Heitjans P., Karger J. (Eds.). Diffusion in Condensed Matter: Methods, Materials, Models

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17 Field-Assisted Diﬀusion Studied by Electrophoretic NMR 721

Fig. 17.1. Basic pulsed ﬁeld gradient (PFG) spin-echo experiment in elec-

trophoretic NMR. The diagram of the pulse program shows the radiofrequency

pulses (π/2- and π-pulse) for the generation of the spin-echo signal at 2τ (a). Pulsed

ﬁeld gradients with a strength g, a duration δ and a distance ∆ are applied for ﬂow

(and diﬀusion) measurements (b). In an ENMR experiment the electric ﬁeld E is

switched on in the sample cell during the spin-echo sequence (c). The acquisition

of the second half of the spin-echo signal starts at 2τ .

we show the eﬀects of diﬀusion and ﬂow in a magnetic ﬁeld gradient g.We

see that in the case of diﬀusion the spins, which were at t = 0 in the plane

0, show at times t>0 a frequency distribution, whereas in the case of ﬂow

(Fig. 17.2(b)) the frequency of all those species is regularly increasing with

t. Diﬀusion as an incoherent motion produces a damping of the spin-echo

signal due to random dephasing of the spin precession about the magnetic

ﬁeld, whereas ﬂow as a coherent motion produces a shift of the spin-echo

signal phase.

Since the NMR diﬀusion experiment is outlined in Chap. 10, we will here

deal only with the basics of NMR ﬂow experiments. We consider a ﬂow or

drift of particles in z-direction along a stationary gradient g = ∂B

/∂z.The

particles shall have a uniform velocity v and then they have at time t the

position z(t)=v · t. The nucleus residing in a particle experiences a time-

dependent magnetic ﬁeld

B(z(t)) = B

+ z(t) · g, (17.9)

where B

represents the constant magnetic main ﬁeld. The resonance fre-

quency ω(t) of such a nucleus is then

ω(t)=γB

+ γgvt. (17.10)

This means that at a time t a frequency diﬀerence with respect to the refer-

ence frequency ω

= γB

occurs. From (17.10) it follows

722 Manfred Holz

Fig. 17.2. The eﬀect of diﬀusion (a), and ﬂow (b) in the presence of a magnetic

ﬁeld gradient g in z-direction. The magnetic ﬁeld B(z) increases from left to right.

The nuclear spins in planes 0, 1 and 2 have diﬀerent NMR resonance frequencies ω

(1)

and ω

(2)

, respectively. In the lower diagrams the frequency distribution P (ω)

is given for the nuclear spins, which were at time t = 0 in the plane 0. In the case

of diﬀusion (a) the random walk of the spin-carrying particles results in a Gaussian

frequency distribution at times t>0, which is the origin of the echo damping. In

the case of plug ﬂow (b) the frequency distribution is a δ-function, which is shifted

to higher frequencies ω with increasing t, resulting in a phase shift of the spin-echo

signal.

∆ω(t)=γgvt. (17.11)

Since ∆ω = ∆φ/t is valid, also a phase angle diﬀerence increasing with t,

with respect to the reference phase, appears. At time t = τ ,wheretheradio

frequency π-pulse in the spin-echo pulse sequence is applied (see Fig. 17.1),

we have a phase diﬀerence

∆φ(τ)=



γgvtdt =

γgvτ

. (17.12)

The π-pulse has the eﬀect (see also Fig. 13.2 of Chap. 13) that it inverts

all nuclear spin precession phases and thus it makes ∆φ(τ)to−∆φ(τ). At

the time 2τ, where the spin echo maximum appears and where the signal is

measured, we ﬁnally have a phase diﬀerence

17 Field-Assisted Diﬀusion Studied by Electrophoretic NMR 723

∆φ(2τ)=−∆φ(τ)+

2τ



γgvtdt = γgvτ

. (17.13)

In an experiment with two pulsed ﬁeld gradients of length δ and distance ∆

(see Fig. 17.1) (pulsed gradients have some important advantages over the

application of stationary gradients [4, 5, 7]), the phase shift at 2τ is simply

∆φ(2τ)=γgvδ∆. (17.14)

The measurement of ∆φ with known g, δ and ∆ then allows the determination

of the velocity v.

In phase sensitive signal detection, the phase shift ∆φ due to the ﬂow

leads to a cosine modulation of the signal S

(2τ) (the subscript v stands for

presence of ﬂow) as a function of g, v, δ or ∆. The total spin-echo signal

intensity in presence of diﬀusion, ﬂow and T

relaxation is then [9]:

= S

cos(γgvδ∆) · exp



−

2τ

− γ



∆ −



. (17.15)

Here we see the three inﬂuences on the spin-echo signal, namely the ﬂow

(cosine term), T

relaxation (ﬁrst term in the exponent) and diﬀusion (second

term in the exponent). If we measure S

(2τ) and then switch oﬀ the ﬂow (in

electrophoretic NMR one can switch oﬀ the electric current) and measure

v=0

(2τ), the simple intensity ratio

(2τ)

v=0

(2τ)

=cos(γgvδ∆) (17.16)

yields the inﬂuence of ﬂow alone and represents the ﬁrst simple method for

the actual measurement of v.

How is ∆φ, the quantity of interest, obtained in practice? In modern NMR

spectrometers the resonance signals are ﬁrst excited and recorded in the time

domain and then transformed mathematically into the frequency domain by

a Fourier transformation (FT) yielding the NMR spectrum. In principle ∆φ

can be measured in the time domain [8, 15] or after FT in the frequency

domain. The latter is today the method of choice. For the introduction and

propagation of this principle, Richard Ernst has been awarded by the 1991

Nobel prize in chemistry [16]. In particular in ENMR there are diﬀerent

possibilities for the practical procedure. ∆φ or cos(∆φ)ismeasuredasa

function of one of the variables g, δ, ∆ or v,wherev canbevariedwiththe

applied electric ﬁeld E and thus by a variation of the electric current I in

the sample cell. As shown in Fig. 17.3 it is possible to observe and evaluate

the cos(γgvδ∆) modulation of the NMR line intensity as a function of E

(or I). By determination of the electric ﬁeld E

, which is required for just

one modulation period, one can directly obtain the mobility by means of

=2π/γgδ∆E

724 Manfred Holz

Fig. 17.3.

H ENMR spectra of an “oil-in water” microemulsion as a function of the

electric current I [10], showing the cos(∆φ) modulation of the lines of electrically

charged species, whereas the line of the solvent water (HOD) is unaﬀected. (S =

ionic surfactant, TIPB = triisopropylbenzene).

Fig. 17.4.

spin-echo phase shifts ∆φ as a function of the PFG duration δ.

The diﬀerent slopes correspond to diﬀerent Li

mobilities in various aqueous LiCl

solutions [15].

With modern NMR instruments a signal phase shift of a line in the spec-

trum can be directly observed and printed out. Thus one can, e.g., vary δ,

resulting in the linear dependence ∆φ(δ)=Kδ with K = γgv∆.Fromthe

slope of this straight line the velocity v is derived (see Fig. 17.4). Since the

really acting magnetic ﬁeld gradient g is the variable, which is less accurately

known compared with the quartz controlled times δ and ∆, often, similar

as in NMR self-diﬀusion studies [5], measurements are performed relative to

a reference system, here with known v. This procedure then allows velocity

measurements by a simple comparison of slopes, without the need of knowing

the actual g values.

17 Field-Assisted Diﬀusion Studied by Electrophoretic NMR 725

17.3 NMR in Presence of an Electric Direct Current.

Technical Requirements, Problems and Solutions

The DCNMR experiments are performed in common NMR spectrometers,

normally equipped with a superconducting magnet. For the PFG experiments

a probe head with gradient coils (actively shielded coils are recommended)

is required together with the corresponding gateable high-current power sup-

ply. Such units are commercially available. Special accessories for DCNMR

are the probe cells with two electrodes, where the electric ﬁeld E can be

applied (see Fig. 17.5) and a gateable constant DC power supply (typically

0 –250 mA, 0 –1 000 V) for the electrophoretic cell. The electric ﬁeld is applied

as a pulsed ﬁeld (see Fig. 17.1) where the gate pulses are usually derived from

the computer system of the spectrometer and are thus synchronized with the

rf excitation pulses and the magnetic ﬁeld gradient pulses. The electrodes

used are Pt or Ag/AgX (X=Cl,Br,J) electrodes. More technical details can

be found elsewhere [8–11].

The electric current ﬂowing during the NMR experiment in the sample

cell is the origin of a number of experimental problems, which shall be brieﬂy

discussed together with the solutions found so far:

Magnetic Fields Induced by the Electric Current

The electric current produces a magnetic ﬁeld in the plane perpendicular to

its direction of ﬂow. This ﬁeld can interfere with the Zeeman ﬁeld B

and

results in undesired B

ﬁeld inhomogeneities and instabilities. However, there

is a relatively easy way to overcome this problem by choosing as the current

direction that of the magnetic ﬁeld B

(z-direction) (see Fig. 17.5). The ad-

ditional magnetic ﬁelds then lie in the x, y-plane and have no components

in the z-direction, which is the relevant direction in the NMR experiment

and thus no signal disturbances are produced. This z-direction for the elec-

tric current is the “natural” direction in superconducting magnets since it is

also the axis of the magnet bore, and cylindrical sample cells are commonly

mounted parallel to this axis.

Fig. 17.5. Three basic DCNMR cell

geometries (a), (b), (c), with the cor-

responding electrode arrangements.

The radio frequency (rf) transmit-

ter/receiver coil is a saddle coil. The

direction of the Zeeman ﬁeld B

and

of the imposed ﬁeld gradient g =

∂B

/∂z is also shown.

726 Manfred Holz

Resistive Heating by the Current in the Sample Cell

More serious problems in ENMR can arise from resistive heating. The un-

desired heat production by the current in the electrolyte solution can result

in a temperature increase, unstable temperature conditions and ﬁnally in a

macroscopic convection in a liquid sample. Any macroscopic motion in a liq-

uid system like convection, vibrations or shock waves can be the source of

dramatic measuring errors in NMR mobility or diﬀusion measurements since

we have to measure small displacements in the micrometer or sub-micrometer

region.

The ﬁrst step to solve the problem of resistive heating is the application of

pulsed electric ﬁelds, so that a current ﬂows only during the relatively short

time of the spin-echo sequence, and the use of eﬀective cooling systems. As

a second step there is the possibility of using stabilizers against convection

as agar-agar [17] or some other gelling agents. In particular agar-agar proved

to be a suitable stabilizer for aqueous solutions without aﬀecting markedly

the diﬀusion and mobility of ions and the diﬀusivity of the solvent water.

However, the experience shows that with low electric currents of  10 mA

also measurement in the free solution is possible. A third possibility to over-

come heat-induced convection is the application of special pulse sequences,

including rf, magnetic gradient and electric ﬁeld pulses, which remove spec-

tral artifacts arising from convection. This novel method is called “Convection

compensated ENMR” [18].

Electro-Osmotic Flow

When ions migrate in the electric ﬁeld there are always collisions with the

neutral solvent molecules which are connected with a transfer of momen-

tum in the direction of the ion ﬂow. However, this eﬀect is cancelled by the

counter-ions, which move in the opposite direction, but the cancellation acts

only under the condition that there is a homogeneous charge distribution,

which means local charge neutrality [19]. On the other hand, an inhomoge-

neous contribution of mobile charges occurs on charged surfaces, since only

the screening counter-ions are mobile. This fact results in a solvent ﬂow at

the surface in the direction of the electrophoretic migration, and in a vertical

sample tube a counterﬂow appears in the centre. In this way electro-osmosis

produces macroscopic ﬂow in the sample, which disturbs the ENMR experi-

ment, resulting in an undesired damping of the NMR signal intensity [10,20].

The solution of this problem is not so easy, however also here a gelling agent

proved to be successful. Johnson and coworkers [10] proposed a coating of

the surface by methylcellulose to reduce electro-osmosis.

Mechanical Disturbances

Due to the fact that in DCNMR experiments electric direct currents are

switched in a strong external magnetic ﬁeld B

, strong forces are acting on

all electrical connections (gradient coils, wires, electrodes etc.) in which the

17 Field-Assisted Diﬀusion Studied by Electrophoretic NMR 727

current does not ﬂow parallel to B

. In this case during switching shock

waves may be generated, which can lead to displacements in the electrolyte

solution. This problem must be solved by a very careful rigid ﬁxing of all

electrical connections and electrodes within the probe head.

17.4 ENMR Sample Cells

We will give two examples of electrolytic cells in use in our laboratory.

They are both cylindrical cells oriented in z-direction (B

direction). The

ﬁrst one [9, 21] is simply cylindric with an upper and a lower electrode (see

Fig. 17.6 (a). The advantage of this cell is a good ﬁlling factor, giving a good

signal-to-noise ratio, which is important for nuclei other than

H, the so-

called heteronuclei. The disadvantage comes from a possible gassing of the

lower electrode, which is not vented to the atmosphere. The second one (see

Fig. 17.6 (b) is a concentric cylindric chamber, where in the inner cylinder

the sample of interest is located. The outer cylinder is ﬁlled with an elec-

trolyte solution as a conductor. In order to be able to use as conductor in

Fig. 17.6. Two examples of ENMR cells. (a) probe head insert consisting of a plexi

glass body on which the gradient coils are mounted. The electrophoresis cell in the

centre is surrounded by a temperature bath liquid [21]. (b) Concentric cylindrical

ENMR cell, where the two electrodes are located in the upper part of the cell.

728 Manfred Holz

the outer cell always the same kind of electrolyte solution as it is observed in

the inner cylinder, in the author’s laboratory a very successful procedure has

been developed, which eliminates the undesired NMR signal contributions

from the ions in the outer cylinder. For this purpose in the outer cylinder

suited small beads are added, which due to magnetic susceptibility and/or

surface relaxation eﬀects extremely broaden NMR signals coming from the

outer cylinder. Thus the undesired signals are suppressed by a “T

-ﬁlter”. The

advantage of this cell is that both electrodes are vented to the atmosphere,

the disadvantage is the lower ﬁlling factor.

17.5 ENMR Experiments (1D, 2D and 3D) and

Application Examples

First we ask the question: which nuclei can be used in ENMR? Whereas

in ordinary NMR more than 100 nuclides are accessible to the experiment,

the number of nuclides, which are suited for ENMR and NMR diﬀusion ex-

periments, is in the range of 10 to 30. The reason for that lies in the PFG

spin-echo experiment, which is applied and where the particles move during

a deﬁned time (∆) and after that time the displacement is measured via

the spin-echo signal at 2τ. Thus, in particular for slowly moving particles,

∆ cannot be chosen too short, since a measurable displacement ∆z has

to be achieved. On the other hand, as can be seen from (17.15), the spin-

echo signal decreases with the nuclear magnetic relaxation (time constant

) and therefore the maximum “observation time” ∆ is limited by the re-

laxation inﬂuence. There are so-called “stimulated-echo” experiments [9–11],

which can be used in those cases where the transverse relaxation time T

much smaller than the longitudinal relaxation time T

and then commonly

is the limiting factor (see also Sect. 10.3.4 of Chap. 10). There is a large

amount of nuclides with nuclear spin I>1/2, which relax by the so-called

quadrupole mechanism and have often very short T

and T

values in the

µs range in liquid solutions so that they are not suited for ENMR measure-

ments. Generally, one can say that all spin-1/2 nuclides with a not too low

NMR receptivity should be usable, as, e.g.,

Si,

113

Cd,

119

Sn and

205

Tl. There are several quadrupolar nuclei, which are candidates

for ENMR, namely for example

Li,

Na,

Al,

Cl,

Vand

133

Cs,

since they have under normal circumstances relaxation times, which are long

enough for the spin-echo experiment.

So far, in the comparatively young ﬁeld of applied ENMR, the nuclei

Li,

Fand

133

Cs have actually been used in experiments, where

of course

H ENMR plays an outstanding role, owing to the high sensitiv-

ity of

H NMR and also as a consequence of the huge number of hydrogen

containing compounds of interest in pure and applied chemistry and in life

sciences.

17 Field-Assisted Diﬀusion Studied by Electrophoretic NMR 729

In NMR, in the last two decades a number of highly sophisticated tech-

niques have been developed allowing the representation of the spectra in a

multi-dimensional form [16]. Thus, we distinguish one- (1D), two- (2D) and

three-dimensional (3D) NMR experiments, corresponding to the representa-

tion of the line intensities as a function of one, two or three axes. In describing

typical ENMR experiments in the following, we will demonstrate their pos-

sible applications. We shall begin with common 1D ENMR.

17.5.1 1D ENMR Applications

The ﬁrst ENMR experiments were performed on simple aqueous solutions

of tetraalkylammonium salts, where e.g. the concentration dependence of

the mobility of the hydrogen containing cations has been measured for the

purpose of demonstrating the agreement with results from classical methods

[8,17]. However, as pointed out above, the main advantage of ENMR lies in

the possibility of studying distinct species in multi-component mixtures.

Mobility Studies in Mixtures

Owing to diﬃculties with classical methods, there exist in the literature only

a few reliable mobility data of simple ions in electrolyte mixtures. Also for

such a simple system like LiX/CsX (X=Cl,Br,J) in aqueous solution there

were no data available.

and

133

are excellently suited for ENMR and

these two cations diﬀer appreciably in their ionic radii. In Fig. 17.7 we see, as

an example for aqueous LiBr/CsBr mixtures at constant ionic strength, the

mobilities u

for both ions [21]. As a by-product one always obtains (with

electric current I = 0) the ionic self-diﬀusion coeﬃcients D

,whicharealso

given in Fig. 17.7. It can be recognized that the mobilities show a curved

composition dependence, with opposite curvature for Li

and Cs

,which

means that the “fast” cation accelerates the migration of the “slow” cation

in the mixture. An opposite result represents the linear dependence of the

ionic diﬀusion coeﬃcients. However, this is not surprising since, in contrast

to u

, D

is an equilibrium quantity. Having measured the mobilities of the

two cations, the transference numbers T

could be calculated. Since the sum

of all transference numbers must be equal to one, T

−

for the common anion

could also be determined (see Fig. 17.7).

Another example is shown in Fig. 17.8 where by high-resolution

HENMR

the electrophoretic mobilities of the amine and the amino acid in an aqueous

mixture could be measured simultaneously [10].

More recently, mixed anionic-nonionic surfactant micelles have been stud-

ied by the same experimental technique with the aim of characterizing the

surfaces of ionic micelles. New insight in counterion binding could be derived,

which cannot be easily gained by other methods. [23]

In the study and application of polyelectrolytes the quantitative determi-

nation of the eﬀective charge density is an important but diﬃcult task. Ap-

plying the combination of ENMR and PFG diﬀusion measurements, Wong

730 Manfred Holz

Fig. 17.7. Results of ENMR mea-

surements on a ternary aqueous mix-

ture of simple ions at constant ionic

strength [21]. System y ×0.5 m LiBr+

(1 − y) × 0.5mCsBrinH

(a) The mobilities u

of Li

and Cs

as a function of mixture composition.

(b) The ionic self-diﬀusion coeﬃcients

of Li

and Cs

the three ionic constituents Li

,Cs

and Br

−

in the mixed electrolyte so-

lution.

and Scheler [24, 25], for the ﬁrst time succeeded in measuring in this way the

electrophoretic mobility of polyelectrolytes in solution and they could de-

termine the eﬀective charge density. The eﬀect of addition of common salts

could also be analysed.

Diﬀusion and Field-Assisted Diﬀusion of Ions in Porous Media

In the literature it is often assumed, that the ionic mobility within a pore

can be approximated by its mobility in the bulk ﬂuid and that therefore the

electric conduction of a porous system, ﬁlled with a liquid electrolyte, is only

restricted by geometrical factors [16]. The validity of this assumption could

not be experimentally checked by classical methods. ENMR, as a non-invasive