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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.
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
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320260
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
ffiffiffiffiffiffiffi15
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
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 261
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
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320262
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 263
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320264
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 265
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,
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320266
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 267
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320268
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:
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 269
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
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320270
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].
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 271
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320272
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 273
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320274
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 275
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320276
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).
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 277
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).
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320278
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 279
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).
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320280
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
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 281
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320282
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 283
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).
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320284
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).
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 285
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320286
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].
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 287
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320288
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
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 289
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].
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320290
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].
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 291
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320292
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 293
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320294
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 295
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320296
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].
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 297
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
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320298
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 299
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320300
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 301
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].
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320302
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].
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 303
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320304
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 305
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
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320306
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
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 307
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
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320308
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 309
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320310
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 311
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320312
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].
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 313
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320314
Y2;1ðtÞ . 2
ffiffiffiffiffi15
8p
sqðtÞexp{iwðtÞ};
Y2;2ðtÞ .
ffiffiffiffiffiffiffi15
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320 315
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.
R. Kimmich, E. Anoardo / Progress in Nuclear Magnetic Resonance Spectroscopy 44 (2004) 257–320316
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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