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570 Hermann Weing¨artner

these cross-correlation functions depend on the reference frame. As deﬁned

here, they depend on the barycentric frame.

To formulate this observation quantitatively, we deﬁne now a distinct-

diﬀusion coeﬃcient for the a–b interaction by [18,33–36]

∞



v

a,α

(0) · v

b,β

dt, (14.32)

where the superscripts α and β indicate that the velocities belong to distinct

particles even if a = b. N is the total number of particles in the system. Note

that, originally, Mills and Hertz [34], adapting an older approach by McCall

and Douglass [37], have used so-called velocity cross-correlation coeﬃcients

related to the coeﬃcents used here by

= D

. (14.33)

In a binary system of components 1 and 2 the deﬁnition used here leads to

four coeﬃcients for 1–1, 2–2, 1–2 and 2–1 interactions, which by symmetry

satisfy the relation

= D

. (14.34)

This symmetry is not shown by the f

. Using (14.32), we see that one can

formally split the L

into self- and distinct contributions according to [18,36]



for a = b

+ L

for a = b

(14.35)

with the deﬁnitions





(14.36)

and





. (14.37)

If these expressions are combined with (14.19), one obtains a set of equa-

tions which allows to determine the three distinct-diﬀusion coeﬃcients D

and D

= D

from the three experimental quantities D

, D

and D

= −

+ x

)

, (14.38)

= −x

− x





, (14.39)

= −x





− x

. (14.40)

14 Mutual Diﬀusion in Mixtures 571

Finally, these equations may be combined to a single one for the mutual-

diﬀusion coeﬃcient

−1

= x

+ x

+ D

− 2D

), (14.41)

but now not all coeﬃcients on the right hand site of this equation are inde-

pendent.

It may be noted that it is straightforward, albeit tedious, to extend this

type of analysis to multi-component systems and to electrolyte solutions.

We omit a detailed discussion of the multi-component case here, and refer

to a paper of Zhou and Miller [38] for a comprehensive and mathematically

compact survey of the resulting expressions.

Of greater interest in the present context is the application of the concept

of distinct-diﬀusion coeﬃcients (or, in Hertz’s notation, velocity correlation

coeﬃcients) to transport processes in electrolyte solutions. This type of work

has been initiated in 1978 by Hertz [39]. Later, the interconnection with

the Onsager coeﬃcients has been worked out in a particularly lucid way by

Miller [40]. Again, the mathematical formalism is quite tedious, and it serves

no purpose to give here details.

The basic procedure is to deﬁne distinct-diﬀusion coeﬃcients for the pair-

wise interactions of cations, anions and water and to relate these to the

ionic l

. Just as in the binary non-electrolyte case the l

are then separated

into self- and distinct contributions. The self-contributions are experimentally

available, because the self-diﬀusion coeﬃcients can be measured for cations,

anions and water. The l

are obtained from conductance, transference and

diﬀusion data, as described earlier. Eventually, one ends up with six distinct-

diﬀusion coeﬃcients. Three coeﬃcients, D

, D

and D

, reﬂect pairwise

interionic interactions between like and dislike ions, respectively. Two further

coeﬃcients, D

and D

, characterize cation–water and anion–water inter-

actions, thus reﬂecting hydration properties. Finally, there is a further coef-

ﬁcient D

characterizing water–water interactions. This type of approach

is quite fascinating, because modern electrolyte theory allows to model these

coeﬃcients, not only in dilute solutions, where the famous Debye-H¨uckel laws

apply [40, 41], but also for concentrated solutions [41].

14.7 Properties of Distinct-Diﬀusion Coeﬃcients

It is seen that mutual diﬀusion is related to velocity cross-correlations be-

tween diﬀerent particles in the same way as self-diﬀusion is related to velocity

auto-correlations. This reﬂects the collective nature of the mutual diﬀusion

process. Moreover, one can reformulate the deﬁnition of the distinct-diﬀusion

coeﬃcients by replacing velocities by displacements of particles. If this is done

correctly, one obtains

= N lim

t→∞

[r

a,α

(t) − r

a,α

(0)] · [r

b,β

(t) − r

b,β

(0)] (14.42)

572 Hermann Weing¨artner

which is the direct equivalent to the Einstein relation (14.1) for self-diﬀusion.

This relation or the equivalent relation (14.32) form the recipe to extract

information on mutual diﬀusion processes from molecular dynamics simula-

tions. In fact, the recent literature has seen some attempts to study along

these lines velocity cross correlations in simple model systems such as mix-

tures of Lennard-Jones particles. We quote for example the work of Padro

and coworkers [42–44]). First attempts of such studies in adsorbate-adsorbent

systems are described in Chap. 23.

Moreover, the application of basic scattering theory yields [35]

= Nπ lim

κ→0

lim

ω→0

(κ, ω), (14.43)

where S

(κ, ω) is the so-called distinct part of the partial dynamic struc-

ture factor, which is responsible for coherent neutron scattering [6] (see also

Chap. 3). Thus, the distinct-diﬀusion coeﬃcient is related to the distinct part

of the partial dynamic structure factor in the same way as the self-diﬀusion co-

eﬃcient is related to the self-part, which justiﬁes the name ‘distinct-diﬀusion

coeﬃcient’.

What are the major features of these coeﬃcients? Experiments show that,

in contrast to self-diﬀusion coeﬃcients, the distinct-diﬀusion coeﬃcients can

be negative. This is most obvious for the coeﬃcient D

which reﬂects corre-

lations between dislike molecules. Here, the minus sign in (14.38) implies that

is negative by deﬁnition, because all other quantities in this equation are

positively deﬁned. On the other hand, it depends on the relative weight of

the self- and distinct terms in (14.39) and (14.40), whether the coeﬃcients for

interactions between like molecules, D

and D

, are positive or negative.

Finally, it should be emphasized again that distinct-diﬀusion coeﬃcients

depend on the chosen reference frame, and the approach used here and in

most other studies of non-electrolyte systems adopts the mass-ﬁxed frame.

However, formulations in other reference frames are now available. A com-

parative analysis of results formulated in diﬀerent frames has been given by

Mills et al. [16]. One may argue that, to some extent, the choice of the ref-

erence frame is a matter of taste. Actually, the comparison with results of

statistical-mechanical calculations suggests, however, the mass-ﬁxed frame

to be the most convenient choice. With electrolyte solutions, the situation is

somewhat diﬀerent. While in the early work the mass-ﬁxed frame has been in

use, it seems that the description of interionic correlations is best performed

in a frame with the solvent being at rest [41].

An interesting idea is, of course, to look for reformulations and combina-

tion of the coeﬃcients, which are independent of the reference frame. In fact,

Trullas and Padro [43] have introduced a set of time correlation functions

based on relative mean molecular velocities that satisfy this requirement.

14 Mutual Diﬀusion in Mixtures 573

14.8 Information on Intermolecular Interactions

Deduced from Diﬀusion Data

To illustrate the formalism, we ﬁrst discuss diﬀusion in ideal systems. To

sharpen the interpretation of diﬀusion data, it is suitable to assess the data

for real systems relative to what is expected for ideal systems where spe-

ciﬁc chemical interactions are absent. This procedure corresponds clearly to

that in equilibrium thermodynamics, where phenomenological laws for ideal

mixtures are used for interpreting data for real mixtures.

In fact, many attempts have been made for predicting ideal diﬀusion

behavior [1]. From these, Darken’s relation has already been mentioned. This

relation is usually considered to represent a simple empirical mixing rule. The

formalism introduced above allows to examine under which approximations

this relation evolves from rigorous theory. It immediately follows from (14.41)

that in this case

+ D

− 2D

=0. (14.44)

Thus, the Darken relation implies that the cross-correlations between mole-

cules of diﬀerent type, 1–2, just compensate the cross-correlations between

molecules of the same type, 1–1 and 2–2. This situation is conceptually analo-

gous to that in equilibrium thermodynamics, where Raoult’s law implies that

eﬀects of 1–2 interactions just compensate those of 1–1 and 2–2 interactions.

Zhou and Miller [38] have discussed the role of some other simple relations

and their extension to multi-component ﬂuids.

To interpret diﬀusion in real systems we re-emphasize that the distinct-

diﬀusion coeﬃcients are measures for the correlation of the velocities of dis-

tinct particles. If we turn on an attractive interaction between these par-

ticles, they will tend to stick together and move in the same direction, so

that their velocities are positively correlated over a long time. In the lat-

ter case one expects positive distinct-diﬀusion coeﬃcients or at least values

that are more positive than expected for ideal systems. In contrast, repul-

sive interactions may cause the distinct-diﬀusion coeﬃcients to become more

negative than found for ideal systems. This scheme, introduced here by intu-

itive arguments, is supported by some more elaborate theories, which indeed

predict the distinct-diﬀusion coeﬃcients to be negative for ideal systems and

to become positive in the case of speciﬁc attractive interactions [19,33]. Such

arguments apply to both non-electrolyte and electrolyte systems.

We postpone a discussion of electrolytes and consider ﬁrst some studies of

binary non-electrolyte mixtures. From the line of arguments given above one

can deduce a simple prediction for direction of the deviations from Darken’s

relation: If molecules of the same species have more tendency to diﬀuse to-

gether than those of diﬀering species, we expect D

+ D

> 2D

, leading

to positive deviations from Darken’s relation. The reverse is true for systems

with speciﬁc interactions between diﬀering species.

574 Hermann Weing¨artner

150

0.1

0.2 0.3

1

0.1

0.2

0.3

100

0.4









0.4

10D/ms

1

10D/ms

1

Fig. 14.6. Distinct-diﬀusion coeﬃcients in the system methanol + carbon tetra-

chloride at 298.15 K. Methanol is component 1.

We illustrate this idea by considering in Fig. 14.6a and Fig. 14.6b some

data for solutions of methanol in carbon tetrachloride [27]. D

and D

are

negative, while D

is large and positive. If Darken’s relation is built into the

system of (14.38), (14.39), and (14.40), D

turns out to be negative. It is

obviously the strong methanol-methanol self-association in this system which

causes the velocities of methanol molecules to be correlated over long times.

Thus, strong methanol-methanol correlations are responsible for the peculiar

diﬀusion behavior in this system [27].

Meanwhile, a lot of similar investigations may be found in the literature,

and only some selected examples can be quoted here. With regard to compar-

atively simple systems, mixtures of hydrocarbons [4,16,17] have been consid-

ered. Already when going to mixtures of hydrocarbons with ﬂuorocarbons,

quite complex behavior is observed, in accordance with known thermody-

namic peculiarities of these mixtures [17]. Much attention has been given

to aqueous solutions of hydrophobic molecules [45–47], where hydrophobic

eﬀects give rise to very subtle behavior of the distinct-diﬀusion coeﬃcients.

Pronounced temperature eﬀects are notable in these systems [16]. Recently

attempts have also been made to extend this sort of analysis to polymer–

solvent systems [48].

Turning to electrolyte solutions, aqueous solutions of simple alkali halides

or alkaline earth halides have been investigated some time ago [49], indicat-

ing the behavior of highly dissociated salts. More interesting is certainly the

application to systems with strong interionic interactions. For example, there

are special problems in modelling the thermodynamic and transport coeﬃ-

cients in solutions of some salts containing two-valent ions such as ZnSO

or Na

, where one expects signiﬁcant electrostatic ion pairing. How-

ever, little is known on the fate of these ion pairs in concentrated solutions.

Comprehensive studies of transport properties and their analysis in terms of

distinct-diﬀusion coeﬃcients have given much insight into the details of inte-

rionic interactions in such systems, where at high salt concentrations there is

14 Mutual Diﬀusion in Mixtures 575

strong evidence for attractive like-ion interactions, predicted by statistical-

mechanical model calculations [50,51].

While all these studies have provided a quite coherent picture of the eﬀect

of attractive and repulsive interactions on distinct-diﬀusion coeﬃcients, there

is, of course, the pressing question, whether predictive theories can be devel-

oped. For electrolyte solutions this is a current problem of modern electrolyte

theory, and there is indeed some obvious success in applying theories to the

problems of interest [41]. With regard to non-electrolytes, molecular trans-

port theories for systems involving strong intermolecular interactions are less

well established. One promising approach has come from hydrodynamic the-

ory. Friedman and Mills [33] showed by hydrodynamic arguments that the

distinct-diﬀusion coeﬃcients D

are proportional to an integral over the pair

correlation function g

that describes the structure of the solution

2kT

3η

∞



(r) − 1] r dr, (14.45)

where η is the shear viscosity, and all other symbols have their usual mean-

ings. This simple relation predicts already some crucial features of the ex-

perimental behavior of distinct-diﬀusion coeﬃcients, namely negative con-

tributions due to excluded volume eﬀects at short distances and positive

contributions from large peaks in g

(r). Also, the factor r in the integrand

tends to increase the contributions at larger separations so that correlations

over comparatively large distances may aﬀect diﬀusion behavior.

This type of analysis appears to be particularly apt, when noting that the

well-known Kirkwood-Buﬀ theory of equilibrium thermodynamics enables the

determination of interaction coeﬃcients

∞



(r) − 1] 4πr

dr, (14.46)

denoted as Kirkwood-Buﬀ coeﬃcients. This integral is suﬃciently similar

to the integral appearing in the hydrodynamic expression (14.45) for the

distinct-diﬀusion coeﬃcient that one may try simple models for the dis-

tribution function g

(r) to interrelate the two integrals. This type of ap-

proach, discussed in some detail by Bender and Pecora [19], seems particularly

promising for predictive purposes. It allows to tune the parameters of a model

distribution function g

(r) to reproduce thermodynamic properties. Subse-

quently, these parameters are then used to predict diﬀusion coeﬃcients. In

fact, it was noted [18] that there is a close connection between Kirkwood-Buﬀ

integrals and distinct-diﬀusion coeﬃcients even in the case of methanol-CCl

mixtures, where particularly strong interactions are present.

On a somewhat more empirical level, one could adopt another well-known

approach from the thermodynamics of mixtures. If intermolecular interac-

tions are strong, it is often proﬁtable to presume in a chemical model the

576 Hermann Weing¨artner

formation of distinct aggregates. For example, a binary system of compo-

nents A and B with strong A-B interactions is modelled as ternary mixture

of A, B and the complex AB, subject to a mass action equilibrium A + B

 AB. By treating the ternary model system as a thermodynamically ideal

system, the deviations from ideality in the real system can be accounted for

by an equilibrium constant. The possibility to extend such chemical models

for dealing with transport properties is evident. In fact, this is a common

approach in the analysis of conductance data for systems exhibiting ion pair-

ing. Comparatively simple approaches have been made to treat diﬀusion in

non-electrolyte mixtures along such lines as well [1, 52], but such ideas still

await a more rigorous analysis, ﬁrmly based in statistical mechanics.

14.9 Conclusion

The mutual diﬀusion process is strongly inﬂuenced by cooperative motions

between distinct particles. Linear response theory allows to develop a formal-

ism in which these cooperative motions are described by a set of coeﬃcients

for the various pairwise interactions in the mixture. As a result, the exper-

imental diﬀusion coeﬃcients D

, D

and D

can be rewritten in terms of

these interaction coeﬃcients, denoted as distinct-diﬀusion coeﬃcients. It is

the speciﬁcity of the distinct-diﬀusion coeﬃcients to the various interactions

which makes them particularly useful for analyzing diﬀusion data. It is now

obvious that diﬀusion theories on the molecular level should be formulated

in terms of the distinct-diﬀusion coeﬃcients. In the future, predictive equa-

tions may result from the incorporation of hydrodynamic theories into the

formalism. Some promising attempts in this direction are already available

from modern electrolyte theory.

Notation

activity of component i

volume concentration of component i

D self-diﬀusion coeﬃcient of pure liquid

self-diﬀusion coeﬃcient of component i in a mixture

mutual-diﬀusion coeﬃcient in the laboratory-ﬁxed and volume-

ﬁxed reference frame

mutual-diﬀusion coeﬃcient in the mass-ﬁxed reference frame

distinct-diﬀusion coeﬃcient for a–b interaction

∗

thermodynamic diﬀusion coeﬃcient

velocity cross-correlation coeﬃcient for a–b interaction

, p

) N-particle correlation function

classical Hamiltonian of the N-particle

ext

perturbation part of the Hamiltonian

14 Mutual Diﬀusion in Mixtures 577

ﬂux density of component i in the cell-ﬁxed frame

ﬂow of component i

ﬂow of component i in the mass-ﬁxed frame

L Liouville-Operator

phenomenological coeﬃcient of the linear transport equations

mass of particle of type i

molecular mass of component i

N total number of particles

Q thermodynamic factor

i,α

(t) location of particle α of component i at time t

(κ, ω) distinct part of the partial dynamic structure factor

(κ, ω) self-part of the partial dynamic structure factor

t time

T temperature

i,α

(t) velocity of particle α of component i at time t

partial molal volume of component i

mole fraction of component i

thermodynamic driving force

chemical potential for component i

∗

speciﬁc chemical potential of component i

partial mass density of component i

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15 Diﬀusion Measurements in Fluids

by Dynamic Light Scattering

Alfred Leipertz and Andreas P. Fr¨oba

15.1 Introduction

Diﬀusion measurements in condensed matter mainly stand for the accurate

determination of transport properties, e. g., of ﬂuids. Due to diﬀerent diﬀusion

processes, these properties are the thermal diﬀusivity a (often also denoted

), the mutual diﬀusion coeﬃcient D

, the sound attenuation D

and the

particle diﬀusion coeﬃcient D

which is directly related to the dynamic vis-

cosity η. Conventional measurement techniques make use of macroscopic gra-

dients of temperature, concentration, pressure, or ﬂuid velocity which must

be large enough to give rise to a measurable eﬀect but small enough to cause

only very little perturbation in the system under investigation. This diﬃculty

is strongly enhanced in the vicinity of critical points where induced gradi-

ents may make any reasonable measurement impossible. However, informa-

tion collected in these regions is of great importance both for a fundamental

understanding of critical phenomena and for setting up correlations for the

individual properties of a substance.

A very successful approach to overcome these limitations is the applica-

tion of light scattering techniques [1–4]. A fundamental advantage of these

methods is given by the fact that these may be used in or – in some in-

stances – close to thermodynamic equilibrium. They make use of microscopic

statistical ﬂuctuations of the local electric susceptibility χ

which may be

related to various diﬀusive processes and thus to the transport properties to

be measured. The basic justiﬁcation for this relation is Onsager’s regression

hypothesis [5], which states that, on a statistical average, microscopic ﬂuctu-

ations are governed by the very same macroscopic transport equations and

which has been conﬁrmed also by light scattering investigations.

Here, an introduction is given to dynamic light scattering (DLS) as a

valuable tool for the measurement of diﬀusion processes. Additionally, var-

ious applications are presented, which especially are related to the deter-

mination of transport and other thermophysical properties in liquids. The

subsequent Chap. 16 shall be devoted to the application of this technique

to colloidal and polymeric systems. In dynamic light scattering (often also

denoted as quasielastic light scattering or light beating spectroscopy) the

temporal behavior of light is analysed. This chapter focuses on “classical”

photon correlation spectroscopy (PCS), using correlation techniques in the