Isotopic Separation of Helium through Nanoporous Graphene Membranes: A Ring Polymer Molecular Dynamics Study
Somnath Bhowmick, Marta I. Hernández, José Campos-Martínez, Yury V. Suleimanov
IIsotopic Separation of Helium throughNanoporous Graphene Membranes: A RingPolymer Molecular Dynamics Study
Somnath Bhowmick, ∗ , † Marta I. Hernández, ‡ José Campos-Martínez, ‡ and YuryV. Suleimanov ∗ , † , ¶ † Computation-based Science and Technology Research Center, The Cyprus Institute, 20Konstantinou Kavafi Street, Nicosia 2121, Cyprus. ‡ Instituto de Física Fundamental, Consejo Superior de Investigaciones Científicas(IFF-CSIC), Serrano 123, 28006 Madrid, Spain ¶ Department of Chemical Engineering, Massachusetts Institute of Technology, 77Massachusetts Ave., Cambridge, Massachusetts 02139, United States
E-mail: [email protected]; [email protected]
Phone: +357 22208721. Fax: +357 22208625
Abstract
Microscopic-level understanding of the separation mechanism for two-dimensional(2D) membranes is an active area of research due to potential implications of thisclass of membranes for various technological processes. Helium (He) purification fromthe natural resources is of particular interest due to the shortfall in its production.In this work, we applied the ring polymer molecular dynamics (RPMD) method tographdiyne (Gr2) and graphtriyne (Gr3) 2D membranes having variable pore sizes forthe separation of He isotopes. We found that the transmission rate through Gr3 is a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n any orders of magnitude greater than Gr2. The selectivity of either isotope at lowtemperatures is a consequence of a delicate balance between the zero-point energy effectand tunneling of He and He. RPMD provides an efficient approach for studying theseparation of He isotopes, taking into account quantum effects of light nuclei motionsat low temperatures, which classical methods fail to capture.
The use of two-dimensional (2D) materials is an open field for many technological applica-tions as well as for basic science.
Graphene, one of the pioneer materials that started thisvery active area of research, acts as an almost impermeable sheet for most atomic and molec-ular species due to its high electron density surrounding the aromatic rings.
Some yearsago, experiments by Lozada-Hidalgo et al. and recently by Creager et al. have shownthat small charged particles such as the proton and its isotopes can penetrate through amonolayer of pristine graphene. There is no complete explanation despite many theoreti-cal calculations, although some kind of chemical interaction seems to be responsible forthis process. Even more recently, it has also be found that hydrogen can permeate pristinegraphene. However, the penetration energy barrier for other neutral atoms is significantlyhigher and therefore supports the widely recognized notion of graphene impermeability. Introducing defects and moderate annealing into the membrane, and nanopores of differentsizes, may enable graphene and other 2D layers to act as atomic and molecular sieves.
The use of 2D materials as a filter at the molecular level is one of the most interestingapplications. Besides using 2D membranes with fabricated nanopores, some compoundscontain nanopores in their structures at different positions and sizes; therefore, they aremore naturally suited for filtering at the molecular level.An interesting class of 2D nanoporous material is graphyne, which is composed of sp − sp hybridized C atoms and can be considered as a graphene derivative. In graphyne,one-third of C − C bonds have been replaced by mono- and poly-acetylenic units ( – C ––– C – ) N .2aughman et al. , theoretically proposed the first stable structures of graphyne, which wastwo decades later synthesized on a copper substrate. N defines the number of acetyleniclinkages and, consequently, the size of the uniformly distributed and repeating sub-nanometertriangular pores. They are termed as graph- N -yne membranes, such as graphdiyne (Gr2),graphtriyne (Gr3), etc ., for N = 2 and 3, respectively. The chemical and mechanical prop-erties of these graphyne membranes have many useful features, such as they are chemicallyinert and stable at ambient temperatures, and flexible enough to withstand deforma-tions induced by high pressures.
The above-mentioned properties of graphynes, coupledwith their unique geometrical structure, make them an excellent candidate to be utilized ingas separation and water filtration technologies. Indeed, one can find many reports on thepurification of gases such as H , N , and O from a mixture of gases and desalinationand filtration of water in the literature.There is a growing worldwide demand for helium purification from its natural resourcesdue to the shortfall in its production. It has numerous industrial and scientific applicationssuch as superconducting magnets, space rockets, arc welding, etc . In particular, the lighterisotope, He, also plays a pivotal role in fundamental research, for example, in neutron-scattering centers, ultracold physics and chemistry, etc . The relative abundance of Heis low ( ≈ × − %) in comparison to its heavier isotope, and its extraction fromnatural gas is usually done by means of expensive cryogenic distillation and pressure-swingadsorption methods. An alternate and more energy-efficient process is to use the 2D porousmembranes in isotopic gas separation since they usually do not involve costly liquefactionof the gases. Many theoretical works have been reported in the last decade on the sep-aration of He isotopes using graphene derivatives, such as: polyphenylene (2D-PP), functionalized graphene pores, nanoporous multilayers,
Gr2,
Gr3, holeygraphene, and graphenylene membranes, etc .The light element He within the vicinity of subnanometer pores is an obvious environmentfor observing the important role of the quantum mechanical effects such as zero-point energy3ZPE) and tunneling effects. A wise approach for an effective isotopic separation maybe is toexploit these quantum properties that could run in the opposite directions. The heavier iso-tope with smaller ZPE will diffuse faster, while quantum mechanical (QM) tunneling favoursthe lighter species. Previously, some of us, using quantum three-dimensional wave packetcalculations have shown that the He/ He selectivity increases with decreasing temperaturefor Gr2 and holey graphene membranes.
It has also been reported that the effect of ZPEis more dominant than tunneling at low temperatures (20 −
40 K). On the contrary, withina low but acceptable gas flux and at low temperatures (10 −
30 K), separation on variousfunctionalized membranes indicates increased selectivity for the lighter isotope due to QMtunneling.
The ring polymer molecular dynamics (RPMD) method, based on the imaginary-timepath integral formalism, is an efficient approach that can accurately and reliably describethe ZPE and deep quantum tunneling effects. RPMD method is essentially a clas-sical molecular dynamics method in an extended ring polymer phase space. It can providereliable estimates of thermal rate coefficients since the RPMD partition function rigorouslyconverges to the QM partition function, the long-time limit of the ring polymer flux-sidecorrelation functions is independent of the choice of the dividing surface that separates reac-tants from products, while its short-time limit is related to various quantum transition statetheories. RPMD was introduced in an ad hoc manner by Craig and Manolopoulos to studythe dynamics of the condensed phase processes, owing to its simplicity and efficiency(scales favorably with the dimensionality of the system). Examples of its successful appli-cation include diffusion in and inelastic neutron scattering from liquid para hydrogen, the translational and orientational diffusion in liquid water, proton transfer in water, diffusion of H and µ atoms in liquid water, hexagonal ice, and on Ni surface, electrontransfer and proton-coupled electron transfer, enzyme catalysis, etc . However, theRPMD method is not restricted to the condensed phases and also found wide applicationin calculating rate coefficients for the gas-phase bimolecular reactions as explored by4uleimanov and co-workers, see for instance, a review by one of us and a recent paper and references mentioned therein. In the permeation of H + and D + on a pristine graphene,the RPMD method also has been used, and the authors point out, again, the importanceof considering both ZPE and tunneling effects for isotopic separation.Coupled with the major scientific and industrial appeal, in this paper, we examine thefeasibility of separation of He isotopes using 2D Gr2 and Gr3 membranes at low temperatures(20 −
250 K) using the RPMD method. A significant part of the success of the RPMDmethod is attributed to the fact that it gives the exact quantum-mechanical rate coefficientfor the transmission through a parabolic barrier, which is advantageous for the presentinvestigation. The primary objective of this study is to provide a reliable estimate of theselectivity and to show that RPMD is a necessary alternative to study these processes wherequantum effects are expected to be very important. To the best of our knowledge, the RPMDmethod has not been applied for the He isotope separation previously and therefore presentsan excellent opportunity to test the accuracy of this method.The paper is organized as follows: in the next section (section 2), we provide the detailsof the RPMD approach along with the PES used in the present study. The results of RPMDrate coefficients and selectivity have been compared with earlier studies in section 3.Concluding remarks are provided in the last section (section 4). We investigate the rate of transmission of both He and He through 2D graphdiyne (Gr2)and graphtriyne (Gr3) membranes using the RPMD method. The unit cell of Gr2 and Gr3has the dimensions (in x − and y − coordinates) of (16.37 Å, 9.45 Å) and (20.82 Å, 12.02Å), respectively. A comprehensive study on the geometry of the membranes can be found inRef. 35. All molecular dynamics simulations have been performed on graphyne membranes5ontaining using same periodic unit as in Ref. 48.The interaction potential between He − graphyne membranes is obtained as an additiveimproved Lennard-Jones (ILJ) He − C pair potentials. The optimized values of the param-eters of the ILJ potentials have been determined from “coupled” supermolecular second-order Møller-Plesset perturbation theory (MP2C) theory using aug-cc-pVTZ and aug-cc-pV5Z basis set for C- and He- atoms respectively. These IJL potentials correspond to thesame potentials as employed in previous studies of He transmission through graphyne mem-branes, and a detailed description can be found elsewhere. -10 -5 0 5 10 z (Å) -20-10010203040 E n e r gy ( m e V ) (a) He - Graphdiyne(b) He - Graphtriyne Figure 1: Variation of the potential energies (in meV) for (a) He atom − graphdiyne and (b)He atom − graphtriyne interaction along the minimum energy transmission pathway (MEP)(in Å). The interaction energies are calculated by using improved Lennard-Jones (ILJ) poten-tials. The MEP corresponds to He atom perpendicularly approaching the geometric centerof graphdiyne and graphtriyne pores with reaction coordinates (0,0, z ).The potential energy curves of He along the transmission pathway ( z − coordinates) toboth membranes are illustrated in Figure 1, while the contour plots in Figure 2(a) and(b) highlight the in-pore displacements of He along x − and y − coordinates. It is quiteevident from both these plots that the minimum energy path (MEP) for an effective Hetransmission may correspond to a straight line perpendicular to the center of the pore. For6he transmission through the Gr2 membrane, the He atom lying at the center of a pore isthe saddle point. The maximum potential barrier height for the MEP is 37.85 meV, which issimilar to the previous studies. In contrast, MEP for the He − Gr3 PEC is devoid ofany potential barrier; instead, a large well (of depth 17.74 meV) appears within the vicinityof the membrane. These results suggest that He penetration of larger Gr3 pores should bemuch easier than that of smaller Gr2 pores, analogous to the report on water permeationby Bartolomei et al. Furthermore, along this MEP and He atom-membrane perpendiculardistance of around 6.0 Å, the interaction potential reaches a plateau and therefore can becharacterized as an asymptotic reactant site. It is interesting to note that the interactionpotential steeply rises to a high value for any movement along the in-pore x − and y − degreesof freedom [see Figure 2(a) and (b)]. − − − − y ( Å ) x (Å) − − − −
50 100 150 200 250 300 350 400 450 500
450 350 250150 50 − − − − y ( Å ) x (Å) − − − − − −
10 0 10 20 30 40 50 60 − (a) (b) Figure 2: Improved Lennard-Jones (ILJ) interaction potentials (in meV) for the displacementof He atom along x and y directions from the center of the pore (0,0,0) on (a) graphdiyneand (b) graphtriyne membranes. The z coordinate of He atom was held at the origin. Theenergy contour step value is 50 meV and 10 meV for (a) and (b), respectively.7 .2 Ring polymer molecular dynamics (RPMD) method The classical Hamiltonian for the system composed of He or He atom under the influenceof external potential arising from M carbon atoms of fixed Gr2 or Gr3 membrane can bewritten in the atomic unit as: ˆ H = | ˆ p | m + V ( ˆ r He , ˆ r C (1) , ..., ˆ r C ( M ) ) , (1)where, ˆ p and ˆ r He (or ˆ r C ( i ) ) are the momentum and position vectors of the i th atom of mass m i . In the RPMD method, the He atom is treated as n classical replicas of the originalparticle, each connected with its nearest neighbor by harmonic springs. The modified ringpolymer Hamiltonian has the form: H n ( p , r ) = H n ( p , r ) + V ( r C(1) , ..., r C(M) ) + n X j =1 V ( r ( j ) He ) , (2)where, H n ( p , r ) = n X j =1 (cid:18) | p ( j ) | m + 12 mω n | r ( j ) − r ( j − | (cid:19) . (3) n is the number of classical beads representing quantum He atom which are connected by aharmonic potential with force constant ω n (= β n ¯ h ). β n ≡ β/n is the reciprocal temperatureof the system, β = 1 / k B T . k B is the Boltzmann constant, T is the system temperature,and ¯ h is the Dirac constant ( ¯ h =1 in atomic units). p ( j ) and r ( j ) are the momentum andposition vectors of the j th bead in the ring polymer necklace of He atom, respectively. Thering polymer trajectory now evolves in f = 3 n total degrees of freedom (in atomic Cartesiancoordinates and including translation and rotational degrees of the freedom of the entiresystem which is a convenient method to propagate RPMD trajectories ).We perform the RPMD simulations at temperatures T below 250 K because, previously,it was observed that the maximum selectivity was obtained at low temperatures. Wechoose seven different T = 20, 30, 50, 100, 150, 200, and 250 K, to study the variation8f selectivities with T and to find an optimal temperature range for isotopic separation.Furthermore, the number of beads is a measure of the resolution of the path integral calcu-lations, i.e. , RPMD calculations scale linearly with n . However, since RPMD calculationsare approximately n times slower than the purely classical calculations, after several trials,we have meticulously chosen the number of beads to be 128. This value of n gives a fair com-promise between computational cost and accuracy (quantum effects of ZPE and tunneling)at low temperatures. The values of n =1, will correspond to a classical calculation.Since RPMD is simply classical molecular dynamics in an extended ( n - bead imaginarytime path integral) phase space, the ring polymer rate coefficient can be expressed as: k ( n )RPMD ( T ) = 1 Q ( n ) r ( T ) ˜ c ( n ) fs ( t → ∞ ) , (4)where, Q ( n ) r ( T ) is the n -bead path integral approximation to the quantum mechanical par-tition function of the reactants per unit volume, and ˜ c ( n ) fs ( t ) is a ring polymer flux-sidecorrelation function ˜ c ( n ) fs ( t ) = 1(2 π ¯ h ) f Z d f p Z d f r e − βH n ( p , r ) δ ( r ) v ( r , p ) h ( r t ) . (5)Here, subscript 0 and t indicates time, δ ( r ) is a delta function centered at r , v ( r , p ) is thevelocity, and h ( r t ) is the Heaviside step function. The RPMD rate coefficient in Equation 4is not straightforward to solve numerically. Therefore, we introduce the Bennett-Chandlerfactorization scheme to simplify Equation 4 that can be solved numerically withoutcompromising its generality. The method has been extensively discussed previously and will not be repeated here. Briefly, in this approach, a reaction coordinate s ( r ) is defined,which monitors the progress of a reaction from the reactant ( s > ) to the product ( s < )site. For the He transmission through 2D membranes, a reasonable reaction coordinate is s ( r ) = ¯ z , where ¯ z is the z - component of the He atom centroid, which follows the minimumenergy path of Figure 1. Within the Bennett-Chandler approach, the RPMD rate coefficient9or a process in which reactants and products separated by a dividing surface at s ‡ (forinstance, center of the pore) can be expressed as a product of two terms: k RPMD ( T ) = k QTST ( T ) κ ( t p ) . (6)Here, the first factor, k QTST ( T ) , is the centroid-density quantum transition-state theory(QTST) rate coefficient. k QTST ( T ) can be calculated from the centroid potential of meanforce (PMF), W ( s ) along the reaction coordinate. If s ∞ is the asymptotic distancein which the He − membrane interaction potential is at the minimum and then introducea dividing surface carefully placed at the TS region, s ‡ , then k QTST ( T ) can be calculatedas: k QTST ( T ) = 1(2 πβm He ) / e − βW ( s ‡ ) R s ‡ s ∞ e − βW ( s ) d s . (7) W ( s ) ’s can be computed by employing the umbrella integration procedure of Kästnerand Thiel. To calculate the PMF profiles, the reaction coordinate s of the He atom hasbeen divided into 130 equally spaced windows (of width 0.05 Å) within the range from 6.00Å (reactant site, s ∞ ) to − W ( s ‡ ) − W ( s ∞ ) = Z s ‡ s ∞ N windows X i =1 " N i P i ( s ) P N windows j =1 N j P j ( s ) (cid:18) β s − ¯ s i ( σ i ) − k i ( s − s i ) (cid:19) d s , (8)with P i ( s ) = 1 σ i √ π exp " − (cid:18) s − ¯ s i σ i (cid:19) . (9)Here, N windows is the number of biasing windows placed along the reaction coordinate.The strength of the force constant ( k i ) of the harmonic biasing potential was chosen tobe 2.72 × − T (K) eV a − . N i is the total number of steps sampled for window i , ¯ s i and10 i are the mean value and variance for the trajectory calculated for the i th window. Ineach umbrella sampling windows, 100 trajectories with different initial configurations werepropagated for 100 ps following an initial equilibration period of 20 ps in the presence of anAndersen thermostat. The ring polymer equation of motion were integrated using velocityVerlet integrator with a step size of 0.1 fs that involves alternating momentum updates andfree ring polymer evolutions. The second term in Equation 6, κ ( t p ) , is the long-time limit of a time-dependent ringpolymer transmission coefficient or the ring polymer recrossing factor and is a dy-namic correction to k QTST . Typically this factor is calculated at the top of the free energybarrier on the PMF profile so as to minimize the time required to reach plateau value. This factor ensures that the final k RPMD value is independent of the choice of the dividingsurface. The mathematical expression for κ ( t p ) can be written as: κ ( t ) = h δ [ s ( r )] ˙ s ( r ) h [ s ( r t )] ih δ [ s ( r )] ˙ s ( r ) h [ ˙ s ( r )] i , (10)where, δ [ s ( r )] constrains the initial configurations to the dividing surface, ˙ s ( r ) is the veloc-ity factor that accumulates the flux through the dividing surface, and h [ s ( r t )] is a Heavisidefunction that gathers trajectories that have crossed over to the product side of the dividingsurface. h [ ˙ s ( r )] is basically a normalization factor that ensures κ ( t → + ) = 1 . To calculatethe recrossing factor, a long “parent” trajectory for He atom of length 2 ns has been carriedout after an initial thermalization period of 20 ps in the presence of Andersen thermostatwith its centroid pinned at the dividing surface using RATTLE algorithms. After each 2ps propagation period of the parent trajectory, 100 trajectories have been generated thathave initial position of the parent trajectory, but their momenta is randomly generated froma Boltzmann distribution. These “child” trajectories are then propagated for 1 ps in theabsence of thermostat and dividing surface constraints.11
Results and discussion
The variation of the RPMD potential of mean force W ( s ) for He and He along the reactioncoordinate s at T = 20 −
250 K are plotted in Figure 3 (Gr2) and Figure 4 (Gr3), andthe corresponding k QTST values are reported in the supplementary information. These PMFprofiles include both potential energy and temperature-dependent entropic contributions.The barrier height at the TS, identified around the membrane plane, increases with increasingtemperature. The free energy profiles enter a shallow well at around s = 3.0 − −
127 meV. ThePMF profiles on Gr3, on the other hand, are more interesting from thermodynamic point ofview. We recall that the He − Gr3 interaction potential does not have any barrier; rather,a well is built around the transmission zone. This behavior is reflected in the free energyprofiles, particularly within the temperature range 20 −
30 K, in which the thermodynamicbarrier in the TS region has a negative (or marginally positive) value compared to thereactant site. This energy barrier gradually takes the shape of a parabola with increasingtemperature and reaches up to a height of 44 meV at 250 K.Since the RPMD method averages points in configurational space on both sides of thepotential energy barrier, modeling the effects of the tunneling and ZPE and leaving a markon the free energy profiles, we can roughly attempt to identify these properties. Comparingthe PMF profiles of He at a particular temperature with the corresponding He’s on Gr2indicate the existence of a prominent ZPE effect; heavier isotope has a lower free energy atthe TS than the lighter ones except at 20 K (see Figure 3(B)), in which quantum tunnelingmay play a crucial role. However, the difference between them at TS decreases with increas-ing temperature. For example, the difference at 30 K is 2.3 meV (see Figure 3(B)), which12ecreases to 0.1 meV at 200 K (see Figure 3(C)). Similarly, the He and He free energy com-parison for Gr3 follows the same trend observed for Gr2, i.e. , they decreases with increasingtemperature. Within 20 −
30 K, He has a smaller TS free energy than He ( ≈ −
150 K, the heavier isotopeprogressively requires more free energy to reach TS, i.e. , PMF profiles for both isotope arealmost identical. Within 200 −
250 K, the free energies of He are consistently larger thanthose obtained for He (see Figure 4(C)). These observations can be notionally interpretedas follows: within 20 −
30 K, the free energy barrier is too broadened for effective tunneling,and consequently, the ZPE effect takes precedence. There is a competition between themat moderate temperatures (50 −
150 K). At still more elevated temperature (200 −
250 K),the greater tunneling probability of He may facilitate a smaller TS free energy comparedto He. Note that this analysis is conceptual because these two quantum effects cannot berigorously separated within the RPMD formalism.There are considerable differences observed between the PMF profiles obtained by theRPMD method with the classical ones at low temperatures (see supplementary information).This, therefore, reinforces our earlier argument of the existence of quantum effects in thesetransmission processes, which the classical calculation fails to capture. For the He trans-mission through Gr2 membrane, the free energy barrier height in the classical calculationsis always lower than those obtained by the RPMD method. The maximum difference for He obtained at 100 K ( ≈
10 meV) and that for He obtained at 20 K ( ≈
12 meV). Notethat the free energy values exponentially contribute to the rate coefficients. As expected,these PMF plots tend to merge with increasing temperature as the quantum effects fade.Comparison between the classical and RPMD method on Gr3 shows that, for both isotopes,the maximum free energy difference at the TS is found at the lowest temperature ( ≈ He within 200 −
250 K, the difference between the RPMD and classical barrier heights becomes 1 meV.The time-dependent transmission coefficients, κ ( t ) , for all temperatures considered in13
20 0 20 40 60 80 100 120−1 0 1 2 3 4 5 (A) W ( s ) / m e V s /Å
20K (i)(ii)30K (i)(ii)50K (i)(ii)100K (i)(ii)150K (i)(ii)200K (i)(ii)250K (i)(ii) −0.4 0 0.4 60 64 68 72 (B) −0.1 0 0.1 (C)
Figure 3: Variation of the RPMD potential of mean force, W ( s ) , (in meV) for (i) He (ii) He atom along the reaction coordinate s (in Å) perpendicular to the graphdiyne membranewithin the temperature range (A) 20 −
250 K, (B) 20 −
30 K, and (C) 200 K. The legendscorrespond to (A), (B), and (C).this study on both membranes are always close to unity (0.98 − κ ( t → ∞ ) , are provided in Table S1. The κ ( t ) value is alwaysclose to unity (0.98 − κ ( t ) value is always 1.00. This implies that most of the He trajectories that reach thecenter of the pore of either Gr2 or Gr3 would overcome the free energy barrier and transportto the other side of the dividing surface. At 20 K, κ ( t ) for He is marginally smaller thanthat obtained for He (0.98 and 0.99 respectively). Note that the classical recrossing factoris always 1, even at low temperatures. 14
10 0 10 20 30 40 50−1 0 1 2 3 4 5 (A) W ( s ) / m e V s /Å
20K (i)(ii)30K (i)(ii)50K (i)(ii)100K (i)(ii)150K (i)(ii)200K (i)(ii)250K (i)(ii) −1 0 1−6−4−2 0 2 (B) −0.4 0 0.4
40 42 44 (C)
Figure 4: Variation of the RPMD potential of mean force, W ( s ) , (in meV) for (a) He (b) He atom along the reaction coordinate s (in Å) perpendicular to the graphtriyne membranewithin the temperature range (A) 20 −
250 K, (B) 20 −
30 K, and (C) 250 K. The legendscorrespond to (A), (B), and (C).
Since the plateau of κ ( t ) is always close to unity, there is practically no significant differencebetween the values of k QTST and k RPMD on both membranes. The variation of k RPMD withtemperature is plotted in Figure 5 and Figure S7 of the supplementary material, while thenumerical values of k RPMD and k QTST are reported in Table S1 (see supplementary infor-mation). It is evident that k RPMD ’s on Gr3 is many orders of magnitude greater than thatobtained on Gr2. The maximum difference was observed at the lowest temperature (bya factor of 10 at 20 K), and with the increase in temperature, this difference decreasesrapidly (by a factor of 10 at 250 K) as the k RPMD vs. T curves start to converge. This isdue to the fact that the value of k RPMD on Gr3 does not change drastically with temperature15confined within 3.63 × s − − × s − ). It starts to decrease slightly with temper-ature up to 50 K and then increases successively with increasing T . On Gr2, however, forboth isotopes, k RPMD increases manifold with temperature, in agreement with the previouscalculations.
This increment in the rate coefficient is more pronounced at the modestrise in T in the low temperature regime than at the high temperature range. For example,the increase in the value of k RPMD is in the order of 10 for the temperature rise from 20 K to30 K or 30 K to 50 K. However, k RPMD increases much less than an order for the consecutive50 K temperature jumps, starting at 150 K. (a) k ( T ) / s − T /K
50 100 150 200 2505.0e+107.5e+101.0e+111.2e+111.5e+111.8e+112.0e+112.2e+11 (b) T /K He (RPMD) He (RPMD) He (classical) He (classical)
Figure 5: Variation of the ring polymer molecular dynamics, RPMD ( k RPMD ) and classical( k cl ) rate coefficients (in s − ) for the transmission of He [red circle (RPMD) and greendiamond (classical)] and He [blue square (RPMD) and orange pentagon (classical)] through(a) graphdiyne and (b) graphtriyne membranes with temperature T (in K). The legendscorrespond to both (a) and (b).The above observations can be explained by inspecting the corresponding PMF profiles.At low temperatures (20 K −
50 K), there is practically no free energy barrier for thetransmission of either isotope on Gr3. However, on Gr2, the free energy barrier is alreadyabove 60 meV at 20 K. The slowly moving He atoms do not have enough kinetic energy toovercome this barrier to reach the other side of the membrane. Therefore, it is not surprisingthat k RPMD ’s on Gr3 is many orders of magnitude greater than those obtained on Gr2, and16he actual He flux through the pores of Gr2 membranes will be extremely slow.
Similararguments can be presented at higher temperatures, although the increased kinetic energyof the incoming He atom will contribute to smaller differences in the k RPMD values. We alsopoint out the analogous quantum wave packet observation reported on the Gr2 and holeygraphene (P7) sheet having a more diffused pore. Finally, when comparing the classicaland RPMD rate coefficients, it is obvious that the classical method overestimates the ratecoefficient within the whole temperature regime studied in this work. This discrepancy isparticularly more apparent at low temperatures, when the quantum effects dominate, andon the Gr2 membrane. For example, at 20 K and for Gr2 membrane, the classical ratecoefficient is more than three orders of magnitude greater than the corresponding k RPMD .However, they do closely follow a similar temperature dependence as obtained by the RPMDcalculations. He/ He selectivity
The He/ He selectivity, defined as the ratio between the RPMD rate coefficient of the heavierHe isotope to the lighter one, k RPMD ( He)/ k RPMD ( He), is plotted as a function of tempera-ture in Figure 6, and the corresponding values are reported in Table S1 (see supplementaryinformation). For Gr2 membrane, He/ He selectivity increases for the temperature rise from20 K to 30 K. At 30 K, the maximum selectivity is obtained ( ≈
2) favouring the heavierisotope and simultaneously indicating pronounced quantum effects as discussed previously.However, this increased selectivity should be accompanied by a decreased permeability. With the increase in temperature, this selectivity ratio progressively becomes smaller andstarts to flatten out, commencing from 100 K. Although at higher temperatures ( >
100 K),the transmission of the lighter isotope is favoured to some extent. We note the analogousselectivity profile was obtained on the P7 sheet. Similarly, for the Gr3 membrane, themaximum He/ He selectivity obtained at 20 K (1.17), which gradually decreases till 50 K,and then remains almost constant (0.87 to 0.83) marginally preferring He for higher tem-17erature regime. The variation of the selectivity is fully consistent with the nature of theirPMF profiles that inherits quantum effects. It is interesting to note that the selectivity plotfor both Gr2 and Gr3 approach each other with increasing T , one from the top and anotherfrom the bottom, and at 250 K, they almost become equal ( ≈ He). This is reasonable since as the quantum effects diminish with temperature and withfewer constraints, the lighter isotope transmission will be slightly favoured. k ( H e ) / k ( H e ) T /K(a) Graphdiyne 1. classical2. RPMD3. WP3D(b) Graphtriyne 1. classical2. RPMD3. WP3D Figure 6: Variation of the He/ He rate coefficient ratio, k ( He)/ k ( He), calculated us-ing 1. classical (triangle) 2. ring polymer molecular dynamics, RPMD (square) and 3.three-dimensional wave packet propagation method, WP3D (circle) for the He transmissionthrough (a) graphdiyne and (b) graphtriyne membranes with temperature T (in K).The selectivity ratios obtained by the RPMD method show good agreement with thoseobtained by the quantum three-dimensional wave packet propagation (WP3D) results ob-tained by Gijón et al. and Hernández et al. for the whole temperature range studiedin this work. In general, the RPMD selectivity profile closely resembles the WP3D results;however, they do overestimate and underestimate to some degree. For example, on Gr2, theRPMD selectivity ratio overestimates the quantum calculations for the whole temperaturerange. The maximum difference between these two methods is found at 30 K (0.2). Sim-ilarly, on Gr3, the RPMD method seems to overestimate the selectivity ratio within 20 K18
30 K and underestimate them for higher temperatures by a maximum of 0.15. Overall,the difference between RPMD and WP3D results for the selectivity ratio lies within 9 − −
15% on Gr3 in comparison to the quantum calculation. On the other hand,the classical selectivity shows no significant change with temperature and remains almostconstant (0.91 − He transmission. This is valid for both membranes.As expected, with the rise in temperature, the classical selectivity follows the RPMD andWP3D ones.
In this work, we have calculated the thermal rate coefficient for the transmission of Heisotopes through the pores of one atom thick graphdiyne (Gr2) and graphtriyne (Gr3) mem-branes using the ring polymer molecular dynamics (RPMD) method within the temperaturerange 20 K −
250 K. Transmission through Gr2 has a substantial free energy barrier evenat 20 K, and this barrier height increases with increasing temperature. On the other hand,transmission through Gr3 can either have marginally negative (up to 30 K) or small positive(T ≥
50 K) free energy barrier. The extent of the barrier height directly impacts the calcu-lated rates, as evident from the fact that the rate coefficient on Gr3 is at least 10 order ofmagnitude greater than on Gr2 at 20 K. The rate coefficient on Gr2 increases rapidly withtemperature and starts to converge, starting from 150 K ( ∼ s − ). However, the ratecoefficients on Gr3 do not vary appreciably with temperature and remain almost constantfor the whole temperature range (3.63 × s − − × s − ). In general, the rate coef-ficient calculated for the transmission through Gr3 is always greater than the correspondingone on Gr2 over the whole temperature regime considered in this work. Moreover, we foundthat the recrossing dynamics have little or no effect on the final value of the rate coefficients.The selectivity ratio, which indicates the preference of either isotope for its permeationthrough the membranes, has been calculated as a function of temperature. From the values19f the selectivity ratio, the quantum effects driving the He transmission, particularly at lowtemperatures, i.e. , the zero-point energy (ZPE) favoring the heavier isotope and the tunnel-ing of the more mobile lighter isotope, has been coarsely deduced. The maximum selectivityratio found in this study ( ≈
2) was obtained on Gr2 membrane at 30 K in which the rate oftransmission of He is almost twice the rate of He seemingly due to a more dominant ZPEeffect. A similar conclusion can be derived on Gr3 at 20 K, where the He/ He selectivityis around 1.2. However, on the Gr3 membrane, the selectivity ratio does not vary consider-ably with the temperature (0.8 − − On the otherhand, the classical method failing to capture either of the quantum effects demonstrated aconsiderable discrepancy with the RPMD and quantum results, particularly at low temper-atures. Therefore, it is of paramount importance to use robust and accurate methods, suchas RPMD, that can correctly describe quantum effects such as ZPE and tunneling whenstudying physical processes for which strong quantum nature is expected.In conclusion, in the present study, we have corroborated the efficient and rigorous na-ture of the RPMD method by determining thermal rate coefficients of physical processesof broad industrial and scientific significance. We also hope that this work will stimulatefuture experimental measurements of the rate coefficients for the separation of He isotopesusing graphene derivatives, taking advantage of quantum effects at low temperatures. As anextension to this work, we would like to investigate the influence of surrounding He atomson the rate coefficient and apply the RPMD method for isotopic He separation on othergraphene derivatives such as polyphenylene, functionalized graphene pores, holey graphene,graphenylene membranes, etc . that were previously reported to serve as excellent atomicsieves. 20 cknowledgement
Y.V.S. and S.B. acknowledge the support of the European Regional Development Fund andthe Republic of Cyprus through the Research Promotion Foundation (Projects: INFRAS-TRUCTURE/1216/0070 and Cy-Tera NEA
YΠO∆OMH / ΣTPATH /0308/31). Y.V.S. wasalso supported by RFBR grant number 20-03-00833. J.C.-M. and M.I.H. were supportedby Spanish MICINN with Grant FIS2017-84391-C2-2-P. We are also indebted to “Centro deSupercomputación de Galicia” (CESGA) and specially to the people in the help desk.
Supporting Information Available
The following files are available free of charge.• supplementary information file: computational details and additional results.
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E-mail: [email protected]; [email protected]
Phone: +357 22208721. Fax: +357 222086251 a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Potential of mean force −20 0 20 40 60 80 100 120−1 0 1 2 3 4 5 6 W ( s ) / m e V s /Å
20K (a)(b)30K (a)(b)50K (a)(b)100K (a)(b)150K (a)(b)200K (a)(b)250K (a)(b)
Figure S1:
Variation of the (a) classical and (b) RPMD potential of mean force, W ( s ) , (inmeV) for He atom along the reaction coordinate s (in Å) perpendicular to the graphdiynemembrane within the temperature range 20 −
250 K. −20 0 20 40 60 80 100 120−1 0 1 2 3 4 5 6 W ( s ) / m e V s /Å
20K (a)(b)30K (a)(b)50K (a)(b)100K (a)(b)150K (a)(b)200K (a)(b)250K (a)(b)
Figure S2:
Variation of the (a) classical and (b) RPMD potential of mean force, W ( s ) , (inmeV) for He atom along the reaction coordinate s (in Å) perpendicular to the graphdiynemembrane within the temperature range 20 −
250 K.2
10 0 10 20 30 40 50−1 0 1 2 3 4 5 6 W ( s ) / m e V s /Å
20K (a)(b)30K (a)(b)50K (a)(b)100K (a)(b)150K (a)(b)200K (a)(b)250K (a)(b)
Figure S3:
Variation of the (a) classical and (b) RPMD potential of mean force, W ( s ) , (inmeV) for He atom along the reaction coordinate s (in Å) perpendicular to the graphtriynemembrane within the temperature range 20 −
250 K. −10 0 10 20 30 40 50−1 0 1 2 3 4 5 6 W ( s ) / m e V s /Å
20K (a)(b)30K (a)(b)50K (a)(b)100K (a)(b)150K (a)(b)200K (a)(b)250K (a)(b)
Figure S4:
Variation of the (a) classical and (b) RPMD potential of mean force, W ( s ) , (inmeV) for He atom along the reaction coordinate s (in Å) perpendicular to the graphtriynemembrane within the temperature range 20 −
250 K.3
Ring polymer transmission coefficient k ( t ) t /fs
20K (a)(b)30K (a)(b)50K (a)(b)100K (a) (b)150K (a)(b)200K (a)(b)250K (a)(b)
Figure S5:
Ring polymer time dependent transmission coefficient, κ ( t ) , in the tempera-ture range 20 −
250 K for (a) He and (b) He atom transmission through the graphdiynemembrane. k ( t ) t /fs
20K (a)(b)30K (a)(b)50K (a) (b)100K (a)(b)150K (a)(b) 200K (a)(b)250K (a)(b)
Figure S6:
Ring polymer time dependent transmission coefficient, κ ( t ) , in the tempera-ture range 20 −
250 K for (a) He and (b) He atom transmission through the graphtriynemembrane. 4
Rate coefficient and selectivity (a) Graphdiyne(b) Graphtriyne k ( T ) / s − T /K He (RPMD) He (RPMD) He (classical) He (classical)
Figure S7:
Variation of the ring polymer molecular dynamics, RPMD ( k RPMD ) and classical( k cl ) rate coefficients (in s − ) for the transmission of He [red circle (RPMD) and greendiamond (classical)] and He [blue square (RPMD) and orange pentagon (classical)] through(a) graphdiyne and (b) graphtriyne membranes with temperature T (in K).5 able S1: Summary of the rate calculations for He and He atom transmission throughthe graphdiyne (Gr2) and graphtriyne (Gr3) membranes at temperatures ( T ) 20, 30, 50,100, 150, 200, and 100 K: k QTST ( k TST ) − centroid-density quantum (classical) transitionstate theory rate coefficient; a κ RPMD ( κ cl ) − ring polymer (classical) transmission coefficient; k RPMD ( k cl ) − ring polymer (classical) molecular dynamics rate coefficient; a He/ He − ratiobetween the He and He rate coefficient. T /K Isotope classical RPMD k TST κ cl k cl 4 He/ He k QTST κ RPMD k RPMD 4
He/ HeGr2 20 He 3.54(-4) 1.00 3.54(-4) 0.91 7.15(-7) 0.98 6.99(-7) 0.89 He 3.21(-4) 1.00 3.21(-4) 6.25(-7) 0.99 6.21(-7)30 He 6.09(0) 1.00 6.09(0) 0.88 4.53(-2) 0.99 4.52(-2) 1.95 He 5.38(0) 1.00 5.38(0) 8.84(-2) 0.99 8.83(-2)50 He 2.00(4) 1.00 2.00(4) 0.84 1.48(3) 1.00 1.48(3) 1.39 He 1.68(4) 1.00 1.68(4) 2.06(3) 1.00 2.06(3)100 He 1.84(7) 1.00 1.84(7) 0.80 4.10(6) 1.00 4.10(6) 1.03 He 1.46(7) 1.00 1.46(7) 4.21(6) 1.00 4.21(6)150 He 2.41(8) 1.00 2.41(8) 0.84 1.65(8) 1.00 1.65(8) 0.93 He 2.02(8) 1.00 2.02(8) 1.54(8) 1.00 1.54(8)200 He 1.03(9) 1.00 1.03(9) 0.81 8.07(8) 1.00 8.07(8) 0.87 He 8.40(8) 1.00 8.40(8) 7.04(8) 1.00 7.04(8)250 He 2.57(9) 1.00 2.57(9) 0.83 2.22(9) 1.00 2.22(9) 0.85 He 2.14(9) 1.00 2.14(9) 1.90(9) 1.00 1.90(9)Gr3 20 He 2.10(11) 1.00 2.10(11) 0.88 7.80(10) 0.99 7.79(10) 1.17 He 1.86(11) 1.00 1.86(11) 9.13(10) 0.99 9.12(10)30 He 1.00(11) 1.00 1.00(11) 0.88 5.19(10) 1.00 5.19(10) 1.08 He 8.77(10) 1.00 8.77(10) 5.62(10) 1.00 5.62(10)50 He 5.41(10) 1.00 5.41(10) 0.85 4.18(10) 1.00 4.18(10) 0.87 He 4.60(10) 1.00 4.60(10) 3.63(10) 1.00 3.63(10)100 He 5.29(10) 1.00 5.29(10) 0.83 4.84(10) 1.00 4.84(10) 0.86 He 4.38(10) 1.00 4.38(10) 4.14(10) 1.00 4.14(10)150 He 6.81(10) 1.00 6.81(10) 0.84 6.46(10) 1.00 6.46(10) 0.86 He 5.74(10) 1.00 5.74(10) 5.53(10) 1.00 5.53(10)200 He 8.58(10) 1.00 8.58(10) 0.84 8.16(10) 1.00 8.16(10) 0.84 He 7.17(10) 1.00 7.17(10) 6.85(10) 1.00 6.85(10)250 He 1.03(11) 1.00 1.03(11) 0.85 1.01(11) 1.00 1.01(11) 0.83 He 8.74(10) 1.00 8.74(10) 8.38(10) 1.00 8.38(10) a The thermal coefficients are given in s − , and the numbers in the parentheses denote powers often., and the numbers in the parentheses denote powers often.