DOI:10.1140/epjd/e2014-40831-2
Regular Article
P
HYSICAL
J
OURNAL
D
Theoretical evaluation of diffusion coefficients of (Al
2
O
3
)
n
clusters in different bath gases
Alexander S. Sharipov1,2, Boris I. Loukhovitski1,2, Chuen-Jinn Tsai3, and Alexander M. Starik1,2,a 1 Central Institute of Aviation Motors, Aviamotornaya st. 2, 111116 Moscow, Russia
2 Scientific Educational Centre Physical-Chemical Kinetics and Combustion, Aviamotornaya 2, 111116 Moscow, Russia 3 Institute of Environmental Engineering, National Chiao Tung University No.1001, University Road, 30010 Hsinchu, Taiwan
Received 26 December 2013 / Received in final form 3 March 2014
Published online 24 April 2014 – c EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2014 Abstract. The binary diffusion coefficients of two low lying isomers of (Al2O3)n, n = 1 . . . 4, clusters in different bath gases, that most frequently met in the nature and in the technical applications: H2, N2, O2, CO, H2O as well as their self-diffusion coefficients have been calculated on the basis of kinetic theory and dipole reduced formalism. The parameters of interaction potential have been determined taking into account the contributions of a dispersion, dipole-dipole and dipole-induced dipole interactions between alumina clusters and bath molecules. The dipole moments, polarizabilities and collision diameters of clusters have been obtained by using quantum chemical calculations of cluster structure. The approximations for temperature dependencies of diffusion coefficients for two low-lying isomers of each considered alumina clusters are reported. It is demonstrated that an account for the contributions of the second for each type of clusters does not affect substantially the value of net diffusion coefficient. The diffusion coefficients of the isomers of small (Al2O3)n clusters can differ notably in the case when their dipole moments are distinct and they interact with strongly dipole molecules.
1 Introduction
For past decade an explosion of interest in the investiga-tion of structure and physical properties of nanoclusters has been driven by the fact that clusters form in vari-ous nature phenomena and that they have wide-range ap-plications for the fabrication of novel nanomaterials for electronics, catalysis, medicine, aerospace vehicles, energy conversion, etc. So, nanoclusters are observed in the up-per atmosphere of different planets [1], they participate in the astrophysical dust formation processes [1,2] and form during the combustion of hydrocarbon and metallized fu-els [3–6] and energetic materials [7,8]. As well, nanoclus-ters play an important role in plasma etching [9], plasma-based technologies [10,11] and chemical vapor deposition upon laser ablation [10,12,13]. Simulations of these pro-cesses require the data on thermodynamic and transport properties of such clusters.
Today, there are no appropriate experimental ap-proaches and diagnostic systems allowing one to measure directly the thermodynamic and transport properties of nanoclusters. At the same time, the modern theoretical approaches make it possible to estimate the needed pa-rameters on the basis of quantum chemical calculations. Such calculations allow one to build the potential energy
a e-mail:star@ciam.ru
surfaces (PESs) for atomic system relevant to clusters and to find the minima on the PESs corresponding to possi-ble isomers of clusters. Therefore, one can determine their electronic energy, frequencies of normal vibrations, rota-tional constants, electronic degeneracy, geometric struc-ture and collision diameter. This information enables to estimate thermodynamic properties such as temperature-dependent enthalpy, entropy, specific heat capacity and Gibbs energy.
In order to estimate the transport properties, such as diffusion, thermal conductivity and viscosity coefficients, it is necessary to find the potential energy of the inter-action of clusters with bath gases or other clusters. In common case, the special efforts are needed to determine the interaction potential for a given type of cluster. How-ever, frequently, it is supposed that nanoclusters inter-act as ordinary molecules via the Stockmayer (SM) 12-6-3 potential for the case of polar clusters and well-known Lennard-Jones (LJ) 12-6 potential for the interaction of nonpolar ones [14,15]. Usually, the parameters of these types of potential are obtained via fitting the predictions and experimental data on such macroscopic parameters as second virial coefficient, coefficients of viscosity and ther-mal conductivity [14].
On the other hand, the parameters of intermolecular potential can be derived from the extensive ab initio cal-culations of the potential energy surface on a large grid of
molecular geometries [16,17]. However, the adequate ab initio description of long-range intermolecular forces re-quires the use of rather expensive post-Hartree-Fock meth-ods for solving the Schr¨odinger equation with large and flexible basis sets, that makes the whole procedure com-putationally expensive, especially, for large molecules and molecular clusters [18]. That is why, the usage of analytic and empiric approaches for finding the PES to estimate interaction potential is desirable even for small clusters.
The present paper addresses the evaluation of diffusion coefficients of (Al2O3)n, n = 1 . . . 4, clusters with differ-ent bath gases most frequdiffer-ently occurring in the applica-tions, such as H2, N2, O2, CO and H2O, on the basis of quantum-chemical calculations of cluster physical prop-erties. It is worth noting that aluminum oxide is one of the most important and widespread oxygen-bearing com-pounds. Alumina is a constituent in a number of ceramic materials. Alumina clusters and particles are formed dur-ing the combustion of Al powder in various environments: air [4], steam [19], carbon dioxide [19,20] as well as dur-ing the combustion of different energetic materials [21]. In addition, (Al2O3)n clusters were observed in astrophysi-cal dust [22]. In order to model these processes and phe-nomena the diffusion coefficients for (Al2O3)nclusters are strongly needed. However, until now there is no informa-tion about transport properties of small alumina clusters.
2 Methodology
2.1 Binary diffusion coefficient and collision integrals
In principle, the traditional kinetic theory allows one to estimate the transport properties of individual species, if the potential energy of particle interaction is known. The binary diffusion coefficients Dijcan be calculated with the
use of known equation obtained on the basis of Chapman-Enskog theory [14,23] and applicable at low and moderate pressures Dij = 3 8 2πMijkT /NA πσ2ij 1 Ω(1,1)∗ RT P02Mij, Mij = MiMj Mi+ Mj. (1) Here Mi(j)is the molar mass of particles of i(j) sort, T is
the temperature, k is the Boltzmann constant, NA is the
Avogadro number, σijis the collision diameter of colliding
particles, P0is the normal atmospheric pressure, Ω(1,1)∗ is
the reduced collision integral normalized by the cross sec-tions for rigid spheres of diameter σij. In fact, the
prod-uct f = σ2ijΩ(1,1)
∗
(T ) is the energy-averaged cross-section that depends on the temperature and interparticle poten-tial ϕij(r). Therefore, in order to estimate the value of Dij
one needs to know the parameters of potential ϕij(r).
As is known, for non-polar molecules the intermolecu-lar interaction can be described by the LJ (12-6) potential
ϕLJij (r) = 4εij σ ij r 12 −σij r 6 , (2)
where the potential well depth can be expressed as εij =C
disp 6
4σ6ij . (3)
The dispersion part of potential ϕdisp is governed by the
Slater-Kirkwood equation [24] in the spherically symmet-ric form: ϕdisp(r) = −C disp 6 r6 , C disp 6 = 3 2Eata 3/2 0 αAαAαB NA + αB NB . (4) Here Eat= 27.211 eV, a0 is the Bohr radius, αA and αB
are the polarizabilities, NA and NB are the numbers of valence electrons (the number of electrons in the outer sub-shell of particle) of particles of A and B sorts, r is the distance between interacting particles.
The collision integrals for LJ potential depend on the reduced temperature Tij∗ = kT /εij, and the
val-ues of Ω(1,1)∗(Tij∗) were calculated and tabulated
else-where [14,25–28]. Note that earlier the phenomenological approach based on the usage of Pirani’s potential [29] was extensively used for the estimation of collision integrals relevant to the transport properties of atmospheric gases for the extremely broad temperature range (T = 102– 105 K) [30,31]. However, in the present work, we used the approximations of collision integrals reported in ref-erence [25] for LJ potential and applicable at the reduced temperature range 0.3 T∗ 100 (T 104K).
Note, that the Slater-Kirkwood equation commonly overestimates slightly the value of the dispersion term C6disp with respect to well-known London one [15].
How-ever, equations (2)–(4) give, strictly speaking, only the es-timates for long-range attractive potential. The exchange effects also should be taken into account at interparticle distance smaller than 15a0[15], and, therefore, the
poten-tial well depth εij can be potentially larger in this case.
One can suppose that slight overestimation of dispersion term by the Slater-Kirkwood equation can be effectively compensated by neglecting the exchange forces.
In the case when molecules or clusters are polar, the angle-dependent dipole-induced dipole and dipole-dipole interactions also must be taken into account [14]. When these contributions are averaged over possible orientations of dipole moments of particles, the attractive potential between neutral species can be expressed as [15,28]
ϕ (r) = −C
summ 6
r6 , (5)
where Csumm
6 coefficient is the sum of the terms:
C6summ= C6el+ C6ind+ C6disp. (6)
Here Cel
6 and C6indspecify the orientation-averaged
tial of electrostatic interaction of dipoles and the poten-tial of polarization interaction. They are governed by the relationships C6el= 2 3kTμ 2 Aμ2B, (7) C6ind= μ2AαB+ μ2BαA. (8)
Thus, the contributions of orientation-dependent dipole-induced dipole and dipole-dipole interactions to the values of collision integrals can be approximately estimated if the effective potential well depth is taken in the form
εij= C el
6 + C6ind+ C6disp
4σ6ij
. (9)
It is worth noting, that the procedure of direct summation of orientation-averaged terms responsible for the dipole-induced dipole and dipole-dipole interactions in the man-ner of equation (6) seems to be somewhat questionable. In practice, SM (12-6-3) potential is frequently used for the approximation of real interaction between two polar molecules of i and j sorts
ϕSMij (r) = 4εij σ ij r 12 − 1 + ξij∗ζd−id(ωij) σij r 6 − δ∗ ijζd−d(ωij) σ ij r 3 . (10)
Here ζd−id and ζd−d are the angle-dependent functions
for dipole-induced dipole and dipole-dipole types of inter-action, ξ∗ij and δ∗ij = μiμj/2εijσ3ij are the mean reduced
polarizability and mean reduced dipole moments of inter-acting particles. In terms of the method of dipole reduced formalism (DRFM) developed by Paul and Warnatz [32], the potential ϕSM
ij subjected to thermally
orientation-averaged procedure can be reduced to the effective LJ potential. This allows one to incorporate the dipole-dipole and dipole-dipole-induced dipole-dipole interaction terms into the methodology of reduced collision integrals developed originally for van der Waals potential. In this case, for the determination of effective potential parameters C6eff and
σeffij the following equations are valid:
C6eff = C6disp 1 + C ind 6 C6disp + Cel 6 4C6disp 2 , (11) σeffij = σij 1 + C ind 6 C6disp + Cel 6 4C6disp −1 6 . (12)
Note that for the realistic values of Cel
6 and C6ind
coef-ficients, the effective C6eff coefficient is smaller than the
C6summone, whereas the relation σeffij σij is valid in any
case.
2.2 Collision diameter of clusters
The collision diameter σij, that specifies the range of
re-pulsive valence forces for the LJ potential, is an impor-tant parameter which used for estimating the collision fre-quency, characteristic time of rotational relaxation, diffu-sion and viscosity coefficients, etc. In view of the absence of data on σij for (Al2O3)n clusters in various bath gases,
it is necessary to estimate the values of σij theoretically.
Table 1. The values of static polarizability of atoms reported in reference [39] and used for estimating the LJ potential pa-rameterσiias well as theσiivalues estimated with the formula of Cambi et al. [37]. Atom α, ˚A3 [39] σii, ˚A Ar 1.66 3.33 H 0.67 2.68 O 0.80 2.80 N 1.10 3.02 C 1.76 3.37 F 0.56 2.56
It is worth noting that, today, there is no common ap-proach for determining the collision diameter for molecules and clusters. The first approach is based on the assignment of effective radius for atoms that correlate with the mean radius of the outermost electronic orbitals [33,34]. Conven-tionally, the values of the radius are selected empirically. Note, that this methodology was applied to polyatomic Al-containing species in reference [35].
The second approach is based on the usage of funda-mental physical properties of colliding particles that are mainly responsible for van der Waals intermolecular in-teraction. It is supposed that the linear scale of electron density of atom is roughly proportional to α1/3 [36]. So, Cambi et al. [37] used this supposition to obtain semi-empirical correlation formula connecting directly the pa-rameter σij with the values of the static average
polar-izability αi and αj of colliding particles. For the general
case of two species i and j, the following relationship for σij was recommended (polarizabilities are in ˚A3):
σij= 1.767 α1/3i + α1/3j (αi+ αj)0.095 1 6 √ 2 ˚ A. (13)
However, it should be mentioned that equation (13) was proposed for atoms and small radicals and it can hardly be applied even to Al2O3 monomer.
The approaches mentioned above [33,35,37] do not take into account the specific spatial structure of molecules and clusters and treat them as structureless particles. In our previous study [38], the other approach for estima-tion of the radius of (Al2O3)n clusters was suggested. It was proposed that the collision diameter of particle i with identical particle is equal to the length of cube edge with a volume of a rectangular parallelepiped of minimal vol-ume circumscribing the aggregate of the van der Waals spheres of atoms, i.e. σii= 3
dxdydz, where dx, dyand dz
are the dimensions of this rectangular parallelepiped. In doing so the diameters of van der Waals spheres, centred on all atoms, are calculated with the use of formula (13) on the basis of experimental values of atomic polarizability reported in reference [39] (see Tab.1).
To validate the prediction ability of proposed method, we compared the collision diameters for a set of some stable species predicted by the approach [38] with the reliable and generally accepted data tabulated else-where [32,40–42]. In addition, the effective diameters σijeff
Ar H2 O2 N2 NO CO H2 O HO 2 CO 2 H2 O2 CF4 CH 4 C2 H4 C2 H6 C3 H8 C6 H6 azu len e n ap h th al en e biph en y l anthra cene ph en an thre ne py re ne co ro n en e 2 3 4 5 6 7 8
this work
this work with DRFM
Paul and Warnatz [32]
Bzowski et al. [40]
Wang and Frenklach [41]
Kee at al. [42]
σ ii
,
A
Fig. 1. The values of collision diameter σii for various in-dividual species i = Ar, H2, O2, N2, NO, CO, H2O, HO2, CO2, H2O2, CF4, CH4, C2H4, C2H6, C3H8, C6H6 and pol-yaromatic hydrocarbons (azulene, naphthalene, biphenyl, an-thracene, phenanthrene, pyrene, coronene) determined in the present work (dotted line) as well as the effective diameters in terms of DRFM (solid lines) and the available data given in references [32,40–42] (symbols).
were also calculated in line with equation (12). The com-parison of the estimated values of σij and σijeff with the
tabulated data is given in Figure 1. One can see an ad-vance of the methodology proposed in reference [38] and applied in the present work for the adequate predictions of the collision diameters of different species (the relative mean deviation does not exceed 6%). Notable discrepancy is observed only for the σii values for the H2O, H2O2
and CF4 molecules and for two polyaromatic hydrocar-bon molecules C12H10 and C24H12. The account for the dipole-dipole and dipole-induced dipole interactions in the manner of Paul and Warnatz (DRFM) [32] makes it possi-ble to improve the agreement between predicted and com-monly accepted values of σii for dipole molecules H2O and
H2O2.
Note that, when calculating the binary diffusion coef-ficient Dij, a few combining rules for the collision
diam-eter σij of different molecules can be applicable. One of
the combining rules, commonly used for the calculation of collision diameter, is the arithmetic mean
σij = 0.5 (σii+ σjj) . (14)
This rule is exactly valid if the interacting particles are hard spheres. More sophisticated rule of Kong [43] was proposed originally assuming LJ potential for noble gases. However, as it was found from fitting the transport prop-erties for binary mixtures [44], both the arithmetic mean and Kong’s rule overpredicts the collision diameter for the particles with considerably distinguishing sizes. This issue is essential for the cases under study, because we consider binary diffusion of rather large clusters in diatomic gases.
Therefore, we used the empirical combining rule suggested by Bastien et al. [44]
σij = 0.43 (σii+ σjj) + 0.49. (15)
This rule provides accurate values of σij for the cases
of substantially distinguishing diameters of molecules (all values in Eq. (15) are in ˚A).
2.3 Electric properties of clusters
In recent study [38] the main electric properties such as dipole moments and static polarizabilities for (Al2O3)n, n = 1 . . . 4, clusters were reported for structures optimized in terms of density functional theory (DFT) with the use of Becke three parameter hybrid functional (B3LYP) [45] combined with the correlation functional [46] with the use of 6-31+G(d) basis set. However, this method does not provide highly accurate values of electric properties [47]. Therefore, in the present work, in accordance with the recommendations [48], the following calculation procedure was applied. At first, the estimations of dipole moments μ and static polarizabilities α were performed by using the UB3LYP/6-311+G(2d) level of theory for the geome-tries of cluster structures reported in reference [38]. The commonly applied notation for this kind of calculations is UB3LYP/6-311+G(2d)//UB3LYP/6-31+G(d) method. Then, the values of static isotropic polarizabilityα esti-mated with this method were scaled by a factor of 1.11, that provides the best coincidence with the known exper-imental data [39] for a test set composed of 21 species (He, Ar, Ne, H, O, N, C, B, Al, Na, H2, O2, N2, CO, CO2, CH4, C2H2, C2H4, C2H6, C3H8, C6H6). Note that all DFT computations were performed by using Firefly QC program package [49] which is partially based on the GAMESS(US) source code [50].
3 Results
We considered each two most stable (Al2O3)n, n = 1 . . . 4 isomers of each type of clusters (monomer, dimer, trimer, tetramer) identified in reference [38]. Their configurations are shown in Figure2.
The calculated values of collision diameter σii,
isotropic polarizability α and dipole moment μ as well as symmetry type and multiplicity for the most stable (Al2O3)n isomers are given in Table2. The values ofα scaled by factor 1.11 are given also in the parentheses. One can see that the majority of the considered clusters are non-polar. Only triplet monomer of C2vsymmetry and singlet trimer of C1symmetry have substantial dipole mo-ments. For comparison the values of α (non-scaled and scaled) calculated in the present work for the most stable clusters are given in Figure3. Theα value for Al2O3 de-rived by using the Clausius-Mossotti relation from the di-electric constant for sapphire (>99.9% Al2O3) [51] is also
monomers a. C2v b. D∞h (Te=0.31 eV) dimers a. Td b. C2h (Te=0.42 eV) trimers a. C1 b. D3h (Te=0.07 eV) tetramers a. D3d b. Oh (Te=0.20 eV) Fig. 2. Considered structures of (Al2O3)n,n = 1 . . . 4, clusters (Al and O atoms are depicted by grey and red colour). The type of symmetry and electronic energy values of isomersTeare also indicated.
Table 2. Collision diameter and electric properties of consid-ered (Al2O3)n clusters.
n Symmetry Multiplicity σii, ˚A α, ˚A3 μ, D 1 C2v 3 5.77 8.60 (9.55) 2.47 1 D∞h 1 6.08 9.14 (10.15) 0.0 2 Td 1 6.91 12.91 (14.33) 0.0 2 C2h 1 6.71 12.99 (14.42) 0.0 3 C1 1 8.02 18.15 (20.14) 1.32 3 D3h 1 7.65 18.85 (20.92) 0.0 4 D3d 1 8.71 24.33 (27.00) 0.02 4 Oh 1 8.20 24.87 (27.60) 0.0 1 2 3 4 0 5 10 15 20 25 30 calculated values scaled values Kim et al. [51] <α >, A 3 n
Fig. 3. The values of α for the most stable (Al2O3)n,
n = 1 . . . 4, clusters estimated in the present work as well as
theα value for Al2O3derived from the dielectric constant of sapphire [51].
strictly valid only for crystals with cubic lattice, the agree-ment ofα values, estimated in this work for the monomer and derived from the data for sapphire [51], is rather good. In addition, one can observe from Figure 3that the vari-ation of polarizability of alumina clusters with increas-ing n number is not absolutely linear. There exists small quadratic dependence:α (n) = 5.81 + 3.22n + 0.519n2.
The analogous data for cluster colliding partners con-sidered in the present study: H2, N2, O2, CO and H2O that were calculated with the use of the methodology of this work are presented in Table 3. Comparing the predicted α values with experimental data, one can conclude that the usage of scaling parameter of 1.11 allows one to com-pensate the effect of underestimation of the polarizability caused by using the limited basis set in quantum chemi-cal computations. The coincidence of predicted dipole mo-ments with experimental data can be also considered as acceptable.
From the plot shown in Figure 4 it also follows that the applied methodology allows us to predict binary diffu-sion coefficients for pairs of nonnonpolar and polar-nonpolar molecules at T = 298 K [52,53] with reasonable accuracy.
One of the most crucial issues important for the problem under study is the temperature dependence of diffusion coefficient for pairs of nonpolar and polar-polar particles. In order to elucidate this topic we com-pared our predictions with the experimental data on self-diffusion coefficient for strongly polar molecule H2O reported in the review [54] and on diffusion coefficient of slightly polar molecule CO in O2 given in reference [53]. Figure5 depicts the dependencies of diffusion coefficients calculated both with DRFM approach and using the ad-ditive scheme (Eq. (6)). One can see that the applied methodology predicts diffusion coefficient for polar gas in non-polar ones very well, irrespectively on the way of account for the orientation-dependent contributions. However, for the H2O self-diffusion coefficient the DRFM
Table 3. The collision diameter and electric properties of considered bath gas molecules. Molecule Multiplicity σii, α, ˚A
3, α, ˚A3 α, ˚A3, μ, Debye, μ, Debye,
2s + 1 ˚A calculations scaled experiment [39] calculations experiment [39]
H2 1 2.91 0.60 0.67 0.79 0.0 0.0 N2 1 3.35 1.63 1.81 1.71 0.0 0.0 O2 3 3.16 1.35 1.50 1.56 0.0 0.0 CO 1 3.63 1.84 2.05 1.95 0.06 0.11 H2O 1 3.40 1.23 1.37 1.45 1.95 1.85 N2 -H 2 N2 -O 2 N2 -C O N2 -H 2 O H2 -O 2 H 2 -C O H2 -H 2O CO-O 2 O2 -H 2 O 2x10-5 4x10-5 6x10-5 8x10-5 1x10-4 Westenberg [52] Babichev et al. [53] this work D ij , m 2 s -1
Fig. 4. Measured [52,53] and estimated binary diffusion coef-ficients for different pairs of bath gases considered in this work at the temperatureT = 298 K and pressure P = 1 atm.
0 500 1000 1500 2000 2500
10-6
10-5
10-4
10-3
CO-O2 exp. (Babichev et al. [53])
H2O-H2O exp. (Fokin and Kalashnikov [54])
H2O-H2O, DRFM approach
H2O-H2O, additive scheme
CO-O2, DRFM approach
CO-O2, additive scheme
D ij
, m
2 s
-1
T, K
Fig. 5. Diffusion coefficients for H2O-H2O and CO-O2 molec-ular pairs vs. temperature calculated in line with DRFM ap-proach and using the additive scheme (Eqs. (5) and (6)) (curves) and the experimental data [53,54] at P = 1 atm (symbols).
approach, at high temperatures, provides a slightly bet-ter agreement of predicted Dijvalues with measurements.
Therefore, the DRFM approach was applied for the eval-uation of Dij values for (Al2O3)n clusters.
In Table4the dispersion potential well depths εdispij for pairs “cluster-bath gas molecule” as well as the effective
potential well depth values εeffij at T = 298 K are
summa-rized. One can see that, as was expected, the values of εdispij and εeffij differ only for the strongly polar molecules (H2O) and clusters possessing the dipole moment.
It is worth noting that preliminary quantum chem-ical calculations of interaction potential for two Al2O3 monomers with the use of unrestricted second order Møller-Plesset perturbation theory (UMP2) method [55] revealed that this potential is highly attractive, and the interaction of monomers leads to the direct formation of highly stable dimer rather than van der Waals complex. That is why the collisions of Al2O3 monomers cannot be treated as elastic. Therefore, the procedure of the deter-mination of self-diffusion and viscosity coefficients for the monomers is meaningless.
The binary diffusion coefficients of (Al2O3)n clusters at atmospheric pressure in various bath gases as well as self-diffusion coefficients for clusters with n = 2 . . . 4 were calculated with the usage of εeffij values. The temperature-dependent diffusion coefficients for (Al2O3)n clusters were approximated by following formula
Dij(T ) = D0ij T 273 a − b exp −c273 T (16)
with an accuracy not worse than ±2% over the tempera-ture range T = 300–6000 K. The D0ij, a, b, and c
param-eters are presented in Table 5. The Dij(T ) dependencies
for j = N2 (non-polar molecule gas), CO (weakly polar molecule gas) and H2O (polar molecule with large dipole moment) as well as self-diffusion coefficients Dij(T ) are
shown in Figures6–9.
The presented data exhibit that an account for the contributions of the second type isomer for each type of clusters (dimer, trimer, tetramer) cannot provide a sub-stantial correction to the net diffusion coefficient, because the isomers possess similar collision diameters and polar-izabilities (see Tab.2). However, the most stable monomer (3C2v) has substantial dipole moment (2.47 D), whereas the singlet monomer has D∞h symmetry, and its dipole moment is equal to zero. This leads to the substantial difference in their diffusion coefficients in the gas of po-lar molecules at low temperature due to distinguishing values of C6el term in the interaction potential
responsi-ble for the dipole-dipole interaction for these isomers. So, at low temperature (T = 200–300 K), for j = H2O we
have Dij(1D∞h) = (1.04 ÷ 1.51) Dij(3C2v). To a lesser
Table 4. The values of dispersion potential well depth εdisp
ij and values ofεeffij calculated in terms of DRFM approach (Eq. (12)) (in parentheses) for different pairs of collision partners atT = 298 K.
εij, K
Mono Mono Dimer Dimer Trimer Trimer Tetramer Tetramer 3C 2v 1D∞h Td C2h C1 D3h D3d Oh CO 95.3 83.4 83.5 93.3 69.3 85.0 67.6 86.8 (100.7) (83.4) (83.5) (93.3) (69.8) (85.1) (67.6) (86.8) H2 42.7 37.0 35.6 40.0 28.5 35.3 27.3 35.5 (45.4) (37.0) (35.6) (40.0) (28.7) (35.3) (27.3) (35.5) H2O 75.4 65.7 65.5 73.2 53.8 66.2 52.1 67.2 (156.6) (82.2) (80.2) (89.7) (70.5) (81.0) (63.7) (82.2) N2 101.7 88.5 87.9 98.4 72.1 88.7 69.8 90.1 (107.2) (88.5) (87.9) (98.4) (72.5) (88.7) (69.8) (90.1) O2 101.1 87.6 86.9 97.5 70.7 87.2 68.2 88.2 (106.2) (87.6) (86.9) (97.5) (71.1) (87.2) (68.2) (88.2) Self – – 78.7 95.0 65.7 92.4 71.7 106.7 (78.7) (95.0) (66.8) (92.4) (71.7) (106.7)
Table 5. Approximation parameters for the temperature dependence for the self-diffusion coefficients Diiand binary diffusion coefficientsDij(in m2 s−1) in various bath gases: CO, H2, H2O, N2, O2 for two structures of (Al2O3)n,n = 1 . . . 4 clusters.
Monomer3C2v Monomer1D∞h partner Dij0, m2 s−1 a b c partner D0ij, m2 s−1 a b c CO 9.86 × 10−6 1.651 1.12 × 10−1 0.2691 CO 9.45 × 10−6 1.652 7.59 × 10−2 0.1966 H2 4.24 × 10−5 1.655 1.73 × 10−2 –0.0519 H2 4.07 × 10−5 1.655 9.09 × 10−3 –0.1631 H2O 1.32 × 10−5 1.641 2.58 × 10−1 0.3381 H2O 1.23 × 10−5 1.652 7.35 × 10−2 0.1908 N2 1.03 × 10−5 1.650 1.26 × 10−1 0.2934 N2 9.87 × 10−6 1.652 8.61 × 10−2 0.2198 O2 1.02 × 10−5 1.651 1.24 × 10−1 0.2897 O2 9.75 × 10−6 1.652 8.41 × 10−2 0.2155
with itself – – with itself – –
Dimer1Td Dimer1C2h CO 7.70 × 10−6 1.652 7.61 × 10−2 0.1972 CO 7.86 × 10−6 1.651 9.58 × 10−2 0.2399 H2 3.48 × 10−5 1.655 7.91 × 10−3 –0.1887 H2 3.55 × 10−5 1.655 1.18 × 10−2 –0.1171 H2O 1.02 × 10−5 1.652 6.97 × 10−2 0.1811 H2O 1.04 × 10−5 1.652 8.85 × 10−2 0.2249 N2 8.03 × 10−6 1.652 8.48 × 10−2 0.2170 N2 8.19 × 10−6 1.651 1.07 × 10−1 0.2605 O2 7.85 × 10−6 1.652 8.28 × 10−2 0.2126 O2 8.01 × 10−6 1.651 1.05 × 10−1 0.2568 with itself 2.01 × 10−6 1.652 6.68 × 10−2 0.1735 with itself 2.08 × 10−6 1.651 9.94 × 10−2 0.2469
Trimer1C1 Trimer1D3h CO 6.46 × 10−6 1.653 5.11 × 10−2 0.1267 CO 6.66 × 10−6 1.652 7.91 × 10−2 0.2041 H2 2.96 × 10−5 1.656 3.31 × 10−3 –0.3630 H2 3.05 × 10−5 1.655 7.73 × 10−3 –0.1930 H2O 8.58 × 10−6 1.653 4.93 × 10−2 0.0628 H2O 8.84 × 10−6 1.652 7.11 × 10−2 0.1848 N2 6.71 × 10−6 1.653 5.58 × 10−2 0.1418 N2 6.93 × 10−6 1.652 8.64 × 10−2 0.2206 O2 6.54 × 10−6 1.653 5.34 × 10−2 0.1342 O2 6.75 × 10−6 1.652 8.34 × 10−2 0.2140 with itself 1.25 × 10−6 1.653 4.61 × 10−2 0.1065 with itself 1.31 × 10−6 1.651 9.40 × 10−2 0.2363
Tetramer1D3d Tetramer1Oh CO 5.77 × 10−6 1.653 4.76 × 10−2 0.1144 CO 6.03 × 10−6 1.652 8.26 × 10−2 0.2122 H2 2.66 × 10−5 1.656 2.62 × 10−3 –0.4138 H2 2.78 × 10−5 1.655 7.87 × 10−3 –0.1898 H2O 7.68 × 10−6 1.653 4.14 × 10−2 0.0909 H2O 8.02 × 10−6 1.652 7.35 × 10−2 0.1909 N2 5.99 × 10−6 1.653 5.12 × 10−2 0.1269 N2 6.26 × 10−6 1.652 8.92 × 10−2 0.2264 O2 5.82 × 10−6 1.653 4.85 × 10−2 0.1175 O2 6.08 × 10−6 1.652 8.54 × 10−2 0.2183 With itself 9.08 × 10−7 1.653 5.43 × 10−2 0.1370 with itself 9.71 × 10−7 1.650 1.25 × 10−1 0.2916
0 1000 2000 10-6 10-5 10-4 n=3 n=2 n=4 n=1 the 1st isomer the 2nd isomer Dij , m 2 s -1 T, K
Fig. 6. Diffusion coefficients of the two energy-lowest isomers of (Al2O3)n(n = 1 . . . 4) clusters with N2.
0 1000 2000 10-6 10-5 10-4 n=4 n=1 the 1st isomer the 2nd isomer D ij , m 2 s -1 T, K
Fig. 7. Diffusion coefficients of the two energy-lowest isomers of (Al2O3)nclusters (n = 1 . . . 4) with CO.
0 1000 2000 10-6 10-5 10-4 n=3 n=2 n=4 n=1 the 1st isomer the 2nd isomer Dij , m 2 s -1 T, K
Fig. 8. Diffusion coefficients of the two energy-lowest isomers of (Al2O3)nclusters (n = 1 . . . 4) with H2O.
temperature, takes place for polar C1trimer and non-polar D3htrimer in H2O bath gas.
0 500 1000 1500 2000 10-6 10-5 10-4 n=4 n=3 n=2 the 1st isomer the 2nd isomer Dii , m 2 s -1 T, K
Fig. 9. Self-diffusion coefficients of the two energy-lowest iso-mers of (Al2O3)n (n = 2 . . . 4) clusters.
4 Concluding remarks
Binary diffusion coefficients for two low-lying pairs of iso-mers of (Al2O3)n clusters (n = 1 . . . 4) in bath gases, H2, CO, N2, O2 and H2O as well as self-diffusion coefficients for clusters with n = 2 . . . 4 were computed over a wide range of temperature using first-principle calculations and reliable interaction potential. The parameters of the po-tential for (Al2O3)nclusters interaction with H2, CO, N2, O2and H2O molecules were calculated taking into account the dispersion, dipole-dipole and dipole-induced dipole in-teraction terms. Special attention was paid to the correct determination of collision diameter of colliding partners (cluster-molecule). A comparison of the computed diffu-sion coefficients for pairs of bath gases with experimental data, obtained at near room temperature showed an ad-vance of applied methodology. It was revealed that the dif-fusion coefficients of the isomers of (Al2O3)nclusters differ notably only in the case when dipole moments of isomers distinguish strongly. So, the diffusion coefficients of two Al2O3structures with3C2v and1D∞h symmetries in wa-ter vapour can differ by a factor of 1.5 at T = 200–300 K. This work was supported by the Russian Foundation for Basic Research (Grants Nos. 12-08-92008 and 14-08-31247).
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![Table 1. The values of static polarizability of atoms reported in reference [ 39 ] and used for estimating the LJ potential pa-rameter σ ii as well as the σ ii values estimated with the formula of Cambi et al](https://thumb-ap.123doks.com/thumbv2/9libinfo/7954982.157953/3.892.549.734.214.332/polarizability-reported-reference-estimating-potential-rameter-estimated-formula.webp)
