E L S E V I E R

29 December 1995

Chemical Physics Letters 247 (1995) 596-600

CHEMICAL PHYSICS LETTERS

Evidence for single file diffusion of ethane in the molecular sieve A1PO4-5

Vishwas Gupta, Sriram S. Nivarthi, Alon V. McCormick, H . Ted Davis *

Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Avenue S.E., Minneapolis, MN 55455, USA

Received 14 August 1995; in final form 11 October 1995

Abstract

We measure the diffusion of ethane in very large crystals of the molecular sieve AIPO4-5 (provided by M. Davis) using pulsed field gradient (PFG) NMR. Thanks to the large crystal size and the long longitudinal relaxation time, the mean square displacement could be monitored over times that arranged over an order of magnitude. The mean square displacement was proportional to the square root of time, providing strong evidence that the ethane molecules are moving in a single file, i.e. they are unable to pass each other. This is the first direct experimental evidence for single-file diffusive motion - a phenomenon that is probably widely prevalent in both industrial catalytic and biological systems.

1. Introduct ion

Fluids confined in nanopores can display diffusive properties that are very different from those of bulk fluids [1,2]. Diffusion in such restricted geometries is of interest because it is frequently encountered in applications such as shape selective catalysis or sep- arations using zeolites. The molecular sieving effect exhibited by nanoporous zeolites is primari ly due to the molecular dimensions of the pores in these mate- rials ( 3 - 1 2 ,~) [3,4]. Diffusion in such pores can result in situations where traditional representations of mass transport break down and it is necessary to postulate alternate definitions for diffusivity.

Single file diffusion refers to motion in one di- mension where the diffusing molecules are unable to pass each other [5-9]. Such motion is different from

* Corresponding author.

unidirectional diffusion in which the molecules move in one dimension but can pass each other (see Fig. 1). An important dist inguishing feature of single-file motion is that the mean square displacement of the molecules, ( z2( t )~ , scales as the square root of the observation time [5-9] ,

( z2 ( t ) ) = 2o~t °'5, (1)

where t denotes the time for diffusion and c~ is the single file analog diffusivity. In contrast, merely unidirectional diffusion scales linearly with observa- tion time obeying Einstein 's equation,

( z2( t ) ) = 2 D t , (2)

where D is the self-diffusion coefficient. The time dependent of the mean square displace-

ment for single-file and unidirectional diffusion (Eqs. (1) and (2)) differs due to the different correlat ion of d i sp lacements from success ive jumps . When molecules are able to pass one another, the displace-

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V. Gupta et al. / Chemical Physics Letters 247 (1995) 596-600 597

(a) Unidirectional tion is responsible for the low sorbate diffusivities measured in zeolite theta [13]. Unexpectedly low diffusivities have been measured by Kiirger et al. [14] for methane and CF 4 in VPI-5, leading to the speculation that single-file motion may be responsi- ble.

Passing allowed

(b) Single File

O No Passing

Fig. 1. Schematic depiction of (a) unidirectional diffusion con- trasted with (b) single-file diffusion. In the case of unidirectional diffusion, the diffusing molecule is small enough to allow passing, whereas in single-file diffusion, passing is not possible. It is analogous to a string of pearls where any pearl may be shifted only if the neighboring ones are also shifted to leave the needed space for displacements. The molecular sieve used in this study (A1PO~-5) can be visualized as a bundle of parallel cylinders (diameter ~ 7.3 ,~) without interconnections. Large crystals of A1PO4-5 are essential in the experiment to eliminate contribution from gas phase diffusion and these were kindly provided to us by Professor M. Davis (Caltech).

ments from successive jumps are uncorrelated; this leads to Eq. (2). When molecules are not able to pass each other, displacements from successive jumps are negatively correlated since a displaced molecule is now more likely to return to its original position than to move further. This negative correlation becomes more stringent as the displacement increases, hence, the mean square displacement increases more slowly at longer times. A similar scaling law at short times for monomer diffusion in a free Rouse chain has been predicted by de Gennes [10] and has been supported by the simulations of Kremer and Grest [11]. Neutron reflection measurements of interface broadening at a hydrogenated and a dehydrogenated polymer interface showed, for times short compared to the repetition time, the mean square broadening going as the square root of time [12]. Moreover, Shen and Rees have speculated that single-file mo-

2. Experimental

In this Letter we present experimental evidence for the single-file diffusion of ethane in the molecu- lar sieve AIPO4-5. AIPQ-5 has straight channels with a diameter of 7.3 A, a repeat unit of 8.5 ,~ along the channel axis, and no interconnections [15]. The A1PO4-5 crystals used in this study were colum- nar hexagonal with an average length of 30 p~m [16]. The channels in A1POn-5 run along the length of these crystals. Following the procedure described elsewhere [16], a loading of =0 .3 ethane per A1PO4-5 repeat unit of 8.5 .~ was obtained. We use the stimulated spin echo pulsed field gradient (PFG) NMR experiment [17-20] to directly measure the mean square displacement of ethane in these long, straight channels.

For the case of diffusion confined to one direc- tion, Callaghan et al. have obtained the following expression for the NMR echo attenuation, A [21]

A = f 0 ] e x p - 7 2 G 2 6 2 x 2 dx, (3)

where G is the gradient magnitude, 6 is the duration of the gradient, 7 is the gyromegnetic ratio for the nucleus and A is the observation time. The integral over the parameter x arises to take account of the random orientation of the crystallites. The expression in Eq. (3) leads to a nonlinear plot of the log of echo attenuation versus 7 2 G2~ 2/2. The distinction between unidirectional (with passing) and single-file diffusion can be made by observing the scaling of the mean square displacement with time.

In contrast, for isotropic diffusion, the echo atten- uation follows [18]

A = exp( - 7 2 G 2 ~ 2 ( r 2 (

6 ) ' (4)

~ ) )

l

598 V. Gupta et al. / Chemical Physics Letters 247 (1995) 596-600

resulting in a liner plot of In A versus 3~2G282/2. Thus, comparison of Eqs. (3) and (4) shows the

unique ability of NMR in detecting the signature of unidirectional diffusion even in a sample of ran- domly oriented crystallites.

3. Results and discussion

Figs. 2a and 2b show the semi-log plots of echo attenuation versus the parameter 72G2~2/2 for two typical observation time used. Also shown in each figure is the least square best fit of Eqs. (3) and (4)

1

0.9

0.8 1

< 0.7

0.6

0.5 (a) , r L , J ~ ~ = 0 1 2 3 4 5 6 7

(-~G262/2)* 10, (p.m) -2

1 0 . 9 ~ A = 169 ms, G = 3.0 Tim

0.70"8 [ (b) ~ = < 0.6

0.5

0.4 0 1 2 3 4

(y2G2~2/2)* 10, (p.m) -2

Fig. 2. NMR spin echo attenuation (A) produced by ethane in A1PO4-5 plotted as a function of the parameter A2GZ82/2 for typical observations times (A): (a) 21.6 ms and (b) 169 ms. Also shown on the plots are the least square best fits to the data using Eqs. (3) and (4) for unidirectional and isotropic diffusion. The mean square displacement is the only adjustable parameter. The ~H spin-spin relaxation time (T:) for ethane was determined to be 2.9+0.2 ms, while the spin-lattice relaxation time (T I) was determined to be 467+1 ms. Approximately 1000 scans were acquired with a delay of 3 s. All data were obtained on a NT-300 FT spectrometer equipped with a field gradient probe from Doty scientific.

12

6

4 v

2

0 - ' ' ' ' I i i i i I I i i , I , , i i

0 50 100 150 200 A, ms

Fig. 3. Ethane mean square displacements in AIPO4-5 plotted as a function of observation time for diffusion (A). Linear and power law best fits to the data are also shown. The linear fit is made to pass through the origin. The error bars are based on the residual of the fits to data of the type shown Fig. 2.

for unidirectional and isotropic diffusion. Clearly the spin echo results indicate the diffusion to be unidi- rectional and not isotropic.

The mean square displacements (the only ad- justable parameter) evaluated from data of the type shown in Fig. 2 (using more observation times) are plotted in Fig. 3 as a function of observation time, A, along with the power law and linear fits to the data. The power law fit describes the data very well in comparison to a linear fit. The exponent deter- mined from the power law fit, 0.48, is in good agreement with the theoretical factor of 0.5 for sin- gle-file diffusion. Thus, the ethane molecules un- dergo unidirectional and single-file motion in A1PO4-5. The maximum ethane rms displacement in these measurements, 3.2 Ixm, is still an order of magnitude smaller than the average length of the A1PO4-5 crystals (30 Ixm); this confirms that we are indeed measuring the diffusion only within the crys- tallites without significant interference from gas phase diffusion [16,22]. Note that it is only with these very large crystals, only recently available from Mark Davis and co-workers, that we have this opportunity to unambiguously detect single-file dif- fusion.

The parameter a which characterizes the single file phenomenon as given in Eq. (1) is found from the data of Fig. 3 to be 1.42 × 10- ] l m 2 s-0.5. The parameter c~ can also be evaluated independently

V. Gupta et al. / Chemical Physics Letters 247 (1995) 596-600 599

using two alternative approaches, one based on a lattice hopping model [5,7] and the other on the dynamics of a one dimensional hard-rod system [6],

1 - 0 0 ( 2 ~ r ) - ' / 2 (5 ) O~lattic e = 12

and

O~hard-rod 1 -- po- 1/2

( O i d / ' r r ) , (6) P

where l is the jump length, r the jump time, 0 the fractional occupancy of sites, Dia the infinite dilu- tion diffusivity, o- the hard rod length, and p the one dimensional density.

The value of a determined experimentally is larger than the estimate based on the lattice model evaluated as follows: potential energy profiles of ethane in a unit cell of a static AIPO4-5 lattice reveal two minima separated by 4.2 ,~, ( = l) with an activa- tion energy of 3.8 k J / m o l [23]. For an estimate of z, we use Eyr ing ' s activated state theory [24],

k T z - ' = -~ -exp( - E a / / R T ) . (7 )

Using this value in the lattice model (Eq. (5)), with 0 = 0.15 (based on two sites per unit cell), we calculate a to be 4 × 10 -J3 m 2 s - °5 . The experi- mentally measured value is also greater than an estimate of a from the hard-rod model (Eq. (6)). Using Did = 3.2 X 10 -7 m 2 s - l (the infinite dilution diffusivity of ethane in A1PO4-5 from molecular dynamics simulations carried out by Keffer et al. [23]), ~ r = 4 . 4 ,~ [25] we get a = 7 . 4 × 10 -13 m 2

s -° 's , which is comparable to the lattice model pre- diction. The underest imation of ct by either lattice model or hard rod theory as compared to experiment may be partly because, as shown by Demontis and Suffritti [26], zeolite lattice vibrations can increase sorbate mobil i ty in these channels. On using the experimental ly determined value of a to extract the infinite dilution diffusivity from Eq. (6), we obtain a value of 10 -4 m 2 s -1, which is approximately 300 times higher than the result from Keffer et al. [23].

It is interesting to compare ethane to methane diffusion in A1PO4-5. The minimum cross-sectional diameters for methane and ethane are the same (3.8 ,~ [3]). However, while we previously [16] observed methane to be undergoing unidirectional diffusion

(with a self-diffusion coefficient of 2.9 × 10 -9 m 2 s - 1 ), no evidence for single-file behavior was found. Here though we find the diffusion of ethane in AIPO4-5 to be unidirectional and single-file. It is clear that when considering a passing event in the pore, we have to account for the fact that the ad- sorbed molecules may be rotating and translating at the same time. Therefore a more appropriate dimen- sion may be closer to an average diameter such as the kinetic (or coll ision) diameter derived from the gas phase properties of the sorbates [25]. The kinetic diameter of ethane (4.4 ,~,) is larger than that of methane (3.8 ,~,) [25], leading to the possibil i ty that passing would be more difficult for ethane as com- pared to methane. For an observation time of = 100 ms, the methane rms displacement, = 24 Ixm, is an order of magnitude grea ter than the value, 3.2 Ixm, that we measure for ethane. It is striking how sensi- tive the onset of single-file diffusion is to molecular size. Our results strongly support the speculation of Caro and co-workers [27], that the reason for the drastic reduction of alkylation yield in AIPO4-5 as compared to VPI-5 is that single-file motion is im- posed in the former but not in the latter.

A c k n o w l e d g e m e n t

We thank Professor Mark Davis (Caltech) for kindly providing the large A1PO4-5 crystals that were essential for this study. This work was supported by the National Science Foundation. VG was supported by a fel lowship from the Chevron Company.

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