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

∗ sharipov@fisica.ufpr.br; http://fisica.ufpr.br/sharipov† vjben@fisica.ufpr.br

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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