Pulsed-field gradient nuclear magnetic resonance as a tool for ...

310
PRICE
tion is sufficiently long so that we still get a
measurable signal and that the measurement is
unaffected by eddy currents or other experimental
complications see, for example, ref. Ž 19 ..
However, in the case of a species diffusing within
a confined space we must be careful to properly
account for the effects of the restricting geometry
on the motion of the species. If a particle is
diffusing within a restricted geometry Žsometimes
referred to as a ‘‘pore’’ . , the displacement along
the z-axis will be a function of , the diffusion
coefficient, and the size and shape of the restricting
geometry. Consequently, if the boundary effects
are not properly accounted for and we analyze
the data using the model for free diffusion
Ž see the next section . , we will measure an apparent
diffusion coefficient Ž D . app and not the true
diffusion coefficient. We illustrate this effect later
in this section.
Before further considering the problem of restricted
diffusion, it is appropriate to briefly consider
what constitutes the true diffusion coefficient.
In a pure liquid Ž e.g., water. the true diffusion
coefficient corresponds to the bulk diffusion
coefficient. However, the situation is rather more
complex in a macromolecular solution Že.g.,
cell
cytoplasm, polymer solutions, protein solutions,
etc. . where the probe molecule Ž e.g., water. has to
skirt around the larger ‘‘obstructing’’ molecules
Ž e.g., proteins, organelles. as well as perhaps interacting
with protein hydration shells e.g.,
Ref.
Ž 67 ..
These effects operate on a time scale much
smaller than the smallest experimentally available
and consequently are well averaged on the time
scale of . For example, if we consider a reasonably
small value of of 5 ms and that at 298 K
water has a diffusion coefficient of about 2.3
9 2 1 Ž .
10 m s 68, 69 , then from Eq. 30 the mean
displacement of a water molecule during is
about 5 m. The true diffusion coefficient will be
an average bulk diffusion coefficient consisting of
all of the interactions that affect the probe
molecule diffusion. The situation can be further
complicated by the effects of exchange through
cell membranes Ž 70 . . The different time scales of
the averaging processes is one of the major reasons
that diffusion probed by using relaxation
studies and diffusion measured using PFG NMR
are essentially different things with the relaxation
based measurements probing motion on the time
scale of the correlation time of the probe molecule
and not on the Ž much longer. time scale of
Ž 21, 71 . . We mention in passing that in a polymer
solution if the diffusion of the polymer itself is
studied there can be additional complications owing
to the entanglement of the polymer molecules
e.g., Ref Ž 19. and references therein .
We now explain the concept of restricted diffusion
and how it relates to PFG NMR diffusion
measurements. Consider two cases where we have
a particle with the same diffusion coefficient; in
one case the particle is freely diffusing Ži.e.,
an
isotropic homogeneous system . , while in the other
case it is confined to a reflecting sphere of radius
R Ž Fig. 4 . . By ‘‘reflecting’’ we mean that the spin
is neither transported through the boundary nor
relaxed by the contact with the boundary. From
Eq. 30 ,
we can define the dimensionless variable
Ž i.e., n 1, t . ,
2
DR , 34
which is useful in characterizing restricted diffusion
as will be seen below. In the case of freely
diffusing particles, the diffusion coefficient determined
will be independent of and the displacement
measured in the z-direction will reflect the
true diffusion coefficient, since the mean-squared
displacement scales linearly with time Žsee
Eq.
33 . . However, for the particle confined to the
sphere, the situation is entirely different. For
short values of such that the diffusing particle
has not diffused far enough to feel the effect of
the boundary Ž i.e., 1 . , the measured diffusion
coefficient will be the same as that observed
for the freely diffusing species. As becomes
finite Ž i.e., 1 . , a certain fraction of the particles
Ži.e.,
in a real NMR experiment there is an
ensemble of diffusing species. will feel the effects
of the boundary and the mean squared displacement
along the z-axis will not scale linearly with
; thus, the measured diffusion coefficient Ži.e.,
D . will appear to be Ž observation. app
time depen-
dent. At very long , the maximum distance that
the confined particle can travel is limited by the
boundaries, and thus the measured mean-squared
displacement and diffusion coefficient becomes
independent of . Thus, for short values of the
measured displacement of a particle in a restricting
geometry observed via the signal attenuation
in the PFG experiment is sensitive to the diffusion
of the particle. At long the signal attenuation
becomes sensitive to the shape and dimensions
of the restricting geometry. The relationships
between the experimental parameters are
further examined in the next section. If the restricting
geometry is spherically symmetric, then
there will be no-orientational dependence with