field-cycling nmr relaxometry

Field-cycling NMR relaxometry

Rainer Kimmicha,*, Esteban Anoardob

aSektion Kernresonanzspektroskopie, Universitat Ulm, Alber Einstein-Allee 11, D-89069 Ulm, GermanybFacultad de Matematica, Astronomıa y Fısica, Universidad Nacional de Cordoba, Ciudad Universitaria, 5000 Cordoba, Argentina

Received 2 February 2004

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

2. Theoretical background of relaxometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

2.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

2.2. Exponential correlation functions and BPP formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

3. Field-cycling relaxation curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

3.1. Low fields ðBr p BpÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

3.2. High fields ðBr ! BdÞ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

4. Signal-to-noise ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

5. Crucial specifications of the field cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

6. Relaxometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

6.1. Conditions for field-cycling magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

6.2. Optimization principles for field-cycling systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

6.3. Diverse magnet designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

6.4. Conditions for field-cycling power supplies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

6.5. Principles of over-damped power supply circuits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

6.6. Principles of sub-damped power supply circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

6.7. Electronic switches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

6.8. Practical solutions for field-cycling magnet current circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

7. Applications to porous media and adsorption phenomena at liquid/solid interfaces . . . . . . . . . . . . . . . . . . . . . . . . . 279

7.1. Two-phase fast-exchange model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

7.2. Bulk-mediated surface diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

7.3. Reorientation mediated by translational diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

7.4. Porous silica glasses and fine particle agglomerates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

7.5. Water/lipid interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

7.6. Water/protein interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

7.7. Water/saponite interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

8. Polymer dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

8.1. The three components of polymer dynamics relevant for NMR relaxometry . . . . . . . . . . . . . . . . . . . . . . . . . . 289

8.2. The different time-scale approach for the NMR correlation function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

0079-6565/$ - see front matter q 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.pnmrs.2004.03.002

Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320

www.elsevier.com/locate/pnmrs

* Corresponding author. Tel.: þ49-7-315-023140; fax: þ49-7-315-023150.

E-mail address: [email protected] (R. Kimmich).

Abbreviations: BMSD, bulk-mediated surface diffusion; BPP, Bloembergen/Purcell/Pound; BWR, Bloch/Wangsness/Redfield; nCB, 40-n-alkyl-4-

cyanobiphenyl (n ¼ 5, 8, 11); DMSO, dimethylsulfoxide; DPL ( ¼ DPPC), dipalmitoyl-lecithin; DPPC ( ¼ DPL), dipalmitoyl-phosphatidyl choline; 8CB,

4-octyl-40-cyanobiphenyl; ENDOR, electron nuclear double resonance; FFC, fast field-cycling; FID, free induction decay; 5CB, 4-pentyl-40-cyanobiphenyl;

GTO, gate turn-off thyristor; HAB, 4-40-bis-hexiloxyazoxy-benzene; HpAB, 4-40-bis-heptyloxyazoxy-benzene; IGBT, insulated gate bipolar transistor; LC,

liquid crystal; MBBA, 4-methoxybenzylidene-40-n-butylaniline; MOSFET, metal oxide semiconductor field effect transistor; N, nematic phase; NFL, non-

freezing surface layer; NMR, nuclear magnetic resonance; NMRD, nuclear magnetic relaxation dispersion; NOE, nuclear Overhauser effect; NQR, nuclear

quadrupole resonance; RF, radio frequency; ODF, order director fluctuations; PAA, para-azoxyanisole; PB, polybutadiene; PDES, polydiethylsiloxane;

PDMS, polydimethylsiloxane; PEO, polyethyleneoxide; PHEMA, polyhydroxyethylmethacrylate; PIB, polyisobutylene; RMTD, reorientation mediated by

translational displacements; SmA, smectic A phase; STELAR, company producing commercial field-cycling relaxometers (see www.stelar.it); TPFE, two-

phase fast-exchange model.

http://www.elsevier.com/locate/pnmrs

http://www.stelar.it

8.3. Evidence for Rouse dynamics ðM , McÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

8.4. The three regimes of spin–lattice relaxation dispersion in entangled polymer melts, solutions and networks

ðM . McÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

8.5. High- and low-mode-number limits (dispersion regions I and II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

8.6. Intra- and inter-segment spin interactions (dispersion region III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

8.7. Mesomorphic phases of polymers without mesogenic groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

8.8. Polymer solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

8.9. Polymer networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

8.10. Chain dynamics in pores (‘artificial tubes’) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

8.11. Cross-over from Rouse to reptation dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

8.12. Protein backbone dynamics and ‘quadrupole dips’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

9. Liquid crystals and lipid bilayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

9.1. Motivation for field-cycling NMR relaxometry experiments in liquid crystals. . . . . . . . . . . . . . . . . . . . . . . . . 303

9.2. Relevant properties of bulk liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

9.2.1. The nematic phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

9.2.2. The smectic A phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

9.3. Order director fluctuations in the nematic phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

9.4. Order director fluctuations in the smectic A phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

9.5. Fluctuations of spin interactions by translational self-diffusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

9.6. Rotational diffusion of individual molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

9.7. Combined action of collective and single-molecule motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

9.8. Field-cycling NMR relaxometry in bulk nematic liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

9.9. Field-cycling NMR relaxometry in bulk smectic and lamellar systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

9.10. Field-cycling dipolar order relaxometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

9.11. Secular dipolar interactions with quadrupole nuclei: ‘quadrupole dips’. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

9.12. The effect of ultrasound on the spin–lattice relaxation dispersion of liquid crystals . . . . . . . . . . . . . . . . . . . . 312

9.13. Surface ordering in porous media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

9.14. Rotating-frame NMR relaxometry in liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

10. A word of caution concerning NMR relaxometry in the kHz regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

11. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

Keywords: Field-cycling NMR relaxometry; NMR relaxometers; Spin-lattice relaxation; Porous media; Lipid bilayers; Proteins; Polymers; Rouse model;

Reptation; Liquid crystals; Molecular dynamics

1. Introduction

Field-cycling NMR relaxometry [1–3] is the preferred

technique for obtaining the frequency (or magnetic field)

dependence of relaxation times (or equivalently of relax-

ation rates). It is therefore also referred to as nuclear

magnetic relaxation dispersion (NMRD). The term ‘relaxo-

metry’ is normally used in the context of measurements of

spin – lattice relaxation times in general. Transverse

relaxation and effects due to residual dipolar couplings

(see Ref. [4], for instance) can also be employed as a source

of useful supplementary information. Fig. 1 shows a

schematic representation of the frequency/time scales

covered by various NMR methods.

Studies employing field-cycling techniques have often

been used for different purposes since the early days of NMR

[7–12]. The principle of a typical field-cycling NMR

relaxometry experiment is illustrated in Fig. 2. The sample

is polarized in a magnetic field with a flux density Bp as high

as technically feasible. The relaxation process takes place in

Fig. 1. Schematic representation of the time ðtÞ and angular frequency ðvÞ scales covered by diverse NMR techniques. The ranges indicated refer to proton

resonance. Descriptions of these methods can be found in the monographs [5,6], for instance.

R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320258

a low-field interval varied with respect to length and flux

density Br: The signal remaining after this relaxation interval

is detected in a field of fixed flux density Bd again as high as

possible. That is, signals are acquired with a radio frequency

(RF) unit tuned to a predetermined frequency irrespective of

the relaxation field chosen. For detection, either the free

induction decay after a simple 908 RF pulse or a spin echo

produced by a sequence of two or more pulses is recorded.

Then an extended recycle delay for the restoration of thermal

equilibrium and polarization follows until the next cycle

begins. Details of the theoretical background and of typical

experimental set-ups will be outlined in Sections 2–6.

The temporal exposure of the sample to a variable

relaxation field (see Fig. 2) can be performed either by

electronically switching the current in a magnet coil

[13–18] or by moving the sample mechanically, normally

pneumatically, between positions of different magnetic flux

densities [19–23]. The latter field-cycling variant is also

referred to as ‘sample shuttle technique’.

Good electronically switched relaxometers have a field

switching and settling time to the required accuracy and

stability in the order of a millisecond, whereas typical

sample shuttling times are of the order 100 ms which

restricts the applicability of the shuttle technique to

correspondingly long relaxation times. On the other hand,

shuttle devices can be used as accessories to ordinary high-

field high-resolution magnets. Signals are then detected in

the homogeneous central field, whereas the sample is moved

in the relaxation interval to a variable position in the fringe

field or to a second satellite magnet of a much lower and

adjustable flux density. Since relaxation times tend to

become shorter with lower fields so that they conflict with

the time needed to transfer the sample, shuttling techniques

are restricted to the range from 1 MHz up to the highest

frequencies achievable with high-field magnets.

The electronically switched field-cycling variant, often

referred to as fast field-cycling (FFC), permits the

measurement of relaxation times down to the local-field

regime corresponding to a time scale where field-gradient

NMR diffusometry already becomes applicable (for a recent

review see Ref. [24]). Above 50–100 MHz, the experimen-

tal frequency scale can be supplemented by spin–lattice

relaxation measurements with sample shuttling devices and

conventional NMR spectrometers operating at up to several

hundred megahertz. The total frequency ranges for proton

and deuteron spin–lattice relaxation covered in this way are

103 Hz , nproton , 109Hz and

102 Hz , ndeuteron , 108Hz;

ð1Þ

respectively.

The high-frequency limits of the frequency ranges given

in Eq. (1) are determined by the available high-field magnets.

At low frequencies several factors may restrict the applica-

bility of the field-cycling technique as will be outlined below

in more detail: (i) The ‘local field’ representing the (residual)

secular part of the spin couplings may exceed the external

flux density in the relaxation interval of the field cycle. (ii)

The compensation of the earth’s field (or other magnetic stray

fields in the lab) at the sample position may be imperfect. (iii)

The low-field spin–lattice relaxation times may be too short

to permit field switching fast enough for reliable measure-

ments. (iv) The low-field spin–lattice relaxation time may be

shorter than the time constant of the longest correlation

Fig. 2. Schematic representation of a typical cycle of the main magnetic field B0 employed with field-cycling NMR relaxometry. Desirable specifications are

given. The magnetization after the relaxation interval is recorded in the form of a free induction decay (FID) after a 908 radio frequency (RF) pulse or a spin–

echo pulse sequence in the detection field. The repetition time amounts to several times the spin–lattice relaxation time in the polarization field. (The most

critical sections of the cycle are ringed.)

R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 259

function component so that the validity condition of the

Bloch/Wangsness/Redfield theory [5,6] is violated (see

Eq. (9)).

The low frequency end of the field-cycling range overlaps

that of spin–lattice relaxation in the rotating frame [25–27]

as illustrated in Fig. 1. For the latter method, the use of off-

resonance variants [28–32] and the so-called spin-lock

adiabatic field-cycling imaging (SLOAFI) technique [33,34]

have been suggested in order to cover the two decades

indicated in Fig. 1. Although not an equivalent alternative to

laboratory frame field-cycling relaxometry techniques, the

advantage of the rotating-frame methods is that they can be

implemented on ordinary pulsed NMR spectrometers. Field-

cycling NMR relaxometers on the other hand need some

more or less sophisticated hardware depending on the fast

field switching rates required (see Section 5).

For sensitivity reasons, most field-cycling relaxometry

applications published so far refer to protons. Nevertheless,

the technique is also applicable to X-nuclei such as 2H

(deuterons) [35–37], 31P [15], 19F [38], 7Li [22], and 111Cd

[22]. The frequency ranges for X-nuclei are shifted relative

to protons toward lower frequencies according to the

different gyromagnetic ratios (for a comparison between

protons and deuterons see Eq. (1)). In view of the—relative

to high-resolution standards—poor spectral resolution

intrinsic to electronically switched field-cycling magnets,

isotope labeling is actually the only way to achieve some

chemical specificity unless the system is so heterogeneous

that different FID time constants or relaxation curve

components can be identified and separately evaluated.

Isotopic labeling in particular refers to partially deuterated

systems studied by deuteron resonance. Several examples

will be discussed in Sections 7 and 8.

Field-cycling relaxometry is the only NMR technique

that permits one to cover several decades of the frequency

with the same instrument. The objective of this article is to

demonstrate that this feature makes field-cycling NMR

relaxometry a most powerful tool for the identification and

characterization of molecular dynamics in complex

systems.

There is already a number of related reviews in the

literature [1,2,39]. The interested reader is also referred to a

periodic conference series devoted to the field-cycling NMR

methodology [40]. Since commercial instruments are now

available and field-cycling relaxometry is getting more and

more popular, it appears to be worthwhile to revisit the

special demands, the theoretical background, and various

applications connected with experiments of this sort.

The main application fields of field-cycling NMR

relaxometry to be considered in the following are: (i)

surface related relaxation processes of fluids in porous

materials (Section 7); (ii) polymer dynamics (Section 8);

(iii) biopolymers and biological tissue (Sections 7 and 8);

(iv) liquid crystals and lipid bilayers (Section 9). These

review fields also include special aspects such as nuclear

quadrupole resonance (NQR) detected via quadrupole dips

in the frequency dispersion of the spin–lattice relaxation

time T1 (or peaks in the dispersion of the corresponding rate,

1=T1) [41,42] (see Sections 8.12 and 9.11), and combined

dipolar-order/field-cycling studies [53] as test experiments

specifically testing relaxation theories (see Section 9.10).

The principal field of interest here is molecular dynamics in

complex media.

On the other hand, there is a number of field-cycling

experiments aimed at objectives other than relaxometry in

general. Examples are low-field nuclear or electron Over-

hauser effect studies with high-field signal detection [20,48,

49], shuttle-based fringe field two-dimensional diffusion

ordered spectroscopy (2D-DOSY) [21], and NQR spec-

troscopy in the proper sense [50–52]. Further applications

of wide interest refer to electron paramagnetic relaxation

agents for contrast enhancement in NMR tomography

[43,44], tunneling processes [45], and superconducting

materials [46,47] to mention only a few.

2. Theoretical background of relaxometry

2.1. Definitions

The studies to be reviewed in this article mainly refer to

nuclei with spin 1/2 (in particular protons) and spin 1 (e.g.

deuterons) in diamagnetic systems. The predominant spin–

lattice relaxation mechanism of ‘like’ spins 1/2 is based on

fluctuating dipole–dipole couplings under these circum-

stances. Spin 1 nuclei, on the other hand, possess a finite

electric quadrupole moment that is subject to quadrupole

couplings to local molecular electric field gradients. Since

this quadrupole interaction is much more efficient than

dipolar interactions among spins of the same species, one

can usually neglect the influence of the dipolar relaxation

mechanism in relaxometry studies of quadrupole nuclei.

Dipolar coupled spins 1/2 tend to form multi-spin systems in

condensed matter whereas spins 1 may be considered as

isolated entities.

Since cross-correlation effects [6] are normally of minor

importance in the context of field-cycling NMR relaxome-

try, spin–lattice relaxation in multi-spin 1/2 systems can be

treated as a sum of two-spin 1/2 relaxation rates of a

reference spin i interacting with all other spins (numbered

by j) in pairs. The effective spin–lattice relaxation rate of

dipolar coupled spins 1/2 thus simply reads

1

T1

¼Xj–i

1

Tði;jÞ1

; ð2Þ

where Tði;jÞ1 is the two-spin 1/2 spin–lattice relaxation time

of the ‘tagged’ spin i interacting with a spin j in an ensemble

of multi-spin 1/2 systems.

The relaxation formalisms of dipolar coupled homo-

nuclear two-spin 1/2 systems on the one hand and of

quadrupolar coupled spin 1 nuclei on the other have much in


common and lead to largely equivalent analytical

expressions [5,6]. In both cases the spin–lattice relaxation

rate is described by the formula given below in Eq. (6). This

facilitates comparisons of experimental results obtained

with either technique.

The reason for this analogy is the fact that the spatial part

of the dipolar as well as of the quadrupolar coupling

Hamiltonians can be described using second order spherical

harmonics Y2;mðq;wÞ with m ¼ 0;^1;^2;

Y2;0ðtÞ ¼

ffiffiffiffiffiffiffi5

16p

s½3cos2 qðtÞ2 1�;

Y2;1ðtÞ ¼ 2

ffiffiffiffiffi15

8p

ssin qðtÞcos qðtÞexp½iwðtÞ�; ð3Þ

Y2;2ðtÞ ¼


32p

ssin2 qðtÞexp½2iwðtÞ�;

where Y2;21ðtÞ ¼ Yp2;1ðtÞ and Y2;22ðtÞ ¼ Yp

2;2ðtÞ: The azi-

muthal and polar angles wðtÞ and qðtÞ; respectively, describe

the instantaneous orientation of the coupling tensor relative

to the magnetic field direction [6].

The only terms in the interaction Hamiltonians relevant

for spin–lattice relaxation in ‘like’ spin systems in the

laboratory frame are those for m ¼ ^1 and ^2 selected by

spin operator terms inducing single-quantum and double-

quantum transitions, respectively. In the case of ‘unlike’

spins, where zero-quantum transitions are also connected

with energy exchange between ‘Zeeman spin energy’ and

thermal ‘lattice’ energy, and in the case of rotating-frame

NMR relaxometry (see Section 9.14) m ¼ 0 terms apply as

well [5,6]. These are the transitions allowed for the two

species of spin couplings predominantly considered here.

The polar angle q and the azimuthal angle w define the

orientation of the internuclear vector and of the orientation

of the principal electric field gradient (i.e. of a molecular

axis) relative to the external magnetic flux density ~B0 for

dipolar and quadrupolar interactions, respectively. In the

latter case, one often anticipates (effectively) rotationally

symmetric electric field gradients so that the perturbation

theoretical treatments of the two relaxation mechanisms

become very similar.

Molecular motions in the sense of reorientations of

molecules or chemical groups lead to fluctuating polar

coordinates, q ¼ qðtÞ;w ¼ wðtÞ: As a consequence, the

dipolar or quadrupolar Hamiltonians become time depen-

dent, and hence, induce spin transitions as predicted by time

dependent perturbation theory. With dipolar coupling, there

is a third variable fluctuating as a result of molecular

motion, namely the internuclear distance r ¼ rðtÞ of a two-

spin 1/2 system as the third polar coordinate [54]. This,

however, matters only with inter-molecular or inter-group

interactions while intra-molecular (intra-group) couplings

can normally be associated with constant r values. The

comparison of proton relaxation (dipolar couplings

dominate) with deuteron relaxation (quadrupole interaction

dominates) of the same molecular species permits one to

distinguish contributions from intra- and inter-molecular

relaxation mechanisms.

In the frame of the Bloch/Wangsness/Redfield (BWR)

relaxation theory [5,6], the fluctuations of the spin

Hamiltonians are represented with the aid of (preferably

normalized) autocorrelation functions of the type

GmðtÞ ¼

Y2;mðq0;w0ÞY2;2mðqt;wtÞ

r30r3

t

* +

lY2;mðq0;w0Þl2

r60

* + ðdipolar couplingÞ;

ð4Þ

GmðtÞ ¼ kY2;mðq0;w0ÞY2;2mðqt;wtÞl ðquadrupolar couplingÞ:

The subscripts 0 and t of the spatial variables indicate the

time at which they are to be taken. Actually, the expressions

given in Eq. (4) are stochastically stationary functions, so

that only the magnitude of the time interval matters rather

than the absolute time. The angular brackets stand for an

ensemble average over all spin systems in the sample.

According to time-dependent perturbation theory, the

transition probability per time unit is proportional to the

spectral density (or intensity function) of the fluctuating

coupling inducing the transition. The spectral density is

given as the Fourier transform of the (even!) autocorrelation

functions,

ImðvÞ ¼ð1

21GmðtÞe

2ivt dt ¼ 2ð1

0GmðtÞcos vt dt: ð5Þ

Provided that the molecular motions considered are

isotropic, the spectral density defined in this way is

independent of the subscript m since it is based on the

normalized autocorrelation functions given in Eq. (4). Note

however, that fluctuations with low amplitudes can cause

remarkable differences depending on the order of the

spherical harmonics. We will come back to this problem

in Section 9.14 in the context of order director fluctuations

(ODFs) in liquid crystals.

The spin–lattice relaxation rate of ‘like’ spins directly

reflects the spin transition probabilities per time unit for

single and double-quantum transitions, and hence is propor-

tional to a linear combination of spectral densities in the form

1

T1

¼ Ccoupl½I1ðv0Þ þ 4I2ð2v0Þ�; ð6Þ

where v0 ¼ gB0 is the resonance angular frequency

depending on the gyromagnetic ratio g of the nuclei and

the external magnetic flux density B0: The analytical form of

Eq. (6) is valid for systems of two ‘like’, dipolar coupled

spins 1/2 as well as for spins 1 quadrupolar coupled to local

electric field gradients. The prefactor Ccoupl is merely a

constant specific for the type of the dominating spin

coupling. The first spectral density term in the brackets on


the right-hand side of Eq. (6) refers to single-quantum

transitions and the second term to double-quantum tran-

sitions. The latter consequently is a function of twice the

single-quantum resonance frequency.

Intermolecular dipolar interactions tend to fluctuate

much more slowly than intramolecular couplings. The

reason is that their time dependence is governed by

translational Brownian motions of whole molecules over

distances appreciably exceeding the dimensions of the

molecule. This is to be compared with rotational diffusion

about molecular axes which tends to be much faster.

Provided that molecular dynamics is anisotropic, spin–

lattice relaxation contributions by intermolecular dipolar

interactions therefore show up only at relatively low

frequencies. An example will be discussed in Section 8.6

in the context of polymer chain modes where intra-segment

spin couplings are distinguished from inter-segment dipolar

couplings. On the other hand, isotropically fluctuating

dipolar couplings do not produce strong modifications of the

spin–lattice relaxation dispersion by inter-molecular con-

tributions [54,212].

The spin–lattice relaxation rate resulting from both

contributions may be written as

1

T1

¼1

T intra1

þ1

T inter1

: ð7Þ

The analytical form of Eq. (7) anticipates stochastic

independence of the two types of fluctuating couplings.

This assumption becomes plausible especially for aniso-

tropic motions where the efficiency of the two contributions

refers to very different time scales. As mentioned before, the

relaxation of quadrupole nuclei such as deuterons is

dominated at any frequency by intra-molecular (intra-

group) interactions with local electric field gradients.

Deuteron relaxation is therefore a favourable means for

the discrimination of intra- from inter-molecular relaxation

mechanisms. A distinction is also possible by diluting

molecules by adding perdeuterated analogs of the same

chemical species (see Fig. 12.1 in Ref. [6], for instance).

The overall correlation time of the fluctuating spin

couplings is defined by

tc ¼ð1

0GmðtÞdt: ð8Þ

The ‘weak collision’ condition anticipated in the frame of

the BWR theory and resulting in Eq. (6) can be defined by

the limit

T1 q tc: ð9Þ

Many fluctuation events take place before the spins

perceptibly take part in the thermal equilibration process.

This is the ordinary case mainly considered in the following.

The situation is termed the ‘weak collision case’ since the

fluctuation amplitude is much smaller than the quantizing

field. The weak-collision condition is violated if the external

field becomes less than the so-called ‘local field’. In this

case, fluctuations directly modify the quantizing field which

is a dipolar or quadrupolar coupling field rather than the

external field of the magnet. It is obvious that any

dependence on the external magnetic field vanishes in this

‘strong collision limit’. The crossover from the weak to the

strong collision case shows up in field-cycling relaxometry

experiments typically in solid or liquid crystalline samples at

flux densities below 10 G and must thoroughly be

distinguished from any low-frequency plateau predicted

by the BWR theory in the ‘extreme narrowing limit’,

v0tc p 1:

If the local field is approached adiabatically (see

Eq. (19)), dipolar (or quadrupolar) order is created, and

the relaxation time measured under such conditions refers to

dipolar (or quadrupolar) order spin– lattice relaxation

[55,56]. In the non-adiabatic case, ‘zero-field spin coher-

ences’ arise in the local field [19] leading to completely

different decay mechanisms that are analogous to those of

high-field spin coherences (‘transverse relaxation’).

2.2. Exponential correlation functions and BPP formulas

The standard formalism with which tentative discussions

of relaxation phenomena are usually started is that

considered first by Bloembergen, Purcell and Pound (BPP)

[57]; it refers to high-field relaxation (based on ‘weak

collisions’) due to isotropic rotational diffusion of mole-

cules and intra-molecular interaction of two-spin 1/2

systems with fixed inter-nuclear distances. The correlation

functions, Eq. (4), is then a monoexponential function,

G1ðtÞ ¼ G2ðtÞ ¼ exp{ 2 ltl=tc}: ð10Þ

The spectral density, Eq. (5), consequently adopts a

Lorentzian (or Debye) form,

I1ðv0Þ ¼ I2ðv0Þ ¼2tc

1 þ v20t

2c

: ð11Þ

The complete expressions for the spin–lattice relaxation

rates in the laboratory and rotating frames and, for

comparison, the transverse relaxation rate including the

full expressions for the dipolar coupling constants of two-

spin 1/2 systems are then

1

T1

¼m0

4p

� �2 3

10r6g4"2 tc

1þv20t

2c

þ4tc

1þ4v20t

2c

" #ð!lab:frameÞ

1

T1r

¼m0

4p

� �2 3

20r6g4"2

�3tc

1þ4v21t

2c

þ5tc

1þv20t

2c

þ2tc

1þ4v20t

2c

" #ð!rot:frameÞ

1

T2

¼m0

4p

� �2 3

20r6g4"2 3tcþ

5tc

1þv20t

2c

þ2tc

1þ4v20t

2c

" #

ð!lab:frameÞ:

ð12Þ


The symbols are defined as follows: m0; magnetic field

constant; g; gyromagnetic ratio; "; Planck’s constant divided

by 2p; r; internuclear (fixed intra-molecular) distance, v0¼

gB0 and v1¼gB1: The magnetic flux densities B0 and B1

refer to the external field and the amplitude of the RF field,

respectively. These expressions are sometimes referred to as

BPP formulae and are discussed in more detail in Ref. [6].

Fig. 3 represents the basic frequency and temperature

dependences characteristic of these expressions.

Mono-exponential correlation functions certainly reflect

strongly idealized situations that are scarcely relevant in

most experimental systems. One of the rare examples is the

rotation of crystal water in gypsum [25] and the rotational

diffusion of cyclohexane in the plastic phase [212] (see also

Ref. [6], Fig. 12.1). It must be emphasized that in the

systems typically investigated with field-cycling NMR

relaxometry this sort of scenario almost never occurs and

must consequently be handled with care. Non-exponential

correlation functions will be exemplified in the following

with complex systems such as surface related motions

(Section 7), polymer chain dynamics (Section 8) and liquid

crystals (Section 9).

3. Field-cycling relaxation curves

3.1. Low fields ðBr p BpÞ

Fig. 2 schematically shows a field cycle typically

employed for measurements of spin–lattice relaxation

parameters as a function of the relaxation flux density Br p

Bp: The time intervals and the flux densities to be discussed

in the following are defined in Fig. 4. The sample is

polarized in the polarization field, Bp; which is chosen as

high as compatible with the cooling device of resistive

magnet coils (with respect to a certain duty cycle. See

Section 6.1). The Curie equilibrium magnetization M0 / Bp

at this particular field value is reached with sufficient

accuracy after a couple of spin–lattice relaxation times.

The magnetic flux density is then switched down to

the preselected relaxation field, Br; at which spin–lattice

relaxation is to be examined. On the one hand, the field

switching rate must be large enough to avoid excessive

relaxation losses of the magnetization during the switch-

ing process. On the other hand, it should be slow enough

to permit adiabatic field changes in case the relaxation

field is perceptibly superimposed by local fields (of

arbitrary directions other than that of the polarizing flux

density).

In the relaxation field which is assumed to be much larger

than the so-called local field by residual secular spin

interactions, the magnetization is aligned along the external

magnetic field direction and is initially equal to the Curie

equilibrium magnetization in the polarization field, Mð0Þ ¼

Mzð0Þ ¼ M0ðBpÞ; where we have, for the moment, ignored

potential relaxation losses during the switching interval. It

then relaxes toward the new Curie equilibrium magneti-

zation of the relaxation field, M0ðBrÞ; so that the magneti-

zation evolves according to the following solution of

Fig. 3. Frequency (a) and temperature (b) dependences of relaxation times calculated according to the BPP expressions given in Eq. (12). The theory is valid for

an ensemble of a two-spin 1/2 ensemble of ‘like’ spins subject to intramolecular dipolar interactions with fixed internuclear distance r: Fluctuations of this

coupling are assumed to originate from isotropic rotational diffusion leading to monoexponential correlation functions. For the temperature dependence of the

correlation time of this process, an Arrhenius law was anticipated: tc ¼ t0c exp{DE=kBT}: All parameter values have been chosen arbitrarily but in typical

orders of magnitude. The condition Dvtc ¼ 1 indicates the cross-over from the (temperature independent) rigid lattice limit to motional averaging. Dv is the

linewidth in the absence of motions. Examples where this simplest form of relaxation behaviour applies are discussed in Ref. [6]. However, relaxation

characteristics of complex systems usually do not look like this in practice.


Bloch’s equation [6]:

MzðtÞ ¼ M0ðBrÞ þ ½M0ðBpÞ2 M0ðBrÞ�exp{ 2 t=T1ðBrÞ}:

ð13Þ

The magnetization remaining after the relaxation interval t

is finally detected with the aid of a 908 RF pulse or a suitable

spin echo sequence in the form of an NMR signal after

switching the magnetic flux density to a fixed detection field

Bd: Since the acquisition of a free-induction signal takes

only a few milliseconds at most, the detection field period

can be kept extremely short which improves the duty cycle

considerably (compare Fig. 2). The detection flux density

can hence be very strong without overloading the magnet

coil.

After having recorded the signal, the flux density is

switched back to the polarization flux density. After a period

of a couple of (high-field) spin–lattice relaxation times the

whole cycle begins again. In order to avoid thermal

instabilities such as drifts of the field in resistive magnets,

the field cycle is often interrupted by an intermittent zero-

current interval immediately after signal detection for the

sake of lower duty cycles [58]. This however is a technical

problem (see Section 6) and does not affect the principle of

the relaxation curves.

Incrementing the relaxation interval t thus permits one to

record the relaxation curve for a given relaxation flux

density Br: The spin–lattice relaxation dispersion is then

scanned point by point by stepping Br through a series of

discrete values spread over the desired flux density (or

frequency) range.

In Eq. (13), the finite switching-down and switching-up

intervals ðDtÞd and ðDtÞu; respectively, have not been

considered explicitly (see Fig. 4). These are taken into

account in the modified relaxation curve formula

Mz½tþ ðDtÞd þ ðDtÞu�

¼ ½ðMz½ðDtÞd�2 Mr0Þe

2t=T1ðBrÞ þ Mr0�e

2cu1 þ cu

2; ð14Þ

where cu1 and cu

2 are constants. The derivation of this

expression can be found in Ref. [6], p. 140. The formula to

be fitted to the acquired raw data, Mdetectedz ðtÞ; of field-

cycling relaxometry experiments thus has the simple form

Mdetectedz ðtÞ ¼ M1

z þ DMeffz e2t=T1 ; ð15Þ

Fig. 4. Field cycle for low relaxation fields ðBr p BpÞ: schematic representation of the external flux density B0; the radio frequency amplitude B1; and the

magnetization Mz in the diverse field-cycling time intervals for polarization fields much larger than the relaxation field. The shaded section indicates the

variable relaxation interval t: The vertical arrow indicates the time when the signal is recorded. The polarization interval tp is typically chosen to be five times

the spin–lattice relaxation time in that field. The detection interval td may be as short as needed for the acquisition of an FID signal. The ‘down’ and ‘up’ field

switching times are indicated by the intervals ðDtÞd and ðDtÞu; respectively. tRF is the RF pulse width. The cycle is repeated periodically for different relaxation

intervals t and different relaxation flux densities Br while all other intervals and flux densities remain constant. A variable zero-field interval just after signal

detection may be necessary in order to ensure a certain duty cycle.


where DM1z ;DMeff

z and T1 are fitting parameters. DM1z and

DMeffz are implicitly defined by Eq. (14).

Relaxation losses in the finite switching intervals

obviously diminish the dynamic range of the variation of

the relaxation decay and hence the experimental accuracy,

but do not cause any systematic experimental error provided

that the passages between the different field levels are

reproducible when incrementing the relaxation interval t for

a given relaxation flux density Br [6]. The limitation of field-

cycling NMR relaxometry with respect to the finite

switching intervals is thus given by the requirement that

DMeffz is large enough to allow for a precise evaluation.

Apart from this condition, the switching intervals need not

to be much shorter than the low-field relaxation times. In

this respect the method is more tolerant than often

anticipated in the literature.

In order to extend the range to higher fields, the field-

cycling NMR relaxometry experiments can be sup-

plemented by ordinary high-field relaxation measurements

employing the inversion-recovery or saturation-recovery

variants. Comparative spin–lattice relaxation experiments

‘in the rotating frame’ ðT1rÞ can be of interest in special

cases particularly in the presence of molecular order (for

details and an example see Section 9.14) and for tests of

potential low-field artifacts (see Section 10).

3.2. High fields ðBr ! BdÞ

If the relaxation flux density Br approaches the

polarization flux density Bp the dynamic range of the

relaxation curve, DMeffz ; becomes too small for accurate

evaluations of spin–lattice relaxation times according to

Eq. (15). In this case it is more favourable to start the cycle

in the absence of any polarization field (as a sort of field-

cycling version of the standard ‘saturation/recovery tech-

nique’) as illustrated in Fig. 5 or even with a negative

polarization (as the field-cycling variant of the standard

‘inversion/recovery method’) prepared with an initial 1808

RF pulse at the end of the polarization interval. That is, the

relaxation curve is a build-up curve starting from a value

close to zero or from a negative value, respectively. The

relaxation curves are then of the type

Mdetectedz ðtÞ ¼ M1

z 2 DMeffz e2t=T1 : ð16Þ

Fig. 5. Field cycle for high relaxation fields ðBr ! BdÞ: schematic representation of the external flux density B0; the radio frequency amplitude B1; and the

magnetization Mz in the diverse field-cycling time intervals for relaxation fields approaching the detection field. The shaded section indicates the relaxation

interval t: The vertical arrow indicates the time when the signal is recorded. The field switching times are indicated by the intervals ðDtÞu1 (from ‘zero’ to the

relaxation field) and ðDtÞu2 (from the relaxation field to the detection field). tRF is the RF pulse width. The recycle delay should be long enough to ensure

complete relaxation of the magnetization before the next cycle begins. The cycle is repeated periodically for different relaxation intervals t and different

relaxation flux densities Br while all other intervals and flux densities remain constant.


Above the field-cycling range, the spin–lattice relaxation

dispersion is usually supplemented by data measured with

ordinary, stationary field NMR spectrometers.

4. Signal-to-noise ratio

One of the crucial limitations of field-cycling appli-

cations is the signal-to-noise ratio. If the detection field is

reproduced in subsequent cycles accurately enough, phase

sensitive detection is feasible, so that signals can be

accumulated [18]. However, averaging a number of

transients, as is standard in high-resolution NMR spec-

troscopy may conflict with the limited stability intrinsic to

field-cycling systems.

The single-transient signal-to-noise ratio can be

expressed by [5,59–63]

S=N / B0j

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihQVs

kBT

n0

Dn

� �s; ð17Þ

where h is the filling factor of the RF coil, Q is the quality

factor of that coil, Vs is the sample volume, kB is the

Boltzmann constant, T is the absolute temperature, n0 ¼

gB0=2p is the Larmor frequency, Dn is the bandwidth of the

receiver filtering and amplification system, and j , 1

represents the reciprocal noise level of the receiver

electronics. Eq. (17) tells us that the S=N ratio increases

proportional to B3=20 : On the other hand, the use of high

polarization and detection flux densities, large samples, high

Q coils, low-noise receivers and narrow RF filters is

favourable as well in this respect.

For field-cycling NMR relaxometry where the polariz-

ation and detection fields often have different flux densities

(compare Fig. 2), Eq. (17) must be modified according to

S=N / Bpj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihQVs

kBT

nd

Dn

� �s; ð18Þ

where nd ¼ gBd=2p is the detection Larmor frequency, Bd is

the detection flux density, and Bp is the flux density of the

polarization field. High polarization and detection flux

densities are crucial for a good sensitivity.

However, it should be kept in mind that Joule heating and

the thermal load of resistive magnet coils is higher for the

polarization field by orders of magnitude relative to those

for the detection field. Sample polarization requires a period

of typically five times the (high-field) spin–lattice relax-

ation time, whereas signal acquisition times are restricted to

the short interval where the FID signal is finite. It is

therefore favourable to use moderate polarization fields not

straining the magnet coil and its cooling device too much,

and compensate the sensitivity loss by correspondingly

larger detection flux densities. To avoid the Joule heat

problem, superconducting coils have been used instead of

resistive magnets [15] at the expense of the ease of system

operation. The duty cycle of the magnet can also be reduced

by intermittent zero-current intervals.

Polarization and detection flux densities up to 1.5 T have

been shown to be feasible with electronically switched

systems equipped with a superconducting [15] or a resistive

copper [18] magnet coil. Switched flux densities up to 3 T

may even be possible with superconducting magnet coils as

a pilot study has demonstrated [64].

The main technical limitation of fast switchable super-

conducting coils is the so-called twist of the superconduct-

ing filaments which should be extremely short while the

filament thickness should be as thin as possible. Unfortu-

nately wires with such specifications are rarely available

commercially in the relatively small quantities needed for

research purposes. Another difficulty is the excessive liquid

helium consumption and, as a consequence, the limited

mechanical magnet lifetime due to frequent room tempera-

ture periods when the system stands idle for some time.

Therefore, field-cycling relaxometers are typically

equipped with resistive copper, aluminium or even silver

magnet coils (see recent developments by the manufacturer

STELAR). Initiated by the IBM group, copper magnets

cooled by liquid nitrogen down to 77 K in a bath cryostat

have been used for some time: this low temperature reduces

the specific resistance relative to room temperature by a

factor of seven so that a good cooling efficiency is

accompanied by a considerably lower Joule heating [65].

5. Crucial specifications of the field cycle

If not concealed by sample internal local fields due to

secular spin interactions, the low-field limit of a field cycle

is determined by the precision with which the earth field and

stray fields from any other magnets or magnetic materials in

the lab are compensated. A set of compensation coils

surrounding the proper field-cycling magnet is mandatory if

fields below 1024 T are to be reached. Actually, flux

densities as low as 10 mT corresponding to a few hundred

Hertz proton resonance could be probed in this way

provided that the relaxation field is reached and settled

with an equivalent precision in a time short enough for

reliable relaxation time measurements.

This latter point is a crucial problem. At low fields,

proton and especially deuteron spin–lattice relaxation times

of viscous systems may easily be less than a millisecond.

That is, coming from the large polarization field, relaxation

fields as low as 1025 T must be reached, settled and

stabilized within a total passage time in the order of

milliseconds with the desired accuracy of better than about

10%. The short settling time is a stringent condition for

short low-field relaxation times and it is not easy to fulfill

this condition practically.

Likewise, the passage from the relaxation field to the

detection field should occur in a transition time of the same

order as the low-field relaxation time. In particular,


the detection flux density needed for magnetic resonance

must be hit and reproduced with an accuracy of about 1025

in subsequent transients. This corresponds to the bandwidth

of the radio frequency system and should be matched to the

field homogeneity within the sample as a further limiting

factor.

Relative field homogeneities between 1025 and 1024 are

feasible with special coil designs and current density

distributions [18,67,68] (see Section 6.3). This is sufficient

for most applications, so that the switching time for the

detection field is defined by the interval needed until the

final level is stabilized with a relative accuracy between

1025 and 1024. Practical switching times defined in this way

should be of the order 1 ms (for an example see Fig. 6).

It must be emphasized that the field homogeneity, the

field reproducibility, and the RF bandwidth are experimen-

tal specifications of the detection field and the detection

system that should match each other. Over-specification of

any of these parameters relative to the other limitations

would unnecessarily increase the technical expenditure.

For a field cycling relaxometer, it is of paramount

importance to check and calibrate the field cycle with the

aid of a test device ensuring the accuracy, the time

resolution and, in particular, the dynamic range required

for this task. A fast field probe placed at the sample

position and connected to a 12–16 bit transient recorder

with a sufficient bandwidth is a safe way to avoid

experimental artifacts by imperfections of the field cycle

[18]. A field cycle recorded in this way is shown in Fig. 6.

Digital teslameters such as the Projekt Elektronik FM210

with a bandwidth of 35 kHz and a resolution of 0.01 mT

are also suitable for the calibration of field cycles [69].

The crucial problem is the time resolution of the field

measurement. That is, the field control and calibration

must be performed with a time resolution better than the

shortest relaxation time to be expected for the samples

under consideration. Field measurements by compensation

techniques using NMR [14,70] are favourable with respect

to accuracy but are hard to fulfill the time resolution

requirement since NMR signal acquisition typically takes

longer than the field interval to be probed.

Modern field-cycling relaxometers reach maximum

switching rates in the order of 1000 T/s ensuring minimal

relaxation losses during the switching intervals. If the total

switching interval, Dt; including the settling times of

the relaxation or detection fields is much less than the

shortest relaxation time in the field-cycling range, T1;min; the

total magnetization at the end of the polarization or

relaxation intervals will be transferred to the subsequent

section of the field-cycle without perceptible losses. Longer

switching intervals, which exceed the shortest spin–lattice

relaxation time, Dt . T1;min; reduce the signal variation

range and, hence, the experimental accuracy of relaxation

time measurements. However, even in this case there is little

danger of systematic errors as outlined in Section 3.1 (see

Eq. (14)).

The field-variation rates may be as high as technically

possible as long as the adiabatic condition [71,72] is fulfilled,

1

B2~B £

d~B

dt

��p gB; ð19Þ

where ~B ¼ ~B0 þ ~Bloc is the total flux density ‘seen’ by the

nuclei. ~Bloc is the local field caused by secular spin

interactions. That is, the local magnetization should always

remain aligned along the quantizing field ~B:The directions of

the local fields are more or less randomly distributed and do

not coincide with that of the external magnetic flux density~B0: At high external flux densities, B0 q Bloc; the local fields

can be neglected and the quantization direction coincides

with the direction of ~B0: Under such conditions, there is no

upper limit of the field variation rate. It remains always

adiabatic.

If B0 , Bloc; the local fields tend to govern the

quantization field. There are two principal ways to conduct

the experiment under such conditions. Firstly, one can vary

the field adiabatically (see Eq. (19)) so that Zeeman order is

converted into dipolar or quadrupolar order. This technique

is called ‘adiabatic demagnetization in the laboratory frame’

[55]. It can be used for measurements of dipolar or

quadrupolar-order relaxation times.

In the opposite limit, when the local fields are

approached non-adiabatically, coherent spin states leading

Fig. 6. Field cycle (left) of a superconducting magnet coil [15] recorded with a Siemens Hall probe RHY placed at the sample position. The data were digitized

with a 12 bit Nicolet oscilloscope. The cross-over to the relaxation field (position A) is represented in enlarged form on the right. The digital noise corresponds

to ^1 kHz.


to finite expectation values of the spin components

transverse to the local field directions are produced. This

permits the so-called ‘zero-field NMR or NQR spec-

troscopy’ [17,19,73–77]. That is, the local-field interval is

taken as the time domain (‘coherence evolution interval’) of

a corresponding Fourier transform spectroscopy procedure.

The interesting feature of this sort of spectroscopy is the fact

that resonance lines can be recorded in powders without

being subject to powder line shape patterns.

6. Relaxometers

The first electronically switched field-cycling instru-

ments were built in the sixties of the past century, and

were designed by Redfield, Fite and Bleich at the IBM

Watson Research Laboratory [13] and by Kimmich and

Noack at the University of Stuttgart [14]. Since then,

although numerous instruments of similar designs have

been made, they have not found the wide spread

acceptance of the commercially available relaxometer

introduced by STELAR some years ago. This relaxometer

has now become widely used in the relaxometry field. The

specifications of various home built systems in use in

various laboratories are partly superior with respect to the

maximum flux density, but these advantages are juxta-

posed against the ease of operation and maintenance

offered by commercial instruments.

Fig. 7 shows a block diagram of a typical relaxometer.

The components that must specifically be designed for field-

cycling purposes are the magnet coil and a switchable power

supply for the magnet current. The performance of both

units largely depends on an efficient cooling system

(see Section 4). The main objective of corresponding

hardware developments is the need to achieve fast switching

and settling times of precise field cycles combined with

strong polarization and detection fields to obtain a signal to

noise ratio ðS=NÞ as good as possible. Typical examples

of up-to-date relaxometer technology can be found in

Refs [18,58].

6.1. Conditions for field-cycling magnets

The energy stored in the field of an air-core magnet coil

is given by

E ¼1

2m0

ðspace

B2 d3r; ð20Þ

where the integral covers the whole space over which the

magnetic field is spread. This is the amount of energy that

has to be cycled into and out of the magnet as fast as

possible. Irrespective of the design of the power supply,

smaller total field energies are easier to cycle fast than larger

ones. It is therefore of paramount importance to minimize

the total field energy while retaining large peak flux

densities in the sample. That is, the magnet coil should be

as compact as possible. On the other hand, this requirement

conflicts with

† a good cooling efficiency permitting the high current

densities needed for high fields

† a good field homogeneity (that is, the relative field

variation in the sample volume should not exceed the

stability and reproducibility of the detection field which

is (or should be) of the order 1025)

† large room temperature bore diameters and large sample

volumes (the signal sensitivity is proportional to the

sample size which is typically 1–2 ml)

The problem is therefore to find an operational compromise

between these factors while avoiding any over-specifica-

tions in this respect.

A further condition of the magnet design is that it must be

capable of fast field variations. On the one hand this requires

some mechanical stability in view of the magnetic impulses

arising during field switching. On the other hand, all

materials used must be non-magnetic. In particular, one

cannot use a ferromagnetic yoke as normally employed in

ordinary electromagnets because of the slow frequency

response intrinsic even to magnetically ‘soft’ iron.

The simplest coil geometry for a field cycling magnet is a

solenoid of cylindrical symmetry (Fig. 8). Assuming

resistive conductors, the efficiency can be expressed in

terms of the field-to-power ratio. The contribution of a

current ring of cross sectional area drdz (see Fig. 8) to the

total magnetic flux density along the solenoid axis is

dB ¼m0

4p

� �jI f ðr; zÞ

r2

ðr2 þ z2Þ3=2

" #dr dz: ð21Þ

Fig. 7. Block diagram of a typical field-cycling relaxometer. The grey

shaded blocks represent units specifically designed for field-cycling

purposes.


The dissipated power can be written as

dW ¼r

l

� �j2

I f 2ðr; zÞr dr dz; ð22Þ

where jI is the ‘current factor’ (proportional to the total

current flowing through the coil), f ðr; zÞ represents the

current density distribution (equal to unity for uniform

windings), l is the specific resistance of the winding

material, r is the fraction of the coil volume occupied by the

conductor (‘coil packing factor’), and the set ðr; zÞ

represents the corresponding coordinates (the variable f is

absent due to the azimuthal symmetry). After integration

these equations read

B ¼m0

4p

� �jI

ðþl

2ldzðr1

r0

drr2f ðr; zÞ

ðr2 þ z2Þ3=2ð23Þ

and

W ¼ j2I

r

l

� �ðþl

2ldzðr1

r0

r dr f 2ðr; zÞ; ð24Þ

where 2l is the coil length, and r0 and r1 are the inner and

outer radii of the cylindrical windings (see Fig. 8).

Eliminating the current factor jI from these equations, we

find [78,79]

B ¼ G

ffiffiffiffiffiffiWl

rr0

s; ð25Þ

where

G ¼m0

4p

� � ðþb

2bdgða

1f ðd;gÞ

d2 dd

ðd2 þ g2Þ3=2ðþb

2bdgða

1dd f 2ðd;gÞd

� 1=2ð26Þ

with g ¼ z=r0; d ¼ r=r0;a ¼ r1=r0 and b ¼ l=r0:

The quantity G represents the geometric properties of the

current distribution and has typical numerical values

between 0.15 and 0.2 [67]. In the literature, it is termed

‘Fabry factor’, ‘geometric factor’ or, simply but mislead-

ingly, ‘G-factor’. The Fabry factor is well known for certain

standard geometries like homogeneous winding, Bitter

radial, Gaume and Kelvin distributions [80,81]. For field-

cycling magnets, the Fabry factor should be made as large as

compatible with the conditions mentioned above.

The optimisation of a given design by simultaneous

consideration of all requirements is a complex problem

usually solved with the aid of empirical parameter iteration.

A favourable possibility is to first optimize the design for

only two magnet characteristics and then to improve the

remaining factors to obtain the best result. This method was

employed for a notch coil with uniform current density [18].

Another procedure is to improve the magnet characteristics

iteratively one by one beginning with a geometry that

minimises the electric power needed for the desired

magnetic flux density.

The Fabry factor for air-core magnets is very low in

comparison with iron-core electromagnets. As a conse-

quence, the current density needed in a field-cycling device

to produce a given magnetic field is much higher than in a

conventional electromagnet. An important aspect in the

design is therefore to assess how the dissipated power will

be distributed in the magnet, and how it can efficiently be

removed by the cooling system.

Liquid nitrogen as a cooling medium is favourable in two

respects. In the first place the temperature is far below room

temperature so that the cooling efficiency is good. Secondly

the specific resistance of the current conducting materials is

strongly reduced so that Joule heating during operation

remains small. In this respect, superconducting magnet coils

are perfect because Joule heat in this case is practically

negligible [15,64]. However, the handling of cryogenic

field-cycling magnets is inconvenient and expensive. The

reason is that, unlike ordinary NMR superconducting

magnets, the coil must be permanently connected to the

power supply by leads that carry not only the current but

also heat from outside into the magnet. The cryogen

consumption is further increased by the electric power

dissipation in the copper matrix of the wires during switched

operation due to local eddy currents.

As a consequence, the field-cycling relaxometers cur-

rently in use are non-cryogenic systems. They are operated

near room temperature and their primary cooling circuits are

filled with suitable liquids. Joule’s heat is finally transferred

Fig. 8. Schematic section of a solenoid (symmetry axis ¼ z-axis) as the

simplest field-cycling magnet coil design. 2l; coil length; r0; inner coil

radius; r1; outer coil radius; dB; flux density contribution along the coil axis

from the current in a winding element of cross-section dr times dz:


in heat exchangers to tap water or cooling water supplies if

available in the lab (see Ref. [18], for instance).

The power transferred from the current leads to the

coolant can be expressed as [79]

Wt ¼ atSDT ; ð27Þ

where at is the ‘heat transfer coefficient’, S the contact area

between the conductor and the coolant,DT is the temperature

difference between the leads and the cooling medium. The

heat transfer coefficient depends on the properties of the

coolant such as viscosity, specific heat, density, and thermal

conductivity. For efficient cooling, it is important to use an

adequate coolant and to maximise the contact area between

the flowing fluid and the coil while keeping the coolant

temperature as low as possible. The use of water as a cooling

medium is unfavourable because of its electrolytic proper-

ties. Other coolants like low-viscosity oils (as typically used

in transformers), liquid Freon or perfluoroheptane turn out to

be more favourable. The latter two media are practically free

of protons so that spurious proton signals on these grounds

are avoided. The contact area can be optimised by allowing

for coolant flow between the winding layers. The current

carrying leads are thus exposed to the flowing coolant

everywhere in the magnet coil.

A general problem of resistive field-cycling magnets is

the time dependent production of Joule heating in the course

of the current/field cycle leading to some temperature

distribution in the magnet and, hence, to mechanical strain.

The peak power to be dissipated in the magnet can reach

40 kW and more. Any mechanical deformation unavoidably

causes drifting field instabilities affecting the measure-

ments. Again, this problem may be solved with cryomagnets

[15] at the expense of the disadvantages mentioned above.

For resistive magnets, thermal drifts can be compensated

with the aid of a current control electronics using a

temperature sensor [58]. It is also favourable to keep the

magnet duty cycle constant in the course of the experiment

by inserting zero-current intervals of varying length

between subsequent field cycles.

6.2. Optimization principles for field-cycling systems

Neglecting the ohmic resistance of the magnet for the

moment, the application of a constant voltage U causes a

linear increase of the current in a period t;

I ¼U

Lt; ð28Þ

where L is the inductance of the magnet coil. This also

means a linear increase of the magnetic flux density

according to

B ¼ ccoilI ¼ ccoil

U

Lt; ð29Þ

where ccoil is a constant specific for the coil geometry.

That is, if the nominal flux density Bn corresponding to

the nominal current In is reached in the total switching time

t ¼ InL=U; we have a switching rate

dB

dt¼

Bn

t¼

UBn

LIn

: ð30Þ

Since both the nominal flux density and the switching rate

should be maximal, we define the quantity

C ; Bn

dB

dt¼

B2n

t¼

UB2n

LIn

¼c2

coilPpeak

L; ð31Þ

as the decisive factor to be maximized in the design of the

magnet and its power supply. The quantity Ppeak ¼ InU is

the peak power the power supply is capable of providing. To

optimize a field-cycling system therefore means to maxi-

mize ccoil and Ppeak; and to minimize L at the same time.

The optimization of power supplies with respect to the

peak power will be discussed below. Apart from the number

of windings per unit length, a small inductance in particular

implies the restriction of the magnetic field to a small

volume in order to keep the total field energy needed for a

certain center magnetic flux density low (see Eq. (20)). This

will be the case if the magnet coil is compact. Both ccoil and

L furthermore depend on the coil geometry and have to be

optimized in this respect [80,81].

One starts with the consideration of the minimum magnet

bore diameter allowing for the accommodation of the probe

for the desired sample size. The next steps are iterative

approaches to optimize C for small outer diameter and coil

length, a large coil packing factor C compatible with the

cooling efficiency needed for the envisaged maximum

effective current density. The geometric magnet shape and

the current distribution are then not yet optimal with respect

to the required field homogeneity corresponding to a relative

field variation of about 1025 in the sample volume. This

value should match the electronic stability and the

reproducibility conditions of the electronic system.

The homogeneity can be improved by modifications of

the solenoid coil shape by a central ‘gap’ [65] or a ‘notch’

[18,67] or most efficiently by spatial optimization of the

current density [68]. From the technological point of view,

the latter method is the most demanding and efficient one,

and will be discussed in more detail below.

6.3. Diverse magnet designs

Fig. 9 shows a cross section of a notch magnet consisting

of a combination of several concentric solenoids sup-

plemented by shorter correction coils. The axial distance

between the outer top and bottom coils defines the notch

width, which is adjusted for an optimum homogeneity. Each

solenoid coil is self-supporting and consists of two winding

layers of wires of rectangular cross section for optimal

packing. The wires are glued together with epoxy resin. The

spacing between the double layers permits the cooling fluid

to circulate so that each section of the wire is directly


and efficiently cooled. This magnet design is relatively

simple and inexpensive to realize and provides detection

flux densities up to 1.5 T of the required homogeneity. For

construction details see Ref. [18].

Another proposal for a compact magnet producing

homogeneous magnetic fields is an (approximately) ellip-

soidal shape of the winding package as illustrated in Fig. 10.

With notch or gap coils, the effective winding (and hence

current) density is reduced in the middle of the magnet so

that the field at the magnet fringes is enlarged at the expense

of that in the center. The result is a homogenized central

field. The same effect is achieved with an ellipsoidal shape

obtained by corresponding variation of the bore diameter.

Actually, an ideal, that is a closed ellipsoid would produce a

perfectly homogeneous field in the interior. A practical

layout [82] can be optimised by varying the winding radius y

along the x axis according to

y ¼ b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2

x2

a2þ c1lxl

3þ c2x4

s; ð32Þ

where a and b are the half axes of an ellipsoid rotationally

symmetric around the x axis. The constants c1 and c2 are

adjusted for the optimisation of the field homogeneity.

These correction parameters are needed to compensate for

the effect of the bore openings on both sides (see Fig. 10).

The best compromise with respect to compactness, field

strength, homogeneity, and ease of practical handling was

certainly found with the optimized current density solenoid

[16,68,83] although this sort of magnet is not easy to

fabricate. The cross section of the current leads is no longer

constant at different positions within the winding package.

Rather the current density is optimized as a function of the

position by varying the lead thickness both along the axis of

the coil and in radial direction. Actually, this sort of magnet

approximates to the so-called Kelvin configuration which

promises the best Fabry factor compared with ordinary

solenoids or Bitter magnets [84]. The magnets consist of a

series of concentric cylinders of copper, aluminium or silver

into which helical gaps are milled, so that each cylinder

represents a solenoid coil with a current lead of varying

cross section. The optimal cross section as a function of the

position is found with the aid of a Lagrange variation

calculation of the current density needed for best homogen-

eity at given outer dimensions. In this way, the power

needed to generate the envisaged magnetic field can be

minimised while simultaneously ensuring a good homogen-

eity and switching properties [68].

The following formalism is used for this purpose. The

current in the coil is subdivided into n elements. One then

considers the magnetic field density contribution ~Bjð~rmÞ at a

position ~rm produced by a current element Ij: The sum

Fig. 9. Section across a ‘notch’ magnet [18].

Fig. 10. Section across an ellipsoidally shaped magnet [82].


Pnj¼1

~Bjð~rmÞ is then brought close to the target function~BTð~rmÞ; while keeping the total dissipated power P ¼Pn

j¼1 RjI2j in the element resistances Ri as low as possible.

The primary problem to be solved with the aid of the

Lagrange variational method is to find the current

distribution corresponding to the minimum of the auxiliary

function

F ¼Xn

j¼1

RjI2j 2 lL

Xn

j¼1

½Bj;zð~rmÞ2 BTzð~rmÞ� ð33Þ

with the Lagrangian multiplier lL: Here we have

restricted ourselves to the z components of the flux

densities for reasons of the symmetry of the magnet set-

up. On this basis several field-cycling magnet versions

have been built in academic laboratories [16,68,83] as

well as in the STELAR company [58]. Fig. 11 shows the

layout of a typical cylinder component of an optimized

current density magnet [83]. The width of the current

leads varies along the cylinder axis. Several such,

individually optimized cylinders are arranged concentri-

cally. A less sophisticated but also less efficient design

can be achieved by cutting uniform loops with different

spacings in metallic hollow cylinders [85].

6.4. Conditions for field-cycling power supplies

The second important and method-specific component

of field-cycling relaxometers is the power supply for

the magnet current. Appropriately this is designated as a

power supply system rather than a single unit. The

problem one is facing in the design of such a system is to

solve partially conflicting requirements (compare Figs. 2

and 4 to Fig. 6):

† The magnet current must be switched fast, i.e. the

control time constant should be short.

† The magnet current should be very stable once a field

level is reached. This in particular refers to the

detection field where a stability of 1025 is needed. A

good stability stipulates a long control time constant.

† An extremely high peak power is needed in order to

energize the magnet in a short time. We are speaking

of tens or even hundreds of kilowatts.

† The magnet current must settle in periods of less than a

millisecond when reaching the relaxation field. Over-

shooting of the desired field level or any slow

exponential approaches to the preset value must be

avoided (compare Fig. 6).

† Magnetic resonance must be met in the detection field

in subsequent cycles with a precision permitting phase

sensitive detection and, hence, signal accumulation

without intermittent manual adjustments. This again

means a relative accuracy of 1025.

Needless to say, such demanding specifications require the

development of sophisticated high-power circuits.

In principle, two basic strategies can be distinguished

for the design of the global circuit consisting of the

inductance of the magnet and the output loop of the

power supply system: each one deals with either an over-

damped or a sub-damped resonance circuit. In the first

case, all current changes tend to be governed by an

exponential time dependence, whereas the latter set-up

intrinsically tends to produce oscillatory current changes

by nature. Both strategies will be discussed in the

following sections.

6.5. Principles of over-damped power supply circuits

Fig. 12 shows the basic current loop that could be used

for cycling between two current levels. The series resonant

circuit is over-damped by resistors. Two current levels can

be chosen by switching between different damping resistors.

It is assumed that Rhigh p Rlow and R p Rlow; and that all

capacitances can be neglected. The currents Ihigh and Ilow

correspond to the ohmic resistors Rhigh and Rlow;

respectively.

Consider now the situation when the circuit is switched

from Rhigh to Rlow; that is, from the high to the low-current

state. The time evolution of the current is described by the

differential equation according to Kirchhoff’s mesh rule

V0 2 LdI

dt¼ IðRlow þ RÞ < IRlow ð34Þ

Fig. 11. Typical cylinder component of an optimized current density

solenoid [83]. Several of such cylinders are arranged concentrically with a

coolant filled gap in between. The current density varies along the coil axis

according to the conducting cross-section of the windings.


with the initial condition Ið0Þ ¼ Ihigh: Integrating Eq. (34)

results in

IðtÞ ¼ ðIhigh 2 IlowÞexp 2t

tdown

� �þ I ð35Þ

with the time constant tdown ¼ L=Rlow: Evidently, for fast

switching we need a magnet with a low inductance. In

addition, the switching rate can be improved by connecting

a damping resistor in series during the transition, in order to

dissipate the magnetic energy stored in the magnet [67].

During the inverse transition we have,

V0 2 LdI

dt¼ IðRhigh þ RÞ; ð36Þ

where the initial condition is now Ið0Þ ¼ Ilow: The solution

is

IðtÞ ¼ ðIhigh 2 IlowÞ 1 2 exp 2t

tup

!" #þ Ilow ð37Þ

with tup ¼ L=ðRhigh þ RÞ: As a consequence we have

tdown p tup: The time derivative of Eq. (37),

dI

dt¼

1

L½V0 2 IðtÞðRhigh þ RÞ�; ð38Þ

represents the slew-rate during switching from the low-

current to the high-current state. Its maximum is reached if

the second term in the brackets becomes negligibly small.

This can be achieved on the one hand by a small resistor R

(assuming that Rhigh is also small) which in particular means

that the magnet is to be wound of a thick and short wire. On

the other hand, the voltage during the switching interval

should be as large as possible.

However, the situation to be discussed in this context is

more complicated. In principle, a low resistance magnet

favours low Joule heating per cycle. As a consequence,

the cooling requirements appear to be less critical, and

would hence permit a larger duty cycle. On the other hand,

low resistance and inductance result in a lower magnetic

flux density for a given current. Larger currents are therefore

needed in order to meet the desired specification. That is, the

cooling problem enters again apart from some technical

inconvenience of handling large currents. So, the system

design must unavoidably lead to a compromise.

The slew rate given by Eq. (38) is more successfully

increased by temporally boosting the voltage V0 during the

switching-up interval. A practical solution of this strategy is

to connect a large, pre-charged boosting capacitor in

parallel with the magnet during the switching time (see

Fig. 13). In such a configuration, an additional high-voltage/

low-current power supply ðVCÞ is used to charge the

capacitor between the switch-up steps. The energy stored

in the capacitor can then be fed into the magnet in order to

energize it in a time much shorter than the time constant tup

given above in context with Eq. (37) [13].

This capacitor-boosted field switching technique can also

be employed for the reduction of tdown: Connecting the pre-

charged boosting capacitor in series (instead of parallel)

with the magnet with a polarity in the sense that a current

opposite to the momentary magnet current is fed in, means

that magnetic field energy is transferred back to the

capacitor [13,18,86]. In this way, the current is actively

forced to subside in a time much shorter than the passive

time constant tdown defined above in context with Eq. (35).

A circuit employing both boosting-up and boosting-

down techniques is shown in Fig. 14. For the sake of

Fig. 13. Over-damped resonant circuit supplemented by a capacitor device

for temporally boosting the voltage during the switching-up interval. The

boosting capacitor C is pre-charged by the voltage VC before it is switched

electronically by the logic to the magnet current loop.

Fig. 12. Principle of an over-damped resonant circuit consisting of a magnet

with the inductance L; a voltage source V0; and resistors R; Rlow; Rhigh:

Switching between Rlow and Rhigh corresponds to two different current

levels in the magnet. R represents all other resistances of the circuit

including the wiring.


simplicity, we assume ideal resistance values here, i.e.

Rhigh ¼ 0; and Rlow ¼ 1: The electronic switches S1 and S2

are controlled by the logic. During the high-current state, S1

is closed whereas S2 is open. For switching down to the low-

current state, S1 opens. The voltage across the magnet coil

reaches a peak value as a result of Lenz’s law. The time-

varying current is allowed to flow into the capacitor via the

diodes D1 and D2: The capacitor is charged by this current

and by the parallel voltage supply VC: For switching up, S1

and S2 are closed, so that the capacitor voltage is applied to

the magnet via the diodes D3 and D4: After reaching the

desired high current state, S2 opens. D5 protects the power

supply V0 against the high voltage of the capacitor.

6.6. Principles of sub-damped power supply circuits

As a potential alternative to the over-damped circuit

strategy, we now discuss sub-damped magnet circuits.

Fig. 15 shows the basic scheme. A capacitor C is added to

the circuit in parallel with a series damping resistor Rs:

The resistor element R represents the total ohmic resistance

of the magnet and the wiring. According to Kirchhoff’s laws

we have

V0 2 LdI

dt¼ IðRhigh þ RÞ þ IRRs; ð39Þ

1

C

ðIC dt ¼ IRRs;

I ¼ IR þ IC:

During the switching-on configuration, the magnet current I

thus obeys the differential equation

LCRs

d2I

dt2þ LþCRsðRhighþRÞh idI

dtþðRhighþRþRsÞI ¼V0:

ð40Þ

The solution contains oscillatory terms if

½L2CRsðRhighþRÞ�2 ,4LCR2s ; ð41Þ

and adopts an exponential (over-damped) form otherwise.

This result suggests that it is possible to generate an

oscillating solution with a certain frequency depending on

the values of C and L provided that C and Rs have adequate

values. If the second term in the brackets can be neglected,

this condition simplifies to

C.L

4R2s

: ð42Þ

Fig. 16 shows the time evolution of the current for different

capacitance values when switching from the low-current to

the high-current state (see Fig. 15). The crossover from

over-damped to sub-damped (i.e. oscillatory) behaviour

with increasing capacitance is obvious. At the same time,

overshooting gets more pronounced. Therefore, working in

the sub-damped limit requires a powerful current control

Fig. 14. Capacitor-boosting device for active acceleration of the switching-

down and switching-up periods. Both electronic switches, S1 and S2; are

operated by the logic unit resulting in the connection of the pre-charged

boosting capacitor C either in parallel to the magnet coil (switching-up

period) or in series (switching-down interval). The diodes D1 to D5 serve to

enforce the desired current pathways.

Fig. 15. Principle of a sub-damped resonant circuit.

Fig. 16. Time evolution of the current when switching from the low- to the

high-current state for different capacitance values in the circuit shown in

Fig. 15. The cross-over from over-damped to sub-damped (i.e. oscillatory)

behaviour with increasing capacitance is obvious.


capable to suppress overshooting efficiently. Probably

because of the problems intrinsic to this field-cycling

variant, it has scarcely been used so far. Anyway, the

realization of the critical limit C¼L=4R2s might improve the

effective slew-rate without overshooting [2].

6.7. Electronic switches

The crucial elements of any field-cycling relaxometry

set-up are the switches turning between the high- and low-

current states. Since electromechanical relays are normally

too slow for field cycling purposes, active semiconductor

devices capable of switching extremely large currents

(typically several hundred Amperes) and withstanding

high voltage peaks (typically up to kilovolts) are normally

employed. Most commonly used are parallel/series combi-

nations of metal oxide semiconductor field effect transistors

(MOSFETs), gate turn-off (GTO) thyristors, and insulated

gate bipolar transistors (IGBTs).

MOSFETs (see Fig. 17) are relatively easy to control, but

are sensitive to overcharges occurring when parallel

arrangements happen to deviate from perfect symmetry.

The current is controlled by the gate voltage which is

applied to the conducting channel across an insulating

material (metal oxide) [87,88]. Due to this function

principle, the input impedance is very high. The schematic

representation of a MOSFET in Fig. 17 shows two n-type

semiconductor regions implanted into a p-type substrate. In

this case, one refers to an n-channel device (p-channel

devices correspond to a reverse scheme). The n-type regions

and the substrate are contacted galvanically. These contacts

are called ‘source’ (S), ‘drain’ (D), and ‘gate’ (G).

MOSFETs are used to control a strong drain current, ID;

by a low gate voltage, VG; with a practically vanishing gate

current, IG:

Fig. 18 shows a family of curves for the drain current, ID;

versus the drain–source voltage, VDS; with the gate–source

voltage, VGS; as a parameter. The curves are linear for low

VDS; so that the device acts as a resistor the value of which is

determined by VGS: With increasing VDS; the width of the

conducting channel is reduced until the ‘pinch-off’ point is

finally reached. The drain current remains then constant for

further increasing VDS; and one speaks of ‘saturation’

operation of the MOSFET. Operation in the linear region

may serve current control applications. On the other hand,

varying VDS between the saturation region and the non-

conducting state ðVDS ¼ 0Þ offers a switching operation

mode. In either case applications for field-cycling purposes

are feasible.

GTO thyristors [88] have bistable characteristics permit-

ting one to switch between high and low impedance states.

They consist of a multilayered p-n-p-n arrangement

specified by low power dissipation in the ‘on’ state.

Fig. 19(a) shows the schematic structure and the symbol

for thyristors, while Fig. 19(b) represents the equivalent

circuit in terms of ordinary transistors. GTO thyristors are

turned on by a positive gate potential, if the anode–cathode

voltage is above a given threshold value. They are turned off

by gate potentials below a threshold level. These devices

have very good current conduction and blocking voltage

capabilities. However, the dynamic characteristics are

relatively poor.

IGBTs (Fig. 20) combine the advantages of bipolar

transistors, that is, high currents and blocking voltages can

be controlled without the need of parallel or serial

arrangements, with the favourable DC current gain offered

by MOSFETs. They are robust switching elements

and highly recommendable for field-cycling purposes.

The conduction state of the device is controlled byFig. 17. Schematic representation of the architecture of a MOSFET switch.

Fig. 18. Performance chart of a MOSFET. On the upper right corner, the

symbol in use for MOSFETs is shown.


the gate-emitter voltage. When the gate-emitter voltage is

less than a threshold value, the device is in the non-

conducting state. In the opposite case when the gate-emitter

voltage is above the threshold, a conducting channel is

formed between the emitter and the collector. The switching

times are mainly determined by internal capacitances and

inductances and the gate input resistance. The performance

in this respect depends also on external ‘parasitic’

inductances of the gate circuit and adequate snubbers and

clamping capacitors to cut overvoltage spikes.

6.8. Practical solutions for field-cycling magnet current

circuits

Fig. 21 shows the circuit suggested by Redfield, Fite and

Bleich [13]. It is based on the boosting capacitor principle

discussed above. The network uses three voltage supplies,

two capacitors and two bipolar transistor banks. The

capacitors C1 and C2 drive the current in the rising and

dropping periods, respectively.

In the low-current state, the voltage supply V can be

adjusted to approach zero current. The two transistor banks

B1 and B2 are then in the non-conducting state. The control

electronics compares the reference voltage rðtÞ with the

feedback signal vðtÞ callipered at the shunt Rs in series with

the magnet. The high voltage supply (H.V.) serves to charge

the capacitor C1.

The crossover to the high-current state is initiated by the

control electronics by transferring the transistor bank B1 to

saturation. As a consequence, the emitter of B2 is practically

grounded, and B2 is therefore set to saturation too so that the

positive sides of C1 and H.V. are grounded. A voltage

V0 þ H.V. develops across the magnet boosting the current.

The circuit effective in this interval is shown in Fig. 22a.

After reaching the desired current level, B2 switches off (the

emitter voltage becomes higher than VQ), and B1 adopts the

control of the current. At stationary operation, the magnet

current flows through D2 and B1 (see Fig. 21).

The opposite switching phase leading from the high-

current to the low-current state is connected with a high

induction peak voltage taken up by the capacitor C2 (see

Fig. 22b). In this case the current flows though D1.

Immediately after the capacitor is discharged by dissipatingFig. 20. Architecture (a) and equivalent circuit and symbol (b) of IGBT

switches. Contacts are ‘gate’ (G), ‘emitter’ (E), and ‘collector’ (C).

Fig. 19. Symbol and schematic representation of the architecture (a) and

equivalent circuit (b) of GTO thyristor switches. The three contacts are

‘gate’ (G), ‘anode’ (A), and cathode (‘K’).

Fig. 21. Magnet current circuit suggested by Redfield et al. [13]. vðtÞ is the

voltage callipered at the shunt resistor RS: rðtÞ is the control voltage for the

field cycle.


the stored energy in the parallel resistor R (which originally

was a Zener diode). The pre-charged capacitor C1 allows for

immediate switching-up without any delay after having the

system switched down.

Fig. 23 shows another field-cycling circuit variant based

on MOSFET transistors and a GTO thyristor [86]. The

MOSFET bank M switches and controls the magnet current

during the steady-state intervals. The control electronics

uses the voltage decaying at the shunt resistor RS as

feedback signal. In the stationary current phases during

the polarisation, relaxation or detection intervals (see Fig. 2),

the magnet current is supplied by the V0 source, and

controlled by the MOSFET bank M. During these intervals,

the high-voltage power supply recharges the capacitor C.

The current pathways effective in the switching-down and

switching-up intervals are displayed in Figs. 24(A) and (B),

respectively. After reaching the desired level the MOSFET

bank starts to control the current. The GTO thyristor merely

serves as a switching element whereas the MOSFET bank

additionally acts as a control instrument. Variants of this

configuration can be found in Refs [16,89,90].

A circuit design on the basis of IGBT switches [18,66] is

shown in Fig. 25. Different power supplies are sequentially

connected to the magnet by using IGBT modules. One of the

salient advantages of these modules is that they can be used

as single, powerful elements without the need to form

thoroughly symmetrized banks as is the case with

MOSFETs. IGBT switches are therefore relatively robust

with respect to malfunctions caused by parasitic voltages.

The control logic receives its steering signals from a pulse

programmer and from voltage comparators. It triggers the

different events of the sequence. The system works under

‘open loop’ conditions, i.e. after the switching intervals,

there is no processing of any feedback signals (see Fig. 25).

The stability needed for signal detection and acquisition is

provided by running the magnet with a bank of car batteries

ðVdÞ during the detection interval. This sort of power supply

turns out to produce extremely precise, reproducible

Fig. 22. Effective current pathways of the Redfield–Fite–Bleich scheme

(Fig. 21) in the cross-over phase from the low-current to the high-current

state (a) and vice versa (b).

Fig. 23. Magnet circuit based on MOSFETs and a GTO thyristor suggested

by Rommel et al. [86]. vðtÞ is the voltage callipered at the shunt resistor RS:

rðtÞ is the control voltage for the field cycle.

Fig. 24. Effective current pathways (bold lines) of the field-cycling circuit

shown in Fig. 23 in the cross-over phase from the low-current to the high-

current state (b) and vice versa (a).


and strong currents on the detection time scale. In the other

intervals of the field cycle, that is the polarisation and

relaxation intervals (see Fig. 2), the required magnet current

is considerably lower and the precision conditions are less

critical. It is provided by a parallel bank of commercial

power supplies Vp: In order to increase the dynamic range in

which the relaxation field can be varied, a further low-

current loop implying a special power supply Vr replaces Vp:

During the polarisation interval, the IGBT switches S1

and S3 are set in the conducting state, and Vp supports the

current through the magnet (Fig. 25). Simultaneously, the

high-voltage power supply recharges the capacitor C1 to

about 600 V. In the transition regime between the

polarisation and the relaxation field, S3 is switched off

while the new value of Vp (or Vr) is set by a digital-to-

analogue converter of the control unit (see Fig. 26a). The

current flows through C2 which in turn controls the time

dependence of the current decay. A feedback signal is

callipered from the shunt Rs; a patron compensated resistor,

for monitoring of the field cycle and for control purposes.

When the feedback signal reaches the control input defining

the desired current in the relaxation interval, S3 is closed

again. This process is automatically controlled by a voltage

comparator. The appropriate choice of the power supplies

Fig. 25. Magnet circuit based on IGBT switches suggested by Seitter et al. [18,66]. The highlighted current circuit corresponds to the situation effective during

the polarization interval.

Fig. 26. Effective current pathways (bold lines) of the field-cycling circuit shown in Fig. 25 in four different phases of the field cycle (see Fig. 2): (A) from the

polarization to the relaxation field; (B) from the relaxation to the detection field; (C) during the detection field; (D) from the detection to the polarization field

(or to an intermittent zero-field interval).


Vp and Vr suited for the control input is automatically set by

the software.

The second phase of the field cycle is the transition from

the relaxation to the detection field (see Fig. 2). The circuit

effective in this interval is shown in Fig. 26b. S2 is closed,

and the pre-charged capacitor C1 boosts the magnet current.

After a delay of about 1 ms, S4 closes so that current

pathway shown in Fig. 26c applies. After reaching the

detection level, S2 opens and the magnet current (about

300 A) is supplied by the battery bank Vd (Fig. 26c). The

switch-off process of S2 is initiated by a voltage comparator,

whose reference is a voltage supply the precision of which

allows for phase sensitive NMR signal accumulation. The

same voltage comparator generates the trigger pulse for the

detection RF pulse.

In order to bring the magnet current from the detection

level back to the polarisation field (or to an intermittent

zero-field interval), S1 is opened and the field energy is

transferred to the capacitor C1. The corresponding current

pathway is illustrated in Fig. 26d. S1 shortcuts the capacitor

after the current approaches zero. After a cooling delay

determined by the applicable duty cycle, a new field-cycle

begins with the current pathway shown in Fig. 26a.

The last field-cycling power supply to be discussed here

is the simplest one in principle: restricting oneself with

respect to the highest field level reached in a field cycle, the

boosting capacitor can be omitted, and the magnet current is

controlled by a MOSFET network capable of the peak

power needed for acceptable switching times. That is, the

current is controlled by a transistor bank during the whole

cycle including the transitions between different field levels

[58]. This is the solution chosen by the only commercial

supplier of this sort of system, STELAR SRL. Fig. 27 shows

the network. It essentially consists of the main power supply

V1 and the MOSFET current control system. A second,

negative voltage supply V2 is used to compensate the current

offset in the magnet. The heart of the system is the control

electronics comparing the reference signal rðtÞ and the

feedback signal vðtÞ callipered at the shunt S. Additional

feedback signals like magnet temperature and AC voltage

from the magnet are additionally used for a precise control

of the field cycle [58].

7. Applications to porous media and adsorption

phenomena at liquid/solid interfaces

Adsorption of molecules at inner surfaces of solid

porous materials or at surfaces of other static or slowly

moving objects such as fine particle agglomerates or

globular macromolecules (proteins) provides some prefer-

ential molecular orientation relative to the local surface.

The consequence is a slowly decaying component of the

autocorrelation function of spin interactions. Total corre-

lation loss occurs only after adsorbate molecules have

escaped from the immediate vicinity of the surface position

where the molecule was initially adsorbed. Actually this

can give rise to correlation function components decaying

eight orders of magnitude more slowly than in the bulk

liquid [36].

The consequence is a tremendous enhancement of low-

field spin–lattice relaxation, whereas the influence on

translational diffusion is comparatively little [91,92]. The

explanation is that relaxometry as well as diffusometry

measure averages of the adsorbed and free phases. In

diffusion measurements [24], the normally large fraction of

molecules in the bulk-like phase tends to dominate, whereas

relaxometry is governed by the often quite small fraction of

molecules that are initially as well as finally adsorbed. Low-

field spin– lattice relaxation is therefore much more

sensitive to surface effects than translational diffusion.

The fact that averages over different molecular phases

have to be taken, requires some consideration of exchange

schemes between those phases. As the most prominent

model, we will consider the two-phase fast-exchange

(TPFE) case in more detail. This is also connected with

the most important diamagnetic low-field relaxation mech-

anism for adsorbate molecules, namely reorientation

mediated by translational displacements (along curved

surfaces), RMTD, which contains both a dynamic and a

topological element [93].

7.1. Two-phase fast-exchange model

In saturated systems, the adsorbate liquid is considered to

coexist in two homogeneous and rapidly exchanging phases

characterized as ‘bulk-like’ and ‘adsorbed’. ‘Fast exchange’

refers to the time scale of the spin–lattice relaxation times

T1 (which is much longer than the molecular reorientation

time constants). Experimentally it manifests itself by

monoexponential relaxation curves. This is the basis of

the two-phase fast-exchange model.

According to Eq. (4), only the molecular orientations at

times 0 and t matter for the decay of the correlation function

irrespective of what happens in between. In the frame of the

TPFE model there are four different situations to beFig. 27. Magnet circuit without boosting capacitor as used in the

commercial STELAR relaxometer.


distinguished and to be characterized by the exclusive

probabilities: fa;aðtÞ; fraction of spins that are initially and

finally located in the adsorbed phase; fa;bðtÞ, fraction of spins

that happen to be initially in the adsorbed phase and finally

in the bulk-like phase; fb;aðtÞ; fraction of spins that happen to

be initially in the bulk-like phase and finally in the adsorbed

phase; fb;bðtÞ; fraction of spins that happen to be initially and

finally in the bulk-like phase. These probabilities are

normalized of course,

fa;aðtÞ þ fa;bðtÞ þ fb;aðtÞ þ fb;bðtÞ ¼ 1: ð43Þ

With field-cycling NMR relaxometry we are probing the

long-time limit t q trot; where trot is the correlation time for

rotational diffusion in bulk or, in restricted form, in the

adsorbed phase. Contributions by rotational diffusion

therefore do not affect the low-frequency relaxation

dispersion monitored with this technique. The two processes

of interest, namely ‘RMTD along surfaces’ and ‘rotational

diffusion’ (either restricted or isotropic) will be indicated by

subscripts ‘RMTD’ and ‘rot’, respectively. The total

correlation function, Eq. (4), can then be analyzed for the

two-phase system into four partial correlation functions:

GmðtÞ ¼ fa;aðtÞkY2;2mð0ÞY2;mðtÞlRMTD;rot

þ fa;bðtÞkY2;2mð0ÞY2;mðtÞlrot

þ fb;aðtÞkY2;2mð0ÞY2;mðtÞlrot

þ fb;bðtÞkY2;2mð0ÞY2;mðtÞlrot: ð44Þ

In the limit t q trot the partial correlation function for

isotropic rotational diffusion (i.e. in the bulk-like phase)

vanishes, and we may write

Gmðt q trotÞ < fa;aðtÞkY2;2mð0ÞY2;mðtÞlRMTD;rot: ð45Þ

Rotational diffusion on surfaces is restricted and leaves a

finite residual correlation that can only decay to zero in the

adsorbed state by RMTD along more or less randomly

curved surfaces. RMTD and rotational diffusion can more-

over be considered to be independent of each other on their

very different time scales. Eq. (45) can therefore be

analyzed according to

kY2;2mð0ÞY2;mðt q trotÞlRMTD;rot

¼ kY2;2mð0ÞY2;mðt q trotÞlRMTDkY2;2mð0ÞY2;mðt q trotÞlrot

ð46Þ

¼ kY2;2mð0ÞY2;mðt q trotÞlRMTD

ðkY2;2mð0ÞY2;mðtÞlrot 2 grotð1ÞÞ|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}!0 for tqtrot

þ grotð1Þ

264

375

< kY2;2mð0ÞY2;mðt q trotÞlRMTDgrotð1Þ

so that

Gmðt q trotÞ < fa;aðtÞgrotð1ÞkY2;2mð0ÞY2;mðtÞlRMTD; ð47Þ

where grotð1Þ ; kY2;2mð1ÞY2;mð1Þlrot ¼ const is the finite

constant residual correlation left over in the long-time limit

of restricted rotational diffusion before RMTD becomes

effective. This is the first example of a treatment of a

complex system in the different time scale limit. Analogous

situations will be encountered in Section 8.2 in context with

polymer dynamics and in Section 9.7, in the treatment of

hydrodynamic modes of liquid crystals.

Eq. (47) tells us, that the correlation function relevant for

the field-cycling frequency range is composed of two time

dependent factors, namely the probability fa;aðtÞ the decay of

which represents exchange losses of the initial surface

population, and the RMTD correlation function in the

proper sense, kY2;2mð0ÞY2;mðtÞlRMTD: This function refers to

the fraction of molecules that are initially and finally in the

adsorbed state.

Relative to the time scales of exchange ðtexÞ and

rotational diffusion ðtrotÞ; two further limits can be

distinguished:

(i) t q tex q trot: The initial and final probabilities to be

in either phase become independent of each other, so that

fa;aðtÞ < f 2a ;

fa;bðtÞ < fb;aðtÞ < fað1 2 faÞ; ð48Þ

fb;bðtÞ < ð1 2 faÞ2;

where fa and ð1 2 faÞ are the (constant) populations of the

adsorbed phase and of the bulk-like phase, respectively. On

the time scale of this limit, the total correlation function thus

reads

Gmðt q trotÞ < f 2a grotð1ÞkY2;2mð0ÞY2;mðtÞlRMTD: ð49Þ

The correlation function is characterized by a square

dependence on fa in this case.

(ii) trot p t p tex: Exchange is then unlikely to occur,

and we may approximate

fa;aðtÞ < fa; ð50Þ

fa;bðtÞ < fb;aðtÞ < 0;

fb;bðtÞ < ð1 2 faÞ:

Under such conditions, the total correlation function

becomes a linear function of fa;

Gmðt q trotÞ < fagrotð1ÞkY2;2mð0ÞY2;mðtÞlRMTD: ð51Þ

Both approximations tend toward an exact result (for long

times) if fa ! 1: In both cases, the RMTD process obviously

dominates the long-time correlation decay, where we are

referring to molecules initially and finally in the adsorbed

phase. The low-frequency relaxation dispersion, T1 ¼

T1ðvÞ; for TPFE systems is thus obtained by combining

Eq. (51) with Eqs. (5) and (6).


7.2. Bulk-mediated surface diffusion

Molecular dynamics of adsorbate dynamics near inner

surfaces of a porous medium is a diffusion/reaction

problem, where ‘reaction’ refers to adsorption and deso-

rption of molecules. In terms of the Bychuk/O’Shaughnessy

formalism [94,95], adsorption of liquid molecules on

surfaces is characterized by a number of characteristic

parameters. In terms of the TPFE model, two adsorbate

phases are assumed to coexist in saturated pores, molecules

adsorbed at the inner surfaces of the matrix on the one hand

and the bulk-like phase filling the interior volume except for

the adsorption layer on the other. The ‘retention time’ thindicates how long it takes until the initial adsorbate

population on a surface is finally replaced by exchange with

the bulk-like phase. This ‘renewal time’ is the maximum

time scale of the processes to be considered in the following.

The retention time is related to the so-called ‘adsorption

depth’ h according to

h ¼ffiffiffiffiffiDth

p; ð52Þ

where D is the bulk diffusivity of the adsorbate. The

adsorption and desorption rates are designated by Qads and

Q; respectively. Defining furthermore a ‘capture range’ b;

that is the distance over which an adsorbate molecule can

directly be adsorbed in a single displacement step, leads to

the relation

h ¼ bQads

Q: ð53Þ

The ‘weak adsorption limit’ is then characterized by

thQ p 1: ð54Þ

In this case the adsorbate molecules most likely escape from

the surface layer to the (much larger) bulk-like phase

immediately after desorption. This is in contrast to the

‘strong adsorption limit’,

thQ q 1; ð55Þ

in which numerous desorption/re-adsorption cycles occur

before an adsorbate molecule finally escapes to the bulk-like

phase. As a consequence, adsorbate molecules are effec-

tively displaced along the surface on a time scale Q21 ,

t , th in a series of desorption/bulk excursion/re-adsorption

cycles where the bulk excursions are thought to be

unrestricted. This surface diffusion mechanism is called

‘bulk-mediated surface diffusion’ (BMSD), and can be

described as a special form of a Levy walk [94,95].

The strong adsorption limit is of particular interest here,

because non-Gaussian propagators arise for BMSD. The

term ‘surface diffusion’ implies that the adsorbate molecule

resides initially as well as finally on the surface irrespective

of any bulk excursions in between. That is, the molecules

are considered to diffuse effectively in an isotropic

topologically quasi two-dimensional surface space. The

displacement within this space is designated by s:

For s pffiffiffiffiDt

p; the surface diffusion propagator is given by

the Cauchy distribution for two dimensions [96],

Pðs; tÞ ¼1

2p

ct

½ðctÞ2 þ s2�3=2: ð56Þ

The constant c is defined by c ¼ D=h and is the dynamic

parameter of this propagator. Note thatffiffiffiffiDt

pcharacterizes

the root mean square displacement in the bulk as a reference

length scale. In the long time limit, t . th; the ordinary

Gaussian propagator for two-dimensional diffusion is

approached:

Pðs; tÞ ¼1

4p

exp{ 2 s2=ð4DtÞ}ffiffiffiffiDt

p : ð57Þ

The superdiffusive displacement behaviour predicted by the

BMSD mechanism was well verified with the aid of a

computer simulation in the limit t , th; s pffiffiffiffiDt

pfor planar

and spherical surfaces [97].

7.3. Reorientation mediated by translational diffusion

Surface related spin–lattice relaxation was studied both

by proton and deuteron resonance. The spin interactions

responsible for the relaxation mechanism are dipolar and

quadrupolar coupling, respectively. The quadrupolar inter-

action is intrinsically intramolecular in nature. In the present

situation, intermolecular contributions to dipolar couplings

also turned out to be negligible [36,98].

The fluctuations of the spin interactions causing spin

relaxation are then exclusively due to molecular reorienta-

tions which are influenced by adsorbate/surface inter-

actions. The correlation function decay reflects rotational

diffusion in a generalized sense. Translations nevertheless

play a crucial role via a process called ‘reorientation

mediated by translational displacements’ (RMTD) [98,99].

The molecular dynamics of interest in this context are

dynamic processes beyond the time and length scales of

ordinary Brownian rotational diffusion. The slow molecular

fluctuations considered here are due to the existence of a

solid adsorbent in the form of pore walls or particle surfaces.

Since the surfaces impose preferential orientations on

adsorbed molecules, the time scale of molecular reorienta-

tions is determined by the total interaction period with the

surface before the molecule escapes to the bulk-like phase.

In contrast to molecular motions in bulk, much slower

processes then appear due to diffusive displacements of

molecules. That is, many elementary diffusion steps are

needed until molecules totally lose the correlation to their

initial orientation.

The RMTD mechanism shows up with polar liquids in

porous silica glasses, for instance, where the longest

orientation correlation components were found to decay 8

orders of magnitude more slowly than in the free liquid

[36,100]. The RMTD process describes molecular reor-

ientation determined by displacements between surface sites


of different orientations as illustrated in Fig. 28a. In the

interval between 0 and t, the molecules may perform more

or less extended and more or less frequent excursions to the

bulk-like medium in the pore space, where no preferential

orientation exists. As soon as molecules return to a surface

site, they adopt the preferential surface orientation intrinsic

to that site. It is then a matter of surface topology how

strongly the ‘final’ orientation is correlated to the ‘initial’

orientation on the surface. The correlation function for the

RMTD process (Eqs. (49) or (51)) thus implies dynamic as

well as geometrical features of the system. The former

refers to translational diffusion and possibly to desorption/

re-adsorption kinetics, the latter to the surface topology of

the pores in the matrix.

Rotational diffusion or tumbling of the adsorbate

molecule about its preferential orientation on the surface

certainly occurs in addition. However, this process is

expected to be restricted to a narrow solid-angle range.

That is, the correlation function initially decays fast to some

residual correlation value, which then is further reduced

owing to the much slower RMTD process (see Eqs. (49) and

(51)). In terms of the spin–lattice relaxation dispersion, this

means that the low-frequency dispersion below the range,

where local processes matter, tends to be dominated by the

RMTD mechanism. This is what one measures with the

field-cycling NMR relaxometry technique.

The RMTD version of the correlation function given in

Eq. (4) (see Eqs. (49) and (51)) can be analyzed in a

dynamic and a geometrical contribution,

Gmðt q trotÞ / kY2;2mð0ÞY2;mðtÞlRMTD

¼ð

gðsÞPðs; tÞ2ps ds: ð58Þ

The propagator Pðs; tÞ for effective displacements s along

the topologically two-dimensional surface represents the

dynamic part which possibly has an anomalous character. It

implies excursions to the bulk-like phase as outlined above.

Exchange losses by molecules having not yet returned to the

surface at time t are taken into account by the proportion-

ality constant given in Eqs. (49) or (51) for the two exchange

limits considered.

On the other hand, the microstructural surface topology

is expressed by the surface correlation function

gðsÞ ¼ kY2;2mð0ÞY2;mðsÞl; ð59Þ

which characterizes how the surface orientation given by

wð0Þ;qð0Þ at the initial surface position of the adsorbate

molecule changes when it is displaced a curvilinear distance

s in the topologically two-dimensional surface layer to a

new surface site with an orientation defined by wðsÞ;qðsÞ:

Instead of representing the RMTD relaxation mechanism

in real space variables it is often favourable to employ a

reciprocal space picture based on the wavenumber k: The

correlation function given in Eq. (58) then reads [98]

GmðtÞ ¼1

ð2pÞ2

ð1

0SðkÞpðk; tÞdk; ð60Þ

with the spatial Hankel transforms of the surface correlation

function, gðsÞ;

SðkÞ ¼ ð2pÞ2kð1

0sgðsÞJ0ðksÞds ð61Þ

(‘(radial) orientational structure factor’), and

pðk; tÞ ¼ 2pð1

0sPðs; tÞJ0ðksÞds; ð62Þ

(‘k space propagator’), where J0ðksÞ is the Bessel function of

zeroth order. In this representation, the k space propagator

takes an exponential form [98,101]

pðk; tÞ ¼ exp{ 2 t=tk}; ð63Þ

Fig. 28. Illustration of surface diffusion of adsorbate molecules at pore

walls (‘solid matrix’). (a) Above the freezing temperature T0; the molecules

exchange between the adsorbed phase and the bulk-like phase. In the strong

adsorption limit, many desorption–re-adsorption cycles occur with bulk

excursions in between before the initial surface population is replaced. This

effectively results in ‘bulk mediated surface diffusion’ (BMSD) within the

retention time. This process can be described as a topologically two-

dimensional Levy walk. Since molecules (represented by open arrows) are

oriented relative to the surface according to the surface topology as soon as

they become adsorbed, NMR relaxation due to ‘reorientation mediated by

translational displacements’ (RMTD) results. That is, components of the

orientation correlation function exist and decay orders of magnitude more

slowly than in the bulk liquid. Between the adsorption events, molecules are

immediately randomized in the bulk-like phase but recover the surface

orientation upon re-adsorption. (b) Below the freezing temperature T0; the

bulk-like phase is frozen and consequently immobilized. Only a thin

interfacial layer at the surfaces is left in the liquid state. Molecular

exchange between the adsorbed phase and the bulk-like phase is hence

excluded. Flip-flop spin exchange between the liquid and solid phases is

also of minor importance as was shown by an isotopic dilution experiment

[98] (see Fig. 30 and also compare with Ref. [103]). Due to the absence of

BMSD, surface diffusion in the topologically two-dimensional interfacial

liquid layer is expected to be of the normal type.


with the time constants

tk ¼ 1=ðDk2Þ ð64Þ

for a Gaussian real space propagator, Eq. (57), and

tk ¼ 1=ðckÞ ð65Þ

for a Cauchy distribution according to Eq. (56). So we have

a common analytical k space propagator form for both types

of diffusion deviating merely in the time constants. The

parameters representing and characterizing the dynamics

are D and c; respectively.

7.4. Porous silica glasses and fine particle agglomerates

The weak and strong adsorption limits can be visualized

with the aid of spin–lattice relaxation dispersion curves

[36]. Fig. 29 shows typical data for six different organic

solvents filled into porous silica glass with a mean pore size

of 30 nm. The different dispersion slopes for polar and non-

polar species are obvious. The data suggest that polar

adsorbates on polar silica surfaces are subject to strong

adsorption in contrast to the non-polar adsorbate species.

The explanation is that the RMTD process based on surface

diffusion is entirely different in the two adsorption limits.

Note that paramagnetic centers such as those discussed in

Ref. [102] are irrelevant in these samples as was shown in

Refs [36,98].

According to Bychuk and O’Shaughnessy [94,95] and

the computer simulation in Ref. [97], the strong adsorption

limit should be connected with anomalous surface

diffusion that can be described as Levy walks along the

surfaces. The basis of this anomaly is BMSD, i.e.

excursions of the adsorbate molecules to the bulk-like

phase. As a consequence, the type of the propagator for

surface diffusion should change when these excursions are

excluded. In this case, ordinary diffusion in the topologi-

cally two-dimensional surface space should occur. In other

words, the two scenarios with and without bulk excursions

are expected to be governed by Cauchy and Gaussian

propagators, respectively.

Bulk excursions can be prevented by freezing the bulk-

like phase and leaving a thin interfacial layer of adsorbate

molecules in a liquid phase as illustrated in Fig. 28b. A

number of suitable adsorbate/adsorbent pairs do exist

indeed that permit field-cycling NMR relaxation [98,105]

and diffusion [91] experiments under such conditions. In

these cases one is dealing with a ‘non-freezing adsorbate

layer’ (NFL) at the matrix interface. The thickness was

estimated to be between one and two molecular diameters.

Note that spin–lattice relaxation as well as translational

diffusion below the bulk freezing temperature exclusively

refer to the NFL molecules under the conditions of usual

NMR experiments.

Fig. 30 shows NMR relaxometry data recorded under

such conditions [98]. Dimethylsulfoxide (DMSO) was filled

into porous silica glass with a mean pore size of 10 nm. A

power law for the T1 frequency dispersion was found in a

wide range ð3 £ 104 , n , 107 HzÞ

T1 / nb: ð66Þ

The exponent changes from b ¼ 0:54 ^ 0:04 at 291 K (i.e.

above the bulk freezing temperature) to b ¼ 0:73 ^ 0:04 at

270 K (i.e. below the freezing temperature of the bulk-like

phase). The RMTD process governing spin–lattice relax-

ation in this frequency range should reflect the conversion of

the surface diffusion propagator from a Gaussian probability

Fig. 29. Frequency dependence of the proton spin–lattice relaxation times of polar and non-polar organic liquids in Bioran B30 porous silica glass [36]. The

mean pore size is 30 nm. The data are given relative to the values measured in the liquids in bulk. Depending on the polar character of the adsorbate molecules,

the ‘strong’ and ‘weak’ adsorption limits can obviously be differentiated as ‘strong’ and ‘weak’ spin–lattice relaxation dispersions, respectively. The solid lines

represent a correlation time distribution analysis providing a formal description of the data. The steep dispersion slope of the polar solvent species, representing

the strong adsorption limit, extends down to the kHz regime. That is, the corresponding components of the correlation function decay almost eight orders of

magnitude more slowly than in the bulk liquid. On the right-hand vertical axis, transverse relaxation time data measured at 90 MHz are given as filled symbols

for comparison.


density function when adsorbate molecules are confined in

the NFLs to a Cauchy distribution in the unfrozen state. This

interpretation is corroborated by the following scaling

argument.

The experimental power law frequency dispersion given

in Eq. (66) suggests power laws for the orientational

structure factor as well:

SðkÞ ¼ bk2x ð0 , x , 1Þ; ð67Þ

where b is a constant. The k space propagators for surface

diffusion, pðk; tÞ; are given in Eq. (63) for Gaussian and

Cauchy probability density functions. Combining Eqs. (5),

(60), (63) and (67) gives the following expressions for the

correlation functions and the spectral densities [98,99]:

GmðtÞ / bD2ð12xÞ=2t2ð12xÞ=2

ImðvÞ / bD2ð12xÞ=2v2ð1þxÞ=2

)Gauss; ð68Þ

GmðtÞ / bc2ð12xÞt2ð12xÞ

ImðvÞ / bc2ð12xÞv2x

)Cauchy:

With Eq. (6), we thus obtain

T1 / b21Dð12xÞ=2vð1þxÞ=2 ðGaussÞ; ð69Þ

T1 / b21cð12xÞvx ðCauchyÞ:

The same orientational structure factor, Eq. (67), yields

different frequency dependences for the spin– lattice

relaxation time depending on the choice of the propagator.

The microstructural topology of the surfaces is commonly

represented by the parameters b and x whereas the

dynamical parameters D and c are specific for the Gauss

(Eq. (57)) and Cauchy (Eq. (56)) distributions, respectively.

Evaluating the experimental power laws given in Eq.

(66) which represent the data in Fig. 30 over a wide range

permits one to evaluate the exponent x for the unfrozen

sample as well as for the non-freezing surface layers

anticipating Cauchy and Gaussian propagators, respect-

ively. The fact that in both cases the same exponent for the

orientational structure factor comes out in the frame of the

experimental accuracy,

SðkÞ ¼ bk20:5^0:04; ð70Þ

corroborates that diffusion along the topologically two-

dimensional surfaces is adequately represented by these two

propagators: the surface geometry in the unfrozen and NFL

cases is the same whereas the dynamical features are

changed. Provided that the bulk-like phase is liquid and

large enough, this in particular implies that surface diffusion

along pore surfaces in porous glasses can be described as

Levy walks on a 1–10 nm length scale in the frame of the

two-phase fast-exchange model in the strong adsorption

limit.

The surface topology represented by the power law given

in Eq. (67) suggests surface fractal properties. In Refs [99,

104] corresponding scaling arguments were established

leading to

SðkÞ / kdf23; ð71Þ

with the surface fractal dimension df : Comparison of Eqs.

(70) and (71) thus suggests a value df ¼ 2:5 ^ 0:04 for

DMSO on surfaces of silica porous glass irrespective of

whether or not the bulk-like phase is frozen.

7.5. Water/lipid interfaces

The RMTD relaxation mechanism becomes effective at

low frequencies at all non-planar liquid/solid interfaces

provided that adsorbate molecules have translational

degrees of freedom. Such a situation can exist not only in

solid porous matrices, but also in quasi-solid substrates as

formed by lipid liposomes forming onion skin like

structures in an aqueous environment [106,107]. The

hydration water separating lipid bilayers can diffuse along

the water/lipid interface shaped according to the liposome

size.

The so-called ripple phase characterized by an undulated

water/lipid interface (see Fig. 31) is of particular interest.

Hydration water molecules diffusing along the lipid bilayer

surface are reoriented according to the spatial variation of

the local surface orientation. This can give rise to a

relaxation mechanism as was demonstrated in Ref. [108]

both with field-cycling NMR relaxometry and a lineshape

analysis. In order to differentiate the water from the lipid

Fig. 30. Frequency dependence of the proton spin–lattice relaxation time of

dimethylsulfoxide (DMSO) in porous silica glass Bioran B10 (mean pore

size 10 nm) above and below the freezing temperature of the bulk-like

liquid. The data for an isotopically diluted sample (80% deuterated DMSO-

d6) demonstrate that spin interactions are governed by intra-molecular

dipolar interactions and that flip-flop spin diffusion across the frozen phase

is negligible. The relaxation times of the partially frozen sample at 270 K

refer to the slowly decaying component of the NMR signal corresponding

to non-freezing surface layers (NFL).


signals, the samples were prepared with heavy water, and

deuteron resonance was employed with each technique.

Fig. 32 shows the water deuteron spin–lattice relaxation

data in the ripple phase for three different concentrations.

The dispersion step in the middle of the frequency range

directly reflects the corrugated surface character. The data

can be described on the basis of the RMTD formalism in

accordance with the deuteron line width/splitting variation

upon transitions to the ripple phase [108]. The undulation

‘wave length’ fitted to the relaxation dispersion and line

width data moreover matches the result concluded from

electron tunnelling microscopy experiments. That is, the

structural surface features are mirrored in the reorientational

dynamics of hydration water as probed by low-frequency

NMR techniques.

It is noteworthy that the ripple/RMTD effect on the spin–

lattice relaxation dispersion is absent for hydration water of

planar substrate surfaces as they occur in synthetic saponites

[109] (see Section 7.7). Note also that chain dynamics of the

lipid molecules themselves (see Section 9.9 and Refs

[110 – 112]) did not perceptibly influence the water

relaxation features.

7.6. Water/protein interfaces

There are numerous field-cycling relaxometry studies of

protein solutions in the literature (see the review in Ref. [3]).

Actually the first studies with this technique in its

electronically switched version were devoted to such

systems [14,113 – 115]. In the following we restrict

ourselves to diamagnetic species. Reviews of paramagnetic

systems can be found in Refs [116,117]. The interaction of

water with biopolymers is considered to be the main source

of relaxation in biological tissue, and is hence of paramount

importance for the interpretation of contrasts in biomedical

magnetic resonance imaging [6,39].

The first questions to be examined deal with the

contributions of the water and the protein phases to spin–

lattice relaxation, and the positions where the main

relaxation processes take place. Fig. 33 shows proton

Fig. 31. Schematic representation of the random walk of a water molecule

diffusing on the corrugated surface of a lipid bilayer in the ripple phase. The

arrows indicate the initial and final orientations of the water molecule

relative to the surface. The ‘wavelength’ of the surface undulation of

dipalmitoylphosphatidylcholine bilayers is reported to be between 12 and

15 nm. The undulation amplitude is 3.8 nm [108].

Fig. 32. Dispersion of the deuteron spin–lattice relaxation rate measured in

heavy hydration water of DPPC liposomes in the ripple phase at three

different water concentrations. The continuous lines correspond to the

RMTD formalism and were fitted to the data in accordance to deuteron line

width/splitting data measured in the same systems [108]. The undulation

wavelength was found to correspond to the value directly evaluated from

electron tunneling micrographs.

Fig. 33. Proton and deuteron spin–lattice relaxation dispersion in a solution

of bovine serum albumin (BSA) in D2O [118]. The protein concentration is

35% by weight. The proton data represent the dynamics of the non-

exchangeable protein hydrogen atoms, whereas motions of water molecules

are characterized by the dispersion of the deuteron relaxation times. The

different dynamics in the two phases are obvious. Note that data points in

the quadrupole dip frequency ranges have been omitted in this experiment

(see Figs. 54a and 55).


and deuteron spin–lattice relaxation curves measured in a

concentrated solution of bovine serum albumin in D2-

O. The deuteron data thus reflect the dynamics of water

molecules (and some exchangeable protein hydrogen

atoms), whereas the proton relaxation times are governed

by the motions of the globular protein molecules. Note that

these data have been recorded on the very same sample. The

different type of molecular dynamics in the two phases is

obvious [118].

Spins inside the macromolecules are relaxed by back-

bone fluctuations, side-group motions and overall tumbling

of the molecule [119,120]. Backbone fluctuations are

largely characterized by power law spin–lattice relaxation

dispersions and will be discussed in the polymer section.

The spin–lattice relaxation in the water phase is partly

affected by these mechanisms too. However, based on

deuteron field-cycling relaxometry experiments the dom-

inating effect was shown to be due to molecular processes in

hydration water at the interface between the aqueous and

protein phases.

Basically there are three competitive mechanisms

contributing to the spin–lattice relaxation of hydration

water [120]:

(i) Restricted rotational diffusion of water molecules

about axes perpendicular to the local surface. An

exponential correlation function can be assumed for

this process with a correlation time not much longer

than that in bulk water.

(ii) Reorientation mediated by translational displacements

(RMTD) along the more or less rugged and curved

surface of the protein. The longest correlation time

limiting this process is designated by tk: It depends

both on the surface topology and the effective

diffusivity along the surface [42,100,120–122] and

can be described by the formalism presented in Section

7.3. Assuming a Gaussian translational diffusion

propagator and an equipartition of wavenumbers

describing the surface topology, the corresponding

correlation function was found to be, [98,100]

GRMTDðtÞ ¼ C

ffiffiffiffiffip

Dt

r½erfðku

ffiffiffiffiDt

pÞ2 erfðkl

ffiffiffiffiDt

pÞ�; ð72Þ

where erfðxÞ is the error function, C is a numerical

coefficient, and D is the diffusivity. The quantities ku

and kl are the upper and lower cutoff values of the

wavenumber equipartition distribution, respectively.

The cutoff correlation time corresponding to kl is

tk ¼ ðDk2l Þ

21: ð73Þ

In the limit Dk2l p vp Dk2

u a square root frequency

dependence is predicted in good coincidence with the

experimental finding (see Fig. 34):

T1 / kuðDvÞ1=2: ð74Þ

(iii) Tumbling of the protein molecule including its

hydration shell. As a correlation function for this

process, an exponential function can be assumed again.

The combined formalism based on these mechanisms

describes the deuteron spin–lattice relaxation dispersion

very well as demonstrated in Fig. 34. Restricted rotational

diffusion matters only at high frequencies above about

10 MHz. The low-frequency dispersion is governed by the

RMTD mechanism of water, and, if not prevented by

mutual sterical hindrance at high protein concentrations,

by tumbling of the hydrated protein molecule. Remark-

ably, the T1 frequency dispersion is qualitatively the same

irrespective of the presence of a bulk-like water phase.

The same dispersion features occur even at water contents

as low as 25% by weight, where the hydration shells are

just saturated and where protein molecules are

immobilized.

The water diffusion coefficient in the hydration shells can

be determined with the aid of field-gradient NMR

diffusometry [24]. Inserting the value obtained in this way

[122–126], a length scale can be estimated from the RMTD

correlation time tk given in Eq. (73). At low water contents

where protein tumbling is excluded, tk can directly be

determined from the inflection frequency (see Fig. 34). The

result for the reorientation length scale is about half the

mean protein circumference as expected for the RMTD

mechanism [100].

Fig. 34. Deuteron spin–lattice relaxation dispersion of D2O solutions of

bovine serum albumin at 291 K for different water contents. The continuous

lines were calculated using the RMTD formalism in combination with

protein tumbling and restricted rotational diffusion of hydration water. The

low-frequency plateau merges into a square root frequency dependence at a

certain ‘inflection frequency’ ni depending on the protein concentration.

The inflection frequencies for 25% D2O (no bulk-like water exists) and 50%

D2O (bulk-like water exists) are the same. This indicates that molecular

tumbling of the protein molecule is not yet effective, and RMTD dominates

the total low-frequency regime. This is in contrast to 75% D2O where

protein tumbling is fast enough to become competitive with RMTD. In this

case, protein tumbling determines the cross-over to the low-frequency

plateau.


This model also accounts well for the weak tempera-

ture dependence of spin–lattice relaxation observed in

concentrated protein solutions. In Refs [127,128] transient

binding of a small percentage of the hydration water at

certain protein sites not permitting any rotational diffusion

was suggested. However, this notion would require

binding energies twice as high as the apparent activation

energies estimated from experimental T1 data. The

problem does not arise in the frame of the RMTD

model because the longest correlation times in this case

are due to the surface topology rather than to high binding

energies. Furthermore, transient binding not allowing for

rotational diffusion would result in dispersion slopes at

intermediate frequencies four times as high as observed in

the experiment (see Fig. 34).

The inflection frequency ni (see Figs. 33 and 34)

depends on the water content cw: This manifests the

competitive nature of macromolecular tumbling and

RMTD at low frequencies. Tumbling can only take place

if the water content exceeds the saturation concentration cs

defined by saturation of the hydration shells. At larger

water contents, bulk-like water is present so that, with a

certain probability, a macromolecule is surrounded by

enough free water for rotational reorientations. For an

illustration see Fig. 35. The ‘free-water volume’ formalism

leads to a macromolecular tumbling correlation time

[120,129]

ttumble ¼ t0tumble exp gpðr 2 1Þ

1 2 cw þ cs

cw 2 cs

! "

ðcw . csÞ:

ð75Þ

t0tumble ¼ hðVp þ VsÞ=ðkBTÞ is the Stokes/Einstein expres-

sion for the tumbling time of a particle of volume Vp þ Vs

(bare protein plus the saturated hydration shell) in a

medium of viscosity h: The quantity r is the ratio of the

volumes of the circumscribing sphere and the hydrated

protein molecule approximated by an ellipsoid (see

Fig. 35); 0:5 , gp , 1 is a numerical constant. The critical

water content c0 at which macromolecular tumbling

becomes competitive to the correlation time tk correspond-

ing to the lower cutoff wavenumber kl of the RMTD

process, is then defined by the condition tk ¼ ttumble: From

this, one finds that

c0 ¼ cs þ~g

1 þ ~g; ð76Þ

where ~g ¼ ðr 2 1Þgp=lnðtk=t0tumbleÞ: For aqueous bovine

serum albumin solutions the following critical concen-

trations (by weight) have been found [120]: cs ¼ 30% and

c0 ¼ 65%; that is ~g ¼ 0:5:

The proton spin–lattice relaxation dispersion found in

diverse biological tissues [120,130,131] and even living

organisms [132] is not so easy to interpret on the basis of a

simple model. Nevertheless, a successful attempt applying

the above model was reported in Ref. [120] for leech and

frog muscles.

7.7. Water/saponite interfaces

Saponites form platelets with essentially planar surfaces

[133]. The water RMTD relaxation mechanism in aqueous

suspensions is therefore of a character somewhat different

from the examples described in Sections 7.4–7.6. As long

as a water molecule is adsorbed on its initial adsorbent

platelet, it is expected to have the same orientation relative

to the external magnetic field. We anticipate here that the

platelets are immobile relative to water motions. Reorienta-

tions of initially adsorbed water molecules are therefore

only possible if the molecules are desorbed or adsorbed at

another platelet of a different orientation, whereas random

walks on the initial surface is preserved. The scenario is

illustrated in Fig. 36. Experimental data and analytical

treatments can be found in Refs [109,134].

8. Polymer dynamics

The three most popular polymer dynamics models are the

Rouse model [135], the renormalized Rouse formalism

[136], and the tube/reptation concept [137]. The Rouse

Fig. 35. Inflection frequency, ni; of the deuteron spin–lattice relaxation

dispersion of bovine serum albumin dissolved in D2O at 291 K as a function

of the water content, cw: This dependence can be described by the ‘free-

water volume model’ of macromolecular tumbling. An illustration is shown

in the inset. A protein molecule (‘protein’) surrounded by hydration water

can only tumble if sufficient free water is available. Otherwise it will be

immobilized. The minimum free-water volume corresponds to the

circumscribing sphere. Below a critical water content c0 ¼ 65% (by

weight), macromolecular tumbling becomes slower than the RMTD

process of water on the protein surface. In this case the inflection frequency

is determined by RMTD. Above c0; the inflection frequency is governed by

the tumbling rate of the macromolecule. The experimental data were

evaluated directly from T1 frequency dispersion curves such as those shown

in Fig. 34. The continuous line was calculated on the basis of formalism

described in the text [120,129].


model is supposed to reflect chain dynamics below a

polymer specific critical molecular weight, Mc; where

neighbouring polymer chains effectively form a homo-

geneous viscous medium (see Fig. 37 and the discussion

below). The other two models are designed to account for

so-called ‘entanglements’ becoming effective above Mc: In

this case, chain dynamics is strongly hindered by topolo-

gical constraints as a consequence of long-living confine-

ments by neighbouring chains (see Fig. 38 for an

illustration). In the renormalized Rouse formalism, entan-

glement effects are taken into account by a generalized

Langevin equation ansatz [4,136,138] applying a so-called

memory function term. The formalism is in contrast to

the more illustrative and heuristic tube/reptation model,

where a fictitious tube surrounding the tagged chain is

assumed to represent the entanglements (compare with

Section 8.10).

In the following we will show that field-cycling NMR

relaxometry faithfully provides evidence for the very

different predictions of each of these models provided that

the model premises are realized in the experiments and on

the time scale probed by the method. These are particularly

convincing examples of the strength of the technique to

elucidate features of different molecular dynamics pro-

cesses. The straightforward connection between theoretical

principles and experimental data characteristic of this

method will be shown to be of paramount importance in

this context.

The experimental frequency/temperature ‘window’

opened by NMR relaxometry is schematically shown in

Fig. 39. Considering temperatures above the glass transition

and below thermal degradation suggests a typical range

Fig. 36. Schematic representation of a saponite platelet of radius R in an

aqueous suspension. The arrows represent the orientation of water

molecules. As long as molecules remain adsorbed (or are re-adsorbed on

the same platelet), their orientation is preserved whereas desorption leads to

total loss of the correlation to the initial orientation (see the ‘pancake’

model formalism presented in Ref. [109]).

Fig. 37. The Rouse model of chain dynamics: a real chain is modeled by

‘beads’ and massless (entropical) ‘springs’.

Fig. 38. Schematic illustration of an ‘entangled’ polymer chain (black line)

in a matrix of neighbouring chains (gray).

Fig. 39. Schematic representation of the experimental temperature/fre-

quency ‘window’ conveniently accessible by field-cycling NMR relaxo-

metry in combination with conventional high-field techniques. It

specifically addresses the chain mode regime of typical polymers, whereas

local segment fluctuations and center-of-mass motions can only be probed

at low temperatures/high frequencies and high temperatures/low frequen-

cies/low molecular weights combinations, respectively.


between 300 and 400 K. The proton/deuteron frequency

range accessible by the field-cycling technique in combi-

nation with high-field relaxometry is within 102 and 109 Hz.

This window almost completely matches the dynamic range

of chain modes of polymers with molecular masses up to

105. Some influence of local segment-internal fluctuations

shows up only at low temperatures and high frequencies. At

high temperatures and low frequencies, center-of-mass

motions of polymers with molecular masses below 105

may manifest themselves in the form of low-frequency cut-

offs of the dispersion curves. For the most part, spin–lattice

relaxation in the experimental window is dominated by

chain modes irrespective of the molecular mass. This in

particular holds in the absence of side group motions that

might be superimposed on chain modes.

8.1. The three components of polymer dynamics relevant

for NMR relaxometry

The chemical composition of a polymer described by a

Kuhn segment chain is exclusively represented by specific

parameters such as the segment friction coefficient and the

Kuhn segment length. That is, no information referring to a

length scale shorter than the Kuhn segment, and hence, to

the specific chemistry of the compound is considered in the

frame of the polymer theories discussed in this context. In

terms of time scales, only molecular motions taking place in

the limit t q ts are regarded, where ts is a time constant

characteristic for local motions occurring inside Kuhn

segments. The question to be examined in this section is to

what extent NMR relaxation experiments are affected by

local motions ðt # tsÞ on the one hand and by chain modes

ðt q tsÞ on the other.

Generally one can distinguish three dynamic components

contributing to motions of a chain [4,139] (see also Fig. 39).

Component A represents restricted fluctuations occurring

within Kuhn segments, that is, on a time scale up to t < ts:

These motions may be supplemented and superimposed by

monomer side-group rotational diffusion if such mobile

groups exist. All reorientations due to component A cover

only a restricted solid angle range of the interdipole vector

(proton resonance) or the electric field gradient principal

axis (deuteron resonance). As concerns the main chain

groups, the motion is predominantly involves rotameric

isomer interconversions and may be interpreted even in

terms of defect diffusion models [140].

The consequence of the restricted nature of reorientations

by component A is that the dipolar or quadrupolar

correlation functions do not decay to zero by these local

and molecular weight independent motions. Rather a

residual correlation remains at long times that decays

further only by chain modes of hierarchically higher order.

In analogy to the argument used for the derivation of

Eq. (46), the correlation functions given in Eq. (4) can

therefore be specified for component A as

GAðtÞ ¼ gAðtÞ þ GAð1Þ; ð77Þ

where gAðt q tsÞ ¼ 0: Experiments suggest that in con-

densed polymer systems GAð1Þ=GAð0Þ ¼ 1023· · ·1022

only. That is, most of the orientation correlation function

decays already due to component A. The slower com-

ponents B and C can therefore refer only to a small residual

correlation GAð1Þ ¼ const: The consequence of the strong

correlation decay caused by component A is that the

minimum in the temperature dependence of T1 is pre-

dominantly determined by this component and indicates the

value of ts via the ‘minimum condition’ vts < 1 (compare

with Fig. 3b). The values deduced from T1 minima

corroborate that the frequency range of field-cycling NMR

relaxometry largely corresponds to the time limit t q ts;

and hence, addresses the chain-mode regime beyond local

segment fluctuations (see Fig. 39).

Component B refers to the hydrodynamic chain-mode

regime which is of particular interest in the context of chain

dynamics models. The time scale is between ts and the

terminal chain relaxation time tt: The chain modes in this

regime are expected to be independent of the molecular

mass and largely govern the spin– lattice relaxation

dispersion data to be discussed in the following.

Component C, the cut-off process of component B,

finally corresponds to the terminal chain relaxation time tt

after which all memory of the initial conformation gets lost.

It may become visible in the experimentally accessible

frequency window of field-cycling NMR relaxometry in the

form of a cross-over to an ‘extreme-narrowing’ plateau.

Since this cross-over is connected with the terminal chain

relaxation time tt; it will be strongly dependent on the

molecular mass. This effect therefore shows up in the

experimentally accessible frequency window of NMR

relaxometry only for relatively small molecular masses.

8.2. The different time-scale approach for the NMR

correlation function

Representing the correlation function decays by com-

ponents A–C by the normalized partial correlation func-

tions GAðtÞ; GBðtÞ and GCðtÞ; respectively, and interpreting

these functions as probabilities that the respective fluctu-

ations of the spin interactions have not yet taken place at

time t, permits one to compose the three partial correlation

function into the total expression

GðtÞ ¼ GAðtÞGBðtÞGCðtÞ: ð78Þ

Combining this function with Eq. (77) and assuming the

different time scale limit, that is gAðt . tsÞ < 0; GBðt #

tsÞ < GBð0Þ ¼ 1; and GCðt # ttÞ < GCð0Þ ¼ 1 leads to

GðtÞ < gAðtÞ þ GAð1ÞGBðtÞGCðtÞ; ð79Þ

where tt in this case generally represents the terminal chain

relaxation time, and GAð1Þ is a constant typically being


equal to less than a few percent of the initial correlation

function value. This different time scale approximation is

completely analogous to the formalism discussed in Section

7.1 for molecular dynamics on surfaces.

For comparison with model theories the chain mode

regime represented by component B is suited best and will

be discussed in detail below. It will be shown that the NMR

relaxometry frequency window of typically 103 , n , 108

Hz (for proton resonance) almost exclusively probes the

influence of chain modes represented by component B

(compare with Fig. 39). That is, the correlation function

relevant in the experimental frequency window for spin–

lattice relaxation dispersion may be identified with com-

ponent B of polymer melts according to

Gðts , t , ttÞ < GAð1ÞGCð0ÞGBðtÞ ¼ constGBðtÞ ð80Þ

provided that M q Mc; T . 300 K; and ts , 1029 s: NMR

relaxometry thus offers a unique way to directly probe

predictions by chain mode model theories.

8.3. Evidence for Rouse dynamics ðMMcÞ

In the frame of the Rouse model, the dynamics of a

‘bead-and-spring’ chain representing a real chain in a

viscous medium without hydrodynamic backflow effects

(Fig. 37) is treated with the aid of the equation of motion for

the nth segment

›

›t~pnðtÞ ¼ Kð2~rn 2 ~rnþ1 2 ~rn21Þ2 z

›~rn

›tþ ~f L

n ðtÞ; ð81Þ

where ~pnðtÞ is the momentum, and ~f Ln ðtÞ is the Langevin

stochastic force acting on this segment. In the continuum

limit, the segment number n can be treated as a continuous

variable ranging from 0 to N; so that Eq. (81) can be

rewritten in approximate form as

›

›t~pnðtÞ < K

›2~rn

›n22 z

›~rn

›tþ ~f L

n ðtÞ: ð82Þ

The effective intramolecular interactions between the

segments are approximated by entropic harmonic inter-

actions, reflecting the Gaussian character of the large-scale

chain conformation. Intermolecular interactions (with the

surrounding viscous medium) are taken into account by

friction and stochastic forces acting on the segments. The

entropic spring constant and the friction coefficient of a

Kuhn segment are given by

K ¼3kBT

b2ð83Þ

Fig. 40. Proton spin–lattice relaxation dispersion under conditions where Rouse dynamics is expected to apply. The theoretical curves have been calculated

with the aid of Eq. (85). The validity of this model is restricted to vp t21s ; that is, below the local segmental fluctuation rate. The positions on the frequency

axes where the condition vts ¼ 1 applies are indicated by arrows for the segment fluctuation time ts fitted to the experimental data. The ts values are in accord

with those derived from the T1 minimum data where applicable [144]. (a) Polyisobutylene (PIB) ðMw ¼ 4700 , Mc < 15; 000Þ; melt at 357 K [138]. (b)

Polydimethylsiloxane (PDMS) ðMw ¼ 5200 , Mc < 20; 000Þ; melt at 293 K [141,145,146]. (c) Solution of 15% polydimethylsiloxane ðMw ¼ 423; 000Þ in

CCl4 at 293 K [141,145,146].


and

z ¼ 6phah; ð84Þ

respectively, where kB is Boltzmann’s constant, T is the

absolute temperature, h is the viscosity of the medium

surrounding the segment, ah is the hydrodynamic radius of

the segments. Rouse dynamics is expected to apply to

molecular weights below the critical value where the so-

called entanglement effects are not yet effective.

Under such circumstances, the following frequency

dependences are predicted [4,141–143]:

1

T1

/2tslnðvtsÞ for t21

R p vp t21s

ts ln N for vp t21R :

(ð85Þ

Eq. (85) directly reflects Rouse chain modes, that is

component B to be probed by field-cycling NMR relaxo-

metry. The other components, A and C, matter only at low

temperatures, high frequencies and high temperatures, low

molecular weights, respectively. Experimental data sets for

the proton spin–lattice relaxation dispersion are shown in

Fig. 40 in accordance with the theoretical frequency

dependence given in Eq. (85).

Very interestingly, the values for the segment fluctuation

time ts fitted to the T1 dispersion data coincide with those

derived from minima of the temperature dependence of T1

(compare Fig. 3b) corrected for the temperature of the field-

cycling measurements [144]. That is, these two independent

methods of obtaining ts lead to consistent results.

The logarithmic Rouse formula given in Eq. (85) is valid

for vp t21s : Deviations of the theoretical curves from the

experimental data points therefore are only expected at

frequencies approaching the ‘minimum condition’ vts < 1;

where component A starts to become perceptible. This

explains why the PDMS data fit better to the model than PIB

with a segment fluctuation time one order of magnitude

longer than that of PDMS.

Entanglement effects expected for large molecular

weights can be reduced by dissolving the polymer even if

its molecular weight is well above the critical value in the

melt. That is, the critical molecular weight in solution is

larger than in the melt [147]. This is demonstrated in

Fig. 40c for PDMS dissolved in CCl4. The data can again be

well described by the Rouse model. The conclusion from

these findings is that the Rouse model is perfectly

corroborated by NMR relaxometry experiments provided

that entanglement effects are excluded.

8.4. The three regimes of spin–lattice relaxation dispersion

in entangled polymer melts, solutions and networks ðMMcÞ

Polymer chains above the critical molecular weight sense

neighbouring chains as such and not just as an unstructured

viscous medium. Fig. 38 shows an illustration. The

consequence is a dramatic change of the dynamic

behaviour. Figs. 41–48 show a series of typical spin–lattice

relaxation dispersion curves for polymer melts above the

critical molecular weight. The field-cycling technique has

been applied to melts, solutions and networks of numerous

polymer species. The parameters varied in the experiments

were the temperature, the molecular weight, the concen-

tration and the cross-link density. For control and

comparison, the studies are partly supplemented by

rotating-frame spin–lattice relaxation data, and, of course,

by high-field data of the ordinary spin–lattice relaxation

time. Furthermore, deuteron spin–lattice relaxation was

Fig. 41. Proton spin–lattice relaxation times in the laboratory system ðT1Þ

and in the rotating frame ðT1rÞ of polyisobutylene (PIB) melts as a function

of the frequency (n; Larmor frequency in the laboratory frame, n1 ¼ gB1=ð2

pÞ; rotating-frame nutation frequency) [145]. The data refer to the

molecular-weight independent chain-mode regimes I (high-mode number

limit) and II (low-mode number limit) [138]. The arrow indicates the cross-

over frequency nI;II between regions I and II.

Fig. 42. Proton spin–lattice relaxation time of polyethyleneoxide (PEO)

melts as a function of the frequency [138]. The data refer to the molecular-

weight independent chain-mode regime II (low-mode number limit) and

regime III influenced by intersegment dipolar interactions [149].


employed for identifying the role that different spin

interactions are playing for relaxation dispersion.

Remarkably, a series of distinct and apparently universal

NMR relaxation dispersion regimes can be identified from

the data in Figs. 41–48 for the diverse polymer species. We

consider the time/frequency window tt p ðt;v21Þp ts

indicative for component B (Fig. 39). The terminal chain

relaxation time, tt; is indicated by a cross-over to a low-

frequency T1 plateau which lies outside of our frequency

window for high enough molecular weights. The segment

fluctuation time, ts; can be determined from T1 minima and

matters only at the highest frequencies if at all. Under such

conditions, there is clear experimental evidence for three

distinct proton dispersion regimes for entangled polymers,

Fig. 43. Proton spin–lattice relaxation times in the laboratory system ðT1Þ and in the rotating frame ðT1rÞ of polydimethylsiloxane (PDMS) melts and solutions

as a function of the frequency (n; Larmor frequency in the laboratory frame, n1 ¼ gB1=ð2pÞ; rotating-frame nutation frequency) [145]. The melt data for

Mw . Mc < 24; 000 refer to the molecular-weight independent chain-mode regimes I (high-mode number limit) and II (low-mode number limit). (a) Melts of

PDMS 250,000 at different temperatures. The cross-over between regimes I and II at nI;II is shifted to higher frequencies with increasing temperature. (b) Melts

of PDMS at 293 K for different molecular weights. For Mw , Mc the chain mode regimes I and II characteristic for entangled dynamics are absent and are

replaced by motions not subject to entanglements. The theoretical T1 dispersion curve expected for the Rouse model is shown for comparison. (c) Temperature

dependence of the ‘cross-over time constant’ tI;II ¼ 1=ð2pnI;IIÞ evaluated from the plot in Fig. 43a. The line represents the Arrhenius law tI;II ¼ 1:1 £ 10210

s exp{ð15:8 kJ mol21Þ=RT}: The deviation of the data point at 213 K indicates the influence of the supercooled state at this temperature. (d) Solutions of PDMS

with Mw ¼ 423; 000 in CCl4 at 293 K for different concentrations. For low concentrations the chain mode regimes I and II characteristic for entangled

dynamics fade more and more. The theoretical T1 dispersion curve expected for the Rouse model is shown for comparison.


Mw q Mc:

"v#T

T1 / M0wv

0:5^0:05 ðregion I; ‘high-mode

number limit’Þ

T1 / M0wv

0:25^0:1 ðregion II; ‘low-mode

number limit’Þ

T1 / M0wv

0:45^0:05 ðregion III; ‘inter-segment

interaction limit’Þ

8>>>>>>>>>>><>>>>>>>>>>>:

ð86Þ

These power laws are in accordance with data for the

frequency dependence of the spin–lattice relaxation time in

the rotating frame, T1r; in the range accessible by this

technique.

The three proton relaxation dispersion regions I (fast

motions), II, III (slow motions) appear in that sequence from

high to low frequencies or from low to high temperatures

(see the up and down arrows in Eq. (86)). The frequency

window of the field-cycling NMR relaxometry technique

(Fig. 39) is often not broad enough to cover all three regions

all at once. Mobile polymers like polyisobutylene (Fig. 41),

polydimethylsiloxane (Fig. 43) and polydiethylsiloxane in

the isotropic phase (Fig. 46) are subject to regions I and II

under the experimental conditions. Less mobile polymers

like polyethyleneoxide (Fig. 42) tend to reveal regions II

and III in the instrumental frequency window.

It is nevertheless possible to demonstrate that all three

proton relaxation dispersion regions are intrinsic to

entangled polymer dynamics: The temperature/frequency

range accessible with NMR relaxometry in polybutadiene

(Fig. 44) permits the observation of all three regions one

after the other in the same sample. At low temperatures,

regions I and II dominate, and at elevated temperatures

region III comes into play. The shift of the cross-over time

between regions I and II,

tI;II ¼ 1=ð2pnI;IIÞ; ð87Þ

for polydimethylsiloxane is plotted in Fig. 43c as a function

of the reciprocal temperature. The Arrhenius-like behaviour

corroborates that the chain mode regimes I and II are subject

to thermal activation.

The power laws given in Eq. (86) may be considered to

be universal in the frame of some scattering of the

exponents found for different polymer species. Slight

deviations from the slope specified in Eq. (86) for region

II were reported in Refs [144,145,148] for proton relaxation

data of polyisoprene melts and in Refs [149,150] for

deuteron relaxation data of polethyleneoxide and polybuta-

diene melts (see Figs. 47 and 48). However, in view of the

model approximations to be considered in the interpret-

ations outlined below, such polymer and spin interaction

specific variations of the experimental exponents relative to

theoretical predictions appear to be of minor importance.

The general trends of the three regimes given in Eq. (86)

remain untouched.

It may be tempting, but these three proton spin–lattice

relaxation dispersion regions must not be identified with the

Doi/Edwards limits (I)DE, (II)DE and (III)DE which are

predicted in the frame of the tube/reptation model for

dynamic ranges with significantly different frequency and

molecular weight dependences (see Table 1) [4]. No Rouse-

like dynamics corresponding to limit (I)DE can be identified

in the shown experimental data sets for entangled polymer

melts (polymer solutions excepted; see Fig. 40c). In the

frame of the tube/reptation model, there is no such thing as

the remarkably weak frequency dependence of region II,

T1 / M0wv

0:25^0:05: The square root molecular weight

dependence predicted in that model for limit (III)DE does

not occur either. However, in Section 8.10 it will be shown

that Doi/Edwards predictions can be verified indeed if

chains are confined to artificial tubes prepared in a solid

polymer matrix [37]. On the other hand, the high- and low-

mode number limits resulting from the renormalized Rouse

model [138] provides a perfect explanation of dispersion

regions I and II, whereas region III can be shown to be due

to inter-segment dipolar interactions.

Fig. 44. Proton spin – lattice relaxation time of a melt of linear

polybutadiene ðMw ¼ 65; 500Þ at different temperatures as a function of

the frequency [149]. By varying the temperature, all three dispersion

regimes show up in the experimental frequency window one by one.


8.5. High- and low-mode-number limits

(dispersion regions I and II)

The appearance of dispersion regions I and II in

experiments (see Eq. (86)) confirms the high- and

low-mode-number, short-time limits predicted by the

renormalized Rouse model [4,138]. The exponents of the

power laws suggested by the experimental data even match

the theoretical predictions almost perfectly. Nevertheless,

the good coincidence of the numerical values of these

Fig. 45. Proton spin–lattice relaxation time of thermoreversible networks of polybutadiene at different temperatures as a function of the frequency [157].

Linear PB ðMw ¼ 51; 000Þ was cross-linked by addition of 4-phenyl-1,2,4-triazoline-3.5-dion (phenylurazole, PU). The cross-over frequencies between

regimes I, II and III are shifted depending on the cross-link density. At the lowest temperatures the dispersion slopes tend to be steeper than in ordinary melts

(see Eq. (86)): (a) 19 phenylurazole groups per chain, (b) 28 phenylurazole groups per chain, (c) 37 phenylurazole groups per chain.


exponents is not necessarily considered to be the decisive

finding backing up the renormalized Rouse model. The

problem is that the theoretical exponents are slightly

affected by the renormalization ansatz that cannot be traced

back to elementary principles and is subject to some

ambiguity [4]. The important result is rather that (i) two

such limiting cases are relevant anyway and indeed become

visible, and that (ii) the exponents change almost exactly as

predicted for the cross-over between the high- and low-

mode-number limits.

The distinction of two such limits with the correct

variation of the power law exponents is a result reflecting

the analytical structure of the generalized Langevin

equation which is thus proven to represent the essential

features of entangled chain dynamics in the limit, ts p t p

tt: The generalized Langevin equation obviously contains

crucial elements determining entangled-chain dynamics

[4,136,138].

8.6. Intra- and inter-segment spin interactions (dispersion

region III)

Intra-segment dipolar interactions fluctuate as a conse-

quence of segment reorientations and conformational

changes. If the interacting nuclear dipoles are residing on

different segments or even different chains, variations of the

inter-nuclear vector are much slower because they are the

consequence of displacements of the dipole hosting

segments by self-diffusion relative to each other. Any

spin–lattice relaxation dispersion affected by inter-segment

dipolar interactions is therefore expected at very low

frequencies. This can be tested by comparing proton with

Fig. 46. Proton spin–lattice relaxation times in the laboratory system ðT1Þ

and in the rotating frame ðT1rÞ of polydiethylsiloxane (PDES) melts in the

isotropic and mesomorphic phases as a function of the frequency (n; Larmor

frequency in the laboratory frame; n1 ¼ gB1=ð2pÞ; rotating-frame nutation

frequency) [154]. The data of the isotropic melt phase refer to the

molecular-weight independent chain-mode regimes I (high-mode number

limit) and II (low-mode number limit). In the (ordered) mesomorphic phase

the dispersion slopes are much steeper and the cross-over frequency is

shifted to a lower value. At very high frequencies, the influence of

component A (including side chain motions) becomes gradually visible.

Fig. 47. Proton and deuteron spin–lattice relaxation times of polyethyleneoxide (PEO) and polybutadiene (PB) melts as a function of the frequency [149]. In

addition to intrasegment dipolar coupling, proton relaxation is also subject to intersegment dipolar couplings leading to the dispersion regime III specific for

this sort of relaxation mechanism. Deuteron relaxation is predominantly due to quadrupole interaction which is of an intrasegment nature. A cross-over from

regime II to regime III therefore does not occur with deuteron resonance. (a) Polyethyleneoxide (for further deuteron relaxation data see Figs. 46 and 50b),

(b) polybutadiene.


deuteron spin–lattice relaxation dispersion of the same

polymer species. Deuteron relaxation is dominated by

(intra-segment) quadrupole coupling with the local electric

field gradient whereas proton relaxation is subject to intra-

as well as inter-segment dipolar interactions.

Fig. 47 shows such comparisons for polyethyleneoxide

and polybutadiene melts. The cross-over between disper-

sions regions II and III observed with proton NMR

obviously disappears in the deuteron studies. Region III

does not occur with deuterons. The different dispersion

slope measured with deuteron NMR was also demonstrated

with samples of lower molecular weights (see Figs. 48 and

52b). That is, dispersion region III must not be considered as

a limit specific for polymer theories. It rather appears to be

mainly an effect intrinsic to NMR relaxation by inter-

segment dipolar interaction.

A schematic representation of the situation one is dealing

with in this context is shown in Fig. 49. We consider the

representative segments k on chain a and l on chain b: The

internuclear vector, ~rklðtÞ; fluctuates because of self-

diffusive displacements ~RrelðtÞ of segment l relative to

segment k: That is, the origin of the reference frame is taken

to be fixed at segment k:

The average correlation function for the inter-segment

dipolar interaction can be expressed as

GðmÞinterðtÞ ¼ rs

ðgðr0ÞG

mklðtÞd

3r0 ðm ¼ 1; 2Þ; ð88Þ

where

GðmÞkl ðtÞ ¼

Y2;mðq0;w0ÞY2;2mðqt;wtÞ

r30r3

t

* +=GðmÞ

kl ð0Þ: ð89Þ

(see Eq. (4)) and gðr0Þ is the radial segment pair correlation

function, and rs is the spin number density. The variables r0

and rt in Eqs. (88) and (89) stand for rklð0Þ and rklðtÞ;

respectively. That is, r0 ; rklð0Þ and rt ; rklðtÞ: For the

analytical treatment the radial segment pair correlation

function may crudely be approximated by (compare Refs

[149,151])

gðr0Þ <0 if r0 # s

1 otherwise:

(ð90Þ

The correlation function in Eq. (89) can be approximated by

the following consideration: For relative displacements

RrelðtÞ (see Fig. 49) much larger than the initial internuclear

distance r0 ; rklð0Þ; the distance and polar angle variation

Fig. 48. Deuteron spin–lattice relaxation times of deuterated polyethyleneoxide (PEO) (a) and polybutadiene (PB) (b) as a function of the frequency [150].

Table 1

Theoretical dependences on time ðtÞ; angular frequency ðvÞ; and molecular mass ðMÞ predicted by the tube/reptation model [4] for the mean squared segment

displacement and the intra-segment spin–lattice relaxation time in the four Doi/Edwards limits

Limit Mean squared segment displacement, kR2l Refs. Intra-segment spin–lattice relaxation time, T intra1 Refs.

(I)DE ts p ðt; 1=vÞp te ð2=p3=2Þb2ðt=tsÞ1=2 / M0t1=2 [137] 2CIðM

0=ðts lnðvtsÞÞ [141,142]

(II)DE te p ðt; 1=vÞp tR b2N1=2e ðt=tsÞ

1=4 / M0t1=4 [137,160] CIIM0v3=4 [160,161]

(III)DE tR p ðt; 1=vÞp td ð2=pÞb2ððNe=3NÞðt=tsÞÞ1=2 / M21=2t1=2 [137,160] CIIIM

21=2v1=2 [160]

(IV)DE td p ðt; 1=vÞ 2ðkBT=zÞðNe=N2Þt / M22t1 [137,160] CIVM2ð1:5· · ·2:0Þv0 [140]

The factors CI; CII; CIII and CIV are frequency and molecular-mass independent constants.


leads to total loss of any correlation to the initial distance

vector, whereas the correlation is completely retained in the

opposite limit. That is,

GðmÞkl ðtÞ! 0 for RrelðtÞq r0 ð91Þ

and

GðmÞkl ðtÞ < 1 for RrelðtÞp r0: ð92Þ

An expression accounting for these limits is

GðmÞkl ðtÞ <

(�� Y2;mð0Þ

r30

��2)

PðtÞ ¼1

r60

PðtÞ; ð93Þ

where PðtÞ is the probability that segment l is in a spherical

volume /r30 around its initial position ~r0 relative to segment

k: This probability is specified by the following limits:

PðtÞ! 1 for kR2relðtÞlp r2

0 ;

PðtÞ!r3

0

kR2relðtÞl

3=2for kR2

relðtÞlq r20 : ð94Þ

The expression

PðtÞ <r3

0

½r20 þ kR2

relðtÞl�3=2

: ð95Þ

complies to both limits given in Eq. (94) and will therefore

be taken as an approximation. Inserting Eqs. (90), (93) and

(95) in Eq. (88) gives

GðmÞinterðtÞ < rs

ln{2kR2labðtÞl=s}

kR2labðtÞl

3=2; ð96Þ

where the mean squared displacement in the laboratory

frame is related to the mean squared displacement in the

frame fixed at segment k by kR2labðtÞl ¼ ð1=2ÞkR2

relðtÞl: Since

the logarithmic term in Eq. (96) varies slowly with time so

that we may approximate further

GðmÞinterðtÞ <

rs

kR2labðtÞl

3=2: ð97Þ

The time dependence of the mean squared segment

displacement in the laboratory frame was derived on the

relevant time scale as low-mode-number, short-time limits

of the renormalized and twice renormalized Rouse models

as kR2labðtÞl/ t2=5 and kR2

labðtÞl/ t1=3; respectively [4,152].

Inserting these power laws in Eq. (97), one obtains after

Fourier transformation (see Eqs. (5)–(7))

1

T1

¼1

T intra1

þ1

T inter1

; ð98Þ

where according to the renormalized Rouse theories given

in Refs [4,152]

T intra1 / v0:20· · ·0:33 ð99Þ

and

T inter1 / n0:4· · ·0:5

: ð100Þ

That is, dispersion region III of proton relaxation is

explained by the dominance of this inter-segment contri-

bution, whereas regions I and II are dominated by intra-

segment spin interactions.

All three dispersion regions occurring in the frame of

component B of chain dynamics can be described in a

consistent way based on the same elementary theory.

Region III very remarkably is determined by the effect of a

certain relaxation mechanism in combination with segment

self diffusion properties as independent phenomena. The

consistency of the interpretation of the three proton

dispersions regions I–III with the aid of the renormalized

Rouse theory is striking.

Fig. 49. Intra- and inter-segment spin couplings. (a) In the frame of the experiments referred to in this article, intrasegment spin couplings can be dominated by

dipole–dipole interactions (1H) or quadrupole couplings (2H). They fluctuate due to segment reorientations relative to the laboratory frame. For component B,

the relevant segment orientation is represented by the chain tangent vector ~bn at segment n: (b) Intersegment interactions (segments on the same or different

chains) are exclusively of a dipolar nature. Spin–lattice relaxation on these grounds originates from reorientation and length variation of the distance vector ~rkl

between segments (in the scheme denoted as k and l). These fluctuations are caused by translational displacements ~Rrel of one segment relative to the other. This

is described best in a frame fixed on one of the segments of the interacting pair. Since self-diffusion is a relatively slow process, spin–lattice relaxation by

intersegment interactions becomes relevant only at relatively low frequencies [149].


The slope of the deuteron spin– lattice relaxation

dispersion curves representing intra-segment interactions

is characterized by the exponent 0.34 (Figs. 47, 48 and 52b).

This is somewhat larger than the exponent 0:25 ^ 0:1 found

in the average for the proton relaxation dispersion in region

II (Eq. (86)) interpreted above as also to be due to intra-

segment interactions alone. The discrepancy may be due to

some rudimentary influence of inter-segment dipolar

interactions on the proton relaxation dispersion in region

II before region III is reached.

8.7. Mesomorphic phases of polymers without mesogenic

groups

In Section 9 we will deal with the spin–lattice relaxation

dispersion of liquid crystals. However, there is another sort

of mesomorphic phenomenon that is not based on the

existence of so-called mesogenic groups. This refers to

totally flexible polymers such as linear polydiethylsiloxane

(PDES) and its higher homologues which form ordered

mesophases in temperature intervals between the solid and

the isotropic melt phases [153]. The results to be described

in the following demonstrate the sensitivity of NMR

techniques to microstructural changes in this respect.

Fig. 46 shows spin–lattice relaxation dispersion data for

PDES both in the isotropic and in the mesomorphic phase.

The dispersion of the isotropic melt are governed by the

same empirical power laws for regions I and II as stated

before (Eq. (86)) and in particular as measured in PDMS

melts. However, PDMS (side groups: –CH3) remarkably

does not show the mesophase seen for PDES (side groups:

–CH2CH3).

In the mesophase of PDES, two power law regimes show

up again, but with larger exponents and a cross-over

frequency shifted by a factor of about 10 (after correction

for the different temperatures) to a lower value. In summary,

the power laws observed for PDES in broad frequency

ranges are (see Fig. 46 and Ref. [154]).

T1 /

M0wv

0:50^0:05 ðregion I ‘isotropic phase’Þ

M0wv

0:73^0:05 ðregion I ‘mesophase’Þ

M0wv

0:25^0:05 ðregion II ‘isotropic phase’Þ

M0wv

0:45^0:05 ðregion II ‘mesophase’Þ

8>>>>><>>>>>:

ð101Þ

The slopes in region II both for the isotropic and for the

mesomorphic phase are in accordance with data for

the spin–lattice relaxation time in the rotating frame, T1r:

The dynamics specific for the mesophase was also examined

with the dipolar correlation effect probing residual dipolar

couplings in ordered phases [155].

The spin–lattice relaxation dispersion results must be

compared with those expected for nematic liquid crystals

where a power law

T1 / v0:5 ð102Þ

was predicted for ODFs at low frequencies (see Section 9).

With nematic compounds, this law representing collective

fluctuations in the ordered state should show up irrespective

of whether it is a monomeric or polymeric substance. This is

in contrast to the mesophase of PDES. The monomers of

this compound do not show any order because they do not

contain any mesogenic groups. That is, the order observed

in the PDES mesophase cannot be traced back to the sterical

packing origin attributed to nematic phases.

It is concluded that the modified power laws for spin–

lattice relaxation dispersion in the mesophase reflect a

modified behaviour of chain modes rather than collective

fluctuations of ensembles of molecules in ordered domains.

This conclusion is corroborated by the identical frequency

dispersion of the spin–lattice relaxation times T1 and T1r in

the laboratory and in the rotating frames, respectively. If the

order in the PDES mesophase would be of a nematic nature

and the fluctuations causing dispersion region II conse-

quently would be of the ODF type, the frequency

dependence of T1r would be absent while that of T1 would

be retained as explained in Ref. [156] and in Section 9.14.

8.8. Polymer solutions

In solutions, the critical molecular weight characteristic

for the cross-over between Rouse-like and entangled

dynamics grows with decreasing polymer concentration

[147]. NMR relaxation measurements turn out to be

particularly suitable for studies of the dilution process.

Spin–lattice relaxation dispersion data for PDMS dissolved

in CCl4 as a proton-free solvent are shown in Fig. 43d.

Compared to the region II dispersion whose occurrence is

one of the characteristics of melts of entangled polymers,

the spin–lattice relaxation dispersion becomes flatter upon

dilution. With a PDMS content of 15% in CCl4 the data can

be described by the Rouse model as the consequence of free-

chain dynamics (see Fig. 43d). This may appear surprising

since the critical molecular weight for this concentration is

below the weight average molecular weight of the polymer

examined, Mw ¼ 423; 000; so that entangled dynamics

should still apply.

On the other hand, all polymer dynamics models

unspecifically predict some short-time/high-frequency

limit where Rouse-like behaviour should dominate. In

melts of entangled polymers such a regime could never be

identified by spin–lattice relaxation dispersion because

component B in the form of the high-mode number limit

(region I) appears to overlap with the local segment

fluctuation regime (manifesting itself as component A).

That is, there is no intermediate time or frequency window

left in entangled melts where Rouse dynamics could

develop in full.

Such a dynamic gap obviously arises in solutions if the

polymer concentration is low enough: Component A

representing local segment fluctuations becomes acceler-

ated upon dilution, whereas the effect of entanglements on


the chain modes is shifted to correspondingly longer

length and time scales. That is, a very important

theoretical requirement is satisfied in this way. We will

come back to the topological constraint problem in

Section 8.10.

8.9. Polymer networks

Permanent or thermoreversible cross-links mediate the

opposite effect on chain dynamics compared with dilution

by a solvent. Instead of releasing topological constraints by

dilution, additional obstruction of chain modes is estab-

lished by the network. With respect to NMR measurements,

relatively large cross-link densities are needed to affect the

chain modes visible in the experimental time/frequency

window. In Ref. [145] spin–lattice relaxation dispersion

data of permanently cross-linked PDMS are reported. With

decreasing mesh length, the chain modes appear to be

shifted to lower frequencies. This is indicated by lower

values of the relaxation times while the dispersion slopes of

regions I and II are retained. On the other hand, the effect on

the cross-over frequency is minor. A field-cycling NMR

relaxometry study on vulcanized natural rubber can be

found in Ref. [158].

The fact that the dispersion regions I–III of proton spin–

lattice relaxation of polybutadiene networks retain their

qualitative appearance in the presence of (in this case

thermoreversible) cross-links is demonstrated in Fig. 45. In

Ref. [157] it is shown that the influence of fluctuating cross-

links on the dispersion curves can be explained by an

effectively modified monomeric friction coefficient. A result

of particular interest is the shift of the cross-over

frequencies between dispersion regions I–III. The shifts

can be explained by the ratios

nI;II

~nI;II

¼nII;III

~nII;III

¼~ts

ts

; ð103Þ

where the quantities with tilde refer to networks. The

friction effect on segment fluctuations equally slows down

chain modes at the lowest frequencies. Field-cycling NMR

relaxometry is also a favourable tool for gel formation

studies in solutions of interacting macromolecules (see Refs

[114,159]).

8.10. Chain dynamics in pores (‘artificial tubes’)

The motivation to study the dynamics of polymer chains

confined in nanoporous materials with more or less rigid

pore walls is twofold. Firstly, there may be important

technological applications requiring knowledge of the

dynamic behaviour of polymers under such conditions.

The second reason making this field intriguing for polymer

science in general is the possibility of studying chain

dynamics under topological model constraints in the form of

artificial tubes. In the following we will focus on this latter

point.

The tube introduced in the frame of the Doi/Edwards

reptation model for the treatment of bulk systems of

entangled polymer systems is a fictitious one [137]. Fig. 50

illustrates the segment displacement phenomena specific for

the reptation model. There are four characteristic time

constants partly already defined before a more general

background in Sections 8.1–8.3: (i) the Kuhn segment

fluctuation time ts which is of an entirely local nature; (ii)

the so-called entanglement time te indicating the time scale

on which chain modes first sense topological constraints;

(iii) the (longest) Rouse relaxation time tR characterizing

the time scale of chain modes along the contour line of the

tube; (iv) the tube disengagement time td after which the

memory to the initial chain conformation gets totally lost

because the chain has escaped from its initial tube and has

adopted an uncorrelated conformation.

Based on these four time constants, four dynamic limits

of chain modes can be defined as listed in Table 1. Doi and

Edwards [137] originally predicted only the laws for the

mean squared segment displacements. These can be

supplemented by equivalent laws for the spin–lattice

relaxation dispersion listed also in Table 1. The formulas

predicted on this basis can provide only a rather crude

picture of chain dynamics in entangled polymer bulk melts

and they fail to account for numerous experimental findings

quantitatively as well as qualitatively. It is therefore helpful

to study chain dynamics in tube-like pores of a physically

real nature.

The diffusion and relaxation behaviour of diverse

oligomers and polymers confined in nanoporous silica

glasses is reported in Ref. [162]. Both measuring techniques

provide clear evidence for modified chain dynamics

compared with the bulk. However, the problem with this

sort of system is that interactions of the polymers with the

pore walls play an important role and must be distinguished

from the geometrical confinement effect. A corresponding

analysis was possible by comparing polymer data with data

obtained for short oligomers of the same chemical species.

In contrast to the polymers, the oligomers were assumed to

be subject only to adsorption but not to modifications of the

chain modes by geometrical confinement. Subtraction of

Fig. 50. Illustration of a polymer chain (the tagged chain) confined in the

fictitious tube of diameter d formed by the matrix. The contour line of the

tube is called the primitive path having a random-walk conformation with a

step length a ¼ d: The four characteristic types of dynamic processes

(dotted arrow lines) and their time constants ts; te; tR; and td defined in the

frame of the Doi/Edwards tube/reptation model are indicated.


the oligomer relaxation rates from those measured with

polymers in the pores suggests a spin–lattice relaxation

dispersion reduced by the influence of geometrical confine-

ment on the chain-mode distribution. Actually a power law

dispersion reproducing that predicted for limit (II)DE of the

tube/reptation model (see Table 1) could be elucidated

this way.

A more direct verification of tube/reptation features was

possible with systems where the solid matrix as well as the

mobile polymer chains confined to nanopores are of similar

organic chemical composition. In the experiments referred

to in the following, linear polymers were confined in a solid,

strongly cross-linked polymer environment. Under such

conditions, the geometry effect can be expected to dominate

whereas the wall adsorption phenomenon is of negligible

influence.

This sort of system was prepared in the form of so-called

semi-interpenetrating networks. Preparation details are

described in Ref. [163]. The matrix consisted of cross-

linked polyhydroxyethylmethacrylate (PHEMA). Linear

polyethyleneoxide (PEO) was incorporated in nanoscopic

pores of this matrix. The molecular weight of the PEO was

chosen to be large enough to ensure that the root mean

squared random coil diameter in bulk exceeds the pore

diameter. Fig. 51 shows electron micrographs of pore

channels in this material having a width of about 10 nm.

In order to distinguish the mobile polymer in the pore

channels from the cross-linked matrix material, perdeuter-

ated polyethyleneoxide was studied with the aid of deuteron

field cycling NMR relaxometry [37,164]. In this way the

exceptional frequency and molecular weight dependence of

the spin–lattice relaxation time expected for limit (II)DE,

T1 / M0v3=4 ðvts p 1 p vteÞ ð104Þ

(see Table 1), was verified as shown in Fig. 52a for different

molecular weights. In the frequency regime, where the

standard Bloch/Wangsness/Redfield relaxation theory [5,6]

is applicable, the law given in Eq. (104) is reproduced as

T1 / M0^0:05w v0:75^0:02

: ð105Þ

The experimental deuteron frequency range in which this

frequency dependence was observed is 5 £ 105 , n ,

6 £ 107 Hz: De Gennes’ prediction [160] for limit (II)DE

has thus been verified for polymers confined in artificial

tubes in full accordance with the theory. This is in contrast

to the bulk behaviour of the very same polymers studied

with deuteron resonance. The bulk polymer melt data shown

in Fig. 52b reproduce features observed with proton

Fig. 51. Transmission electron micrograph of a replica of a freeze-fractured surface of polyethyleneoxide Mw ¼ 6000 in polyhydroxyethylmethacrylate

(PHEMA) [166]. The pore channels have a width of about 10 nm.


and deuteron resonance with bulk melts of other polymer

species as discussed above (see Figs. 47 and 48). Polymer

chains in bulk melts obviously possess additional degrees of

freedom not existing under pore confinements.

Recently, the so-called ‘corset effect’ on polymer melts

confined in nanopores was discussed in detail in Refs [69,

167]. Based on proton spin–lattice relaxation data, the

features of all three polymer dynamics theories considered

above were comparatively shown for a single polymer

species in a single T1 dispersion plot.

8.11. Cross-over from Rouse to reptation dynamics

The theoretical background of the confinement effect in

(artificial) tubes has been examined in detail with the aid of

an analytical theory as well as with Monte Carlo simulations

[161]. The analytical treatment referred to a polymer chain

confined to a harmonic radial tube potential. The computer

simulation mimicked the dynamics of a modified Stock-

mayer chain [165] in a tube with ‘hard’ pore walls. In both

treatments, the characteristic laws of the tube/reptation

model were reproduced. Moreover, the cross-over from

reptation (tube diameter equal to a few Kuhn segment

lengths) to Rouse dynamics (tube diameter ‘infinite’) was

demonstrated by varying the tube diameter. Fig. 53 shows

data obtained in this way. The frequency dispersion most

specific for the tube/reptation model, namely T1 / v0:75

predicted for limit (II)DE was perfectly reproduced at low

frequencies. With increasing pore diameter the cross-over to

Rouse dynamics becomes visible.

In Refs [69,167], the cross-over was attempted to be

detected with the aid of field-cycling NMR relaxometry.

Melts of linear polyethylene oxide were studied in strands of

variable widths embedded in a solid methacrylate matrix

similar to the one described in Section 8.10. Confined

dynamics, that is reptation, was interestingly observed in the

whole range of pore diameters examined, namely from 8 to

about 60 nm. This ’corset effect’ was attributed to the low

compressibility of polymer melts. Furthermore it was

concluded that the ‘tube’ effective under such conditions

is merely equal to the mean nearest neighbour distance of

the chains. Based on theoretical considerations, a cross-over

to bulk dynamics, that is Rouse ðMw , McÞ or renormalized

Rouse ðMw . McÞ dynamics, can be expected for pore

Fig. 53. Spin–lattice relaxation dispersion for a chain of N ¼ 1600 Kuhn

segments (of length b) confined to a randomly coiled tube with a harmonic

radial potential with varying effective diameters d: The data were

calculated with the aid of the harmonic radial potential theory [161]. ~c is

a constant. At low frequencies the curves show the cross-over from

reptation (T1 dispersion proportional to v3=4 characteristic for the

Doi/Edwards limit (II)DE) to Rouse dynamics for increasing effective

tube diameter.

Fig. 52. Frequency dependence of the deuteron spin–lattice relaxation time of perdeuterated PEO confined in 10 nm pores of solid PHEMA at 80 8C (a) and in

bulk melts (b) [37,164]. The dispersion of the confined polymers verifies the law T1 / M0wv

0:75 at high frequencies as predicted for limit (II)DE of the

tube/reptation model (see Table 1). The low-frequency plateau observed with the confined polymers indicates that the correlation function implies components

decaying more slowly than the magnetization relaxation curves, so that the Bloch/Wangsness/Redfield relaxation theory is no longer valid in this regime. The

plateau value corresponds to the transverse relaxation time, T2; for deuterons extrapolated from the high-field value measured at 9.4 T.


widths much larger than 10 times the mean end-to-end

distance of the polymer chains.

8.12. Protein backbone dynamics and ‘quadrupole dips’

Proton spin–lattice relaxation in protein solutions and

tissue is normally dominated by the relaxation rate in water

enhanced by the water/biopolymer interface processes

discussed above in Section 7.6. However, if the water

content is relatively low or absent [120] or if the sample is

prepared with deuterated water so that water proton signals

are negligible, protein internal motions become relevant in

the spin–lattice relaxation dispersion of protons [118].

Protein and polypeptide internal dynamics can be sub-

divided into side-group motions (e. g. methyl group

rotations or phenyl ring flips) and backbone fluctuations.

Side-group motions are important at high frequencies above

the megahertz regime, whereas backbone fluctuations

govern the low-frequency field-cycling window.

A direct indication of backbone fluctuations are 14N1H

and (if deuterated water is used) 2H1H quadrupole dips

originating from relaxation sinks formed by quadrupole

nuclei in amide groups, that is 14N and, after deuteron

exchange, 2H. The additional quadrupole interactions

experienced by these nuclei produce a much tighter

coupling to the lattice and consequently cause enhanced

spin–lattice relaxation. Fig. 54a shows a typical example

for the occurrence of quadrupole dips arising at proton

frequencies where the low-field quadrupole resonances of14N or 2H cross the proton Larmor frequency (see Fig. 55).

For a theoretical description see Ref. [41].

The condition for the occurrence of quadrupole dips is

that molecular motions are restricted in the sense that

motional averaging is incomplete on the time scale of the

experiment. This applies to dry or hydrated, but rotationally

immobilized proteins and other compounds such as liquid

crystals, drugs and explosives containing quadrupole nuclei

dipolar coupled to protons. Quadrupole dips have also been

observed in tissue such as leech and frog muscle [42,120,

132] where macromolecular motions are restricted due to

microstructural constraints.

Another finding specific for backbone fluctuations of

proteins and polypeptides is that spin–lattice relaxation is

universally subject to a power law after elimination of

Fig. 54. Proton spin–lattice relaxation dispersion of a-chymotrypsin

hydrated with 16% D2O (a) and dry polyglycine (molecular weight

M ¼ 10; 000) (b) at 15 8C [42,119]. The quadrupole dips in (a) arise from

dipolar coupling of protein protons with 14N in the amide groups (three dips

at 680 kHz, 2.13 and 2.81 MHz; anisotropy parameter h ¼ 0:4) and from

dipolar interaction of non-exchangeable protons with amide hydrogen

exchanged by 2H (single dip at 148 kHz; anisotropy parameter h ¼ 0). The

resonance crossing mechanism is illustrated in Fig. 55. In the case of

polyglycine (b), the frequency ranges of the 14N quadrupole dips have been

omitted in order to demonstrate the power law spin–lattice relaxation

dispersion outside the dips.

Fig. 55. Origin of quadrupole dips in the case of 14N1H amide groups of

poly-L-alanine. (a) Magnetic field dependence of the 1H ðnHÞ and the three14N ðn

ð1ÞN ; n

ð2ÞN ; n

ð3ÞN Þ resonance frequencies in amide groups (anisotropy

parameter h ¼ 0:4). The hatched areas indicate the range covered in a

powdered sample [41]. The solid lines within these areas represent powder

averages. The magnetic flux density B0 is expressed in units of the proton

Larmor frequency. (b) Proton spin–lattice relaxation dispersion of poly-L-

alanine at 21 8C. The quadrupole dips arise at the resonance crossings of1H and 14N [42].


potential contributions from side-group motions [39,42,119,

168,169]

T1 / nb; ð106Þ

where the exponent b is typically in the range 0.65–0.85 at

room temperature and decays to lower values upon

temperature reduction [118]. Examples of this behaviour

are shown in Fig. 54a and b. Note that the polypeptide

polyglycine (Fig. 54b) does not contain any mobile side-

groups so that all relaxation mechanisms occur in the main

chain of the macromolecule.

The power law given in Eq. (106) for backbone

fluctuations can be explained on the basis of a multiple

trapping defect diffusion mechanism [119]. Another

interpretation is the assumption that spin–lattice relaxation

is mediated by (direct) spin–phonon coupling. Taking into

account the fractal nature of the backbone conformation,

Korb et al. were able to derive this power law in a

straightforward way [170,171].

9. Liquid crystals and lipid bilayers

Liquid crystalline mesophases are orientationally

ordered phases between the solid crystalline and the

isotropic amorphous liquid states. Typical molecules

capable of the formation of liquid crystalline phases in the

conventional sense contain mesogenic groups, i.e. rodlike,

more or less rigid sections such as linear aromatic arrays.

Liquid crystals of this type may be identified as ‘class I’

systems (apart from the distinction of ‘thermotropic’ and

‘lyotropic’ liquid crystals). Nematic and smectic examples

of this sort will be discussed in Sections 9.8 and 9.9.

However, there are two more classes of ordered fluid

systems that should be discussed in this context as well.

The order in the class II and III mesophases is not due to

mesogenic groups in the sterical sense mentioned above. In

the mesophases of class II, one is rather dealing with flexible

molecules capable of numerous rotationally isomeric and

energetically equivalent conformations. Under such con-

ditions, phase space may be covered more extensively in the

mesophase than in the isotropic melt. Orientational order is

then thermodynamically more stable. The theory of the class

II mesophases is however still in its infancy. Examples for

class II are the flexible mesomorphic polymers referred to in

Section 8.7. These polymers are not to be confused with

liquid crystalline side or main-chain polymers which

contain mesogenic groups and behave with respect to

ordered phases just as conventional liquid crystals.

The third class refers to amphipilic molecules in an

aqueous environment. Typical compounds are soaps and

lipids, and have often a lyotropic character. Amphiphilic

molecules in water tend to be packed in a way that

minimizes the contact of the hydrophobic parts of the

molecules with water, and maximizes the interaction of

water with the hydrophilic groups. The hydrophobic moiety

of the molecules often consist of fatty acid residues with

hydrocarbon chains 12–20 carbons long, whereas the

hydrophilic groups are of a polar nature. In Section 9.9,

we will discuss field-cycling NMR relaxometry of lipid

bilayers in liposomes as a typical example.

Liquid crystals are systems of special interest for several

reasons, ranging from important and increasing technologi-

cal applications to the impact they have for basic research

[172,173]. Structural features were considered from the very

beginning due to the observable changes in nematics under

the influence of external magnetic or electric fields and

surfaces [174–176]. More recently, increasing interest in

the physical properties of microconfined mesogens was

triggered by the discovery of polymer-dispersed liquid

crystals [177].

The dynamics and order of liquids and liquid crystalline

matter confined in porous systems is more complicated than

without confinement. On the other hand, important new

features arise that are not observable in the same substances

in bulk. In addition to the fundamental questions related to

molecular dynamics, molecular order and phase transitions,

the increasing interest of this topic may essentially be

associated with the strong impact on optoelectronics,

photonics and information display technology.

9.1. Motivation for field-cycling NMR relaxometry

experiments in liquid crystals

The study of molecular dynamics in liquid crystals was

originally promoted by the interest in collective dynamic

processes associated with orientational fluctuations of the

local molecular order [178,179]. NMR relaxation due to

ODFs as a manifestation of collective dynamics was first

worked out for the nematic phase in a pioneering paper by

Pincus [178]. He predicted a characteristic n1=2 Larmor

frequency dependence for the spin–lattice relaxation time

T1: Experimentally this was first verified in the kHz regime

by Wolfel et al. [180]. Since then, field-cycling relaxometry

has been applied to many different liquid crystalline

compounds. A good reason for the use of field-cycling in

studies of bulk liquid crystals lies in the strong relaxation

dispersion that characterises these materials, and the strong

differences in the dispersions corresponding to different

mesophases. A second good reason is the remarkably good

sensitivity to spatial confinements, even at temperatures

corresponding to the bulk isotropic state [156].

The examination of NMR relaxation properties using

rotating frame techniques is often considered as an

alternative way of studying slow motions without the need

of any complex instrumental accessories like field-cycling

relaxometers. However, a direct comparison between these

two experiments is only feasible in selected cases like

isotropic liquids [34]. In Section 9.14, we will show that the

information provided by the two techniques refers to

different dynamic processes [156].


Field-cycling proton T1 relaxometry has been used for

the study of molecular dynamics and order in different bulk

and confined liquid crystal compounds. Experiments were

mainly carried out on the isotropic, nematic (N) and smectic

A (SmA) phases in the bulk and in confined N and isotropic

states. The field-cycling method gives evidence for

significant changes in molecular dynamics induced by the

presence of confining surfaces, even if the surface-to-

volume ratio is not very high.

9.2. Relevant properties of bulk liquid crystals

In the following we briefly discuss relevant properties of

the nematic and smectic phases that are crucial for the

interpretation of relaxation dispersion experiments. We will

neither deal with the basic properties of these materials nor

with a detailed classification of the different mesophases.

For further details the reader is referred to one of the

specialised books about the matter [172,173,181–186].

9.2.1. The nematic phase

In contrast to the isotropic state, molecules in nematic

mesophases are orientationally correlated over macroscopic

distances with rotational invariance about the alignment

direction. The director field nðr; tÞ indicates the molecular

average orientation at point r at time t (Leslie–Ericksen

hydrodynamic theory) [188]. The free energy of the nematic

phase is modified by elastic terms (continuum theory or

Frank–Oseen theory) [187,189]. In the static limit, it can be

expressed in terms of the spatial derivatives of the director

field

Fe ¼1

2

ðdr3{K11ð7·nÞ2 þ K22½n·ð7 £ nÞ�2

þ K33½n £ ð7 £ nÞ�2 2 ðK22 þ K24Þ

� ½ð7·nÞ2 2 7inj7jni�}; ð107Þ

where the coefficients Kij are elastic constants (or elastic

modules). The first term on the right-hand side is associated

with the splay deformation, the second with twist (torsion)

and the third with bend (flexion). The last term does not

contribute to the bulk volume free energy, but can be

relevant at the boundaries [190]. Furthermore the free

energy of a nematic includes the standard terms typical for

isotropic liquids (velocity, temperature and pressure fields).

In the static limit, the hydrodynamic Leslie–Ericksen

theory converges to the elastic Frank–Oseen theory. Both

are useful for describing macroscopic phenomena but

present limitations at the microscopic level.

As a consequence of the complexity of the molecular

interactions taking place in a nematic, quantitative relations

between the macroscopic elastic and the molecular proper-

ties are only possible based on empirical arguments.

Relationships have been found between the elastic constants

and molecular properties such as the relative size and

flexibility of the alkyl chains [191–193], the ratio K33=K11

being sensitive to these properties. While the inequality

K33 . K11 is generally valid in nCB (40-n-alkyl-4-cyanobi-

phenyl) and K11 . K33 in alkenyl compounds, the ratio

tends toward the value of unity when the nematic–isotropic

transition is approached. This behaviour suggests that

elastic properties are severely affected by pre-transitional

fluctuations. Other properties at the molecular level that

strongly affect the elastic energy are the curvature of the

nematic domains and the general topology of the molecular

arrangements (calamitic, discotic, etc.).

Important pre-transitional effects can appear in the

nematic phase close to the transition from the nematic to

the smectic phase. In the case of cyanobiphenyls (40-n-alkyl-

4-cyanobiphenyl), the elastic constants of 5CB ðn ¼ 5Þ and

8CB ðn ¼ 8Þ have very different temperature dependences

within the nematic phase. In contrast to 5CB, 8CB also

forms a SmA phase. This fact is reflected within the nematic

phase of 8CB by a critical divergence of K22 and K33 when

approaching the N–SmA transition (which is absent in

5CB). The mere addition of three methyl groups to the alkyl

chain with an identical molecular core is obviously

sufficient to give rise to SmA order.

The magnetic anisotropy and the interaction of a uniaxial

nematic director with an external magnetic field can be

described by a symmetric second rank tensor of the

diamagnetic susceptibility according to

M ¼1

m0

xB; ð108Þ

where, if the z axis (magnetic field) is parallel to the director,

x ¼

x’ 0 0

0 x’ 0

0 0 xk

0BB@

1CCA: ð109Þ

The parallel (k) and normal ( ’ ) susceptibility tensor

principal components are negative. The anisotropy of the

diamagnetic susceptibility is defined by Dx ¼ xk 2 x’: If

Dx . 0; the director points along the magnetic field. Hence,

if the nematic is exposed to a magnetic field, a new

contribution to the free energy density must be considered

[172]:

fM ¼ 21

2

x’

m0

B2 21

2

Dx

m0

ðn·BÞ2: ð110Þ

The first term on the right-hand side is only of a magnetic

nature and is not influenced by the director orientation

relative to the magnetic field. It is therefore not of further

interest in this context. The second term represents the

torque exerted by the magnetic field on the director. The free

energy obviously adopts a minimum when the director and

the field are parallel. This term thus represents a field

alignment tendency which competes with thermal fluctu-

ations of the director.

There may be further contributions favouring or oppos-

ing the alignment of the director along the magnetic field.


These can be electric or acoustic fields, surface or flow

induced alignments.

Solid surfaces interacting with the liquid crystal

introduce a contribution to the free-energy that can be

expressed in terms of the order parameter tensor [195].

Randomly oriented solid surfaces consequently perturb the

nematic order by favouring preferential director orientations

relative to the surface. A surface ordered, interfacial layer of

thickness j is formed between the surface and the bulk

liquid crystal. This phenomenon is called ‘anchoring’. The

contribution to the free energy can be defined in terms of the

interfacial energy, that depends in turn on the degree of

adsorption of the molecules at the substrate [196,197]. In

competition between magnetic and surface torques, the

surface governs the director alignment within a layer

thickness given by [172]:

jB ¼1

B

ffiffiffiffiffik

Dx

s; ð111Þ

where k is a mean elastic constant (sometimes ascribed

to K22).

Flow properties of a nematic may be described in terms

of three viscosity coefficients h1;h2;h3 depending on the

orthogonal orientation of the director with respect to the

flow velocity v [194]:

h1 : nk7v ð112Þ

h2 : nkv

h3 : n ’ v; n ’ 7v

A viscous torque is applied to a director under rotation about

an axis perpendicular to n;

G ¼ 2g1

df

dt; ð113Þ

where f is the rotation azimuthal angle and g1 is the

rotational viscosity coefficient. Like the elastic constants,

the viscosity coefficients show critical pretransitional

behaviour when approaching the smectic A phase.

9.2.2. The smectic A phase

Smectic phases are characterized by layer structures.

They generally occur at lower temperatures than nematic

phases. The translational order and dynamical properties

within the layers may range from liquid-like to solid-like.

Examples of liquid-like layered structures are the smectic A

(SmA) and the smectic C (SmC) phases (see Fig. 56). The

SmA phase is usually described to consist of equidistant

layers of molecules with an equilibrium director perpen-

dicular to the layer planes.

The line integral over the director along a closed loop in

an incompressible layered structure free of dislocations is

known to comply to the conditionþn·dl ¼ 0: ð114Þ

This means that 7 £ n ¼ 0: Twist and bend elastic

deformations do not occur in a perfect smectic A crystal.

That is, the elastic constants K22 and K33 diverge [198].

Therefore, only the splay term of the nematic elastic free

energy survives. The elastic free energy density thus

becomes [199]:

fe ¼b

2s2

z þ1

2K11ð7·nÞ2: ð115Þ

The first (‘smectic’) term on the righthand side represents

the compression energy of the layers, where b is the

compression elastic constant and sz represents the layer

strain along the layer normal. The second (‘nematic’) term

is the splay energy.

The molecular orientational order parameter in the

nematic phase can be described by the temperature

dependent average distribution of assumed rigid, cylindri-

cally symmetric molecules around the director:

SðTÞ ¼ k1

2ð3cos2 u2 1Þl; ð116Þ

where u is the polar angle between the assumed symmetry

axis and the director. The translational order parameter for

molecules in a SmA phase is defined as the amplitude of a

one-dimensional density wave whose wave vector points

along the average director of the layer normals [172,198]:

rðrÞ ¼ r0 1 þ1

2lClcosðq0z þ wÞ

� : ð117Þ

In this equation lCl is a measure of the strength of the

smectic order, r0 is the average density, q0 ¼ 2p=d is the

wave vector of the density wave, d is the interlayer distance,

and w is an arbitrary phase angle. Note that the order

parameter is a complex number.

Close to the N–SmA transition the compression elastic

constant tends to be zero. As a consequence, critical

fluctuations of lCl become important. The interpretation is

a coupling between smectic order fluctuations and the

nematic director fluctuations. A corresponding ‘smectic–

nematic term’ should therefore be included in the free

energy. The smectic ðfSmÞ and nematic–smectic ðfSm–NÞ

terms can be written in the Landau–Ginzburg form as [172]

fSm ¼ aðTÞlCl2 þ bðTÞlCl4 þ · · ·; ð118Þ

Fig. 56. Schematic representation of nematic, smectic A and smectic C

molecular order in liquid crystals.


fSm–N ¼ ð7þ iq0dnÞCp 1

2Mð72 iq0dnÞC;

where a and b are the coefficients of the expansion, M is a

mass tensor with two principal components (along and

normal to the layers), and dn ¼ nðrÞ2 n0 with n0 the

unperturbed director. In the pretransitional regime, the SmA

free energy density is thus composed as

fSmA ¼ fe þ fSm þ fSm–N; ð119Þ

where fe now represents the nematic contribution with the

splay term. In the presence of a magnetic field, an additional

term fM applies (see Eq. (110)). Further terms due to the

presence of surfaces and flow may occur but are not

considered here.

9.3. Order director fluctuations in the nematic phase

ODFs are a process of a collective nature. They are

treated as a superposition of normal modes of the elastically

coupled entity of molecules in a liquid crystalline domain.

NMR relaxation due to ODF modes was first treated for the

nematic phase by Pincus [178]. The relaxation process is

considered to be due to intramolecular dipolar interactions

that are modulated by the hydrodynamaic order fluctuation

modes. In competition to molecular reorientations on these

grounds, displacements of molecules by self-diffusion to

positions of different director orientation were assumed

[179,200,201].

The last term of the nematic elastic free energy (see

Eq. (107)) does not contribute to the bulk volume free

energy. The free energy density including elastic and

magnetic torque contributions can thus be written as

fN ¼1

2{K11ð7·nÞ2 þ K22½n·ð7 £ nÞ�2

þ K33½n £ ð7 £ nÞ�2} 21

2

Dx

m0

ðn·BÞ2: ð120Þ

Expanding the director components in terms of the wave

vectors q of the fluctuations, and transforming the normal

components ðnx; nyÞ to a new system of two uncoupled

modes ðn1; n2Þ; the last equation may be written as [172,202]

fN ¼1

2V

Xq

X2

a¼1

KaðqÞlnaðqÞl2; ð121Þ

where

KaðqÞ ¼ Kaaq2’ þ K33q2

z þDx

m0

B2 ð122Þ

with q2’ ¼ q2

x þ q2y : Using the equipartition theorem we find

on this basis for the mean square amplitude of the modes

klnaðqÞl2l ¼

kBTV

KaðqÞ; ð123Þ

where kB is the Boltzman constant, T is the absolute

temperature, and V is the considered volume.

The local time correlation function for director fluctu-

ations can be expressed as

GODFðtÞ ¼3

2

1

V2

Xq;q0

kn1ðq; tÞnp1ðq; t þ tÞl

24

þkn2ðq; tÞnp2ðq; t þ tÞl

: ð124Þ

The transverse modes relax exponentially with a time

constant taðqÞ ¼ haðqÞ=KaðqÞ depending exclusively on the

viscoelastic properties of the media while the magnetic term

is usually neglected. The correlation function thus becomes

GODFðtÞ ¼3

2

1

V2

X2

a¼1

Xq

kln2aðq; 0Þl

2lexp 2t

taðqÞ

� : ð125Þ

The correlation function for intramolecular spin inter-

actions, i.e. for constant internuclear distances in the case

of dipolar couplings, is defined by (see Eq. (4))

GmðtÞ ¼ kY2;mð0ÞYp2;mðtÞl: ð126Þ

The second order spherical harmonics for m ¼ 0; 1 and 2 are

given by Eq. (3). The azimuthal and polar angles wðtÞ and

qðtÞ; respectively, describe the instantaneous orientation of

the coupling tensor relative to the magnetic field direction.

In the continuum limit valid for large volumes, the

normal mode sum in Eq. (125) can be replaced by an

integral. The spectral density then reads [203,204]

IODF1 ðvÞ ¼

3kBT

16p3

X2

a¼1

ðq

ha dq3

½KaðqÞ�2 þ h2

av2: ð127Þ

A promising approach for analyzing experimental field-

cycling data in terms of distributions of mode related terms

contributing to the spin – lattice relaxation rate was

suggested in Ref. [205].

If the anisotropic nature of the viscoelastic coefficients

has to be taken into account, the integration over the normal

modes can be performed over a cylindrical volume [206]. In

the case of isotropic coefficients, i.e. the single elastic

constant approximation, the integration can be calculated

over a spherical volume [178] or an ellipsoidal volume

[203]. The integration is limited on the one hand by an upper

cut-off wavenumber qzh ¼ p=d determined by the molecular

size d and a lower cut-off wavenumber qzl accounting for the

finiteness of the nematic order correlation length

[207–209]. In terms of the spectral density, Eq. (127),

these limits correspond to high and low frequencies,

respectively. The low-frequency cut-off is determined by

the domain (or sample) size and is usually much lower than

that corresponding to the lowest accessible frequency.

Between these two limits, Pincus’ law for nematic ODFs is

expected to be valid assuming isotropic viscoelastic


properties, [178]

IODF1 ðvÞ / v21=2

; that is T1ðvÞ / v1=2: ð128Þ

9.4. Order director fluctuations in the smectic A phase

The assumptions used for the derivation of Eq. (128) are

in contrast to the conditions to be expected in smectic A

phases where elastic constants tend to be strongly

anisotropic. The limit K33 p K11 ¼ K22 corresponds to a

pseudo two-dimensional system where bending defor-

mations are suppressed whereas splays and torsions occur

[203,206]. Under this assumption the spectral density

becomes

IODF1 ðvÞ / v21

; that is T1ðvÞ / v: ð129Þ

Another approach is the assumption that the splay elastic

term dominates the free energy. Hydrodynamic modes are

restricted to the ðx; yÞ plane of the smectic layers, that is

qz ¼ 0: Between the upper and lower cut-off limits one then

finds [206]

IODF1 ðvÞ ¼

S2kBT

4K11jz

v21; ð130Þ

where S is the molecular order parameter and jz is the

smectic layer orientation correlation length. It should again

be mentioned that spectral densities based on hydrodynamic

modes can be deduced from experimental field-cycling data

with the aid of a formalism presented in Ref. [205].

9.5. Fluctuations of spin interactions by translational

self-diffusion

Molecular self-diffusion is a mechanism that can give

rise to fluctuations of spin interactions in two quite different

ways. The first sort of influence corresponds to the RMTD

mechanism described in Section 7.3. Diffusion of molecules

along the normal modes of the crystal leads to reorientations

as a consequence of the spatial director fluctuations. Under

the assumption that the collective modes and self-diffusion

are independent processes, this competitive action of

temporal and spatial fluctuations was already taken into

account by Pincus in his pioneering paper [178]. Omitting

the anisotropy subscripts, the correlation time for the mode

with wavenumber q (see Eq. (125)) is given by

tq ¼1

K

hþ D

� �q2

; ð131Þ

where D is the self-diffuion coefficient of the molecules. The

spatial fluctuations are probed by self-diffusion, whereas the

temporal fluctuations directly affect the local molecular

orientation.

The RMTD process predominantly refers to intramole-

cular spin interactions. The second source of a potential

influence of self diffusion is associated with intermolecular

(i.e. dipolar) spin interactions. Distance fluctuations of the

internuclear vector affect the decay of the dipolar corre-

lation function given by Eq. (4). This was discussed by

Torrey [211] and later by Sholl [54]. In ordinary low-molar

mass liquids the influence of translational diffusion on the

spin–lattice relaxation dispersion is very weak because

intramolecular interactions normally dominate [212]. For

such systems the low-frequency T1 dispersion due to

diffusive modulations of intermolecular dipolar interactions

is of the type

T1 / ½a 2 bffiffiv

p�21

; ð132Þ

where a and b are constant coefficients. Adaptations of the

theory to anisotropic conditions such as in liquid crystals

have been reported in Refs [210,213–215].

It should be noted that intermolecular dipolar inter-

actions modulated by translational diffusion form a very

slow relaxation mechanism contributing mainly at the

lowest frequencies. It becomes perceptible as soon as the

system involves some strong motional anisotropy such as

expected for liquid crystals and polymers. In the latter

system this sort of relaxation mechanism was demonstrated

by comparison of deuteron (solely intramolecular inter-

actions) and proton (intra- and intermolecular couplings)

relaxation dispersions [144].

9.6. Rotational diffusion of individual molecules

Fluctuations of spin interactions due to rotational

diffusion of molecules are much faster than the collective

modes discussed so far. Therefore, collective modes show

up in spin–lattice relaxation dispersion only if rotational

diffusion is strongly anisotropic or even restricted. In this

respect there is a complete analogy to the anisotropic

component A with polymer chain dynamics (see Section 8)

and the anisotropic rotational diffusion expected for

adsorbed molecules (see Section 7).

Mesogenic molecules typically are elongated in shape

permitting fast rotational diffusion only about the long axis

when embedded in an ordered medium. Apart from

rotational diffusion of the whole molecule, there may be

contributions from internal degrees of freedom leading to

conformational fluctuations. Since all these fluctuations are

relatively fast, they merely show up at the highest

frequencies of NMR relaxometry and are often represented

by correlation functions consisting of a single exponential or

a sum of a few exponentials.

9.7. Combined action of collective and single-molecule

motions

In a liquid crystal all motions discussed so far occur

simultaneously so that the question arises what effective

relaxation rate results from the combined action. A fallacy


ubiquitous in the field-cycling literature about liquid

crystals is the belief that one can just add up the relaxation

rates of the individual processes calculated in the absence of

all other motions. However, rate sums of this sort are only

permitted if different motions refer to different spin

interactions. Examples are the combined action of intra-

and intermolecular couplings potentially being subject to

different fluctuations (see Eq. (7)) or the two-spin system

approximation of multi-spin systems (see Eq. (2)). If a given

spin interaction that dominates spin–lattice relaxation is

modulated by different types of molecular motions, the

effect of these motions must be combined on the level of the

correlation function of that spin interaction (see e.g. Eq. (4))

rather than on the level of relaxation rates.

Let us assume that molecular motions in liquid crystals

can essentially be analyzed into the two independent and

superimposed components described above, namely collec-

tive order director fluctuations (subscript ODF) and

individual rotational diffusion (subscript RD). According

to the relaxation theory of Pincus [178], ODF are treated in a

way not only implying hydrodynamic modes but also self

diffusion of molecules relative to these modes, that is, self

diffusion of the RMTD type. Both types of motion are

anisotropic, that is, they cover a restricted solid angle range

on the time scale of the experiment.

For protons the relaxation mechanism is predominantly

based on intramolecular dipolar interaction. Intermolecular

contributions tend to be negligible at short times [149].

Spin–lattice relaxation is then based on the time evolution

of intramolecular dipolar correlation functions for two

independent processes, GRDðtÞ and GODFðtÞ (compare with

Ref. [216]). These functions are adequately interpreted as

probabilities that the dipolar fluctuation has not yet taken

place in a time t: The total correlation function reads then

GðtÞ ¼ GRDðtÞGODFðtÞ: ð133Þ

The (anisotropic and restricted) RD correlation function can

be analyzed into

GRDðtÞ ¼ gRDðtÞ þ GRDð1Þ; ð134Þ

where the residual correlation of RD at long times is

represented by GRDð1Þ ¼ const: Since ODFs as a collective

phenomenon are much slower than individual rotational

diffusion, we can assume that GODFðtÞ < GODFð0Þ on the

time scale on which gRDðtÞ decays to zero. That is, the total

correlation function can be expressed in the ‘different-time-

scale limit’ as

GðtÞ < gRDðtÞGODFð0Þ þ Grotð1ÞGODFðtÞ: ð135Þ

Note that this analysis is analogous to the different time

scale treatments in Section 7.1 for surface related relaxation

and in Section 8.1 for polymer chain dynamics.

Taking the Fourier transforms of Eq. (135) in order to

obtain the spectral densities according to the BWR theory,

we thus are dealing with an effective relaxation rate

consisting of a sum of two terms:

1

T1

<aRD

TRD1

þaODF

TODF1

; ð136Þ

where aRD and aODF are constants representing the relative

contribution of each process. TRD1 and TODF

1 are the spin–

lattice relaxation times for the two processes in the absence

of the other motional component. Note that the different

time scales anticipated for Eq. (136) imply a stochastic

quasi-independence of the two processes even though they

rigorously speaking depend on each other [216]: rotational

diffusion steps occur many times before reorientations by

collective modes become effective for spin – lattice

relaxation.

9.8. Field-cycling NMR relaxometry in bulk nematic

liquid crystals

There are numerous field-cycling NMR relaxometry

studies of bulk nematic liquid crystals in the literature [180,

217–223]. The typical n1=2 spin–lattice relaxation dis-

persion law was first observed in a broad frequency range

for protons in PAA ( p-azoxyanisole) [180]. In a subsequent

work, the relaxation dispersion of PAA, MBBA

(4-methoxybenzylidene-40-n-butylaniline) and HAB (4-40-

bis-hexyloxyazoxy-benzene) was extended to extremely

low frequencies [217]. Note however, that in all those

papers unfortunately no detailed information is provided on

how the low fields were calibrated with the required

accuracy and time resolution (see the discussion in Section

5). The low-frequency dispersion plateaus found in these

studies remain therefore of unclarified origin although they

were discussed in terms of a low-frequency ODF cut-off or,

more realistically, by the influence of unaveraged local

fields. A discussion of factors potentially affecting the low-

frequency dispersion behaviour will follow in Section 10.

In Ref. [225], the spin–lattice relaxation dispersion was

studied as a function of the angle u between the mean order

director and the external magnetic field. The samples were

5CB (40-n-pentyl-4-cyanobiphenyl) and its homologue

8CB. The direction of the magnetic field was tilted during

the relaxation interval by the angle u assuming that the

rotational viscosity is high enough to prevent any percep-

tible orientational relaxation of the nematic domains on this

time scale.

This reorientation of the external field was achieved by

switching on a second, auxiliary field Bn perpendicular to

the main field of the magnet during the relaxation interval

(see Fig. 57). The magnetic field was varied adiabatically in

terms of Larmor precession (see Eq. (19)) in order to

prevent the generation of any transverse magnetization

components, but fast with respect to orientational domain

relaxation as well as to spin–lattice relaxation.

The fact that finite tilt angles affect the absolute values of

the spin–lattice relaxation times, but not their dispersion


slope corroborates the dominance of ODF modes as a

relaxation mechanism at low fields. For instance, the

modulation of spin interactions by ODF is more efficient

when the tilt angle is 908 compared to the case where the

mean director is aligned along the external field. This is the

direct consequence of the angle dependence of the spherical

harmonics given in Eq. (3).

The key message of the field-cycling NMR relaxometry

studies of nematic samples mentioned above is the

verification of the square root low-frequency dependence

given by Eq. (128), whereas the high-frequency behaviour

above about 1 MHz tends to be dominated by motions of

individual molecules. However, in several cases flatter low-

frequency T1 dispersions than predicted by Pincus’ law were

observed. The origin of these phenomena is not yet clear

[219,223–225]. It should also be kept in mind that

intermolecular dipolar interaction leads to additional

relaxation mechanisms at very low frequencies (below

about 100 kHz). This was demonstrated and interpreted with

polymers in Ref. [149] (see Fig. 47). A corresponding study

with liquid crystals has not yet been reported.

9.9. Field-cycling NMR relaxometry in bulk smectic

and lamellar systems

In contrast to the square-root law for the low-frequency

dispersion of bulk nematic liquid crystals, the linear

frequency dependence predicted for bulk smectic A phases

according to Eq. (129) so far has not been convincingly

verified. This is partly due to the fact that field-cycling

experiments were carried out without taking care of the

adiabatic condition for the cross-over from the high-field

limit (B0 dominates) to the low-field case (Bloc dominates).

The ‘artifacts’ arising from non-adiabatic conduct of the

field cycle are demonstrated in Fig. 58 [226] and will be

discussed in more detail in Section 10.

Lyotropic systems such as potassium laurate phases in

water and lipid bilayers can form ordered layer structures

(lamellar phases) and are therefore expected to show

collective modes similar to smectic liquid crystals. Field-

cycling data of lamellar potassium laurate systems were

analyzed in Ref. [205].

Lipid bilayers formed of di-palmitoyl or di-myristoyl

lecithin were studied in Refs [110–112]. Fig. 59 shows

proton T1 dispersion data in the diverse phases of

dipalmitoyl lecithin (DPL) bilayers in liposomes prepared

in D2O [110]. The La phase at 45 and 60 8C is liquid

crystalline. That is, fast motions of individual molecules

as well as slow collective modes occur. The gel phases

Pb0 and Lb0 below 41 8C are still ordered but collective

modes are more or less suppressed, whereas motions of

individual molecules are retained practically in the same

way as in the liquid crystalline phase [111].

Motions of individual molecules cannot be explained

by mere rotational diffusion in this case, because lipid

molecules mainly consist of (pairs of) flexible hydro-

carbon chains unlike the rigid rod character of typical

mesogenic groups found in class I liquid crystals. The

motions in individual molecules can however be

described by one-dimensional diffusion of structural

defects along the hydrocarbon chains between two

reflecting barriers formed by the polar headgroup layers

of the DPL molecules [229]. A ‘defect’ can be a rotamer

such as a kink or a torsion that is able to diffuse back

and forth on the hydrocarbon chains. A schematic

representation of the situation is shown in Fig. 60.

Assuming a defect diffusion coefficient D; a defect width

b and a distance d between the polar headgroups, we are

Fig. 58. Demonstration of apparent spin–lattice relaxation dispersions

arising from non-adiabatic field cycles. The slew rates and the polarization

flux densities are varied as indicated [226]. The data are for 11 CB in the

smectic A phase at 328 K.

Fig. 57. Variation of the tilt angle, u; between the external magnetic field

and the mean domain director. The magnetic field is varied (both with

respect to magnitude and orientation) during the relaxation interval. The

tilting rate is adiabatical with respect to Larmor precession and fast relative

to both orientational relaxation of the domains and spin–lattice relaxation.


dealing with two time constants, namely

tb ¼b2

2Dand td ¼

d2

2D: ð137Þ

These are the mean diffusion times a defect needs to travel a

distance b and d; respectively. The spectral density arising

from the defect diffusion process can be represented by

the following limits [110,111,227,228]:

IðvÞ ¼

ð2=3Þðt1=2b t1=2

d 2 tbÞ for vtd p 1

tdt1=2b 2 2tbt

1=2d

ðt1=2d 2 t1=2

b Þ21

v1=2for vtb p 1 p vtd

t1=2d

t1=2d t1=2

b 2 tb

1

v3=2for vtb q 1:

8>>>>>>><>>>>>>>:

ð138Þ

Note the square root frequency dependence predicted for the

intermediate frequency range. The complete (but complex)

expression can be found in Ref. [227]. A numerical analysis

is discussed in Ref. [229]. For DPL, the distance between

the polar headgroups is d < 3:8 nm: Assuming a gt�g

rotational isomer (i.e. a kink) as diffusing defect, we have

a defect width of b < 0:25 nm: The ratio between the two

characteristic time constants consequently is td=tb < 230:

An Arrhenius law is assumed in order to accunt for the

temperature dependence:

tb ¼td

230¼ t1exp{E=RT}: ð139Þ

R is the general gas constant. The apparent activation

energy, E ¼ 27:8 kJ=mol; is taken from the known value for

polyethylene. The pre-exponential factor is taken to be

t1 ¼ 1:7 £ 10214 s in the gel phases and t1 ¼ 5:9 £ 10215

s in the liquid crystalline phase. With these parameters, the

experimental spin–lattice relaxation dispersion shown in

Fig. 59 and the temperature dependence of the spin–lattice

relaxation time shown in Fig. 61 can commonly be

described as far as concerns the motions in the fatty acid

residues of individual molecules.

This analysis nicely demonstrates the interplay of

individual and collective motions: There is little influence

Fig. 60. Schematic representation of the limited defect diffusion model used

for the description of motions in the hydrocarbon part of individual

molecules in lipid bilayers. xd is the instantaneous position of a structural

defect (kink, torsion). r is the position of the reference nucleus, and b is the

width of the diffusing structural ‘defect’ [227–229].

Fig. 61. Temperature dependence of the proton spin–lattice relaxation time

in a dispersion of 60% DPL in D2O at 40 MHz. The experimental data are

from Ref. [230]. Tt indicates the gel-to-liquid-crystalline phase transition.

The curve was calculated for the limited defect diffusion model described in

the text (compare Fig. 60). An Arrhenius law was assumed for the

temperature dependence of the characteristic time constants with a slight

change of the preexponential factor at the gel-to-liquid-crystalline phase

transition. Note that the same parameter set was used as for the description

of the frequency dependences in Fig. 59 [110].

Fig. 59. Frequency dependence of the proton spin–lattice relaxation time in

a dispersion of 40% dipalmitoyl lecithin (DPL) in D2O [110]. The curves

represent motions in the hydrocarbon chain part of individual molecules

and were calculated for the limited defect diffusion model described in the

text (compare Fig. 60). An Arrhenius law was assumed for the temperature

dependence of the characteristic time constants with a slight change of the

pre-exponential factor at the gel-to-liquid-crystalline phase transition. Note

that a parameter set common to all curves was used which also accounts for

the temperature dependence given in Fig. 61.


of the gel-to-liquid-crystalline phase transition on motions

of individual molecules mainly due to some minor

increase of the free volume. Collective modes, on the

other hand, clearly show up in the liquid crystalline phase at

low frequencies, but are strongly suppressed in the gel phase

(see Figs. 59 and 61). Combining relaxation data of

undeuterated and chain deuterated DPL liposomes even

permitted the elucidation of processes specifically occurring

in the fatty acid residue polar headgroup parts (see Fig. 62).

Molecular motions within the hydration water phase

have already been discussed in Section 7.5. It should be

noted that lamellar systems of this sort can again be subject

to experimental low-field artifacts due to non-adiabatic field

cycles (see Section 10).

9.10. Field-cycling dipolar order relaxometry

Spin–lattice relaxation due to dipolar interaction is

normally treated under the assumption that the system can

be represented by an ensemble of uncorrelated two-spin

systems 1/2 [5,6] (compare with Eq. (2)). The adequacy of

this assumption can be checked by comparison with

deuteron, i.e. quadrupole relaxation which is dominated

by single-spin interactions. In systems largely subject to

motional averaging, it was shown that the deuteron and

proton relaxation dispersions are equivalent [36,98].

However, if motional averaging is incomplete so that

secular local fields arise, the situation can be different. This

was demonstrated by relating the frequency dependence of

the (Zeeman order) spin–lattice relaxation time T1 with that

of the dipolar order spin–lattice relaxation time T1d

[53,231]. Based on the standard two-spin 1/2 relaxation

theories one expects a ratio

T1

T1d

# 3; ð140Þ

which is much less than found experimentally in liquid

crystals (see Fig. 63). The pulse sequences used for

recording these data are a combination of the Jeener/

Broekaert sequence [232] for the generation of dipolar order

and a field cycle as shown in Fig. 64.

Fig. 63. Ratio of the proton Zeeman order and dipolar order spin–lattice

relaxation times, T1 and T1d ; respectively, as a function of the frequency for

the nematic liquid crystals 4-octyl-40-cyanobiphenyl (8CB) and 4-40-bis-

heptyloxyazoxy-benzene (HpAB) [53,231]. The ratio is much larger than

expected from the standard two-spin 1/2 relaxation theories. The

conclusion is that these theories are not adequate for multispin systems in

the incomplete motional averaging limit, i.e. when finite secular dipolar

local fields exist.

Fig. 62. Frequency dependence of the proton spin–lattice relaxation time in

a dispersion of undeuterated (a) and chain perdeuterated (b) DPPC

( ¼ DPL) in D2O [111]. The curves indicate the average dispersions from

which the selective behaviour of the hydrocarbon chains (c) was derived

according to the fast cross-relaxation relationship 1/T1(hydrocarbon) ¼ 1/p

[1/T1(undeuterated) 2 (1 2 p)/T1(chain deuterated)], where p is the frac-

tion of hydrogen atoms in the hydrocarbon chains.


9.11. Secular dipolar interactions with quadrupole nuclei:

‘quadrupole dips’

Many mesogenic compounds contain quadrupole nuclei

such as 14N. Incomplete motional averaging on the time

scale of NMR is intrinsic to liquid crystalline phases. That

is, secular dipolar interactions between spins in general and

between protons and quadrupole nuclei in particular are

important. Since quadrupole nuclei are more strongly

coupled to the lattice, they relax faster and remain

essentially always in thermal equilibrium. When the

resonances of the two nuclear species cross each other as

a function of the external magnetic field, fast exchange of

spin energy occurs. Quadrupole nuclei therefore act as

efficient relaxation sinks for protons dipolar coupled to them

(see Fig. 55). The consequence is the appearance of

quadrupole dips as already explained in Section 8.12 for

the equivalent situation of slowly tumbling or immobilized

proteins (see Fig. 54) [42].

There are numerous studies of 14N1H quadrupole dips in

the spin–lattice relaxation dispersion of liquid crystalline

phases [233–242]. Since quadrupole dips occur when the

low-field nuclear quadrupole resonance (NQR) frequencies

coincide with the NMR frequency of protons, relatively

precise information on temperature and structure dependent

NQR frequencies can be obtained. Quadrupole dips can also

be taken as a measure of liquid crystalline order on the NMR

time scale which is a prerequisite for strong secular dipolar

interaction.

9.12. The effect of ultrasound on the spin–lattice relaxation

dispersion of liquid crystals

Coupling of ultrasound waves to hydrodynamic modes in

liquid crystals, that is the competition between ordering and

disordering tendencies, is a tantalizing problem considered

many times in the literature [243–246]. Field-cycling NMR

relaxometry was shown to probe ODF modes over a wide

frequency range. It should therefore be suitable for the

detection of this sort of coupling.

Corresponding experiments [226,247,248] were carried

out with the aid of a glass ultrasound sonotrode directly

immersed in liquid crystalline samples in a field-cycling

relaxometer (see Fig. 65). Any galvanic contact between the

electro-acoustic device and the field-cycling instrument was

avoided in this way. Dissipation of sound energy tends to

increase the sample temperature, of course. In order to avoid

experimental artifacts on these grounds, the temperature

was measured inside the sample during sonication, and the

temperature control was ‘misadjusted’ correspondingly to

compensate for heating by the sound.

Fig. 66 shows spin–lattice relaxation dispersion curves

in the nematic phases of PAA, 5CB and 8CB with and

without sonication [248]. The sonication powers were

13.5 W/cm2 and 22.5 W/cm2. The sound frequency was

30 kHz. The main sonication effect in cyanobiphenyls is a

change of the dispersion slope. In PAA, the relaxation rate is

also strongly enhanced by the sound.

In Ref. [246] an increase of ODF fluctuations by

ultrasound was suggested. In Refs [247,248] the expression

for the free energy given by Eq. (121) was therefore

supplemented by one more term accounting for the coupling

between sound waves to the hydrodynamic modes. On this

basis the enhanced relaxation rates as well as the flatter T1

dispersion could be well described.

An interesting finding is, that no specific effect occurs

when the proton resonance frequency crosses the sound

frequency of 30 kHz. The interpretation is that the sound

energy is immediately dissipated among the whole spectrum

of collective modes. It appears that there is no way to

enhance a certain mode by coupling it to a sound wave of

corresponding frequency. A discussion of the theory behind

can be found in Ref. [204].

9.13. Surface ordering in porous media

Bulk liquid crystals can be ordered by external fields.

The situation changes in the presence of solid surfaces such

as occurs in the pore space of porous media or in the

presence of embedded constituents like Aerosil particles or

polymer networks forming large sample internal surfaces.

The (anisotropic) interaction of the liquid crystal molecules

with the surfaces induces local order relative to the surface,

that is, in competition with the ordering tendency of external

fields. Liquid crystal molecules confined in pore spaces can

thus be considered to be subdivided into two phases, namely

a bulk-like phase affected merely by the external field, and a

surface-ordered phase close to the surfaces. The surface

ordered phase has a finite thickness determined by the

relative strengths of the two ordering mechanisms. Com-

pared to single-phase, bulk systems, two-phase systems in

porous media are subject to further dynamic processes such

as molecular exchange between the two phases (see Section

7.1) and RMTD processes (see Section 7.3).

Studies have been performed in nematic droplets

dispersed in an epoxi polymer matrix [249]. The droplet

Fig. 64. Combination of the Jeener/Broekaert RF pulse sequence with a

field cycle. The first two RF pulses generate dipolar order which relaxes in

the relaxation interval tM: Signals are acquired in the detection field, Bd;

with the aid of a 458 RF pulse converting dipolar order to detectable spin

coherences.


size ranged from 0.1 to 10 mm. The surface induced order

and its effects on the molecular dynamics were studied by T1

relaxometry at frequencies between 8 and 270 MHz

supplemented by T1r experiments in the kHz regime. It

was found that NMR relaxation in the MHz region is mainly

determined by cross-relaxation between protons in the

liquid crystal and in the solid matrix at the liquid crystal–

polymer interface. In the kHz region, both cross relaxation

and relaxation by RMTD processes in a generalized sense

were found to be the dominant mechanisms. The latter

mechanism was attributed not only to molecules directly at

the surface but also for more distant particles diffusing in

orientationally distorted liquid crystal domains in the bulk-

like phase.

Wavelengths of director fluctuations within a droplet

cannot be larger than the pore dimensions. Therefore, a cut-

off frequency at about 40 kHz occurs and must be taken into

account in the description of low-frequency relaxation data

[250]. In Ref. [251], the generalized RMTD process for

different pore geometries was treated theoretically. The first

field-cycling experiment corroborating surface induced

order and the cross-relaxation mechanism is reported in

Ref. [252]. The relaxation dispersion was analysed at

different temperatures for two different droplet sizes.

The confinement of 5CB or 8CB in nanoporous silica

glasses leads to surface ordering at inner surfaces even

above the bulk nematic–isotropic transition temperature.

This strongly affects the T1 dispersion [253–255]. For

example, the values of T1 for protons in isotropic 8CB in

bulk do not show any perceptible frequency dependence

in the kHz and MHz regime. However, when confined in

Bioran porous glass with a mean pore dimension of 200 nm

the T1 dispersion becomes steep and approaches the square

root law given in Eq. (128). Data for another example,

Fig. 65. Schematic cross-section through the experimental set-up used for the field-cycling/sonication experiments described in Refs. [247,248].


namely 8CB in an Aerosil network are shown in Fig. 67.

Note that the Aerosil content in this case was only 3%. The

T1 dispersion is nevertheless changed dramatically relative

to the bulk isotropic liquid.

Field-cycling studies of the wavenumber cut-off due to the

finite size of the pores tend to be concealed by the strong

influence of local fields at frequencies below 100 kHz. In this

respect, techniques directly probing local fields arising from

residual dipolar couplings are superior and can be considered

as supplementing field-cycling data most favourably with

respect to the total dynamic range (see Fig. 1). The so-called

dipolar correlation effect on the stimulated echo turned out to

be particularly useful for the detection of the pore size

dependence of the cut-off wave length [256].

9.14. Rotating-frame NMR relaxometry in liquid crystals

Another experiment supplementing the information on

the dynamics in liquid crystals obtained from field-cycling

NMR relaxometry ðT1Þ is rotating-frame NMR relaxometry

ðT1rÞ: Actually, the frequency range covered by rotating-

frame experiments coincides with the low-frequency section

of the field-cycling ‘window’ (see Fig. 1) so that no

extension can be expected in this sense. However, the

rotating-frame spin–lattice relaxation rate depends on a

different set of spectral densities particularly sensitive to

small-angle fluctuations which are characteristic of ODFs.

The rotating-frame analog to the laboratory expression

given by Eq. (6) is [25–27,156]

1

T1r

¼Ccoupl

2½3I0ð2v1Þ þ 5I1ðv0Þ þ 2I2ð2v0Þ�; ð141Þ

where the rotating-frame angular frequency v1 ¼ gB1 is

determined by the RF amplitude B1; and v0 ¼ gB0 is the

ordinary Larmor frequency (compare with the BPP

expressions given in Eq. (12)). Eq. (141) is again valid in

the frame of the two-spin 1/2 approximation in the weak-

collision limit [6]. Since rotating-frame experiments are

usually carried out in a fixed main magnetic field B0;

Eq. (141) can be simplified as

1

T1rðv1Þ¼

3Ccoupl

2½I0ð2v1Þ þ a�; ð142Þ

where a is a constant depending on B0: The spectral density

determining the frequency dependence is now I0 instead of

I1 and I2 in the case of the laboratory frame expression given

by Eq. (6).

Normally this difference is of minor importance as was

demonstrated in Section 8 with various polymer systems.

However, mesogenic molecules have more or less rigid

rodlike cores forming ordered domains with directors

preferentially aligned along the external magnetic field.

Ordered collectives of molecules are subject to thermally

activated hydrodynamic modes. The alignment of the

director along the external magnetic field and the collective

nature of such ODFs suggest that both the polar angle

between the magnetic field and typical internuclear vectors

on phenyl rings of the mesogenic groups and the polar-angle

fluctuations are small on the time scale of field-cycling

experiments. This is in contrast to isotropic phases where no

restriction of the polar angle exists. In liquid crystalline

phases of mesogenic molecules, the spherical harmonics

given by Eq. (3) are therefore relevant in the low-polar-

angle limit only which reads in first-order approximation

Y2;0ðtÞ .1

2

ffiffiffiffi5

p

s; ð143Þ

Fig. 66. Proton spin–lattice relaxation dispersion of para-azoxyanisole

(PAA), 4-pentyl-40-cyanobiphenyl (5CB) and 4-octyl-40-cyanobiphenyl

(8CB) in the nematic phase with and without sonication [248]. The

sonication powers were 13.5 (filled circles) and 22.5 W/cm2 (filled stars).

The sound frequency was 30 kHz. The data represented by open circles

were recorded without sonication.

Fig. 67. Comparison of the proton spin–lattice relaxation dispersions of

8CB in bulk and confined in a network of Aerosil particles [156]. The

Aerosil content was 3%. The temperature was 323 K, i.e. above the

nematic–isotropic transition temperature for an unconfined sample (‘8CB

bulk’). The frequency dependence appearing in the confined system is due

to surface ordering.


Y2;1ðtÞ . 2

ffiffiffiffiffi15

8p

sqðtÞexp{iwðtÞ};

Y2;2ðtÞ .


32p

sq2ðtÞexp{2iwðtÞ}:

The average polar angle between the director and the main

magnetic field vanishes in bulk liquid crystals, that is

kqðtÞlt ¼ 0:

The spectral densities are obtained from the correlation

functions of the spherical harmonics by Fourier transform

according to Eq. (5). The expressions given in Eq. (143)

suggest that the spectral densities I1ðvÞ and I2ðvÞ contribute

to the spin–lattice relaxation dispersion by ODF processes

whereas I0 tends to vanish for finite frequencies. The

rotating-frame spin–lattice relaxation is therefore insensi-

tive to the ODF relaxation mechanism and merely reflects

local molecular reorientations of a non-collective nature, so

that no frequency dependence arises at frequencies typical

for rotating-frame experiments (see Fig. 68). On the other

hand, this fact nicely demonstrates and corroborates that

field-cycling relaxometry indeed probes the ODF spectrum.

The above argument refers to bulk samples where the

mean director is aligned along the main magnetic field so

that kqðtÞlt ¼ 0: The situation may be different in surface

ordered systems discussed in the previous section. Since the

director is then determined by the local surface orientation,

any angle with the main magnetic field can occur, and the

low-angle approximation given by Eq. (143) no longer

applies. The consequence is that the spectral density of

zeroth order, I0ðvÞ; becomes a function of the frequency due

to ODFs. In this case, T1 as well as T1r are suitable to probe

ODF modes more or less sensitively [156].

In other words, any mechanism lifting the averaging of

the polar angle between the magnetic field and the director,

so that kqðtÞlt – 0; will cause some T1r frequency

dependence due to ODFs. Surface ordering is not the only

reason for such an effect. If the local field starts to dominate

at very low frequencies, an equivalent situation arises. This

is demonstrated again in Fig. 68, where some T1r frequency

dispersion arises below about 15 kHz just when local fields

are known to become relevant.

10. A word of caution concerning NMR relaxometry

in the kHz regime

Above about 10 kHz proton resonance, field-cycling

measurements of spin–lattice relaxation times are relatively

reliable compared to the conventional inversion/recovery,

saturation/recovery and progressive saturation RF pulse

techniques [6]. In principle there is little danger of

experimental artifacts. The reason is that the magnetic

field in the sample normally is much more homogeneous

than the amplitude of RF pulses. The distortion of the

thermal equilibrium of the spin level populations, that is the

primary step of any spin–lattice relaxation experiment, is

therefore also very homogeneous. Detection of the magne-

tization at the end of the relaxation interval, on the other

hand, is not critical in this respect because one always gets a

signal proportional to the magnetization provided that the

RF bandwidth is sufficient to cover all signal components

and the field cycle is not subject to any time dependent

drifts. Also, as outlined in Section 3, the finite switching

times may lead to signal losses, but do not affect the

relaxation time measurement directly in samples with

monoexponential relaxation curves.

The main source of experimental errors refers to

frequencies around or below 10 kHz proton resonance. In

this range, there are three frequent reasons for experimental

low-field artifacts or misinterpretations: (i) unsettled

relaxation field levels, (ii) local fields, and (iii) violation

of the basic prerequisites of the relaxation theory

considered.

(i) Unsettled relaxation field levels: Spin–lattice relax-

ation times below about 10 kHz can become extremely short

and are often less than 1 ms especially for deuteron

resonance (see for example Fig. 52). In such short periods,

the relaxation field may not have come to a stationary value

after an extremely steep decay from the polarization field

(see Fig. 2). As outlined in Section 5, it is therefore

mandatory to measure the relaxation field in the whole

variation range of an experiment. That is, magnetic flux

densities just after the polarization interval must be

determined in less than a millisecond with an accuracy of

10% or better.

(ii) Local fields: Spurious fields ‘seen’ by the spins in

addition to the nominal field adjusted in the relaxometer can

be of an external and/or a sample internal origin. External

Fig. 68. Laboratory frame proton spin–lattice relaxation times ðT1Þ of 8CB

in isotropic and nematic phases as a function of the frequency in

comparison to rotating-frame spin–lattice relaxation times (T1r; see

inset) in the nematic phase of the same sample [156]. The different

dispersion slopes are due to the different spectral densities effective in the

low-polar-angle limit of the director. The T1r data below 15 kHz are already

influenced by local fields so that the low-angle limit does no longer apply.


fields such as the earth field and stationary stray fields from

any other magnetic equipment in the lab can readily be

compensated for by correction coils. The consequences of

fields by residual spin interactions incompletely averaged

by molecular motions are much more serious. Adiabatically

conducted field cycles lead to dipolar order in the kHz

regime (or quadrupolar order in the case of quadrupole

nuclei), so that the relaxation times measured are of a

different nature [111]. If the field cycle violates the

adiabicity condition given by Eq. (19), so-called ‘zero-

field coherences’ are generated in the local field. It is then no

longer spin–lattice relaxation that determines the evolution

of the magnetization. The local magnetization rather adopts

a transverse character in this case and decays by transverse

relaxation mechanisms [5,6]. Examples of the dramatic

‘artifacts’ that can arise on these grounds are shown in

Fig. 58. It appears that they often occur in lamellar-like

systems (smectic liquid crystals [226], lipid bilayers [66,

108], surface ordered nematic liquid crystals [253]). The

remedy used to prevent pitfalls of this sort is to slow down

the field switching rates (see Fig. 58). In unfavourable cases,

the switching times needed to preserve the adiabaticity

could however be of the order or even longer than the

spin–lattice relaxation times at low relaxation fields. Such a

situation is a principal limitation for the application of the

field-cycling technique indeed. On the other hand,

local fields can be the basis of very powerful tools for

non-field-cycling studies of slow molecular motions [257].

Quasi stationary ‘local fields’ arise due to incomplete

motional averaging of spin couplings. Motional averaging

can however be artificially induced by sonication of the

sample during the experiment [258]. This has been

demonstrated for smectic liquid crystals where the low-

field spin– lattice relaxation dispersion dramatically

changes with the power of the sound irradiaded to the

sample [226] (see Fig. 69). The influence of local fields

can strongly be reduced in this way. Provided that the

smectic order is not destroyed by the sound, so that the

characteristic hydrodynamic modes still exist, the ‘true’

low-field spin– lattice relaxation dispersion can be

measured with this sort of experiment. On the other

hand, sonication does not affect the isotropic phase of the

same liquid crystal species where averaging by Brownian

motion is sufficiently efficient.

(iii) Relaxation theory is not applicable: Strongly

anisotropic dynamics as occurs for polymers for instance

is connected with an extremely wide and continuous

spread of the decay time constants of the correlation

function. At the same time, spin–lattice relaxation times

of such systems tend to become very short at low

frequencies. The situation may arise that the longest time

constants of the correlation decay get longer than the low-

field spin–lattice relaxation times. The consequence is

that the standard BWR theory is no longer adequate to

describe the effective low-field relaxation behaviour

[37,111]. An example is shown in Fig. 52.

It is not difficult to identify low-field phenomena that are

not due to true spin–lattice relaxation in terms of molecular

motions. In cases of doubt, comparative T1r and (high-field)

T2 measurements are helpful to identify the effects

mentioned before. The potential generation and influence

of zero-field coherences can be checked by variation of the

slew rate as far as possible: true spin–lattice relaxation is

not affected by the slew rate, whereas the generation of zero-

field coherences is.

11. Outlook

When one of the authors wrote the first review article on

field-cycling NMR relaxometry 24 years ago [1], it was

relatively easy to mention most of the work that had been

done so far in the field. Nowadays this appears to be a task

impossible to cope with. At that time a quarter of a century

ago there were only a handful of home built relaxometers in

use in a few laboratories around the world with correspond-

ingly moderate scientific output. The situation did not

change very much when Seymour Koenig and Rodney

Brown at the IBM research center at Yorktown Heights

started to market their (and A. Redfield’s original) design in

the eighties and early nineties. Only after Gianni Ferrante at

STELAR began to exploit the technique commercially

almost ten years ago, did the number of applications soar

dramatically.

Fig. 69. Proton spin–lattice relaxation dispersion in the smectic phases of

8CB and 11CB. The apparent low-field dispersions are flattened or even

vanish upon sonication with 30 kHz sound at various sound powers. The

interpretation is that these low-field dispersions are local-field effects more

or less averaged out by sonication.


The present review describes the basic principles, the

theory and the pitfalls of the field-cycling NMR relaxometry

technique. We have restricted ourselves to applications

involving selected diamagnetic material classes where

mature theoretical descriptions or numerical simulations

are available. That is ‘adsorbate dynamics near surfaces’,

‘polymer chain dynamics’ and ‘fluctuations in ordered

systems like liquid crystals’. We have totally omitted a field

that is most important from the point of view of practical

applications, that is relaxation by interactions with para-

magnetic complexes, free radicals or ions [38,44,260–264].

The main value of field-cycling relaxometry studies is

that a straightforward connection to molecular dynamics

theories exists. Theoretical predictions can be checked

decisively in a way that other methods scarcely permit. The

combination of analytical model formalisms, numerical

simulations and field-cycling experiments possibly sup-

plemented by diffusometry and residual spin interaction

data forms an unsurpassed, most powerful research tool.

There is growing interest in applying the field-cycling

relaxometry technique to more and more complex systems.

The problem is then that often no suitable model theories or

simulations are available. The interpretation of field-cycling

data in a purely empirical way without having sound model

theories at hand is however problematic. Unlike NMR

spectroscopy, there is little stand-alone information that can

directly be evaluated from relaxation dispersion data

(leaving NQR or zero-field spectroscopy applications

aside). Further progress will be achieved in the future to

the extent that model theories are developed in parallel.

The problem is partly due to the relatively poor

selectivity in the experimental investigations. Spin–lattice

relaxation in complex systems tends to reveal information

averaged by spin diffusion and cross-relaxation over all

phases or components included. Attempts to identify

assignable exponentials in multi-exponential relaxation

curves of slowly exchanging multi-phase or multi-com-

ponent systems are scarcely crowned with success or may be

subject to large uncertainties.

There are two ways to achieve a better degree of

selectivity in complex systems. Firstly one can label certain

groups isotopically. For sensitivity reasons, this mainly

refers to the replacement of protons by deuterons, while

other typical labelling nuclides such as 13C or 15N are not

suitable for low-field NMR studies. Deuteron field-cycling

NMR relaxometry is possible even in very selectively

deuterated systems of high viscosity provided that the

detection field is larger than 1 T. Future developments

leading to much stronger field-cycling magnets are expected

to facilitate such experimental protocols (see Section 6.3).

Raising the detection field strength considerably higher than

nowadays is possible should be the primary goal of

hardware developments for promoting applications of the

field-cycling NMR relaxometry technique.

Selective examination of phases or components so to

speak the other way round, that is, by proton NMR with

the undesired phases or components being perdeuterated

and hence off-resonance can also be employed. However,

some caution is recommended since deuteration is never

100%. The signal of the residual protons in the perdeuter-

ated phase or component may be of significant strength and

must be identified by a gradual isotopic dilution series.

The other strategy used to improve selectivity, namely

the shuttle principle applied to high-resolution/high-field

magnets, simultaneously enhances the sensitivity at the

expense of the relaxation time resolution and the accessa-

bility of very low fields. At first sight, the combination of

field-cycling relaxometry with high-resolution spectroscopy

appears to be a paradox because there is little motional

narrowing of spectral lines in systems of interest for field-

cycling studies. However, the very high fields available in

ordinary NMR spectrometers partly overcomes this spectral

resolution problem as is demonstrated by the most

successful spectroscopy application to biological macro-

molecules. Even more important is the advantage of a much

better detection sensitivity: this should even permit studies

of 13C or 15N labeled systems, i.e. nuclides promising the

utmost selectivity if the chemical or biological sample

preparation can be managed. Another isotope of interest in

biological systems which is difficult to study is 17O [259]. In

this case, the problem is the low degree of enrichment

normally available and the short relaxation times due to the

finite quadrupole moment.

The combination of field-cycling with dynamic polariz-

ation techniques of nuclei in the presence of free radicals or

of NOE enhancement of X-nuclei appears also to be a

technology with bright prospects to be advanced in the near

future. Other signal enhancement strategies such as laser

polarization of noble gases or the use of para-hydrogen can

be considered, but are expected to be not ‘robust’ enough for

exploitation in a field-cycling apparatus. Potentially there

are also promising ENDOR experiments combined with

field-cycling [265].

Acknowledgements

Grants by the Deutsche Forschungsgemeinschaft, the

Alexander von Humboldt foundation, DAAD, Foncyt-

ANPCyT and CONICET are gratefully acknowledged.

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