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PHYSICAL REVIEW E 00, 002100 (2017)1
Nonlinear diffusion in multicomponent liquid solutions2
Vyacheslav V. Obukhovsky,1,*Andrii M. Kutsyk,1Viktoria V. Nikonova,1and Oleksii O. Ilchenko2
3
1Taras Shevchenko National University of Kiev, 01601 Kiev, Ukraine4
2D. F. Chebotaryov Institute for Gerontology, National Academy of Medical Sciences of Ukraine, 04114 Kyiv, Ukraine5
(Received 19 September 2016; published xxxxxx)6
Mutual diffusion in multicomponent liquids is studied. It is taken into consideration that the influence of
complex formation on the diffusion process may be substantial. The theory is applied to analyze mass transfer
in an acetone-chloroform solution. The molecular complex concentration was obtained from the analysis of
Fourier transform infrared spectra of this solution. Taking into account molecular complex formation allows one
to explain the experimental dependence of diffusion coefficients on the composition (components concentration).
The accuracy of experimental and theoretical data descriptions in the frame of our model is compared to the
accuracy for some other approaches.
7
8
9
10
11
12
13
DOI: 10.1103/PhysRevE.00.00210014
I. INTRODUCTION15
Standard Fick laws [1] are usually used to describe exper-16
imental results of mutual diffusion, i.e., transport of solute17
from regions with high concentrations to low concentrations18
in nonuniform solutions. In this way, the velocity of matter19
transfer is determined with DF, the mutual diffusion (inter-20
diffusion) coefficient. Numerous experimental data indicate21
that in many solutions this coefficient DFis not constant22
but sufficiently depends on the solute concentration (see, for23
example, Refs. [2–4]).24
Many methods have been developed for the theoretical25
description of diffusion effects in liquid solutions by now (see,26
for example, Refs. [2–18] and references therein). The Darken27
equation [7] and its modifications [5,8,10] are widely used.28
Also the Vignes equation [13] is the basis of many empirical29
equations [5,14–16]. Fluctuation theories of diffusion arouse30
interest due to attention paid to the random spatial distribution31
of molecules [17,18]. The interconnection between mutual32
diffusion and viscosity of solutions allows us to predict some33
physical characteristics [12].34
In some papers diffusion is considered a random walk35
of particles through free positions (vacancies) (see, e.g.,36
Refs. [3,19]). But the concept of “vacancy” is very difficult37
to introduce in the multicomponent liquid mixture, in which38
molecules have significantly different sizes. Besides that, many39
liquids have low compressibility. This can be interpreted as an40
absence (or a very small quantity) of free volume.41
Another direction of diffusion theories’ development is42
connected with the application of probability theory for43
random walks of atoms in the nodes of a crystal lattice [20].44
Note that in the framework of a two-component lattice gas45
model new nonlinear diffusion effects can be found: the drag46
effect, the formation of the drifting spatial structure, the effect47
of “negative” mass transport, etc. [20–26].48
But the results of the latter papers are not adapted for49
the description of diffusion in liquid solutions. The reason50
is that mutual diffusion cannot be separated from the process51
of dissolving, and mixing of initial components is accompa-52
nied by the energetic process (change of enthalpy, heating53
*vobukhovsky@yandex.ua, vvo@univ.kiev.ua
and cooling). New components can be formed as the result of 54
dissolution, and they are called “complexes” or “heteroasso- 55
ciates” [27]. In experiments [28–30] the average quantity of 56
such structural formations was large enough (for example, it 57
can exceed 50%), despite their instability.158
An essential step was made in Refs. [31–34],where the 59
hypothesis of the influence of solution structure on the 60
diffusion process was successfully realized. The impact 61
of association or complex formation on interdiffusion and 62
intradiffusion in multicomponent systems was investigated 63
in the frame of traditional thermodynamic theories [35–39]. 64
Peculiarities, which complicate their practical applications, 65
include the necessity of taking into account (in some theories) 66
the concentration dependence of auxiliary functions, e.g., 67
thermodynamic factor, intradiffusion coefficients of solution 68
components, etc. 69
The purpose of this work is the application of probability 70
theory to the description of a mutual diffusion process in a 71
liquid multicomponent solution with associates, or complex 72
generation. As an example of our theoretical application, 73
diffusion in an acetone-chloroform solution is analyzed. 74
II. NONLINEAR FLOW UNDER 75
NORMALIZATION CONDITIONS 76
It is known that in an ideal binary mixture, where interac- 77
tions between components are not taken into account, transport 78
of particles Aand Bis governed by a linear diffusion law [2,3]. 79
The simplest model, which implies nonlinear flow behavior 80
(relative to concentration characteristics),2must include more 81
than two types of particles (“particles” may include individual 82
molecules, associates, and complexes). 83
Hereinafter a mixture of three components A,B, and C84
is considered. Diffusion takes place only along axis x.In 85
the frame of our model construction, all the particles are 86
located in planes xi(i=0,±1,±2,±3,...). The distance 87
L between all adjacent planes is small. In any plane the total 88
1The small lifetime of a molecular complex has to be compared
with the small time of free motion of individual molecules.
2In our theory, diffusion flow relating to the corresponding thermo-
dynamic force is considered in linear approximation.
2470-0045/2017/00(0)/002100(11) 002100-1 ©2017 American Physical Society
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OBUKHOVSKY, KUTSYK, NIKONOVA, AND ILCHENKO PHYSICAL REVIEW E 00, 002100 (2017)
amount of particles, N0, is fixed (vacancies are not available).89
If a particle Ais located in the plane xiand a particle B90
is located in the plane xi+1(near the molecule A), then the91
correlated process when Amoves into the plane i+1 and B92
moves to the vacant place in plane ican occur. Therefore, this93
process AB →BA is called the “exchange of positions.”94
In fact, exchange of positions can be performed in a more95
difficult way (displacement of several neighboring particles96
in the closed loop). However, the flux density is determined97
only by the difference between initial and final molecular98
distributions and does not depend on the method of particle99
transfer. The exchange of positions of particles of one type100
is excluded from consideration because it does not result in101
experimentally registered changes.3
102
We denote pab ,pbc, and pac the probabilities of exchange of103
the positions (per unit time) for the processes A↔B,B↔C,104
and C↔A, respectively. The number of particles of type “s”105
in the plane xiis denoted NS(xi,t). The probability to find106
the particles with s1= s2in the neighboring positions with the107
coordinates (xi,yj,zk), (xi+1,yj,zk) will be proportional to the108
product (NS1(xi,t )
N0)( NS2(xi+1,t)
N0). In any case, the normalization109
condition must be satisfied:110
Na(xi,t)+Nb(xi,t)+Nc(xi,t)=N0.(1)
The particle flux is determined by the number of exchanges
111
A↔B,A↔C(per unit time) in the direction of the axis x112
and equal to113
Ja
x(x,t)=1
N2
0
{pab[Na(xi,t )Nb(xi+1,t )
−Nb(xi,t),Na(xi+1,t)]+pac[Na(xi,t)Nc(xi+1,t)
−Nc(xi,t)Na(xi+1,t)]}.(2)
If smooth spatial variations for functions NS(x) take place,114
it can be spread into a Taylor series:115
NS(xi+1)∼
=NS(xi)+∂NS
∂x xi
L (3)
(L =xi+1−xi). Substituting Eq. (3)inEq.(2), we obtain116
for the three-component case117
Ja
x(xi,t)=pabL
N2
0Na∂Nb
∂x −Nb∂Na
∂x xi
+pacl
N2
0Na∂Nc
∂x −Nc∂Na
∂x xi
.(4)
It is easy to generalize Eq. (4) for continuous distribution
118
of equal size particles, and then we obtain119
Ja(r,t)=
q
Q(a,q)[Na(r,t)∇Nq(r,t )
−Nq(r,t)∇Na(r,t)].(5)
Here we introduce the following notations: aand qare
120
components of the system, Jais the total flux of the particles121
3This theory is oriented at macroscopic diffusion experiments,
for example, with optical registration of spatial distributions of
concentrations.
a, and the value Q(a,q) is determined by the speed of position 122
interchange for the substances (a,q). The flow (5) is a quadratic 123
function of the concentration. Therefore, diffusion has to be 124
classified as a second order nonlinear effect. 125
To generalize results above, it is necessary to take into 126
account the differences in sizes of the diffusing particles. 127
According to the principles of the phenomenological theo- 128
ries [40], the diffusion flux of the matter should be considered 129
as the averaged value over “physically infinitesimal volume.” 130
Following Ref. [40], we select a physically infinitesimal 131
volume4V0(r) in the mixture (centered at the point r) and 132
define the volume fraction ϕn(r) as the relative portion of this 133
volume occupied by all molecules of the type n:134
ϕn(r)=Vn(r)
V0(r),
n
ϕn(r,t)=1,(6)
where Vn(r) is the part of the volume V0(r) occupied by the 135
substance n.136
The flow (5) was defined earlier as the number of particles 137
that transit through a unit area per unit time. We can replace 138
this flow by the ϕstream. It is the volume of substance nwhich 139
is carried over a unit area per unit time. It is easy to check that 140
in this way the nonlinear diffusion flows can be written as 141
follows: 142
ji=
j
bij [ϕi∇ϕj−ϕj∇ϕi].(7)
Here the indices iand jdenote the components of the liquid 143
mixture, ϕiis the volume fraction of the ith component, ji144
is volume flow of the ith substance, and bij is the symmetric 145
matrix of position interchange coefficients (bij =bji), that can 146
be interpreted as “nonlinear diffusion coefficients.” Contrary 147
to the mutual diffusion coefficient DFthe values of bij are 148
constant and do not depend on component concentration. Here 149
and below, the volume-fixed frame of reference is used. 150
The nonlinearity of flow similar to Eq. (7) appears in other 151
tasks. For example, it can be considered a generalization of 152
the mass transfer law for the model [41] that was created 153
for multicomponent photopolymers. Also the quadratic non- 154
linearity similar to Eq. (7) was met in the investigations of 155
impurity diffusion in cubic crystals [20–26] and in monolayers 156
of reagents on the surface of a catalyst [42,43]. 157
Generally, the diffusion of liquid components has to be 158
described as a macroscopic phenomenon. So, all physical 159
quantities, related to mass transfer, have to be averaged 160
over physically infinitesimal volume. As a result, all other 161
equations could be formulated in terms of “partial volumes” 162
too. Besides Eq. (7), the following laws (generalization of 163
standard formulas [41,44]) can be used in investigation of 164
diffusion: 165
(a) the equations of continuity 166
∂ϕi
∂t +div ji=Si,(8)
4According to the definition of physically infinitesimal volume
(PIV), its sizes are much smaller compared to the precision of space
coordinate measuring (in diffusion experiment), but the PIV contains
a large number (N1) of particles.
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NONLINEAR DIFFUSION IN MULTICOMPONENT LIQUID . . . PHYSICAL REVIEW E 00, 002100 (2017)
(b) and the conditions of conservation167
i
ji=0,
i
Si=0.(9)
Here Siis the function of sources, which depends on the168
processes of i-component formation (decomposition).169
For the case of the negligibly small influence of the170
shrinkage and swelling phenomenon on the processes of171
diffusive transport, we take into account the law of volume172
conservation:5
173
i
ϕi=1.(10)
Equations (7)–(10) are valid in the range 0 ϕi1, (i=174
1,2,...).175
It is obvious that macroscopic diffusion flux of any176
component must be accompanied by reverse flow of other177
components in the case of volume conservation. Hence the178
problem of macroscopic diffusion can be considered an analog179
of two-body problems (taking into account two related or180
interconnected flows). In the general case, the many-body181
problem can be described by nonlinear equations [46]. And182
it is not surprising that simultaneous consideration of two183
correlated subsystems, “direct flow–reverse flow,” leads to184
quadratic dependence in the equations of motion.185
First of all, it can be checked that the nonlinear form of186
the flow in Eq. (7) in the particular case does not contradict187
the linear Fick law. Indeed, consider a mixture of two188
noninteractive molecular liquids with the flow189
j1=b12[ϕ1∇ϕ2−ϕ2∇ϕ1].(11)
The condition of volume conservation ϕ1+ϕ2=1 leads to190
ϕ2=1−ϕ1,∇ϕ2=−∇ϕ1.(12)
Substituting Eq. (12) into Eq. (11), we find the expression
191
for the volume flow:192
j1=−b12 ∇ϕ1.(13)
The diffusion coefficient in Eq. (13) is constant. The volume
193
flow j1and molar flow J1are interconnected by the following194
relation:195
j1=¯
V1J1,(14)
where ¯
V1is the molar volume of substance 1. Volume flow (13)196
can be rewritten as197
J1=−DF∇c1,(15)
where the coefficient DF=b12 is constant. Thus, in the case198
of two noninteractive components the diffusion flow is a linear199
function of concentration.200
5Shrinkage and swelling effects can be taken into account if
an additional component (free space) is added. See, for example,
Ref. [45].
III. DIFFUSION IN LIQUID SOLUTION 201
Consider diffusion in the molecular solutions that are 202
formed by mixing two liquid substances Aand B.The203
theoretical description of this system is considered in the frame 204
of ideal associated solution model (IASM) [27,47]. This model 205
treats nonideal mixtures of associated components as ideal 206
mixtures of free molecules and molecular complexes [48–51]. 207
Interaction between molecules of the original components can 208
lead to formation (with some probability) of complexes of the 209
AnBmtype. Below we discuss the system with the following 210
interactions: 211
nA +mB →AnBm(formation of complex),
AnBm→nA +mB (dissociation of complex).(16)
(To simplify the calculations, the intermediate steps of reac- 212
tions are not considered.) 213
Within the framework of our model, this kind of mixture 214
consists of three components: A,B, and AnBm. Hereinafter 215
these components are denoted as 1, 2, and 3. The reactions (16)216
determine the form of the functions Si(sources). In this case 217
S1=β1ϕ3−α1ϕn
1ϕm
2,
S2=β2ϕ3−α2ϕn
1ϕm
2,
S3=−S1−S2.⎫
⎬
⎭
(17)
The form of the functions (17) is similar to the description 218
of generation and decay processes for nonstable products of 219
chemical reactions [52]. Below, the reaction rates are assumed 220
to be rapid compared to diffusion so that chemical equilibrium 221
exists locally [31]. Then the approximation 222
Si∼
=0 (18)
can be used in Eqs. (17). In other words, diffusion occurs 223
under conditions of a local chemical equilibrium. In this case, 224
the volume fraction of a complex component can be found 225
from Eqs. (17) and (18): 226
ϕ3∼
=Kϕϕn
1(1 −ϕ1−ϕ3)mKϕ=α1
β1
=α2
β2.(19)
Below we take into account the following peculiarity. The 227
transport of molecules in the diffusion process occurs via two 228
mechanisms: (i) individually and (ii) as a part of the complex 229
AnBm. Therefore, the total flow of the matter “A” (measurable 230
in experiments) is defined as a linear combination, 231
jtot
1=j1+¯α1j3¯α1=n¯
V1
n¯
V1+m¯
V2,(20)
where α1is the volume fraction of substance Ain the complex 232
AnBm, and ¯
V1and ¯
V2are the molar volumes of components 1 233
and 2. 234
After substitution of Eq. (7)inEq.(20)wehavethe 235
following relationship: 236
jtot
1=b12[ϕ1∇ϕ2−ϕ2∇ϕ1]+b13 ¯α2[ϕ1∇ϕ3−ϕ3∇ϕ1]
+b23 ¯α1[ϕ3∇ϕ2−ϕ2∇ϕ3].(21)
As we highlighted above, there are two states of substance 237
1 in the solution: free and bonded (in molecular complex 238
composition). Therefore, its total partial volume is 239
ϕtot
1=ϕ1+¯α1ϕ3.(22)
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OBUKHOVSKY, KUTSYK, NIKONOVA, AND ILCHENKO PHYSICAL REVIEW E 00, 002100 (2017)
Similarly, we can determine the total partial volume for the240
second component, ϕtot
2:241
ϕtot
2=ϕ2+¯α2ϕ3¯α2=m¯
V2
n¯
V1+m¯
V2.(23)
In this case the normalization condition is identically
242
satisfied:243
ϕtot
1+ϕtot
2=1(¯α1+¯α2=1).(24)
It is easy to verify the following assertion: the value ϕtot
1
244
completely determines all other functions of diffuse flows [i.e.,245
ϕi=ϕi(ϕtot
1), i=1,2,3] if conditions defined by Eq. (18)are246
valid. Therefore, the total flux of substance 1 also can be247
represented as the function of ϕtot
1only:248
jtot
1=−Def
1ϕtot
1∇ϕtot
1.(25)
It is obvious that Eq. (25) has the same form as the first
249
Fick law of diffusion; however, it contains the generalized (ef-250
fective) coefficient of diffusion, Def
1. In fact, the “coefficient”251
Def
1is not constant but is a function which depends on the252
“concentration” of interacting substances:253
Def
1ϕtot
1=w12ϕtot
1b12 +w13ϕtot
1b13 +w23ϕtot
1b23.
(26)
Here wij determines the relative contribution of the interac-254
tions between different components {i,j}={1,2,3}into the255
effective (measured) diffusion coefficient. In the general case6
256
w12 =ϕ2
∂ϕ1
∂ϕtot
1
−ϕ1
∂ϕ2
∂ϕtot
1
,
w13 =¯α2ϕ3
∂ϕ1
∂ϕtot
1
−ϕ1
∂ϕ3
∂ϕtot
1,
w23 =¯α1ϕ2
∂ϕ3
∂ϕtot
1
−ϕ3
∂ϕ2
∂ϕtot
1.(27)
It is not difficult to verify that
257
w12 +w13 +w23 =1.(28)
All functions wij in Eqs. (26) and (28) must be expressed258
through ϕtot
1.259
The results obtained above [Eqs. (25)–(27)] describe260
diffusion in a “binary” liquid solution with complex AnBm
261
generation in the process of mixing. The material parameters262
of this system are Kϕand bij . In Appendix Ait is proved263
that in the simplest case of solutions with 1:1 a complex these264
parameters can be found from the experimentally measured265
values on the borders of an area of measuring (in the case266
of infinite dilution of component Aor B)Def
1(ϕtot
1=0), Def
1
267
(ϕtot
2=0), ∂Def
1/∂ϕtot
1(ϕtot
1=0), and ∂Def
1/∂ϕtot
1(ϕtot
2=0).268
Therefore, parameters Kϕand bij do not need to be considered269
mathematically adjustable.270
Unfortunately, precise experimental measuring of diffusion271
coefficients under conditions ϕtot
1=0orϕtot
2=0 usually is272
not realized (in practice they are found by extrapolation of the273
6For an arbitrary type of third component, particularly for any n,m.
nearest points7). This is the reason why another way could 274
be more preferable. Indeed, the equilibrium constant Kϕcan 275
be determined from other experimental data (for example, by 276
vibrational spectroscopy or NMR techniques [53,54]). As a 277
rule, use of additional data leads to improvement of calculation. 278
IV. MOLECULAR COMPLEX FORMATION IN 279
ACETONE-CHLOROFORM MIXTURE 280
The presence of the C-H ···O hydrogen bond in the 281
acetone-chloroform mixture causes the formation of the 282
molecular complex which consists of one acetone molecule 283
and one chloroform molecule. Equimolecular (1:1) complex 284
formation is indicated by inelastic neutron scattering [55], 285
low-Raman, far- [56], mid- [57], and near-infrared, and 1H286
NMR [53] spectroscopies. On the other hand, the oxygen 287
atom in acetone contains two electron lone pairs, so in fact 288
two types of complexes may exist. The indirect evidence 289
of 1:2 complex existence is the slight asymmetry of excess 290
thermodynamic functions (excess Gibbs energy, enthalpy, and 291
entropy) [58,59]. Due to the small number of 1:2 complexes 292
fixed experimentally [54,60], hereinafter we consider only 293
equimolecular 1:1 complex formation. 294
Before using Eqs.(26) and (27), we need to obtain the 295
equilibrium constant Kϕof complex formation which allows 296
determination of the volume fractions ϕ3. It can be obtained 297
from an infrared (IR) absorption spectroscopy, because it is 298
well known that IR spectra are very sensitive to any structural 299
changes which occur in the investigated system [61]. 300
Fourier transform infrared (FTIR) absorption spectra, be- 301
tween 3750 and 6200 cm−1(Fig. 1), were used to study in- 302
termolecular interactions between components of the acetone- 303
chloroform liquid solution. FTIR spectra were measured using 304
a Thermo Scientific Nicolet iS50 FTIR spectrometer with 305
maximum spectral resolution of 0.125 cm−1. Heating of the 306
sample almost did not occur during the measurements due to 307
the small value of the absorption coefficient at the excitation 308
frequency. The temperature of liquid samples was 25 ±0.2◦C. 309
Chloroform and acetone with purity 99.9% were used in this 310
research. The concentration of components was changed from 311
0% to 100% (in vol %) with a step of 10%. Spectra recordings 312
were repeated 32 times for each sample. Thereafter the average 313
spectrum was calculated for every concentration and used in 314
further analysis. 315
Considering the liquid solution as multicomponent, which 316
contains unbonded (“pure”) and bonded (molecular complex) 317
species, the IR data matrix may be written as 318
D=CST+R.(29)
Here Dis the measured IR absorption spectra matrix of 319
solution, its rows containing spectra measured at different 320
concentrations; Cis the matrix of concentrations, its columns 321
containing concentration profiles of solution components; Sis 322
the matrix, which contains spectral profiles of each solution 323
component; and Ris the residuals matrix. 324
7In many diffusion experiments the minimal step of concentrations
is near 5–10 %.
002100-4
NONLINEAR DIFFUSION IN MULTICOMPONENT LIQUID . . . PHYSICAL REVIEW E 00, 002100 (2017)
4000 4500 5000 5500 6000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
ν(cm−1)
IR absorption (arb. units)
ϕtot
1=0.0
ϕtot
1=0.1
ϕtot
1=0.2
ϕtot
1=0.3
ϕtot
1=0.4
ϕtot
1=0.5
ϕtot
1=0.6
ϕtot
1=0.7
ϕtot
1=0.8
ϕtot
1=0.9
ϕtot
1=1.0
FIG. 1. FTIR absorption spectra of acetone-chloroform solutions between 3750 and 6200 cm−1;ϕtot
1is the initial volume fraction of acetone
(before mixing).
The goal of multivariate curve resolution (MCR) techniques325
is the determination of the matrices Cand Susing the326
experimental data matrix D[62,63]. The basic principle of327
MCR is to seek a bilinear model that gives the best fit to the328
matrix D.329
Use of the liquid solution structural model may simplify330
greatly the decomposition of spectra matrix D[63]. The331
concentration matrix C(in our case it contains volume332
fractions) may be found by using a mass balance equation333
at fixed value of Kϕ. The solution of Eq. (29)atfixedDmay334
be estimated in sense of least squares:335
S=(CTC)−1CTD=C+D.(30)
Here C+=(CTC)−1CTis Moure-Penrose pseudoinverse ma-336
trix. By using Eqs. (29) and (30), the residuals matrix Rmay337
be written as338
R=D−CC+D.(31)
Matrix Cdepends on the equilibrium constant Kϕonly;339
thus, matrix norm of Rmust be minimized for the estimation340
of the equilibrium constant optimal value:341
R=D−C(Kϕ)C+(Kϕ)D→min,K
ϕ>0.(32)
The three-component model of the solution, which was
342
proposed for the description of mutual diffusion, was used343
for decomposition of spectra matrix D. Using measured IR344
absorbance data, solution of Eq. (32) gives the following345
optimal value of the equilibrium constant:346
Kopt
ϕ=2.5±0.2.(33)
With this value of equilibrium constant Kopt
ϕwe can347
numerically solve Eq. (19) and calculate volume fractions of348
mixture components ϕ3,ϕ2, and ϕ1as a function of initial349
volume ϕtot
1. The results are shown in Fig. 2.350
The complex fraction ϕ3arises as the result of interaction351
between molecules in the dissolution process and can occupy352
in our case up to 30% of the total volume.353
V. ANALYSIS OF DIFFUSION IN 354
ACETONE-CHLOROFORM MIXTURE 355
We applied the modified system of diffusion equations 356
(7)–(10) to analyze the mass transfer processes in a liquid 357
mixture of acetone (C3H6O, component 1) with chloroform 358
(CHCl3, component 2). These substances are completely 359
mutually soluble. The molar volumes of molecules 360
¯
V1=74.00 ml/mol,¯
V2=80.64 ml/mol,(34)
for acetone and chloroform, respectively (which are needed 361
for the calculation of ¯α1and ¯α2), were found from Ref. [27]. 362
Experimental data on diffusion in an acetone-chloroform 363
mixture were obtained in Refs. [64,65]at25◦C. These data 364
have been restated to build the curve of the diffusion coefficient 365
Dexpt
12 as a function of the acetone partial volume ϕtot
1(Fig. 3). 366
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
ϕtot
1(vol. fraction of acetone)
ϕi(vol. fraction)
acetone
chloroform
complex
FIG. 2. Calculated volume fractions of mixture components. ϕtot
1
is the initial volume fraction of acetone (before mixing).
002100-5
OBUKHOVSKY, KUTSYK, NIKONOVA, AND ILCHENKO PHYSICAL REVIEW E 00, 002100 (2017)
0 0.2 0.4 0.6 0.8 1
2
2.5
3
3.5
4
ϕtot
1(vol. fraction of acetone)
Def
1·10−9(m2/s)
experimental
calculated
FIG. 3. Concentration dependence of the effective diffusion
coefficient for the acetone-chloroform solution: points, experimental
data [64]; solid curve, theoretical calculations; ϕtot
1, initial volume
fraction of acetone (before mixing).
It is known that the concentration dependence of the excess 367
enthalpy of mixing has a parabolic form with a minimum 368
when the component concentration ratio is 1:1 [58,66]. It 369
is an indirect evidence of equimolecular complex formation 370
(C=A+B). This conclusion is confirmed by many other 371
experimental results (see, for example, Refs. [53,55–57]). In 372
our case, such a complex formation process corresponds to the 373
following set of coefficients: n=1, m=1. 374
The concentration of components 1 and 2 after mixing can 375
be written as 376
ϕ1=ϕtot
1−¯α1ϕ3,ϕ
2=ϕtot
2−¯α2ϕ3.(35)
In this case the equation for the volume fraction of the 377
component C≡[A1B1] can be obtained from Eq. (19): 378
ϕ3=Kϕϕtot
1−¯α1ϕ3ϕtot
2−¯α2ϕ3,(36)
which is a quadratic form of ϕ3. The solution of Eq. (36) can 379
be found as 380
ϕ3(ϕtot
1,ϕ
tot
2)=1+Kϕ¯α1ϕtot
2+¯α2ϕtot
1−1+Kϕ¯α1ϕtot
2+¯α2ϕtot
12−4¯α1¯α2K2
φϕtot
1ϕtot
2
2¯α1¯α2Kϕ
.(37)
We can rewrite Eqs. (27) for the case of 1:1 complex381
formation in the form382
w12 =(1 +ϕ3),
w13 =Kϕ¯α2ϕ1(ϕ1+ϕ3),
w23 =Kϕ¯α1ϕ2(ϕ2+ϕ3),(38)
where =[1 +Kϕ(¯α1ϕ2+¯α2ϕ1)]−1.383
The explicit form for the concentration dependence of384
functions wnm is obtained by use of Eqs. (35)–(37):385
w12 =1+ϕ3
1+Kϕ¯α1ϕtot
2+¯α2ϕtot
1−2¯α1¯α2Kϕϕ3
,
w13 =Kϕ¯α2(ϕtot
1−¯α1ϕ3)(ϕtot
1+¯α2ϕ3)
1+Kϕ¯α1ϕtot
2+¯α2ϕtot
1−2¯α1¯α2Kϕϕ3
,
w23 =Kϕ¯α1(ϕtot
2−¯α2ϕ3)(ϕtot
2+¯α1ϕ3)
1+Kϕ¯α1ϕtot
2+¯α2ϕtot
1−2¯α1¯α2Kϕϕ3
.(39)
Experimental data Dexpt
1can be compared with the386
theoretical dependence of Eqs. (26) and (29). Fitting of387
bnm can be realized by minimization of the difference388
|Dexpt
1(ϕtot
1)−Dcalc
1(ϕtot
1)|2by the least-squares method. It389
leads to the following values of material parameters:390
b12 =3.86 ×10−9m2/s,
b13 =3.32 ×10−9m2/s,
b23 =1.28 ×10−9m2/s.(40)
A graphical comparison of the theoretical results with the
391
experimental data is presented in Fig. 3. Good agreement392
of theory and experiment (average relative deviation 1.4 %)393
supports the assumption that the coefficients bij are really 394
constants (material parameters) and do not depend on concen- 395
trations. 396
Despite the low share of complexes (does not exceed 30% 397
of the volume), its influence on the diffusion is significant. 398
This is manifested in the nonlinear dependence of the diffusion 399
coefficient on the concentration of components. 400
VI. COMPARISON WITH OTHER APPROACHES 401
The mutual diffusion phenomenon has been investigated 402
since Fick proposed his equation [1], but there is no unique 403
theory for its description8[4,5]. The Darken equation9[7]404
is one of the earliest equations which takes into account the 405
concentration dependence of the mutual diffusion coefficient. 406
First, it was proposed for binary metal alloys, but now some 407
modifications of it are used for description of diffusion in 408
liquid solutions as well. 409
Concentration dependencies of tracer diffusion coefficients 410
are needed for application of the Darken equation, but 411
sometimes they cannot be measured directly. In such cases 412
a modified Darken equation is formulated, in which mutual 413
diffusion coefficients at infinite dilution are used instead of 414
tracer diffusion coefficients [68]. The Vignes equation [13]415
is another widely used equation; it is based on Eyring’s 416
kinetic theory and theoretically substantiated by Cullinan [69]. 417
Darken and Vignes equations are the basis for many empirical 418
8For more details, see Appendix B.
9Sometimes it is called the Hartley-Crank equation (Hartley and
Crank derived a similar equation independently in Ref. [67]).
002100-6
NONLINEAR DIFFUSION IN MULTICOMPONENT LIQUID . . . PHYSICAL REVIEW E 00, 002100 (2017)
TABLE I. Comparison of different approaches for acetone-chloroform solution.
Number of Number of
specific nonspecific
Version parametersaparametersbCDAFcARDd(%)
Group 1 (measuring of auxiliary functions is needed)
Moggridge +NRTL [10,11]032×Np2.2
Zhu et al. +NRTL [11]032×Np2.8
Li et al. [8]451×Np3.3
Group 2 (theories with adjustable nonspecific parameters)
UNIDIF [15] 2 6 0 1.5
Modified Darken +UNIQUAC [15] 2 7 0 1.9
Modified Darken +NRTL (version 2) [15] 2 3 0 2.1
Vignes +NRTL [15] 2 3 0 2.1
Vignes +UNIQUAC [15] 2 7 0 2.1
Medvedev and Shapiro [73] 0 8 0 1.4–7.4e
Group 3 (theories without adjustable nonspecific parameters)
Yan et al. +Wilson [70] 4 2 0 2.3
Yan et al. +NRTL [70] 4 3 0 4.2
Zhou et al. +NRTL [14] 2 3 0 5.5
Bosse and Bart +Wilson [16] 2 2 0 6.0
Zhou et al. +Wilson [14] 4 2 0 6.2
Modified Darken +Wilson [70] 2 2 0 17.0
Modified Darken +NRTL (version 1) [14] 2 3 0 17.0
Vignes +Wilson [14] 2 2 0 18.4
Obukhovsky et al. 3 1 0 1.4
aCoefficients of self-diffusion D0
1and D0
2and/or mutual diffusion at infinite dilution D0
12 and D0
21 (material constants).
bAny other parameters (from nondiffusion experiments).
cConcentration-dependent auxiliary function (number of functions ×Np,whereNpis the number of experimental points). Tracer diffusion
coefficients D∗
ior viscosity of mixture, η, can be used as auxiliary functions. In this table, Np=9.
dAverage relative deviation.
eARD depends on the chosen thermodynamic model and the way of the determination of penetration length Z.
equations [5,8,10–12,14–16]. Nowadays, most popular419
modifications are based on the local composition con-420
cept [8,12,14,70]. All these equations contain a thermody-421
namic correction factor, but there is no unique theory for its422
determination [71].423
Hsu and Chan proposed the UNIDIF model [15], which424
combines the lattice theory of liquids and absolute reac-425
tion rate theory. Shapiro proposed the fluctuation theory426
of diffusion [17] based on thermodynamics of irreversible427
processes. The main idea of this approach is the fact that the428
matrix of Onsager phenomenological coefficients is related429
to the product of three matrices: kinetic (which accounts430
for the rates of molecular motion), thermodynamic (which431
is connected with the second order derivatives of the entropy),432
and resistance (which accounts for the resistance to molecular433
motion by other molecules) matrices. Determination of the434
thermodynamic matrix requires knowledge of the equation of435
state. Also it is difficult to determine the resistance matrix from436
first principles [72]; thus, empirical formulas with variable437
numbers of adjustable parameters are used [18,73].438
Calculation of the concentration dependence of the mutual439
diffusion coefficient D12(c1) requires information (e.g., tracer440
diffusivities and activity coefficient) about the investigated441
mixture. Such information can be obtained (1) directly442
from diffusion coefficient data or (2) from other available443
experimental data (for example, the concentration dependence444
of the activity coefficient is often obtained from vapor-liquid 445
equilibrium data [68]). In some theories the concentration 446
dependence of accessory functions (tracer data and, viscosity) 447
is needed [8,10,11]. Thus, all theoretical descriptions of 448
the mutual diffusion coefficients mentioned above can be 449
separated into three groups (see Table Iand Appendix B)450
depending on the presence of auxiliary functions and specific 451
and nonspecific parameters. 452
The acetone-chloroform solution is a popular system for 453
diffusion theories. Below in Table Ithe results [8,10,11,14–454
16,70,73] are processed and classified. 455
In Table Ithe average relative deviation (ARD), 456
ARD =1
Np
Np
i=1
Dexpt
i−Dcalc
i
Dexpt
i
,(41)
was used to define the difference between experimental 457
and theoretical data. Values for ARD were taken from the 458
respective cited papers. For the main formulas of different 459
approaches see Appendix B.460
Insertion of adjustable parameters into theoretical formulas 461
leads to decreasing of ARD. Thus, minimization of ARD can 462
be achieved by using three or four adjustable parameters (in 463
the formula of penetration length Z) in the fluctuation theory 464
of diffusion [73]. But there is no rigorous basis for the formula 465
of penetration length Z, and its choice is quite arbitrary. 466
002100-7
OBUKHOVSKY, KUTSYK, NIKONOVA, AND ILCHENKO PHYSICAL REVIEW E 00, 002100 (2017)
TABLE II. Some approaches to the description of the mutual diffusion coefficient.
Parameters of model
Approach Equation Specific Nonspecific CDPCa
Group 1 (measuring of auxiliary functions are needed)
Moggridge +NRTL [10,11]D12 =(x1D∗
2+x2D∗
1)˜α,˜α≈0.64; =(τ12,τ21 ,¯
a;x1)τ12,τ21 ,¯
aD
∗
1(x1),D
∗
2(x1)
Zhu et al. +NRTL [11]D12 =(x11D∗
2+x22D∗
1)˜α,˜α≈0.64; =(τ12,τ21 ,¯
a;x1)τ12,τ21 ,¯
aD
∗
1(x1),D
∗
2(x1)
Li et al. [8]D=φ22 ¯
V
¯
V2D∗
1+φ11 ¯
V
¯
V1D∗
2;=(12,21 ;x1), D0
12,D
0
21,12 ,
21,η(x1)
D∗
i=D0
iηi
ηn0
i
1+(n0
i−1)xi˜
β,i=1,2D0
1,D
0
2η1,η
2˜
β
n0
1=η2D0
21
η1D0
11/˜
β,n0
2=η1D0
12
η2D0
21/˜
β
˜
β=1/2
Group 2 (theories with adjustable nonspecific parameters)
UNIDIF [15]
ln D12 =x2ln D0
12 +x1ln D0
21 +2x1ln x1
φ1+x2ln x2
φ2
+2x1x2φ1
x11−λ1
λ2+ϕ2
x21−λ2
λ1
+x2q1[(1 −θ2
21)lnτ21 +(1 −θ2
22)τ12 ln τ12 ]
+x1q2[(1 −θ2
12)lnτ12 +(1 −θ2
11)τ21 ln τ21 ]
D0
12,D0
21,
a12,a21
r1,r2
q1,q2
θji =θjτji
lθlτli ,θ
j=xjqj
lxlql,τ
ji =exp −aji
T,
φi=xiλi
lxlλl,λ
i=(ri)1/3,i =1,2
Modified Darken D12 =D0
12x2+D0
21x1;=(τ12 ,τ21,r1,r2,q1,q2,Nc;x1)D0
12,D
0
21 τ12,τ
21,q
1,q
2,N
c
+UNIQUAC [15,68]r1,r
2,
Modified Darken D12 =D0
12x2+D0
21x1;=(τ12 ,τ21,¯
a;x1)D0
12,D
0
21 τ12,τ
21,¯
a
+NRTL(version 2) [15,68]
Vignes +NRTL [13,15]D12 =D0
12x2D0
21x1;=(τ12 ,τ21,¯
a;x1)D0
12,D
0
21 τ12,τ
21,¯
a
Vignes +UNIQUAC [13,15]D12 =D0
12x2D0
21x1;=(τ12 ,τ21,r1,r2,q1,q2,Nc;x1)D0
12,D
0
21 τ12,τ
21,
r1,r
2,q
1,q2,N
c
Medvedev and Shapiro [73]D12 =LDMm
M1M2T1
x1M2
∂ln μ2
∂c2+1
x2M1
∂ln μ1
∂c1,A, B1a1,a
2,
LD=GLTrGT, LTr =1
2LTr +LT
Tr,B2,B
12 b1,b
2
LTr =1
4LKLTLR.L
K,ij =δij 8RT
πMj,
LT,ij =−fij ,f =F−1Fij =∂2S
∂ci∂cj,
Fi,3=F3,i =∂2S
∂ci∂U ,F
3,3=∂2S
∂U2,
LR,ij =δij Z−ci
∂Zi
∂ci,L
R,i3=−ci
∂Zi
∂U ,
Zi=Mi
Mmix A1−B1c1−B2c2−B12 c1c2
c1+c2,(i,j =1,2)
Group 3 (theories without adjustable nonspecific parameters)
Yan et al. +Wilson [70]D12 =x2φ21
D0
12
+φ11
D0
1+x1φ12
D0
21
+φ22
D0
2−1;D0
12,D
0
21,12 ,
21
=(12,21 ;x1)D0
1,D
0
2
Yan et al. +NRTL [70]D12 =x2φ21
D0
12
+φ11
D0
1+x1φ12
D0
21
+φ22
D0
2−1
;D0
12,D
0
21,τ12 ,τ
21,¯
a
=(τ12,τ21 ,¯
a;x1)D0
1,D
0
2
Zhou et al. +NRTL [14]D12 =D0
12¯
Vφ
22/¯
V2D0
21¯
Vφ
11/¯
V1;=(τ12,τ21 ,¯
a;x1)D0
12,D
0
21 τ12,τ
21,¯
a
Zhou et al. +Wilson [14] D12 =D0
12¯
Vmφ22/¯
V2D0
21¯
Vmφ11/¯
V1;=(12,21 ;x1)D0
12,D
0
21 12,21
Bosse and Bart +Wilson [16]D12 =D0
12x2D0
21x1e−GE
RT ;GE=GE(12,21 ;x1); D0
12,D
0
21 12,
21
=(12,21 ;x1)
Modified Darken D12 =D0
12x2+D0
21x1;=(12 ,21;x1)D0
12,D
0
21 12,
21
+Wilson [68,70]
002100-8
NONLINEAR DIFFUSION IN MULTICOMPONENT LIQUID . . . PHYSICAL REVIEW E 00, 002100 (2017)
TABLE II. (Continued.)
Parameters of model
Approach Equation Specific Nonspecific CDPCa
Modified DarkenbD12 =D0
12x2+D0
21x1;=(τ12 ,τ21,¯
a;x1)D0
12,D
0
21 τ12,τ
21,¯
a
+NRTL (version 1)c[14,68]
Vignes +Wilson [13,14]D12 =D0
12x2D0
21x1;=(12 ,21;x1)D0
12,D
0
21 12,
21
aConcentration-dependent physical characteristic. Such functions are determined experimentally without theoretical description in correspond-
ing cited works.
bOriginal Darken equation is D12 =(D∗
1x2+D∗
2x1).D∗
iis the tracer diffusivity of the ith component (it depends on the concentration of the
solution), but it is often replaced by a value at infinite dilution (which leads to the modified Darken equation).
cThe modified Darken + NRTL (version 1) model differs from the modified Darken + NRTL (version 2) model only by the way of extraction
of interaction parameters a12,a21 in the thermodynamic factor model. They may be regressed from mutual diffusion data (as in version 2) or
obtained from other experimental data (as in version 1).
As follows from Table I, our theory produces good results.467
At the current stage of developing this model, the material468
parameters b12,b13 , and b23 cannot be determined from469
independent data. But they can be obtained using diffusion470
characteristics at infinite dilution (see Appendix A). Thereby,471
there are no adjustable parameters in this model.472
VII. CONCLUSIONS473
The proposed approach can be applied to the description474
of mutual diffusion in nonideal liquid solutions. Based on the475
obtained results, we can conclude the following:476
(a) Intermolecular interactions between mixture compo-477
nents can be taken into account as a formation of molecular478
complexes. Thus an initially binary (before mixing) solution479
has to be modeled as a multicomponent system (three or more480
components).481
(b) For acetone-chloroform mixtures, the three-component482
model can be successfully used to explain the specifics of the483
coefficient of mutual diffusion.484
(c) The concentrations of the complexes are needed to485
explain the nonlinearity of diffusion. These values can be486
obtained from other experiments (optical spectroscopy and487
NMR) or calculated from experimental mutual diffusion data.488
(d) The proposed theory could be generalized for other489
multicomponent mixtures (two types of complexes or more).490
ACKNOWLEDGMENT491
The authors are grateful to Professor Yu. M. Volovenko492
for discussion of problems of complex formation in molecular493
liquids.494
APPENDIX A495
The case of the simplest complex (1:1) formation is496
considered below. It corresponds to n=1, m=1inEq.(19).497
Experimentally measured data for Dexpt
1and their derivatives498
∂Dexpt
1/∂ϕtot
1at the limiting points10 (ϕtot
1=0, or ϕtot
2=0) may499
be used for determination of the unknown parameters b12,b13 ,500
10Note that the volume conservation law in Eqs. (10)and(24) should
be kept.
b23, and Kϕ. The following system of four equations can found 501
from Eqs. (26) and (39): 502
B1≡Dexpt
12 ϕtot
1=0=b12 +(Kϕ¯α1)b23
1+(Kϕ¯α1),(A1)
B2≡Dexpt
12 (ϕtot
2=0) =b12 +(Kϕ¯α2)b13
1+(Kϕ¯α2),(A2)
B3≡dDexpt
12
dϕtot
1ϕtot
1=0
=+2(Kϕ¯α1)(1 +Kϕ)(b12 −b23)
(1 +(Kϕ¯α1))3,(A3)
B4≡dDexpt
12
dϕtot
1ϕtot
2=0
=−2(Kϕ¯α2)(1 +Kϕ)(b12 −b13)
(1 +(Kϕ¯α2))3.(A4)
The solution of Eqs. (A1) and (A2) allows the determination 503
of all the necessary parameters: bnm =bnm(B1,B2,B3,B4). 504
Thus, in this case, the theory which uses a three-component 505
model of the liquid solution does not have any adjustable 506
parameters. The problem of calculation accuracy is still 507
open, because a big step in concentration change (typical for 508
experimental conditions) may produce big uncertainty of some 509
coefficients (in particular, Kϕ). 510
To improve the precision of the calculation, data from 511
independent experiments11 can be used, as it was demonstrated 512
above for the case of acetone-chloroform solutions. 513
APPENDIX B 514
Table II represents some of the most widely used ap- 515
proaches for description of the concentration dependence of 516
the mutual diffusion coefficient in binary liquid mixtures. 517
Many of them require additional thermodynamic models for 518
determination of excess Gibbs energy GEand, thus, for 519
determination of the thermodynamic factor . These models 520
are represented in Table III. Classification in Table II was made 521
not only by the model of mutual diffusion used but also by 522
11Constant Kϕcan be obtained, for example, from NMR or optical
experiments.
002100-9
OBUKHOVSKY, KUTSYK, NIKONOVA, AND ILCHENKO PHYSICAL REVIEW E 00, 002100 (2017)
TABLE III. Some thermodynamic models of the solution.
Model Excess Gibbs energy, GEParameters
Wils on [74]GE
RT =−k
i=1xiln k
j=1xjij ,12,21
ii =jj =1
NRTL [75]GE
RT =k
i=1xik
j=1τjiGjixj
k
l=1Glixl,τ12 ,τ21,¯
a
Gij =exp(−¯
aτij )τ12,τ21,¯
a
UNIQUAC [76]
GE
RT =ixiln φi
xi
−NC
2xiqiln φi
θi−xiqiln kθjτjiτ12,τ
21,
τii =τjj =1,θ
i=xiqi
xiqi+xjqj=xiqi
qr1,r
2,q
1,
φi=xiri
xiri+xjrj=xiri
rq2,N
C=10
the thermodynamic models used. Additionally, classification523
of the required parameters of the model was made. All524
parameters were divided in two parts: specific and nonspecific.525
Specific parameters of the model can be determined from526
diffusion experiments only. Nonspecific parameters may be527
determined from other available experiments. Some models528
need additional values of physical quantities which depend 529
on solution concentration (e.g., viscosity of mixture); such 530
types of quantities are aggregated in the CDPC column of 531
Table II.532
The thermodynamic factor is connected with excess 533
Gibbs energy in the following way: 534
=1+x1x2
RT ∂2GE
∂x2
1
+∂2GE
∂x2
2
−2∂2GE
∂x1∂x2.(B1)
Local composition models (Wilson, NRTL,UNIQUAC)are 535
widely used for description of the thermodynamic factor in 536
mutual diffusivity equations. These models do not have rigor- 537
ous theoretical backgrounds and are semiempirical [71], but 538
incorporation of more advanced thermodynamics models leads 539
to significant complication of expressions for the description of 540
mutual diffusion coefficients. Thermodynamic models, which 541
are regarded in Table II, are represented in Table III. Parameters 542
12 and 21 (Wilson model) and τ12 and τ21 (NRTL and 543
UNIQUAC models) represent interaction between the mixture 544
components and are regressed from experimental data (for 545
more details see Ref. [71]). 546
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