Hyperloop-like diffusion of long-chain molecules under confinement

The ultrafast transport of adsorbates in confined spaces is a goal pursued by scientists. However, diffusion will be generally slower in nano-channels, as confined spaces inhibit motion. Here we show that the movement of long-chain molecules increase with a decrease in pore size, indicating that confined spaces promote transport. Inspired by a hyperloop running on a railway, we established a superfast pathway for molecules in zeolites with nano-channels. Rapid diffusion is achieved when the long-chain molecules keep moving linearly, as well as when they run along the center of the channel, while this phenomenon do not exist for short-chain molecules. This hyperloop-like diffusion is unique for long-chain molecules in a confined space and is further verified by diffusion experiments. These results offer special insights into molecule diffusion under confinement, providing a reference for the selection of efficient catalysts with rapid transport in the industrial field.


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Based on van der Waals (vdW) interaction potential (Supplementary Fig. 1) analysis 1 2 , it shows the minimum value was 3.95 Å, as well as the potential function firstly decreases and then increases with the increase of the distance (r) between the oxygen atom of zeolite and the guest molecule.The interactions between subject and object is calculated by equation (1) 3 Corresponding areas of attraction (blue) and repulsion shown in Supplementary Fig. 1.
In our simulations, all the nano-channel models were composed of oxygen atoms.
Channel poresize were selected ranging from 7 to 12 Å, the lattice of each model were

Diffusion coefficient
In this work, the mean square displacement (MSD) of adsorbates is defined via where Nm represents the number of gas molecules, Nτ is the number of time origins used in calculating the average, and ri is the coordinate of the i-th molecule.In addition, the slope of the MSD as a function of time determines the self-diffusion coefficient (Ds) defined according to the Einstein relationship (equation In which, n is the dimension of frameworks (n = 1 for 1-D diffusion).The diffusion coefficients were calculated by fitting the linear region of MSD using a least-square fit.The Ds values were obtained as the average of three dependent MD trajectories.
The influence of active site did not take into account in this work, studies have shown that the acidity of zeolite contributes the diffusion of alkenes in zeolites, but makes little influence on alkanes 5 .Furthermore, The flexibility of zeolite framework is also considered, and the conclusion is consistent with that of rigid one (Supplementary Table 3).

Non-bonded interactions 1
In this paper, the guest−host interactions are described by pairwise-additive LJ (Lenard-Jones) 12−6 term, as following: Where r is the distance between two atoms, ε is LJ well depth, which reflects the strength of the interaction between two atoms, and σ is LJ diameter means the distance between atoms when the potential energy is equal to zero 6 .This is consistent with the parameter setting of force field above-mentioned (Fig. 3 a-h and Supplementary Fig. 1).

Density map
The density map (Supplementary Fig. 4, 9 and 10) is defined as Where a is the number of C atoms in each gird (the radial plane of the zeolite channel is divided into grids) during the whole MD simulation time, and b is the number of total C atoms present over the simulation time.
Interaction energy 7 The interaction energy (preferred adsorption sites and interaction energy barriers, Supplementary Fig. 12) is strongly correlated with the zeolite framework.Firstly, a C12 molecule was placed into the center of 1D channel in zeolite, then systematically moved from one end of the channel to the other following the diffusion path with 28 equi-spaced steps.The interaction energy between framework and molecule at each point was calculated, and the energy barrier for molecule diffusing in the center of 1D channel was determined by the difference between the lowest and highest energy along the diffusion pathway.All the interaction energy were calculated by the same force field and software of MD simulations.

Ab initio molecular dynamics (AIMD)
The periodic density functional theory (DFT) as well as the advanced AIMD simulation were performed using the CP2K 8-10 package.The TON, MTW and VFI frameworks were obtained from the IZA database 11 .The zeolite-dodecane (C12) complexes were cell optimized, and then, 30 ps AIMD simulations in NVT ensemble were carried out for zeolite-C12 complex at 298 K.A Nosé-Hoover 12 thermostat with a time constant of 100 fs and a time step of 0.5 fs was employed during the AIMD process.All the calculations were carried out using the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional 13 in conjunction with Grimme's D3 correction 14 with zero damping to account for dispersion interactions.The DZVP basis set and GTH pseudo potentials 15 were chosen for all elements.The plane wave cutoff energy and relative cutoff were 650 Ry and 60 Ry, respectively.
Uptake measurements implemented by infrared microscope (IRM) Using Eq. ( 6) to decouple the surface barriers from overall mass transport and calculate surface permeability 16 .
where mt/m∞, t , l and α is the relative uptake loading of guest molecules, the uptake time, and the half thickness of the plane sheet (i.e., characteristic length of the intracrystalline diffusion) and the surface permeability, respectively.
Based on the obtained surface permeability, using dual resistance model (DRM) 17 to fit uptake curves can calculate the intracrystalline diffusivity D, where D and L = αl/D is the intracrystalline (transport) diffusivity and the ratio of characteristic time of intracrystalline diffusion to that of surface barriers.

Reduced density gradient (RDG) analysis
Confinement effect in zeolite can be assessed by qualitative method, the scatter plot of the reduced density gradient (RDG) in real space is an effective and widely used tool to visualize non-covalent interaction between adsorbates and zeolite 18 .A random frame during the molecular dynamic simulation was selected to plot the RDG scatter as well as structures and dominant intermolecular interactions for the description of the host-guest interaction between the adsorbate and framework.The non-covalent interaction was performed in the region with low density and the RDG is defined as 18) ) ( together with the electron density ρ, which is used to distinguish the interaction (covalent and non-covalent).Hessian can be used to distinguish bonded (λ2 ˂ 0) from non-bonded (λ2 ˃ 0) interaction for the sign of the second largest eigen-value (λ2) of the electron density.This helps to distinguish different types of interactions ((λ2)ρ ˂ 0, strong intramolecular interaction; ((λ2)ρ ≈ 0, weak van der Waals (vdW) interaction; (λ2)ρ ˃ 0, strong repulsive interaction).To reveal the confinement effect more preciously, the inter-molecular interaction was adapted for the RDG analysis.The RDG function was calculated by Multiwfn 19 .

Distribution of deformation angle 20
The deformation angle of the molecule is calculated from the deviation of the angle along the (i-2)th, (i)th and (i+2)th carbon atoms from equilibrium angle 180°.
As is depicted in Supplementary Fig. 3, the deviation of the angle ∠1 is following equation (9), where ∠2 represents the variable angle for the (i-2)th, (i)th and (i+2)th carbon atoms.Then, the number of occurrences for a maximal ∠1 of each molecule at various temperatures is counted, which indicates the degree of molecular bending and reflects the flexibility of the molecule in the confined channel at different temperatures.