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

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17 Field-Assisted Diﬀusion Studied by Electrophoretic NMR 731

Fig. 17.8. (a)

H ENMR (250 MHz)

spectraofamixtureof1mMethylene

diamine with an amino acid (1 mM L-

Ala-Gly-Gly) in D

O [10].

(b) The intensity of the

H NMR lines

in Fig. 17.8a versus current I.From

the diﬀerent cosine modulations of the

amine line and the amino acid lines,

the diﬀerent mobilities can be derived

[10].

technique, which is also independent of optical properties, is a unique tool for

the investigation of the transport of ionic charge carriers in disordered media.

Thus, it is not surprising that this ﬁeld of ENMR application developed

comparatively fast in the last years. The ﬁrst successful measurements of

ionic mobilities in porous systems, namely in gels, have been performed in

the author’s laboratory [26, 27]. It could be demonstrated that in the ﬂuid-

ﬁlled pore space both the observed ionic self-diﬀusion coeﬃcient D

and

the ionic mobility u

in an external electric ﬁeld show a dependence on the

observation time ∆ (see Fig. 17.9). As previously observed for liquid molecules

conﬁned in porous media the phenomenon of “anomalous diﬀusion” occurs

(see Chaps. 10, 19 and 22). Now, this phenomenon has been also observed

for self-diﬀusion of charged species and for the ﬁrst time it has been shown

that also the analogous phenomenon of “anomalous ﬁeld-assisted diﬀusion”

or “anomalous electric mobility” exists [26], which means that there is a

time regime, where the average displacement z(∆) of an ion by migration

along the electric ﬁeld is a non-linear function of time. At long observation

times ∆ the quantities D

and u

become independent of ∆ and reach their

eﬀective values D

eﬀ

and u

eﬀ

in the porous system, the quantities D

and u

in Fig. 17.9 thus reach plateau values. (The index “0” refers to

quantities in the non-conﬁned state.) The inverse plateau value T (D

eff

delivers an important characteristic value of the porous medium,

namely its tortuosity T (see, e. g., [28]; cf also Fig. 10.13 in Chap. 10). T of a

porous medium is usually deﬁned as the squared ratio of the true migration

path length of a ﬂuid particle and the straight-line distance between the

732 Manfred Holz

(a)

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.2

0.4

0.6

0.8

1.0

 [s]

(b)

0 200 400 600 800 1000 1200

0.2

0.4

0.6

0.8

1.0

"'[Vm

-1

Fig. 17.9. (a) The relative ionic self-diﬀusion coeﬃcient D

of the (C

)

cation in aqueous solution conﬁned in the gel Sephadex LH-20 as a function of the

observation time ∆. D

=0.49 × 10

−9

−1

is the ionic self-diﬀusion coeﬃcient

in the non-conﬁned bulk solution [26].

(b) The relative ionic mobility u

as a function of the product of observation

time ∆



and the externally applied electric ﬁeld E. u

=1.22 × 10

−8

−1

All other details as in (a).

starting and ﬁnishing point of the particle’s motion in this medium. Since,

in general, the translational motion can have its origin in diﬀerent processes

as ﬂow, diﬀusion or electric ﬁeld-assisted diﬀusion, one sometimes speaks

of “hydraulic”, “diﬀusional” or “electric” tortuosity, respectively [29]. From

the plateau value T (u

)=u

eﬀ

again the tortuosity of the system is

17 Field-Assisted Diﬀusion Studied by Electrophoretic NMR 733

available, now measured via the electric mobility. The comparison of the

behaviour of the ionic self-diﬀusion coeﬃcient and the electric mobility of

the same ion in the medium with internal boundaries allows a ﬁrst direct

experimental check of the validity of a very important relation in porous

media, namely the Einstein relation (D

∝ u

, see (17.1)). Fig. 17.9 reveals

that T (u

)=T (D

) is found, which means that indeed in the porous gel

system under investigation the Einstein relation remains valid [26,27]. In the

paper by Holz et al. [27], also a ﬁrst attempt has been undertaken to derive

the speciﬁc surface S/V

of the porous medium from the time-dependence

of the ﬁeld-assisted diﬀusion, similarly, as it has been performed, e. g., by

means of the observation time-dependence of the self-diﬀusion coeﬃcient in

Sect. 10.5.3 of Chap. 10 [30–32]. The present author and his co-workers also

discussed basic advantages of time-dependent electric mobility measurements

for morphology studies of porous media [27].

Very important special porous media in life science and in technical appli-

cations are membranes, and it has been pointed out that ENMR might be a

powerful technique for the study of ion transport in those media [9]. Polymer

electrolyte membranes are of outstanding interest in connection with fuel

cells. Ise et al. [33] succeeded in measuring by ENMR the electro-osmotic

drag coeﬃcient K

drag

in polymer membranes. This drag coeﬃcient is deﬁned

as the number of water molecules transferred through the membrane per H

ion, and the authors could determine K

drag

as a function of water content

and temperature for diﬀerent membrane materials as Naﬁon and sulfonated

polyetherketones.

Anion and cation (Li

) transference numbers have been studied in com-

posite polymer electrolytes with fumed silica as an inorganic ﬁller by

Fand

Li ENMR, respectively, [34, 35]. In this way the ionic transport properties

could be investigated as a function of ﬁller content and Li salt concentration

without the usual need of assumptions. The power and validity of ENMR

could be demonstrated by the independent measurement of the anion and

cation transference numbers for diﬀerent salts, where within the experimen-

tal error these transference numbers sum to unity. Also in lithium polymer

gel electrolytes, used in lithium secondary batteries, the individual ionic mo-

bilities of cations and anions could be measured by

Li and

FENMRexper-

iments [36]. The authors also observed by

H ENMR a solvent drift in the gel

electrolyte when the strength of the applied electric ﬁeld exceeds a distinct

value, where also an anomalous change of the Li

mobility occurs. They dis-

cuss as a possible reason a ﬂow of lithium, compensating lithium deposition

at the electrode surface, possibly causing a counter ﬂow of solvent.

Detection and Identiﬁcation of Charged Species by ENMR Phase

Diﬀerence Spectroscopy

We saw that for resonance lines of coherently moving particles a typical phase

shift ∆φ occurs in the spectrum (Sect. 17.2.2). If a mixture contains both

charged and uncharged species, in an ENMR experiment only the lines of

734 Manfred Holz

Fig. 17.10. “Mobility ﬁlter” ENMR experiment:

CNMRspectraofanaqueous

mixture of ethanol (EtOH), acetonitrile (AN) and sodium formate [40]. (a) Normal

(undecoupled)

CNMRspectrum(I = 0); (b)

CENMRspectrum(I =0);

(HCOO

−

)remain.

charged species are phase-shifted while the lines of uncharged particles are

not aﬀected. This fact can be utilized for the detection and identiﬁcation

of charged species in multi-component mixtures, provided that these species

contain the nucleus under observation. As an example of such a procedure,

we show ﬁrst in Fig. 17.10 (a) the normal

C spectrum (I =0)ofanaque-

ous mixture of ethanol (EtOH), acetonitrile (AN) and sodium formate [40].

When the same spectrum is then taken in presence of an electric ﬁeld (I =0)

(Fig. 17.10(b)) only the formate ion lines are phase-shifted and in the diﬀer-

ence spectrum of the spectra with I =0andI = 0 the lines without a phase

shift disappear and only the lines of the ion remain (see Fig. 17.10 (c)).

In this way, in a simple 1D ENMR experiment a “mobility ﬁlter” can

be generated, which is able to ﬁlter from a complex spectrum only the lines

of migrating species, which in this way can be detected and simultaneously

identiﬁed via their NMR lines [40].

17.5.2 2D and 3D Experiments

In 2D NMR a two-dimensional data set is generated and the signal intensities

S(F

) are given as a function of two variables F

and F

. One advantage

of 2D NMR is the fact that the spectrum is stretched in two dimensions and

much more lines can be resolved. However, the main advantage is that in

the 2D spectrum “correlations” between certain values of F

and F

can be

detected and assigned.

17 Field-Assisted Diﬀusion Studied by Electrophoretic NMR 735

Fig. 17.11. ENMR signal intensity S at a given frequency F

,(a)when∆ = t

(stepwise) varied, (b) when the electric ﬁeld E (current I)isvaried.

The 2D spectrum is obtained by recording the NMR signals in the time

domain as a function of the acquisition time t

,asinnormal1DNMR,

however, this is performed k times, whereby another time t

(or another

variable) is stepwise increased in k increments resulting in k signals with 2N

data points:

(1)

),S

(2)

), ··· ,S

(k)

Thereafter, a ﬁrst Fourier transformation with respect to t

is performed,

followed by a second Fourier transformation with respect to t

, yielding the

2D spectrum S(F

)withk × N data points [2, 16].

In 2D ENMR, F

is the chemical shift axis, which allows the identiﬁ-

cation of the chemical species and F

is the mobility axis. In this way the

observed species are “correlated” with their mobility in the electric ﬁeld. This

kind of experiments, developed by Johnson and co-workers [10, 11], is called

“mobility-ordered spectroscopy” (MOSY) and it should be mentioned that

also very interesting analogous 2D “diﬀusion-ordered spectroscopy” (DOSY)

experiments have been developed, where the second axis is the diﬀusion

axis [11, 41].

In practical 2D ENMR two quantities can be incremented, namely ∆ in

the PFG experiment (Fig. 17.11) or the velocity v, via a stepwise increase of

the electric ﬁeld E and thus of the current I.Ifwechooset

= ∆ and stepwise

increase ∆, the inﬂuence of normal diﬀusion on the decay of the transverse

magnetization S also increases (see (17.15) and Fig. 17.11 (a)), leading to a

736 Manfred Holz

Fig. 17.12. Mobility-ordered (MOSY) 2D ENMR spectrum (stacked plot) of

(CH

)

(TMA) in the presence of mixed micelles [11]. [Sodium dodecyl sulfate

(SDS) and octaethylene glycol dodecyl ether (C

)]. Note the diﬀerent sign for

the mobilities of TMA and C

line broadening in the 2D spectrum. Thus, it proved to be more favourable

to hold constant all the timing parameters and to increment the ﬁeld E since

in this case the intensity oscillations are not damped (Fig. 17.11 (b)) [10,11].

2D ENMR experiments require the observation (and Fourier transformation)

of several intensity oscillations as a function of ∆ or E, that means in these

experiments high ∆φ-values must be achieved. Since ∆φ is proportional to

the magnetogyric ratio γ of the nucleus, 2D ENMR is much easier performed

with high-γ nuclides as

F than with nuclides having a lower γ as

Li or

C [40]. So far, 2D experiments have only been performed with

HENMR.

In Figs. 17.12 and 17.13 two examples of MOSY experiments are given,

where in Fig. 17.12 the 2D spectrum is shown as a stacked plot and in

Fig. 17.13 as a contour map, which is the mostly used representation of

spectra in 2D NMR. The two examples show the most interesting ﬁeld of

application of 2D ENMR, namely mobility measurement of molecular aggre-

gates, as micelles and vesicles. In Fig. 17.12 the

HMOSYspectrumofa

mixture of (CH

)

ions and micelles is shown. This spectrum also demon-

strates the important fact that in ENMR not only the amplitude but also

the sign of the mobility is obtained, allowing the determination of the sign of

charge of the species of interest. In the second example (Fig. 17.13) glucose,

as a good NMR active label, has been entrapped inside vesicles, for which in

this way the vesicle mobilty in water is found to be u =2×10

−8

−1

A very interesting extension of ENMR, namely 3D ENMR for the study

of protein mixtures, has been demonstrated by He and co-workers [37, 38].

With two frequency axes a 2D spectrum of the protein mixture is generated.

However, in protein mixtures severe signal overlapping occurs and conven-

tional NMR methods have diﬃculties in characterising structures or struc-

tural changes of multiple protein components in biochemical reactions. Now

17 Field-Assisted Diﬀusion Studied by Electrophoretic NMR 737

Fig. 17.13. Example of a MOSY contour map. 2D

H ENMR spectrum of glucose

entrapped inside vesicles [11]. The 1D NMR spectrum of glucose is shown as a

projection at the top. At the left the 1D mobility spectrum is given.

the overlapping signals can be separated in a 3D experiment, where the third

axis is the electrophoretic mobility, allowing the simultaneous measurement

of the diﬀerent proteins due to their diﬀering mobility. The otherwise needed

physical separation is no more necessary. “Capillary array ENMR”, devel-

oped by the same group [39], allows the study of protein mixtures at high

ionic strength in buﬀer solutions.

17.5.3 Mobility and Velocity Distributions. Polydispersity and

Electro-Osmotic Flow

In Fig. 17.11 (b) we saw that in an ENMR experiment the intensity oscil-

lations are not damped if we vary the electric ﬁeld E. However, there it

was supposed that all observed particles have the same velocity. In polydis-

perse samples, on the other hand, we have a distribution of mobilities g(u)

of particles with the same chemical shift (of the same chemical kind) but

of diﬀerent size and mass. In such a case, in (17.14), (17.15) we have to

replace cos(γgvδ∆)=cos(γguEδ∆)by

+∞

−∞

g(u)cos(γguEδ∆)du and this

results in a damping of the intensity oscillations as a function of E (or I)

(see Fig. 17.14(a). However, from an inverse FT of S(E,F

) with respect to

E (or I) the mobility distribution g(u) can be derived (Fig. 17.14 (b) [10,42].

Even for monodisperse solutions the presence of electro-osmotic ﬂow leads

to mobility (velocity) distributions. Such a mobility distribution was studied

in the just described manner in an oil-in-water microemulsion [43]. The aim

was the study of the development of the electro-osmosis proﬁle in a vertical

738 Manfred Holz

Fig. 17.14. (a)

HENMRsig-

nal intensity of sucrose entrapped in

charged phospholipid vesicles versus

current I in a polydisperse system,

showing the damping of S(E, F

) due

to a mobility distribution [42]. (b) The

mobility distribution g(u) obtained by

a Fourier transformation of the data in

(a) (solid line). The dashed line repre-

sents a ﬁt curve [42].

cylindrical tube as a function of ∆ and as a function of t

,i.e.ofthetime

during which the current ﬂowed before the PFG experiment was started.

The result (Fig. 17.15) shows the development of the mobility distribution

g(u) with time, when both electrophoretic and electro-osmotic mobilities are

present. The electrophoretic migration velocity is then superimposed by an

opposite ﬂow at the wall and a counterﬂow in the centre of the tube. After

a time of ca 400 ms in this example the system reached equilibrium and, as

theoretically expected, the ﬂow proﬁle, that is the mobility as a function of

thetuberadiusr

, becomes purely parabolic [42].

For the study of the electro-osmotic ﬂow proﬁle in capillaries also highly

sophisticated NMR micro-imaging techniques have been applied, which as

well deliver the velocity distributions and their time dependence [44].

The DCNMR studies described in this section might be a very interest-

ing tool for future developments in capillary electrophoresis (CE), where the

electro-osmotic ﬂow is the driving force in these extremely important sepa-

ration techniques in biochemistry.

17.6 Conclusion

The possibility of performing NMR experiments in presence of an electric

current in the sample under investigation opens a novel way of using NMR

17 Field-Assisted Diﬀusion Studied by Electrophoretic NMR 739

Fig. 17.15. Mobility (velocity) dis-

tributions g(u) in presence of electro-

osmotic ﬂow for various electric ﬁeld

durations t

and ﬂow times ∆,show-

ing the development of the electro-

osmotic ﬂow proﬁle in an “oil-in-

water” microemulsion [43].

techniques in the wide ﬁeld of electrochemistry and its applications. The

electric charge or the electric current is thus introduced as a new parameter in

the NMR experiment. Electrophoretic NMR combines in-situ electrophoresis

with the selectivity of NMR, a fact, which is of particular importance for

the study of transport properties in complex multi-component ﬂuid systems

with charge-carrying species, such as simple ions, charged macromolecules,

and charged supramolecular systems. In mixtures of organic compounds the

single components can be simultaneously observed, however, separated by

CNMR.

The pulsed ﬁeld gradient NMR methods allow the non-invasive measure-

ment of electrophoretic mobilities u

and one always obtains the self-diﬀusion

coeﬃcient D

of the distinct charged particles as a by-product. This oﬀers

the possibility of a systematic qualitative comparison of transport diﬀusion

(here ﬁeld-assisted diﬀusion) with self-diﬀusion, that means transport in the

non-equilibrium state with transport in the equilibrium state. Furthermore,

being independent of the optical properties of the sample, the ENMR meth-

ods may also be applied to porous systems. This ﬁeld of application is just

in its beginning, but seems to be very promising for basic electrophoretic

mobility studies in systems with conﬁned geometry.

Apart from mobility and diﬀusion studies, DCNMR also oﬀers a new en-

trance to the investigation of electro-osmotic ﬂow and phenomena related

with it. In particular in capillaries and porous systems those electro-osmotic

740 Manfred Holz

ﬂow studies, also combined with NMR imaging techniques, represent a pow-

erful new tool.

On the other hand, ENMR has certain limitations owing to the compar-

atively low sensivity of radiofrequency spectroscopy and owing to problems

with nuclides or systems with short nuclear magnetic relaxation times T

and

. However, it should be pointed out that the low energies connected with

electromagnetic radiofrequency waves have the advantage that investigations

can be performed without any disturbance of thermal equilibrium.

ENMR is a comparatively young and novel technique and the required

accessory units are partly not yet commercially available and therefore so far

we cannot speak of an NMR “routine technique”. On the other hand, the

power of ENMR and DCNMR methods has been impressively demonstrated

[9–11] and it is clear that a further development of special techniques and a

further extension of applications will take place in the coming years.

Notation

A cross section of the electrophoretic cell

B magnetic ﬁeld

magnetic ﬁeld, corresponding to the NMR reference frequency ω

c concentration of an electrolyte in the solution

D self-diﬀusion coeﬃcient

∗

ionic self-diﬀusion coeﬃcient at c → 0

ionic self-diﬀusion coeﬃcient in the non-conﬁned state

DCNMR direct current NMR

DOSY diﬀusion ordered spectroscopy

E electric ﬁeld

ENMR electrophoretic NMR

F Faraday constant

g magnetic ﬁeld gradient

g(u) mobility distribution

I electric current

ionic strength

drag

electroosmotic drag coeﬃcient

MOSY mobility ordered spectroscopy

PFG pulsed ﬁeld gradient

P (ω) NMR frequency distribution

r radius of a particle

rf radio frequency

S NMR signal intensity

NMR signal intensity at the kth step in a 2D experiment

NMR signal intensity in presence of ﬂow

initial NMR signal intensity

S/V

surface to pore volume ratio, also called “speciﬁc surface”