Field Cycling in NMR Relaxation Spectroscopy:
Applications in Biological, Chemical
and Polymer Physics
Rainer Kimmich
Sektion Kernresonanzspektroskopie
UniversitSt Ulm
D-7900 Ulm, West Germany
I.
II.
INTRODUCTION
PRINCIPAL ASPECTS OF NUCLEAR MAGNETIC RELAXATION
A.
B.
C.
D.
III.
FIELD CYCLING EXPERIMENTS
A.
B.
C.
D.
IV.
Why field cycling?
Longitudinal Relaxation
Transverse Relaxation
Experimental Requirements and Limits
APPLICATIONS
A.
B.
C.
D.
E.
V.
General Remarks
Relaxation Mechanisms
Correlation and Intensity Functions
Heterogeneously Relaxing Systems
195
195
195
197
198
200
201
201
203
204
204
207
Biological Systems
Cells and Tissues
Liquid Crystals
<
.
Synthetic Oligomers and Polymers
Non-macromolecular Systems with Electronparamagnetic Centers
FINAL REMARKS
207
211
211
212
215
216
References
216
I. INTRODUCTION
Nuclear magnetic relaxation spectroscopy has the potential to be a very versatile method for the investigation
of molecular dynamics in combination with structural
details. This fact has been well known since the early
days of NMR (1). The successful exploitation of many
possible applications, however, required technical
progress which has only occurred in the last few years.
This article deals with the special technique of field
cycling in which the NMR signal is detected in a reasonably high field after a relaxation period during which the
magnetic field strength was switched to a lower or higher
value. (Typical field cycles are described in section III.)
Thus, the relaxation can be studied in variable magnetic
fields without varying the frequency of the spectrometer.
In other words, the cyclic variation of the external magnetic flux density Bo increases the available range by
several orders of magnitude. Advances in high power
electronics and the technology of cryomagnets allow the
measurement of relaxation rates, often including individual intensity functions, in the 10- 5 to 1.0 T field
range, corresponding to a proton frequency range of 103
to 108 Hz with any kind of proton-containing systems.
For X-nuclei the frequency range is shifted according to
the different magnetogyric ratio. Thus, the necessary
tool is now available to distinguish different types of
molecular dynamics.
On the other hand, theoretical models of molecular
dynamics have been developed, which indicate significant differences between relaxation spectra. Several
authors have established catalogues of correlation functions appropriate for the discussion of many problems.
Thus, both the apparatus and the theoretical background are available to promise solutions of problems in
molecular dynamics.
The following two sections review experimental and
theoretical fundamentals and describe the limits of the
method at the present state of the art and developments
expected in the future. In section IV, the applications
performed to date by several groups are classified and
discussed. We shall restrict ourselves to applications
relevant to biological, chemical, and polymer physics.
II. PRINCIPAL ASPECTS OF NUCLEAR
MAGNETIC RELAXATION
A. General Remarks
The theory of the nuclear magnetic relaxation times T-,
and T2 has been treated in different ways with different
ranges of validity. Reviews are given in several textbooks (2-4). In this section, we will restrict ourselves to
those parts of the relaxation theories predominantly
relevant to the applications to be subsequently outlined.
With regard to the applicability of perturbation theory,
one distinguishes between the cases of "weak" and
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Vol. 1,No. 4
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"strong" collisions (5). The papers reviewed in this article
mainly concern the weak-collision case where the
Zeeman field-splitting is large compared with the jnfluence of the perturbing field.
There are two types of weak-collision theories. The
first implies that the spin system reaches a spin temperature (i.e., a Boltzmann distribution of the level populations combined with a completely random distribution of
the Larmor precession phases) in a time short compared
with 7Y This condition is usually fulfilled in solids, for
which the Gorter-Hebel-Slichter equation holds in the
high-temperature limit (reference 3, p. 143).
Wmn (Em - Enf
1 _1
T-, 2
£
(B)
(A)
Figure 1. Transition probabilities between eigenstates of the
spin operators: (A) longitudinal components, (B) transverse
components. (6)
E2
where Wmn is the probability per second that a transition
takes place from the state m to the state n characterized
by the spin energies Em and En, respectively. The disadvantage of this type of theory is that it is restricted to
systems which rapidly reach a spin temperature, but it
has the advantage that the quantum mechanical features of the lattice can be taken into account. This plays
an important role with relaxation mechanisms due to the
coupling of nuclei to conduction electrons (ref. 2, p. 355)
or to spin-phonon coupling (ref. 2, p. 401).
In other cases, a theoretical perturbation treatment
with no explicit consideration of the lattice quantum
mechanics is preferable. Then, a semi-classical treatment is performed taking the lattice globally into account. Several modifications of this type of theory have
been outlined in the literature and lead to the same results. A straightforward derivation of the most important
standard case (dipolar coupling in two-spin systems) has
been given (6) to which most of the following applications
are related.
For nuclei of spin 1/2, we have to deal with the pattern
of transitions between the eigenstates of the longitudinal
and transverse components of the spin operators as
given in Figure 1. The relaxation rates are then given for
like spins, i.e., for equal magnetogyric ratios (y, = ys), in
the high-temperature limit by (6)
^
w2)
(2a)
and in the motional-narrowing case by
1
-zr= 2(Ui + U2)
(2b)
In the derivation of equations 2c and 2d, it was assumed that the nonresonant spin species stay always in
equilibrium as a consequence of an additional interaction of the S-spins with the lattice. This condition is often
fulfilled for unpaired electrons (zero-field-splitting) and
for quadrupole nuclei (coupling to the electric-field
gradient of the lattice).
The transition probabilities in equations 2a-2d can be
calculated with the aid of first-order perturbation theory
(6) or its equivalent modifications (ref. 2, p. 264 and ref.
3, p. 137). The essential result is that the probabilities
can be expressed by the intensity functions of the fluctuating interaction. In the standard case mentioned
above, the probability w^ (Figure 1), for instance, is given
by (6)
Ah2
(3)
where a = - -^ - ^ h2 y, ys.
li (cc) is the intensity function to be taken at the resonance frequence w and defined by the WienerKhintchine relation
where G^ (T) is the autocorrelation function of the part of
the dipolar interaction which induces the flip of the single
spin
Gi(r) = F,(0) F](T)
(5)
'2
For unlike spins (7, =£ 7S)
• ~ w0 + 2M
w2
(2c)
' 1
and (again in the motional-narrowing case)
—= u0 + 2u1 + u2
>2
(2d)
R, <f>, 6 are the time-dependent spherical coordinates of
the spin-spin vector. The bar denotes the ensemble
average over all pairs. The expressions for the other
transition probabilities are summarized elsewhere (6).
Thus, the correlation functions play a central role in all
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weak-collision cases. Instead of the correlation function
in equation 5, it is more convenient to introduce the
reduced, or normalized, correlation function
G(r) =
F,(0) F{{r)
-
(6)
with the mean square deviation
<T;=
\F/ 2 _
so that G(0) = 1 and G(°°) = 0.
The Fourier transform of G(r) will be designated by
l(w). For isotropic systems defined by the condition F, =
0, equation 6 can be simplified further. Equating the ensemble average with the time average for a single spin
pair, it becomes clear that the term "isotropic system"
means that during the relaxation times each interaction
situation has occurred with the same probability.
Frequently one has to deal with more than two interaction partners. It is then a good approximation to calculate the effective relaxation rates (7"12~1)eff as a sum
over the two-spin rates r,^" 1 of all spins interacting with
a reference nucleus, even if the spin states are perturbed
in a correlated manner (2,7,8)
—
)= pairs
7"i,2eff
M,rans( t) = MUans(0) exp { - M/ j (t - r) gu (T) df]
(8)
: O
where M2r is the second moment of the rigid lattice and
gjj) is the autocorrelation function of the fluctuating
part of the resonance frequency.
When the externally applied magnetic field Bo approaches zero, the strong collision case becomes
relevant. The spin-lattice relaxation time is then (5)
.2(1 -P)
£BO (9)
where T* is the mean time between two "collisions" (i.e.,
atomic rearrangements), p is a function of the dipolar
fields changed with a collision (p = 1, if no change occurs; p = 0 if there is any correlation to the fields before
the collision is lost). BL is the local field caused by the
magnetic dipoles.
B. Relaxation Mechanisms
1
E
narrowing effect is incomplete, and this may be the case
even for very rapid but anisotropic motions, the Blochdecay of the transverse magnetization can be described
by the Anderson-Weiss formula (9,10)
(7)
This two-spin approximation is generally assumed in the
literature.
Transverse relaxation in highly viscous or solidlike
systems implies some complication. If the motional-
The standard case of dipolar interaction is the relaxation mechanism most frequently assumed in NMR applications. There is, however, a series of quite different
interactions leading to nuclear magnetic relaxation
which should always be kept in mind. Table 1 gives a
Table 1. Nuclear Magnetic Relaxation Mechanisms
Spin of the
Relaxing Nucleus
/ > y2
/> 1
Coupled
by
Modulation
Source
Examples
to
fluctuations of
dipolar
interaction
equal d i p o l e ^
unequal d i p o l e ^
1
scalar interaction
(Fermicontact and
indirect
coupling)
dipole/I 2 ofa
quadrupole
nucleus
1H<-14N
chemical
shift
tensor
local magnetic
field
13
spin rotation
tensor
Coulomb
interaction
angular momentum of the
molecule
electric field
gradient
?12
H**Mn 2 +
fluctuations of
T12.
fluctuations of
fi2.
C**e0>67
M2
References
2,6
2,11,12
13,14
M2
molecular reorientation
2,15,16
gases
atomic collisions
2,17
14
molecular reorientation (lattice
vibrations)
2,18
N in asymmetric molecules
Note: Quantities \x^ and M2 a r e * n e dipole moments of the interacting particles; r12 is the distance vector.
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Table 2. The models are suitable throughout for dipolar
interaction. Some of the more generalized models can
also be applied to the other relaxation mechanisms.
A frequently assumed but not necessarily justified
standard case is that of an exponential correlation
function
brief survey of the different mechanisms and the references in which they are described and quantitatively
explained.
C. Correlation and Intensity Functions
The most important aim of field-cycling relaxation
spectroscopy is to obtain information about intensity
and correlation functions of microscopic fluctuations. In
order to show the potential differences in the types of
fluctuations occurring in molecular dynamics, we will
review some simple model cases distinguishable by the
method. The high and low frequency limits are given in
G(r) = e x p ( - | T | / T C )
(10a)
(10b)
1 +
where TC is the correlation time. There are two main
situations for which these functions hold. The first is the
isotropic and continuous rotational diffusion of a dipole
Table 2. Limiting Behavior of Several Fluctuation Models*
Type of fluctuation
Low frequency/
high temperature
High frequency/
low temperature
WT; « 1
Rotational diffusion
(a) isotropic
(b) anisotropic
Markov jump models for
discrete interaction
states and environments (c)
Translational diffusion
(a) continuous
(b) stepwise
modulating intermolecular
interactions
Reorientation by
(a) continuous
(b) stepwise
diffusion of stable
defects with restoring
capability
(c)in 1 dimension,
unlimited
(d) in 3 dimensions,
unlimited
(e) in 1 dimension,
limited
(f) on a crystal
lattice
Reorientation by continuous,
unlimited diffusion of stable
defects without restoring
capability
(a) in 1 dimension
(b) in 3 dimensions
Reorientation by curvilinear
chain diffusion
References
C0Ti» 1
(a) 1,2
(b)19
(c) 20,21
-1
7", -v E ax T;
i
i
_
i
3/2 1/2
(a) r, -v a) T 0
22
(b) T-i ~ a;2 T 0
1/2 -1/2
(C) / , -v O) T o
3/2 1/2
(a) Tf ~ o) T O
(d.e.or^To"1
(a,b)T, -VT O " 1
(f) 25,26
3/2
(a) T, -v o)
(b) T-i ~ 0)2
1/2 -1/2
' 1 ~ W To
(a-e)23,24
1/2
24,27
TO
T0
3/2 1/2
/ 1 ~ 0) T o
28
*Time parameters T-, and TO indicate the time scales of molecular reorientations and displacements. The same symbols are
used for all models for simplicity. Exact definitions vary with the model and can be found in the references. Quantities a, are
weighting factors.
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pair with a fixed distance (29 and ref. 2, p. 298). The
second case appears with Poisson-like jumps between a
discrete set of perturbation states of any source, characterized by a single jump rate (3,20,21,30-32). While it is
hard to verify the first case experimentally because
molecules are either not spherical or not rigid (implying
internal motions), the second situation occurs, for instance, in solid benzene (31), with NH3 groups (33), and
for crystal water in gypsum (32). All examples showed
excellent agreement with the theory. Nevertheless, the
applicability of this simple correlation function remains
an exception.
The anisotropic, continuous, and rotational diffusion
of ellipsoidal molecules bearing the interacting dipoles
leads to a linear combination of several exponentials
(19). Also compare with (34). The same holds for internal
rotational diffusion about an axis fixed on a tumbling
molecule (35-37), for the exchange in two-phase
systems (38,39), and for Markovian jumps between a
discrete set of perturbation states of any source with
different jump rates (21). The latter situation can generally be treated as a Markovian chain, the mathematics of
which can be found (40). In this-context, the influence of
environmental fluctuation (Figure 2) is of special interest
(41). In cases where the analytical treatment fails, a
numerical calculation can be used even for rather complicated models (23).
Bessel function. A modified theory for liquid crystals has
been given (45). The application of these formulae is restricted to cases where the influence of intramolecular
interaction is weak. An example would be molecules
bearing a single spin. An experimental distinction
between intra- and intermolecular interactions is possible with the use of isotopically diluted systems.
In solid materials, especially if they are amorphous,
the reorientation of molecules or of parts of molecules
can be coupled to the arrival of structural defects. Examples are vacancies or those defects discussed in the
polymer section. Generally this type of fluctuation is
called "defect diffusion." A non-Markovian behavior is
the consequence because then the probability for the
reorientation of a molecular group at a definite time
depends on the positions of defects at a previous time.
Defect diffusion in crystalline materials has been
treated as a random walk on a special type of lattice
(22,25,26). For the continuous diffusion of defects in
one- and three-dimensional disordered systems, a
nearest-neighbor approximation has been suggested by
Glarum (46) and by Hunt and Powles (47). This approximation is inconsistent with the assumption that there is
no mutual hindrance of the defects. Perturbation by the
non-nearest neighbor defects can be excluded only in
the case of mutual hindrance, where the defects can not
permeate each other. In fact, different results have been
obtained by taking into account all defects and not only
nearest neighbors (24,27). Assuming that the arrival of a
defect leaves an interaction state completely uncorrelated with the initial state, gives in the case of unlimited
one-dimensional diffusion (24,27)
(12a)
Figure 2. Two-state model showing the effect of fluctuation
between different environments on the kinetics and degree of
molecular reorlentation. (21)
Intermolecular interactions between dipoles can be
modulated by translational diffusion. If this contribution
to nuclear magnetic relaxation is dominant, a quite
different type of correlation function occurs (22,42-44,
ref. 2, p. 300). In the limit of continuous translational diffusion and of a continuous distribution of spins, we have
^j
=J^)i/2f^
du
+(o>2To2/l/4)
(11a)
(11b)
with T 0 = d2/2D. D is the translational diffusion coefficient, n is the number of spins per unit volume, d is the
minimum distance of the interacting nuclei, and J3/2 is a
.,3/2
1 - C2(a)]
[ l - S2(a)]}
(12b)
with a = (4a)Tp)-1 and re = ir £0 (16 D)-\ where £o is the
mean nearest defect distance, D is the diffusion coefficient, and C2 and S2 are the Fresnel integrals. In three
dimensions, an exponential correlation function again
occurs (27).
A different situation arises when the defects have restoring capability. This means that after the passage of
the defect the original interaction state is restored instead of an uncorrelated state as in the previous model.
A molecular group then suffers a series of uncorrelated
"pulses" due to the passing defects. Therefore, the rate
of the pulses influences only the absolute value of the
relaxation rate but not the shape of the intensity function, in clear contrast to the "total correlation loss"
models.
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For unlimited one-dimensional diffusion of noninteracting defects with restoring capability we obtain
(24)
JJL
(13a)
1TTb
, reflecting barriers
b
i"
i
(13b)
/(«>) =
- exp(-
sin
where rb = bz/2D. The quantity b is the width of a defect
and D is the diffusion coefficient. The symbol erf designates the error function. The treatment has been extended to three dimensions (24).
Mutual hindrance of the defects alters the situation
especially in the one-dimensional case where defects
cannot "overtake" each other. One has then to consider
the fluctuations of the whole ensemble of defects. This
can be done by a computer simulation (48,49). An analytic approximation can be achieved by treating the two
neighboring defects as reflecting barriers limiting the
mean square displacements (Figure 3). This has been
performed for continuous (50) and for stepwise (23)
diffusion.
The exact expressions (23,50) are lengthy. We want
therefore to present only the limiting expressions for the
intensity function derived from a 6th order approximation (51). Two characteristic times play a role: the diffusion time rb fora mean square displacement b2 (b is the
length of a defect) and the diffusion time ra for a mean
square displacement d2 (d is the distance between the
reflecting barriers). With these quantities we have
for wTd < 1
(14a)
for 0)Tb <3C 1 <K U>Td
M
= \/TbTd
1
-7= (14b)
1
O,3/2T-( / I £ - 1 )
V T/,
Another application of the limited diffusion model will be
described in the model membrane section.
The defect diffusion models discussed so far assume
that the defect has an infinite lifetime, more precisely, a
lifetime long compared with the relevant diffusion times.
At the opposite limit, where defects can be frequently
created and annihilated at any position in the system,
the appearance of a perturbation by a defect at a
Figure 3. Schematic representation of defect diffusion
between reflecting barriers at x = 0 and x = d. Reference nucleus is at x = r. Relaxation formulae must be averaged over all
possible values of r. Defect has width b. (50)
molecular group is then no longer diffusion controlled.
The consequence is a Poisson-like fluctuation with an
exponential correlation function (see section 3.1 of ref.
21).
Molecular dynamics in polymer melts and concentrated solutions are strongly influenced by a wormlike chain
diffusion process. This mechanism is described by (28)
IT
G(T) = exp(^J
-erf
(15a)
(15b)
with Tj> = C 2 /20. D is the curvilinear chain diffusion coefficient, £ the orientation correlation length of the diffusion path. This model is relevant for all systems where
the reorientation of molecular groups is coupled to a displacement along a randomly curved diffusion path.
Oscillator models can be relevant in solids with
frozen-in molecular reorientation or diffusion. Stochastic excitations of vibrations by random pulses may then
describe nuclear magnetic relaxation (13,30, chap. X of
ref. 51). It is clear that the diverse oscillator models have
to be strictly distinguished from spin-phonon coupling
processes (52).
This section indicated the broad variety of correlation
and intensity functions which can be studied by the field
cycling method in NMR relaxation spectroscopy. There
is, however, an important condition for this type of study,
namely the homogeneity of the relaxation in the system.
This question will be discussed in the following section.
D. Heterogeneously Relaxing Systems
Samples with structural heterogeneity do not
necessarily relax in an inhomogeneous manner. The
question is whether the structural heterogeneity is stationary or fluctuating. In the latter case, we have to take
into account the time scale of relaxation itself. Spins
experiencing the same type of fluctuation during the lifetime of their states, relax equivalently. This means that
there is either no structural heterogeneity or the
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structural inhomogeneity is fluctuating with a mean lifetime essentially shorter than the lifetime of the spin
states (i.e., the relaxation times), so that all spins "see"
the same type of fluctuation.
Spins in different chemical groups of stable compounds are permanently in different structural environments, and therefore relax heterogeneously with different correlation fuctions. This behavior can easily be detected under high-resolution conditions (16,53). Spins in
rapidly varying structural environments, implying, for instance, weak bonds which are frequently broken up,
relax homogeneously.
Homogeneous relaxation means that a single correlation function is relevant, while heterogeneous relaxation
requires a discrete or continuous distribution of correlation functions. Several types of such distributions have
been reviewed (54).
A problem may be that distributions of correlation
functions do not always reveal themselves by a distribution of relaxation times. Magnetization transfer by spin
diffusion in solid-like substances (2,55-58) and crossrelaxation (59,60) can lead to average exponential relaxation curves. Distributions of correlation functions can
nevertheless be revealed by isotopic dilution of the
resonant nuclei, so that the averaging mechanisms are
prevented. Rapid material exchange, on the other hand,
should be considered as a special case of environmental
fluctuation leading to a single correlation function
(21,38,39,61).
It is clear that the occurrence of distributions of correlation functions, whether or not combined with a distribution of relaxation times, causes a severe complication of the analysis of relaxation data, apart from the appearance of additional effects (62). The relatively long
time scale of nuclear magnetic relaxation times, however, often allows the exclusion of such distributions in
chemically homogeneous systems, even in cases where
no special experimental test is possible. In connection
with the applications discussed below we will occasionally return to this question.
III. FIELD CYCLING EXPERIMENTS
A. Why Field Cycling?
The conventional method to determine nuclear magnetic relaxation times uses appropriate radio frequency
(RF) pulse sequences and stationary magnetic fields
(16). This method fails at low magnetic field strengths
because it becomes more and more difficult to detect
NMR signals in decreasing magnetic fields due to the
weaker
nuclear
induction, the
lower
Curiemagnetization, and the narrower bandwidth of the RF
pulses at low frequencies. While it might be possible to
compensate for the poor sensitivity by larger sample
volumes and by signal accumulation, one cannot
produce high-power RF pulses short enough for the detection of solid state signals at frequencies below several MHz.
The problem can partially be overcome by performing
relaxation experiments in the rotating frame (16,63-65).
These methods, however, merely cover a frequency
range below about 100 kHz, leaving a gap of two orders
of magnitude for solids between rotating-frame and conventional laboratory-frame experiments. Moreover, the
theory of the relaxation in the rotating frame is, though
similar, not identical to that of the laboratory frame,
leading to some inconvenience in the simultaneous interpretation of both experiments.
A more appropriate solution to the problem of measuring relaxation times in the whole range down to the
kHz region is the field-cycling method. Typical field
cycles are plotted in Figures 4 and 5. The general principle is to combine the high sensitivity and bandwidth in
high magnetic fields, where the sample is polarized and
the signal detected, with the relaxation effect in variable
relaxation fields.
Moreover, the increase in the available frequency
(field) range is accompanied by two important advantages in the general measuring technique of relaxation
times. First, the method is very rapid and allows automatization because, independent of the relaxation field, the
detection frequency is always the same, so that no time
consuming adjustments of the RF apparatus are necessary for a varying relaxation field. Second, this technique
is insensitive to systematic errors in 7^ caused by inhomogeneous RF fields (chap. 3 of ref. 16). The perturbation of the sample occurs by a rapid jump of the external field Bo and not by an RF pulse. As Bo is very
homogeneous compared with RF fields, a homogeneous
perturbation of the sample is guaranteed. In contrast to
the perturbation, the detection of free induction decays
by inhomogeneous RF pulses is uncritical and causes at
most a loss in signal intensity but no systematic error.
Before outlining the practical aspect of the technique,
let us first classify the general aims of the possible applications into three main groups.
1. Relaxation Dispersion
Since molecular dynamics are characterized by certain correlation functions leading to a variety of intensity
functions, identification of a special type of fluctuation
requires a measurement of the intensity function over as
wide a range as possible. This can be done by relaxation
spectroscopy in two ways. One can shift the intensity
function by changing the temperature to vary the fluctuation time constants while observing the relaxation
rates at a certain frequency given by the spacing of the
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Vol. 1, No. 4
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Figure 4. Typical field cycle (idealized) for the determination of 7"v The z-component of the magnetization is increased to the equilibrium value MXP in the polarization field Bof>. In the relaxation field Bor, the magnetization relaxes with T^B,/) toward Mxr. Detection occurs in the detection field 6od(here chosen equal to fio") with the aid of a 90° RF pulse or a spin-echo sequence.
Bo
P Rd
So-^O
Bo
t
Mt
90°
\
\
\
\
180°
A
90°
\
180°
A
Figure 5. Typical field cycle (idealized) for the determination of T2 by Hahn's spin-echo method. The lower plot shows the time
development of the transverse magnetization Mt. Carr-Purcell sequences can be used with analogous field cycles.
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Zeeman levels. Alternatively, one can keep the temperature constant and vary the frequency "window" to monitor the intensity function (Figure 6).
The second method, using field cycling, is preferable
in many cases because it avoids the limitations or complications introduced by phase transitions and unknown
temperature dependences of the parameters. Various
activation laws that show such complications are
reviewed elsewhere (66).
2. Relaxation Anisotropy
Anisotropic systems show anisotropic motions and
therefore anisotropy in the relaxation times, which can
be detected in oriented samples. This fact offers the
possibility of distinguishing different types of motions,
especially different types of defects. (Figure 7 shows anisotropy curves calculated for dipolar interaction in alkane chains.) Thus, relaxation anisotropy provides
structural information to add to the kinetic features
studied by relaxation dispersion.
According to equations 2-6, the anisotropic dipolar
relaxation rate can be written as
=-
+ C2 <T2)
J (w)
/\
Figure 6. The intensity function /(o) can be scanned point by
point by varying theZeeman-field splitting h y Bo.
(16)
1
1
where
and
a-t and <J2 a r e the mean square deviations of the dipolar
interaction functions as in equation (6).
We now define the orientation angle 0 as the angle
between the external flux density Bo and the axis of the
molecular motion. For polymers, this axis is identical
with the chain axis. If the molecules are incompletely
oriented, the angle /3 refers to the symmetry axis of the
orientation distribution. We then obtain the ratio
C2 o-2(0)
770)
(17)
which depends only on C2. At fixed frequencies and
temperatures, C2 is a constant, weighting the contributions of ff! and o2. The range of Cz for Debye-functions
(equation 10b) is limited to 0.25 for O>TC » 1 and to 1 for
WTC < 1. This means that different relaxation anisotropies result for different frequencies. As shown in
Figure 7, the strongest anisotropy holds for C2 = 1 or o>rc
«. 1. In other words, the experimental investigation of
the anisotropy is easier at the low frequencies accessible
by the field-cycling technique.
3. Quadrupolar Dip
Cross-relaxation (68,69) from dipole to quadrupole
nuclei leads to a quadrupolar dip in the ^-dispersion
curves (13,14,70,71). The reason is a crossing of the
Zeeman and quadrupole splittings as shown in Figure 8.
The frequency range in which this effect occurs depends
on the quadrupole splitting. The corresponding frequencies are often only accessible by the field-cycling
technique.
The shape of the quadrupolar dip reveals the spectrum of the quadrupole splittings and, by circumstance,
the intensity function of the interaction with quadrupole
nuclei (13). The determination of the cross-relaxation
rates from the quadrupolar dip allows the study of the
specific interactions between the dipole and the quadrupole nuclei.
The equivalent effect of "spin mixing in the laboratory
frame" is also useful in a double-resonance technique
for the detection of pure nuclear quadrupole resonances
(72-74). However, this type of application of field cycling
is beyond the scope of this article.
B. Longitudinal Relaxation
The standard field-cycling program for the measurement of the longitudinal relaxation time r, is shown in
Figure 4. The sample is polarized in as high a field as
possible. After reaching equilibrium, the field is rapidly
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Vol. 1,No. 4
203
20
oscillation
rotation
40
60
Figure 7. Orientation dependence of intramolecular mean square deviations of the
dipolar interaction functions F,m<1> and F/m<2>
of the N proton pairs with numerical indices /
and m in an isolated alkane chain. /} is the
angle between the chain axis and the flux
density. Calculation includes the effect of the
sixth neighbor methylene group (omission of
interaction with a neighbor group increases
the apparent anisotropy). Models refer to
whole-chain motion around the chain axis or
diffusion of local defects along the chain.
Kink defects (gtg sequences) have an assumed concentration of 0.15 kinks per
methylene group. "Oscillation" around the
chain axis is assumed to be classical with an
amplitude of 18.5°. The curves show the
different efficiencies and anisotropies of the
functions (67). Also see Figures 13a and 13b.
80 0
jumping kink
jumping kink*rotation
switched to the variable relaxation field, leading to a
non-equilibrium magnetization, provided that the field
was altered fast enough (see section III. D). After a variable period T, the field is switched to the detection field
according to the resonance condition u> = y Bo. After
reaching resonance, the magnetization is detected by a
90° pulse or an echo sequence. Alternatively, a signal
can be obtained by a fast adiabatic passage of the
resonance field. The evaluation is performed according
to
Mz(r) -
= [MM -
(18)
/W2(°°) is the Curie magnetization in the relaxation field,
MZ(Q) is the non-equilibrium magnetization at the beginning of the period in the relaxation field, and 7"., is the
longitudinal relaxation time in the relaxation field.
For higher relaxation fields, the difference from the
polarization field may be too low to cause a sufficient
deviation from equilibrium. In this case, a 90° pulse
which sets Mz = 0 immediately before switching to the
relaxation field is useful. The evaluation formula is then
identical to that of the conventional 90 ° /90 ° method (16)
with Mz = 0.
In appropriate systems, the polarization of the sample
can also be enhanced by taking advantage of any kind of
dynamic polarization (2). A considerable increase in sensitivity can be expected (75).
C. Transverse Relaxation
The method of Hahn as well as the CarrPurcell-Meiboom-Gill sequence (16) can be combined
with field cycling. Figure 5 shows the field program appropriate for Hahn's method. The possibility of extension for a Carr-Purcell sequence is obvious. The decay of
the echoes directly yields the time constant T2 as a function of the relaxation field.
D. Experimental Requirements and Limits
The field programs shown in Figures 4 and 5 are
idealized. In reality, finite switching times between polarization, relaxation and detection fields, respectively, are
unavoidable. The problem is to find the limits of the
switch intervals that will allow one to obtain reliable
measurements.
In some field-cycling experiments described in the
literature, the direction of the external field is changed
during the field program. Then the transitions have to be
slow enough to satisfy the adiabatic condition (76,77)
(19)
In all cases, where the direction of B remains stationary,
this condition becomes trivial. Switching to zero external
field Bo, however, means that B approaches the local
field BL with no definite direction. A dipolar order is
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204
Bulletin of Magnetic Resonance
=
V\ B o
(1)
Q
X
= )XB°
B o /T
Figure 8. Frequency survey for 35 pl and 1H resonance crossing. u>A is the Uarmor frequency of 1 H; yx is the
gyromagnetic ratio of 35CI; OJXQ is the zero-field and cos the low-field resonance frequency of 35CI. The crosshatched
area indicates the entire range for powder samples. (71).
produced and the condition given by equation (19) is
again relevant (77).
On the other hand, the transitions between the diverse
fields have to be fast enough to avoid equilibration during the switch intervals. This is certainly the case in the
"fast transition limit" of the switch time AT
AT « 7"i,2(Sor)
(20)
where, as the strongest restriction, the relaxation times
in the relaxation field Bor have been chosen. In reality,
relaxation times frequently rise with Bo, so that the restriction is less severe than it may seem.
If the course of the field during the switch intervals is
reproducible from field cycle to field cycle, it can be
shown that the condition given by equation (20) is far too
restrictive (78). The longitudinal relaxation curve, measured after the time r + AT, is given by
MZ(T + AT) - Mz{°°)= k, A/Wzexp(- T / T^BJ)
(21)
where AMZ is the difference between the magnetization
after reaching the relaxation field and the Curie magnetization Mzor in this field, and k^ is a constant for
reproducible field transitions. There is then no systematic error, even for Tf < AT. The restriction is given rather
by the condition that k^AMz must be of a measurable
size, i.e. by the sensitivity requirement. For the apparatus referred to in ref. 78, this led to the condition
TABor)^Ar/6
(22)
for samples with an overall proportionality of approximately T-i ^ Bo. AT is the total switching period from triggering the field rise to resonance. (Note also that the actual slope of the field rise often is initially extremely steep
and becomes flatter near resonance, so that the "critical" region with short relaxation times is rapidly
traversed.)
A further requirement is a sufficiently large magnetic
flux density Bod of the detection field. The reliable detection of solid state signals is related to the bandwidth of
the RF apparatus. Therefore, detection frequencies of at
least 10 MHz should be used. For liquids, the bandwidth,
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Vol. 1, No. 4
205
and therefore the detection frequency, may be lower. In
this case, detection is possible even in the earth's magnetic field (79,80).
There are two ways to produce the field cycle. Either
one transports the sample between positions with different magnetic flux densities, or one switches the field
electronically between the desired values.
The first applications of the sample-transport method
were reported by Pound (81) and Ramsey and Pound
(82). The method was then improved by moving the sample between the middle of the gap of an iron magnet and
a predetermined point in the fringe field (83). Compressed air has been used to transport the sample
through a tube between two magnets (69,84,85). Thus,
considerably shorter transport times have been
achieved than with a manual moving device (86).
A detailed description of a compressed-air machine is
available (87) with a reported traverse time of 250 ms.
Though this is certainly not the limit of the compressed
air method, it shows that applications are possible only
for samples with relatively long relaxation times. The direction of Bo may differ along the path of the sample as a
consequence of field inhomogeneities, in which case the
adiabatic condition equation (19) might be relevant.
These disadvantages can be avoided by an electronic
field-switching arrangement. The use of ironless magnet
coils helps to obtain reproducible fields and prevent
losses and residual magnetism in the material. For a
solenoid of n windings, length C, and cross section A
much smaller than C2, the instantaneous rate of field variation is
Bo(t) =
U{t) - l(t) R(t)
nonA
(23)
where U is the applied voltage, / the current, and R the
effective ohmic resistance of the whole circuit. The
number of windings n can be expressed by the magnetic
flux density Bod of the detection field and the current ld
needed to produce it. The instantaneous velocity of field
variation is then
^
v
(24)
where the solenoid volume V = A9.. From equation (24) it
becomes obvious that in order to rapidly leave the
critical low-field range (where the relaxation times are
short), one should provide a peak power ld Umm as high
as available, while the ohmic resistance and the solenoid
volume should be kept as small as possible. This means
that U(t) should be initially higher than the voltage
required later for steady state operation.
Equation (24) also indicates that rapid decrease of the
flux density requires the rapid dissipation of the field
energy, and, therefore, an ohmic resistance R(t) as large
as possible. In other words, the ohmic resistance should
be switchable. One example of a practical variable resistance is a well cooled parallel circuit of high-voltage transistors. The applied voltage should be zero or, if possible, even negative during the field decay. The volume of
the solenoid should again be as small as compatible with
homogeneity requirements.
In practice, the combination of high-power programmable power supplies and small solenoids leads to
cooling problems. Further limiting factors are the extent
of homogeneity in the sample volume and the stability
and reproducibility of the detection field. Various suggested compromises are discussed below.
In the first proper field-cycling experiments (79,80,88),
a high polarizing field was non-adiabatically switched off
after a certain period leaving the earth's field as the only
magnetic field. The polarizing field was adjusted perpendicular to the earth's field so that the non-adiabatic transition led to a free precession signal without any RF excitation. A similar method, but with adiabatic field transitions and 90°-pulse detection, has been described (89).
Considerably higher sensitivity can be achieved with
higher detection fields. An apparatus has been described (90) which allows polarization of the sample in a
field ranging from 0.05 to 1 T while the signal detection is
performed at 4 MHz (i.e., 0.1 T for protons). Liquidnitrogen-cooled copper solenoids or a superconducting
coil have been used to reduce or eliminate the ohmic resistance of the solenoid wires. This machine has since
been improved to allow automatic computer-controlled
operation (91) for which the field-cycling method is particularly suited.
A relatively simple but effective field compensating
method has been described (92-94). Two power supplies
have been used. One non-switchable precision power
supply produces the polarization and detection field in a
rather large pair of Helmholtz coils, thus insuring high
stability and homogeneity. The relaxation field is
produced by a second concentric pair of Helmholtz coils
of considerably smaller diameter, together with a commercial programmable power supply, which generate an
additional field during the relaxation period. The superposition of the two Helmholtz fields allows a partial compensation or addition depending on size and polarity. As
the volume of the compensation coils is small, short
switching times can be achieved without deterioration of
the detection field. The inhomogeneity of the relaxation
field produced in this way is not important at higher magnetic flux densities, but can restrict application at the
lowest fields, where the total flux density becomes undefined as a consequence of the difference in the
homogeneities of the two pairs of coils.
Several other field-cycling machines have been used
(78,95). The present apparatus in our own laboratory
was designed especially for applications with solids and
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206
Bulletin of Magnetic Resonance
neither frozen nor denatured. Thus, in contrast to many
other systems, the only possible way to study the protein
motions by relaxation spectroscopy is to vary the
frequency.
Early relaxation studies at a single frequency (96,97)
showed that water protons relax faster in protein solutions than in pure water even for diamagnetic proteins.
Relaxation dispersion measurements were therefore desirable to learn something about the protein properties
that cause this effect.
Many proteins have been studied by this method and
many
interpretations
have
been
suggested
(91,92,98-108). A typical relaxation dispersion plot is
shown in Figure 9. The fact that the dispersion is spread
over almost the entire frequency range (103 to 108 Hz)
leads to the conclusion that the relaxation of the water
protons somehow reflects the protein motion. All authors agree to this but suggest several quite different
models to explain how water molecules, which normally
tumble four or five orders of magnitude faster than protein molecules, can reveal the motions of these
molecules. A relevant question is: In which part of the
system and by which interaction partners are the nuclei
of water molecules relaxed?
for samples yielding weak signals (51). Polarization and
detection fields are in the range 0.5 to 1.2 T and we use
liquid-nitrogen-cooled copper coils or a superconducting solenoid wound of multifilament niobium-titanium
wires. The homogeneity in the sample volume (1 to 3
cm3) is better than 5 x 10- 5 . The stability of the detection field is about 5 x 10~5. Starting from zero, a flux
density of 0.1 T is passed within 4 ms with the copper coil
and within 2 ms with the superconducting solenoid. As
the course of the field rise is reproducible, TA values
down to 5 ms can be measured provided T^ has sufficient field dependence. The peak power of the pulsed
power supply is about 18 kW, and the inductance of the
field solenoids ranges from 80 to 100 mH.
IV. APPLICATIONS
A. Biological Systems
1. Protein Solutions
The most frequent application of field cycling in NMR
relaxation spectroscopy concerns protein solutions.
This is simply because the variation of the temperature is
restricted to the range where the protein solution is
1O C
Aqueous BSA solutions
A
>A
DA
A
x
X
8
A
A
0 A
A A
8
•
0
. 8°
10-
-
#
•0
•
0
0
•
§
o* #8
°
0
H20
•
H 2 O/D2O 50%, L-signal
x
D2 O , L- signal
D
.8
•
A
1 I 11
10 4
10 5
1
1
1 1 [ i 1 1 1
10°
1
i
l
L- signal
0
D 2 O, S + L - signal
d r y BSA, S + L-signal
l
10'
10°
1O9 v/Hz
Figure 9. Longitudinal proton-relaxation times versus frequency for various bovine serum albumin (BSA) solutions at 0°C (100).
Concentrations are 0.391 g BSA/cm 3 solvent. L is the signal detected about 200 us after the 90° pulse; S + L is the signal detected
10 ixs after the start of the 90° pulse. Close agreement of the data for the various systems led to the model shown schematically in
Figure 10.
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Vol. 1,No.4
207
Several authors have suggested that more or less
irrotationally-bound water molecules in the hydration
shell relax the bulk water by a fast material-exchange
process (98,99,107), so that the rotational correlation
time of the macromolecule becomes relevant (two-site
exchange model). Koenig and Schillinger (98) have
pointed out that the water molecules need not be bound
rigidly on the protein surface for this argument to be
valid. Rather, it would be sufficient if the motion of the
bound water was restricted to rotational diffusion
around one axis fixed on the macromolecule. Clearly,
this type of model includes many variants concerning
more or less restricted and continuously or discretely
distributed degrees of motional freedom within the
hydration shell. The discussion concerning all two-site
exchange models can be concentrated on two crucial
points: the mean residence time rM of a water proton in
the bound state and the contributions of intra- and intermolecular interaction to the water relaxation.
The mean residence time TM of the water protons and
the number of water molecules in the bound state within
the hydration layer have been dealt with by Koenig et al
(105). Assuming a two-site exchange formula, these
authors estimated the limits of rM on the basis of 1 H, 2H
and 17O relaxation data. The argument used in this estimation was that rM should be longer than the correlation time relevant for the rotational diffusion of the protein molecule, but shorter than the measured relaxation
times. This follows from the fact that the protein rotational diffusion is visible in the ^-dispersion of the water
protons and also from the observation of exponential
relaxation curves characteristic of the fast exchange
limit. The main advantage of the use of 2H and 17O relaxation data for this estimation is that these isotopes show
very short relaxation times, so that the upper limit of TM
could be reduced by two orders of magnitude.
According to Koenig et al (105), the ranges of possible
values of TM estimated in this way for solutions of a small
protein (lysozyme, molecular weight 14,700) and a large
protein (hemocyanin, molecular weight 9 x 106) are not
compatible with a common rM value or a common distribution of rM values. This conclusion depends on how
much reliance is placed on the estimated limits of rM.
Another objection against all simple two-site exchange models concerns the order of magnitude of TM.
The tentative explanation of the relaxation data by such
a model leads to values of rM five orders of magnitude
larger than is compatible with the binding energy of
hydrogen bonds (105), which could be assumed to be
the main reason for the formation of the hydration layer.
On the other hand, it is well known that rM values in the
required order of magnitude occur, for instance, in neutral solutions of small ions (12,98). Therefore, the electrostatic interaction with a few charged groups on the
protein may explain such long residence times in the
case of protein solutions.
Furthermore, one has to consider exchangeable protons of the protein molecule itself, so that in the case of
the hydrogen isotopes TM has to be considered rather as
an exchange time between protein and water than as a
residence time of water molecules in the hydration shell.
This explanation is supported by the observation that
the number of protein-associated exchanging nuclei is
essentially smaller than the number of water molecules
required to form a single-layered, closed hydration shell
(105). Below, the term "exchange" will be generalized to
mean non-material spin exchange, leading to a combined material- and magnetization-transfer model,
which avoids the problems mentioned above.
If dipolar interaction among the water protons were
the only mechanism of relaxation, i.e., if the two sites assumed in this model were isolated with respect to interactions with other parts of the system, an isotopic dilution experiment should show a strong lengthening of the
relaxation times according to the degree of dilution.
Experiments of this type (100,101) showed almost no
isotopic-dilution effect for serum albumin solutions of a
sufficiently high concentration (Figure 9). This led to the
unambiguous conclusion that the water protons are
relaxed by an interaction with spins in the protein which
are not affected by the isotopic dilution. Clearly, in this
type of investigation one must take into account the
small but not negligible increase of the water viscosity
upon deuteration (109), and a sufficiently high protein
concentration, depending on the individual protein, is
necessary for reliable results. The conclusions for
lysozyme solutions (110,111) were drawn without considering these conditions.
A hydrodynamic ordering of unbound water surrounding the protein molecules has been suggested
(91,105). The motion of the water molecules should then
be anisotropic, leaving a small but slowly reorienting
component. However, the hydrodynamic water-ordering
model again contradicts the finding that the water protons are relaxed via /ntermolecular water-protein interaction especially at higher protein concentrations.
The models mentioned so far do not take explicitly
intouaccount the protons of the protein. This oversimplifies the system, which is in fact highly heterogeneous
in the sense of section II.D. Observations with serum albumin indicated that the free induction decay of the protein protons is composed of at least two contributions,
namely a signal behaving like a solid and another
component decaying like a liquid, and also that the longitudinal relaxation curve is exponential, except for proteins like gelatin. This led to a combined materialexchange and magnetization-transfer model (100,101).
Figure 10 gives a schematic representation of the model.
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Bulletin of Magnetic Resonance
protein
"mobile" protons
solvent
Dl, MT
* L-signal of the protein
hydration
CE
unbound
CE
MT
shell
rigid " protons
= S -signal
Dl,
water
MT
V//////X.
Figure 10. Schematic representation of the exchange processes that influence proton relaxation in aqueous protein solutions (101). MT indicates
magnetization transfer, Dl dipolar interaction, and CE chemical exchange.
We have used the term "magnetization transfer" instead of the terms "spin diffusion" or "cross relaxation"
usually used for extended solids and direct interaction
partners, respectively. It is rather a matter of taste
whether one considers a more or less rigid macromolecule as a solidlike structure showing spin diffusion due to mutual flip-flop processes, or one emphasizes the heterogeneous mobility of the protein protons
so that the relevant scheme is a complex multinuclei
cross-relaxation which includes more than flip-flop transitions. Both views are justified in a certain sense. The
term "magnetization transfer", thus far not associated
with any special system, may therefore be preferable.
The essence of the model shown in Figure 10, or in
variants of it, has now been adopted by several authors
(108,112-118). Campbell and Freeman have treated the
population dynamics of coupled spins with different
relaxation rates, due to additional specific interactions,
on the basis of Solomon's theory (6). They find that cross
relaxation nevertheless can lead to the exponential
relaxation curves most frequently observed. A similar
application of Solomon's theory has been given by Kalk
and Berendsen (114). They believe that rapidly rotating
methyl groups act as relaxation sinks which relax the
total protein at sufficiently high frequencies. Edzes and
Samulski (113, 117) and Koenig et al (108) have given
systems of rate equations for the global solvent and protein magnetizations. Thus, it seems to be well established that magnetization transfer plays an important
role in the complex "protein water", albeit the correct
analytical description is still under discussion.
A special question concerns the existence of relaxation sinks within the protein molecules. These sinks
should fulfil some essential requirements. They should
be accessible to all spins in the sense of the exchange
scheme of Figure 10. Deuteron dilution should also dilute
the number of protons directly accessible to the sinks,
so that the solvent relaxation is unaffected while the nonexchangeable protein protons may have even longer
relaxation times (see ref. 113 and Figure 9). Furthermore, different mobilities are required (100).
As a suitable candidate for the sinks, we have suggested the protons in the NH groups within the protein
(13,119,120). It can be shown that NH protons dispersed
in a "bath" of the other protons account for the experimental facts. Furthermore, the quadrupole dip (section
III.A.3) due to NH cross relaxation has been observed in
dry bovine serum albumin, directly verifying this type of
relaxation sink at low frequencies (121). Of course, there
may be competing sinks depending on the system and
the experimental conditions, for example, the paramagnetic heme group in hemoglobin (96,102) or methyl
groups at high frequencies (114)!
The explanation of the special frequency dependence
of r, observed in protein solutions is difficult. As the protein molecule contains the sinks of the total relaxation,
one has to deal with a heterogeneous system not describable by a single correlation function. At least two
differently relaxing components occur, as is revealed by
the wide-band shape of the protein-free induction decay
(100). The problem then is that the type of the individual
correlation function is known only for groups suffering
the overall tumbling of the molecule alone, but not for
those showing internal motions, if no special assumptions are made (see section II.C). Furthermore, the influence of spin diffusion itself on r, dispersion must be
considered (101).
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Vol. 1,No. 4
209
Nevertheless, information can be derived concerning
the correlation time of the rotational diffusion of the
whole molecule and, with less assurance, of the motions
of the flexible groups. For example, the former correlation time could clearly be related to the molecular weight
of the protein (91). Thus, an empirical distribution of correlation functions has been used simply to get a quantitative description of the experimental data. In the same
sense, the intermolecular interaction between hemoglobin molecules at very high concentrations could be
studied (104,106).
The relaxation enhancement of water protons by paramagnetic ions or radicals bound to enzymes or other
macromolecules is an appropriate tool for studying
binding sites (122-133). The relaxation enhancement
upon addition of macromolecules to a solution of paramagnetic ions or radicals is due to the different relaxation parameters in the bound state of the paramagnetic
substances. Most frequently, Mn 2 + has been used,
which interacts with the surrounding protons by dipolar
and scalar coupling (Table 1). A number of parameters
determine these interactions because the electron relaxation time must be taken into account as a function of
the magnetic flux density. It was therefore necessary to
investigate a large frequency range by the field-cycling
method to allow less ambiguous fits of all parameters.
Examples have been discussed (134-138) in terms of the
proton exchange behavior between the paramagnetic
center and the solution; the exchange behavior of the
paramagnetic ion between the binding sites on the protein and the free water; the electron-proton distance;
and the correlation times for the dipolar and, if relevant,
scalar interaction. This is treated in section IV.D in connection with solutions of paramagnetic ions.
1
2. Model Membranes
Lipid bilayers are model structures for biological
membranes (139). The type of molecular motion in these
model membranes is of special interest with respect to
the permeability properties of membranes. One
proposed model for passive transport assumes diffusing
defects of the lipid structure to be "carriers" for small
permeants (140). In an alternative model, the diffusion of
vacancies provides the free volume required for a diffusion step of the permeant. To test these views, it is
necessary to verify the special type of fluctuation connected with defect diffusion (section II.C). Apart from
such experimental tests, general investigations of flexibility help to characterize and provide a physical understanding of the different thermodynamic phases of
membranes.
Many papers (141-144) treat molecular dynamics in
model membranes. Field-cycling measurements in both
the gel and crystalline phase of dipalmitoyl-lecithin
bilayers have been reported (145). Two frequency
regions in the 7, dispersion can be distinguished (Figure
11). The low-frequency process, clearly visible in the
liquid-crystalline state, must be attributed to cooperative motions of many chains. Such "order fluctuations"
will be discussed in section IV.B.
The high-frequency process has been interpreted in
terms of defect diffusion limited by two reflecting barriers, namely the polar headgroups terminating the lipid
molecules (23,51; also see section II.C and Figure 3).
Figure 11 shows the comparison between theory and
experiment. It was assumed that the transition from the
gel to the liquid crystalline state affects only the preexponential factor of the Arrhenius-activation law for the
defect motion. Another simplification was the neglect of
40% DPL in D2O
s
Figure 11. Frequency dependence of the longitudinal protonrelaxation time Ti of a dipalmitoyl lecithin (DPL) aqueous
dispersion (143). Phase was a gel at 35°C, a liquid crystal at
60°C. Solid lines indicate calculations for the limited defectdiffusion model (equation 15 and Figure 3) with r^Tb = 227.
The pre-exponential factor of the Arrhenius activation law for
defect-diffusion times was assumed to decrease by a factor
0.35 at the gel-liquid crystal transition. The dashed line represents TJTO =113 and a ratio 0.25 of the pre-exponential factors. Arrows indicate the positions wTd = 1. Order fluctuations
are believed to cause the decay at low frequencies in the liquid
crystalline phase.
.-2
10
10"
104
105
10 e
-i—i
i i n i l .
10'
v/Hz
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Bulletin of Magnetic Resonance
the so-called flexibility gradient along the alkyl chains of
the lipids (37,143). An attempt has been made to take
this gradient into account (146), using the numerical
Markov chain method (23) and assuming that the diffusion step rate varies along the alkyl chains while the
defect concentration is constant. The result is that the
dispersion of the non-selective proton-relaxation times
is almost the same as expected in the case of constant
defect-diffusion step rates.
B. Cells and Tissues
The relaxation times of cell and tissue water is remarkably sensitive to alterations of the cell state. For
example, it has been found that water relaxation times
are affected by mitosis (147), tissue malignancy
(148,149), and ischemic damage (149,150). Even the
conversion of the spin state of a single protein (cytochrome P-450 complexes) influences water relaxation
times (151). The relevant relaxation mechanisms are
therefore of considerable interest.
Unfortunately, cells and tissues are extremely heterogeneous materials. Apart from the problems mentioned in the previous sections in connection with protein solutions and membranes, one has to distinguish
between intra- and extracellular water. Several, somewhat controversial papers deal especially with heterogeneity in the water phase and the exchange mechanisms (117,152-158). In general, one can state that the
cell proteins are the centers responsible for the total
relaxation behavior.
Reported 7Vdispersion data for erythrocytes and
whole blood (101,103) confirm this statement. These
sources also concluded that encapsulation of hemoglobin within cells has little effect on its molecular motion, while oxygenated sickle cell hemoglobin forms aggregates even in cell-free solutions, as indicated by a
more hindered molecular tumbling.
Held et al (159) have measured the r, dispersion of
muscle water over four decades. The fact that no final
low-frequency plateau could be observed in the avarlable frequency range can be explained by the large protein aggregates within muscle cells.
Fung (160) has also investigated the frequency dependence of Tf for protons and deuterons in muscle water.
An important result is that (as in solutions of globular
proteins) isotopic dilution of cell water by D2O has almost no effect on the proton relaxation rates. This is a
clear indication that the relaxation sinks are not located
in the water phase. Probably, a relaxation scheme
analogous to that shown in Figure 10 is relevant (101).
The magnetization transfer from water protons to protein protons can also be deduced from the fact that
7"1(2H)/ Ti(nO) in muscle water is close to that in pure
water, while f ^ H ) / F ^ O ) is 2.1 times greater in pure
water than in muscle water (153). Magnetization transfer
requires dipolar or scalar interaction between the participating nuclei as a dominant relaxation mechanism. 2H
and 17O, however, are quadrupole nuclei relaxed mainly
by the interaction with electric field gradients (Table 1).
C. Liquid Crystals
7"rdispersion measurements by the field-cycling
technique play a crucial role in the physics of liquid
crystals. As molecular dynamics in liquid crystals are
anisotropic, differences in the fluctuation rates around
the different axes of the molecules are expected. Order
fluctuations of the local director are of special interest
with respect to liquid crystalline properties (161-163).
The problem is experimental distinction between the
diverse components of molecular motion.
As order fluctuations and rotational or translational
diffusion (45,165) of the molecules are relevant at different frequencies, it may be possible to use the frequency
dependence of 7", to distinguish between the various
components of motion. Experimental investigations not
using the field-cycling technique have been described
(164-166). The relaxation behavior of liquid crystalline
solutions of a polypeptide (poly-7-benzyl-L-glutamate)
has been reported (101,167), but the frequency range of
the conventional technique used was not adequate for a
decisive analysis of all components of molecular motion.
Noack and co-workers performed field-cycling measurements of p-azoxyanisole and p-methoxybenzylidene-p-butylaniline (168,169) and interpreted their results at low frequencies (about 104 Hz) according to the
theory of order fluctuations (161). For the 7"-, dispersion
at higher frequencies, they assumed isotropic translational diffusion (see equations 11a, b) and anisotropic
rotational diffusion (19). In their "fast exchange" formula
(ref. 169, equation 1a), the contributions of different proton phases are mixed with contributions of different
components of motion in the same proton phase. The
latter can occur for motions separately affecting intraand intermolecular interactions, i.e., different proton
pairs. For a particular proton pair, however, different
components of molecular motion must be combined on
the level of the correlation function. Otherwise, the result
is valid only under the condition that the correlation
times are sufficiently different from each other. The
problem can be partially circumvented by studying
separately the intra- and intermolecular contributions
and the motion of definite groups with specifically
deuterated compounds. Thus, the ring protons in pazoxyanisole have been investigated at several frequencies with the use of partially and perdeuterated
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Vol. 1.NO.4
211
molecules (166). From this, one may conclude that fieldcycling data specified in this sense would help to separate the different components so that a final picture
could be obtained of molecular motions in liquid
crystals.
A useful parameter for the distinction between different types of molecular motion is the angle between the
director and the external magnetic field (see section
III.A.2). By this means, the anisotropy of the longitudinal
relaxation time 7", has been investigated experimentally
(170) and theoretically (45).
Field-cycling data were reported for the liquidcrystalline phase of aqueous dispersions of dipalmitoyllecithin, (see Figure 11 and ref. 145). Though the
molecular structure of this substance is characterized by
long fatty-acid residues and differs therefore from the
class of substances usually referred to in connection
with liquid crystals, the ^-dispersion was quite similar
to that found for p-methoxybenzylidene-p-butylaniline
(169). This fact indicates that the low-frequency dispersion is again influenced by order fluctuations. The upperfrequency range was assumed to be dominated by
structural defects diffusing up and down the alkyl chains
as described in section IV.A.2. The frequency dependence of 7^ caused by this process contradicts the "extreme narrowing" case (2) occasionally assumed for this
frequency-temperature range (171).
D. Synthetic Oligomers and Polymers
1. Solutions and Melts
The problems investigated with synthetic polymer
solutions are similar but less complicated than with protein solutions (section IV.A. 1). Again, questions such as
exchange kinetics and interactions in the solvation shell
(172-174) and the mobility and conformationai changes
of the macromolecules themselves (175-180) are of
interest. The latter is also the central subject studied
with polymer melts (181,182). It is clear that special emphasis must be laid on the anisotropy of molecular
dynamics in these systems.
Both solutions and melts of polyethylene oxide have
been studied with the field-cycling technique (183). It
was shown that melts and solutions underlie different
types of processes of molecular motion. These are
revealed at low frequencies where a pronounced dependence of the relaxation times on the chain length was observed. The data were interpreted in terms of the translational diffusion model for intermolecular interaction
(equation 11). However, such an interpretation is doubtful because intermolecular interaction is of minor importance compared with the intrachain contributions and
has no influence at low frequencies. This was demonstrated for polyethylene melts by dissolving a few un-
deuterated chains in a perdeuterated matrix (184,185)
so that the intermolecular interaction was suppressed.
As a result, the relaxation can be understood by the
same processes as in the undiluted case.
A different interpretation has been given with the
three-component model of molecular dynamics in
polymer melts (186,187). The components are anisotropic segment reorientation by defect rearrangements,
longitudinal chain diffusion due to defect diffusion, and
the conformationai fluctuation of the environmental tube
incorporating the reference chain. The latter can be interpreted as a consequence of chain diffusion, so that all
components are deduced from elementary processes
accessible to thermal activation. The model successfully
describes the molecular weight and frequency dependences of the relaxation times (Figure 12) and also verifies predictions concerning the behavior of mixtures of
polymers with different chain lengths (184,185). The
relaxation formula used in these papers is an extension
of equation (16) from the longitudinal chain diffusion
process alone to the case where all three components
are relevant. The extended correlation function has also
been used to calculate the shape of the free induction
decays according to equation (8) (188).
For sufficiently high molecular weights and low
frequencies, the contribution of the longitudinal chain
diffusion process dominates, and the simpler equation
(15) can be used. This has been done to derive information about the correlation length of the environmental
tube surrounding the reference chain (37). For this purpose, data of the self-diffusion coefficient have been
combined with the characteristic time constant for the
longitudinal chain-diffusion. A similar treatment has
been given for polyethylene (185). The close relationship
of nuclear magnetic relaxation in polymer melts with
self-diffusion has also been demonstrated by a computer simulation of the model (48).
From a different point of view, namely by considering
conformationai jumps of a chain described in a tetrahedral lattice, Valeur et al (189) obtained the same expression given in equation (15a) (also see ref. 190). However,
the parameters introduced in these calculations have a
different meaning. As no interchain effects are included,
these models as well as Ullman's treatment, based on
the Zimm-Rouse theory, (179,180) are relevant for
polymers in solutions. Unfortunately, no comparison
with field-cycling data has been reported so far. Such a
comparison would be a valuable test of the models,
especially for long chain lengths.
2. Solids
Polymers and oligomers in the solid state are either
crystalline, partially crystalline, or amorphous. The common feature of the relaxation processes in all three
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212
Bulletin of Magnetic Resonance
T,
sec
M=1OOO
Figure 12. Frequency dependence of
the longitudinal proton-relaxation time
7", of polyethylene oxide melts at 70 °C.
M is the molecular weight. The data
(183) closely fit the solid lines derived
from the three-component model for
polymer melts (186,187).
M=4OOO
»"'•
0=7OC
M=27OOO
1010'
io7
_J_
rad/sec
states is that they are local; the chain length has no direct influence. The anisotropy of molecular dynamics is
again of central interest. Models assuming any kind of
"molecular tumbling" are questionable in this light. Instead, local segment reorientation, assumed to result in
part from the diffusion of chain defects (for example, gtg
sequences or torsions), is the proper starting point for
discussing these problems, as was realized in early
papers (191-195).
The development of an improved field-cycling technique for solids with short relaxation times (78,95) allowed an experimental test of the defect-diffusion
models discussed in section II.C. The 7"i dispersion of
partially crystalline polyethylene has been investigated
in the ranges 104 to 108 Hz and 77 to 373 K (51,78,196).
Apart from end-group rotation, which becomes visible at
lower molecular weights, two discrete chain processes
attributable to the amorphous regions were detected.
The homogeneity of the amorphous regions with respect
to nuclear magnetic relaxation was demonstrated by an
isotopical dilution experiment in which spin diffusion is
interrupted. Taking advantage of the relaxation enhancement by (paramagnetic) oxygen, which is dissolved only in the amorphous regions even at rather high
pressures, it was shown that the amorphous parts virtually behave as a homogeneous phase (51). This means
that a single correlation function holds for the amorphous methylene groups and that there is no distribution
of correlation functions (section II.D).
The assumption that methyl-group rotation is initiated
by local density fluctuations as a consequence of
vacancy diffusion led to the suggestion that threedimensional diffusion of defects with restoring capability
(equation 14), or one-dimensional diffusion of defects
without restoring capability (equation 12), might be
relevant for the kinetics of this process. In fact, these expressions describe the T-, dispersion at liquid-nitrogen
temperature where methyl-group rotation is dominant,
at least for short chain lengths. Alternatively, a ColeDavidson distribution of correlation times can be assumed (51,78). In this case, the dilution experiment described above could not be carried out because no appropriate samples were available.
At about - 50° C, the so-called y process dominates
the T-, dispersion. Assuming diffusing kinks (gtg
sequences) reflected at each other, this process could
be explained by equation (15) for the limited onedimensional defect-diffusion case (196). At higher
temperatures (about 30° C), the so-called /? process
finally becomes relevant. This process was also thought
to originate from limited one-dimensional defect diffusion, but, in the presence of a second type of defect, was
presumed to diffuse independently between the crystallites which terminate at both sides of the amorphous
parts of the chain. Extended chain torsions may be an
appropriate example of such a defect (196).
Field cycling is suitable for studying the physical nature of the defects in addition to their kinetic properties.
As outlined in section III.A.2, the anisotropy of nuclear
magnetic relaxation, apparently enhanced at low
frequencies, is a potential tool for investigating the
structural rearrangements which cause the relaxation,
^-anisotropy investigations found only a small effect,
probably because the experimental frequency (30 MHz)
was too high (197). Our group has performed the first
experiments (unpublished) at frequencies down to 104
Hz on polyethylene and polymethylmethacrylate with the
aim of distinguishing among the various models presented in Figure 13. Though the broad distribution of the
chain orientations has not yet allowed us to make a final
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Vol. 1,No. 4
213
1.7
T,(P) •
Ti(0)
2.4
oscillatton
rotation
2.2
1.4
2.01.3
1.8
1.2
1.6
1.1
1.4
0.25
0.25
1.0 •
1.2Q9
1.00.8
20
40
60
80 P
20
1.7
40
60
80 P
1.7-,
1
T,(3)i -6
Tito)
1.5
T,(B)
jumping kink
1,0-
r
T,(P) 0.9
T,(0)
jumping kink
rotation
1.4
1.4
0,8-
1.3
1.3-
0,7-
1.2
1.2
0,6-
1.1
1.1
0,5
1.0
1.0-
0,4-
0.9
0.9
0,3-
0.8
0.8
0,2-
0
20
40
60
80
0
20
40
60
80 P
flip-flop
20
40
60
80 B
Figure 13. Orientation dependence of the anisotropy quotient T-,(fi)/ 7,(0) for different kinetic models, (a) Molecular motion in alkane chains (see Figure 7 caption). Solid lines represent the intramolecular contribution; small, dotted segments indicate combined intra- and crystalline intermolecular contributions, (b) 180° flip-flop motions in an alkane crystal. Data include combined
intra- and intermolecular contributions in a crystalline arrangement. Curves are labeled with C2 values. (67)
decision, the superposition of diffusing kinks and torsions does seem to be compatible with the T, anisotropy
of polyethylene at room temperature (67). In polymethylmethacrylate, methyl-group rotation is a dominant
process even at room temperature (198). A 7, anisotropy of a factor 2 was observed below 105 Hz (199).
Crystalline paraffins have been studied by the fieldcycling technique (95). A more-or-less hindered endgroup rotation was characterized. The methylene group
dynamics in paraffins is of special interest because, in
contrast to polyethylene, it occurs here in a pure crystalline phase. Certain mobility changes appear at the socalled rotator transition, a solid state phase transition
below the melting point. Unfortunately, no information
was given about the history of the samples.
Samples crystallized from the melt do not necessarily
behave exactly like samples crystallized from solution
(146). It is still not clear whether the processes observed
in the crystalline phase originate from a homogeneously
relaxing system or from local perturbations, for example, at the inner grain surfaces characteristic of meltcrystallized material. The former was assumed by the
authors in their use of a defect-diffusion model to describe the r, dispersion. The Hunt and Powles expression (47) has also been applied, but, as mentioned in
section II.C, the assumptions for this model are somewhat inconsistent. Reconsideration of modified defectdiffusion models is desirable.
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214
Bulletin of Magnetic Resonance
I
E. Non-macromolecular Systems with
Electron-paramagnetic Centers
The presence of paramagnetic centers often causes
dominating enhancement of the nuclear magnetic relaxation rates as a consequence of dipolar and scalar interactions with these centers (Table 1). The investigation of
this frequency-dependent enhancement allows conclusions concerning the properties and dynamics of these
interactions. Examples for such centers are paramagnetic ions (12), free radicals (200), and excitons (201). A
short review of applications in connection with enzymes
was given in section IV.A.1. In the following, we consider
particularly systems of small molecules suitable for
studying the basic interactions.
Manganese ions in solution are an example of a
system in which scalar interaction is revealed by a lowfrequency step in the 7", dispersion. In addition to the
coupling constant, the exchange time in the solvation
sphere and/or the electronic relaxation time can be investigated. The latter is possible for scalar interaction, in
contrast to dipolar coupling, because the correlation
time for scalar interaction depends solely on the relaxation time of the unpaired electron and the exchange time
of the solvation shell. If the electronic relaxation time is
long enough, the exchange time can even dominate the
scalar correlation time (11,12).
Field cycling has been applied to the study of manganese solutions (88,89,93). The transverse relaxation time
r2 has also been investigated with this technique, using a
Carr-Purcell-Gill-Meiboom pulse sequence (ref. 93 and
Figure 14). The use of different solvents in these studies
modified the parameters of the solvation complex.
An example in which scalar interaction plays no role is
solutions of Gd 3 + (202). The exchange behavior of the
solvation sphere is then not generally deducible from
r r dispersion data, i.e., from the height of the dispersion
step at WSTC = 1, where cos is the Larmor frequency of the
unpaired electron and r c is the effective correlation time
of the dipolar interaction. This step height is reduced in
the so-called hindered exchange case TM <« T 1M , where
TM is the mean residence time of a solvent proton and
T 1M is the longitudinal proton relaxation time in the first
solvation sphere (203,204). The same effect can also be
derived from an assumption of a field-dependent correlation time, i.e., electronic relaxation times short
enough to be comparable with the rotational correlation
time of the solvation complex. An extensive discussion
of this ambiguity is given in ref. 202.
In the last few years, nitroxyl radicals have been of
special interest because they are used to spin label
macromolecules (133). Such radicals have been studied
in various solvents with the field-cycling technique
(205,206). Scalar interaction is again present in these
10
-
10
10
-
Figure 14. Frequency dependence of the molar protonrelaxation rates cT^ -1(0) and cT2~ \H) of a solution of Mn(CI04)2
in dimethylsulfoxide (DMSO) at various temperatures. The
contribution of bulk-phase DMSO has been subtracted. (93,
reprinted by permission).
systems. A complete set of structural and kinetic parameters of the solvation complexes was derived.
Excitons in molecular crystals are a quite different
type of paramagnetic center. Naphthalene and anthracene crystals irradiated with light have been investigated
(201,207,208). Because the non-paramagnetic contribution to the relaxation rate is rather weak, the effective
relaxation time is given solely by the exciton-proton
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Vol. 1,No. 4
215
interaction. In these investigations, the relaxation times
were so long that the field was varied by manual upand-down switching. Information concerning exciton migration via a hopping mechanism could thus be derived.
We finish this review with this application of the fieldcycling technique to a system which could also be considered a subject of solid state physics. Further applications of solid state physics are considered beyond the
present scope. It must be emphasized, however, that important studies with this method have been performed in
that field too. Applications to superconductors could
even be considered as crucial experiments for Cooper
pairing and other phenomena (209-211). Here one exploits the possibility of detecting NMR signals in the normal state by field cycling across the critical field
strength.
V. FINAL REMARKS
The aim of this review article was to outline the
capabilities of the field-cycling technique in nuclear
magnetic relaxation spectroscopy. At the present state
of the art, the method allows the investigation of the
frequency dependence of relaxation times of liquids and
solids in a broad range (103 to 108 Hz for protons) but
without resolution into individual resonance lines. Several applications are reported which have clarified ambiguities left by conventional relaxation studies. On the
other hand, studies would be more detailed if the experiments could be carried out under high-resolution conditions. Whether this will ever be possible is doubtful
because of extreme technical difficulties. The most
problematic requirement for such experiments is the
generation of homogeneous and stable detection fields
within a few milliseconds. Thus, it cannot be expected
that the resolution of the method will be greatly improved in the near future. Rather, the sensitivity will increase so that the technique will be applicable to
chemically or isotopically labeled substances, i.e., to low
concentrations of the resonant nuclei.
The field-cycling technique is most frequently applied
to the study of molecular dynamics. Non-NMR methods,
for example, quasi-elastic light or neutron scattering and
dielectric relaxation, are available for the detection or
derivation of correlation and intensity functions. These
methods can have advantages, such as lower cost or
more desirable frequency ranges. The main advantages
of the NMR field-cycling technique are its applicability to
almost all kinds of systems and a straightforward connection between theory and experiment worked out for
many models. Field cycling can thus be considered a
favorable compromise with respect to frequency bandwidth and broad range of applications.
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