Hydrogen diffusion in potassium intercalated graphite studied by quasielastic neutron scattering

Transcription

1 Supporting Information for Hydrogen diffusion in potassium intercalated graphite studied by quasielastic neutron scattering Justin Purewal *, J. Brandon Keith, Channing C. Ahn and Brent Fultz California Institute of Technology, W. M. Keck Laboratory , Pasadena, CA Craig C. Brown and Madhu Tyagi NIST Center for Neutron Research, 100 Bureau Drive, Gaithersburg, MD Figure S1 Figure S2 Figure S3 Figure S4 Figure S5 Figure S6 Figure S7 Figure S8 Fits of KC 24 (H 2 ) 0.5 QENS spectra at 80 K to the honeycomb net (HC) model Fits of KC 24 (H 2 ) 0.5 QENS spectra at 90 K to the HC model Fits of KC 24 (H 2 ) 0.5 QENS spectra at 110 K to the HC model Fits of KC 24 (H 2 ) 0.5 QENS spectra to powder-averaged function for continuous twodimensional diffusion Fits of KC 24 (H 2 ) 0.5 QENS spectra at 80 K to the Chudley-Elliott (CE) model Fits of KC 24 (H 2 ) 0.5 QENS spectra at 90 K to CE model. Fits of KC 24 (H 2 ) 0.5 QENS spectra at 100 K to CE model Fits of KC 24 (H 2 ) 0.5 QENS spectra at 110 K to CE model Figure S9 Powder X-ray diffraction pattern of KC 24 Figure S10 Figure S11 Figure S12 Neutron diffraction pattern of KC 24 (H 2 ) x for a series of H 2 loadings Hydrogen adsorption and desorption measurements in KC 24 at 77 K and 87 K Isosteric heat of KC 24 as a function of H 2 adsorption amount Figure S13 Figure S14 Bibliography Hydrogen trajectories from molecular dynamics (MD) simulations of KC 24 (H 2 ) 0.5 at 50 K and 70 K. Mean-squared-displacements and intermediate scattering functions from MD simulations of KC 24 (H 2 ) x system. *Present address: HRL Laboratories LLC, Malibu, California 90265, United States

2 FIG. S1 Measured QENS spectra of KC 24 (H 2 ) 0.5 at 80 K itted to the honeycomb net jump diffusion model 2/17

3 FIG. S2 Measured QENS spectra of KC 24 (H 2 ) 0.5 at 90 K itted to the honeycomb net jump diffusion model. 3/17

4 FIG. S3 Measured QENS spectra of KC 24 (H 2 ) 0.5 at 110 K itted to the honeycomb net jump diffusion model. 4/17

5 FIG. S4 Fits of the lowest Q-value QENS spectra for KC 24 (H 2 ) 0.5 at temperatures K to the powderaveraged model for two-dimensional continuous diffusion. In the low-q limit the QENS linewidth approaches D s Q 2, which is the expected value for continuous diffusion following Fick s law. For purely two-dimensional diffusion, the incoherent scattering function for continuous diffusion within a single crystallite is given by S 2D (Q, E) = 1 D(Q sin θ) 2 πħ [D(Q sin θ) 2 ] 2 + E 2 where D is the self-diffusion coef icient, and θ is the angle between the scattering vector Q and the normal to the diffusion plane. The powder average S 2D (Q, E) = 1/2 S 2D (Q, E) sin θθθ has been calculated analytically in Refs. [1,2]. This expression for S 2D (Q, E) was numerically convolved with the instrument resolution and itted to the Q = 0.31 A spectra ( its shown above). Fitted values for the self-diffusion coef icients at each temperature are listed. Because S 2D (Q, E) is more sharply pointed around E = 0 compared to a Lorentzian, it is not necessary to include a delta function term to account for the elastic-like component. π 0 5/17

6 FIG. S5 (a) Fits of the QENS spectra of KC 24 (H 2 ) 0.5 to a sum of the elastic peak (delta function) and quasielastic peak (Lorentzian) convolved with instrument resolution S(Q, E) = I ee δ(e) + I qq π f(q) f(q) 2 + E2 R(Q, E) where f(q) is the Lorentzian half-width-at-half-maximum (HWHM), I ee and I qq are the elastic and quasielastic intensities, respectively, and R(Q, E) is the instrument resolution. (b) Values of experimental HWHM are itted to a well-known expression for isotropic jump diffusion with variable jump length. From the Chudley-Elliott model, f(q) = (ħ/nn) j 1 e iq l j where l j are jump 6/17

7 vectors between NN sites on a Bravais lattice, τ is the residence time, and n is the number of NN atoms [3]. In the isotropic approximation, f(q) is averaged over orientation (as opposed to averaging S(Q, E) over orientation) to give f(q) = ħ/τ[1 (1/QQ) sin QQ]. This is averaged over the jump length distribution a(l) = le l/l 0 to account for variable hopping lengths[4]. The resulting expression is given by f(q) = ħ τ Q 2 l 0 2 which is then itted to the experimental HWHM. Fitted values for τ and l 0 are listed. The 3D diffusion coef icient is given by D s = l 0 2 /τ. (c) The elastic incoherent structure factor (EISF), de ined as the fraction of elastic intensity (I ee ) relative to the combined quasielastic (I qq ) and elastic intensity, is plotted versus Q. 7/17

8 FIG. S6 (a) Fits of the QENS spectra of KC 24 (H 2 ) 0.5 measured at 90 K to a sum of the elastic peak (delta function) and quasielastic peak (Lorentzian), convolved with instrument resolution. (b) Experimental Lorentzian HWHM values itted to the isotropic, variable-jump-length model. (c) Experimental EISF versus Q. 8/17

9 FIG. S7 (a) Fits of the QENS spectra of KC 24 (H 2 ) 0.5 measured at 100 K to a sum of the elastic peak (delta function) and quasielastic peak (Lorentzian), convolved with instrument resolution. (b) Experimental Lorentzian HWHM values itted to the isotropic, variable-jump-length model. (c) Experimental EISF versus Q. 9/17

10 FIG. S8 (a) Fits of the QENS spectra of KC 24 (H 2 ) 0.5 measured at 110 K to a sum of the elastic peak (delta function) and quasielastic peak (Lorentzian), convolved with instrument resolution. (b) Experimental Lorentzian HWHM values itted to the isotropic, variable-jump-length model. (c) Experimental EISF versus Q. 10/17

11 Fig. S9 Powder X-ray diffraction pattern of KC 24 measured at room temperature with Cu Kα radiation. 11/17

12 Figure S10 Neutron diffraction pattern of KC 24 (H 2 ) x measured at 4 K for a series of H 2 loadings. Hydrogen was loaded at 60 K between each measurement. 12/17

13 Figure S11 (a) Measured H 2 adsorption and desorption isotherms for KC 24 at 77 K (circles) and 87 K (triangles). Due to the slow sorption kinetics in KC 24 (H 2 ) x at x > 1.5 (as described in Sec. IV-A in the main article), it is likely that the measured desorption points are not equilibrium values. This kinetic limitation may be the reason for the apparent hysteresis loop between the adsorption and desorption curves in the 77 K isotherm. We also note that the hysteresis between the adsorption and desorption curves is much smaller at 87 K, where the sorption kinetics are faster. (b) Adsorption cycling test for KC 24 at 77 K in which three consecutive adsorption measurements at 77 K were performed. After each isotherm measurement, the H 2 was removed by allowing the KC 24 sample to warm to room temperature while continuously pulling vacuum (oil-free diaphragm pump). The good repeatability between the 77 K adsorption isotherms (even at low pressure) indicates that no H 2 remains chemisorbed following the mild evacuation. 13/17

14 Figure S12 Isosteric heat KC 24 as a function of H 2 adsorption amount. (a) The measured KC 24 /H 2 adsorption isotherms were itted to a modi ied Langmuir equation described by Terai et al. in Ref. 5. This modified Langmuir equation is expressed as a function of adsorption amount (n), as p = n K(n max n) where A is the adsorption coefficient, p is the equilibrium pressure and n max is the maximum adsorption capacity. The equilibrium coefficient A is related to the adsorption enthalpy ( ΔH ads ) by K = K 0 exp H ads RT where the pre-factor (K 0 ) is an entropic contribution (assumed constant here), and where the value of ΔH ads is equivalent to the isosteric heat [6]. Typically ΔH ads is assumed to be constant over the complete range of loading. In this case, however, ΔH ads is allowed to increase initially (due to the work used up in expanding the KC 24 interlayer spacing) before leveling off to a constant amount. (b) We compare our estimates of the isosteric heat of KC 24 versus hydrogen adsorption amount to values from the literature. First, we applied the standard isosteric method to the 77 K and 87 K isotherms, in which the equilibrium pressure was interpolated at fixed intervals of adsorption amount. The isosteric heat was then calculated by the Clausius-Clapeyron equation from the slope of ln(p) versus 1/T along lines of constant adsorption amount. This method is susceptible to experimental errors (particularly due to the low pressures present here, and to the fact that only two temperatures are used), which is the likely reason for the large fluctuations between 8 kj/mol and 10 kj/mol observed here. Second, we fitted the modified Langmuir equation to obtain ΔH ads as a function of adsorption amount. These estimates are compared with published values. Values obtained by Terai [5] are plotted in panel (b) as the solid green line. The isosteric heat (9 kj/mol) measured by Watanabe [7] for a single point at half-coverage is also shown. In contrast to what is observed for most adsorbents, the isosteric heat for KC 24 does not seem to decrease as the hydrogen loading is increased. 14/17

15 FIG. S13 In-plane trajectories of hydrogen spheres from molecular dynamics simulations of KC 28 (H 2 ) 1 at 50 K (top) and 90 K (bottom). Total simulation time is 500 ps. Representative trajectories of four different hydrogen spheres are shown at 50 K, while trajectories of two different hydrogen particles is shown for 90 K. Solid circles indicate the inal in-plane positions of the potassium atoms (in the 90 K simulations, several potassium atoms have left their initial 7 7 structure). The graphite network is not shown. 15/17

16 FIG. S14 (a) Plots of the total mean square displacement (MSD, A 2 ) of hydrogen as a function of time (ps) for KC 28 (H 2 ) 1 at three different temperatures. (b) Intermediate scattering function at Q = 1.0 A 1 calculated from simulated trajectories of hydrogen in KC 28 (H 2 ) x, plotted as a series of H 2 /K concentrations. 16/17

17 Bibliography 1 R. E. Lechner, Solid State Ionics 77, (1995). 2 A. J. Dianoux, F. Volino and H. Hervet, Mol. Phys. 30(4), 1181 (1975). 3 C. T. Chudley and R. J. Elliott, Proc. Phys. Soc. London 77, 353 (1961). 4 P. A. Egelstaff, An Introduction to the Liquid State, (Academic Press, London) Chap T. Terai and Y. Takahashi, Synthetic Metals 34, (1989). 6 A. Myers, AIChE J. 48, (2002). 7 K. Watanabe, T. Kondow, M. Soma, T. Onishi and K. Tamaru, Proc. R. Soc. Lond. A 333, (1973). 17/17