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

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PRICE
In terms of the concentration in number of
particles per unit volume, cŽ r, t . , the flux of a
particle is given by Fick’s first law of diffusion to
be for example, see Ref. Ž 2, 62 .,
Ž . Ž .
Jr,t Dc r,t . 18
Ž .
The minus sign indicates that in isotropic media
the direction of flow is from larger to smaller
concentration. Because of the conservation of
mass, the continuity theorem applies, and thus,
Ž .
c r,t
t
Ž .
Jr,t . 19
In other words, Eq. 19 states that cŽ r, t. t is
the difference between the influx and efflux from
the point located at r. Combining Eqs. 18 and
19 we arrive at Fick’s second law of diffusion
e.g., Refs. Ž 4, 51, 62 .,
Ž .
c r,t
t
2 Ž .
D c r,t 20
So far in our mathematical descriptions of
diffusion, we have, perhaps simplistically, assumed
that the diffusion process is isotropic and
can therefore be described by the isotropic diffusion
coefficient D Ž i.e., a scalar . . More generally
the diffusion process is represented by a cartesian,
or rank two, tensor Ž i.e., a 3 3 matrix.
Ž 51 . , D ŽD
where and take each of the
Cartesian directions . ; thus, written more generally,
Eq. 18 can be written as
which is shorthand for
Ž . Ž .
Jr,t Dc r,t , 21
Ž .
c x,t
D D D
x
JŽ x,t. xx xy xz
cŽ y,t.
JŽ y,t. Dyx DyyDyz .
y
JŽ z,t. Dzx Dzy Dzz
cŽ z,t.
z
22
We note that the diagonal elements of D, Že.g.,
. scale concentration gradients and fluxes
in the same direction, the off-diagonal elements
Ž e.g., . couple fluxes and concentration gra-
dients in orthogonal directions, and similarly, Eq.
20 becomes
Ž .
c r,t
t
Ž .
Dc r,t . 23
For simplicity in most of what follows, we are
concerned only with isotropic diffusion. However,
in the section on anisotropic diffusion, we will
consider in detail the significance of anisotropic
diffusion in PFG diffusion measurements.
In the case of self-diffusion, there is no net
concentration gradient, and instead we are concerned
with the total probability, PŽ r , t. 1 of finding
a particle at position r1 at time t. This is given
by
H
PŽ r ,t. Ž r . PŽ r ,r ,t. dr 24 1 0 0 1 0
where Ž r . is the particle density Ž
0
the formal
definition of the particle density is considered in
detail below . , and thus, Ž r . PŽ r , r , t. 0 0 1 is the
probability of starting from r 0 and moving to r1 in
time t. The integration over r 0 accounts for all
possible starting positions. Similar to concentration,
PŽ r , t. 1 describes the probability of finding
a particle in a certain place at a certain time.
PŽ r ,t. 1 is a sort of ensemble-averaged probability
concentration for a single particle, and it is thus
reasonable to assume that it obeys the diffusion
equation Ž 41 . . Because the spatial derivatives in
Fick’s laws refer to r 1,
we can rewrite Fick’s laws
in terms of PŽ r , r , t. with the initial condition,
0 1
Ž . Ž .
P r ,r ,0 r r 25
0 1 1 0
Žn.b., in Eq. 25 is the Dirac delta function, not
the length of the gradient pulse . . Thus, if in Eq.
18 PŽ r , r , t. is substituted for cŽ r, t .
0 1
, J becomes
the conditional probability flux. Similarly,
in terms of PŽ r , r , t. Eq. 20 becomes
0 1
PŽ r ,r ,t.
0 1 2 D PŽ r ,r ,t . . 26 0 1
t
In the case of anisotropic diffusion, Eq. 26 can
be changed similarly to Eq. 23 . PŽ r , r , t. 0 1 is
commonly termed the Green’s function or diffusion
propagator Ž 55, 63 . .