Journal of Engineering Physics and Thermophysics, Vol. 86, No. 3, May, 2013

INCORPORATION OF GAS MOLECULES INTO NANOSIZED PARTICLES V. V. Levdanskii,a J. Smolik,b V. Zdimal,b and P. Moravecb

UDC 541.182

The influence of size effects on the incorporation of gas molecules into nanosized aerosol particles has been investigated theoretically. Keywords: size effects, adsorption, desorption, absorption. Introduction. In recent years, great attention of researchers has been given to questions associated with incorporation of gas molecules (atoms) into nanosized aerosol particles. This is related to both processes in the atmosphere and those realizable in the production and use of nanoobjects in a number of the areas of modern nanotechnology. The process of incorporation of gas particles into a condensed phase is characterized by the so-called mass accommodation coefficient γ, which is dependent on the processes of molecule adsorption, desorption, and absorption [1, 2]. It is pertinent to note that in a number of works the mentioned coefficient is calculated with the use of the parameters that characterize the surface processes for a massive sample. It should be noted that the course of the processes on the surface of nanosized particles can differ from their course on the surface of a massive sample. Some questions relative to the influence of the nanoparticle size on the absorption of molecules (atoms) adsorbed on the nanoparticle surface were considered in [3]. However, the size dependence of the sticking coefficient of gas molecules incident on the surface of a nanosized particle, as well as the size dependence of molecule desorption, were not considered there. In the present paper, the joint influence of the size dependence of the mentioned factors on the probability of incorporation of gas molecules (atoms) into a nanoparticle is investigated. Results and Discussion. The above-mentioned mass accommodation coefficient, which represents a part of collisions of gas molecules (atoms) with a particle resulting in their incorporation into the latter, at sufficiently small concentrations of the considered molecules both in the adsorbed state and in the absorbed one takes the form [1, 2]:

γ =

α s k abs . k des + k abs

(1)

The sticking coefficient of a molecule adhering to a nanosized particle (cluster) generally depends on the particle size. For example, it is known that the probability of formation of a dimer in collision of two molecules (i.e., the probability of "adhering" of one molecule to another) is well below unity. At the same time, the sticking coefficient of a gas molecule in its collision with a sufficiently large particle can be very close to unity. Thus, generally the sticking coefficient in some particle size range should depend on their diameter. To obtain the size dependence of the sticking coefficient, we use its connection with the adsorption rate constant [1]: αs =

4 k ads . v

(2)

According to [4, 5], the dependences of the adsorption and desorption rate constants on the diameter of a spherical nanoparticle can be presented as ⎛ 4α ads σVm ⎞ k ads = k ads,∞ exp ⎜ − ⎟, dRT ⎠ ⎝

(3)

⎛ 4(1 − α ads )σVm ⎞ k des = k des,∞ exp ⎜ ⎟, dRT ⎝ ⎠

(4)

a

A. V. Luikov Heat and Mass Transfer Institute, National Academy of Sciences of Belarus, 15 P. Brovka Str., Minsk, 220072, Belarus; email: [email protected]; bInstitute of Chemical Process Fundamentals AS CR, v.v.i., 135 Rozvojova Str., Prague, 16502, Czech Republic. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 86, No. 3, pp. 516–519, May–June, 2013. Original article submitted December 18, 2012. 0062-0125/13/8603-0547 ©2013 Springer Science+Business Media New York

547

Fig. 1. Dependence of γ1* on d*: 1) φ = 0.2 and ψ = 25; 2) 0.2 and 20; 3) 0.5 and 25; 4) 0.5 and 20. Fig. 2. Dependence of γ *2 on d*: 1) χ = 0.15; 2) 1; 3) 2; 4) 4. where kads,∞ and kdes,∞ are the adsorption and desorption rate constants, respectively, for a massive sample and αads is the Polanyi parameter. The transition of adsorbed molecules (atoms) into the particle (their absorption) is an activated process where the activation energy depends on the concentration of vacancies in the nanoparticle. We can write the absorption rate constant as ⎛ Q ⎞ k abs = k 0 exp ⎜ − abs ⎟ , ⎝ RT ⎠

(5)

where Qabs is the absorption activation energy, which generally depends on the nanoparticle size due to the size dependence of the activation energy of vacancy formation and, correspondingly, of the vacancy concentration, and k0 is the pre-exponential factor, which is assumed constant. In [6], an expression is suggested that characterizes the relationship between the cohesion energies, as well as between the activation energies of physicochemical processes (in particular, the activation energies of the formation of vacancies), and the melting temperatures for a nanoparticle and massive sample. Considering [6, 7], we can write the expression for the dependence of Qabs on the nanoparticle size: 4δ ⎞ ⎛ Qabs = Qabs,∞ exp ⎜ − ⎟. δ + d⎠ ⎝

(6)

The absorption rate constant kabs for a nanoparticle is obtained from the equation ⎧Qabs,∞ k abs = k abs,∞ exp ⎨ ⎩ RT

4δ ⎞⎤⎫ ⎡ ⎛ ⎢1 − exp ⎜ − δ + d ⎟⎥⎬ . ⎝ ⎠⎦⎭ ⎣

(7)

where kabs,∞ is the absorption rate constant for a massive sample. We consider the cases limiting in reference to kabs/kdes for d >> δ. 1. At kabs << kdes with regard to the size dependence of the surface tension [7, 8], we have the following equation for the mass accommodation coefficient obtained from Eqs. (1)–(7): ⎧⎪ 4Qabs,∞ γ = γ1 = γ1∞ exp ⎨ * ⎩⎪ RTd

⎡ σ ∞Vm ⎢1 − Q abs,∞ δ ⎣

4 ⎞⎤⎫⎪ ⎛ ⎜1 − * ⎟⎥⎬ , d ⎠⎦⎪⎭ ⎝

where γ1∞ = 4kads,∞kabs,∞/(vkdes,∞), d* = d/δ, and σ∞ is the surface tension for a massive sample.

548

(8)

Figure 1 gives the dependence of the quantity γ1* = γ1/γ1∞ on the dimensionless particle diameter d* at different values of the parameters φ = σ∞Vm/(Qabs,∞δ) and ψ = Qabs,∞/(RT). It is seen from the figure that γ1* increases with decrease in the particle size and parameter φ and with increase in the parameter ψ. 2. At kabs >> kdes, we obtain the equation for γ with consideration for Eqs. (1)–(3) and the size dependence of the surface tension:

⎡ 4α ads σ ∞Vm γ = γ 2 = α s = γ 2∞ exp ⎢− d * RT δ ⎣

4 ⎞⎤ ⎛ ⎜1 − * ⎟⎥ , d ⎠⎦ ⎝

(9)

where γ2∞ = 4kads,∞/v = αs∞. Figure 2 shows the dependence of γ *2 = γ2/γ2∞ on d*. It follows from the figure that γ *2 decreases with the nanoparα σ V ticle size and increases with decrease in the parameter χ = ads ∞ m . RT δ It is self-evident that the case where kabs >> kdes occurs more rarely than the above-considered case with kabs << kdes. As an example we refer to the aluminum adsorption on the rhenium surface at a temperature below 1200 K [9]. In [9], it is noted that the aluminum desorption from the rhenium surface is not observed under the above-mentioned condition for the temperature, and the decrease in the surface concentration of aluminum can be explained only by its dissolution. In the general case, i.e., for an arbitrary ratio between the quantities kabs and kdes, at d >> δ, we have the following equation for the mass accommodation coefficient γ based on Eqs. (1)–(7) with the size dependence of σ: ⎛ 4Qabs,∞ δ ⎞ 4δ ⎞⎤ ⎡ 4α ads σ ∞Vm ⎛ k ads,∞ exp ⎢− ⎟ ⎜1 − ⎟⎥ k abs,∞ exp ⎜ dRT d ⎠⎦ 4 ⎝ ⎣ ⎝ RTd ⎠ . γ = v ⎛ 4Qabs,∞ δ ⎞ 4δ ⎞⎤ ⎡ 4(1 − α ads )σ ∞Vm ⎛ k des,∞ exp ⎢ ⎟ ⎜1 − ⎟ + k abs,∞ exp ⎜ dRT d ⎠⎥⎦ ⎝ ⎣ ⎝ RTd ⎠

(10)

The following can be noted regarding the numerical values of the parameters appearing in the expression for γ. The Tolman length δ for a specific substance is difficult to find with reasonable accuracy. The paper [8] gives the range of variation of δ from 0.096 to 0.35 nm. The surface tension for different substances varies within wide limits and has been studied rather thoroughly for many of substances. As is noted in [4], the value of the surface tension for a number of metals lies in the range 1–2 J/m2. The value of the absorption activation energy (Qabs,∞) depends both on the absorbed substance and on the substance into which the molecules (atoms) enter in their transition from the adsorption state to absorption one; this value can vary appreciably with the use of different substances. In a number of cases, the determination of the value of Qabs,∞ is rather a challenge. The scarcity of data on this parameter, which is of great importance for describing a number of processes proceeding in heterogeneous systems, was pointed in [10]. In particular, according to the latter work, the value of the activation energy for the transition of carbon atoms from the (100) surface of molybdenum into its volume is equal to 3.9 eV. Conclusions. We have analyzed the size dependence of the mass accommodation coefficient γ in the gas–nanoparticles system with consideration for the dependence of the adsorption, desorption, and absorption rate constants on the nanoparticle size. It has been shown that γ can both increase and decrease with reduction in the nanoparticle size depending on the ratio between the desorption and absorption rate constants. The derived equations can be used for estimating the probability of incorporation of gas molecules (atoms) into aerosol particles in the studies of physicochemical processes occurring both in atmosphere and in various fields of chemical technology which are related to obtaining and using nanosized particles. The work was carried out with partial support from GAAVCR (project IAA200760905) and Belarusian Republican Foundation for Fundamental Research (project T12K-018).

NOTATION d, nanoparticle diameter; kabs, kads, and kdes, rate constants of absorption, adsorption, and desorption of molecules, respectively; Qabs,∞, activation absorption energy for a massive sample; R, gas constant; T, temperature; v, mean thermal velocity of gas molecules; Vm, molar volume of a nanoparticle substance; αs, sticking coefficient of a molecule adhering to a particle surface; σ, surface tension. Indices: abs, absorption; ads, adsorption; des, desorption; s, surface; ∞, massive sample.

549

REFERENCES 1. D. R. Hanson, Surface-specific reactions on liquids, J. Phys. Chem. B, 101, 4998–5001 (1997). 2. J. N. Crowley, M. Ammann, R. A. Cox, et al., Evaluated kinetic and photochemical data for atmospheric chemistry: Volume V — heterogeneous reactions on solid substrates, Atmos. Chem. Phys., 10, 9059–9223 (2010). 3. V. V. Levdanskii, J. Smolik, and P. Moravec, Influence of size effects on gas absorption by nanoparticles, Inzh.-Fiz. Zh., 81, No. 5, 944–947 (2008). 4. D. Yu. Murzin, Thermodynamic analysis of nanoparticle size effect on catalytic kinetics, Chem. Eng. Sci., 64, 1046– 1052 (2009). 5. X. Zhou, W. Xu, G. Liu, D. Panda, and P. Chen, Size-dependent catalytic activity and dynamics of gold nanoparticles at the single-molecule level, J. Am. Chem. Soc., 132, 138–146 (2010). 6. S. C. Vanithakumari and K. K. Nanda, A universal relation for the cohesive energy of nanoparticles, Phys. Lett. A, 372, 6930–6934 (2008). 7. S. Sh. Rekhviashvili and E. V. Kishtikova, On the melting temperature of nanoparticles and nanostructure substances, Pis’ma Zh. Tekh. Fiz., 32, Issue 10, 50–55 (2006). 8. R. C. Tolman, The effect of droplet size on surface tension, J. Chem. Phys., 17, 333–337 (1949). 9. N. R. Gall’, E. V. Rut’kov, and A. Ya. Tontegode, Interaction of aluminum with the surface of rhenium: adsorption, desorption, growth of surface compounds, Fiz. Tverd. Tela, 44, Issue 7, 1332–1336 (2002). 10. N. R. Gall’, E. V. Rut’kov, and A. Ya. Tontegode, Diffusion of carbon between the volume and (100) surface of molybdenum, Zh. Tekh. Fiz., 72, Issue 4, 113–119 (2002).

550