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710 Gerhard N¨agele, Jan K. G. Dhont, and Gerhard Meier

scattering (FRS), where a refractive index grating is created by means of an

interference pattern, from which grating scattered intensities are measured

(see Sect. 15.5.2 in Chap. 15). FRAP and FRS are specialized to measure long-

time translational (and rotational) self-diﬀusion coeﬃcients of various kinds

of macromolecules (also non-spherical molecules). As far as we are aware,

FCS has not yet been applied to synthetic macromolecular systems. For very

large macromolecules (larger than the dimensions of the confocal volume),

FCS probes a combination of translational and rotational motion.

The concentration dependence of translational/rotational diﬀusion coeﬃ-

cients characterizing the various diﬀusion mechanisms, of sedimentation co-

eﬃcients, and the time dependence of mean square displacements, Van Hove

functions and dynamic structure factors have been analyzed for colloidal sus-

pensions in Sect. 16.5. The performances of the theoretical methods which

have been described in this section (i.e., short-time cluster expansion, MCT,

and BD and SD simulations) have been scrutinized through a detailed com-

parison with experimental results for diﬀusional properties obtained from

DLS, DDLS, FRAP and FCS measurements. Our focus was geared towards

qualitative diﬀerences in the dynamics of suspensions with short-range par-

ticle interactions (i.e., hard spheres), and of systems with long-range repul-

sion of particles like charge-stabilized dispersions and magnetic particles with

dipolar repulsion. In particular, it was shown that charge-stabilized suspen-

sions at low salinity are characterized by peculiar non-linear density depen-

dencies of their (short-time) transport properties. For long times, dynamic

scaling is observed, with interesting implications on the equivalence of certain

static and dynamic freezing criteria.

For systems of monodisperse colloidal spheres in the ﬂuid phase, mean-

while a quantitative level of understanding has been reached, at least with

regard to translational diﬀusion. As yet, a ﬁrst-principle inclusion of many-

body HI into the theoretical description of concentrated suspensions remains

as a major theoretical challenge. For mixtures of colloidal spheres, a semi-

quantitative level of accuracy has been reached in existing theoretical meth-

ods. Far less is known, however, about diﬀusion in systems of non-spherical

particles like, to mention the most simple case, colloidal hard rods. In these

systems, translational particle motion is coupled to the rotational one even at

short times. Recent theoretical eﬀorts are devoted to understand diﬀusion in

conﬁned geometries like in colloidal matrices and gels, to electrokinetic eﬀects

on charged colloidal macro-ions arising from the dynamics of the neutralizing

micro-ion clouds, and to anisotropic diﬀusion in suspensions under shear.

Our conclusions regarding the dynamics in binary polymer blends in the

homogeneous one-phase region are as follows: The derivation of the kinetic

interdiﬀusion coeﬃcient on the basis of the two-component dynamic RPA

assumes an incompressible system. This is probably a rather crude assump-

tion since, if it were true, theory would predict a single relaxation process

in the time evolution of density autocorrelation function. However, a single

16 Diﬀusion in Colloidal and Polymeric Systems 711

mode is usually not observed in DLS experiments on binary blends, although

in some cases the slow-mode result is nevertheless found to be valid. On the

other hand, if the fast-mode model of interdiﬀusion for a compressible system

would apply, two modes should be experimentally observable [28]. Yet in the

siloxane model system where the fast-mode form holds true, a second mode

has never been observed. The ANK formula constitutes certainly a signiﬁcant

progress in our understanding of polymeric interdiﬀusion, but the conﬂict-

ing results between experiment and theory regarding the observed scattering

modes remain to be understood and resolved in future work.

Notation

a sphere radius (polymer segment length)

short-time collective diﬀusion coeﬃcient

long-time collective diﬀusion coeﬃcient

translational short-time self-diﬀusion coeﬃcient

translational long-time self-diﬀusion coeﬃcient

rotational short-time self-diﬀusion coeﬃcient

interdiﬀusion coeﬃcient

) collective long-time diﬀusion coeﬃcient at q

) mean collective diﬀusion coeﬃcient at q

0α

center-of-mass diﬀusion coeﬃcient of an α-type polymer

translational free diﬀusion coeﬃcient

rotational free diﬀusion coeﬃcient

(t) self-dynamic Van Hove function

(r, t) distinct Van Hove function

(t) electric ﬁeld autocorrelation function

H(q) hydrodynamic function

(t) interdiﬀusion current

M mass of a Brownian sphere

irr

(q, t) irreducible collective memory function

irr

(q, t) irreducible self memory function

number of α-type spheres (monomers)

polymerization index of α-type homopolymers

(q, t) collective dynamic structure factor

(q) static structure factor

(q, t) interdiﬀusion autocorrelation function

(t) rotational autocorrelation function

(q, t) self-dynamic structure factor

U sedimentation velocity

W (t) mean square displacement

kinetic factor of interdiﬀusion divided by geometric factor

γ friction coeﬃcient

kinetic factor of interdiﬀusion

712 Gerhard N¨agele, Jan K. G. Dhont, and Gerhard Meier

ρ number density

Φ volume fraction

(t) velocity autocorrelation fuction

volume fraction of A monomers

(t) angular velocity autocorrelation function

χ Flory-Huggins parameter

Flory-Huggins parameter at the spinodal

BD Brownian dynamics (simulation)

DDLS depolarized dynamic light scattering

DLS dynamic light scattering

EACF electric ﬁeld autocorrelation function

FCS ﬂuorescence correlation spectroscopy

FRAP ﬂuorescence recovery after photobleaching

GSE generalized Smoluchowski equation

HI hydrodynamic interactions

IACF intensity autocorrelation function

MCT mode coupling theory

MSD mean square displacement

RPA random phase approximation

SD Stokesian dynamics (simulation)

VAF velocity autocorrelation function

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17 Field-Assisted Diﬀusion Studied by

Electrophoretic NMR

Manfred Holz

17.1 Introduction

Nuclear magnetic resonance (NMR), a low-energy radiofrequency (rf) spec-

troscopic technique, provides a huge number of diﬀerent experimental meth-

ods allowing energy, time and spatially resolved measurements [1,2] (see also

Chaps. 9 and 10). In the present chapter an introduction into an NMR method

shall be given, which allows the study of transport properties of distinct elec-

trically charged species in condensed matter, mainly in the liquid state. Typ-

ically, those charged species can be certain simple, monoatomic ions like Li

, or polyatomic ions as, e.g., alkylammonium ions, or charged macro-

molecules, like polyelectrolytes as well as charged molecular aggregates, like

micelles.

The most interesting transport quantity of charged particles is their mo-

bility u

,measuredinm

−1

, that is their drift velocity in an electric

ﬁeld of unit ﬁeld strength. But how is this transport quantity related to diﬀu-

sion, the main subject of this book? In a liquid, the ions share in the general

Brownian motion and have randomly-directed instantaneous velocities of the

order of 10

−1

, of course with an extremely short mean free path. In an

electric ﬁeld of say 10

−1

, the velocity of ions in the ﬁeld direction is

of the order of 10

−4

to 10

−6

−1

resulting in a very small perturbation

of the random ionic motions and this represents only a small biasing of the

Brownian motion in a given direction. We can speak of a “directed diﬀusion

process” (similar as in the case of mutual or transport diﬀusion in a concen-

tration gradient) or of “ﬁeld-assisted diﬀusion”. So, it is understandable that

the ionic self-diﬀusion coeﬃcient D

and the mobility u

must be somehow

related and this relation can be expressed at least for the limiting quantities

at zero ion concentration D

∗

and u

∗

by the well-known “Einstein relation”

∗

= RT

∗

≡ k

∗

, (17.1)

where z is the ion valency, F is the Faraday constant, R is the gas constant

and T is the temperature and where the equivalence of the second and third

terms results from the fact that R ≡ k

and F ≡ eN

, with the Avogadro

constant N

and the elementary charge e. However, it should be pointed out

718 Manfred Holz

that self-diﬀusion belongs to the class of incoherent motions, whereas the

uniform migration of ions in the ﬁeld direction is a coherent motion, a fact,

which will later be reﬂected in the NMR experiment. The distinction between

coherent and incoherent motion is a topic, which is extensively discussed also

in Chaps. 3, 13, and 14.

The most famous classical methods for the measurement of mobilities

and transference numbers T

of ions are the “moving boundary” and the

“Hittorf” method (see, e. g., [3]). However, these methods fail or are connected

with great diﬃculties in the case of porous media and in the case of mixtures

of electrolytes, where the mobility of a distinct kind of ions is the quantity

of interest. The simplest electrolyte solution contains three constituents, the

anions, the cations and the solvent. Thus, mixed electrolyte solutions consist

of a markedly higher number of components and are typical multi-component

mixtures. Consequently, a highly selective method of observation is required

for obtaining information about a distinct kind of species in the solution.

Such a selective method is NMR, since one can observe selected nuclides

as, e.g.,

Li,

Si,

133

Cs etc. or even the resonance lines of the

same nucleus, but with diﬀerent chemical shifts according to diﬀerent chemi-

cal species or groups [1,2]. This frequency (energy) resolved, selective observa-

tion technique can be combined with spatially resolving pulsed ﬁeld gradient

(PFG) NMR spin-echo experiments [4,5], allowing diﬀusion studies (as out-

lined in Chap. 10) and ﬂow studies [6, 7]. There are further advantages of

NMR, namely (i) the measurement is non-invasive and thus non-destructive,

(ii) low-energy electromagnetic ﬁelds are applied, which do not disturb the

thermal equilibrium of the system and (iii) the systems under investigation

can be solid or liquid and there is no need for optical permeability.

Taking all these advantages into consideration it was tempting to use

PFG NMR ﬂow measurement techniques to observe the slow drift velocity

of distinct charged species in an external electric ﬁeld E. In 1982, in the

present author’s group, the ﬁrst successful experiment of this kind could be

performed, namely the measurement of the drift velocity of (C

)

ions

in water [8]. This new type of experiment thus represented a combination

of in-situ electrophoresis with the selectivity of NMR spectroscopy. From

1982 until today a steady methodical and instrumental development in this

ﬁeld of NMR occurred [9]. Johnson and co-workers introduced the name

“electrophoretic NMR” (ENMR) [10], where the term “electrophoretic” is

used in a general sense for the migration of all kinds of charged particles

in an electric ﬁeld. In general, for all types of NMR experiments, which are

performed in presence of an electric direct current (DC) in the sample under

investigation, the abbreviation DCNMR [9] is used.

Now, DCNMR experiments are performed in a number of laboratories,

but we cannot yet speak of an NMR routine experiment, due to the fact that

the ﬁeld is still under development and that the required accessory units for

these experiments are not yet commercially available. On the other hand,

17 Field-Assisted Diﬀusion Studied by Electrophoretic NMR 719

the potential of ENMR has been clearly demonstrated [9,11] and these novel

NMR techniques can ﬁnd many ﬁelds of interesting applications, in particular

in electrochemistry, biochemistry and in technical sciences.

In the following sections we will brieﬂy discuss the basics of ENMR, the

general problems and technical requirements of DCNMR and ﬁnally examples

as well as applications of actual ENMR experiments will be presented.

17.2 Principles of Electrophoretic NMR

17.2.1 Electrophoresis

The characteristic mobilities of charged solutes in a given solvent can be

used to separate the ions on the basis of their diﬀerent drift velocities in

an electric ﬁeld E. Shortly after the application of the electric ﬁeld the ions

reach a constant drift velocity

= u

· E. (17.2)

In an electric ﬁeld E of 10

–10

−1

, these drift velocities are typically in

the range of 10

−6

to 10

−4

−1

(As we will see, the observation times in

ENMR are typically ≤ 1 s and thus the observed spatial displacement of the

particles of interest is small compared with the length of the NMR receiver

coil, which is typically 1 to 3 cm, and the migrating particles cannot leave

the observation volume during the experiment.).

In the case of strong electrolytes, from the single-ion velocity v

we can

derive another important quantity, namely the transference number T

,via

the relation

A · F

, (17.3)

where c is the ion concentration in moles per liter and A and I denote,

respectively, the cross section of the sample cell and the electric current. The

knowledge of the mobility of charged species in a given environment is of

high interest since u

depends upon the properties of the species and of the

particular environment and can thus yield information on both of them. The

theoretical treatment of u

is diﬃcult [12] and one can only derive expressions

for u

on the basis of approximations. There are two important cases for

which we will give the corresponding relations. For ions with a small radius

r and at low ionic strength I

, we obtain from (17.1) and from the Stokes-

Einstein equation D = RT/6πη r (see also Sect. 6.3 of Chap. 6), where η is

the viscosity, the “small particle limit”

z · e

6πηr

, (17.4)

where z · e is the electric charge of the ion.

720 Manfred Holz

Thus (17.2) is applicable in cases where the Stokes-Einstein equation is

valid and where the ionic radius r is also small compared to κ

−1

,theDebye

length, that means κ · r  1. The latter is given by

κ =



2ρF





, (17.5)

where ρ is the density and 

, 

are the dielectric constant of the vacuum

and the relative dielectric constant of the electrolyte solution, respectively.

If κr  1, that means large charged particles and a small Debye length,

one obtains in this “large-particle limit”:

4πr

η · κ

ηκ

. (17.6)

σ = ze/4πr

is the surface charge density. In the large-particle limit, the

mobility u

can also be related to the electrokinetic potential ζ at the surface,

the so-called zeta potential, by [13]:



. (17.7)

Thus, in this limit the measurement of u

gives valuable information about

σ and ζ of large molecules or molecular aggregates.

17.2.2 Pulsed Field Gradient NMR for the Study of Drift

Velo citi es

The basis of NMR diﬀusion and ﬂow studies is spatially resolved NMR (see

also Chap. 10). Since in a homogeneous magnetic ﬁeld the space points are

equivalent, the superimposition of known magnetic ﬁeld gradients g produces

an inequivalence and allows the gain of information from the now spatial

dependence of the NMR frequency. If we apply, e.g., a gradient g = ∂B

/∂z

in z-direction, the NMR signal frequency ω is a function of z:

ω(z)=γB(z) (17.8)

with γ = magnetogyric ratio of the nucleus, B = magnetic ﬁeld strength.

This spatially resolving NMR is also the basis of all NMR imaging tech-

niques, like NMR tomography and microscopy [6]. If a particle, which carries a

nucleus under observation, performs a translational motion and consequently

changes its position during the observation time, its NMR frequency changes.

This frequency change is observed in a “spin-echo experiment” (Fig. 17.1) al-

lowing very elegant NMR diﬀusion and ﬂow studies, which were developed by

Stejskal [14]. Since translational diﬀusion and ﬂow are diﬀerent kinds of mo-

tion they have diﬀerent inﬂuence on the NMR spin-echo signal. In Fig. 17.2