Size and Shape Dependent Catalytic Activation Energy of Different Nano Structures

The size and shape dependent properties of nanomaterials have been extensively studied during the past few years. Once the size of the matter approaches to atomic scale, it exhibits fascinating electrical, optical, magnetic, vibrational, thermo dynamical and catalytic properties than its bulk counterpart resulting from salient features like large surface-to-volume ratio and quantum effect (Alivisatos et al., 1998, Alivisatos, 1996, Gupta et al., 2009, Gupta et al., 2008, Gupta and Jha, 2009, Mankad et al., 2011, Talati and Jha, 2005, Gupta et al., 2010, Jha et al., 2006). Researchers have shown that the size and shape of a particle are key factors to influence the various phenomena including melting, interface energy, surface diffusion, catalysis, etc (Bhatt et al., 2012, Mishra et al., 2012, Gleiter, 2000, Zhang et al., 2000, Jiang and Lu, 2008, Jiang et al., 2004, Zhou et al., 2010, Gupta et al., 2008). Abstract


Introduction
The size and shape dependent properties of nanomaterials have been extensively studied during the past few years.Once the size of the matter approaches to atomic scale, it exhibits fascinating electrical, optical, magnetic, vibrational, thermo dynamical and catalytic properties than its bulk counterpart resulting from salient features like large surface-to-volume ratio and quantum effect (Alivisatos et al., 1998, Alivisatos, 1996 In recent years, enormous efforts have been devoted to study the size and shape dependence of nanoparticles to understand the catalytic and electro catalytic performances (Narayanan and El-Sayed, 2004, Burda et al., 2005, Li et al., 2000).Nanoparticles of different materials act as efficient catalysts for oxidation of hydrocarbons, C-C coupling, hydrogenationdehydrogenation, redox and other chemical transformations (Zhou et Kung et al., 2007).Past studies show that the titanium dioxide (TiO2) is a promising candidate for the decomposition of wide variety of organic and inorganic moieties in both liquid and gas phases (Mankad et al., 2011, Gupta et al., 2010, Jha, 2013, Desai et al., 2011).The nano scale cerium dioxide (CeO2) has also advantages as a support for catalysis since it can enhance the reactivity for the oxidation reaction and CO oxidation when supported to gold nanoparticles (Guo et al., 2010, Carrettin, 2004).Although nanoparticles of tin dioxide (SnO2) are of less interest particularly for catalytic applications, Sharghi et al., (2006) have found that the SnO2 nanoparticles are highly reactive in the Knoevenagel condensation process.
Catalytic activation energy (EC) is one of the important kinetic parameters, which measures the chemical activity of the catalyst to catalyze different reactions (Narayanan and El-Sayed, 2004).In the present paper, we have calculated and compared the catalytic activation energy (EC) of three different nanostructures i.e., titanium dioxide (TiO2), cerium dioxide (CeO2) and tin dioxide (SnO2).The size and shape dependent catalytic activation energy of TiO2, CeO2 and SnO2 nanostructures has been calculated using a simple model based on analytical approach.Further, to analyze the dimension dependency on the catalytic activity of these nanostructures, we have also calculated the activation energy for-0-, 1-and 2-dimensional TiO2, CeO2 and SnO2 nanostructures that correspond to spherical nanoparticles, nanowires and thin films respectively.We observe that the different shapes have considerable effect on the catalytic activation energy.The paper is divided into four main sections.In section 2, brief description of theoretical model is presented.Section 3 describes the results and discussion followed by a summary of the present work in section 4.

Methodology and Computation
The catalytic activation energy in the present study is determined using a theoretical model based on an analytical approach which includes the shape factor (Lu and Meng, 2010).The size and temperature dependent rate constant of any chemical reaction as per Arrhenius equation is expressed as, Where, K is a rate constant of the reaction, K0 is the pre-exponential factor, Ea is activation energy (here considered as catalytic activation energy), R is a universal gas constant and T is the temperature.
At the melting temperature (Tm), if rate constant (K) is independent of melting temperature and size of the particle, the rate constant will be the same for all the nanoparticles of a given system.Therefore,

By rewriting
The effect of pre-exponential factor K0(D) on K(D,T) is negligible in comparison to the exp[-Ea(D)/(RT)] hence using first order approximation, we can assume K0(D)≈[K0(∞)].This leads to the expression, The liquid-drop model and (BOLS) mechanism suggest that both melting temperature Tm and cohesive energy EC, are related to the bond energy of crystalline atoms hence From that consideration, one can get EC as a function of size and shape, Here, Sb=Eb/Tb is the bulk coherent entropy of crystals with Tb being the bulk solid-vapor transition temperature and R is the ideal gas constant.Due to the instability in semiconductor compounds, it is difficult to find the value of Sb value therefore we have used the value of Sb equal to the 13R in our calculations (Yang and Jiang, 2006).The shape factor, λ depends on the ratio of surface atoms to total atoms.For the different shapes, the value of the shape factor (λ) is used from reference (Lu and Meng, 2010).In addition to the shape effect, the dimension dependence of EC for the different size of TiO2, CeO2 and SnO2 nanoparticles is also reported.In eqn. 5, D denotes the diameter of nanoparticles while D0 is the diameter at which all atoms are located on surface.D0 can be expressed as, Where h and d denote the value of atomic diameter and different dimensions of nanoparticles, i.e. d=0 for spherical nanoparticles, d=1 for nanowires and d=2 for thin films (Li et al., 2004).

Results and Discussion
Figure 1[a] presents the variation of EC(D)/EC(∞) (ratio of size dependent activation energy to its bulk counterpart) of rutile TiO2 nanoparticles of different dimensions which shows that the catalytic activation energy decreases as the size of the TiO2 nanoparticle decreases.The value of activation energy rapidly decreases below 5 nm.It is clear from this figure that after 15 nm, the ratio of catalytic energies becomes almost constant.The figure also depicts that the catalytic activation energy is minimum for spherical nanoparticles while it is maximum for the two dimensional case.This indicates that the spherical nanoparticles can act as better catalysts compared to one and two dimensional nanostructures.The decrease in the catalytic activation energy with the decreasing size reveals efficiency of nanoparticles to catalyze any chemical reaction.Fig. 1[b] and Fig. 1[c

Conclusion
In summary, we have calculated the shape and size dependent catalytic activation energy of TiO2, SnO2 and CeO2 nanostructures.The catalytic activation energy decreases with the decrease in particles' size.It becomes almost size independent when size of the particle exceeds 10 to 15 nm.We observe that zero dimensional (i.e., spherical) nanoparticles have minimum catalytic activation energy than the two and one dimensional structures.The considerable difference is found for EC in the case of different shapes of nanoparticles, and tetrahedral shaped particles showed maximum efficiency for catalysis.In the case of TiO2, CeO2 and SnO2, TiO2 has minimum value of catalytic activation energy hence considered as better catalyst.Our study will find its implications in catalytic activity based applications.
Eqns. 1 and 2, the following expression can be obtained, _____________________________________________________________________________ ______________ Purvi A. Bhatt, Arun Pratap and Prafulla K. Jha (2015), Journal of Research in Nanotechnology, DOI: 10.5171/2015.557871. 0 Nanda et al., 2002, Sun, 2007).If the nature of the chemical bonds remains the same, this relationship can be written for nano scaled region i.e. modified the thermodynamic model proposed by Yang et al.,(2008)  and include the shape factor (λ) to calculate size and shape dependent catalytic activation energy.

Figure 2
Figure 2 shows the variation of EC(D)/EC(∞) with size and different shapes of TiO2 nanoparticles.For a particular size, a tetrahedral shaped particle with (111) facet exhibits maximum value of EC compared to octahedral and cubic shaped nanoparticles.It can be observed from the figure that the cubic nanoparticles are having the minimum value of EC for a given size.This nature of catalytic activation energy indicates that the tetrahedral shaped nanoparticles are the most catalytically active.For CeO2 and SnO2

Figure 3
Figure3shows comparison of all the three tetrahedral shaped nanostructures (i.e., TiO2, CeO2 and SnO2) in terms of size dependent catalytic activation energy.It is clear from the figure that the TiO2 nanoparticles have minimum catalytic activation energy which implies that the nano structured TiO2 can be considered as an efficient candidate for catalytic activities.Although nano sized SnO2