Thermally-driven isotope separation across nanoporous graphene

Chemical Physics Letters 521 (2012) 118–124

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Thermally-driven isotope separation across nanoporous graphene Joshua Schrier a,⇑, James McClain a,b,1 a b

Department of Chemistry, Haverford College, Haverford, PA 19041, USA Department of Chemistry, Princeton University, Princeton, NJ 08544, USA

a r t i c l e

i n f o

Article history: Received 14 October 2011 In final form 24 November 2011 Available online 2 December 2011

a b s t r a c t Quantum tunneling contributes to the transmission of atoms through nanoporous graphene barriers, even at room temperature. In a temperature gradient, the mass-dependence of tunneling leads to isotope separation, in contrast to the classical transmission case where no separation can occur. Using transition state theory, we show that zero-point and tunneling contributions enrich the isotopes in opposite directions with respect to the temperature gradient. Zero-point energy differences dominate around room temperature. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Experiment [1] and theory [2] indicate that a single graphene sheet is impermeable to gases even as small as helium; pores are required for transmission of atoms or molecules. The synthesis of two-dimensional polyphenylene (2D-PP), a porous analogue of graphene, by Bieri et al. [3] (Figure 1b) has stimulated computational predictions that this material has unparalleled capabilities for the separation of He and H2 from a variety of industrially-relevant gas mixtures [4–6]. Most of this work, and related computations on gas [7] and liquid [8] transport through hypothetical porous graphenes, considered only classical transmission. However, quantum tunneling plays a role in transmission of atoms through 2D-PP even at room temperature [4], motivating us to study how the mass-dependence of tunneling might lead to isotope separation. Moreover, previous transition-state theory calculations of atom-passage through 2D-PP [5,6] have not considered the role of the zero-point energy difference of the activated complex, which is also mass-dependent. To study these two effects, we consider a 2D-PP barrier separating two gas reservoirs containing a mixture of gaseous helium isotopes (Figure 1a) at different temperatures. In principle, there are no impediments to constructing this in the laboratory, since 2D-PP has been synthesized [3] and calculations indicate that it will adhere to a nanoporous alumina support structure [6]. Placing a graphene material between two reservoirs of gases of different composition and temperature could be performed using an apparatus similar to that used by Bunch et al. [1] to demonstrate the impermeability of graphene to helium. Enrichment (or depletion) of the isotope mixture can be determined either using a gas-density balance to determine the mean molecular weight of the gas ⇑ Corresponding author. Fax: +1 610 896 4963. 1

E-mail address: [email protected] (J. Schrier). Present address.

0009-2614/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2011.11.069

(to precisions of 0.01 amu [9]) or by more sensitive methods of He isotope detection, e.g., mass spectrometry and atom trap trace analysis methods [10], which can determine the isotopic ratios to parts-per-million precision. From a practical standpoint, this type of device might be useful for separating 3He from 4He, motivated by shortages of the former isotope [11]. Like classical thermodiffusion, this separation method involves no moving parts, no energy-intensive phase changes, and operates under steady-state conditions. From a fundamental standpoint, such a device would also serve as a simple experimental test of chemical dynamics theories. The role of atom tunneling (typically of hydrogen atoms) on chemical reaction rates is also widely studied [12], and the typical evidence for tunneling is a deviation from classical Arrhenius behavior of the temperature dependence. However, Schatz noted that tunneling contributes more than 75% of the total rate constant in the H + H2 ? H2 + H reaction (and its isotopic variants) even at room temperature where there is no significant deviation from the Arrhenius behavior [13,14]. Even when deviations are present, they are often due to zero-point energy differences, either of the reactants or of the transition state, which can be difficult to distinguish from tunneling effects. This motivated us to study what conditions give rise to a definitive signature of tunneling in this type of experiment.

2. Theory Consider the system depicted in Figure 1a, where a barrier, e.g., 2D-PP, separates ‘cold’ (C) and ‘hot’ (H) compartments, a. External reservoirs with absolute temperature Ta maintain the gradient; it is convenient to define the inverse-temperatures ba = 1/kBTa, where kB is the Boltzmann constant. Each compartment contains a mixture of two isotopes of a classical ideal gas, where Pa,i is the partial pressure of isotope i in compartment a. We will compute the relative partial pressures of the isotopes once the system reaches a

J. Schrier, J. McClain / Chemical Physics Letters 521 (2012) 118–124

TC

(a)

Below we calculate the isotope concentrations predicted by each of these theories.

TH 3

3

4

4

4

3

2.1. Classical barrier transmission

4 4

3

From the kinetic theory of gases, the frequency of molecule collisions with the membrane is [16],

4

4

3

3

ca;i

3 4

(c) Probability Density

TC

Transmission probability

TH

tclassical ¼ a;i

Energy 1.0

rffiffiffiffiffiffiffiffiffiffiffi Z 1 pffiffiffiffiffiffiffiffiffiffiffi mi ba 1 2 eba mi v x =2 dv x ¼ Erfc½ ba V 0 ; p ffiffiffiffiffiffiffiffiffiffiffi 2 2p 2V 0 =mi

cC;i tC;i  cH;i tH;i ¼ 0:

0.6

ð2Þ

ð3Þ

Rearranging to find the ratio of the partial pressure of species i on each side,

0.4

PC;i ¼ PH;i

0.2

0.0 0.48

0.50

0.52

Energy

( e)

0.54

0.56

eV

1.15

sffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi bH Erfc½ bH V 0  pffiffiffiffiffiffiffiffiffiffiffi : bC Erfc½ bC V 0 

ð4Þ

Eq. (4) is mass-independent, demonstrating that classical barrier transmission cannot lead to a relative isotopic enrichment. Conceptually, this is because classical transmission is only a function of kinetic energy, and since the two isotopes have identical kinetic energy distributions at a given temperature, they are transmitted identically. 2.2. Quantum barrier transmission

1.10

t α,3 t α,4

ð1Þ

where Erfc[] is the complementary error function. After a sufficient time, the system reaches a steady-state where there is no net particle flux,

3He 4He Classical

0.8

sffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼A Pa;i b1=2 a ; 2pmi

where A is a geometric constant due to the area of the membrane which is the same for a = C, H, and mi is the isotope mass. Labeling the direction normal to the membrane as the x-axis, the probability of transmission through the barrier, si[Ex], is a function of the xcomponent of the kinetic energy of the particle, Ex ¼ mi v 2x =2, where vx is the x-component of the velocity. Classical barrier transmission is only a function of kinetic energy and not mass. Particles with Ex greater than the maximum potential energy of the barrier, V0, are always transmitted, and particles with Ex < V0 are never transmitted. The thermally-weighted transmission probability, ta,i, is determined by integrating the product of si[Ex] with the Maxwell– Boltzmann distribution of velocities, yielding

(b)

(d)

119

In contrast, quantum particles have a mass-dependent si[Ex], as shown in Figure 1d. Low-mass particles behave less classically— greater si[Ex] for Ex < V0 and lower si[Ex] for Ex > V0—than high-mass particles. The crossover between the transmission of the two masses at V0 is common to all single-barrier quantum transmission problems, e.g., the square-barrier problem [17]. As a consequence,

1.05

1.00 0

200

400

600

800

1000

Temperature (K) Figure 1. (a) Proposed experimental setup of two reservoirs separated by a barrier of (b) two-dimensional polyphenylene (2D-PP) [3]; (c) Maxwell–Boltzmann distribution; (d) transmission probability as a function of kinetic energy, si[Ex]; (e) 3 He/4He thermally-weighted quantum transmission ratio, ta,3/ta,4. Both (d) and (e) are computed using the MP2/cc-pVTZ potential energy surface for passage of He atoms through 2D-PP from Ref. [4].

steady state. For kinetic theories that explicitly consider the potential energy surface, one can either follow the passage of the system over the potential energy surface using classical or quantum mechanics, or else make the assumption of equilibrium between the reactants and an activated complex (‘transition state’) [15].

ta;i ¼

 1=2 Z 1   mi ba 2 si mi v 2x =2 eba mi v x =2 dv x ; 2p 0

ð5Þ

is a function of mass and must be evaluated numerically, and then substituted into Eq. (3) to obtain the partial-pressure ratio under steady-state conditions. Unlike the classical case, the mass dependence of si in the quantum case (Figure 1d) leads to a mass dependence of ta,i and Pa,i, in contrast to Eq. (4). Conceptually, the ‘hot’ a has a greater fraction of particles with Ex > V0 particles and the ‘cold’ a has a greater fraction of particles with Ex < V0 (Figure 1c). The ‘hot’ and ‘heavy’ particles have a higher probability of exiting the hot reservoir, while the ‘light’ and ‘cold’ particles have a higher probability of exiting the cold reservoir, resulting in a relative isotopic enrichment/depletion due solely to the different quantum tunneling

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probabilities. Thus in the quantum case, the separation factor, defined by the ratio

ðPC;4 =PC;3 Þ ; ðPH;4 =PH;3 Þ    t H;4 t C;3 ; ¼ t H;3 t C;4

r¼

ð6Þ ð7Þ

has a value of unity when no isotope concentration occurs (e.g., in the classical case using Eq. (4)), and deviations from unity indicate isotope enrichment or depletion. Figure 1e shows a plot of ta,3/ta,4 computed using the MP2/cc-pVTZ potential barrier for passage of He atoms through 2D-PP from Ref. [4]. We note two properties: (i) For smooth, single potential energy barriers, ta,3/ta,4 > 1 for all Ta, because low mass particles are more prone to tunneling; (ii) ta,3/ta,4 is a monotonically decreasing function of Ta with an asymptote of unity. Consequently, r > 1 for TC < TH, i.e., tunneling leads to an enrichment of heavy mass particles in the cold reservoir and a depletion of light mass particles in the hot reservoir. This isotopic enrichment arises solely from the barrier transmission, and not any non-ideality of the gas.

Figure 2. Optimized geometry of the BLYP/6-31G⁄ transition state for helium passing through a finite molecular analogue of 2D-PP.

2.3. Transition state theory Transmission across the barrier can be considered as a chemical reaction, where the reactants are the barrier and the gaseous atom in one temperature reservoir, and the product is the barrier and atom in the other temperature reservoir. In transition state theory, the activated complex is assumed to be in equilibrium with the reactants [15,16,18]. The rate constant for the reaction of a particle of mass i leaving reservoir a, is given by [16,18]

ka;i ¼

Q zi

kB T a eðV 0 þDi Þba ; h Q b Q a;i

ð8Þ

where h is the Planck constant, Qb is the partition function of the isolated barrier (e.g., 2D-PP), Qa,i is the partition function of the gaseous isotope i at temperature Ta, and Q zi is the partition function of the activated complex. Since the barrier is between the two reservoirs, it must have the same temperature for the H ? C and C ? H directions of the reaction, so Qb and Q zi are both independent of the temperature reservoir index a. Since the barrier is attached to a surface, it has no translational or rotational degrees of freedom [18], so Q zi consists only of the vibrational degrees of freedom, some of which depend on the mass of the gas atom. As in Section 2.1, V0 is the maximum potential energy barrier height, but because of the different vibrational normal modes of the activated complex, the two isotopes may have a different zero-point energy, Di, which makes the effective activation barrier height mass dependent. At the steady state,

kC;i ½iC   kH;i ½iH  ¼ 0;

ð9Þ

where [ia] is the concentration (number density) of i in reservoir a. Using the ideal gas law,

½ia  ¼

Ni;a Pi;a ba ¼ ; Va NA

ð10Þ

where the last equality follows from the ideal gas law (NA is Avogadro’s constant). Substituting Eqs. (8) and (10) into Eq. (9) and rearranging for the ratio of partial pressures in the two compartments yields

 1=2 Pi;H Q H;i eðV 0 þDi ÞbC bC ¼ ¼ eðV 0 þDi ÞðbC bH Þ ; ðV þ D Þb P i;C Q C;i e 0 i H bH

ð11Þ

where Eq. (11) follows from substitution of the (translational) partition function for the ideal monoatomic gas,

2pmi

Q a;i ¼

!1=2 ð12Þ

:

2

ba h

Eq. (11) can then be used to obtain the separation factor (Eq. (6)), yielding

rTST ¼

eðV 0 þD3 ÞðbC bH Þ ¼ eðD3 D4 ÞðbC bH Þ : eðV 0 þD4 ÞðbC bH Þ

ð13Þ

When D3  D4 = 0, this recovers the classical result of Section 2.1. Recalling the quantum harmonic oscillator problem [16], the vibrational frequencies scale inversely with particle mass so D3 > D4, and thus D3  D4 > 0. Choosing TC < TH, then bC  bH > 0, and consequently rTST < 1, in contrast to the quantum tunneling result of Section 2.2. To determine the magnitude of D3  D4 we performed BLYP/6-31G⁄ optimization of the transition state geometry, followed by a harmonic vibrational frequency calculations for the finite molecular analogue of 2D-PP used in Ref. [4] using GAUSSIAN 09, Revision B.01 [19]. The optimized transition state geometry is shown in Figure 2. The difference between the zero-point energies of all the modes of the two isotopic transition states is D3  D4 = 47.3647 cm1 = 5.87247  103 eV. See Supporting information for optimized coordinates and a full list of vibrational frequencies. BLYP/6-31G⁄ overestimates the vibrational frequencies by about 6% [20], which in turn overemphasizes the role of the zero-point energy. The simplest way to incorporate quantum tunneling effects into transition state theory is to include a ‘tunneling factor’ [13,21],

ja;i ¼

t a;i ; t classical a;i

ð14Þ

in terms of Eqs. (2) and (5), as a multiplicative constant in the rate expression Eq. (8). Incorporating ja,i into the rate constant and solving the steady-state condition, jC,ikC,i[iC]  jH,ikH,i[iH] = 0, leads to a multiplicative factor of jC,i/jH,i in Eq. (11), and a separation factor

rTSTþQ ¼



jC;3 jC;4







jH;4 ðD3 D4 ÞðbC bH Þ tH;4 e ¼ jH;3 tH;3



 tC;3 r TST ; tC;4

ð15Þ

where the last equality follows from the mass-independence of tclassical (Eq. (2)). The only difference between this expression and a;i Eq. (7), is the effect due to the zero point energy of the activated complex. As described in Section 2.2, the product of the first two terms is greater than unity. Since rTST < 1 was demonstrated above for TC < TH, then rTST+Q > 1 only occurs due to tunneling effects and

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not zero-point differences. Tunneling leads to an enrichment of heavy mass particles in the cold reservoir and a depletion of light mass particles in the hot reservoir, whereas the zero-point contributions lead to isotopic enrichments in the opposite direction. The latter does not indicate the absence of tunneling, but the former is a definitive indication of tunneling.

1000 0.98

0.998 800

0.99 1.00

1.002

0.96

TC K

A possible complication is classical thermodiffusion, in which the long-range interactions of a non-ideal gas (independent of the barrier) in a temperature gradient give rise to isotope separations [22]. In the absence of convection currents, this leads to a steady-state separation factor of

600

1.005

1.01 400 1.02 1.04 200

ð16Þ

where experimental values of the thermodiffusion factor, d, for 3 He/4He mixtures are between 0.061 and 0.065 for temperatures between 273 and 373 K [23]. For d > 0, this results in the low-mass particles becoming enriched on the ‘hot’ side and the high-mass particles becoming enriched on the ‘cold’ side, just like the isotopic enrichments due to the tunneling effects in Section 2.3. The opposite occurs for d < 0, e.g., UF6 discussed in Section 3.3. Since the tunneling isotopic enrichment is due to the nature of the barrier (computed for an ideal gas), and the classical thermodiffusion isotopic enrichment is due to the non-ideality of the gas (computed without a barrier), the phenomena have independent origins and may be considered as additive.

1.14 200

400

600

2.22 800

1000

1000 1.14 800

(b)

1.02 1.04

1.55

1.08 1.00 0.98

1.25

TC K

 d TH ; TC

0.88 0.45

2.4. Classical thermodiffusion

q¼

(a)

0.995

600

0.96

0.93 400

3. Results and discussion

0.88

3.1. Magnitude of the quantum tunneling and zero-point effects 0.80

200

0.65 200

400

600

800

1000 200 160

400

1.01

0.94 120 0.83 0.99 1 0.99 1.01 100 1.06 1.20 80 80 100 120 140 160 180 200

1.20

1.00

1.03

800 140 0.94 1.01 1

600

1000

(c)

1.06

180

TC K

Having established the basic theory, we used the previously reported [4] MP2/cc-pVTZ potential barrier for passage of He atoms through 2D-PP, and resulting si[Ex] (Figure 1d), and ta,i, to demonstrate the feasibility of this proposal. The 3He/4He separation factor, computed from Eq. (7) with varying reservoir temperatures, is shown in Figure 3a. The diagonal line indicates TC = TH, where there is no temperature gradient between the two compartments and no relative change in the isotope concentrations. To the right of this line, there is an enhancement of 3He in the H compartment, and an enhancement of 4He in the C compartment. Moving to the left of this line produces the opposite effect, and the separation factor is the inverse of that of the opposite side of the TC = TH divide. The largest isotope enrichment occurs when the temperature difference is greatest. However, there is a diminishing increase in the separation ratio as TH is increased, because the first term in Eq. (7) asymptotically approaches unity with increasing TH, as seen in Figure 1e. A larger separation ratio can be achieved by decreasing TC, with the caveat that ta,i can become very low, requiring an impractical duration of time to reach the steady state. For the sake of discussion, we will consider a conservative TC = 273 K. For the conditions of TC = 273 and TH = 373 K, the separation factor is 1.017. For the conditions of TC = 273 K and TH = 1000 K, the separation factor is 1.036. The separation factor according to transition state theory (without tunneling), Eq. (13), is shown in Figure 2b. As proved in Section 2.3, rTST < 1 for TC < TH, leading to isotopic separation in an opposite direction to that done by tunneling effects. For the conditions TC = 273 K and TH = 373 K the ratio is 0.935, and for TC = 273 K and TH = 1000 K the ratio is 0.834. As in the tunneling case, the largest separation occurs when the temperature gradient is greatest.

0.99 0.97

1.06

1.11 0.94 0.90

200

0.83

200

400

600

800

1000

TH ( K ( Figure 3. Isotope separation factor (Eq. (6)) as a function of thermal reservoir temperatures, using the MP2/cc-pVTZ potential barrier for passage of He atoms through 2D-PP and the BLYP/6-31G⁄ transition state and vibrational frequency data. (a) Quantum transmission, r (Eq. (7)); (b) classical transition state theory, rTST (Eq. (13)); (c) transition state theory with tunneling correction, rTST+Q (Eq. (15)); the inset shows the same data at low temperature.

J. Schrier, J. McClain / Chemical Physics Letters 521 (2012) 118–124

3.2. Effect of potential energy barrier: the symmetric Eckart barrier In this section, we consider only the purely-tunneling separation factor, r (Eq. (7)), and study the role of the potential barrier height and width, using the symmetric Eckart barrier, 2

V½x ¼ U Sech ½ax;

ð17Þ

due to its well-known analytical solution for si[E] [17]. Model parameters fit to the atomistic potential of Ref. [4] correspond to U = U0 = 0.0197258 hartrees and a = a0 = 0.651209 inverse bohr radii. Under conditions of TC = 273 K and TH = 373 K, this model potential yields a separation factor of 1.021, and for TC = 273 K and TH = 1000 K, a separation factor of 1.043. Thus the Eckart barrier is a reasonable approximation of the atomistic potential based on its agreement with the separation factors calculated in the previous section. Figure 4 shows the relative separation factor of 3He/4He using the Eckart barrier with varying height, U, and width, a, at reservoir temperatures TC = 273 K and TH = 373 K. To make the connection to barrier width more intuitive, the full-width at half-maximum (FWHM) in Ångstroms corresponding to the particular a values is also indicated on the bottom axis. The parameters corresponding to 2D-PP are indicated by the square. The dashed line leading away from it shows how the potential parameters scale with increasing pore size, e.g., by drilling holes in graphene using an electron-beam [24–26]. This is obtained by treating the edge of the pore as a ring that interacts with the atom by a Lennard–Jones potential, where the potential energy of the approach atom can be decomposed into a function of the He-plane distance, x, interaction parameters, r

log a a0 0.8 1.06

1.0

0.6

0.4

0.2

0.0

0.2 0.8

3.0 2.4 1.12 1.22 1.8

0.4

1.2 1.42

2.50 2.00 1.67

0.6

1.04 0.0 1.01 1.06

1.22

0.2

0.2

1.02

1.12

0.4

log [U/U0 ]

The separation factor according to transition state theory with tunneling, Eq. (15), is shown in Figure 2c. For the conditions TC = 273 K and TH = 373 K the separation factor is 0.952. For comparison, classical thermodiffusion (Eq. (16)) with this temperature difference leads to a separation factor of 1.02, and will thus be exceeded by the zero-point energy dominated isotope effect studied here. For TC = 273 K and TH = 1000 K the separation factor is 0.865. The competition between zero-point energy barrier hopping and tunneling leads to more complicated behavior than the previous cases. First, the zero-point and tunneling contributions can cancel when TC – TH, giving rise to a second contour of unity, that runs with an opposite slope to the one that runs along the diagonal. This second contour of unity arises because the separation factor due to tunneling (Eq. (7)) increases much faster than the separation factor due to zero-point energy differences (Eq. (13)) at low temperatures. These two contours cross at 121 K, as shown in the low-temperature inset in Figure 3c. Second, at low temperatures increasing the temperature gradient no longer leads to the largest separation factor, due to the opposite behavior with respect to temperature of the last term and the first two terms in Eq. (15). Both the crossover to the tunneling-dominated behavior (r > 1) and the non-monotonic behavior can be studied using a liquid nitrogen bath at atmospheric pressure to set TC = 77 K, which is still sufficiently high of a temperature that helium behaves as a classical gas. In general, because the zero-point energies raise the height of the barrier, the classical transition state process should occur more slowly than the quantum tunneling process. Consequently, at short times, one might expect to see isotopic enrichment similar to the tunneling-only calculation (i.e., Figure 3a), which only reach the final steady-state (i.e., Figure 3c) over a long time. Calculating the time-scales involved with these two processes is beyond the scope of the current work, since it requires additional knowledge of the system, such as the temperature of the barrier and the barrier partition function.

U eV

122

0.4

0.7

1.0 1.3 1.6 1.9

0.8

FWHM ( ) 3

4

Figure 4. He/ He separation factor, r (Eq. (7)), as a function of barrier height, U, and full-width at half maximum (FWHM) of the symmetric Eckart barrier (Eq. (17)) between thermal reservoirs at 273 and 373 K. The square corresponds to parameters corresponding to 2D-PP. The cross-hatched region at the top of the graph shows parameters with flux rate <1 lmol cm2 s1 at P = 1 atm, T = 273 K.

and , and an effective radius of the pore, R, as V[x] = 4 {r12/ (x2 + R2)6  r6/(x2 + R2)3}. Model parameters fit to the MP2/ccpVTZ potential correspond to  = 0.0192151, r = 1.86703, R = 2.14299, all in atomic units. In general, as the pore size is increased, the barrier height rapidly decreases and the barrier width slowly increases. A pore that is 40% larger (R  3) than that of 2DPP would still give rise to a 1% concentration enhancement in the tunneling-only case. If the flux rate is too small, the time needed to reach the steady-state may exceed experimental patience. For a gas at room temperature and pressure, there are approximately 1023 collisions cm2 s1 [16]. Thus, ta,i  1017, corresponds to a lmol cm2 s1 flux rate under those conditions. The parameters within the cross-hatched region at the top of Figure 4 are where ta,i < 1017 for 3He at 273 K. Performing the experiment with electron-beam generated pores would take advantage of migration and coalescence of the induced defects [27]; the pore size is proportional to electron beam dose. Starting with an impermeable (no hole) graphene barrier, there would be no mixing either classically or via tunneling [1]. A low electron beam dose would lead to primarily small holes, allowing for both quantum and classical effects. Continued electron beam exposure would lead to large holes, eliminating the quantum tunneling contribution and lead to only purely classical thermodiffusion. In this way, one could ‘turn on’ tunneling by opening a pore, and then ‘turn off’ the tunneling enhancement by making the pore so large that the barrier height is negligible, all in the same sample. In summary, the isotopic enrichment is maximized when: (1) The potential barrier is wide enough for exponential decay of the wavefunction to make a difference for the two isotopes, but narrow enough so that the exponential decay does not cause the non-classical transmission to become too small compared to classical transmission. (2) The potential barrier is high enough to make the exponential decay of the wavefunction different for the two isotopes, yet not so large that the thermally-weighted transmission probability is so low as to make reaching the steady-state take

J. Schrier, J. McClain / Chemical Physics Letters 521 (2012) 118–124

log a a0 0.8

1.0

0.6

0.4

0.2

0.0

3.0 2.4

0.2 0.8

1.8

U eV

1.001 0.6

1.0001

0.0

log [U/U0 ]

0.4

1.2

1.00001

0.4

0.2

0.2

0.4

0.7 FWHM ( )

1.0 1.3 1.61.9

0.8

Figure 5. 235UF6/238UF6 separation factor, r, as a function of barrier height and width of the symmetric Eckart barrier, between thermal reservoirs at 330 and 477 K.

123

Dirac distribution for electrons or the Maxwell–Boltzmann distribution for the classical gas here). In the thermoelectric case, the particles are electrons (or holes), so this flux is electrical current. Heat transfer occurs in both cases, mediated by lattice vibrations and electrons in a semiconductor, and the kinetic energy of the gas particles here. Voltage is more generally an electrochemical potential difference, which is just the chemical potential (molar Gibbs free energy) in the atomic case [16]. With this generalized understanding of the quantities involved, the various thermoelectric properties, e.g., the Seebeck coefficient (volts per Kelvin) can be understood as a special case of a more general thermoelectrochemical property (e.g., Gibbs energy per Kelvin). Instead of using the thermoelectric effect to perform electric work, here it performs separation work (i.e., undoing the Gibbs energy of mixing of the two isotopes). Unlike classical thermodiffusion (which relies on the fixed intermolecular interactions), the form of the transmission barrier can be changed, e.g., by chemical modifications, to optimize the separation. For example, a sharply-peaked si[E] is known to maximize the thermoelectric figure of merit [30], and by analogy would maximize the separation efficiency in the chemical case. We are currently investigating how this could be implemented using the resonant-tunneling transmission through a 2D-PP bilayer.

4. Conclusion too long (the specific criterion will depend on the application). (3) A large temperature gradient (especially decreasing TC) is beneficial, subject to the constraint of a sufficiently large ta,i to allow reaching the steady-state. From Figure 4, a small increase in the barrier height and a decrease in the barrier width would lead to greater isotope separation. We speculate that this could be achieved by using atoms with spatially-localized lone electron pairs on the interior of the pore. 3.3. Extension to heaver isotope separation: UF6 It is natural to ask whether this scheme can be used to enrich heavier species, e.g., 235UF6/238UF6. For simplicity, we ignore internal degrees of freedom and treat the molecules as point masses interacting with a symmetric Eckart barrier. The temperatures of the reservoirs are TC = 330 K (the lowest temperature at which UF6 is a gas at 1 atm pressure) and TH = 477 K (a standard process temperature in UF6 production [28]). The results shown in Figure 5 are less promising than the previously considered 3He/4He separation. The thermally-weighted transmission probabilities are low, due to the high mass, thus ruling out most of the potential parameters as ‘too slow’ according to the criterion posed above. To achieve a relative isotopic enhancement of 0.01% requires a potential with FWHM of <0.4 Å, which will be difficult for a large molecule. Increasing the FWHM reduces the possible enhancement to <0.001%. The classical thermodiffusion factor of gaseous UF6 is d = 2.8  105 at the mean temperature (TC + TH)/2 = 403 K [29], leading to a classically-induced enrichment of 0.01%. Because d < 0, the light isotope is enriched in the cold region by classical diffusion, counteracting the smaller quantum effect. Thus we conclude that the tunneling scheme is impractical for 235UF6/238UF6 enrichment. 3.4. Thermochemical analogy to the thermoelectric effect Finally we note that this scheme is the chemical generalization of a thermoelectric device. A temperature gradient across the device leads to a particle flux due to an energy-selective coupling between unbalanced thermal reservoirs (described by the Fermi–

Because a nanoporous graphene barrier is thin, quantum tunneling plays a role in the transmission of atoms through the barrier, even at room temperature. We have shown that the mass-dependence of the tunneling, combined with a temperature gradient, can separate isotope mixtures, in contrast to the classical transmission case where no isotopic separation is possible. Using transition state theory, we demonstrated that the zero-point and tunneling contributions lead to isotopic separations in opposite directions with respect to the temperature gradient. Like classical thermodiffusion, the separation occurs under steady state conditions, but differs in not relying on the non-ideality of the gas. Using ab initio calculations for the barrier, we examined the feasibility of an experiment to separate 3He/4He across an existing nanoporous graphene membrane under modest temperature and pressure conditions. The tunneling contribution, as a function of potential shape, is investigated using an Eckart-barrier model. Similar calculations indicate that this scheme is impractical for 235UF6/238UF6 separation. Acknowledgments We thank Prof. David Lippel for help with GridMathematica, Prof. David Cahill (UIUC) for pointing out the classical thermodiffusion effect, the participants of the George Coleman Lecture Series for comments on a preliminary presentation of this work, and Anna Brockway and William McClain for a careful reading of the Letter. This work was partially supported by the Donors of the American Chemical Society Petroleum Research Fund and Research Corporation for Science Advancement’s Cottrell Scholar grant. This work used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the US Department of Energy under Contract No. DE-AC02-05CH11231.

Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.cplett.2011.11.069.

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