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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

Phys.Rev.E. v.95, #2, 0022133, 1Feb. 2017.

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)

002100-3

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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