Nuclear Quantum Effects in the Layering and Diffusion of Hydrogen Isotopes in Carbon Nanotubes Piotr Kowalczyk*1, Artur P. Terzyk2, Piotr A. Gauden2, Sylwester Furmaniak2, Katsumi Kaneko3 and Thomas F. Miller III4 [1] School of Engineering and Information Technology, Murdoch University, Perth, Western Australia 6150 [2] Faculty of Chemistry, Physicochemistry of Carbon Materials Research Group, Nicolaus Copernicus University in Toruń, Gagarin Street 7, 87-100 Toruń, Poland [3] Center for Energy and Environmental Science, Shinshu University, Nagano 380-8553, Japan [4] Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA
Corresponding author footnote (*To whom correspondence should be addressed): Dr Piotr Kowalczyk Tel: +61 8 9360 6936 E-mail:
[email protected]
S1
Number of Pages: 9 Number of Figures: 2 Number of Tables: 1
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RPMD Simulations
We perform RPMD simulations of pure and binary H2, D2, and T2 mixtures confined in relatively wide carbon nanotube (2.72 nm pore diameter) at 25 K in NVT ensemble1-3. The total interactions are computed using4,5
U = U int + U ext
U int =
U ext =
(S1)
N
mP
P
∑∑ r 2(βh ) 2
(k ) i
(k +1)
− ri
(S2)
i =1 k =1
( )
( )
P 1 1 N P ϕ ff rij(k ) + ∑∑ ϕ sf ri (k ) ∑∑ P i < j k =1 P i =1 k =1
(S3)
k k where rijk = r i − r j , P = 32 is the number of ring polymer beads is imaginary-time
path-integral discretization, N is the number of hydrogen isotopes in the nanotube,
β = (k B T )−1 is the inverse of thermal energy, m denotes the mass of hydrogen isotope, and h = h / 2π is Planck’s constant. Owing to the cyclic condition of the ring polymer, if k = P , then k + 1 = 1 4. Interaction between hydrogen isotopes are computed from spherically symmetric Silvera-Goldman (SG) potential6:
C C6 C8 C10 + 8 + 10 f c (r ) + 99 f c (r ) 6 r r r r
ϕ ff (r ) = exp(α − δr − γr 2 ) −
(S4)
where the first term on the right-hand side (RHS) accounts for short-range repulsive interactions, the second set of terms on the RHS account for long-range attractive dispersion interactions, and the last term on the RHS is an effective three-body correction. The last two terms are multiplied by a damping function, which turns off these interactions at short distances and is given by6 f C (r ) = e − (rc / r −1) θ (rC − r ) + θ (rC − r ) 2
(S5)
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where θ (r ) is the Heaviside function. The parameters for the SG potential are given in Table S1. The interactions between hydrogen isotopes and an infinitely long, cylindrically symmetric carbon nanotube are computed using7
(
)
(
)
63 F − 4.5,−4.5,1.0; δ 2 F − 1.5,−1.5,1.0; δ 2 − 3 10 4 32 1− δ 2 ζ 1− δ 2 ζ
ϕ sf (r ) = π 2 ρ s ε sf σ sf2
[(
)]
[(
)]
(S6)
where F (α , β , γ ; χ ) is a hypergeometric function, R = 13.6 Å is the radius of the nanotube (this pore size corresponds to (20,20) armchair single-walled carbon nanotube), ρ s = 0.382 Å-2 is the density of carbon atoms in the nanotube wall, δ = r / R , and ζ = R / σ sf
7,8
. The parameters σ sf = 3.179 Å and ε sf / k B = 32 .056 K are
obtained via mixing rules from self-interaction parameters for carbon atoms and molecular hydrogen8. The initial configurations of pure and binary mixtures of hydrogen isotopes in carbon nanotube (29.58 Å long) are generated from path integral grand canonical Monte Carlo simulations9,10. The system consists of 30-250 hydrogen molecules, depending on the studied pore density. In RPMD, we apply periodic boundary conditions and minimum image convention for computing the interactions between hydrogen isotopes in the longitudinal direction11. We truncate all interactions between neighboring ring-polymers at centroid-to-centroid distance of 14.79 Å. We equilibrated the system for 50 ps, and then calculated ensemble averages, mean square displacement, ring-polymer radius of gyration, and ring-polymer centroid density for 10 ps (production time) by averaging over 1000 consecutive 4 ps ring polymer trajectories with a time step of 0.5 fs11. For selected densities, we performed 10 independent RPMD runs with longer production time of 25 ps. The averages computed for 10 and 25 s production runs are within statistical error of RPMD simulations (see Figs. 1-S1). The equations of motion were integrated using a symplectic integrator based on alternating free energy harmonic ring polymer and external force steps11,12. The temperature was controlled by resampling the ring polymer momenta from the Maxwell distribution at inverse temperature between each trajectory11,12.
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References
(1) Hata, K.; Futaba, D. N.; Mizuno, K.; Namai, T.; Yumura, M.; Iijima, S. WaterAssisted Highly Efficient Synthesis of Impurity-Free Single-Walled Carbon Nanotubes. Science 2004, 306, 1362-1364. (2) Furmaniak, S.; Wisniewski, M.; Werengowska-Ciecwierz, K.; Terzyk, A. P.; Hata, K.; Gauden, P. A.; Kowalczyk, P.; Szybowicz, M. Water at Curved Carbon Surfaces: Mechanism of Adsorption Revealed by First Calorimetric Study. J. Phys. Chem. C 2015, 119, 2703-2715.
(3) Kagita, H.; Ohba, T.; Fujimori, T.; Tanaka, H.; Hata, K.; Taira, S.-I.; Kanoh, H.; Minami, D.; Hattori, Y.; Itoh, T.; Masu, H.; Endo. M.; Kaneko, K. Quantum Molecular Sieving Effects of H2 and D2 on Bundled and Nonbundled Single-Walled Carbon Nanotubes. J. Phys. Chem. C 2012, 116, 20918-20922. (4) Ceperley, D. M. Path Integrals in the Theory of Condensed Helium. Rev. Mod. Phys. 1995, 67, 279-355.
(5) Kowalczyk, P.; Gauden, P. A.; Terzyk, A. P. Cryogenic Separation of Hydrogen Isotopes in Single-Walled Carbon and Boron-Nitride Nanotubes: Insight Into the Mechanism of Equilibrium Quantum Sieving in Quasi-One-Dimensional Pores. J. Phys. Chem. B 2008, 112, 8275-8284.
(6) Silvera, I. F.; Goldman, V. V. The Isotropic Intermolecular Potential for H2 and D2 in the Solid and Gas Phases. J. Chem. Phys. 1978, 69, 4209-4213. (7) Tjatjopoulos, G. J.; Feke, D. L.; Mann Jr., J. A. Molecule-Micropore Interaction Potentials. J. Phys. Chem. 1988, 92, 4006-4007. (8) Tanaka, H.; Kanoh, H.; Yudasaka, M.; Iijima, S.; Kaneko, K. Quantum Effects on Hydrogen Isotope Adsorption on Single-Wall Carbon Nanohorns. J. Am. Chem. Soc. 2005, 127, 7511-7516. (9) Wang, Q.; Johnson, J. K.; Broughton, J. Q. Path Integral Grand Cannonical Monte Carlo. J. Chem. Phys. 1997, 107, 5108-5117. (10) Kowalczyk, P.; Gauden, P. A.; Terzyk, A. P.; Bhatia S. K. Thermodynamics of Hydrogen Adsorption in Slit-like Carbon Nanopores at 77 K. Classical Versus PathIntegral Monte Carlo Simulations. Langmuir 2007, 23, 3666-3672. (11) Miller III, T. F.; Manolopoulos, D. E. Quantum Diffusion in Liquid paraHydrogen from Ring-Polymer Molecular Dynamics. J. Chem. Phys. 2005, 122,
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184503-1-184503-7. (12) Miller III, T. F. Isomorphic Classical Molecular Dynamics Model for an Excess Electron in a Supercritical Fluid. J. Chem. Phys. 2008, 129, 194502-1-194502-10.
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Parameter
Numerical value
α
1.713
δ
1.567
γ
0.00993
C6
12.14
C8
215.2
C9
143.1
C10
4,813.9
rc
8.321
Table S1. Parameters of the Silvera-Goldman model potential in atomic units6.
S7
Figure S1. Pore density variation of H2 (top) and D2 (bottom) ring-polymer radius of gyration, obtained using RPMD trajectories. Open and filled symbols correspond to 10 and 25 ps production runs, respectively.
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Figure S2. Self-diffusion coefficients for H2 and the heavy isotope as a function of H2 mole fraction in the H2/D2 (top) and H2/T2 (bottom) mixtures at a fixed pore density of 16.4 mmol/cm3 and 25 K.
S9
