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Size-dependent nanoparticle reaction enthalpy: Oxidation of aluminum nanoparticles
Journal of Physics and Chemistry of Solids 72 (2011) 719–724
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Size-dependent nanoparticle reaction enthalpy: Oxidation of aluminum nanoparticles Stephen W. Chung a, Elena A. Guliants b, Christopher E. Bunker c,n, Paul A. Jelliss a,n, Steven W. Buckner a,n a
Saint Louis University, Department of Chemistry, 3501 Laclede Avenue, St. Louis, MO 63103, USA Department of Electrical and Computer Engineering, University of Dayton Research Institute, Dayton, OH 45469, USA c Air Force Research Laboratory, Propulsion Directorate, Wright-Patterson Air Force Base, OH 45433, USA b
a r t i c l e i n f o
abstract
Article history: Received 11 October 2010 Received in revised form 14 February 2011 Accepted 27 February 2011 Available online 9 March 2011
Here we present a model describing the particle size dependence of the oxidation enthalpy of aluminum nanoparticles. The model includes the size dependence of the cohesive energy of the reactant particles, the size dependence of the product lattice energy, extent of product agglomeration, and surface capping effects. The strongest effects on aluminum nanoparticle energy release occur for particle diameters below 10 nm, with enhanced energy release for agglomerated oxide products and decreased energy release for nanoscale oxide products. An unusual effect is observed with all nanoparticle reaction enthalpies converging to the bulk value when agglomeration of the products approaches the transition between nanoparticle-nanoparticle and nanoparticle-bulk energetics. Optimal energy output for Al NP oxidation should occur for sub-10-nm particles reacting with significant agglomeration. & 2011 Elsevier Ltd. All rights reserved.
Keywords: A. Metals A. Nanostructures D. Thermodynamic properties
1. Introduction Micron-sized aluminum particles are used in high energy thermite reactions and utilized as a propellant in fuels [1–4]. This is due to the high enthalpy of oxidation for aluminum: 2Al þ3=2O2 -Al2 O3
DH1rxn ¼ 1675:7kJ=mol Al2 O3 Aluminum’s low density gives aluminum particles a much higher energy density than the most energetic organic compounds. Currently, there are efforts to optimize the physical properties of aluminum particles by reducing their dimensions to the nanoscale [5–7]. While this presents clear challenges for synthesis and stabilization due to the high reactivity of aluminum nanoparticles (Al NPs) with water and oxygen under ambient conditions, there are advantages in the combustion properties of Al NPs. Some of the physical changes with Al NPs relative to larger particles include an increased combustion rate [8–10], increased reactivity [11–13], and lower melting point [14–19]. The melting point depression that accompanies the decrease in particle size can be described with the Gibbs–Thomson equation for spherical particles [14–15]. The Al NP combustion rate and reactivity are observed to increase as the particle size decreases. The origin of n
Corresponding authors. Tel.: þ 1 314 977 2850; fax: þ 1 314 977 2521. E-mail address: [email protected] (S.W. Buckner).
0022-3697/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2011.02.021
this increased reactivity is the increased surface area-to-volume ratio of a particle with decrease in particle diameter. Other nanostructure thermodynamic properties have been modeled for a range of sizes. The heat of formation of nanocompounds is modeled for various sizes [20]. In addition to spherical particles, the melting point of metallic nanowires has been studied and modeled [21]. Thermodynamic models for small systems in the nano-range have been developed [22,23]. Basic equations of nanothermodynamics have been derived by changing the ensemble level to account for N small systems. Nanothermodynamics have been used to treat systems such as onedimensional adsorbed lattice gas. In this paper, we consider the size dependence of the enthalpy of reaction. We consider the effects of increased surface area and the corresponding decrease in cohesive energy of the material on the enthalpy of oxidation of Al NPs as a function of particle diameter.
2. Model The energetics of the aluminum oxidation reaction can be visualized using Hess’s law diagram, as shown in Fig. 1. Three reactions are shown. One reaction consists of the bulk Al oxidizing under O2 to make bulk Al2O3. Here we define bulk based on macroscopic thermodynamic values. Particles with micron diameters will have thermodynamic properties matching those of macroscopic materials, within experimental error. The other two reactions correspond to oxidation of Al NPs. We consider the
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for the Al2O3 bulk product Eq. (2) becomes
DH1rxn,np,bk ¼ DH1rxn,bk 2DSDCEAl
Fig. 1. Hess’s diagram is used to illustrate the Al NP oxidation energetics and to show DH1rxn,np, including DH1rxn,np,np and DH1rxn,np,bk.
Fig. 2. Energy diagram shows the thermodynamic potentials of (a) DSDCEAl and cohesive energy and (b) DSDLEAl2 O3 and lattice energy.
reaction enthalpies for Al NP conversion to nano-sized Al2O3 (DH1rxn,np,np) and Al NP conversion to bulk Al2O3 (DH1rxn,np,bk) if the nano-Al2O3 agglomerates. Following the thermodynamic cycle for this system we can obtain the reaction enthalpies for the Al NP oxidation reaction to produce Al2O3 NPs, DH1rxn,np,np, and the reaction enthalpy for the Al NP oxidation reaction to produce bulk Al2O3, DH1rxn,np,bk. For DH1rxn,np,np, the relationship is derived from the thermodynamic cycle: DH1rxn,np,np þ DH1rxn,bk 2DSDCEAl þ DSDLEAl2 O3 ¼ 0 kJ=mol Al2 O3 ð1Þ where DH1rxn,bk is the reaction enthalpy of the bulk reaction ( 1675.7 kJ/mol), SDCE is the size-dependent cohesive energy, SDLE is the size-dependent lattice energy, DSDCEAl is the change in cohesive energy from the bulk Al to nano-Al, and DSDLEAl2 O3 is the change in lattice energy from the bulk Al2O3 to nano-Al2O3. DSDCE and DSDLE are direct measures of the change in thermodynamic potential of the Al NPs relative to bulk due to loss in surface binding. Fig. 2 shows an energy diagram illustrating DSDCE and DSDLE. Solving for DHrxn,np,np, the equation becomes
DH1rxn,np,np ¼ DH1rxn,bk 2DSDCEAl þ DSDLEAl2 O3
ð2Þ
In both equations there is a stoichiometric coefficient of 2 for
DSDCEAl, and a stoichiometric coefficient of 1 for DSDLEAl2 O3 . For the reaction of nano-Al-bulk Al2O3, the DSDLEAl2 O3 is 0 because there is no reduction in lattice energy for the bulk product. Thus,
ð3Þ
Additional factors must be included. First, in the nano-Alnano-Al2O3 reaction, the resulting Al2O3 particle has a diameter that is 9% greater than the initial Al particle. A factor, F, is added to Eq. (2), in which F is the ratio of reactant and product NP diameters. It is equal to 1/1.09 in this reaction and is multiplied to the DSDLEAl2 O3 . Both DSDCEAl and DSDLEAl2 O3 are inverse functions of the diameter of the nanoparticle. This factor is necessary to account for the change in lattice energy with changing NP diameter. Second, we need to include surface passivation effects in the reactant NP. All Al NPs require a passivation scheme. Bare Al NPs are thermodynamically and kinetically unstable toward grain growth. In addition, bare Al NPs are pyrophoric. Most simply, Al particles may be capped with an oxide layer. This effect is described by inclusion of the enthalpy contribution consisting of the interfacial energy of Al-Al2O3, gAl-Al2 O3 ,enth . The aluminum oxide layer is present on aluminum nanoparticles when the surface aluminum atoms react with O2 or H2O. The gAl-Al2 O3 ,enth term is also multiplied by 2 because it is in terms of moles of Al atoms. In addition, another factor that we must consider is the extent of agglomeration. The product Al2O3 NPs tend to agglomerate [24], effectively decreasing the DSDLEAl2 O3 . Agglomeration occurs when the surface atoms of a particle bond with the surface atoms of another particle, decreasing the fraction surface atoms with a dangling bond (d). The d term can range from 0 to 1, in which 0 corresponds to products when full agglomeration of the particles occurs. d ¼ 1 corresponds to products when no agglomeration of the NP products occurs. With these three factors amending the model, Eq. (2) becomes
DH1rxn,np,np ¼ DH1rxn,bk 2DSDCEAl þ F dDSDLEAl2 O3 2gAlAl2 O3 ,enth ð4Þ For the nano-Al-bulk-Al2O3, the values of F and d are obviously not amended since DSDLEAl2 O3 ¼ 0, but the interfacial energy contribution is amended. So, the relationship for the sizedependent enthalpy of reaction of Al NPs to produce bulk Al2O3 is
DH1rxn,np,bk ¼ DH1rxn,bk 2DSDCEAl 2gAlAl2 O3 ,enth
ð5Þ
In order to calculate the size dependence of the reaction enthalpy using DH1rxn,np,np and DH1rxn,np,bk, the DSDCE and DSDLE terms must be calculated. The DSDCE and DSDLE terms are a function of size of the nanoparticle. As the particle size decreases, the cohesive and lattice energy also decreases as well because the small diameter leads to a large surface area-to-volume ratio. At high specific surface areas, a large fraction of the Al atoms has dangling bonds. The increase in surface area-to-volume ratio also means that there are more surface atoms, and the surface atoms have fewer bonds with neighboring atoms than interior atoms, causing a decrease in the cohesive or lattice energy. There are many models for estimating the size dependence of the cohesive energy for metallic NPs. These include the liquid drop model [25], bond energy model [26], surface area difference model [27–29], Qi’s model based on surface relaxation [30], and Qi’s model dealing with shape effects [31]. The liquid drop model considers the surface tension of nanoparticles, in which the surface tension decreases with decrease in size of NPs. The bond energy model takes into account the total bond energy of interface and the total bond energy of interior atoms. In the current study we only consider spherical particles. All the Al NPs synthesized and reported to date are spherical.
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From the parameters and properties of Al and Al2O3 that are well known, we use Nanda’s model to calculate the cohesive energy [32]. This bond geometry model considers the number of bonds of the surface atoms and interior atoms and the fraction of surface atoms within a nanoparticle. The equation for the Nanda model is rffiffiffiffiffi Ns zs av,d ¼ av 1 1 ð6Þ N zb in which av,d is the SDCE or SDLE, av is the cohesive energy of an interior atom (bulk), zs is the coordination number of a surface atom, zb is the coordination number of an interior atom, and Ns/N is the fraction of surface atoms in the NP. For a spherical particle, Ns/N ¼8ra/d, where ra is the atomic radius and d is the diameter of the particle. Nanda’s model can be modified to calculate the SDLE for Al2O3 NPs, as shown below: av,d,AlNP ¼ ð2=5Þav,d,Al þ ð3=5Þav,d,O
ð7Þ
¼ av ½0:4ð1fNs,Al =Ngf1ðzs,Al =zb,Al Þ1=2 gÞ þ 0:6ð1fNs,O =Ngf1ðzs,O =zb,O Þ1=2 gÞ
ð8Þ
Thus, the same model can be used for reactants and products, resulting in the same type of curve for the lattice energy of Al2O3 nanoparticles. In Eq. (8), the factor [0.4(1 {Ns,Al/N}{1 (zs,Al/ Zb,Al)1/2}) þ0.6(1 {Ns,O/N}{1 (zs,O/zb,O)1/2})] decreases the value of av,d,AlNP and indicates the degree that the surface atoms of both Al and O affect the lattice energy of Al2O3 NPs. The calculation for these particles is different because they are composed of two constituent atoms. The av,d is calculated for both the Al and O atoms, and the av,d for Al is multiplied by 2/5 and the av,d for O is multiplied by 3/5 because two-fifths of the atoms in Al2O3 are made up of Al and three-fifths are made up of O, which includes those atoms on the surface. Separate coordination numbers are also used for Al and for O. The values of av for both the Al and O constituent atoms are the same as the bulk lattice energy of Al2O3 (15,270 kJ/mol). Table 1 shows the parameters that were used to calculate the DSDCE and DSDLE for Al and Al2O3, respectively. The cohesive energy decreases as the particle size decreases. Also, the cohesive energy asymptotically approaches the bulk cohesive energy of Al (330 kJ/mol). Next we consider the enthalpy contribution to the interfacial energy, gAl-Al2 O3 ,enth . As mentioned before, this term is needed to represent the interface that occurs when a native oxide layer passivates an Al particle. gAl-Al2 O3 ,enth was determined to be 0.068 J m 2 at 298 K [33]. This value was calculated from
g/MSfMOx g,enth ¼ pHfuse,/MS =A/MS
ð9Þ
where g/MS {MOx},enth is the enthalpy contribution to the interfacial energy between the crystal phase of M and the amorphous phase of the oxide layer (MOx), p is the fraction of the total surface area of M in contact with MOx phase, Hfuse,/MS is the enthalpy of fusion of crystal phase of M, and A/MS is the area occupied by one mole of M atoms at the interface. g/MSfMOx g,enth is calculated for Table 1 Parameters for Al and Al2O3 are used to calculate the SDCE and SDLE. Particle
av (kJ/mol)
zs/zb
Atomic radius (pm)
Al
330
8/12
125
Al2O3
15,270
Al: 5/6 O: 3/4
Al: 125 O: 60
721
the three metal surfaces: {1 1 1}, {1 1 0}, and {1 0 0}. The values were 0.083, 0.051, and 0.072 J m 2, and the average of these values was taken because it was assumed that a spherical nanoparticle contains all three surfaces equally. This value is in terms of surface area; so it must be converted into terms of moles of Al atoms. Appendix 1 shows the derivation of the enthalpy contribution to the interfacial energy in terms of kJ per mole of Al atoms. The equation for gAl-Al2 O3 ,enth is
gAlAl2 O3 ,enth ¼
4:1 kJ nm mol1 d
ð10Þ
where d is the diameter of the Al nanoparticle (in nm). This equation shows an inverse relationship between gAl-Al2 O3 ,enth and the diameter of the Al NP. A decrease in the diameter of an Al particle increases the fraction of surface Al atoms, and an increase in the fraction of surface Al atoms for an Al particle increases the interfacial energy contribution, which is expressed in kJ per mole of Al atoms. The origin of this effect is that the surface Al atoms are the only atoms that may interface with the aluminum oxide monolayer. Since the fraction of surface Al is near 0 for bulk Al, the bulk Al has a negligible value ( 0 kJ/mol) for the interfacial energy contribution. However, even for the nanoparticles with a diameter of 3 nm, gAl-Al2 O3 ,enth is 1.4 kJ/mol Al. When compared with the bulk enthalpy of reaction of 1675.7 kJ/mol Al2O3, the gAl-Al2 O3 ,enth term is small. After calculating the SDCE for Al particles and SDLE for Al2O3 particles, the DH1rxn,np,np may be calculated using Eq. (4). Fig. 3a shows DH1rxn,np,np as a function of size assuming no product agglomeration. As the particle size decreases, DH1rxn,np,np decreases in exothermicity. Thus, the reaction is less exothermic for very small Al NPs. The effect only becomes significant for Al NPs with diameters less than 10 nm. This decrease in reaction enthalpy may appear counterintuitive. The decrease in enthalpy is due to the much greater effect of the product DSDLEAl2 O3 relative to the reactant cohesive energy term, DSDCEAl. As the particle size increases, DH1rxn,np,np asymptotically approaches the bulk enthalpy of reaction. Significant effects are only present below 10 nm, where the fraction of surface atoms becomes a dominating factor. Next, we consider oxidation of Al NPs to produce bulk Al2O3. As just mentioned, during oxidation of Al NPs the product Al2O3 particles have a tendency to agglomerate. Large agglomeration leads to a bulk-phase Al2O3 product. Fig. 3b shows a graph of DH1rxn,np,bk as a function of size. In contrast to DH1rxn,np,np, DH1rxn,np,bk increases in exothermicity as the particle size decreases. This occurs because DSDLEAl2 O3 is not present when bulk Al2O3 is produced. As for the NP-NP reaction, significant effects on the reaction enthalpy are only seen below Al NP diameters of 10 nm. This leads us to consider the effects of agglomeration for the nano-Al-nano-Al2O3 reaction. Since we established that significant effects in enthalpy are only seen below 10 nm, we consider the effects to the enthalpy of reaction with agglomeration of these NPsr10 nm. Fig. 4 shows the enthalpy of reaction for 2, 4, 6, 8, and 10 nm particles as d is varied from 0 to 1. As expected, decreasing the fraction of surface atoms with dangling bonds for a given size of Al NP increases the enthalpy of reaction. This is because DSDLEAl2 O3 is decreasing. Also, as the particle size decreases, the effect of increase in extent of agglomeration on the reaction enthalpy increases, according to the increasing slope of the line when the particle size decreases. Smaller particles have larger deviations from the bulk enthalpy, and agglomeration effectively makes these smaller particles approach bulk behavior. It is also interesting that all enthalpies for all nanoparticles as well as the bulk value for the reaction enthalpy converge at d ¼0.126. A detailed mathematical analysis of this is found in the
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relative to the bulk enthalpy as the particle size decreases. As the agglomeration increases, the enthalpies for each of the particle sizes converge. As the agglomeration increases, the Al NPs enter the regime of nano-Al-bulk Al2O3. In this regime, the Al NP reaction exothermicity increases relative to the bulk value as the particle size decreases, and the enthalpies diverge from one another {vide infra}. It is at the transition between these two regimes that we see convergence to the bulk value, when a fraction of 0.126 surface atoms of Al2O3 is present. At this convergence point, the heat of reaction is size independent and is equal to the bulk reaction enthalpy. There are recent reports of Al NP oxidation in which water is found to be much more reactive than O2 toward Al NPs [3]. This is consistent with results on gas phase aluminum cluster ion reactivity. Small Al ionic clusters, consisting of 2–50 atoms, have been shown to react much faster with H2O than O2. While Al NPs are known to generate H2 as a product in reaction with H2O, the Al NPs also incorporate oxygen and hydroxide groups from water in the process. The resulting products are boehmite (g-AlO(OH)) and bayerite (Al(OH)3) when Al is oxidized under moist conditions, as opposed to corundum, which is made with oxidation of Al with pure oxygen [34]. Al reacts with water vapor according to the following reactions: 2Alþ4H2O-2AlO(OH)þ3H2 2Alþ6H2O-2Al(OH)3 þ3H2
Fig. 3. (a) The plot shows the enthalpy of reaction for the nano-Al-nano-Al2O3 (DH1rxn,np,np) as a function of diameter of Al particle and (b) the enthalpy of reaction for the nano-Al-bulk Al2O3 (DH1rxn,np,bk) as a function of diameter of Al particle.
This can significantly affect the thermodynamics of the oxidation reaction. We have considered the oxidation of Al NPs to bulk boehmite and to bulk bayerite using our model. The formation of bulk boehmite and bulk bayerite results in less heat release than corundum: the bulk reaction enthalpies for formation of boehmite and bayerite from aluminum are 1021 kJ/mol g-AlO(OH) and 1094.4 kJ/mol Al(OH)3, respectively. Fig. 5 includes the curves for DH1rxn,np,bk for the reaction of Al NPs to produce boehmite and bayerite as a function of reactant NP diameter Fig. 5 also includes experimental data obtained from Sun et al. [11]. This is the most reliable size-dependent thermodynamic data for mid-range Al NPs with narrow size distribution particles. The experimental data are for Al NPs with a narrow size distribution and passivated with an oxide layer. The experimental data show an energy release between that expected for
Fig. 4. NP enthalpy of reaction is plotted as a function of the fraction of surface atoms with dangling bonds (a measure of extent of product agglomeration) for a series of different NP diameters.
Appendix. The Al NP oxidation energetics occupies two regimes: nano-Al-nano-Al2O3 and nano-Al-bulk Al2O3. Initially, without agglomeration, the Al NPs are in the nano-Al-nano-Al2O3 regime. Within this regime, the reaction exothermicity decreases
Fig. 5. Plot compares the reaction enthalpy curves for the reactions of nano-Al oxidizing to the product bulk boehmite (DH1rxn,boe), bulk bayerite (DH1rxn,bay), and bulk corundum (DH1rxn,cor). It is compared to Pantoya’s experimental data of Al NPs with a narrow size distribution.
S.W. Chung et al. / Journal of Physics and Chemistry of Solids 72 (2011) 719–724
production of bayerite/boehmite and corundum. Possibly, traces of water give a mixed-phase reaction product during Al NPs oxidation.
723
where A ¼ 16av ra ð1ðzs =zb Þ1=2 Þ
ðeÞ
B ¼ Fav ½ð2=5Þð8ra,Al Þ ð1ðzs,Al =zb,Al Þ1=2 Þ þ ð3=5Þð8ra,O Þð1ðzs,O =zb,O Þ1=2 Þ
3. Conclusion The enthalpy of reaction for the oxidation of Al NPs is dependent on the diameter of the NP. The enthalpy depends heavily on the size-dependent cohesive energy of the Al nanoparticles and lattice energy of the resulting Al2O3 nanoparticles. Two reaction products are considered when Al nanoparticles are oxidized with oxygen: bulk and nano-Al2O3. Each reaction product results in two different reaction enthalpy curves. For the reaction of AlNP-nano-Al2O3, the reaction enthalpy decreases as the particle size decreases. In contrast, the reaction of AlNP-nano-Al2O3 has an increasing reaction enthalpy with decrease in particle size. The optimal system for maximum energy release should occur for sub10-nm Al NPs reacting to produce substantially agglomerated products. Eq. (4) should be a generally applicable equation for any NP reaction with the appropriate substitution of appropriate stoichiometric and other constants. Comparison of experimental data with that of boehmite and bayerite shows that the product of oxidized Al NPs may contain boehmite and bayerite, as well as corundum. There is a clear need for reliable thermodynamic data for very small Al NPs in the sub10-nm range.
ðfÞ
and C ¼ 12Mð6:8 105 kJm2 Þ=r
ðgÞ
Eq. (d) is rearranged to remove the dependence on nanoparticle diameter, d, to give A þB dC ¼ 0
ðhÞ
For the current oxidation reactions of Al NPs the values for the constants return the values for A, B, and C as follows: A Að1=dÞ ¼ 2ð330av ½1ðNs =NÞð1ðzs =zb Þ1=2 ÞÞ ¼ 2ð330330þ av ð8ra =dÞð1ðzs =zb Þ1=2 ÞÞ ¼ 2ðav Þð8ra =dÞð1ðzs =zb Þ1=2 Þ ¼ 16ra av ð1=dÞð1ðzs =zb Þ1=2 Þ A ¼ 16ra av ð1ðzs =zb Þ1=2 Þ ¼ 121:11 B Bdð1=dÞ ¼ FdðSDLEbk,Al2 O3 SDLEnp,Al2 O3 Þ ¼ Fd 15270-½ð2=5Þav,d ðAlÞ þ ð3=5Þav,d ðOÞ ¼ Fdf15270-½0:4av
Acknowledgments
0:4av ð1=dÞð8ra,Al Þð1ðzs,Al =zb,Al Þ1=2 Þ
We gratefully acknowledge the Air Force Research Laboratory Nanoenergetics Program for supporting this work. We would like to thank Dr. Michelle Pantoya and Dr. Sindee Simon for their experimental data.
½0:6av 0:6av ð1=dÞð8ra,O Þð1ðzs,O =zb,O Þ1=2 Þg ¼ Fd½0:4av ð1=dÞð8ra,Al Þ ð1ðzs,Al =zb,Al Þ1=2 Þ þ 0:6av ð1=dÞð8ra,O Þð1ðzs,O =zb,O Þ1=2 Þ ¼ Fdav ð1=dÞ½0:4ð8ra,Al Þ ð1ðzs,Al =zb,Al Þ1=2 Þ þ 0:6ð8ra,O Þð1ðzs,O =zb,O Þ1=2 Þ
Appendix Fig. 4 in the main text illustrates an interesting convergence in the reaction enthalpy where reactions for all particle sizes (including nano all the way to bulk) have the same reaction enthalpy at a particular extent of agglomeration. While this was not initially intuitive to us it can be shown to be true starting from Eq. (4) in the main text:
DH1rxn,np,np ¼ DH1rxn,bk 2ðSDCEbk,Al SDCEnp,Al Þ þ FdðSDLEbk,Al2 O3 SDLEnp,Al2 O3 Þ2gAl2 O3 ,enth
ðaÞ
B ¼ 8Fav ½0:4ðra,Al Þð1ðzs,Al =zb,Al Þ1=2 Þ þ 0:6ðra,O Þð1ðzs,O =zb,O Þ1=2 Þ ¼ 1028:782 C Cð1=dÞ ¼ ð2Þð6:8 105 kJ m2 Þ ðSÞ=ðrV=MÞ 2
3
¼ ½ð2Þð6:8 105 kJ m2 Þðpd Þ=½ðrÞðpd =6Þ=M ¼ 12 Mð6:8 10 S ¼ surface area ¼ pd
5
2
kJ m
Þ=rd
2
3
From Figs. 3 and 5 it is apparent that the nanoparticle reaction enthalpy may be larger or smaller than the bulk enthalpy, depending on the state of the products. At the extent of agglomeration at which DH1rxn,np,np is equal to the bulk value:
DH1rxn,np,np ¼ DH1rxn,bk
1
M ¼ molar mass ¼ 26:981 g mol C ¼ 12 Mð6:8 10
5
kJ m
2
Þ=r
¼ ½12ð6:8 105 kJ m2 Þ ð26:981 g mol ¼ 8:2 kJ nm mol
ðcÞ
Eq. (c) can be rearranged to separate the dependence on the reactant nanoparticle diameter, d, to give Að1=dÞ þ Bdð1=dÞCð1=dÞ ¼ ð1=dÞ½A þ BdC ¼ 0
r ¼ density ¼ 2:7 106 g m3
1
ðbÞ
and Eq. (a) reduces to 0 ¼ 2ðSDCEbk,Al SDCEnp,Al Þ þ FdðSDLEbk,Al2 O3 SDLEnp,Al2 O3 Þ 2gAl2 O3 ,enth
V ¼ volume ¼ pd =6
ðdÞ
Thus, at the point defined by Eq. (b) above we see that the NP particle diameter dependence can be removed from the equation,
Þ=ð2:7 106 g m3 Þ
1
Solving for d for Eq. (h) 121:11 þ1028:782d8:2 ¼ 0 d ¼ 0:126
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Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
Size-dependent nanoparticle reaction enthalpy: Oxidation of aluminum nanoparticles Stephen W. Chung a, Elena A. Guliants b, Christopher E. Bunker c,n, Paul A. Jelliss a,n, Steven W. Buckner a,n a
Saint Louis University, Department of Chemistry, 3501 Laclede Avenue, St. Louis, MO 63103, USA Department of Electrical and Computer Engineering, University of Dayton Research Institute, Dayton, OH 45469, USA c Air Force Research Laboratory, Propulsion Directorate, Wright-Patterson Air Force Base, OH 45433, USA b
a r t i c l e i n f o
abstract
Article history: Received 11 October 2010 Received in revised form 14 February 2011 Accepted 27 February 2011 Available online 9 March 2011
Here we present a model describing the particle size dependence of the oxidation enthalpy of aluminum nanoparticles. The model includes the size dependence of the cohesive energy of the reactant particles, the size dependence of the product lattice energy, extent of product agglomeration, and surface capping effects. The strongest effects on aluminum nanoparticle energy release occur for particle diameters below 10 nm, with enhanced energy release for agglomerated oxide products and decreased energy release for nanoscale oxide products. An unusual effect is observed with all nanoparticle reaction enthalpies converging to the bulk value when agglomeration of the products approaches the transition between nanoparticle-nanoparticle and nanoparticle-bulk energetics. Optimal energy output for Al NP oxidation should occur for sub-10-nm particles reacting with significant agglomeration. & 2011 Elsevier Ltd. All rights reserved.
Keywords: A. Metals A. Nanostructures D. Thermodynamic properties
1. Introduction Micron-sized aluminum particles are used in high energy thermite reactions and utilized as a propellant in fuels [1–4]. This is due to the high enthalpy of oxidation for aluminum: 2Al þ3=2O2 -Al2 O3
DH1rxn ¼ 1675:7kJ=mol Al2 O3 Aluminum’s low density gives aluminum particles a much higher energy density than the most energetic organic compounds. Currently, there are efforts to optimize the physical properties of aluminum particles by reducing their dimensions to the nanoscale [5–7]. While this presents clear challenges for synthesis and stabilization due to the high reactivity of aluminum nanoparticles (Al NPs) with water and oxygen under ambient conditions, there are advantages in the combustion properties of Al NPs. Some of the physical changes with Al NPs relative to larger particles include an increased combustion rate [8–10], increased reactivity [11–13], and lower melting point [14–19]. The melting point depression that accompanies the decrease in particle size can be described with the Gibbs–Thomson equation for spherical particles [14–15]. The Al NP combustion rate and reactivity are observed to increase as the particle size decreases. The origin of n
Corresponding authors. Tel.: þ 1 314 977 2850; fax: þ 1 314 977 2521. E-mail address: [email protected] (S.W. Buckner).
0022-3697/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2011.02.021
this increased reactivity is the increased surface area-to-volume ratio of a particle with decrease in particle diameter. Other nanostructure thermodynamic properties have been modeled for a range of sizes. The heat of formation of nanocompounds is modeled for various sizes [20]. In addition to spherical particles, the melting point of metallic nanowires has been studied and modeled [21]. Thermodynamic models for small systems in the nano-range have been developed [22,23]. Basic equations of nanothermodynamics have been derived by changing the ensemble level to account for N small systems. Nanothermodynamics have been used to treat systems such as onedimensional adsorbed lattice gas. In this paper, we consider the size dependence of the enthalpy of reaction. We consider the effects of increased surface area and the corresponding decrease in cohesive energy of the material on the enthalpy of oxidation of Al NPs as a function of particle diameter.
2. Model The energetics of the aluminum oxidation reaction can be visualized using Hess’s law diagram, as shown in Fig. 1. Three reactions are shown. One reaction consists of the bulk Al oxidizing under O2 to make bulk Al2O3. Here we define bulk based on macroscopic thermodynamic values. Particles with micron diameters will have thermodynamic properties matching those of macroscopic materials, within experimental error. The other two reactions correspond to oxidation of Al NPs. We consider the
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for the Al2O3 bulk product Eq. (2) becomes
DH1rxn,np,bk ¼ DH1rxn,bk 2DSDCEAl
Fig. 1. Hess’s diagram is used to illustrate the Al NP oxidation energetics and to show DH1rxn,np, including DH1rxn,np,np and DH1rxn,np,bk.
Fig. 2. Energy diagram shows the thermodynamic potentials of (a) DSDCEAl and cohesive energy and (b) DSDLEAl2 O3 and lattice energy.
reaction enthalpies for Al NP conversion to nano-sized Al2O3 (DH1rxn,np,np) and Al NP conversion to bulk Al2O3 (DH1rxn,np,bk) if the nano-Al2O3 agglomerates. Following the thermodynamic cycle for this system we can obtain the reaction enthalpies for the Al NP oxidation reaction to produce Al2O3 NPs, DH1rxn,np,np, and the reaction enthalpy for the Al NP oxidation reaction to produce bulk Al2O3, DH1rxn,np,bk. For DH1rxn,np,np, the relationship is derived from the thermodynamic cycle: DH1rxn,np,np þ DH1rxn,bk 2DSDCEAl þ DSDLEAl2 O3 ¼ 0 kJ=mol Al2 O3 ð1Þ where DH1rxn,bk is the reaction enthalpy of the bulk reaction ( 1675.7 kJ/mol), SDCE is the size-dependent cohesive energy, SDLE is the size-dependent lattice energy, DSDCEAl is the change in cohesive energy from the bulk Al to nano-Al, and DSDLEAl2 O3 is the change in lattice energy from the bulk Al2O3 to nano-Al2O3. DSDCE and DSDLE are direct measures of the change in thermodynamic potential of the Al NPs relative to bulk due to loss in surface binding. Fig. 2 shows an energy diagram illustrating DSDCE and DSDLE. Solving for DHrxn,np,np, the equation becomes
DH1rxn,np,np ¼ DH1rxn,bk 2DSDCEAl þ DSDLEAl2 O3
ð2Þ
In both equations there is a stoichiometric coefficient of 2 for
DSDCEAl, and a stoichiometric coefficient of 1 for DSDLEAl2 O3 . For the reaction of nano-Al-bulk Al2O3, the DSDLEAl2 O3 is 0 because there is no reduction in lattice energy for the bulk product. Thus,
ð3Þ
Additional factors must be included. First, in the nano-Alnano-Al2O3 reaction, the resulting Al2O3 particle has a diameter that is 9% greater than the initial Al particle. A factor, F, is added to Eq. (2), in which F is the ratio of reactant and product NP diameters. It is equal to 1/1.09 in this reaction and is multiplied to the DSDLEAl2 O3 . Both DSDCEAl and DSDLEAl2 O3 are inverse functions of the diameter of the nanoparticle. This factor is necessary to account for the change in lattice energy with changing NP diameter. Second, we need to include surface passivation effects in the reactant NP. All Al NPs require a passivation scheme. Bare Al NPs are thermodynamically and kinetically unstable toward grain growth. In addition, bare Al NPs are pyrophoric. Most simply, Al particles may be capped with an oxide layer. This effect is described by inclusion of the enthalpy contribution consisting of the interfacial energy of Al-Al2O3, gAl-Al2 O3 ,enth . The aluminum oxide layer is present on aluminum nanoparticles when the surface aluminum atoms react with O2 or H2O. The gAl-Al2 O3 ,enth term is also multiplied by 2 because it is in terms of moles of Al atoms. In addition, another factor that we must consider is the extent of agglomeration. The product Al2O3 NPs tend to agglomerate [24], effectively decreasing the DSDLEAl2 O3 . Agglomeration occurs when the surface atoms of a particle bond with the surface atoms of another particle, decreasing the fraction surface atoms with a dangling bond (d). The d term can range from 0 to 1, in which 0 corresponds to products when full agglomeration of the particles occurs. d ¼ 1 corresponds to products when no agglomeration of the NP products occurs. With these three factors amending the model, Eq. (2) becomes
DH1rxn,np,np ¼ DH1rxn,bk 2DSDCEAl þ F dDSDLEAl2 O3 2gAlAl2 O3 ,enth ð4Þ For the nano-Al-bulk-Al2O3, the values of F and d are obviously not amended since DSDLEAl2 O3 ¼ 0, but the interfacial energy contribution is amended. So, the relationship for the sizedependent enthalpy of reaction of Al NPs to produce bulk Al2O3 is
DH1rxn,np,bk ¼ DH1rxn,bk 2DSDCEAl 2gAlAl2 O3 ,enth
ð5Þ
In order to calculate the size dependence of the reaction enthalpy using DH1rxn,np,np and DH1rxn,np,bk, the DSDCE and DSDLE terms must be calculated. The DSDCE and DSDLE terms are a function of size of the nanoparticle. As the particle size decreases, the cohesive and lattice energy also decreases as well because the small diameter leads to a large surface area-to-volume ratio. At high specific surface areas, a large fraction of the Al atoms has dangling bonds. The increase in surface area-to-volume ratio also means that there are more surface atoms, and the surface atoms have fewer bonds with neighboring atoms than interior atoms, causing a decrease in the cohesive or lattice energy. There are many models for estimating the size dependence of the cohesive energy for metallic NPs. These include the liquid drop model [25], bond energy model [26], surface area difference model [27–29], Qi’s model based on surface relaxation [30], and Qi’s model dealing with shape effects [31]. The liquid drop model considers the surface tension of nanoparticles, in which the surface tension decreases with decrease in size of NPs. The bond energy model takes into account the total bond energy of interface and the total bond energy of interior atoms. In the current study we only consider spherical particles. All the Al NPs synthesized and reported to date are spherical.
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From the parameters and properties of Al and Al2O3 that are well known, we use Nanda’s model to calculate the cohesive energy [32]. This bond geometry model considers the number of bonds of the surface atoms and interior atoms and the fraction of surface atoms within a nanoparticle. The equation for the Nanda model is rffiffiffiffiffi Ns zs av,d ¼ av 1 1 ð6Þ N zb in which av,d is the SDCE or SDLE, av is the cohesive energy of an interior atom (bulk), zs is the coordination number of a surface atom, zb is the coordination number of an interior atom, and Ns/N is the fraction of surface atoms in the NP. For a spherical particle, Ns/N ¼8ra/d, where ra is the atomic radius and d is the diameter of the particle. Nanda’s model can be modified to calculate the SDLE for Al2O3 NPs, as shown below: av,d,AlNP ¼ ð2=5Þav,d,Al þ ð3=5Þav,d,O
ð7Þ
¼ av ½0:4ð1fNs,Al =Ngf1ðzs,Al =zb,Al Þ1=2 gÞ þ 0:6ð1fNs,O =Ngf1ðzs,O =zb,O Þ1=2 gÞ
ð8Þ
Thus, the same model can be used for reactants and products, resulting in the same type of curve for the lattice energy of Al2O3 nanoparticles. In Eq. (8), the factor [0.4(1 {Ns,Al/N}{1 (zs,Al/ Zb,Al)1/2}) þ0.6(1 {Ns,O/N}{1 (zs,O/zb,O)1/2})] decreases the value of av,d,AlNP and indicates the degree that the surface atoms of both Al and O affect the lattice energy of Al2O3 NPs. The calculation for these particles is different because they are composed of two constituent atoms. The av,d is calculated for both the Al and O atoms, and the av,d for Al is multiplied by 2/5 and the av,d for O is multiplied by 3/5 because two-fifths of the atoms in Al2O3 are made up of Al and three-fifths are made up of O, which includes those atoms on the surface. Separate coordination numbers are also used for Al and for O. The values of av for both the Al and O constituent atoms are the same as the bulk lattice energy of Al2O3 (15,270 kJ/mol). Table 1 shows the parameters that were used to calculate the DSDCE and DSDLE for Al and Al2O3, respectively. The cohesive energy decreases as the particle size decreases. Also, the cohesive energy asymptotically approaches the bulk cohesive energy of Al (330 kJ/mol). Next we consider the enthalpy contribution to the interfacial energy, gAl-Al2 O3 ,enth . As mentioned before, this term is needed to represent the interface that occurs when a native oxide layer passivates an Al particle. gAl-Al2 O3 ,enth was determined to be 0.068 J m 2 at 298 K [33]. This value was calculated from
g/MSfMOx g,enth ¼ pHfuse,/MS =A/MS
ð9Þ
where g/MS {MOx},enth is the enthalpy contribution to the interfacial energy between the crystal phase of M and the amorphous phase of the oxide layer (MOx), p is the fraction of the total surface area of M in contact with MOx phase, Hfuse,/MS is the enthalpy of fusion of crystal phase of M, and A/MS is the area occupied by one mole of M atoms at the interface. g/MSfMOx g,enth is calculated for Table 1 Parameters for Al and Al2O3 are used to calculate the SDCE and SDLE. Particle
av (kJ/mol)
zs/zb
Atomic radius (pm)
Al
330
8/12
125
Al2O3
15,270
Al: 5/6 O: 3/4
Al: 125 O: 60
721
the three metal surfaces: {1 1 1}, {1 1 0}, and {1 0 0}. The values were 0.083, 0.051, and 0.072 J m 2, and the average of these values was taken because it was assumed that a spherical nanoparticle contains all three surfaces equally. This value is in terms of surface area; so it must be converted into terms of moles of Al atoms. Appendix 1 shows the derivation of the enthalpy contribution to the interfacial energy in terms of kJ per mole of Al atoms. The equation for gAl-Al2 O3 ,enth is
gAlAl2 O3 ,enth ¼
4:1 kJ nm mol1 d
ð10Þ
where d is the diameter of the Al nanoparticle (in nm). This equation shows an inverse relationship between gAl-Al2 O3 ,enth and the diameter of the Al NP. A decrease in the diameter of an Al particle increases the fraction of surface Al atoms, and an increase in the fraction of surface Al atoms for an Al particle increases the interfacial energy contribution, which is expressed in kJ per mole of Al atoms. The origin of this effect is that the surface Al atoms are the only atoms that may interface with the aluminum oxide monolayer. Since the fraction of surface Al is near 0 for bulk Al, the bulk Al has a negligible value ( 0 kJ/mol) for the interfacial energy contribution. However, even for the nanoparticles with a diameter of 3 nm, gAl-Al2 O3 ,enth is 1.4 kJ/mol Al. When compared with the bulk enthalpy of reaction of 1675.7 kJ/mol Al2O3, the gAl-Al2 O3 ,enth term is small. After calculating the SDCE for Al particles and SDLE for Al2O3 particles, the DH1rxn,np,np may be calculated using Eq. (4). Fig. 3a shows DH1rxn,np,np as a function of size assuming no product agglomeration. As the particle size decreases, DH1rxn,np,np decreases in exothermicity. Thus, the reaction is less exothermic for very small Al NPs. The effect only becomes significant for Al NPs with diameters less than 10 nm. This decrease in reaction enthalpy may appear counterintuitive. The decrease in enthalpy is due to the much greater effect of the product DSDLEAl2 O3 relative to the reactant cohesive energy term, DSDCEAl. As the particle size increases, DH1rxn,np,np asymptotically approaches the bulk enthalpy of reaction. Significant effects are only present below 10 nm, where the fraction of surface atoms becomes a dominating factor. Next, we consider oxidation of Al NPs to produce bulk Al2O3. As just mentioned, during oxidation of Al NPs the product Al2O3 particles have a tendency to agglomerate. Large agglomeration leads to a bulk-phase Al2O3 product. Fig. 3b shows a graph of DH1rxn,np,bk as a function of size. In contrast to DH1rxn,np,np, DH1rxn,np,bk increases in exothermicity as the particle size decreases. This occurs because DSDLEAl2 O3 is not present when bulk Al2O3 is produced. As for the NP-NP reaction, significant effects on the reaction enthalpy are only seen below Al NP diameters of 10 nm. This leads us to consider the effects of agglomeration for the nano-Al-nano-Al2O3 reaction. Since we established that significant effects in enthalpy are only seen below 10 nm, we consider the effects to the enthalpy of reaction with agglomeration of these NPsr10 nm. Fig. 4 shows the enthalpy of reaction for 2, 4, 6, 8, and 10 nm particles as d is varied from 0 to 1. As expected, decreasing the fraction of surface atoms with dangling bonds for a given size of Al NP increases the enthalpy of reaction. This is because DSDLEAl2 O3 is decreasing. Also, as the particle size decreases, the effect of increase in extent of agglomeration on the reaction enthalpy increases, according to the increasing slope of the line when the particle size decreases. Smaller particles have larger deviations from the bulk enthalpy, and agglomeration effectively makes these smaller particles approach bulk behavior. It is also interesting that all enthalpies for all nanoparticles as well as the bulk value for the reaction enthalpy converge at d ¼0.126. A detailed mathematical analysis of this is found in the
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relative to the bulk enthalpy as the particle size decreases. As the agglomeration increases, the enthalpies for each of the particle sizes converge. As the agglomeration increases, the Al NPs enter the regime of nano-Al-bulk Al2O3. In this regime, the Al NP reaction exothermicity increases relative to the bulk value as the particle size decreases, and the enthalpies diverge from one another {vide infra}. It is at the transition between these two regimes that we see convergence to the bulk value, when a fraction of 0.126 surface atoms of Al2O3 is present. At this convergence point, the heat of reaction is size independent and is equal to the bulk reaction enthalpy. There are recent reports of Al NP oxidation in which water is found to be much more reactive than O2 toward Al NPs [3]. This is consistent with results on gas phase aluminum cluster ion reactivity. Small Al ionic clusters, consisting of 2–50 atoms, have been shown to react much faster with H2O than O2. While Al NPs are known to generate H2 as a product in reaction with H2O, the Al NPs also incorporate oxygen and hydroxide groups from water in the process. The resulting products are boehmite (g-AlO(OH)) and bayerite (Al(OH)3) when Al is oxidized under moist conditions, as opposed to corundum, which is made with oxidation of Al with pure oxygen [34]. Al reacts with water vapor according to the following reactions: 2Alþ4H2O-2AlO(OH)þ3H2 2Alþ6H2O-2Al(OH)3 þ3H2
Fig. 3. (a) The plot shows the enthalpy of reaction for the nano-Al-nano-Al2O3 (DH1rxn,np,np) as a function of diameter of Al particle and (b) the enthalpy of reaction for the nano-Al-bulk Al2O3 (DH1rxn,np,bk) as a function of diameter of Al particle.
This can significantly affect the thermodynamics of the oxidation reaction. We have considered the oxidation of Al NPs to bulk boehmite and to bulk bayerite using our model. The formation of bulk boehmite and bulk bayerite results in less heat release than corundum: the bulk reaction enthalpies for formation of boehmite and bayerite from aluminum are 1021 kJ/mol g-AlO(OH) and 1094.4 kJ/mol Al(OH)3, respectively. Fig. 5 includes the curves for DH1rxn,np,bk for the reaction of Al NPs to produce boehmite and bayerite as a function of reactant NP diameter Fig. 5 also includes experimental data obtained from Sun et al. [11]. This is the most reliable size-dependent thermodynamic data for mid-range Al NPs with narrow size distribution particles. The experimental data are for Al NPs with a narrow size distribution and passivated with an oxide layer. The experimental data show an energy release between that expected for
Fig. 4. NP enthalpy of reaction is plotted as a function of the fraction of surface atoms with dangling bonds (a measure of extent of product agglomeration) for a series of different NP diameters.
Appendix. The Al NP oxidation energetics occupies two regimes: nano-Al-nano-Al2O3 and nano-Al-bulk Al2O3. Initially, without agglomeration, the Al NPs are in the nano-Al-nano-Al2O3 regime. Within this regime, the reaction exothermicity decreases
Fig. 5. Plot compares the reaction enthalpy curves for the reactions of nano-Al oxidizing to the product bulk boehmite (DH1rxn,boe), bulk bayerite (DH1rxn,bay), and bulk corundum (DH1rxn,cor). It is compared to Pantoya’s experimental data of Al NPs with a narrow size distribution.
S.W. Chung et al. / Journal of Physics and Chemistry of Solids 72 (2011) 719–724
production of bayerite/boehmite and corundum. Possibly, traces of water give a mixed-phase reaction product during Al NPs oxidation.
723
where A ¼ 16av ra ð1ðzs =zb Þ1=2 Þ
ðeÞ
B ¼ Fav ½ð2=5Þð8ra,Al Þ ð1ðzs,Al =zb,Al Þ1=2 Þ þ ð3=5Þð8ra,O Þð1ðzs,O =zb,O Þ1=2 Þ
3. Conclusion The enthalpy of reaction for the oxidation of Al NPs is dependent on the diameter of the NP. The enthalpy depends heavily on the size-dependent cohesive energy of the Al nanoparticles and lattice energy of the resulting Al2O3 nanoparticles. Two reaction products are considered when Al nanoparticles are oxidized with oxygen: bulk and nano-Al2O3. Each reaction product results in two different reaction enthalpy curves. For the reaction of AlNP-nano-Al2O3, the reaction enthalpy decreases as the particle size decreases. In contrast, the reaction of AlNP-nano-Al2O3 has an increasing reaction enthalpy with decrease in particle size. The optimal system for maximum energy release should occur for sub10-nm Al NPs reacting to produce substantially agglomerated products. Eq. (4) should be a generally applicable equation for any NP reaction with the appropriate substitution of appropriate stoichiometric and other constants. Comparison of experimental data with that of boehmite and bayerite shows that the product of oxidized Al NPs may contain boehmite and bayerite, as well as corundum. There is a clear need for reliable thermodynamic data for very small Al NPs in the sub10-nm range.
ðfÞ
and C ¼ 12Mð6:8 105 kJm2 Þ=r
ðgÞ
Eq. (d) is rearranged to remove the dependence on nanoparticle diameter, d, to give A þB dC ¼ 0
ðhÞ
For the current oxidation reactions of Al NPs the values for the constants return the values for A, B, and C as follows: A Að1=dÞ ¼ 2ð330av ½1ðNs =NÞð1ðzs =zb Þ1=2 ÞÞ ¼ 2ð330330þ av ð8ra =dÞð1ðzs =zb Þ1=2 ÞÞ ¼ 2ðav Þð8ra =dÞð1ðzs =zb Þ1=2 Þ ¼ 16ra av ð1=dÞð1ðzs =zb Þ1=2 Þ A ¼ 16ra av ð1ðzs =zb Þ1=2 Þ ¼ 121:11 B Bdð1=dÞ ¼ FdðSDLEbk,Al2 O3 SDLEnp,Al2 O3 Þ ¼ Fd 15270-½ð2=5Þav,d ðAlÞ þ ð3=5Þav,d ðOÞ ¼ Fdf15270-½0:4av
Acknowledgments
0:4av ð1=dÞð8ra,Al Þð1ðzs,Al =zb,Al Þ1=2 Þ
We gratefully acknowledge the Air Force Research Laboratory Nanoenergetics Program for supporting this work. We would like to thank Dr. Michelle Pantoya and Dr. Sindee Simon for their experimental data.
½0:6av 0:6av ð1=dÞð8ra,O Þð1ðzs,O =zb,O Þ1=2 Þg ¼ Fd½0:4av ð1=dÞð8ra,Al Þ ð1ðzs,Al =zb,Al Þ1=2 Þ þ 0:6av ð1=dÞð8ra,O Þð1ðzs,O =zb,O Þ1=2 Þ ¼ Fdav ð1=dÞ½0:4ð8ra,Al Þ ð1ðzs,Al =zb,Al Þ1=2 Þ þ 0:6ð8ra,O Þð1ðzs,O =zb,O Þ1=2 Þ
Appendix Fig. 4 in the main text illustrates an interesting convergence in the reaction enthalpy where reactions for all particle sizes (including nano all the way to bulk) have the same reaction enthalpy at a particular extent of agglomeration. While this was not initially intuitive to us it can be shown to be true starting from Eq. (4) in the main text:
DH1rxn,np,np ¼ DH1rxn,bk 2ðSDCEbk,Al SDCEnp,Al Þ þ FdðSDLEbk,Al2 O3 SDLEnp,Al2 O3 Þ2gAl2 O3 ,enth
ðaÞ
B ¼ 8Fav ½0:4ðra,Al Þð1ðzs,Al =zb,Al Þ1=2 Þ þ 0:6ðra,O Þð1ðzs,O =zb,O Þ1=2 Þ ¼ 1028:782 C Cð1=dÞ ¼ ð2Þð6:8 105 kJ m2 Þ ðSÞ=ðrV=MÞ 2
3
¼ ½ð2Þð6:8 105 kJ m2 Þðpd Þ=½ðrÞðpd =6Þ=M ¼ 12 Mð6:8 10 S ¼ surface area ¼ pd
5
2
kJ m
Þ=rd
2
3
From Figs. 3 and 5 it is apparent that the nanoparticle reaction enthalpy may be larger or smaller than the bulk enthalpy, depending on the state of the products. At the extent of agglomeration at which DH1rxn,np,np is equal to the bulk value:
DH1rxn,np,np ¼ DH1rxn,bk
1
M ¼ molar mass ¼ 26:981 g mol C ¼ 12 Mð6:8 10
5
kJ m
2
Þ=r
¼ ½12ð6:8 105 kJ m2 Þ ð26:981 g mol ¼ 8:2 kJ nm mol
ðcÞ
Eq. (c) can be rearranged to separate the dependence on the reactant nanoparticle diameter, d, to give Að1=dÞ þ Bdð1=dÞCð1=dÞ ¼ ð1=dÞ½A þ BdC ¼ 0
r ¼ density ¼ 2:7 106 g m3
1
ðbÞ
and Eq. (a) reduces to 0 ¼ 2ðSDCEbk,Al SDCEnp,Al Þ þ FdðSDLEbk,Al2 O3 SDLEnp,Al2 O3 Þ 2gAl2 O3 ,enth
V ¼ volume ¼ pd =6
ðdÞ
Thus, at the point defined by Eq. (b) above we see that the NP particle diameter dependence can be removed from the equation,
Þ=ð2:7 106 g m3 Þ
1
Solving for d for Eq. (h) 121:11 þ1028:782d8:2 ¼ 0 d ¼ 0:126
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