Transport coefficients of multi-component mixtures of noble gases based on ab
initio potentials. Diffusion coefficients and thermal diffusion factors.
Felix Sharipov∗ and Victor J. Benites†
Departamento de Fısica, Universidade Federal do Parana, Curitiba, 81531-990, Brazil
Diffusion coefficients and thermal diffusion factors of binary, ternary and quaternary mix-
tures of helium, neon, argon, and krypton at low density are computed for wide ranges of
temperature and molar fractions, applying the Chapman-Enskog method. Two definitions
of the diffusion coefficients are discussed and a general relation between them is obtained.
Ab initio interatomic potentials are employed in order to calculate the Omega-integrals be-
ing part of the expression of the reported quantities. The relative numerical errors of the
diffusion coefficients do not exceed the value of 5 × 10−5 being even smaller in some case.
The uncertainties of diffusion coefficients due to the interatomic potential varies between
4 × 10−4 and 6 × 10−3. The numerical error and uncertainty due to the potential of the
thermal diffusion factors are estimated as 10−4 and 3 × 10−3, respectively. It is shown that
the present results for binary mixtures are more accurate than any other available in the
literature, while the results for ternary and quaternary mixtures are reported for the first
time.
Key words: multi-component gaseous mixture, diffusion coefficient, thermal diffusion
factor, ab initio potential.
I. INTRODUCTION
Diffusion and thermal-diffusion processes
play an important role in many technological
and scientific fields such as: microfluidics [1, 2],
gas separation [3, 4], plasma reactors [5], vacuum
equipment [6, 7], combustion [8], shock waves
[9, 10], velocity slip [11–13], etc. A general the-
ory of mass transfer via diffusion phenomena
is well described in several textbooks, see e.g.
[14–21]. The transport coefficients such as diffu-
sion coefficients (DC) and thermal diffusion fac-
tors (TDF) are important parameters determin-
ing mass transfer phenomena in multicomponent
∗ [email protected]; http://fisica.ufpr.br/sharipov† [email protected]
mixtures. Computation of these coefficients is
based on the kinetic Boltzmann equations solved
by the Chapman-Enskog method [14–16]. Alter-
natively, the stochastic algorithm for simulating
gas transport coefficients [22] can be used.
At the moment, the transport coefficients are
well known for many kinds of binary mixtures.
In this case, we have only one DC and only one
TDF. Kestin et al. [23] reported semi-empirical
data on both DC and TDF for all binary mix-
tures of noble gases. In fact, they used the
transport coefficient expressions based on the
Chapman-Enskog method. Then the Omega-
integral being part of these expressions were ac-
curately determined by a complex numerical fit
to the best measurements that could be per-
2
formed in 1984. No significant advance have
been done in measuring of the DCs and TDFs
from that time till now so that the data reported
in [23] can be considered as a compilation of the
best experimental results available in the open
literature. The papers [24–28] provided the DCs
and TDFs calculated departing from the kinetic
theory of gases [14–16] employing ab initio po-
tentials. Such ab initio potentials for all no-
ble gases and for most of their mixtures were
proposed in many works, see e.g. Refs. [28–
37]. They are widely used in the kinetic theory
of gases [24–28, 38] and in rarefied gas dynam-
ics [39–41]. In fact, several phenomenological
models of intermolecular interaction were elabo-
rated in order to describe correctly the diffusion
processes by the direct simulation Monte Carlo
(DSMC) method [42]. Such models have a num-
ber of unknown parameters usually extracted
from some experimental data. The look-up ta-
bles of intermolecular interactions [43] allowed to
implement ab initio potentials into the DSMC
method [44]. As a result, a modelling of the
transport phenomena through gaseous mixtures
became possible without any adjustable param-
eters [45].
Diffusion phenomena in multicomponent
mixtures are more complicated. First of all,
there are more transport coefficients determining
the diffusion fluxes. For instance, in case of mix-
ture composed of K species, we need K(K−1)/2
DCs and (K−1) TDFs to determine the diffusion
fluxes. To reduce the number of DFc, Wilke [46]
proposed an effective DC for each species flowing
through a stagnated mixture. However, several
disadvantages of this concept were pointed out
in the book [18], namely, the effective DCs are
not system properties, but they are dependent
on the diffusion fluxes. Second, besides the high
number of the DCs, there are several definitions
of the DCs. Only two of them will be consid-
ered here: the definition in the form by Fick and
that in the form by Maxwell-Stefan (MS). Cur-
tiss [47] expressed the Fick DCs of multicompo-
nent mixtures via the MS coefficients of binary
mixtures. However, this expression is valid in
the frame of the first order of expansion of the
distribution function with respect to the Sonine
polynomials. It is expected that the contribu-
tion of the higher order terms in this expansion
is small, but it is still unknown. In order to es-
timate this contribution, the MS-DCs must be
calculated for ternary and quaternary mixtures
and then compared to those of the corresponding
binary mixtures.
The thermal DCs are much sensitive to many
factors including the chemical composition of
mixtures. As a consequence, it is difficult to ex-
press these coefficients for multicomponent mix-
tures via TDFs of binary mixtures. Some at-
tempts to obtain such a relation, see e.g. Refs.
[48, 49], were successful only in case of isotope
mixtures of the same gas. In this case, the
mass of all mixture components are close to each
other, but is impossible to propose a similar re-
lation for mixtures with quite different atomic
masses, for instance, for helium-neon-krypton
mixture. Till now, there are only general ex-
pressions of the thermal DCs for multicompo-
nent mixtures, see e.g. Refs.[14–16, 50–52], but
numerical values of these coefficients have never
been reported.
In the present paper, a general relation of
the Fick DCs to those defined in the Maxwell-
3
Stefan form based only on their definitions is ob-
tained for arbitrary multicomponent mixtures.
The MS-DCs for ternary and quaternary mix-
tures of helium, neon, argon, and krypton are
obtained using ab initio potentials [28–37]. A
comparison of these coefficients with the corre-
sponding coefficients for binary mixtures is per-
formed. A new definition of the TDF for mul-
ticomponent mixtures is proposed and justified.
Numerical results on the TDF of binary, ternary
and quaternary mixtures of helium, neon, argon
and krypton in a wide range of the temperature
are reported. The numerical errors and uncer-
tainties related to the potentials are estimated.
An influence of small quantity of one species on
the transport coefficients is analysed.
II. MAIN DEFINITIONS
Here, we consider a mixture of K monatomic
gases at a temperature T and pressure p. The
number density of each species is denoted as ni
(1 ≤ i ≤ K). The chemical composition of the
mixture can be characterized by the mole frac-
tion defined as
xi = ni/n, n =
K∑
i=1
ni. (1)
Some expressions are more compact in terms of
the mass fraction given as
yi = ρi/ρ, ρi = mini, ρ =K∑
i=1
ρi, (2)
where mi and ρi are the atomic mass and mass
density of species i, respectively. It is easily ver-
ified that
K∑
i=1
xi = 1,K∑
i=1
yi = 1. (3)
The mixture pressure is assumed to be so low
that the state equation corresponds to ideal
gases, i.e. p = nkBT , where kB is the Boltzmann
constant.
The diffusion driving forces di due to a chem-
ical composition non-uniformity and external
forces are defined as
di = ∇xi+(xi − yi)∇ ln p−ρip
F i −K∑
j=1
yjF j
,
(4)
with F i being an external force per unit mass
acting on an atom which is independent of the
particle velocity. The definition (2) and condi-
tions (3) lead to the relation between the forces
K∑
i=1
di = 0. (5)
In other words, there are only (K − 1) indepen-
dent forces.
Let us denote the diffusion velocity of species
i relatively the mean mass velocity of the mix-
ture as V i, which obey the relation
K∑
i=1
yiV i = 0. (6)
The multi-component DCs denoted as Dij and
thermal DCs denoted as DTi are defined via the
Fick law as, see Eq(6.3-32) from Ref.[16],
V i = −K∑
j=1
Dijdj −DTi∇ lnT. (7)
The coefficients Dij and DTi obey the following
general relations [16]
K∑
i=1
yiDij = 0,
K∑
i=1
yiDTi = 0. (8)
Moreover, The DCs are symmetric
Dij = Dji, (9)
4
i.e. they are consistent with the Onsager recip-
rocal relation [17, 53, 54]. Thus, the relations (8)
and (9) reduce the number of independent DCs
to K(K − 1)/2 and that of independent thermal
DCs to (K−1). A knowledge of these coefficients
allows us to calculate all diffusion velocities by
Eq.(7).
III. METHOD OF CALCULATION.
The expressions of the DCs for multi-
component mixtures are derived in the book by
Ferziger & Kaper [16] by the Chapman-Enskog
method applied to the kinetic Boltzmann equa-
tion. These expressions obtained in terms of
bracket integrals are used here with slightly dif-
ferent notations. Each bracket integral contains
information about only two gaseous species so
that the expressions of the bracket integrals ob-
tained in Refs. [55–58] can be used here for
multi-component mixtures too.
Following the Chapman-Enskog method [16],
the coefficients Dij and DTi are expressed as
Dij =3kBT
2nd(0)ij , (10)
DT i = −15kBT
4n
K∑
j=1
xjd(1)ij . (11)
The quantities d(p)ij are calculated from the (K−
1) systems of algebraic equations
K∑
j=1
N−1∑
q=0
Λ(pq)ij d
(q)kj = (ykδik − yi) δp0, (12)
where 1 ≤ i ≤ K, 0 ≤ p ≤ N − 1, δik is the Kro-
necker delta, and N is the order of approxima-
tion with respect the Sonine polynomials. Thus,
each system corresponds to a fixed value of k
and contains K ×N equations. The coefficients
Dij and DTi converge to their exact values in
the limit N → ∞. The matrix Λ(pq)ij is given in
terms of the bracket integrals as
Λ(pq)ii = mi
K∑
j=1j 6=i
xixj
[
S(p)3/2,iC
CC i, S(q)3/2,iC
CC i
]
ij
+ mix2i
[
S(p)3/2,iC
CC i, S(q)3/2,iC
CC i
]
i, (13)
and
Λ(pq)ij =
√mimjxixj
[
S(p)3/2,iC
CC i, S(q)3/2,jC
CC j
]
ij,
(14)
where i 6= j. In order to satisfy the relation (8),
the expressions (13) and (14) with the subscripts
satisfying the conditions p = 0 and i = k must
be substituted by
Λ(0q)kj = yjδq0. (15)
The functions S(p)3/2,i are the Sonine polynomials
with the argument C 2i , i.e.
S(p)3/2,i =
p∑
n=0
Γ(5/2 + p)
(p− n)!n!Γ(5/2 + n)
(
−C2i
)n,
(16)
with Γ being the gamma-function. The dimen-
sionless molecular velocity CCC i is defined for each
species as
CCC i =
√
mi
2kBT(ci − u), (17)
where ci is the molecular velocity of species i
and u is the hydrodynamic velocity of the mix-
ture. The general expressions of bracket inte-
grals for arbitrary orders p obtained in the pa-
pers [57] for binary mixtures can be used here.
The brackets integrals[
S(p)3/2,iC
CC i, S(q)3/2,iC
CC i
]
ij,
[
S(p)3/2,iC
CC i, S(q)3/2,iC
CC i
]
i, and
[
S(p)3/2,iC
CC i, S(q)3/2,jC
CC j
]
ij
5
are given by Eqs. (117), (119), and (115) from
Ref.[57], respectively. To generalize the expres-
sions given in Ref. [57] to a multi-component
mixture, the subscripts “1” and “2” are replaced
by “i” and “j”, respectively. In case of i > j, the
symmetry relations Λ(pq)ij = Λ
(qp)ji are employed.
The brackets integrals are expressed in terms
of the Ω-integrals defined as
Ω(n,r)ij =
√
kBT
8πmij
∫ ∞
0Q
(n)ij εr+1e−ε dε, (18)
where ε is the dimensionless energy of interact-
ing particles
ε =E
kBT, E =
1
2mij |ci − cj |2, (19)
mij = mimj/(mi + mj) is the reduced mass
of interacting particles, Q(n)ij are transport cross
sections depending on ε and determined by the
interatomic potential. Here, we used the same
transport cross sections calculated in our previ-
ous work [38] where the reader can find all details
of the numerical scheme and analyses of numer-
ical error sources.
The potentials used in the present work for
the main calculations have been taken from the
following references: A concise description of the
potentials He-He [35] and Ne-Ne [32] is given
in Appendix to Ref.[25]; The Ar-Ar potential is
given by Eqs.(2) - (4) and Table XI from Ref.[33];
The Kr-Kr potential is presented by Eq.(8) and
Table VI from Ref.[37]; The potentials for He-
Ne, He-Ar, Ne-Ar interactions are given by Ta-
ble 2 from the paper [31] with an expression in
its caption; The potential He-Kr is computed by
Eq.(6) and Table III from [28]; The potentials
Ne-Kr and Ar-Kr are reported by Eq.(1) and
Table XV from the work [30].
IV. THERMAL DIFFUSION FACTORS
In practice, it is more convenient to use the
thermal diffusion ratios instead of the thermal
DCs, which are related as
K∑
j=1
DijkTj = DTi,
K∑
i=1
kTi = 0. (20)
Then, the Fick law Eq.(7) takes the form
V i = −K∑
j=1
Dij (dj + kTj∇ lnT ) . (21)
In case of binary mixture, we have only one DC,
namely, D12 and only one thermal diffusion ratio
given as
kT1 = −kT2 = −y1DT1
D12. (22)
A mixture with K species has (K − 1) thermal
diffusion ratios.
The TDF denoted as αTi is also frequently
used in practice and well defined in case of binary
mixture
kT1 = x1(1 − x1)αT1, K = 2. (23)
However, this concept is not well established for
multicomponent mixtures. Some books, see e.g.
Refs.[15, 16], propose to define a factor matrix
such as kTi =∑K
j=1 αTijxixj or something simi-
lar, see e.g. Refs.[59, 60]. Indeed, this definition
is reduced to Eq.(23) in case of K = 2, but it
creates K(K − 1)/2 independent coefficients de-
parting from (K−1) ones so that the matrix αTij
is not uniquely defined. Moreover, each term of
the matrix αTij is chemical composition depen-
dent. It would be more reasonable and justified
to define (K − 1) new quantities via (K − 1) al-
ready defined quantities. In fact, the thermal
diffusion ratio kTi vanishes in the limits xi → 0
6
(the ith species disappears) and xi → 1 (the
mixture becomes a single gas composed only of
the ith species), i.e. the quantity kTi strongly
depends on the molar fraction xi. The advan-
tage of αT1 against kT1 in Eq.(23) is its weak
dependence on the mole fraction x1. However,
the matrix αTij proposed in Refs.[15, 16] does
not have this property. A more reasonable defi-
nition of the TDF for multicomponent mixtures
is as follows
kTi = xi(1 − xi)αTi, 1 ≤ i ≤ K. (24)
In this way, we have (K − 1) independent coef-
ficients which do not vanish in the limits xi → 0
and xi → 1. The second equality in Eq.(20)
leads to
K∑
i=1
xi(1 − xi)αTi = 0, (25)
so that only (K− 1) coefficients can be reported
here.
V. MAXWELL-STEFAN EQUATION
The Fick law (7) determines the diffusion ve-
locities V i in terms of the driving forces di. The
Maxwell-Stefan equation expresses the driving
forces di as functions of the relative velocities
V i − V j of two species in a mixture. Here, we
include the temperature gradient in the driving
force, therefore, the Maxwell-Stefan equations
are written in a more general form [51, 52, 60]
as
di + kTi∇ lnT =
K∑
j=1j 6=i
xixjDij
(V j − V i) ,
(26)
where Dij are the Maxwell-Stefan diffusion co-
efficients (MS-DC). In case of binary mixture
(K = 2), we have just one independent coeffi-
cient D12. Combining Eqs.(7), (8), and (26) for
K = 2, we obtain a simple relation between the
Fick DC and MS-DC
D12 = −x1x2y1y2
D12, K = 2. (27)
Usually, the MS-DC D12 is reported in the lit-
erature, see e.g. Refs. [23–28], but not the Fick
DCs. Thus, a general relation between these two
definitions is needed. In Appendix, it is shown
that the MS-DCs are related to the Fick ones by
K∑
j=1j 6=i
xixjDij
(Dik −Djk) = δik − yi, (28)
where only the definitions (7),(26), and proper-
ties (6), (8) have been used.
It is common to refer the MS-DCs Dij as the
binary DCs, see e.g. Refs.[15, 16, 18, 19]. The
origin of this term is the derivations by Curtiss
[47] showing that an approximate solution of the
Boltzmann equation, namely N = 0 in Eq.(12),
leads to the relation (28) where Dij are chemi-
cal composition independent and equal to those
calculated for binary mixtures, while Dik are the
Fick DCs for multicomponent mixtures. Accord-
ing to Curtiss [47], once the MS-DCs for binary
mixtures are known, the Fick DCs for any mul-
ticomponent mixture is know too via (28). In
other words, the MS-DCs Dij for any multicom-
ponent mixture do not depend on its chemical
composition and are equal to those for binary
mixtures. Below, it is shown that it is not true.
In fact, the MS-DCs Dij are independent from
the chemical composition only in the zero order
approximation N = 0 in Eq.(12). The contri-
bution of the higher order terms can be small
7
but not negligible. Relative variations of the
DCs of binary mixtures due to the mole frac-
tion are shown in Figure 1 using the data from
Refs.[25, 26] and those of the present work. The
smallest variation, about 0.1%, is observed for
the Ar-Kr mixture and the largest one, about 4
%, corresponds to the He-Kr mixture so that it
depends on the atomic mass ratio: the higher
ratio the larger variation. It is expected that
the dependence of the MS-DCs of ternary and
quaternary mixtures on their chemical composi-
tion will be of the same order as that of binary
mixtures.
First, the Fick DCs will be calculated nu-
merically via Eq.(10), then the MS-DCs will be
calculated by Eq.(28) and reported here. For a
ternary mixture (K = 3), we have three inde-
pendent coefficients D12, D13, and D23. Then
Eq.(28) leads to the explicit expression of the
MS-DCs
D12 =x1x2 (y1D12D13 + y3D23D13 + y2D12D23)
y1y2 [y3D12 − (y2 + y3)D23 − (y1 + y3)D13]. (29)
-3
-2
-1
0
1
0 0.2 0.4 0.6 0.8 1
(∆D
12 /D
12)
× 10
0
x1
m2 /m1=1.98 (Ne- Ar)2.10 ( Ar- Kr)4.15 (Ne- Kr)5.04 (He-Ne)9.98 (He- Ar)20.9 (He- Kr)
FIG. 1. Relative variation of Maxwell-Stefan diffu-
sion coefficient ∆D12(x1) = D12(x1) − D12(0.5) vs.
mole fraction x1.
The expressions for D13 and D23 are obtained by
simple permutations of the subscripts 1 → 2 →3 → 1. For a quaternary mixture, the analyt-
ical relations of Dij to Dij are cumbersome so
that it is easier to solve the system of algebraic
equations (28) numerically.
VI. UNCERTAINTIES
In the present calculations, we distinguish
two types of uncertainties: numerical errors and
that related to the interatomic potentials used as
input data. Both uncertainties vary significantly
from one coefficient to another so that the uncer-
tainties obtained for viscosity and thermal con-
ductivity [38] cannot be adopted for the coeffi-
cients calculated here. Moreover, the uncertain-
ties depend on the mixture temperature and its
chemical composition. Theoretically, both nu-
merical and potential uncertainties can be cal-
culated for each value of the coefficients Dij and
αTi that will increase significantly the quantity
of the reported data. Instead, the maximum rel-
ative uncertainty will be given for Dij and αTi
over all temperatures and chemical compositions
considered here.
The expressions Dij contain many kinds of
the Ω-integrals for all possible pairs (ij) of the
8
species composing the mixtures. However, not
all of them contribute equally into the DCs. Ac-
tually, the main term of Dij calculated for N = 0
contains only the integral Ω(1,1)ij so that the un-
certainty of Dij is determined only by collisions
between species i and j even in the presence of
other species. If the relative uncertainty ur of
Dij is calculated for a binary mixture composed
of species i and j, then the same uncertainty can
be adopted for the coefficients Dij correspond-
ing to the same two species composing a multi-
component mixture. Thus, it is enough to es-
timate the relative uncertainty of the DCs only
for all binary mixtures that can be composed
of the gases considered here. The main terms
of the TDFs αTi correspond to the first order
approximation (N = 1) which contain the Ω-
integrals for all possible pairs of a mixtures. In
this case, the uncertainty of a multicomponent
mixture cannot be reduced to those of several
binary mixtures, but the uncertainty should be
calculated for each specific mixtures. In case of
the TDF αTi, it is more reasonable to work with
the absolute numerical error u(αTi), because the
quantity αTi can be equal to zero under some
conditions so that the relative error has a singu-
larity.
The sources of the numerical errors are,
mainly, the order of approximation N in Eq.(12)
and the numerical scheme to calculate the Ω-
integrals (18). The main calculations have been
carried out for the order approximation N = 10
and additional test results have been obtained
for N = 12 in order to estimate the correspond-
ing error. All sources of the numerical errors to
calculate the Ω-integrals were analysed in the
previous paper [38]. The total relative error
ur(Dij) due to the numerical scheme of the DCs
is given in Table I for all possible pairs of the
gases considered here. The largest error ur equal
to 5 × 10−5 corresponds to the pair He-Kr. In
this case, the order of approximation N domi-
nates in the total budget of errors. In fact, the
larger atomic mass ratio the slower convergence
with respect to the order N . The relative error
ur = 2 × 10−5 corresponding to the pair He-Ar
is also determined mainly by the approximation
order N . The numerical error of the same order
ur = 2 × 10−5 corresponds to the pair Ar-Kr.
In this case, the main contribution into the to-
tal error is due to the number of the nodes to
calculate the Ω-integrals. As shown in [38], the
transport cross section Q(1)ij has many peaks in
case of heavy gases that leads to a larger error of
the integration in Eq.(18). For the other pairs,
He-Ne, Ne-Ar, Ne-Kr, the relative numerical er-
ror ur has the order 10−6.
The absolute numerical errors u(αTi) of the
TDFs for all mixtures considered here are given
in Table II. Again, the mixture He-Kr has the
largest error with the main contribution due to
the approximation order N . All other mixtures
containing helium and krypton have the slightly
smaller error equal to 7 × 10−5. The mixtures
without helium and/or without krypton have the
error of the order 10−5.
The ab initio potentials [28–37] used in the
present work have different accuracies. The co-
efficients Dij and αTi calculated here are more
sensitive to the heterogeneous collisions [28, 30,
31] and less sensitive to the homogeneous ones
[32, 33, 35, 37] so that only the contribution of
the formers will be analyzed here. The uncer-
tainties of the potentials of He-Ne, He-Ar, and
9
Ne-Ar collisions are estimated by comparing the
results on Dij and αTi based on the potentials
proposed in [31] with those based on the poten-
tials obtained in [29]. The uncertainties due to
the potential He-Kr obtained in [28] are used in
the present work. The uncertainties of the po-
tentials Ne-Kr and Ar-Kr obtained in [30] were
not estimated previously. We assume that they
are the same as that of the dimmer Ne-Ar de-
rived also in [30]. The uncertainty of the lat-
ter is estimated comparing the results based on
the potential Ne-Ar obtained in [31] with those
based on the Ne-Ar potential derived in [30].
The relative uncertainties ur(Dij) of the MS-
DCs due to the potentials for all possible pairs
considered here are given in Table I showing that
they are orders of magnitude larger than the cor-
responding numerical errors. The less accurate
potentials are those of the Ne-Kr and Ar-Kr dim-
mers leading to the DC uncertainty of 0.8%. The
other pairs have the relative uncertainties of the
order 10−4.
The uncertainties u(αTi) of the TDF for all
mixtures considered here are given in Table II.
All mixtures containing krypton have the uncer-
tainty of the order 10−3, while the mixture He-
Ne-Ar has the uncertainty one order of magni-
tude smaller.
Note all uncertainties reported here are max-
imum values over all temperatures and all chem-
ical compositions so that they can be smaller for
some combinations of these characteristics.
TABLE I. Relative uncertainty ur of Maxwell-Stefan
diffusion coefficients Dij due to numerical error and
due to potential.
ur(Dij)
i j numerical potential
He Ne 3×10−6 4×10−4
He Ar 2×10−5 5×10−4
He Kr 5×10−5 5×10−4
Ne Ar 5×10−6 6×10−4
Ne Kr 3×10−6 6×10−3
Ar Kr 2×10−5 6×10−3
TABLE II. Uncertainty u of thermal diffusion factors
αT due to numerical error and due to potential
u(αT)
mixture numerical potential
He-Kr 1×10−4 2×10−3
Ne-Kr 1×10−5 3×10−3
Ar-Kr 2×10−5 3×10−3
He-Ne-Ar 4×10−5 4×10−4
He-Ne-Kr 7×10−5 2×10−3
He-Ar-Kr 7×10−5 2×10−3
Ne-Ar-Kr 1×10−5 3×10−3
He-Ne-Ar-Kr 7×10−5 3×10−3
VII. RESULTS AND DISCUSSIONS
A. Binary mixtures
Some binary mixtures, namely, helium-neon,
helium-argon, neon-argon were considered in our
previous papers [25, 26], where the MS-DCs and
TDFs were calculated with a high numerical ac-
curacy using the quantum approach to the inter-
atomic collisions. The authors of [28] proposed
an ab initio potential and reported numerical
results on the same coefficients for the helium-
10
krypton mixture employing the classical theory
to the Kr-Kr collisions and the quantum one to
the He-He and He-Kr interactions. Moreover,
they took into account only fourth order (N = 4)
of the approximation in Eq.(12). Some equimo-
lar binary mixtures were considered by Song et
al. [27] in the frame of the second order (N = 2)
in Eq.(12). They used the potentials reported in
Refs.[30, 31]. The database by Kestin et al. [23]
based on semi-empirical expressions of the trans-
port coefficients will be also used for comparison
of the results obtained here. Below, numerical
results on the binary mixtures not considered
in our previous papers [25, 26], namely, helium-
krypton, neon-krypton, and argon-krypton are
presented and compared with those reported in
the other works [23, 27, 28].
The numerical values of the MS-DCs and
TDFs for the helium-krypton mixture are re-
ported in Table III. Previously, the similar re-
sults were published by Kestin et al. [23] with
the uncertainty of 1.5 % and 5 % for the MS-DC
and TDF, respectively. A comparison of these
data with the present results is given in Figure
2 showing that the discrepancies of both MS-
DC and TDF slightly exceed the uncertainty de-
clared in [23], but they are significantly larger
than the uncertainties of the present work given
Tables I and II. The theoretical results by Song
et al. [27] are also compared to the present ones
in Figure 2. The disagreement of the MS-DC
equal to 2 % is explained by two factors: the au-
thors of [27] used the less precise potential ob-
tained in [30] and lower approximation (N = 2)
of Eq.(12).
Till now, the numerical data on the trans-
port coefficients of the helium-krypton mixture
reported by Jager & Bich [28] are most exact
among all data on this mixture available in the
open literature. The uncertainty estimated by
them is adopted here as that due to the poten-
tial, see Tables I and II. However, the results
obtained here using the same potential are more
precise because of the higher order N in Eq.(12).
The deviations of the results by Jager & Bich
[28] from the present ones are plotted in Figure
3 showing that the discrepancies of both MS-DC
and TDF significantly exceed the numerical un-
certainties ur(D12) and u(αT) given in Tables I
and II, respectively. Both coefficients D12 and
αT are undervalued by Jager & Bich [28]. In
fact, the authors of [28] checked the convergence
with respect to the order N in Eq.(12) only at
T = 300 K where it is rather fast. However, this
convergence is much slower at lower (T ≈ 70
K) and higher (T ≈ 2000 K) temperatures, i.e.,
the contributions of the high order terms into
these two coefficients are significant. As a re-
sult, the discrepancies between the present re-
sults and those reported in [28] reach the values
ur(D12) = 8 × 10−4 and u(αT) = 2.5 × 10−3 in
spite of the same potential used in both works.
The numerical values of the MS-DC and TDF
for the neon-krypton mixture are reported in Ta-
ble IV. Semi-empirical data for this mixture are
published by Kestin et al. [23] with the uncer-
tainty being 1 % and 3 % for the MS-DC and
TDF, respectively. A comparison of these data
to the present results is performed in Figure 4
showing that the discrepancies of both MS-DC
and TDF exceed the corresponding uncertainties
declared in [23] and, consequently, they exceed
the uncertainties of the present work. The de-
viations of the results by Song et al. [27] from
11
TABLE III. Maxwell-Stefan diffusion coefficients D12 at the standard pressure (p = 101325 Pa) and thermal
diffusion factor αT vs. temperature T and molar fraction x1 of helium for He-Kr mixture.
D12 × 106 (m2/s) −αT
T (K) x1 = 0.25 0.5 0.75 x1 = 0.25 0.5 0.75
50 2.77541 2.77272 2.76824 0.12377 0.15876 0.22178
100 9.86872 9.83905 9.78882 0.25881 0.32803 0.44848
300 65.7797 65.3685 64.7292 0.34544 0.42966 0.57164
500 156.237 155.180 153.580 0.35058 0.43419 0.57473
1000 505.847 502.514 497.554 0.34039 0.42086 0.55616
2000 1653.74 1644.04 1629.63 0.31943 0.39532 0.52324
5000 8109.84 8072.82 8017.51 0.28055 0.34833 0.46312
-2
-1
0
1
40 400 4000 100 1000
(∆D
12 /D
12)
× 10
0
T(K)
x1=0.25 0.5 0.75
-4
-2
0
2
40 400 4000 100 1000
∆ α T
× 1
00
T(K)
FIG. 2. Deviation of diffusion coefficients D12 (left) and thermal diffusion factors αT (right) of helium-
krypton mixture reported in other papers (subscript “O”) from those calculated in the present work (sub-
script “P”), ∆C = (CO −CP), C = D12, αT: solid lines with symbols - results by Kestin et al. [23]; dashed
lines with symbols - results by Song et al. [27]; point-dashed lines - uncertainty due to potential given in
Tables I and II
the present data are also depicted in Figure 4.
The discrepancies of these data are larger than
the corresponding uncertainties of the present
work because of the low order of approximation
N used in [27]. Moreover, the authors of [27]
used a slightly different potential given by Eq.(1)
and Table IX from Ref.[30].
The numerical values of the MS-DCs and
TDF for the argon-krypton mixture are reported
in Table V. A comparison the semi-empirical
data for this mixture by Kestin et al. [23] with
the present results is performed in Figure 5. The
discrepancies depicted in Figure 5 are within
the uncertainties of D12 and αT estimated in
[23], i.e., 2 % and 4 %, respectively. However,
the discrepancies exceed the uncertainties of the
present work. The deviations of the results by
Song et al. [27] from the present data shown in
Figure 5 are slightly larger than the correspond-
ing uncertainties given Tables I and II. The rea-
12
-1
-0.8
-0.6
-0.4
-0.2
0
400 4000 100 1000
(∆D
12 /D
12)
× 10
3
T(K)
x1=0.20.50.8
-3
-2
-1
0
400 4000 100 1000
∆ α T
× 1
03
T(K)
FIG. 3. Relative deviation of diffusion coefficient D12 (left) and deviation of thermal diffusion factor αT
(right) of helium-krypton mixture reported by Jager & Bich [28] (subscript “J”) from those calculated in
the present work (subscript “P”): ∆C = (CJ − CP), C = D12, αT.
TABLE IV. Maxwell-Stefan diffusion coefficients D12 at the standard pressure (p = 101325 Pa) and thermal
diffusion factor αT vs. temperature T and molar fraction x1 of neon for Ne-Kr mixture.
D12 × 106 (m2/s) −αT
T (K) x1 = 0.25 0.5 0.75 x1 = 0.25 0.5 0.75
50 0.931187 0.931033 0.930843 0.02682 0.03176 0.03920
100 3.59762 3.59617 3.59392 0.08590 0.10359 0.13056
300 26.5487 26.4683 26.3482 0.23505 0.27835 0.34236
500 64.0063 63.7405 63.3557 0.26677 0.31342 0.38180
1000 207.312 206.318 204.919 0.27825 0.32484 0.39269
2000 669.812 666.678 662.320 0.27045 0.31485 0.37914
5000 3200.98 3188.10 3170.31 0.24743 0.28749 0.34505
sons of these disagreements are the same as those
for the neon-kryptom mixture: low order of ap-
proximation in Eq.(12) and a different potential
used in [27].
B. Ternary mixtures
The numerical data on the three MS-DCs,
namely, D12, D13, D23, of ternary mixtures He-
Ne-Ar, He-Ne-Kr, He-Ar-Kr, and Ne-Ar-Kr are
given in Tables VI, VII, VIII, and IX, respec-
tively. Since the variations of these coefficients
with the chemical composition are small, only
the equimolar mixtures (x1 = x2 = x3) are pre-
sented in these Tables. Some other chemical
combinations are considered in Supplementary
Material to the present paper. The numerical
data presented in Tables VI-IX show that the
MS-DCs of the ternary mixtures are close to the
values of the MS-DCs of the corresponding bi-
nary mixtures. For instance, the coefficient D23
of the He-Ne-Kr mixture, see Table VII, is close
13
-2
-1
0
1
2
40 400 4000 100 1000
(∆D
12 /D
12)
× 10
0
T(K)
x1=0.25 0.5 0.75
-2
-1
0
1
2
3
40 400 4000 100 1000
∆ α T
× 1
00
T(K)
FIG. 4. Deviation of diffusion coefficient D12 (left) and thermal diffusion factor αT (right) of neon-krypton
mixture reported in other papers (subscript “O”) from those calculated in the present work (subscript “P”),
∆C = (CO − CP), C = D12, αT: solid lines with symbols - results by Kestin et al. [23]; dashed lines with
symbols - results by Song et al. [27]; point-dashed lines - uncertainty due to potential given in Tables I and
II
TABLE V. Maxwell-Stefan diffusion coefficients D12 at the standard pressure (p = 101325 Pa) and thermal
diffusion factor αT vs. temperature T and molar fraction x1 of argon for Ar-Kr mixture.
D12 × 106 (m2/s) −αT
T (K) x1 = 0.25 0.5 0.75 x1 = 0.25 0.5 0.75
50 0.440815 0.440487 0.440105 0.08039 0.08491 0.09028
100 1.66701 1.66684 1.66664 0.02104 0.02218 0.02353
300 14.0405 14.0328 14.0234 0.07746 0.08297 0.08964
500 35.6712 35.6205 35.5588 0.12453 0.13294 0.14311
1000 119.789 119.486 119.124 0.16111 0.17104 0.18303
2000 390.853 389.698 388.343 0.16956 0.17935 0.19108
5000 1861.66 1856.54 1850.62 0.15970 0.16845 0.17884
to the coefficient D12 of the Ne-Kr mixture, see
Table IV. The relative differences between the
corresponding coefficients for binary and ternary
mixtures are depicted in Figure 6 showing that
the differences are within 1 % for the mixtures of
He-Ar-Kr and Ne-Ar-Kr. In case of the mixture
He-Ne-Ar, the difference slightly exceed 1 % and
it reaches 2 % for the mixture He-Ne-Kr.
Numerical data on the two TDFs, αT1 and
αT2, of ternary mixtures He-Ne-Ar, He-Ne-Kr,
He-Ar-Kr, and Ne-Ar-Kr are given in Tables X,
XI, XII, and XIII, respectively. Since this co-
efficient is much sensitive to the chemical com-
position, several mole fraction combinations are
presented. First, the equimolar mixtures (x1 =
x2 = x3) are considered, then three situations
are reported when one species has a small frac-
tion equal to 0.1, while two other species have
the same fractions equal to 0.45. The first TDF
αT1 of all considered mixtures is negative in the
14
-2
-1
0
1
40 400 4000 100 1000
(∆D
12 /D
12)
× 10
0
T(K)
x1=0.25 0.5 0.75
-1
0
1
40 400 4000 100 1000
∆ α T
× 1
00
T(K)
FIG. 5. Deviation of diffusion coefficient D12 (left) and thermal diffusion factor αT (right) of argon-krypton
mixture reported in other papers (subscript “O”) from those calculated in the present work (subscript “P”),
∆C = (CO − CP), C = D12, αT: solid lines with symbols - results by Kestin et al. [23]; dashed lines with
symbols - results by Song et al. [27]; point-dashed lines - uncertainty due to potential given in Tables I and
II
-2
-1
0
1
50 300 5000 100 1000
He-Ne-Ar
(∆D
ij /D
ij)
× 10
0
T (K)
D12D13D23
-2
-1
0
1
50 300 5000 100 1000
He-Ne-Kr
(∆D
ij /D
ij)
× 10
0
T (K)
-2
-1
0
1
50 300 5000 100 1000
He-Ar-Kr
(∆D
ij /D
ij)
× 10
0
T (K)
-2
-1
0
1
50 300 5000 100 1000
Ne-Ar-Kr
(∆D
ij /D
ij)
× 10
0
T (K)
FIG. 6. Relative deviation of Maxwell-Stefan diffusion coefficinet for ternary mixture D(3)ij from that for the
corresponding equimolar binary mixture D(2)ij , ∆Dij = (D(3)
ij −D(2)ij ): solid lines - x1 = x2 = 0.45, x3 = 0.1;
dashed lines - x1 = 0.1, x2 = x3 = 0.45.
15
TABLE VI. Maxwell-Stefan diffusion coefficients
D12, D13, and D23 at the standard pressure (p =
101325 Pa) vs. temperature T for equimolar ternary
mixture of He-Ne-Ar.
Dij × 106 (m2/s)
T (K) D12 D13 D23
50 5.19628 3.24555 1.16942
100 17.4060 11.4110 4.48220
300 112.156 75.1986 32.2506
500 266.128 178.312 77.2382
1000 866.121 577.364 249.425
2000 2856.00 1890.08 806.532
5000 14190.0 9288.64 3865.61
TABLE VII. Maxwell-Stefan diffusion coefficients
D12, D13, and D23 at the standard pressure (p =
101325 Pa) vs. temperature T for equimolar ternary
mixture of He-Ne-Kr.
Dij × 106 (m2/s)
T (K) D12 D13 D23
50 5.19415 2.77516 0.93020
100 17.4081 9.85814 3.58653
300 112.270 65.4892 26.2899
500 266.439 155.422 63.2413
1000 867.142 503.118 204.664
2000 2859.06 1645.51 661.812
5000 14202.0 8077.30 3169.62
considered range of the temperature. The mag-
nitude of the second coefficient αT2 is smaller
than that of the first one. The sign of αT2
can be both positive and negative depending on
the temperature and chemical composition. The
third coefficient αT3 is not reported in Tables
X-XIII, because it can be obtained from Eq.(25)
and checked that it is always positive. The chem-
ical composition with the small mole fraction of
TABLE VIII. Maxwell-Stefan diffusion coefficients
D12, D13, and D23 at the standard pressure (p =
101325 Pa) vs. temperature T for equimolar ternary
mixture of He-Ar-Kr.
Dij × 106 (m2/s)
T (K) D12 D13 D23
50 3.24395 2.77475 0.440143
100 11.4128 9.85826 1.66443
300 75.4131 65.5988 13.9852
500 178.921 155.753 35.4676
1000 579.405 504.276 118.889
2000 1896.29 1649.08 387.803
5000 9313.53 8091.88 1849.21
TABLE IX. Maxwell-Stefan diffusion coefficients
D12, D13, and D23 at the standard pressure (p =
101325 Pa) vs. temperature T for equimolar ternary
mixture of Ne-Ar-Kr.
Dij × 106 (m2/s)
T (K) D12 D13 D23
50 1.17074 0.93097 0.44022
100 4.49760 3.59645 1.66620
300 32.5506 26.4732 13.9967
500 78.0877 63.7416 35.4850
1000 252.280 206.281 118.904
2000 815.074 666.494 387.745
5000 3899.02 3187.18 1848.43
one species shows us the influence of small quan-
tity of one gas on the TDF. In the limit x3 → 0,
the coefficients αT1 and αT2 obey the relation
αT1 → −αT2, at x3 → 0 (30)
according to (25). If we compare the coeffi-
cients αT1 and αT2 of the He-Ne-Kr mixture at
x1 = x2 = 0.45, see Table XI, they are still far
to obey the relation (30) even though the mole
16
fraction of krypton is equal to 0.1. At the same
time, the coefficients αT1 is close to αT of the He-
Ne mixture reported in Table X of the previous
paper [25]. Thus, a small quantity of a heavy gas
added to a binary mixture slightly increases the
coefficient αT1 and strongly decreases the coef-
ficients αT2. Now, let us consider the He-Ar-Kr
mixture with the small mole fraction of helium,
i.e. x1 = 0.1. In this case, we have the relation
similar to (30), namely,
αT2 → −αT3, at x1 → 0. (31)
The coefficient αT3 can be calculated from
Eq.(25) using the data from Table XII. For in-
stant, αT3 = 0.19362 at T = 5000 K, while
αT2 = −0.10471 for the same temperature so
that they do not obey Eq.(31), neither approxi-
mately. Comparing these values of αT2 and αT3
to that of αT from Table V, we conclude that a
small quantity of a light gas added to a binary
mixture strongly decreases the coefficients αT2
and slightly increases the coefficient αT3.
C. Quaternary mixture
The values of the MS-DCs for the quater-
nary mixture of helium, neon, argon and kryp-
ton are reported in Table XIV. Since these co-
efficients weakly depend on the chemical com-
position, only the equimolar mixture is consid-
ered. Other combinations of the mole fractions
can be found in Supplementary Material to the
present paper. A comparison of the MS-DCs for
the quaternary mixture with those of the corre-
sponding equimolar binary mixtures is shown in
Figure 7. Two chemical compositions are con-
sidered: (i) equal fractions of helium, neon and
argon (x1 = x2 = x3) with a small fraction of
krypton (x4 = 0.1); (ii) equal frations of neon,
argon and krypton (x2 = x3 = x4) with a small
fraction of helium (x1 = 0.1). Figure 7 shows
that the discrepancies between the MS-DCs of
the quaternary mixture and those of the corre-
sponding binary mixtures only slightly exceed 1
%.
The three TDFs, αT1, αT2, αT3, of the He-
Ne-Ar-Kr mixture are reported in Table XV. Ac-
cording to these data, the coefficient αT1 is al-
ways negative, the coefficients αT3 is mostly pos-
itive except one combination of the temperature
T = 50 K and chemical composition x1 = 0.1,
x2 = x3 = x4 = 0.3. The coefficient αT2 can be
both positive and negative. Its sign depends on
the temperature and chemical composition. The
coefficient αT4 is not reported in Table XV, but
it can be easily calculated by Eq.(25). It can be
verified that the value of αT4 is always positive.
The magnitudes of αT1 and αT4 are close to each
other and always larger than those of αT2 and
αT3. Comparing the values of all coefficients αTi
for the quaternary mixture with the mole frac-
tion x1 = 0.1 and x2 = x3 = x4 = 0.3 with
those of the coefficients for the ternary equimo-
lar mixture Ne-Ar-Kr, we conclude that a small
quantity of helium added to this ternary mixture
slightly decreases its coefficient αT1, significantly
changes the coefficient αT2, and slightly increases
the coefficient αT3. Let us consider the equimo-
lar ternary mixture of He-Ne-Ar presented by
Table XIII. A small addition of krypton into this
mixture practically does not change the coeffi-
cient αT1, significantly change the coefficient αT2
and significantly decreases the coefficient αT3. In
general, a small quantity of additional gas into
17
TABLE X. Thermal diffusion factors αT1 and αT2 vs. temperature T and molar fractions x1 of helium and
x2 of neon for ternary mixture of He-Ne-Ar.
αT1 αT2
x1 = 1/3 0.1 0.45 0.45 1/3 0.1 0.45 0.45
T (K) x2 = 1/3 0.45 0.1 0.45 1/3 0.45 0.1 0.45
50 -0.17021 -0.14554 -0.16826 -0.20956 0.07295 0.01128 0.07110 0.16558
100 -0.27527 -0.23850 -0.30168 -0.29444 0.06630 -0.02983 0.06566 0.20163
300 -0.32571 -0.28617 -0.37104 -0.32505 0.01979 -0.11223 0.02194 0.19244
500 -0.32331 -0.28463 -0.37065 -0.31914 0.00833 -0.12673 0.01090 0.18307
1000 -0.30910 -0.27212 -0.35625 -0.30291 -0.00020 -0.13144 0.00243 0.16934
2000 -0.28792 -0.25306 -0.33319 -0.28112 -0.00525 -0.12809 -0.00283 0.15426
5000 -0.25202 -0.22070 -0.29323 -0.24523 -0.01070 -0.11867 -0.00867 0.13110
TABLE XI. Thermal diffusion factors αT1 and αT2 vs. temperature T and molar fractions x1 of helium and
x2 of neon for ternary mixture of He-Ne-Kr.
αT1 αT2
x1 = 1/3 0.1 0.45 0.45 1/3 0.1 0.45 0.45
T (K) x2 = 1/3 0.45 0.1 0.45 1/3 0.45 0.1 0.45
50 -0.16076 -0.13542 -0.15857 -0.20474 0.05555 -0.00455 0.05283 0.15412
100 -0.27736 -0.23645 -0.30890 -0.29685 0.03746 -0.05832 0.03770 0.18272
300 -0.34069 -0.29468 -0.39694 -0.33417 -0.05292 -0.20327 -0.04487 0.15210
500 -0.34060 -0.29541 -0.39985 -0.32924 -0.07421 -0.23323 -0.06498 0.13865
1000 -0.32731 -0.28410 -0.38659 -0.31329 -0.08647 -0.24467 -0.07706 0.12378
2000 -0.30566 -0.26499 -0.36250 -0.29114 -0.08936 -0.23902 -0.08046 0.11013
5000 -0.26758 -0.23130 -0.31881 -0.25403 -0.08785 -0.22048 -0.08003 0.09073
a ternary mixture can change significantly some
its TDFs.
VIII. CONCLUSIONS
In the present work, two different definitions
of diffusion coefficients are discussed. A rela-
tion between the Fick and Maxwell-Stefan diffu-
sion coefficients has been obtained on the basis
of their definitions without any assumption on
the method of their calculation. The thermal
diffusion factors have been defined by a manner
different from usually defined in the open litera-
ture [15, 16]. The proposed definition is reduced
to the tradition one in case of binary mixture,
but it needs a smaller number of independent
coefficients that proposed in the books [15, 16].
The Maxwell-Stefan diffusion coefficients and
thermal diffusion factors of multi-component
mixtures composed from helium, neon, argon,
18
TABLE XII. Thermal diffusion factors αT1 and αT2 vs. temperature T and molar fractions x1 of helium
and x2 of argon for ternary mixture of He-Ar-Kr.
αT1 αT2
x1 = 1/3 0.1 0.45 0.45 1/3 0.1 0.45 0.45
T (K) x2 = 1/3 0.45 0.1 0.45 1/3 0.45 0.1 0.45
50 -0.13826 -0.11409 -0.15178 -0.15791 0.01164 -0.05455 0.01340 0.10370
100 -0.27654 -0.23027 -0.30996 -0.30485 0.11259 0.01898 0.12005 0.23538
300 -0.35951 -0.30380 -0.40489 -0.38687 0.11154 -0.02187 0.12027 0.28302
500 -0.36276 -0.30755 -0.40893 -0.38853 0.08387 -0.06388 0.09244 0.27230
1000 -0.35088 -0.29782 -0.39608 -0.37482 0.05679 -0.09778 0.06482 0.25277
2000 -0.32888 -0.27886 -0.37178 -0.35123 0.04180 -0.10805 0.04899 0.23221
5000 -0.28920 -0.24451 -0.32739 -0.30948 0.02952 -0.10471 0.03539 0.20168
TABLE XIII. Thermal diffusion factors αT1 and αT2 vs. temperature T and molar fractions x1 of neon and
x2 of argon for ternary mixture of Ne-Ar-Kr.
αT1 αT2
x1 = 1/3 0.1 0.45 0.45 1/3 0.1 0.45 0.45
T (K) x2 = 1/3 0.45 0.1 0.45 1/3 0.45 0.1 0.45
50 -0.02226 -0.01952 -0.02829 -0.01926 -0.04456 -0.07111 -0.04566 -0.00853
100 -0.08123 -0.07065 -0.09547 -0.07909 0.01703 -0.00951 0.01710 0.05213
300 -0.21054 -0.18581 -0.25370 -0.19394 0.02422 -0.04764 0.02489 0.11640
500 -0.23539 -0.20893 -0.28506 -0.21385 0.00336 -0.08745 0.00423 0.11798
1000 -0.24270 -0.21633 -0.29499 -0.21836 -0.01568 -0.11873 -0.01473 0.11241
2000 -0.23467 -0.20952 -0.28570 -0.21029 -0.02287 -0.12649 -0.02204 0.10514
5000 -0.21404 -0.19128 -0.26078 -0.19143 -0.02354 -0.11947 -0.02295 0.09474
and krypton have been calculated on the basis of
ab initio potentials over the temperature range
from 50 K to 5000 K. The Chapman-Enskog
method with the 10th order of approximation
has been employed. The relative numerical error
of the diffusion coefficients varies from 3×10−6 to
5×10−5 depending on the mixture composition.
The relative uncertainty of the diffusion coeffi-
cients due to the potential varies from 4 × 10−4
to 6 × 10−3. The absolute numerical error of
the thermal diffusion factor is in the range from
10−5 to 10−4, while the uncertainty due to the
potential does not exceed 3 × 10−3.
The reported numerical values of the
Maxwell-Stefan diffusion coefficients are weakly
sensitive to the chemical composition of mix-
tures. The maximum variation of these coef-
ficients because of the mole fractions is 3 %.
The diffusion coefficients calculated for ternary
and quaternary mixtures are very close to those
19
TABLE XIV. Maxwell-Stefan diffusion coefficients Dij at the standard pressure (p = 101325 Pa) vs. tem-
perature T for equimolar quaternary mixture of He-Ne-Ar-Kr.
Dij × 106 (m2/s)
T (K) D12 D13 D14 D23 D24 D34
50 5.19471 3.24574 2.77606 1.17000 0.930387 0.439984
100 17.4240 11.4238 9.86776 4.48901 3.58982 1.66435
300 112.459 75.4547 65.6411 32.4129 26.3471 13.9638
500 266.898 178.983 155.819 77.7196 63.3854 35.3847
1000 868.538 579.505 504.380 251.101 205.094 118.530
2000 2863.08 1896.40 1649.20 811.656 663.009 386.585
5000 14217.4 9313.23 8091.57 3886.17 3173.99 1844.02
-1
0
1
50 300 5000 100 1000
(∆D
ij /D
ij)
× 10
0
T (K)
D12D13D14
-1
0
1
50 300 5000 100 1000
(∆D
ij /D
ij)
× 10
0
T (K)
D23D24D34
FIG. 7. Relative deviation of Maxwell-Stefan diffusion coefficient for quaternary mixture D(4)ij from that for
the corresponding equimolar binary mixture D(2)ij , ∆Dij = (D(4)
ij − D(2)ij ): solid lines - x1 = x2 = x2 = 0.3
x4 = 0.1; dashed lines - x1 = 0.1 x2 = x3 = x4 = 0.3.
of the corresponding binary mixtures with the
maximum discrepancy of 2 %.
The thermal diffusion factors are very sensi-
tive to the interatomic potential and chemical
composition so that it is impossible to express
these coefficients for multicomponent mixtures
via those of binary mixtures. Even small quan-
tity of a third gas added into a binary mixture
can significantly change its thermal diffusion fac-
tor.
The results reported in the present work to-
gether with those published in Refs.[25, 26, 35,
61] represent the complete database of the diffu-
sion coefficients and thermal diffusion factors of
all possible mixtures composed of helium, neon,
argon, and krypton over wide ranges of tem-
peratures and chemical composition. More de-
tailed data on the transport coefficients are given
in Supplemental Materials to this paper where
more values of the temperature and of the mole
fraction are considered.
20
TABLE XV. Thermal diffusion factors αT1, αT2, and αT3 vs. temperature T and molar fractions x1 of
helium, x2 of neon, and x3 of argon for quaternary mixture of He-Ne-Ar-Kr.
x1 = 0.25 0.1 0.3 0.3 0.3
x2 = 0.25 0.3 0.1 0.3 0.3
T (K) x3 = 0.25 0.3 0.3 0.1 0.3
αT1
50 -0.14477 -0.12980 -0.14058 -0.15394 -0.15892
100 -0.25982 -0.23455 -0.26908 -0.27016 -0.26938
300 -0.32453 -0.29565 -0.34442 -0.33421 -0.32673
500 -0.32526 -0.29683 -0.34663 -0.33448 -0.32573
1000 -0.31296 -0.28574 -0.33460 -0.32160 -0.31232
2000 -0.29234 -0.26671 -0.31321 -0.30037 -0.29131
5000 -0.25608 -0.23317 -0.27501 -0.26304 -0.25508
αT2
50 0.02792 -0.00287 0.02676 0.04315 0.05170
100 -0.00155 -0.04995 -0.00039 0.01996 0.03404
300 -0.09029 -0.16230 -0.08411 -0.06978 -0.03257
500 -0.10917 -0.18454 -0.10227 -0.09002 -0.04739
1000 -0.11810 -0.19241 -0.11110 -0.10081 -0.05600
2000 -0.11765 -0.18746 -0.11098 -0.10223 -0.05844
5000 -0.11140 -0.17271 -0.10542 -0.09862 -0.05841
αT3
50 0.01278 -0.02227 0.01201 0.01372 0.05693
100 0.10414 0.05063 0.10869 0.10963 0.15979
300 0.13394 0.06690 0.12012 0.14066 0.22554
500 0.11930 0.04870 0.09788 0.12578 0.22364
1000 0.10153 0.03032 0.07474 0.10762 0.21247
2000 0.08915 0.02112 0.06092 0.09477 0.19813
5000 0.07582 0.01542 0.04830 0.08075 0.17560
SUPPLEMENTARY MATERIAL
See supplementary material for the complete
data of diffusion coefficients and thermal diffu-
sion factors. The name of each file corresponds
to the mixture, e.g., the file He-Kr.xlsx contains
data for the He-Kr mixture.
ACKNOWLEDGMENTS:
One of the authors (F.S.) acknowledges the
Brazilian Agency CNPq for the support of his
21
research, Grant No. 304831/2018-2.
DATA AVAILABILITY STATEMENT
The data that supports the findings of this
study are available within the article and its Sup-
plementary Material.
Appendix A: Relation of Fick diffusion
coefficients Dij to Maxwell-Stefan ones Dij .
Using Eq.(21), we obtain
V i − V j = −K∑
k=1
(Dik −Djk) (dk + kTk∇T ) .
(A1)
A substitution of (A1) into (26) leads to
di + kTi∇T =K∑
k=1
K∑
j=1j 6=i
xixjDij
(Dik −Djk)
× (dk + kTk∇T ) . (A2)
Since the vectors di and coefficients kTi have
the constrains (5) and (20), respectively, we con-
clude that
K∑
j=1j 6=i
xixjDij
(Dik −Djk) = δik + αi, (A3)
where αi is unknown constant. The property (8)
and the symmetry (9) lead to the relation
K∑
k=1
yk(Dik −Djk) = 0. (A4)
Multiplying (A3) by yk and summarizing it with
respect to the subindex k, we have
yi + αi = 0, (A5)
where the relations (3) and (A4) have been used.
Thus, Eq.(28) is derived.
22
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