs energy)

\(\Delta \bar{H}_{{ [ {\text{O]Sn(l)}}}}^{\infty }\) :

The partial molar enthalpy of oxygen dissolution in liquid tin

n i :

The number of moles of the species i

\(n_{i}^{o}\) :

The initial number of moles of the species i

p :

Pressure

R :

The gas constant

r :

Radius of spherical nanoparticle

T :

Temperature

T F :

Temperature of fusion

V m :

Molar volume

W ad :

Work of adhesion

δ :

Thickness of surface layer

γ :

Surface or interfacial energy

μ i :

The chemical potential of the species i

\(\mu_{i}^{o}\) :

The standard chemical potential of the species i (at pressure po = 100 kPa)

ν i :

The stoichiometric coefficient of the species i

ξ :

Extent of reaction

bulk:

Related to bulk properties (bulk contribution)

np:

Related to nanoparticle

surf:

Related to surface properties (surface contribution)

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Leitner, J., Sedmidubský, D. Thermodynamic Modeling of Oxidation of Tin Nanoparticles. J. Phase Equilib. Diffus. 40, 10–20 (2019). https://doi.org/10.1007/s11669-018-0686-4

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Keywords

  • surface energy
  • thermodynamic modeling
  • tin nanoparticles
  • tin oxides