Letter pubs.acs.org/JPCL
Xenon in L‑Alanyl‑L‑Valine Nanochannels: A Highly Ideal Molecular Single-File System Muslim Dvoyashkin,† Aiping Wang,‡ Sergey Vasenkov,*,‡ and Clifford R. Bowers*,† †
Department of Chemistry, University of Florida, Gainesville, Florida 32611, United States Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611, United States
‡
ABSTRACT: Xenon-129 pulsed field gradient NMR and hyperpolarized xenon129 spin tracer exchange NMR experiments were performed on polycrystalline LAla-L-Val (AV) and L-Val-L-Ala (VA) nanochannels under identical conditions of temperature and Xe pressure. Displacements of up to 3 μm were measured in specimens with mean crystallite lengths of around 50−100 μm. The combination of the two NMR techniques yielded the most definitive evidence for molecular single file diffusion to date, in which the time-scaling of the mean squared displacement is proportional to the square-root of time. The PFG-NMR echo attenuation data yield single-file mobility factors of 6 ± 0.7 × 10−13 m2 s−1/2 and 4.4 ± 0.2 × 10−13 m2 s−1/2. The results establish Xe in AV nanochannels, in particular, as an ideal experimental model system for fundamental studies of single-file diffusion dynamics. SECTION: Physical Processes in Nanomaterials and Nanostructures displacements are too large to be consistent with SFD in a finite single-file system with blocked boundaries. An alternate interpretation of the data in terms of normal diffusion in curvilinear channels has been debated.27,28 Evidence for singlefile diffusion has also been obtained in several types of molecular crystals using hyperpolarized (HP) Xe-129 NMR.22−24 However, in some cases this evidence relied on multiparameter fitting, and up to now no PFG-NMR measurements have been reported for any of these materials. Here we employ two very different NMR techniques to characterize Xe diffusion dynamics in the channels of the dipeptide L-Ala-L-Val (AV) and its retroanalogue L-Val-L-Ala (VA): (1) stimulated echo PFG-NMR, which detects intrachannel diffusion, and (2) HP spin tracer exchange (HSTE) NMR, which probes displacements associated with molecular exchange at the channel openings. In both AV and VA, helical channels are formed by hydrogen bonded assembly of the dipeptides into a spiral in hexagonal crystals (P61 space group).29−31 The winding spacing in AV and VA is nearly the same (ca. c = 10 Å). The radius of the channel center from the 61 axis is 0.60 and 1.05 Å, respectively.31 The Xe atom, with a diameter σXe = 0.44 nm, satisfies the single-file criterion in AV and VA, where the mean channel diameters are 0.51 and 0.49 nm, respectively.31,32 The single-file nature of Xe in AV or VA is evidenced by the loading dependence of the 129Xe chemical shift tensors.23,32,33 Scanning electron microscopy on our AV and VA specimens, as received from MP Biomedicals, revealed whisker-shaped crystals with a distribution of lengths ranging from roughly 10−100 μm. At room temperature, the 129Xe spectra at 3 bar exhibit two well-
D
iffusive dynamics in single-file channel-particle systems has been the subject of numerous experimental studies over the past few decades and is among the key topics of modern transport theory.1−12 Universal characteristics of diffusion in single-file channel-particle systems of all length scales include conservation of the sequential order of the particles and suppressed transport rate due to steric interactions. Instead of the normal proportionality of the mean squared displacement (MSD) to the diffusion time (i.e., ⟨z2⟩ = 2Dt) that is observed for Fickian diffusion (FD), the MSD in single-file diffusion (SFD) increases as the square-root of time:13−15 ⟨z 2⟩ = 2Ft 1/2
(1)
The single-file mobility F is a function of the fractional occupancy θ and temperature. While eq 1 has been validated in various macroscopic single-file channel-particle contrivances,16−19 observations of molecular diffusion in single-file systems are rare,20−25 and in some cases controversial.20,21,25−28 The first reported observation of molecular SFD utilized pulsed field gradient (PFG) NMR to measure the MSD of CH4 and CF4 loaded into several types of zeolites with unidimensional channel structure.20,21,26 Inconsistent observations on nominally the same materials were attributed to the occurrence of high defect densities and imperfections at the channel openings, allowing mutual particle passages to occur. Quasielastic neutron scattering (QENS) and the zero length column (ZLC) tracer exchange were also applied to study CH4 diffusion in AlPO4-5 channels. These techniques, which operate on very different length-scales than does PFG-NMR, revealed FD for observation times in the 10−13−10−9 s and 0.1−1 s time windows, respectively. PFG-NMR echo attenuation data indicative of SFD of H2O in single-wall carbon nanotubes was recently reported,25 but as noted in ref 27, the measured © 2013 American Chemical Society
Received: August 6, 2013 Accepted: September 12, 2013 Published: September 12, 2013 3263
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be neglected. At longer times, exchange at the channel openings can lead to a crossover from SFD to the Fickian center-of-mass (CM) diffusion regime.3,4 The MSD at the crossover can be estimated analytically in the simple random-walk model. Assuming simple interparticle exclusion interactions, free exchange at both file boundaries, uniform channel length l, and elementary displacement λ, the crossover MSD for tagged particles initially located at the center of channels is estimated to be3,4
resolved peaks: a bulk gas phase signal near 0 ppm and an adsorbed phase signal centered at chemical shifts of 140 and 156 ppm in AV and VA (referenced to Xe gas extrapolated to zero pressure34). Under these experimental conditions, the adsorbed phase 129Xe NMR peaks in both polycrystalline dipeptide specimens exhibit a symmetric line shape with a full width at half-maximum (fwhm) of ≈7 ppm. This wide separation of the adsorbed and free gas peaks facilitates selective excitation and detection of Xe atoms located within the channels. Using the adsorption isotherm data given in ref 31, the fractional loadings are estimated to be θ = 0.35 and θ = 0.7 in AV and VA, respectively. Adsorption and desorption of Xe at channel openings can be observed as distinct processes by HP 129Xe 2D exchange spectroscopy,35 proving that a substantial fraction of the channels are at least singly openended. Advantageously, the longitudinal nuclear spin relaxation time T1c of 129Xe adsorbed in AV or VA nanochannels is relatively long, making it possible to follow Xe displacements over correspondingly long time-scales. Using thermally polarized saturation-recovery NMR, we measured 36 s in AV and 87 s in VA at 300 K. These T1c measurements are not significantly affected by exchange because, as will be shown below, Xe displacements on the T1 time-scale are short compared to typical crystallite dimensions. In addition, the ability to hyperpolarize 129Xe gas by spin exchange optical pumping (SEOP) provides signal enhancements of around 104, facilitating recording of HSTE-NMR signals for repolarization times down to 10 ms. Stimulated echo PFG-NMR is an established method for measuring self-diffusion in porous media.36 We performed PFG-NMR experiments in both AV and VA in a 17.6 T Bruker BioSpin NMR spectrometer operating at 208.6 MHz. NMR sample tubes containing about 0.1 g of AV and VA were attached to a vacuum manifold. Following overnight evacuation at 373 K, the 5 mm tubes were pressurized to 3 bar with Xe gas and then flame-sealed. As in ref 37, magnetic field gradients of up to 30 T/m were applied using a Diff60 diffusion probe and Great60 gradient amplifier (Bruker BioSpin). This combination of high static field and high field gradients provided sufficiently high PFG-NMR sensitivity to carry out measurements of slow Xe translational dynamics in AV and VA. Under certain conditions, discussed below, the PFG-NMR echo attenuation due to one-dimensional diffusion in straight channels is expected to follow20 ψ (q , t ) =
1 × 2
+1
∫−1
dr exp[−q2⟨z 2(t )⟩r 2/2]
⟨z 2(tc)⟩ = (2λl /π )(1 − θ )/θ
For other initial locations, deviations from the t time-scaling will occur at shorter diffusion times, depending on proximity to the opening and boundary conditions. For the helical channels in AV and VA, the MSD and elementary displacement in eq 4 is taken to be that of the displacement distribution along the curvilinear channel path. Our 129Xe spin echo attenuations for AV and VA are plotted in Figure 1 as a function of q2t1/2 for a series of diffusion times
(2) Figure 1. Stimulated echo PFG-NMR diffusion attenuation data for 129 Xe at 297 K in (a) AV and (b) VA at the observation times indicated in the legends. Best fits of data to eq 3 are shown with dashed curves. (c) MSDs extracted from panels a and b. Solid and dashed lines represent least-squares fits for VA and AV. Dotted line represents FD (∼ t).
where q = γgδ, γ is the gyromagnetic ratio, δ denotes the duration of the applied gradient pulse, g is the gradient amplitude, and ⟨z2⟩ is the MSD along the channel axis at the diffusion time t (i.e., the z-axis). For helical topology and observation times such that the displacement distribution spans many windings of the helix, the MSD in eq 2 is interpreted as the variance of the displacement distribution with respect to the helix axis. In the case of SFD, eq 2 becomes ψSFD(q , t ) =
(4) 1/2
between 15 ms and 12 s. The data in Figure 1a,b conform very well to eq 3 at all diffusion times studied. Figure 1c presents the fitting results on a double log scale. The dependence is welldescribed by straight lines with slopes of 0.47 ± 0.05 and 0.49 ± 0.02 for AV and VA, and intercepts yielding mobility factors F of 6 ± 0.7 × 10−13 m2 s−1/2 and 4.4 ± 0.2 × 10−13 m2 s−1/2 for AV and VA, respectively, with respect to the channel helix axis. The single-file mobility along the channel path is estimated as follows. In both AV and VA, full loading corresponds to a
π × erf[(q2F t )1/2 ] × [q2F t ]−1/2 2 (3)
Equation 3 is appropriate for short diffusion times when the displacement probability distribution is still relatively narrow compared to the channel length and all file boundary effects can 3264
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capacity of ∼3 Xe atoms per c-translation of the unit cell.32 Taking λ = σXe, the single-file mobility with respect to the channel path would be Fhelix ≈ (3σXe/c)2F = 1.7F. The selfdiffusivity D of an isolated Xe atom in the channel can be related to the corresponding single-file mobility in the same channel38 using D = πF2helix/l2Xe−Xe, where lXe−Xe = λ(1/θ − 1) is the mean free distance between adjacent Xe atoms in the single-file channel. We estimate D = 5 × 10−6 and 5 × 10−5 m2/ s for a single isolated Xe atom along the channel path in AV and VA, respectively. The relevance of CM diffusion to our experimental conditions can be addressed using eq 4. In 100 μm crystals spanned by unobstructed channels, the root MSD at the crossover is estimated to occur at ∼0.2 μm and ∼0.1 μm in AV and VA, respectively. These values are comparable to the minimum displacements and about a factor of 10 smaller than the maximum displacements probed by our PFG-NMR measurements. Thus, our PFG-NMR experiments are performed entirely in the regime where eq 4 predicts CM diffusion to dominate. Nevertheless, we detect no change in slope that might indicate a crossover to Fickian time-scaling (i.e., log⟨z2⟩ ∝ t) stemming either from mutual passages within the nanochannels or as a consequence of displacements correlated to diffusion of the CM of all particles within the channel. Displacements associated with Xe exchange at channel openings in AV and VA were probed using HSTE-NMR, a technique tantamount to continuous-flow HP Xe-129 selective saturation-recovery NMR.22,35 As described in ref 35, 15−20 mg of the polycrystalline AV and VA were packed loosely into a PEEK NMR sample holder and evacuated for more than 8 h at 100 °C. The samples were immersed in a continuous stream of HP 129Xe produced by Rb SEOP at a Xe flow rate of ∼100 cm3/min and Xe pressure of 3 bar. The 129Xe signal was recorded at a series of time delays following selective RF saturation of the 129Xe spin polarization in the nanochanneladsorbed Xe phase. Recovery of the adsorbed phase NMR signal occurs via diffusion of depolarized atoms to the channel openings where exchange occurs with HP Xe atoms in the bulk gas phase. Assuming (i) Langmuir adsorption dynamic equilibrium; (ii) sufficiently long channels such that the root MSD for diffusion on the T1c time-scale is short compared to the file length l, i.e., ⟨z(T1)2⟩ ≪ l2; and (iii) large excess of HP Xe in the surrounding gas phase, the HSTE-NMR tracer exchange functions for SFD and FD, normalized to the steady state adsorbed gas signal, are as follows:24,35,39
Figure 2. Hyperpolarized spin tracer exchange NMR data for 129Xe at 297 K in (a) AV and (b) VA. Dashed and solid curves show the analytical expressions for single-file (eq 5) and FD (eq 6), respectively. Blue symbols represent the gas phase signals and are associated with the scales on right-hand ordinate (shown in blue). Insets show an expanded view of the same data.
values. For VA, the data converge to pure SFD at long τ but are somewhat intermediate between SFD and FD at short τ, where the gas phase polarization is also reduced relative to the steadystate value. This could be indicative of a violation of assumption (iii) above; i.e. the polarization of the gas surrounding the channels is significantly reduced by desorption of unpolarized Xe. A crossover to FD of the center-of-mass in the HSTE-NMR data is not observed in either specimen. To summarize, the PFG and HSTE-NMR results presented in Figures 1 and 2 represent definitive experimental evidence of single-file diffusion spanning 3 orders of magnitude variation in the observation time. Since tracer exchange is not affected by curvature or helical topology of the channel, interpretation of the PFG-NMR echo attenuation data in terms of normal diffusion in curved unidimensional channels is completely ruled out. No indication of any transition to Fickian time-scaling of the MSD is detected for diffusion times in the range 15 ms −12 s. This could be due to one or more factors, including the existence of desorption barriers, channel blockages, or particle− particle interactions. Based on the similar channel geometries of AV and VA, one might expect the retroanalogs to exhibit similar self-diffusivities and equally good compliance to the SFD model, but this was not the case. The self-diffusivities differ by approximately an order of magnitude in AV and VA, reflecting substantial differences in Xe dynamics in the two systems. The extremely high values of the self-diffusivities could be taken as an indication of concerted motions or Xe clustering effects, which would be highly sensitive to channel-particle versus particle− particle interactions.40,41 Violation of assumption (iii) of the HTSE-NMR model resulting from higher Xe loading has been
Sc(τ ) Sc(τ → ∞) τ ⎧ −1 (Γ(1/4)T11/4 t −3/4e−t / T1c dt SFD c ) ⎪ ⎪ 0 =⎨ ⎪ (πT )−1/2 τ t −1/2e−t / T1c dt FD ⎪ 1c ⎩ 0
γNMR (τ ) =
∫
∫
(5,6)
As mentioned earlier, T1c intrinsic to the nanochannels was measured in separate thermally polarized NMR experiments. Therefore, the model is fully determined with no unknown fitting parameters. Figure 2 presents the measured HSTE-NMR data for AV and VA plotted together with both tracer exchange functions given in eqs 5 and 6. In AV, the data are in near perfect compliance to the SFD model (eq 5) over the entire range of experimental τ 3265
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suggested as a possible explanation for the deviations in the HTSE-data from the SFD model at short exchange times in VA. On the other hand, the experimental HSTE-NMR data in AV exhibits excellent agreement with the idealized random-walk model. The results of our study suggest that this material in particular is highly suitable for fundamental studies of molecular single-file diffusion dynamics.
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AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS PFG NMR data were obtained at the Advanced Magnetic Resonance Imaging and Spectroscopy (AMRIS) facility in the McKnight Brain Institute of the University of Florida. This material is based upon work supported by the National Science Foundation under Grant No. CHE-0957641. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
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