Separation of carbon isotopes in methane with nanoporous materials

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Thermodynamics, Transport, and Fluid Mechanics

Separation of carbon isotopes in methane with nanoporous materials Yun Tian, Weiyang Fei, and Jianzhong Wu Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b00364 • Publication Date (Web): 21 Mar 2018 Downloaded from http://pubs.acs.org on March 22, 2018

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Industrial & Engineering Chemistry Research

Separation of carbon isotopes in methane with nanoporous materials Yun Tian1, Weiyang Fei2 and Jianzhong Wu1* 1

Department of Chemical and Environmental Engineering, University of California, Riverside, CA 92521, United States 2

Department of Chemical Engineering, Tsinghua University, Beijing 100084, China ABSTRACT Traditional methods for carbon isotope separation are mostly based on macroscopic

procedures such as cryogenic distillation and thermal diffusion of various gaseous compounds through porous membranes. Recent development in nanoporous materials renders opportunities for more effective fractionation of carbon isotopes by tailoring the pore size and the local chemical composition at the atomic scale. Herein we report a theoretical analysis of metal organic frameworks (MOFs) for separation of carbon isotopes in methane over a broad range of conditions. Using the classical density functional theory in combination with the excess-entropy scaling method and the transition-state theory, we predict the adsorption isotherms, the gas diffusivities, and the isotopic selectivity corresponding to both adsorption and membrane-based separation processes for a number of MOFs with large methane storage capacity. We find that nanoporous materials enable much more efficient separation of isotopic methanes than conventional methods and allow for operation at ambient thermodynamic conditions. MOFs promising for adsorption- and membrane-based separation processes have also been identified according to their theoretical selectivity for different pairs of carbon-isotopic methanes.

*

To whom correspondence should be addressed. Email: [email protected] 1 ACS Paragon Plus Environment

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1. INTRODUCTION Naturally occurring carbon consists of two stable isotopes, and a minuscule amount of radioactive carbon

12

C (98.89%) and

13

C (1.11%),

14

C (e.g., 1 part per trillion in the atmosphere).

Among different molecular forms of carbon isotopes, labeled methane isotopes 13CH4 and 14CH4 are widely used in organic and analytical chemistry as a powerful agent to detect trace compounds in various chemical and biochemical systems1-7. Highly specific radioactivity is a prerequisite for applications such as medical diagnosis or in vivo detection of receptor bindings5, 8, 9

. Isotopic methanes have virtually identical molecular characteristics and chemical properties.

The isotopic effect on carbon-substituted methanes,

13

CH4,

14

CH4 and

12

CH4, is even more

diminutive in comparison with that introduced by isotopic hydrogen substitutions. Especially, the molar volumes of hydrocarbon compounds with peripheral deuterium atoms are substantially smaller than those with skeletal

13

C or

14

C atoms of the same molecular weight1,

10, 11

.

Conventional methods for isotopic separation are mostly based on macroscopic processes that utilize small differences in the thermodynamic or transport properties of bulk systems8,

12-17

.

Whereas processes such as thermal diffusion, gas-liquid chromatography or cryogenic distillation have been well established for industrial production of carbon isotopes, these traditional techniques have extremely low overall separation efficiency and often entail intensive energy consumption and high operation cost5, 9, 18, 19. Relatively inexpensive methods such as chemical isotope exchange are not suitable for industrial-scale applications because there are extra energy and material costs affiliated with the recovery of the isotopic products20-22. Despite its superior separation efficiency, the laser-induced plasma method cannot be scaled up to

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industrial operations due to high cost and technical difficulties related to the conversion of the direct chemical products to those widely-used isotopic carbon compounds5. Crystalline nanoporous materials such as metal-organic frameworks (MOFs) and covalentorganic frameworks (COFs) have unique pore structures and chemical compositions that can be exploited for industrial-scale gas storage and separation. These materials have attracted significant attention in recent years and are promising for a wide range of applications including efficient fractionation of hydrogen isotopes9, 18, 19, 23, 24. However, we are unaware of previous experimental or computational studies on isotopic methane separation with nanoporous materials. As a matter of fact, reliable experimental data are scarce for the thermodynamic properties of methane with different carbon isotopes. Existing experimental results from the literature are mostly limited to scattering bulk properties of isotopic methanes at few isolated conditions. For example, the experimental data for the vapor pressure isotope effect (VPIE) and the virial coefficient isotope effect (VCIE) are usually not measured in the same range of temperatures25. Because experimental data are indispensable for calibration of semi-empirical force field parameters, the lack of reliable experimental results impedes direct investigations of isotope fractionation with molecular simulations. On the one hand, extremely accurate data are needed for calibration of computational models due to the similarity in the chemical properties of isotopic molecules26. On the other hand, first principles prediction of isotopic effects on thermophysical properties is computationally demanding for most practical applications. From a computational perspective, the transport properties of gas molecules in nanostructured materials are difficult to calculate at high accuracy with standard methods such as molecular dynamic (MD) simulation. The numerical uncertainty inherent to conventional simulation methods is on the



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order of 10% ~ 25%, which is much too large to discern the transport properties of different carbon-isotopic methanes. Recently we developed a theoretical procedure for efficient and comprehensive evaluation of gas adsorption isotherms, isosteric heat of adsorption, and gas diffusivity in nanoporous materials using the classical density functional theory (cDFT)27-29. In comparison to conventional simulation methods, cDFT is computationally much less demanding and not subject to sampling errors or thermal fluctuations. Importantly, cDFT provides analytical expressions that can be readily utilized for rapid and self-consistent prediction of both thermodynamic and transport properties as required for high-throughout screening of nanoporous materials for methane or hydrogen storage. Leveraging the new theoretical capability established in our earlier publications, the present work explores the potential application of MOFs for efficient separation of different carbon isotopes in a methane gas. To facilitate theoretical investigations, we first recalibrate the one-site model for three carbon-isotopic methanes by fitting the Lennard-Jones (LJ) parameters with the saturation-pressure data from the literature30. In combination with the transition-state theory and the excess-entropy scaling method, cDFT calculations are carried out to predict the adsorption isotherms, self-diffusivity at infinite dilution and finite loadings, and both adsorption selectivity and membrane selectivity for different pairs of isotopic methanes in MOFs with large adsorption capacity. As demonstrated in our previous work27, 28, these nanoporous materials provide the highest methane storage capacity at conditions set by the US Department of Energy. Our theoretical results indicate that certain MOFs are able to separate carbon-isotopic methanes with much improved selectivity in comparison to conventional isotope fractionation methods. MOFs for most efficient isotopic separations can be identified through ranking their adsorption 4 ACS Paragon Plus Environment

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selectivity or the membrane selectivity. These nanoporous materials are promising for highly efficient isotope separation at relatively mild conditions thereby reducing equipment and operation costs. 2. THEORETICAL METHODS 2.1 Molecular model One of the major challenges in theoretical study of isotopic effects is to establish a reliable molecular model that can accurately represent both the thermodynamic and the transport properties of different isotopic methanes at conditions of practical interest. A number of semiempirical models, ranging from the single-site Lennard-Jones (LJ) model to fully atomistic representations, have been proposed to describe the pair potential between methane molecules31, 32

. In addition to two-body interactions, analytical expressions are available to represent effective

many-body forces, which make significant contributions to the macroscopic properties of bulk fluids33, 34. Because experimental results for different isotopic methanes are scarce and their behavior in a confined geometry depends not only on the molecular model for methane but also on the crystalline structure and the force-field parameters for the underlying porous material, here we adopt the simple one-site LJ model to represent the effective pair interaction between various isotopic methane molecules. The energy and size parameters for the pairwise additive potential are customized such that the model can best reproduce existing experimental data. Application of the LJ model to normal methane has been well established27, 35. It has been shown that the one-site model is able to reproduce the phase diagram of bulk methane including the saturation pressure versus temperature in excellent agreement with experiment36. Importantly, existing experimental data on the isotopic effects are mostly related to cryogenic distillation of isotopic methanes. As the isotopic effects manifest only in terms of the molecular weight and the 5 ACS Paragon Plus Environment


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LJ parameters in our model, the thermodynamic properties of different isotopic methanes can be correlated with the principle of corresponding states. Such correlation allows us to extract the LJ parameters for different isotopic methanes from the experimental data for the saturation pressure. 2.2 Classical density functional theory Application of the classical density functional theory (cDFT) to gas adsorption in nanoporous materials has been reported in our previous publications27, 29. Here we recapitulate only the key equations. More extensive discussion of the different versions of cDFT functionals, their comparisons with simulation results, and the numerical procedure for solving the density profiles can be found in our earlier work29. Within the one-site LJ model, cDFT predicts that the local number density of methane molecules in a nanostructured material is determined by bulk density ρ b , external one-body potential V ext (r) , and excess local chemical potential ∆µ ex (r) :

ρ (r) = ρb exp  − βV ext (r) − β∆µ ex (r) 

(1)

where β = 1/ (k BT ) and k B is the Boltzmann constant. The external potential accounts for the interaction of each gas molecule with the adsorbent, i.e., the one-body energy arising from interaction of each gas molecule with all atoms from the porous material. The last term on the right side of Eq. (1) is defined as ∆µ ex (r) = µ ex (r) − µbex , which represents deviation of the local excess chemical potential, µ ex (r) , from that corresponding to the bulk phase, µbex . The thermodynamic properties of the bulk system, including bulk density ρ b and excess chemical potential µbex as functions of T and P , can be determined from the modified Benedict-WebbRubin (MBWR) equation of state37.


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Intuitively, Eq.(1) may be understood as the Boltzmann distribution for gas molecules in the porous material. While this equation is formally exact, approximations are inevitable in calculation of the local excess chemical potential, µ (r) . In this work, the local excess chemical ex

potential is derived from the functional derivative of the intrinsic excess Helmholtz energy,

F ex (r) ,

µ ex (r) = δ F ex (r) / δρ (r) .

(2)

Similar to typical excess properties in classical thermodynamics, F ex (r) accounts for the thermodynamic non-ideality due to intermolecular interactions. It is an intrinsic property of the gas molecules, independent of the external potential or specific properties of the adsorbent. At low density or bulk pressure, the interaction between gas molecules is negligible. In that case, we may neglect the intrinsic excess Helmholtz energy, and Eq.(1) reduces to the distribution of ideal-gas molecules inside the porous material as predicted by Henry’s law. To determine the distribution of gas molecules inside the porous material, we need a quantitative relationship between local chemical potential µ ex (r) and local density profile ρ (r ) . Highly accurate expressions are available to compute F ex (r) for LJ systems38, 39. However, little is known about the performance of various functionals at conditions most relevant to industrial processes. In a previous work29, we examined four versions of classical DFT for predicting hydrogen adsorption in various MOF materials. These functionals are all based on the modified fundamental measure theory (MFMT)40 to describe molecular excluded volume effects. The attractive part of the intrinsic excess Helmholtz energy functional is represented by the meanfield approximation (MFA)41, the first-order mean-spherical approximation (FMSA)39, and two slightly different forms of weighted density approximation (WDA-Y42 and WDA-L43). The 7 ACS Paragon Plus Environment


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WDA methods were found most accurate at 77 K, a temperature typically used for characterization of porous materials with nitrogen gas. In the present work, all cDFT calculations are based on the WDA-Y functional42. From the excess Helmholtz energy functional, we can readily calculate the excess entropy from the thermodynamic relation

 ∂F ex  S = −  ∂T  ex

.

(3)

βµ ,V

The excess entropy is used as an input to evaluate the self-diffusivity of gas molecules in MOFs at finite loadings. The quantity can also be utilized to calculate the isosteric heat of adsorption28. 2.3 Transition-state theory We use the transition-state theory (TST) to describe the hopping kinetics of individual gas molecules in a nanoporous material at infinite dilution. The self-diffusion coefficient is related to the transmission from one cage to another by

1 D0 = ϖ a 2 2

(4)

where a is the distance between the equilibrium positions of the gas molecule in two neighboring cages (the initial and final states of the transmission), and ϖ stands for the hopping rate. The latter is calculated from44

k BT ϖ= 2π m

exp  − V ext (s* ) / k B T 

∫

1

0

exp  − V ext (s) / k B T  ds

(5)

where m denotes the molecular mass, the integral is performed along the reaction coordinate of gas hopping, and superscript * represents the transition state for the hoping of gas molecules between neighboring cages. It is noted that MOF materials of our interests in this work consist of 8 ACS Paragon Plus Environment

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micropores ranging in 10~20 Å with relatively less confined channels. We can identify the diffusion pathway for a gas molecule hopping between neighboring pores without using the geometrical analysis to detect accessible cages. One major advantage in application of TST at infinite dilution is that the energy landscape can be directly calculated from external potential V ext (r) . To identify a multidimensional minimum-energy path, we use the string method28. Specifically, we define s as a normalized reaction coordinate, i.e., 0 ≤ s ≤ 1 , and p(s) as a minimum energy pathway that connects the initial ( s = 0 ) and final ( s = 1) states of the gas molecule hopping between neighboring cages. The saddle point, which has a maximum energy along the diffusion pathway, corresponds to the transition state. 2.4 Excess entropy scaling For gas adsorption at finite loadings, we need to account for the effect of gas-gas interactions. In our previous publications27, 45, we have demonstrated that the self-diffusivity can be predicted by a linear combination of the corresponding result at the infinite dilution and the excess due to interaction between gas molecules:  λσ 3  λσ 3 ln Ds =  1− ln DE  ln D0 + v free  v free  where D0 is calculated from TST as discussed above, molecular packing density,

(6)

λ is a constant related to the maximum

v free represents the total accessible volume per gas molecule in the

porous material. The mixing parameter, λσ 3 / v free , accounts for the relative contribution of gasgas interactions to the self-diffusivity. While this parameter changes slightly for different systems, it is independent of pressure and temperature45.



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The contribution to self-diffusivity due to gas-gas interactions, DE , is calculated from the excess entropy scaling method45:

DE =

(

0.585 kBT exp 0.788S ex / Nk B 1/3 m ρ

)

(7)

where ρ is the average number density of gas molecules inside the porous material, and

N = ρV is the total number of gas molecules. 2.5 Selectivity Adsorption selectivity and membrane selectivity are commonly taken as benchmarks to compare the effectiveness of different porous materials for gas separations4. The former is most relevant to gas separation by pressure swing adsorption (PSA) or temperature swing adsorption (TSA) processes, and the latter is used for gas permeation through a membrane. In this work, we consider the potential of MOFs for the separation of isotopic methanes using both adsorptionand membrane-based processes. The adsorption selectivity is defined as the ratio of the adsorption amounts of different species in the porous material normalized by the corresponding concentration ratio in the bulk phase46. For a binary mixture of components 1=light methane and 2=heavy methane, the adsorption selectivity is given by

α=

Γ 2 / y2 Γ1 / y1

(8)

where Γ i is the adsorption amount for component i

Γi =

1 ρ (r) dr V∫ i

(9)


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and yi is the corresponding mole fraction in the bulk phase. According to Eq.(8), α > 1 means preferential adsorption of the heavy methane, and the greater deviation of the selectivity from unity, the easier separation of the mixture by gas adsorption. At low pressure, we can predict the local density or the distribution of gas molecules in the porous material from the Boltzmann law. In that case, the adsorption selectivity can be calculated from the ratio of Henry’s constants:

α IM =

∫ρ ∫ρ

exp  − β Vext ,2 (r)  dr y2 K h,2 / = y K h,1   exp − β V (r) dr 1 b,1 ext ,1  

b,2

(10)

where superscript IM stands for ideal mixture, ρb,i denotes the bulk density, and Kh stands for Henry’s constant

Kh =

1 exp  − β Vext (r)  dr RTV ∫

(11)

where R is the gas constant, and V is the system volume. For comparison with conventional separation methods, the adsorption selectivity amounts to the relative partition coefficient in chromatography or the relative volatility in distillation. The thermodynamic quantity reflects the degree of enrichment in separation of different compounds. With the assumption of Raoults’ law for vapor-liquid equilibrium, the relative volatility is related to the compositions of liquid and vapor phases as

x2 / y2 P1sat α= = x1 / y1 P2sat

(12)

where xi and yi represent the mole fractions of component i in the coexisting liquid and vapor phases, respectively.



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For membrane-based separation processes, the effectiveness is often discussed in terms of the permeation or membrane selectivity. Conventionally, the membrane selectivity is defined as the ratio of self-diffusivity DS multiplied by the adsorption selectivity46, 47:  D Γ / y  k =  s,2   2 2   Ds,1   Γ1 / y1 

(13)

The higher the membrane selectivity, the easier is the separation of different compounds by membrane permeation. In Henry’s law region (viz., at low pressure), the membrane selectivity can be written as the ratio of Henry’s constants multiplied by the ratio of the self-diffusivity at infinite dilution46, 48: k IM =

K h,2 D0,2 K h,1 D0,1

(14)

Unlike adsorption selectivity, membrane selectivity depends on both equilibrium and transport properties of individual compounds.

3. RESULTS AND DISCUSSIONS 3.1 LJ Parameters We extract the LJ parameters for three carbon isotopic methanes from the saturation pressures of the corresponding pure liquids. Relative to the saturation pressure of the normal methane (12CH4), P0sat , the experimental data for the saturation pressure of heavier isotopic methanes, P sat , can be empirically correlated with respect to absolute temperature T and second virial coefficient B2 30,

 P0sat   v  sat  2  ln P sat  1+ P  B2 − RT   = A / T − B / T   

(15)


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where v refers to the molar volume of the saturated liquid, A and B are isotopic-specific fitting parameters. Assuming that the isotopic effects on v and B2 are negligible, we can readily obtain the saturation pressures of 13CH4 and 14CH4 liquids at different temperatures from Eq.(15) based on the corresponding experimental results for normal methane49. With the saturation-pressure data, we can then fit the LJ parameters using the corresponding-state relation50

ln P* = 1.2629T * − 4.8095 / T * − 0.15115 / T *4

(16)

where P* ≡ P satσ 3 / ε and T * ≡ k BT / ε . As shown in Figure 1, the LJ model provides an excellent fitting of the saturation pressure curves for different isotopic methanes. Table 1 presents the energy and size parameters, ε and

σ , for 13CH4, 14CH4 and 12CH4 obtained from the saturation pressure. It is worth noting that the LJ parameters shown in Table 1 reflect those corresponding to an effective pair potential that accounts for multi-body interactions implicitly. In other words, the one-site potential is not necessarily the same as the pair potential between two isolated methane molecules in vacuum; these parameters depend on the particular properties and the range of thermodynamic conditions that were used in calibration of the methane model. Figure 2 compares the LJ parameters for normal methane with those corresponding to the standard one-site models51. As expected, the difference between different one-site models is small. Nevertheless, noticeable discrepancies can be identified for both size and energy parameters. Even a small difference could be significant in theoretical investigation of isotopic separation. We use the new set of the LJ parameters in this work because, as shown in Figure 1, they provide a more faithful description of limited experimental data for different isotopic methanes and, for normal methane, are slightly more accurate than alternative one-site models for representing the saturation pressure (e.g., RMSD=0.006 compared with 0.04 for TraPPE52). 13 ACS Paragon Plus Environment


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3.2 Adsorptive and diffusive behavior With the LJ parameters for different carbon isotopic methanes, we can now investigate their adsorption isotherms and self-diffusivity in various nanoporous materials. In a previous work53, we identified a number of MOF materials from a hypothetical online library constructed by Snurr’s group35 that provide the best methane storage capacity at conditions specified by the US Department of Energy (DOE). The crystalline structures of these MOFs are available online35. The calculations in this work are focused on the selectivity of 10 materials from this subset. Because of their large adsorption capacity, we expect that these MOF materials are most promising for efficient separation of different pairs of isotopic methane gases. For simplicity, we assume that these framework structures are rigid and the LJ parameters for each solid atom can be taken from the UFF model (with a cutoff of 12.9 Å). To be consistent with our previous work, we use the WDA-Y functional in all cDFT calculations. Figures 3 and 4 show, respectively, the adsorptive and diffusive properties of different isotopic methanes in a representative MOF material (NO.1007). From panel A of Figure 3, we see that Henry’s constants for different isotopic methanes are virtually identical and fall rapidly as temperature increases. Because the properties of isotopic methanes are close to each other, we plot the adsorption amounts and the self-diffusion coefficients for

13

CH4 and

14

CH4 relative to

those corresponding to the normal methane in panel B of Figures 3 and 4, respectively. At low pressure, the nanoporous material shows preferential adsorption of the lighter methane, while the trend is reversed as the gas loading increases. As shown in Table 1, the lighter isotopic methane has larger size but smaller energy parameters, which result in a larger Henry’s constant in comparison to that for the heavier isotopic methane. To facilitate a better understanding of this unintuitive result, we may consider 14 ACS Paragon Plus Environment

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gas adsorption in carbon slit pores at different temperatures. A comparison of the external energies of

14

CH4/12CH4 pair in the idealized system shown in Figures S1-3 indicates that the

wall exerts a less repulsive energy for the heavier isotopic methane near the surface (the blue line) but a stronger attraction for the lighter isotopic methane in the region where the surface energy is attractive (the red line). Henry’s constant for the lighter methane is larger than that of the heavier methane at low temperature and in large pores (Tables S1-3). Unlike isotope fractionalization by distillation, the adsorption behavior depends on the interactions of methane molecules with the substrate at low pressure and intermolecular attractions at high loadings. Because self-diffusivity is inversely proportional to the square root of the molecular weight (see Eqs.(5) and (7)), the value for the heavier methane is always smaller than that of the lighter methane. The difference is diminishing as the temperature falls or the pressure increases due to the exponential dependence of the self-diffusivity on thermodynamic quantities.

3.3 Adsorption and membrane selectivity Figure 5 shows the ideal adsorption selectivity versus temperature for different pairs of isotopic methanes by the same MOF material discussed above. For comparison, also shown in this figure are the relative volatility of the corresponding methane pairs (solid lines) and the selectivity from chromatography measurement at a single temperature (symbol). Interestingly, the adsorption selectivity shows an opposite trend compared with the relative volatility. The relative volatility is larger than 1, which means that the heavier methane is enriched in the condensed phase. By contrast, the adsorption selectivity is less than one, indicating that the MOF material preferentially adsorbs the lighter methane. Our theoretical results indicate that adsorptive separation with MOF is much more efficient than the conventional methods. While 15 ACS Paragon Plus Environment


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cryogenic distillation is applicable only to the subcritical region (

Separation of carbon isotopes in methane with nanoporous materials

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