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Size-Dependent Phase Transformations in Bismuth Oxide Nanoparticles. II. Melting and Stability Diagram Gerrit Guenther,†,§ Ralf Theissmann,‡ and Olivier Guillon*,§ †

Technische Universitaet Darmstadt, Institute of Materials Science, Petersenstr. 23, 64287 Darmstadt, Germany Universitaet Duisburg-Essen, Nanostructures and Technology, Department of Engineering, 47057 Duisburg, Germany § Friedrich Schiller University of Jena, Otto Schott Institute of Materials Research, Loebdergraben 32, 07743 Jena, Germany ‡

S Supporting Information *

ABSTRACT: Melting of nanocrystalline bismuth oxide particles between 6 and 50 nm was investigated in situ in the transmission electron microscope (TEM). It revealed a size-dependent melting behavior with a strong melting point reduction (−55% at 6 nm). One reason is a −230 K offset in the bulk melting temperature which is apparently caused by the β-phase in which the nanomaterial resides. As a second reason, a strong size dependency was observed from which an approximate solid-surface energy of 0.3 J/m2 was determined. Yet, the conditions in the TEM could cause a lowering of the transition temperatures compared to chemically neutral conditions, although theoretical considerations predict reduction in the solid state to be negligible. Everything indicates that no stable, liquid surface layer forms prior to melting. In spite of the covalent-ionic bonds in this oxide material the qualitatively same size dependence shows like in metals. Combined with size-dependent evaporation examined in a companion study [Guenther et al. J. Phys. Chem. C 2014, 10.1021/jp412531t], a sizedependent phase diagram is proposed for this oxide material.

1. INTRODUCTION Size-dependent melting point reduction was a phenomenon first described by Pawlow1,2 in 1909. Since then a number of models have evolved.3−10 Principally they are all based on the surface-to-volume ratio which increases drastically with decreasing size. Describing the particles as perfect spheres facilitates mathematical modeling and renders formulas for sizedependent melting temperatures of the following kind Tm 1 =1−X T0 D

of suitable characterization methods. For one, the synthesis of oxide particles has to be performed under an oxygen-rich atmosphere where most suitable characterization methods fail. For another, vacuum conditions which are necessary for the usually applied probe-beam methods could affect the stability of the compounds. All this considerably complicates the experiments and requires new methods and procedures to be developed. Nevertheless, a better knowledge of structural propertiesincluding the determination of phase stability domains as a function of sizeis of the utmost importance for the further development of reliable nanodevices. Bismuth oxide was exemplarily chosen for this work. The size effect of phase transition temperatures in bismuth oxide nanoparticles will be systematically shown. The present study means to answer the question about the qualitative and quantitative differences of size effects in compounds with covalent/ionic bonds compared to metals. Together with transformations in copper sulfide and cadmium sulfide,13,14 it is the first report for inorganic compounds, and it is the only one for oxide materials. The study consists of two parts, separating the particle synthesis and evaporation measurements performed with an aerosol evaporation−condensation apparatus on the one hand and the melting experiments performed in situ in the transmission electron microscope (TEM) on the other hand. The present second part is structured as follows: It first

(1)

Tm is the melting temperature of the nanoparticle; T0 is the bulk melting temperature of the substance; D is the diameter; and the factor X differs from model to model. It depends on different bulk-material properties. In the basic homogeneous growth model (HOG) of Pawlow, X is X=

⎛ ⎛ vm(l) ⎞2/3⎞ 4vm(s) ⎜ γ − γ ⎟ ⎟ lv ⎜ Δm H ⎜⎝ sv ⎝ vm(s) ⎠ ⎟⎠

(2)

vm(s) and vm(l) are the molar volumes of the solid/liquid phase. γsv and γlv are the surface energies between gas and the solid/ liquid phase. The enthalpy of melting is ΔmH. Such effects were experimentally and theoretically shown for metals.10−12 However, no systematic investigation of phase transformations including melting has been undertaken for oxides yet. This is due to experimental challenges and the lack © 2014 American Chemical Society

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describes the experimental procedure for measuring melting of the size-selected nanoparticles. Afterward the results about sizedependent melting are presented and discussed. Finally, a sizedependent stability (phase) diagram is constructed followed by a comprehensive discussion of the results from both parts.

2. EXPERIMENTAL SECTION The in situ TEM experiments were carried out at a FEI Tecnai F20 with a water-cooled Gatan 652 heatable transfer sample holder. The synthesis and deposition of the monodisperse Bi2O3 nanoparticles were described in part one of this article.15 Most samples were deposited on 15 nm thick, amorphous silicon nitride support films on Si wafers with 0.1 × 0.1 mm (Ted Pella, Inc.). For testing the influence of substrate materials, a few samples were deposited on nickel TEM grids with a holey 10 nm thick carbon support film. No surfactants were used for any of these preparations to avoid surface contamination. All experiments were performed at a vacuum of approximately 10−5 Pa and an acceleration voltage of 200 kV. Before and after every experiment the sample was characterized at room temperature by taking bright field (BF) and selected area electron diffraction (SAED) as well as high-resolution (HR) images for selected regions. The heating experiments were performed by stepwise heating and recording of SAED and BF images at each step. Figure 1 shows a selection of SAED images together with the corresponding bright field images as they were captured for each sample at each temperature. SAED gave information about the evolving structure and loss of crystallinity at the melting transition, Tm, averaged over some thousand particles (Figure 1, left column). BF showed the morphological evolution (Figure 1, right column). As a special feature they were taken without an objective aperture in slight under focus so that the superimposed dark field image served as an indicator of crystallinity of single particles (see Supporting Information). The temperature step size was decreased when approaching Tm, and heating was stopped 30−50 K above that temperature. Subsequent cooling was performed in the same stepwise manner to observe freezing as well. For some samples, a second, similar heating cycle was added to melt and freeze the particles a second time. Five different particle sizes were investigated (see part I15):

Figure 1. Series of SAED images and corresponding BF images taken without objective aperture for a 29.7 nm sample. The diffraction rings are well visible at 573 K, and numerous particles show diffraction halos. 768 K is very near to the transformation temperature. At 778 K the diffraction rings have completely vanished, and no particle shows any diffraction halo anymore.

(Figure 3). These patterns were analyzed with the software FullProf16 for phase identification.

6.4 nm: σGSD = 1.05, area fraction = 13.5%, magnification = 43k × 8.7 nm: σGSD = 1.05, area fraction = 4.5%, magnification = 43k ×

3. RESULTS The crystalline structure of all particles was tetragonal β-Bi2O3 (see part I15). The SAED signal intensity was too weak, and the adjustments were not optimized for more detailed analyses like evolution of lattice constants (as a function of particle size and temperature) or even profile matching. During heating the signal intensity gradually decreased until it vanished eventually (e.g., Figures 1 and 3). In BF some particles appeared darker than others due to diffraction contrast. Approaching the transformation temperature the diffraction halos in BF started to disappear. For each particle the change was an abrupt switching from crystalline to noncrystalline structure. These transitions took place over a temperature range of approximately 30−40 K and correlated with the vanishing diffraction intensity in SAED. As combined transformation criterion any diffraction reflex had to be vanished in the diffraction pattern, and any diffraction halo had to disappear in the bright field images. For the illustrated example in Figures 1 and 3 the

16.0 nm: σGSD = 1.07, area fraction = 22.6%, magnification = 29k × 29.7 nm: σGSD = 1.09, area fraction = 3.9%, magnification = 9.9k ×

48.8 nm: σGSD = 1.18, area fraction = 2.5%, magnification = 9.9k ×

TEM micrographs are shown in Figure 2. The population densities (expressed as area fraction of the substrate covered by particles) were a result of the synthesis conditions and the deposition time. The density was chosen low enough that no significant particle interaction could take place during heating. Samples 4 and 5 do not purely consist of spherical primary particles: Sample 4 also contains a limited amount of smaller agglomerates made up of even smaller primary particles. Sample 5 consists of rather aspherical, agglomerated particles. For further analysis the integrated intensity of the rotational average of each SAED image was calculated which gave diffraction patterns similar to common powder diffractograms 27021

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Figure 2. Exemplary pictures of the five particle sizes used for the melting experiments. More detailed characterization in part I.15

carbon substrate instead of SiNx. Carbon is known to act as a reducing agent, but the analyzed melting temperature of 678 K agreed well with the value on the SiNx substrate. A striking observation was that approximately 20 K (29 nm sample) or approximately 50 K (8 nm sample) below the transformation temperature the SAED pattern degraded by e-beam illumination. This started after a period of several seconds, and after several minutes the sampled region appeared less crystalline. Adjacent regions which were not illuminated still showed crystallinity, though. Therefore, images close to the transformation temperature were always taken at “fresh” spots of the sample. Waiting for a long time (up to 15 min) close to Tm did not change the SAED and BF images, so time was not a relevant parameter as long as the region was not illuminated. During the whole heating procedure the particles were not observed to migrate or rotate on the substrate. The exterior shape hardly changed during the course of the experiment not even in the second heating cycles (e.g., after repeated melting and crystallization). Only the edges of the particles had become softer, and agglomerates had grown together at the necks. Yet, after a heating cycle the coarser diffraction images and the uniform diffraction halos of former agglomerates showed that inside crystallites had coalesced. The effect of the e-beam together with temperature steps of 10−20 K near Tm and the error in temperature measurement of approximately 10 K are summarized in an estimated error of 20−30 K as indicated by the different error bars in Figure 4. Cooling back to room temperature revealed two phenomena. First, the crystallization of the separate particles did not occur until below 398 K. Second, analysis of the diffraction patterns of the resolidified particles revealed that they had become metallic bismuth. The size-dependent melting temperatures of Bi2O3 are shown in Figure 4 plotted reciprocally together with the measured melting temperatures of bismuth nanoparticles and literature values. A least-squares fit of a linear function was

Figure 3. Exemplary diffractograms of 29.7 nm Bi2O3 nanoparticles from RT up to Tm. The transition occurred at 783 ± 15 K, when the (222) reflex disappeared and no crystalline particles were visible in BF anymore. A standard background pattern was subtracted.

melting temperature was analyzed to be 783 K. The other samples transformed at the Tm shown in Table 1. For testing the influence of other substrate materials one experiment was performed with 16.0 nm particles on a 10 nm Table 1. Tm of the Four Investigated Particle Sizes of βBi2O3 size [nm]

Tm [K]

6.4 8.7 16.0 29.7 48.8

483 525 673 783 810 27022

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nanoparticles as well.3 The resulting value of 0.3 J/m2 can be judged by comparing with other oxide materials. The value is of course subject to a high uncertainty caused by the propagation of uncertainties of the other estimated parameters. It is a very rough estimate that should not be mistaken as a real material property but rather considered as a quantity for further discussion. The used parameters are summarized in Table 2, and the validity and accuracy of the chosen values will be discussed. Table 2. Analyzed or Estimated Properties of β-Bismuth Oxide Nanoparticles Which Were Used for the Thermodynamic Model Calculations

Figure 4. Size-dependent melting temperatures of bismuth oxide and metallic bismuth nanoparticles in the in situ TEM heating experiments. The cyan triangles show literature values of metallic bismuth particles, and the blue, filled triangles are values from the second heating cycles of this study (with a broad size distribution). X-error bars are dominated by the size distribution and Y-error bars by the transformation region of the particles. The respective best linear fit is shown by solid lines. According to the HOG model the slope of the Bi2O3 curve corresponds to a γsv of approximately 0.3 J/m2, and the hypothetical T∞ of bulk β-Bi2O3 would be 867 K.

property

value

unit

obtained from

T∞ Δ mH γsv γlv vm(s) vm(l)

867 14700 ∼0.3 0.13 ± 0.04 47.28 × 10−6 53.07 × 10−6

K J/mol J/m2 J/m2 m3/mol m3/mol

this work (part II) ref 17 (from δ-Bi2O3) this work (part II) this work (part I15) calculated from refs 18−20 ref 22, extrapolated

4. DISCUSSION: MELTING In Figure 4 the melting points coincide reasonably well with the linear fit of the HOG model. As no negative deviation from linearity is observable for smaller sizes, there is no indication for a liquid surface layer, which was confirmed in addition by BF observations. A peculiarity is the intercept which usually denotes the bulk melting temperature: With 867 K it is 230 K below the usual bulk value. However, in contrast to the known metal studies these oxide nanoparticles reside in a different phase than the common bulk structure (β-Bi2O3 instead of δBi2O3). Therefore, the intercept at T∞ marks a hypothetical value. Particles larger than the critical size Dc adopt the usual αstructure. After this size-dependent change from β to α, the particles will follow the usual bulk transformations to the δphase and melt from there. The transition around Dc is treated elsewhere.20 No value for the absolute stability limit is given.20 Hence T∞ would be equal to T0 for particles larger than the critical size, and a jump in the size-dependent melting curve would occur. As no bulk counterpart exists there are no reference values for the β → liquid transformation. This is also true for most other thermodynamic parameters required for the models. Therefore, the widest possible range of the common fit parameter γsv (approximately 0.3 J/m2) will now be shown by using maximal and minimal values for the parameters ΔmH, vm(l), vm(s), and γlv: The chosen value for ΔmH is the one of the bulk δ → liquid transformation. It served as a lower limit because the βstructure with its lower symmetry will have a higher enthalpy of melting. The literature values for ΔmH scatter strongly.23 Still, an upper limit would be the combined change ΔtrH(α → δ) + ΔmH(δ → l) = 29.8 + 14.7 = 44.5 kJ/mol.17 The solid molar volume could be as high as 52.9 × 10−6 m3/ mol due to changes in Bi:O ratio up to stoichiometric Bi2O3 and a higher thermal expansion coefficient reported by Levin et al.24 (3.57707 × 10−9 K−1). As liquid bismuth oxide tends to reduce to bismuth under vacuum conditions, γlv of liquid bismuth (0.375 J/m2) can be estimated as an upper limit. The lowest value of 0.217 J/m2 was found for bulk Bi2O3 in air at 1123 K.25

performed which is indicated by the solid lines. The intersection at 1/D = 0 gave a T∞ of 867 K for the β-Bi2O3 phase under the given conditions in the TEM. The common melting temperature known from the bulk cubic δ-phase would be 1097 K. While the first data points can be fitted by the straight line very well, the smaller sizes, especially the smallest particles with 6.4 nm, only accord with the line at the border of the error range. This might be a hint for a deviation from linearity for the smallest sizes. The basic HOG model (eq 2) was applied to test the validity of the melting curve. T0 was replaced by T∞. The particles resided in the β-phase, but no enthalpy or free energy of this usually metastable phase is known. Thus, the value of the bulk transition from δ-Bi2O3, 14.7 kJ/mol,17 was used for ΔmH. This estimate is justified because the stoichiometry is the same and the structure is very similar. The solid molar volume was calculated at 673 K from two quantities: (a) The average stoichiometry in bismuth oxide under vacuum conditions,18,19 Bi2O2.29, and (b) the unit cell volume of 0.34 nm3 known from X-ray diffraction characterization.20 The thermal expansion coefficient of the unit cell was measured with HT XRD in technical vacuum on a commercial Bi2O3 nanopowder (Alfa Aesar, NanoArc):20 1.24656 × 10−9 K−1. The unit cell of stoichiometric Bi2O3 would bare 20 atoms (8 bismuth atoms and 12 oxygen atoms), but the present cell of Bi2O2.29 only bares 17.16 atoms (in average). Hence the final value for vm(s) is 4.7 × 10−5 m3 /mol. Like in most common models the molar volume was assumed to be temperature-invariant. The error which is introduced by this simplification is discussed elsewhere.21 For the same temperature the liquid molar volume was extrapolated from Hwang et al.22 to be 5.307 × 10−5 m3/ mol in air and ambient pressure. The γlv was analyzed in part I15 of this paper: γlv = 0.13 J/m2. No literature values for γsv are known, so it was used as a fit parameter to match the black line in Figure 3. This is the usual procedure for metallic 27023

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With the smallest combination of the estimates of ΔmH, vm(l), vm(s), and γlv the solid surface energy (γsv) would be 0.25 J/m2. On the other hand, with the largest combination of the estimates of ΔmH, vm(l), vm(s), and γlv the solid surface energy (γsv) would be 0.75 J/m2. So 0.3 J/m2 appears to be a sensible evaluation. The main cause of uncertainty is the enthalpy of melting. Nevertheless, 0.3 J/m2 as well as the relative difference between γlv and γsv are plausible values when compared to the surface energies of, e.g., solid titania nanoparticles (0.6−1.0 J/ m226) and liquid titania (0.38 J/m227). Titania is one of the few oxides which is well investigated at the nanoscale. The particles in the biggest samples show clear agglomerated nature and asphericity (Figure 2). This raises the question whether such shapes alter the transition temperature. The topic has often been discussed and used as one point of criticism about the thermodynamic melting models which all assume a spherical crystallite shape. Chang et al.28 conceived experiments where aluminum needles with different apex angles were melted, and thus curvature and molten volume could be changed independently. Their results showed that the overall melting temperature only depends on the surface to volume ratio and not on the curvature. The present results confirm this finding as the melting of the 29.7 and 48.8 nm particles would be drastically reduced if the local curvature had been of importance. In fact the 29.7 nm sample seems to be exactly in line with the two smaller sizes. Because of the smaller size and volume of the agglomerates in this sample they diffract much less than the big, spherical particles. Thus, they do not influence the analyzed T m . They were not considered for the determination of the average size either. On the other hand, Tm of the 48.8 nm sample seems a bit lower than expected, though still within the error range. This also accords with the predictions: All particles consist of agglomerated and coalesced primary crystallites. They strongly deviate from a perfect spherical shape. So the surface to volume ratio is increased, and thus Tm should be lower. Although still within the error range, the smallest particle size seems to deviate from linearity. Definite conclusions cannot be drawn, and more measurements at even smaller sizes would be necessary. Still this potential deviation cannot be ascribed to geometric reasonsespecially because the value is higher instead of lower. It is known that the material properties as ascribed to a bulk material start to change (nanomaterials) and even cease to apply (clusters of up to approximately 1000 atoms). This end of the continuum regime could be reached at 6.4 nm as spherical particles of this size contain less than ten thousand atoms. Another explanation could be that the reduced (bismuth-rich) surface region causes a convergence to the melting curve of bismuth as the influence of a more oxidic core diminishes. The fact that the molten particles do not form larger droplets and that the exterior shape hardly changes has already been observed for metallic nanoparticles, too.29 No explanation has been found in the literature. A possible reason could be strong adhesion forces or even a superficial chemical bonding with the substrate. Lee et al.30 observed the dissolution of one atomic layer of carbon in molten Au on the particle−substrate interface, and they also discuss the influence of different substrates on the melting of nanoparticles.6 The produced deepening could have a mechanical anchoring effect: It could keep the particle in position and hinder its free movement which is necessary for the coalescence of droplets.

The observed compositional change from solid Bi2O3 to liquid Bi will be discussed now. It is the consequence of the reducing conditions in the transmission electron microscope. In the present experiments the diffraction reflexes of β-Bi2O3 did not change to metallic bismuth before melting occurred (see Figure 2), and melting appeared abruptly for the individual particles (see Figure 1). This shows that no gradual reduction took place. Calculations of the size-dependent redox reaction Bi2O3 ⇌ 2Bi + (3/2)O2(g) showed that the oxide is stable under the experimental conditions (Supporting Information). The CALPHAD simulations of Arefin et al.31 confirmed this result for bulk Bi2O3, and it was experimentally supported by Takahashi et al.32,33 They performed electromotive force (emf) measurements in electrochemical cells with varying p(O2) up to 973 K (solid, α-Bi2O3). Furthermore, it is known that in contrast to its solid form bismuth oxide melts are highly reactive and possess a low thermodynamic stability under nonoxidizing conditions.34−36 One can conclude that the transformation in the Bi2O3 nanoparticles is a melting process which is followed by oxygen loss in the liquid phasejust like in the bulk. However, a shift of Tm by the experimental conditions is quite possible. This possible shift could have different causes: First, it could be caused by the stoichiometry changes, due to incongruent evaporation in vacuum (part I15): From a thermodynamic point of view, loss of oxygen by evaporation will inevitably change the chemical potential of bulk as well as of nanoparticles. Thus, the phase equilibrium (transition) temperature for which μ(s) = μ(l) would shift. Due to the defects in the structure, the chemical potential must be higher, and the transition temperature would be decreased. Second, an influence of the electron beam was observed here. Sample heating has often been discussed as an influence in TEM studies, especially for organic samples.37 Calculating the temperature rise under typical experimental conditions of the present work according to ref 38 results in a maximum of +6 K in the center of the irradiated region. Other mechanisms eligible for affecting the bismuth oxide sample are electron beam sputtering and radiolysis.39 Both mechanisms would deplete the surface region from oxygen. Furthermore, the large number of electrons provided by the electron beam can facilitate any reduction process. One can conclude that the electron beam further supports the oxygen depletion of bismuth oxide and therefore also might play a role in shifting the melting point. It is not clear whether one or both effects decrease Tm. Moreover, if the surface region is depleted of oxygen, γsv will be altered.40 It is even conceivable that a metallic surface layer forms. Assuming that one of these influences applies, T0 and T∞ would be decreased, and it is possible that the sizedependent melting curve is offset compared to one under oxidizing conditions. A complementary, experimental method like high-temperature chip calorimetry which is able to measure in variable atmospheres could validate and quantify this assumption.41 The obtained values for the melting point depression of bismuth oxide nanoparticles are stronger than the effect for any known metal (Figure 5). The reason is 2-fold. For one thing, T∞ is drastically reduced compared to T0 which is caused by the elevated ground state of the nanoparticles in the metastable β-phase and probably by the conditions in the TEM which tend to deplete bismuth oxide of oxygen. For another, the size 27024

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Figure 6. Size-dependent melting and boiling temperatures of bismuth oxide nanoparticles. The blue points are measured melting temperatures in the TEM (under vacuum), and the blue line comes from the HOG melting model. The red crosses with arrows are the particle sizes used to measure and calculate evaporation at lower temperatures (part I15 under 1 bar total pressure with 90% N2 and 10% O2). From that data their theoretical boiling point was extrapolated (red line).

Figure 5. Comparison of the relative size-dependent melting point reduction between gold and bismuth oxide. The curves were calculated with the HOG model on the basis of experimental values from this work and literature values. The Bi2O3 curve was only plotted for the ascertained size range. Causes for the stronger depression in Bi2O3 are (a) an offset of the T∞ value and (b) a steeper slope because of a larger Δγ.

at lower temperatures. The line is the boundary between the liquid and the gaseous state. The schematic stability diagram of bismuth oxide is shown in Figure 7. The melting and boiling information from Figure 6 is

dependence is stronger because of a bigger Δγ = γsv − γlv as shown with the HOG model.

5. SIZE-DEPENDENT STABILITY DIAGRAM AND GENERAL DISCUSSION All findings from the experimental chapters about sizedependent phase transformations of Bi2O3 nanoparticles in part I15 and part II are brought together to form a schematic, size-dependent stability diagram for bismuth oxide. The in situ heating experiments in the TEM performed on five particle sizes (6, 9, 16, 30, and 49 nm) showed a very strong melting point depression which is in accordance with a 1/D melting behavior. However, the high vacuum and the electron beam are not chemically neutral conditions for oxides. Hence, a shift of the absolute values of the melting temperatures was discussed. The HOG model was fitted to the data points in the reciprocal plot of Figure 4. The same curve was plotted in Figure 6 over D together with the data points. They are represented as a blue line and blue squares, respectively. These curves represent the boundary line between the region of stability of the solid β-phase and the melt. The combined measurement and calculation of the dynamic evaporation was performed on five initial sizes as well (6, 8, 17, 29, and 47 nm) in part I.15 The resulting size-dependent boiling temperature was discussed. Since the evaporation rate at the actual boiling point would be much too high, the vapor pressure curve was determined between 850 and 1175 K. With the Clausius−Clapeyron equation (eq 7, part I15) this curve was extrapolated to the boiling point. A ΔvH of 250 kJ/mol42 was used for this purpose. The main problem of the evaporation experiments was the unknown, complicated evaporation behavior of bismuth oxide. Its vapor consists of many different gas species whose vapor pressures depend on temperature and atmosphere. The curve of Figure 9 in part I15 has been reproduced in Figure 6. The red line is the calculated line according to eq 9 in part I,15 and the arrows mark the initial particle sizes which were used to determine the vapor pressure

Figure 7. Schematic T−D stability diagram of Bi2O3. The dashed lines indicate unknown lines and regions. The squares and arrows are measured values in this work. The break in the abscissa marks the change from the macro-scale to the nano-scale. The different stable phases at room temperature are probably responsible for the gap between limr→∞ Tm(r) in the nano-range and Tm,0 in the macro-range. Note: The melting values and the evaporation values were measured at different pressures.

reproduced. Additionally the transformations of bulk Bi2O3 are plotted as horizontal (size-independent) lines in the macroscale region. Squares and arrows indicate measured sizes. Regions where the stability boundary is unknown are marked with dashed lines. More information about the solid−solid transition from nano β-Bi2O3 to bulk α-Bi2O3 can be found in ref 20. The conversion between the nano-scale and the macroscale is not characterized and understood, yet. The boiling temperature of the bulk, Tv,0, is known with a high inaccuracy. Consequently the absolute value of Tv(r), which is based on this value, is uncertain as well. The axes are not scaled since the 27025

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absolute values of the phase transitions are subject to the previously mentioned uncertainties. All size-dependent phase transformations of this work are summarized in this scheme. Because of the pronounced polymorphism of bismuth oxide further phase transformations depending on the oxygen content and the atmosphere are possible. This diagram, showing the regions and borders of stability for the different phases in bismuth oxide, has one remarkable difference from traditional phase diagrams: The abscissa is not an intensive variable (T or P or composition n) but the size. Size is representative for the specific surface energy contributing to the free energy of the system as an excess energy. It thus changes the phase boundaries with increasing amount of surface. Along that line Kaptay reports attempts to include size/surface energy in the calculation of thermodynamic systems.43

AUTHOR INFORMATION

Corresponding Author

*Tel.: +49 2461 615181. Fax: +49 2461 619866. E-mail o. [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge financial support of the Deutsche Forschungsgemeinschaft (DFG) within the frame of the Emmy Noether program (GU 993/1-1). We are also very grateful to R. Schmechel and G. Schierning and the whole NST group for the possibility to produce the nanoparticles and perform melting experiments and the numerous scientific discussions.

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6. CONCLUSIONS A comprehensive study was conducted to investigate the size dependence of phase transformations in binary oxide materials in general and bismuth oxide in particular because previous publications were restricted to metallic materials. The study is published in two parts. In this second part the size-dependent melting point depression was described, and a conclusive sizedependent stability diagram was drawn. Monodisperse, monocrystalline β-Bi2O3 nanoparticles were synthesized and directly deposited on TEM grids with amorphous SiNx support films. Five different sizes were heated in the TEM, and with BF and SAED analyses the solid (crystalline)−liquid transition was determined. The influence of the high vacuum and the electron beam on the melting temperatures was discussed, but no quantitative conclusion could be drawn. The linear dependence of Tm on 1/D, known from metallic systems, was validated for these oxide nanoparticles with covalent/ionic bonds. This indicates that the effect is independent of bond nature and is purely caused by the geometry-dependent surface/volume ratio. A surface energy of γsv ≈ 0.3 J/m2 was roughly determined. Together with γlv ≈ 0.13 J/m2 (part I15) this caused a size effect which is stronger than any melting point depression known in metallic systems. Furthermore, the melting curve is offset probably because the particles already reside in an elevated state at RT. For example 10 nm gold nanoparticles melt approximately 10% lower than the bulk, whereas Bi2O3 at the same size melts approximately 45% lower. The surface/volume ratio proves to be responsible for the phase shifts in any kind of nanomaterial. Therefore, size, representative for the specific surface energy, can be used as an abscissa in stability diagrams showing the stability ranges of the different phases at the nanoscale. This is similar to traditional phase diagrams with P or T as intensive variables and could be an important tool for nanoscience.

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S Supporting Information *

Details on BF images without objective aperture, analysis of metallic bismuth nanoparticles, calculation of the size- and pressure-dependent redox equilibrium. This material is available free of charge via the Internet at http://pubs.acs.org. 27026

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