Heitjans P., Karger J. (Eds.). Diffusion in Condensed Matter: Methods, Materials, Models

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10 PFG NMR Studies of Anomalous Diﬀusion 435

the assumption that molecular diﬀusion proceeds by stochastic jumps be-

tween adjacent adsorption sites of separation λ with a mean time τ between

subsequent jump attempts. Jump attempts are only successful if the sites

where the jumps are directed to are unoccupied. The occupation probability

is denoted by θ. It could be shown by MD simulations [49] that the validity of

(10.27) and (10.28) is not conﬁned to the case of activated jumps but holds

quite generally if the quantity θ is understood as a suitably deﬁned pore-

ﬁlling factor. Moreover, it could be shown [50] that as in the case of normal

diﬀusion the propagator is given by a Gaussian, albeit with a mean square

displacement increasing in proportion to the square root of the observation

time (cf. (10.27)) rather than with the observation time itself (cf. (10.8)).

By use of the PFG NMR technique, zeolites with one-dimensional chan-

nels served as the ﬁrst model systems where the validity of (10.27) could

be demonstrated experimentally [51–54]. As an example, Fig. 10.7 shows the

molecular mean square displacement of CF

in AlPO

-5 at 180 K as a func-

tion of the observation time for diﬀerent concentrations. At least for the

larger concentrations, the validity of (10.27) could be conﬁrmed over up to

two orders of magnitude of the observation time. The displacement z(t)ofa

molecule in a single-ﬁle system may be shown to be correlated to the displace-

ment s(t) of the same molecule without any neighbours (i.e. of an isolated

molecule) by the equation [55, 56]

z

(t) = l |s(t)| , (10.29)

where l denotes the mean free distance (clearance) between adjacent mole-

cules in the single-ﬁle system.

Table 10.1 summarizes the values for F taken from Fig. 10.7, together

with the values for the mean free distance l calculated from the sorbate

Fig. 10.7. Molecular mean square displacement of CF

in AlPO

-5 at 180 K as a

function of the observation time at sorbate concentrations of 0.005 (3), 0.05 (

),

0.2 (2), and 0.4 () molecules per unit cell, respectively [54].

436 J¨org K¨arger and Frank Stallmach

Table 10.1. Results of the PFG NMR self-diﬀusion studies with CF

adsorbed in

AlPO

-5 at 180 K, represented in terms of the single-ﬁle mobility factor F and the

limiting diﬀusivity. The errors of the experimental values are about 50%.

clFD

(mol./unit cell) (nm) (m

−1/2

)(m

−1

)

0.4 1.63 0.7 · 10

−12

0.6 · 10

−6

0.2 3.73 1.5 · 10

−12

0.5 · 10

−6

0.05 16.3 4.5 · 10

−12

0.3 · 10

−6

0.005 167 1.0 · 10

−10

1.1 · 10

−6

concentration c and the diﬀusivities D, which an isolated molecule in the

channel would have. The latter quantity follows from (10.29) by using the

expression

|s(t)| =

√

t, (10.30)

which may be easily derived from the propagator for normal diﬀusion (cf.

(10.6) and (10.8)). The estimated single-particle diﬀusivities are found to

be by two orders of magnitude larger than the largest diﬀusivities so far

observed in zeolitic adsorbate-adsorbent systems [3]. In contrast to the three-

dimensional pore network of “ordinary” zeolites, in one-dimensional channels

the molecular momentum is preserved over much longer time intervals, so

that the large values for the single-particle diﬀusivities are not unexpected.

The direct measurement of such diﬀusivities by PFG NMR is excluded due

to the extremely low concentration, which the diﬀusants should have in such

experiments.

For practical applications, one has to take account of the ﬁnite size of the

zeolite crystallites of single-ﬁle type. While under non-equilibrium conditions

– i.e. during molecular adsorption or desorption – ﬁnite single-ﬁle systems do

not exhibit any peculiarities in comparison with the case of ordinary diﬀusion

[48,57,58], tracer exchange and chemical reactions are found to be controlled

by the single-ﬁle conﬁnement. Since any arbitrarily chosen pair of molecules

in single-ﬁle systems cannot be completely uncorrelated in its movement the

boundary conditions relevant for ﬁnite single-ﬁle systems aﬀect the transport

behaviour of all individual diﬀusants in the single-ﬁle system. The inﬂuence

of the boundary conditions on single-ﬁle diﬀusion is far from being solved,

and recent theoretical studies [59–63] try to handle the diﬃculties arising

from these complications.

As a consequence of the ﬁnite extension of real single-ﬁle systems, parti-

cle displacements within the ﬁle are subjected to a second mechanism, which

is caused by the particle exchange between the marginal sites and the sur-

10 PFG NMR Studies of Anomalous Diﬀusion 437

rounding atmosphere. Since subsequent events of molecular adsorption and

desorption on the marginal sites may occur independently from each other,

the associated shifts of the particles in the ﬁle are also independent from

each other. The corresponding molecular displacements are therefore subject

to normal rather than to anomalous diﬀusion. As a consequence, with in-

creasing observation time molecular displacements are very soon dominated

by this second mechanism. The ratio between the corresponding diﬀusivity

and the diﬀusivity of a sole (i.e. isolated) molecule in the ﬁle is of the order of

the reciprocal value of the site number [60, 64]. Molecular exchange between

single-ﬁle systems and the surrounding atmosphere is therefore dramatically

slowed down in comparison with adsorbents undergoing normal diﬀusion.

In [65] this mechanism has been identiﬁed as the key process for ensuring

reactivity enhancement by “molecular traﬃc control” [66]. Reactivity en-

hancement by “molecular traﬃc control” has been postulated to occur if the

reactant and product molecules enter and leave pore networks along diﬀerent

diﬀusion paths. The possibility of substantial diﬀerences in the accessibility

of diﬀerent ranges of the intracrystalline pore system by the diﬀerent com-

ponents in multi-component adsorbate-adsorbent systems has been recently

demonstrated by MD simulations [67]. If the involved molecules are accom-

modated by diﬀerent channels, reactant molecules will enter and product

molecules will leave the catalysts along separated channels, i.e. without be-

ing inhibited by the presence of the other component. Microscopically, this

situation appears in the existence of overall concentration gradients along

both channel types, viz. into the catalyst particle for the reactant and out

of the catalyst particle for the product molecules. From the above considera-

tions it is well known that under such conditions, i.e. under the existence of

macroscopic concentration gradients, there is no additional transport inhibi-

tion due to the single-ﬁle eﬀects. In the conventional case, i.e. for coinciding

accommodation probabilities of the two components, however, the overall

concentration is constant over the total channel network, so that molecular

displacements within the system are subject to the additional transport in-

hibition of single-ﬁle conﬁnement [68]. Obviously, reactivity enhancement by

molecular traﬃc control is based on the suppression of this inhibition mecha-

nism by caring for diﬀerent accommodation probabilities of the reactant and

product molecules in the two channel systems [69, 70].

10.4.6 Diﬀusion in Ordered Mesoporous Materials

The propagation rate of guest molecules in porous materials depends strongly

on the pore structure of the host system. As a consequence, measurement

of intraparticle diﬀusion is able to provide information about structural fea-

tures, which are not easily accessible by conventional techniques of structural

analysis such as scattering and diﬀraction methods. This is in particular true

for structural eﬀects such as pore blockages and leakages in pore walls. As

an example, investigations of this type have substantially contributed to an

438 J¨org K¨arger and Frank Stallmach

Fig. 10.8. Dependence of the parallel (2) and perpendicular ()componentsof

the axisymmetrical self-diﬀusion tensor on the inverse temperature for water in

MCM-41 as measured at 10 ms observation time with PFG NMR. The dotted lines

maybeusedasaguidefortheeyes.Forcomparison,thefulllinerepresentsthe

self-diﬀusion coeﬃcients of super-cooled bulk liquid water [74].

improvement of the understanding of the real structure of mesoporous ma-

terials of the MCM-41 type. In comparison with zeolites, diﬀusion measure-

ments with this type of material are complicated by the polydispersity of

the sample. As a consequence of the irregular shape of the adsorbent parti-

cles, which is quite common for mesoporous materials, analysis of transient

adsorption/desorption measurements is not free from some ambiguity. Be-

ing sensitive to molecular shifts within the individual adsorbent particles,

PFG NMR (like QENS, cf. Chaps. 3 and 13) oﬀers better possibilities for the

measurement of intraparticle diﬀusivities. Some of these measurements, how-

ever, have been complicated by the fact that during the observation time of

the PFG NMR experiment a substantial fraction of the adsorbate molecules

leaves the individual adsorbent particles [71–73].

In [74] this complication has been circumvented by monitoring the wa-

ter diﬀusivity in MCM-41 samples overloaded with water at temperatures

below 0

◦

C. In this way it is possible to ﬁnd experimental conditions under

which the phase of intraparticle water is still mobile, while the ice formed

in the space between the particles deﬁnitely excludes any exchange of wa-

ter molecules between diﬀerent adsorbent particles. The signal attenuation

observed in PFG NMR is found to be in excellent agreement with the theo-

retical ﬁt based on the assumption that intraparticle diﬀusion is described by

an axisymmetrical diﬀusion tensor. Fig. 10.8 shows the main elements of the

diﬀusion tensor in comparison with the diﬀusivities of free water. The larger

component, which represents the diﬀusivity in the direction of the axis of

symmetry, has obviously to be attributed to diﬀusion in the direction of the

10 PFG NMR Studies of Anomalous Diﬀusion 439

1.3

1.0

0.7

∆ / s

0.01

0.1

-2

-1

(∆)> / µm

Fig. 10.9. Dependence of the parallel (2, ) and perpendicular (◦, •)components

of the mean square displacement on the observation time for water in two MCM-

41 samples at 263 K. The mean square displacements were calculated via (10.8)

from the components of the axisymmetrical self-diﬀusion tensor. The horizontal

lines indicate the limiting values for the axial (full lines) and radial (dotted lines)

components of the mean square displacements for restricted diﬀusion in cylindrical

rods. The lengths l and diameters d of the rods are written in micrometers on

the lines. The oblique lines, which are plotted for short observation times only,

represent the calculated time dependences of the mean square displacements for

unrestricted (free) diﬀusion with D

par

=1.0 × 10

−10

−1

(full line) and D

perp

2.0 × 10

−12

−1

(dotted line), respectively [74].

channels of the MCM-41 structure. Fig. 10.8 allows the following conclusions:

(i) Diﬀusion along the channels is slower than in the free liquid by about one

order of magnitude, it is obviously inhibited by partial channel blockages.

(ii) Though being much smaller than the diﬀusivity in axis direction, there is

also a ﬁnite probability of molecular propagation perpendicular to the chan-

nel axes. As the most obvious explanation, the channel walls should therefore

be considered as being somewhat permeable to the water molecules.

In the PFG NMR studies, molecular conﬁnement within the intraparticle

space was used as an independent check of the data compatibility (Fig. 10.9).

With increasing observation times the diﬀusion path lengths along the main

axes of the diﬀusion tensor were found to be in satisfactory agreement with

the particle dimensions.

10.5 Anomalous Diﬀusion by External Conﬁnement

It has been explained in Sect. 10.2 that anomalous diﬀusion following the

time dependence of (10.4) with a time exponent κ<1 is most likely to

440 J¨org K¨arger and Frank Stallmach

be observed under the conﬁnement by self-similar structures, where the dis-

placements must be in the space interval between the lower and upper cut-oﬀ

of self-similarity. In the regular pore-network of zeolites, anomalous diﬀu-

sion due to this type of conﬁnement can be excluded if the pore network is

not additionally structurized, e.g. by coke depositions [39]. Good prospects

for the observation of anomalous diﬀusion should be provided, however, by

amorphous porous materials. While in ﬁrst detailed PFG NMR studies with

amorphous carbonaceous adsorbents molecular displacements were found to

follow ordinary diﬀusion over orders of magnitude [39], polymeric matrices

turned out to be most eﬀective host systems for anomalous diﬀusion. Two

examples of molecular diﬀusion in such systems will be considered. In the

third subsection, the beneﬁt of restrictions for elucidating structural details

of the samples under study shall be illustrated.

10.5.1 Restricted Diﬀusion in Polystyrene Matrices

Polystyrene (PS) matrices with inclusions of polydimethylsiloxane (PDMS)

droplets turned out to be a useful material for the production of self-

lubricating components [75]. PFG NMR measurements with these systems

may serve as a model for dynamic imaging. As an example, Fig. 10.10 shows

the spin-echo attenuation of the PDMS as a function of the observation time,

with the gradient intensity (in terms of the generalized scattering vector

q = γδg) as a parameter. It turns out that with increasing observation time

the spin-echo attenuation approaches a ﬁnite value Ψ

∞

, corresponding to

the EISF of QENS (cf. (10.19)). From the q-dependence of the spin-echo at-

tenuation for each individual observation time, via (10.18) and (10.9), one

may deduce both the eﬀective diﬀusivity and the mean square displacement.

Fig. 10.11 shows the resulting time dependence of the eﬀective diﬀusivity.

It turns out that for suﬃciently small observation times the eﬀective dif-

fusivity approaches the value of the free PDMS (average molecular weight

28 000 g/mol), while with increasing observation time the eﬀective diﬀusivi-

ties approach the reciprocal proportionality required by (10.21). The situation

is now the same as for the small crystallites at the lowest temperature con-

sidered in Fig. 10.1 where molecular propagation as observed by PFG NMR

was controlled by the crystal size as well as for water diﬀusion in MCM-41

(cf. Sect. 10.4.6), where in Fig. 10.9 for the largest observation times molecu-

lar displacements were controlled by the geometrical shape of the adsorbent

particles rather than by their intrinsic diﬀusivity. The mean droplet radii

obtained in this way were found to be in excellent agreement with the micro-

scopic measurements [76]. In contrast to these measurements, however, PFG

NMR permits a non-destructive determination of the droplet sizes where the

external conditions such as pressure and/or temperature may easily be varied.

The described experiments revealed that PDMS diﬀuses out of the cavities

into the PS matrix at elevated temperatures (the cavity radius decreases).

This process was entirely reversible.

10 PFG NMR Studies of Anomalous Diﬀusion 441

Fig. 10.10. Spin-echo attenuation of PDMS conﬁned to cavities of mean diameter

1.1 µm in a polystyrene matrix in dependence on the observation time [76].

q/m

−1

=1.13 · 10

(), 1.71 · 10

(), 2.27 · 10

(2)and3.41 · 10

().

10.5.2 Diﬀusion in Porous Polypropylene Membranes

The total conﬁnement of molecular propagation considered in Sect. 10.5.1

clearly excludes the possibility of anomalous diﬀusion. For this purpose,

molecular propagation has to proceed in a network of interconnected pores,

with small pores being the replica of larger ones. Commercially available

polypropylene membranes of type “Accurel” [77, 78] proved to be an excel-

lent host matrix for such studies. As an example, Fig. 10.12 shows the time

dependence of the eﬀective diﬀusivities of PDMS (22.5kg/mol) in this matrix.

In contrast to Fig. 10.11, the time regime exhibiting a constant diﬀusivity is

442 J¨org K¨arger and Frank Stallmach

Fig. 10.11. D

eﬀ

in dependence on the observation time t for PDMS conﬁned

to cavities with a mean diameter of 0.6 µm. The diﬀusivity of the bulk PDMS is

indicated as the dashed line [76].

Fig. 10.12. Eﬀective diﬀusivities of PDMS (22.5kg/mol) in a polypropylene host

matrix at pore ﬁlling factors as indicated [78].

now followed by a less pronounced decay, indicating that molecular prop-

agation – though being progressively hindered with increasing observation

time – is not completely restricted. It is interesting to note that the slope

of the representation, i.e. the eﬀectiveness of the hindrance with increasing

observation time, increases with decreasing pore-ﬁlling factors. This corre-

10 PFG NMR Studies of Anomalous Diﬀusion 443

Fig. 10.13. Eﬀective diﬀusivity of PDMS (22.5kg/mol) in a polypropylene host

matrix at a pore ﬁlling factor of 100 % at 293 K as a function of the root mean

square displacement of PDMS during the NMR experiment. The ﬁlled circles refer

to eﬀective diﬀusivities extrapolated from measurements at 343 K to room temper-

ature by applying the well-established temperature-time shift principle of polymer

dynamics [78].

sponds to the fact that the inﬂuence of the matrix on molecular propagation

is the more pronounced, the larger the relative amount of molecules that

are in immediate contact with the matrix. On the basis of (10.4) and (10.9)

or (10.10), from the representation of Fig. 10.12 the time exponents κ are

found to be κ =0.83, 0.72 and 0.55 in the sequence of decreasing pore-ﬁlling

factors. From the representation of the eﬀective diﬀusivities as a function of

the displacement (Fig. 10.13), it becomes obvious that anomalous diﬀusion as

reﬂected by a dependence of the diﬀusivity on the diﬀusion time and hence

on the diﬀusion path length occurs for displacements between about 100 and

600 nm. It is interesting to note that these lower and upper cut-oﬀs are of the

order of the diameters of the smallest and largest pores of the host matrix

which has a nominal pore width of 200 nm [77]. The reader will easily ﬁnd

that transfering Fig. 10.13 into a r

(t) –vs.–t plot leads to the pattern of

Fig. 16.6 in Chap. 16, with the slopes for short and large times given by the

large and small diﬀusivities (i.e. the “short-range“ and “long-range“ diﬀusivi-

ties) of Fig. 10.13. In the present context, the ratio between these diﬀusivities

is an exclusive function of the architecture of the pore space. It is referred to

as the tortuosity factor (cf. Sect. 17.5.1 of Chap. 17).

444 J¨org K¨arger and Frank Stallmach

10.5.3 Tracing Surface-to-Volume Ratios

As an example, Fig. 10.4 has illustrated the diﬀerent patterns of time de-

pendence of the molecular mean square displacements in zeolitic adsorbate-

adsorbent systems, which result as a function of the considered space scales

and temperature ranges. For the smallest temperatures, the adsorbate mol-

ecules were found to be essentially conﬁned to the intracrystalline space.

This can be attributed to the fact that the thermal energy of the molecules

is far too small to permit molecular escape into the intercrystalline space

with its correspondingly higher potential energy. Molecular propagation in

the vicinity of the surface clearly bears features of anomalous diﬀusion since

subsequent displacements are predominantly oriented in opposite directions.

Since under the given conditions molecular passage through the surface is

essentially excluded, displacements towards the surface are most likely fol-

lowed by reversed displacements. The resulting conﬁnement in molecular dis-

placements clearly aﬀects their mean values and is therefore reﬂected in the

PFG NMR diﬀusivities as their direct measure. Fitting the PFG NMR spin

echo attenuation under these conditions to (10.18) yields an eﬀective, time-

dependent diﬀusivity D(∆). In the limiting case of suﬃciently small obser-

vation times ∆, D(∆) may be expanded as [79, 80]

D(∆)

=1−

√

(

∆ + R(

(

∆). (10.31)

where D

denotes the unperturbed diﬀusivity and S

= S/V

stands for

the surface-to-volume ratio of the conﬁning pore geometry, i.e., the ratio be-

tween the surface area (S) of the conﬁning wall and the total (pore) volume

) in which the diﬀusants are contained. The value R(

√

∆) accounts for

higher-order terms in

√

∆, which become increasingly important with in-

creasing observation time. (10.31) may be rationalized by conceiving that the

fraction of diﬀusants, which are subjected to conﬁnement eﬀects, is propor-

tional to the ratio between the total surface area of conﬁnement multiplied

by the mean diﬀusion path during the observation time (S

√

∆)andthe

total volume occupied by the diﬀusants (V

),whereitisassumedthatthe

diﬀusant density over this volume is constant. This dependence is reﬂected

by the second term on the right-hand side of (10.31), which represents the

ﬁrst-order inﬂuence of the conﬁnement on the eﬀective diﬀusivity D(∆) with

respect to the unperturbed value D

. It has to be kept in mind that this

ﬁrst-order correction is only justiﬁed over a rather modest range of eﬀective

diﬀusivities (D

≥ D(∆) ≥ 0.85D

). In this range, the plot of D(∆)/D

ver-

sus

√

∆ yields a straight line with a slope determined by the S/V

ratio

(cf. (10.31)). With further increasing observation times (viz. as soon as the

root mean square displacements in the unconﬁned space get into the order of

the curvature radii of the conﬁning areas), the inﬂuence of the higher-order

term in

√

∆ on the right-hand side becomes dominating and there is no