Quest for Inexpensive Hydrogen Isotopic Fractionation: Do We Need

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On the Quest for Inexpensive Hydrogen Isotopic Fractionation: Do We Need 2D Quantum Confining in Porous Materials or Are Rough Surfaces Enough? The Case of Ammonia Nanoclusters Massimo Mella, and Emanuele Curotto J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b08005 • Publication Date (Web): 26 Sep 2016 Downloaded from http://pubs.acs.org on October 4, 2016

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The Journal of Physical Chemistry

On the Quest for Inexpensive Hydrogen Isotopic Fractionation: Do We Need 2D Quantum Confining in Porous Materials or Are Rough Surfaces Enough? The Case of Ammonia Nanoclusters

†

Massimo Mella∗,‡ and E. Curotto∗,¶ Dipartimento di Scienza ed Alta Tecnologia, Universit` a degli Studi dell’Insubria, via Valleggio 9, 22100 Como (I), and Department of Chemistry and Physics, Arcadia University, Glenside, Pennsylvania, 19038-3295 E-mail: [email protected]; [email protected] Phone: +39 0312386625. Fax: +39 0312386630

†

EC carried out the cluster optimisations, the fitting of the ammonia–H2 potential, and contributed to the writing of the manuscript. MM carried out the diffusion Monte Carlo simulations, implemented the algorithms for computing expectation values analysing the results, and contributed to the writing of the manuscript. ∗ To whom correspondence should be addressed ‡ Università dell’Insubria ¶ Arcadia University

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Abstract We study the adsorption energetics and quantum properties of the molecular hydrogen isotopes H2 , D2 , and T2 onto the surface of rigid ammonia nanoclusters with quantum simulations and accurate model potential energy surfaces (PES). A highly efficient diffusion Monte Carlo (DMC) algorithm for rigid rotors allowed us to accurately define zero–point adsorption energies for the three isotopes, as well as the degree of translational and rotational delocalisation that each affords on the surface. From the data emerges that the quantum adsorption energy (Eads ) of T2 can be up to twice the one of H2 at 0 K, suggesting the possibility of exploiting some form of solid ammonia to selectivity separate hydrogen isotopes at low temperatures (≃ 20 K). This is discussed focusing on the structural motif that may be more effective for the task. The analysis of the contributions to Eads , however, surprisingly indicates that the average kinetic energy (Ekin ) and rotation energy (Ekin rot ) of T2 can also be, respectively, twice and twenty times higher than the one of H2 ; this finding markedly deviates from what is predicted for hydrogen molecules inside carbon nanotubes (CNT) or metallic–organic frameworks (MOF), where Ekin and Ekin rot is higher for H2 due to the unavoidable effects of confinement and hindrance to its rotational motion. The rationale for these differences is provided by the geometrical distributions for the rigid rotors, which reveal an increasingly stronger coupling between rotational and translational degrees of freedom upon increasing the isotopic mass. This effect has never been observed before on adsorbing surfaces (e.g. graphite) and is induced by a strongly anisotropic and anharmonic bowl–like potential experienced by the rotors.

1

Introduction

With deuterium and tritium presenting a growing volume of important applications in technology 1 and in the production of energy 2 or health–related substances, 3 the quest for a convenient and unexpensive approach to separate them from protium has been pursued relentlessly over the last few years. As cryogenic distillation 4 at 24 K and electrolysis of heavy 2


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water 5 show very low separation factors and high energetic costs, attention has been directed toward exploiting differences in quantum physical properties that more markedly depend on the isotopic mass. Quantum effects reveal themselves in both static and dynamic properties. As to the former, we recall solubility, 6 adsorption energy 7,8 and phase transition temperature. 9 Dynamic effects are instead apparent in chemical reactions involving hydrogen transfer, 10 diffusion through narrow pores, 11–14 viscosity 15 and evaporation kinetics. 16 In principle, all the mentioned properties may be exploited to separate the isotopic variants of hydrogen; however, the costs and practicality associated to any approach are key factors in defining whether or not it could be pursued, especially in view of the sheer amount of material needed for the applications mentioned above. Among the possibilities, the separation of hydrogen isotopes could be accomplished exploiting the dissimilarity in dynamic and energetic properties shown when interacting with porous materials (quantum sieving, QS). Notice, however, that differences between, e.g., H2 and T2 begins to emerge only when some geometrical feature of the material (e.g. the pore diameter) becomes comparable with the de Broglie wave length associated to each species, and this is independent of the specific property exploited for the separation. It is, in fact, only in this condition that quantum interference begins to play a role in defining the quantum molecular states and behaviour. With respect to static (or energetic) properties, the three isotopes usually present different sorption energies inside pores, 13,17 interstices between carbon nanotubes (CNT) or fibers, 13 and in the vicinity of open metal sites located in the interior of metal organic frameworks (MOF). 18,19 This difference impacts on the sorption isotherms and desorption temperature, 13,17–19 so that the isotope with the highest sorption energy (usually the heaviest) is preferentially adsorbed and has a longer transit time across a chromatographic column compared to the others. When it comes to exploiting dynamic isotopic effects, the recent literature has evidenced

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the presence of differences between H2 and D2 in both the diffusion of absorbed species 14,20–22 and the rates of sorption or desorption; 11–13,23–25 since deuterium transport is faster, perhaps contradicting classical expectations, both findings have been rationalised as due to a smaller “quantum” size compared to H2 . 20 As a result, D2 needs to surmount lower free energy barriers than H2 to diffuse across or enter porous materials. 22,25 If present, substantial differences in surface adsorption energies Eads between isotopologues 7,26,27 can also be exploited for separation purposes 8,28,29 despite the lack of material porosity. Notice that such energetic differences, primarily due to different zero point energies (ZPE), may also induce kinetic effects during adsorption and desorption; 30,31 nevertheless, the main effect due to the disparity in mass should be a shift in the adsorption isotherms. For such effect to appear, some form of confinement must however be imposed, at least, onto one degree of freedom; 32–35 the latter may involve either perpendicular and parallel translations of a rigid H2 isotopologue with respect to the surface, or its orientation. 36,37 Whether or not coupling between a few degrees of freedom may also play a role depends uniquely on the details of the molecule–surface interaction. 38 In general terms, increasing the atomic mass should induce an increase in adsorption energy as the molecule acquires a more classical nature; a decrease in the average kinetic energy should also be expected in going from H2 to T2 . 39 Compared to using nanoporous materials, surface–based separation may be less expensive and technically demanding, especially if the substances employed to prepare the active material are already produced in bulk or nearly freely available. In this respect, the ability of water ice to preferentially bind ortho–H2 (o–H2 ) compared to para–H2 (p–H2 ) or o–D2 has already been highlighted and its relevance to astrophysics discussed. 8,27,28 Specifically, Eads for ortho–H2 onto amorphous ice seems to be roughly 30 K higher than for p–H2 , 28 Eads for p–D2 is 16 K higher than for o–D2 , 7 and p–D2 requires an energy 30 K higher than o–H2 to desorb. 8 Importantly, the mentioned differences seem to already suffice when it comes to separating isotopes. For instance, the molar fraction of p–H2 adsorbed onto porous ice at

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12 K is found four times lower than in the overlaying gas. Similarly, the p–D2 /o–D2 ratio of adsorbed deuterium may be enhanced by a factor of 5 at 10 K. Another simple and largely available substance that may be exploited for separation purposes is ammonia, which may also freeze into an amorphous solid. 40 Albeit models for porous amorphous ammonia ice are currently not available as they are for water, 27,28 indications on possible adsorption locations and Eads values for H2 onto frozen ammonia have been already provided in a detailed study of the interaction between H2 and rigid (NH3 )n . 41 The latter involved classical optimization and quantum simulations on rigid-body (5D) and a rotationally averaged (3D) potential energy surfaces (PES) for H2 –NH3 . 42 In particular, it has been shown that rhomboidal surface tetramers composed of two double donor and two double acceptor ammonia molecules bind H2 more strongly than exposed nitrogen atoms (Eads =21 cm−1 , modelled with a single ammonia) or cluster edges (Eads =8–13 cm−1 ). Interestingly, the rhomboidal motifs are characterized by a bowl–like structure with four dangling hydrogen atoms forming a rim; both translational and rotational degrees of freedom may thus be simultaneously subjected to some form of confinement (or even forced to couple), a feature that may induce into solid ammonia the ability of separating isotopologues. 25 Bearing in mind the discussion provided above, the present work investigates the possibility of exploiting solid ammonia species in order to separate hydrogen isotopologues via selective surface adsorption. Here, the emphasis is mainly placed on determining important quantities such as Eads and (theoretical) separation ratios for various adsorption sites; we thus hope to understand what is, if any, the key geometrical parameter controlling the separation performance of frozen ammonia. Clearly, it may be possible that also dynamic effects (e.g. the ones highlighted by Lemaire et al 31 ) play a role in defining the separation capability of ammonia surfaces. This notwithstanding, we consider sensible to approach the mentioned issue gradually and tackle what is likely to represent the key factor (Eads ) in the first place. The organisation of the Manuscript is therefore as follows. Section 3.1 discusses results

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for Eads obtained adsorbing H2 , D2 and T2 onto (NH3 )n clusters selected to represent possible features of frozen ammonia surfaces. Among the latter, it is useful to recall the presence of trimeric and tetrameric rings characterizing both ammonia ice and species formed via very low temperature quenching, 43 as well as ad–molecules and rhomboidal tetramer often appearing in annealed clusters 41 and emerging from the primitive cell of ammonia ice. A single immobile NH3 is also included in the set of investigated species as a minimalistic model for under–coordinated nitrogen atoms, which may be present in amorphous ice 27,28 but never appear in clusters independently of the formation process. Section 3.2 discusses, instead, the theoretical estimates for the selectivity S(X2 /H2 ) =

xX2 /xH2 yX2 /yH2

(X2 =T2 or D2 , and

x/y are the mole fraction of each species on the surface and in the bulk) obtainable via simulations at 0 K, single–site Langmuir adsorption isotherms for the different isotopes, as well as the variations induced in S(X2 /H2 ) by raising the temperature. The quantitative methodologies, approximations and interaction potential surfaces employed in our work are presented in Section 2. Finally, Section 4 provides additional comments, our conclusions and future avenues of explorations.

2

Methods

Before discussing the details of our methodologies, it seems useful to explicitly indicate the analytical form of the Hamiltonian operator describing the systems (atomic units). This can be written as X ∇2CoM Λ2 H=− + + V5D (riX2 , ΩiX2 ) 2mX2 2IX2 i=1,n

(1)

with CoM indicating the dimer center of mass coordinates, mX2 its molecular mass, Λ2 the total angular momentum operator, I = µX2 re2 the dimer momentum of inertia (µ being the reduced mass, half of the nuclear mass), riX2 and ΩiX2 being the relative position and orientation of the dimer with respect to the i–th (of n) ammonia molecule in the cluster. 6


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2.1

Diffusion Monte Carlo simulations

Previous work of ours 41 has shown that the quantitative investigation of the quantum systems described in this study with the interaction PES developed in Ref. 42 (and here reoptimized for the sake of efficiency and accuracy) must be carried out with a methodology capable of accurately describing anharmonicity in all degrees of freedom. Among the techniques capable of doing so, Diffusion Monte Carlo (DMC) 44,45 is a particularly convenient one thanks to its recently improved ability in treating systems such as He droplets 46 and other molecular clusters. 47–51 Specifically, DMC is a ground state technique that allows one to obtain both energetics (i.e. the ground state energy E0 ) and structural details at a limited computational cost, as the version commonly employed when treating molecular species scales linearly with the number of configurations sampled during the simulation and with the cost associated with the PES evaluation. The particular implementation of DMC employed in this study has previously been described in detail 41,52 and compared with alternative schemes. 48,51 Thus, we shall discuss only the features that are relevant for this work for the sake of brevity. In essence, DMC is a projection method employed to purify from the excited states the distribution of points in configuration space providing a discrete representation for the wave function of a system; in the end, it will distribute the points proportionally to either ψ0 or ΨT ψ0 , depending of the chosen algorithm. The projection is carried out with a series of alternated diffusion and branching steps applied to all configurations; the branching is carried out using the difference between the value of the potential at each point and its average over the configuration ensemble. In the case of molecules representable as rigid linear rotors, the diffusion takes place in Cartesian space for the center of mass, and onto the S 2 manifold for the orientation. The diffusion in Cartesian space can be simulated exactly; instead, the same is not generally possible in curved manifolds. One has thus to rely upon an approximate diffusion matrix, which we have originally taken from the work by Faraudo. 53 The latter provides one with a robust approximation to the diffusion Green’s function that has an error proportional to the square of the time step δτ 2 , so that the overall order of the 7



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δτ

δτ

algorithm matches the one deriving from the symmetric splitting e− 2 V e−δτ T e− 2 V commonly employed with molecular clusters such as H+ (H2 O)n . 54,55 Here, V and T are respectively the potential energy and translational plus rotational kinetic energy operators. Notice that the δτ 2 dependence of the error for the algorithm allows one to extrapolate to third order in δτ either E0 a posteriori, 56 or the branching weights “on the fly”. 51 All simulations were run employing a time step of 100 a.u. and 10000-20000 configurations to minimise both time step and population biases, in conjunction with a slightly modified version of the full–dimensional (5D) potential energy surface developed earlier 42 (vide infra). Ammonia clusters were kept rigid; thus, we used DMC only to sample ψ0 for the hydrogen molecule. The possible impact of introducing quantum effects only for H2 is expected to be minimal, as minor changes are induced in the optimal structure of pure ammonia clusters 57 by adding H2 . 41 H2 equilibrium distance was set as 1.401 bohr for all isotopes, while their nuclear masses were set as 1836.2, 3669.6 and 5495.1 atomic units. Rotational constants (or inertia moments) for all species have been computed using the mentioned data. The initial position and orientation of the hydrogen molecules were sampled uniformly inside a cube of side length 30 bohr and over φ and cos θ to avoid any bias. Notice that X2 molecules initially presenting a marked overlap with ammonia clusters are automatically eliminated during the initial steps of a DMC simulation. As previously done, we investigated the geometrical features of ψ0 either by a direct visualisation of the replica ensemble or by collecting density histograms. Albeit alternative approaches to produce ψ02 related information are available, 58,59 the chosen method suffices for the task at hand. 54,55 To analyse the DMC energy results, we decomposed E0 into average translational (Ekin tr ), rotational (Ekin rot ) and potential (hV i) energies for the adsorbed species. To do so, we employed the perturbation theory approach implemented by Sandler, Buch and Clary. 60 This is based on linear response theory, i.e. the idea that E0 changes linearly with respect to the magnitude of a perturbation P added to the Hamiltonian H if P is small. In this regime, the aspectation value for observables that do not commutes with H can be estimated via a

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two–point difference or linearly fitting a few E0 values. Kinetic energies can be estimated using the inverse of the system mass and/or the rigid rotor inertia moment as perturbation parameters. The average potential value hV − Vmin i is, instead, computed as a difference kin between E0 and the total kinetic energy Ekin tr +Erot .

2.2

Full–dimensional (5D) potential energy surfaces for NH3 –H2

The PES employed in this work to describe the NH3 –H2 interaction were generated in Ref. 42 by fitting high level ab initio energies obtained with a dual MP2/MP4 scheme and large basis sets. The model is a sum of sources of electrostatic, dispersion and repulsion interactions located on all atoms in ammonia and in the H2 center of mass. No polarization–dependent terms were included in the model PES given the low static polarisability of H2 and its non– polar nature. An effective description of the inductive interaction (mainly, dipole–induced dipole) is nevertheless introduced while fitting the coefficients of the multipolar expansions employed to represent the electrostatic components of the PES. For the present investigation, we reoptimise all the parameters of the full–dimensional (5D) surface to improve the efficiency of its computation and its accuracy. Consequently, we have conducted a new characterization of the PES by locating all its minima. The 5D surface for NH3 –H2 has a global minimum with a De ≃ 199 cm−1 (286 K) and the H2 molecule hydrogen bonded to the nitrogen of ammonia, the H2 bond axis being aligned with the C3v axis of NH3 . A second minimum with De ≃ 157 cm−1 (226 K) is also found; this is characterized by a N–H bond pointing nearly perpendicularly toward the middle of the H2 axis. The structures of these minima and the associated values of the energy are in good agreement with previously reported, 42 and are in better agreement with the ab initio data. To describe the PES for the pure ammonia clusters, we have used a many–body rigid model developed by us 57 that includes point charges and dispersion/repulsion terms on all atoms, as well as a description of dipole polarisability implemented following the single–step 9



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“charge on spring” approach. 61 This PES has been used to successfully model the MP2/aug– cc-pVTZ energetics of small ammonia clusters, 62 their formation probability in ultra cold dissipating environment, 43 as well as their quantum 49 and thermal properties. 63 The total potential energy for a (NH3 )n –H2 system is written as a sum of the ammonia cluster and NH3 –H2 contributions, albeit the former is not needed as the ammonia moiety is kept frozen with the structure of the global minimum found by optimising the pure ammonia aggregates 41 (vide supra Section 2.1). Notice that the minimum energy structure of pure and H2 –doped (NH3 )n clusters closely match even at the end of a global search via a genetic algorithm. The minima found for the (NH3 )n –H2 species (n = 9, 12, 16 and 17) studied in this work are characterized by H2 being adsorbed on a surface binding motif built by cyclic ammonia trimers (henceforth triangle, see Figure S1 in the Supporting Information) for n = 9, 16 and 17, and rhomboidal ammonia tetramers for the dodecamer. The latter contains two double donor (DD) and two double acceptor (AA) molecules. 41 Table 1 provides the adiabatic binding energies of H2 on the optimised clusters. For all the minima, the dominant interaction mode appears to be the global minimum of the 5D PES, i.e. the one in which H2 points toward the negative region of the ammonia quadrupole (“lone pair”), and the small differences in potential energy being due to the details of the local structure and orientation of H2 . Notice that all cyclic binding motives always feature a “rim” of hydrogen atoms pointing upward with respect to the plane containing the ammonia nitrogens, effectively making these surface patterns the equivalent of “molecular bowls”. The width of the latter may be expected to play a role in defining the location of the quantum distribution of the isotopic species as much as the magnitude of the binding energy De , as a rim ought to impose some form of spatial confinement to X2 . Indeed, the results presented in Sections 3.1 and 3.2 support this view. The difference in quantum binding motif for the various clusters, however, makes impractical discussing any correlation between E0 and width.

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Table 1: Adiabatic binding energy for rb–H2 onto (NH3 )n (n = 9, 12, 16 and 17). Energies in cm−1 . n 9 12 16 17

2.3

De (rb–H2 ) 446.7 427.1 424.0 581.2

Low–temperature approximations for the partition coefficients and selectivity gauges

The adsorption selectivity between two species A and B is defined as

S(A/B) =

xA /xB yA /yB

(2)

where x and y are the mole fraction of each species on the surface and in the bulk. Indicating with Q the partition function of a single molecule, at low pressure one can simplify the formula obtaining (β ≡ (kB T )−1 ) QfBree QA

P −β∆E(l,0) (A) mB 3/2 Qrot B −β∆E0 1 + l=1 e =( e ) S0 (A/B) = f ree P mA Qrot 1 + l=1 e−β∆E(l,0) (B) QA QB A

(3)

where Qrot is the free rotor partition function, ∆E0 = E0 (A) − E0 (B) the difference in ground state (l = 0) energy for the two adsorbed species, and ∆E(l,0) = El − E0 the energy gap between the ground state and the l–th excited state. Unlike what is usually proposed discussing the case of CNT selectivity, 39,64,65 we assumed that all translational degrees of freedom are quantised for the adsorbed molecules, as the shape of the interaction potential is such that restrains are imposed onto the center of mass motion both perpendicularly and B 3/2 tangentially to the cluster surface. Hence, the ( m ) factor, related to the motion of the mA

free molecules, remains unmodified. Focusing onto Equation 3, one notices that ∆E0 is likely to play the key role in defining the selectivity, especially at the low temperatures (∼ 20 K) commonly investigated for separation

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purposes. For instance, we expect

Qrot B Qrot A

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≃ 1, a conclusion easily reached remembering that

the energy gap between the J = 2 and J = 0 states are ∼525 and 175 K for para–H2 and T2 ; in fact, thermal excitation would increase the rotational partition function, respectively, only by 2.0 × 10−11 and 7.8 × 10−4 at 20 K. To progress further, one may assume that the translational and rotational degrees of freedom of adsorbed molecules are only weakly coupled, so that El can be written as the sum of a center of mass and a rotational (or librational) energy terms. If this was the case, it would be easy to conclude that the rotationally excited states of adsorbed molecules should not contribute to the two sums in Equation 3 as any hindrance in the rotational motion due to the surface is likely to increase the energy gap between states. 36,37 It seems, in fact, that only the instance of a molecule restrained to rotate parallel to the surface 37 may lead to a 20–25% maximum decrease in the gap between the rotation ground state and one of the states correlating with the J = 2 level of the free molecule, so that the only contribution to the two sums in the last term of Equation 3 may come from excitations into states connected to CoM motion. In this case,

P −β∆E(l,0) (A) 1+ l=1 e P −β∆E(l,0) (B) 1+ l=1 e

≥ 1 due to the different

isotopic mass (mA > mB ), which means that omitting the contribution due to the sums altogether would provide a lower bound to S0 . Indeed, the latter conclusion is reached even in the presence of coupling between rotational and CoM degrees of freedom, so that, albeit approximated and underestimated, a sensible gauge of the zero–pressure selectivity would B 3/2 −β∆E0 ) e term. The latter is easily estimated with DMC be provided simply by the ( m mA

simulations on the title systems.

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3

Results and discussion

3.1

Energetics and structure of quantum (NH3 )n –X2 (X=H, D and T)

Table 2 presents the energy results obtained simulating the ground state of (NH3 )n –X2 with DMC employing the 5D (rb–X2 ) PES (see Section 2.2). Worth remembering it is the fact that simulations with the 3D surface 41 strongly supported a nearly free–rotor ground state behaviour for adsorbed rb–H2 , as expected on the basis of the energy gap between the J = 0 and 2 rotational states (∆Erot 2,0 ) of the free species. The same situation is found with the newly fitted PES as the probability distribution p(| cos(θ)|) sampled in this work (see Figure 1, top panel), where θ is the angle between the vector joining the geometrical center of the nitrogen atoms in a cluster to the center of mass of H2 and its bond vector, indicate the light molecule to have almost uniformly distributed orientations independently of the specific cluster investigated. Such independence can be taken as an indication for the decoupling between rotational and translational degrees of freedom (DoF), an idea also supported by the average values of the rotational kinetic energy (Ekin rot ), which are only a small fraction −1 or 525 K). (< 2/100) of the aforementioned energy gap (∆Erot 2,0 ≃ 365 cm

In reality, a weak mixing between the two states has to be allowed for rb–H2 to have a non–zero Ekin rot . This effect is made apparent, albeit weakly, by the slightly higher values of p(| cos(θ)|) in the range | cos(θ)| ≥ 0.7, particularly for the largest cluster. As a corollary of such mixing, the energy gap between the ground and first rotationally (or librationally) excited states of the adsorbed rb–H2 should also increase compared to ∆Erot 2,0 , an idea that bears high importance when discussing the macroscopic adsorption properties of the ammonia clusters (vide infra Section 3.2). Turning to the heavier isotopes, we always find their E0 to be lower than for H2 , as it would be expected due to the heavier masses. In spite of this, the trend of E0 versus n seems to be generally well conserved for all isotopes. With a deeper meaning in term of the general 13



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Figure 1: Distribution of | cos(θ)| for rb–X2 over the (NH3 )n clusters employed in this work. θ is the angle between the vector joining the geometrical center of the nitrogen atoms in (NH3 )n to the R center of mass of rb–H2 , and its bond vector. The distributions are normalised so that p(| cos(θ)|)d| cos(θ)| = 1. 14


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Table 2: Energetic properties (including Zero Point Energy, ZPE) of the linear rotor X2 (X=H, D, and T) adsorbed onto (NH3 )n (n = 1, 9 12, 16 and 17). Energies in cm−1 . Molecule H2 H2 H2 H2 H2 D2 D2 D2 D2 D2 T2 T2 T2 T2 T2

n 1 9 12 16 17 1 9 12 16 17 1 9 12 16 17

E0 -25.5 -73.0 -66.7 -79.6 -120.0 -36.4 -102.2 -95.2 -108.0 -166.5 -42.9 -130.5 -119.4 -123.4 -206.6

ZPE 173.5 373.6 360.4 344.4 461.2 162.6 344.5 331.9 316.0 414.7 156.1 316.2 307.7 300.7 374.6

hVi-Vmin 173.5 334.2 322.7 299.8 402.5

Ekin rot

Ekin tr

0.6 2.3 2.2 8.0

38.9 35.4 42.4 50.7

279.0 270.0 276.4 322.0

20.3 14.3 2.7 38.5

38.5 38.6 37.0 54.2

228.4 233.3 261.6 263.3

47.4 37.0 4.8 57.3

40.4 37.4 34.3 54.0

behaviour for the studied species, we notice, instead, the strongly anharmonic behaviour displayed by ZPE values versus the molecular mass mX2 , the ZPE decreasing at much slower p pace than expected on the basis of the harmonic relationship mH2 /mX2 . In principle, this can be rationalized remembering that ∆Erot (X2 )2,0 (262 and 175 K, respectively for D2 and T2 ) decreases upon increasing the masses. While increasing X2 mass should always lead to decrease in E0 , ZPE and hV i, the net effect of a decrease in ∆Erot 2,0 , according to fundamental rules in state coupling, should be a stronger propensity for the rotational states to mix due to the presence of an external potential. This, of course, increase the flexibility of the wave function of the adsorbed X2 in adapting to the local potential nuances, an adaptation, however, that happens at the cost of an increase in Ekin rot . That this is just the case is clearly represented by the behaviour shown by the Ekin rot data provided in Table 2 (see also Figure 2) upon increasing the isotopic mass. The validity of such an interpretation is also neatly supported by the distribution p(| cos(θ)|) for the two heavy isotopes (Figure 1, middle and bottom panels); the latter, in fact, present clear maxima indicating, sometimes markedly,

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a preferential orientation for D2 and T2 so to have an atom pointing toward the cluster. Given the uniform distribution expected for a freely rotating quantum X2 , the presence of such maxima at | cos(θ)| = 1 can only be due to the rotational state mixing induced by the interaction potential. Perhaps also worth noticing, it is the precise positive correlation between Ekin rot and the maximum values in Figure 1 for the heavy isotopes. Figure 3 provides a visual representation of the spatial localization of the CoM density for the adsorbed X2 . Noteworthy, there is the difference in location between the X2 CoM “clouds” sampled during the DMC simulations and the position of X2 in the global energy minimum structure (see Supporting Information, Figure S1) in X2 (NH3 )9 (X2 moves from over a triangle to spread over the only square and its fused rhombus if X2 =H2 , or over a rhombus opposite to the square in case of the heavier isotopes) and X2 (NH3 )16 (from over a triangle to a square in all cases). Juxtaposing the results shown in Figures 1 and 3 suggests that the dominant binding motif, especially for the heavier isotope, ought to be the attraction between the positive part of X2 quadrupole and the negative side of the ammonia dipole. For the specific case of X2 (NH3 )17 , such binding interaction is supplemented by the interaction with the dangling H–N bond of a isolated ammonia on the surface of a distorted (NH3 )16 core. From the data in Table 2, it is clear that the presence of the latter interaction has important consequences from the energetic point of view as it substantially lowers E0 kin and De (respectively, by 58 and 130 cm−1 for H2 ), while it increases both Ekin rot and Etr .

To resume our analysis on the quantum energetics of the (NH3 )n –rb–X2 systems, we highlight the fact that the average translational kinetic energy Ekin tr does not conform to the expectation of its decrease upon increasing mX2 (e.g. see n = 9 and 16 in Table 2, for which Ekin tr is higher for T2 than for D2 and H2 ). Such an expectation would, however, be truly realistic for rb–X2 only if complete decoupling between translational and rotational degrees of freedom was possible despite the presence of the external potential, an hypothesis clearly contradicted by our energy results and, at least to some extent, by the results shown in Figures 1–3 and Table 2.

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kin Figure 2: Rotational (Ekin rot , dotted lines, right axis) and translational (Etr , solid lines, left axis) kinetic energy for rb–X2 .

kin Despite what is discussed on the behavior of Ekin tr and Erot versus mX2 , the decrease in E0

upon increasing mX2 remains so substantial (at least 20 cm−1 or 31 K for the D2 –T2 couple onto (NH3 )16 ) to allow one speculating on the possibility of separating isotopes using frozen ammonia as adsorbing chromatographic medium instead of CNT. Notice, however, that the mechanism for separating isotopic species via adsorption onto (NH3 )n or solid NH3 may be somewhat different from what may happen employing CNT. In fact, CNT–absorbed X2 are always tightly bound despite the strong quantum confinement 39,64 as the interaction with the CNT internal surface seems to always retain an attractive nature independently of the rb–X2 orientation, at least with the currently employed model potentials. In this situation, kinetic trapping of H2 inside pores clogged up by heavier isotopes may markedly reduce the separation capability. In the case of the adsorption of rb–H2 on ammonia clusters, instead, the latter interact weakly with the surface (vide infra 2.2) due to the quantum averaging, and it may have a short lifetime as adsorbed species in the range of temperatures (∼ 20 K)

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Figure 3: Density distribution of rb–X2 onto (NH3 )n (from left to right: 9, 12, 16, 17; from top to bottom X=H, D and T).

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appropriate for the separation task. It is thus only thanks to the capability of the heavier species to distort their wave function (see Figure 1) that the adsorption energy substantially increases, the positive consequence of this difference being that H2 may not compete at all with the heavy isotopes for the adsorption sites if the chromatographic conditions are properly tuned to substantially keep the light isotope in the vapour phase. Somewhat indirect indications that this may be a practical possibility also come from the difference in adsorption location onto (NH3 )9 between H2 and the other isotopes (Figure 1), as well as the fact that the centroid of the rb–H2 CoM distribution is located further away from the surface than the one of heavier species (see Figure 3) facilitating the displacement of the former over an adsorption site despite the increase in Ekin tr upon increasing the mass. As for the latter, we suggest that it may be due to the narrowing of the average attractive well width sampled by the more localised distributions of the heavy isotopes.

Figure 4: Distribution function R2 p(R) for the distance R between the X2 center of mass and the geometrical centerR of the nitrogen atoms in (NH3 )17 and (NH3 )9 (inset). The graphs are normalised such that R2 p(R)dR = 1. Distributions collected simulating X2 adsorption onto the other ammonia clusters show similar behaviours.

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The distributions shown in Figures 1 and 4 can also be used to improve our understanding of the statistical mechanics of the adsorbed isotopomers, at least at low temperature (∼ 20 K). As for H2 , the uniform distribution shown by cos(θ) and the maximum in p(R)R2 shifted rot ≃ at larger distances strongly support the view that it behaves as a free rotor, so that ∆E2,0

525 K for all (NH3 )n . In the case of the heavier species, it is instead clear that some form of angular confinement is present with a relative strength and orientation that may differ depending of the cluster or the isotopic variant adsorbed onto it, and it is thus of interest for our investigation to gauge its impact on the low lying states of the rotational energy spectrum. As studying low lying excited states with DMC may be cumbersome, to gather such information one can exploit the results provided by Shih at al. 36,37 on the dependence of the quantum rotational state energies on the angular width (α) and depth (BV0 ) of restraining conical potentials. These were meant to simulate the potential experienced by surface adsorbed diatomics of rotational constant B allowing for a finite amount of tunnelling inside the potential walls. As to the results shown in Figures 2 and 3 of Ref. 36 for a molecule adsorbed vertically onto a surface, it is immediately evident that any form of confinement imposed onto θ increases the energy gap between the ground rotational state and the states rot correlating with the J = 2 level up to twice ∆E2,0 . In turn, this means that fundamentally no

role should be played by the thermally induced rotational excitation of adsorbed molecules. A slightly different behaviour for the energy gap was instead found for horizontally adsorbed molecule, 37 a geometry that, however, does not seem relevant for hydrogen isotopes onto ammonia clusters.

3.2

Adsorption isotherms and theoretical isotopic selectivities

The differences in ground state energy E0 (i.e. the adsorption energy at 0 K) shown in Table 2 for the three isotopes are expected to translate into dissimilar adsorption equilibria, the key factor in the mass–selective chromatographic separation of gas mixtures at low temperatures. To show that this is exactly the case, we estimated the adsorption isotherms for 20


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the three isotopic variants of hydrogen onto the investigated ammonia clusters adapting the standard derivation of the Langmuir isotherm (N non–interacting molecules adsorbed onto M>N independent sites from a perfect monoatomic gas 66 ) to the case where the adsorbed molecules are linear rotors. From the derivation it emerges that the fraction θa of populated sites can be written as

θa =

pχr (T ) 1 + pχr (T )

(4)

where p is the fugacity of the molecule in the gas phase. Here, χr (T ) = eµ0 /(kb T ) qads (T )/qr (T )

(5)

B T 3/2 with µ0 = −kB T ln[ kΛB3T ] = −kB T ln[( 2πmk ) kB T ] being the standard chemical potential h2

for the fictitious point–like species with the same mass as the diatomic, qr (T ) the rotational partition function for the free rotor, and qads (T ) the partition function for the adsorbed linear rotor. The latter describes the relative vibration of the molecular center of mass with respect to the cluster, as well as the molecular libration (hindered rotation). As discussed at some length in Section 2.3, it is possible to simplify the expression in Equation 5 considerably provided that the system temperature is substantially lower than the one required to populate the rotationally excited levels of the free molecules and the vibrational/librational states of the adsorbed rotors. In this case, one obtains χr (T ) ≃ exp[(µ0 − E0 )/(kB T )], where E0 is the ground state energy of the adsorbed molecule. Notice that such approximation provides a lower bound to the low pressure selectivity derived in Section 2.3. Figure 5 shows the behaviour of θa at 20 K for the two clusters with highest and lowest value for ∆E0 (AB) = E0 (A) − E0 (B) (see Figure 6), the temperature being sufficiently low so that the hypothesis used to derive our approximation are fulfilled. We begin by noticing that the impact produced by a different mass on E0 and, hence, on the adsorption isotherm is fairly evident for both clusters. In fact, the adsorption isotherm for

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Figure 5: Adsorption isotherms of H2 , D2 and T2 onto (NH3 )16 (top) and (NH3 )17 (bottom) at 20 K. Notice the difference in scale of the fugacity axes.

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Figure 6: Ground state energy difference E0 (X2 )-E0 (H2 ) (X=D and T). the lightest and heaviest species are separated by at least one order of magnitude in fugacity even for the hexadecamer case, which produced the smallest ∆E0 (AB) = E0 (A) − E0 (B) energy differences. Notice that the behavior of ∆E0 (AB) versus n markedly depends on the mass of the heavy isotope considered; thus the D/H couple shows a nearly constant ∆E0 (AB) for the three small clusters, whereas it varies by as much as 12 cm−1 for the T/H pair. Notice, also, that the isotherms for the heptadecamer are all shifted toward lower fugacities, as it would be expected based on the higher adsorption energy found for all isotopes. Interestingly, the separation between curves as a function of the molecular mass is lower for the couple T2 /D2 than for D2 /H2 in the case of (NH3 )16 , while the shift of the curves upon increasing mX2 is comparable for both couples when (NH3 )17 acts as adsorbant. While the former finding is in agreement with what is found in the CNT case, 39 which showed the D2 isotherm to be markedly closer to the T2 one than to the θa curve for H2 , as easily predictable on the basis of the relative increase in mass, the behavior of θa versus mX2 for (NH3 )17 suggests that ad–molecules may also boost selectivity between heavier isotopes. Finally, we point out that 23



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the adsorption isotherms for the ammonia clusters are all shifted toward higher fugacities than in the case of the (2,8) and (3,6) CNT due to the stronger binding of the X2 species with the carbonaceous material suggested by the employed potential models. Assuming the validity of the Langmuir isotherm model (i.e. considering the adsorption sites as independent) allows one also to estimate the low–pressure isotopic selectivity S0 , which is clearly independent of the gas pressure as there are no side–interaction between the adsorbed molecules. Employing Idealized Adsorption Solution Theory (IAST), 18,67 it is easy (0)

H2 3/2 −β∆EX2 H2 2 to derive that S0 (X2 /H2 ) = χX e , whose behaviour as r (T )/χr (T ) ≃ (mH2 /mX2 )

a function of T in the interval 15–35 K is shown in Figure 7 for n = 16 and 17. As expected 2 from the exponential nature of χX r (T ), the selectivity rapidly decreases upon increasing T .

In spite of this, the separation capability of the ammonia clusters seems to remain high.

Figure 7: Low pressure selectivity of heavy isotopes with respect to H2 onto (NH3 )16 and (NH3 )17 as a function of T . A comparison with previous results employing CNT as sieving material may help in placing the estimate for S0 (A/B) in the correct reference frame. We make specific references 24


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to the results published by Garberoglio and Johnson 39 which gave S0 (T2 /H2 ) = 1.7 × 108 and 50.6 at 20 K for the (3,6) and (2,8) CNT, respectively. Our approximation suggests that all ammonia clusters affords a relative low–pressure selectivity in the range [0.2–4] compared to the (2,8) CNT one at 20 K, a performance that is sustained even at slightly higher temperature in the case of the heptadecamer. In fact, S0 (T2 /H2 ) for (NH3 )17 at 25 K is still comparable with the one of (2,8) CNT at 20 K. The (3,6) CNT, instead, presents a S0 (T2 /H2 ) value at 20 K that is six orders of magnitude larger than the highest one found for the ammonia aggregates (at 20 K). Such finding is due to the extremely narrow channel inside which molecules are inserted (the potential increases by roughly 740 K upon rotating the diatom from being aligned to the CNT axis to being perpendicular to it), which exacerbates the effects due to quantum nature of the particles. A “vis a vis” comparison between the different components of the total energy of the isotopes inside the CNT and adsorbed onto the ammonia clusters reveals, however, that the rotational modes play a more important role in defining the selectivity for the latter species than for the CNT, as they are key to the onset of a strong attraction with the cluster surface. As a final comment on the relative separation performances of ammonia clusters and CNT, we mention that S(T2 /H2 ) is predicted to increase twofolds for the latter case upon increasing the external pressure 39 due to the intermolecular interactions, which favour even more T2 adsorption. This, of course, is due to the weaker quantum nature of the heavy isotope, which tends to concentrate its wave function in the region of the potential minimum more that the protium dimer. 68–70 Notice that the finite pressure estimates provided 39 were obtained adding an alchemical transformation step (T2 ↔H2 ) during constant chemical potential simulations, which allowed to circumvent possible broken ergodicity issues due the strong binding energy of X2 with the CNT’s. An increase in S(T2 /H2 ) may, at least in principle, be expected also inside ice pores, provided that the average distance between adsorption sites is comparable with the equilibrium distance between to hydrogen molecules. Unlike the CNT case, however, it is quite difficult to provide even at rough approximation for

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the magnitude of such effect without a direct simulation, as this may be markedly dependent on the relative population and spatial distribution of the different adsorption sites inside the pores.

4

General considerations and conclusions

In this work, we have investigated the quantum energetic and structural properties of H2 , D2 and T2 adsorbed onto frozen ammonia clusters at 0 K. This is intended as first step in exploring whether solid ammonia species may be useful in separating hydrogen isotopes. Albeit the behaviour of the adsorption energy and quantum distribution as a function of the isotopic mass matches general expectations, their detailed analysis reveals, instead, that the interaction mode characterizing the adsorption of the heavy isotopes with solid ammonia is profoundly different from that of protium. In particular, the larger mass of D2 and T2 foster a marked raise in the adsorption energy Eads thanks to their increased ability in adapting the angular distribution to the local nuances of the interaction potential. In parallel with the increase in Eads , we also observed and increase in both the average translational and rotational kinetic energies, which should instead decrease. This finding is a clear indication of a strong coupling between degrees of freedom, which may be exploited for separation purposes, and of an adsorption mechanism that differs from what suggested in the case of CNT. To investigate if the coupling between the degrees of freedom of the adsorbed species may indeed translate into the capability of separating hydrogen isotopes, we have exploited the Langmuir adsorption theory to estimate adsorption isotherms and low–pressure selectivities S0 (X2 /H2 ) for single molecule adsorption onto binding sites in (NH3 )n . The latter approach is justified as an initial approximation on the basis of the marked separation between adsorption sites introduced by the rim of dangling hydrogens forming the typical cyclic structure onto which X2 adsorbs. Form the latter analysis, it turned out that ammonia aggregates may

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perform somewhat better than, or at least as well as, (2,8) CNT as, for instance, S0 (T2 /H2 ) ≃ 200 and S0 (D2 /H2 ) ≃ 20 for the heptadecamer, while S0 (T2 /H2 ) = 50.2 and S0 (D2 /H2 ) ≃ 25 71 for the (2,8) CNT at 20 K. Albeit S0 (D2 /H2 ) for the other ammonia clusters is smaller by a factor of three than for the heptadecamer due to the lower ∆E0 (AB) (Figure 6), they may still maintain some degree of separation capability compared to methods currently in use thanks to the substantial preference for adsorbing deuterium dimers rather than H2 . In developing the real potential of, for instance, amorphous ammonia ice in separating hydrogen isotopes, the discussion just presented suggests that a key factor would clearly be represented by the total amount of ad–molecules with dangling N–H bonds, as well as the relative distributions of the rim width for the vicinal cyclic motif over which the quantum density may be localized. Thus, it would be important to investigate under which preparation conditions the population of the strongest adsorbing sites can be maximized. Notice that this aspect is of particular importance given the possibility that kinetically trapped local structures may present a somewhat higher number of such features, which, instead, tend to be minimized in the clusters studied in this work due to energetic reason. If it is so, one may hope to further increase the selectivity for the heaviest isotope, an intriguing possibility especially in view of the fact that materials capable of high S0 such as (3,6) CNT are yet to be synthesized and thus represent only a theoretical possibility at this moment in time. Apart from the difficulties that may be encountered in the preparation of extremely narrow CNT, a possible additional advantage afforded by ammonia solids may be due to the (p,T) conditions under which they can be effectively operated. As an example, we recall here that H2 adsorption isotherm into (3,6) CNT seems to saturate already at fugacities of the order of 10−20 –10−19 bar at 20 K. 39 Albeit it may be that such low values are due to an interaction potential that is too strongly binding, 39,72 one could nevertheless be forced to operate the process at very low pressures in order to allow the onset of the equilibria involved in the separation. Given the high Eads shown by all isotopes in the case of CNT, there is, in fact, the risk that H2 may quickly fill the pores, at least partially, due to its higher average

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speed and large abundance. 12 If this happens during the initial stages, the heavier isotopes could be thus effectively prevented from entering the CNT, or they could only form a top layer impeding H2 from exiting the pore as the latter is not wide enough to allow molecules exchanging positions. As there is no need for a narrow structure in order to differentiate adsorption properties on amorphous ice, one may be able to generate sufficiently large pores so that a blockade effect due to fast filling by the lighter isotope may be prevented. Assuming to operate a pulsed gas flow, any adsorbed H2 would be easily displaced by the D2 present in the next incoming batch of gas mixture until the material is saturated with the heavy isotope. Alternatively, one may exploit the similarity in value between Eads for H2 (Table 2) and the surface chemical potential of H2 in medium size clusters 68–70 choosing a temperature around or slightly above 20 K in order to completely prevent the protium dimer from competing for surface adsorption. In this way, the latter would be continuously transported across the stationary phase provided by the ammonia ice leaving behind adsorbed D2 . Concluding our presentation, we would like to indicate the avenues of exploration that appear worth considering. First, it seems important to develop models for amorphous ammonia ice as the ones already available for water ice 27,28,73 and with the specific intention of understanding the possible relationship between the parameters of the formation process and the structural details (e.g. site distribution, pore volume) of the solid species. Possible differences in aggregate structures due to the inclusion of quantum effects 63 with, e.g., Ring Polymer Dynamics 74 seem also worth investigating. With such models made available, it would be then possible to directly investigate adsorption isotherms and isotopic separation performances at finite T and p, as well as possible contributions to quantum sieving due to dynamic effects. 20–22,25 Here, we specifically consider extending a recently proposed Gaussian–based time dependent Hartree dynamics. 75 A “vis–à–vis” comparison with similar data obtained with water ice would also help to foster a better understanding of all variable that play a role in these complicate phenomena.

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In conclusion, it seems worth noting that much of the future work requires simulations with more than a single hydrogen molecule, so that an appropriate description for the H2 –H2 interaction need to introduced. In view of reducing the system complexity and computational costs, one may wish to avoid a full–dimensional 4D PES relying instead on the newly developed Adiabatic Hindered Rotor (AHR) approximation 1D curves; these seem capable of accurately constructing intermolecular densities, hence computing shifts in Raman spectra, of clusters composed of H2 , D2 and T2 . 76 Maintaining consistence between levels of description for intermolecular interactions would, however, require to employ a similar approximation also for the ammonia–H2 PES, which is likely to perform somewhat less well as it happened in the water–protium dimer case. 77 Indeed, the performances of AHR PES may worsen upon increasing the isotope mass as the spacing between rotation levels reduces, a fact that may foster preferential orientations between two dimers due to their interaction with the same or nearby adsorption sites. As the H2 –H2 PES is anisotropic, disregarding such possibility may lead to an unbalanced description of the adsorption energetics at non–zero pressure, whose accuracy may be key in properly gauging separation capabilities. Whether or not effects due to the relative orientation of H2 –H2 would be weaken thanks to the juxtaposition of quantum delocalisation and repulsive quadrupole–quadrupole interaction has never been investigated previously and is presently difficult to gauge. We leave such task for future work on medium and large size hydrogen isotope aggregates.

Acknowledgement EC acknowledges support from the ACS–PRF grant number 55264–UR6, and Arcadia University’s Faculty Development Funds. M.M. acknowledges funding from the Università degli Studi dell’Insubria under the scheme “Fondi di Ateneo per la Ricerca–FAR2014”.

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Supporting Information Available Images presenting the putative global minima for the (NH3 )n –rb–X2 systems studied in this work. This material is available free of charge via the Internet at http://pubs.acs.org/.

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Quest for Inexpensive Hydrogen Isotopic Fractionation: Do We Need

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