Monte Carlo Potential Energy Sampling for Molecular Entropy in

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Monte Carlo Potential Energy Sampling for Molecular Entropy in Zeolites Mikkel Jørgensen, Lin Chen, and Henrik Grönbeck J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b05382 • Publication Date (Web): 09 Aug 2018 Downloaded from http://pubs.acs.org on August 10, 2018

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The Journal of Physical Chemistry

Monte Carlo Potential Energy Sampling for Molecular Entropy in Zeolites Mikkel Jørgensen, Lin Chen, and Henrik Grönbeck∗ Department of Physics and Competence Centre for Catalysis, Chalmers University of Technology, 412 96 Göteborg, Sweden E-mail: [email protected]

∗

To whom correspondence should be addressed

1

ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract Zeolites are widely applied as molecular sieves and porous host materials for active sites in heterogeneous catalysis. Adsorption and reaction kinetics depend critically on the molecular entropy in the zeolite. In this work, we introduce a method to calculate the entropy of molecules in zeolites using Monte Carlo integration of the semi-classical partition function. The method is demonstrated for N2 and CH4 in Chabazite and MFI silicalites. We find that the molecular entropy is lowered by a factor between 1/3 and 1/2 with respect to the gas-phase value. The results are corroborated by explicit molecular dynamics simulations revealing the active molecular degrees of freedom. The possibility of accurate entropy estimations opens up for an improved description of catalytic reactions and sorption phenomena in zeolites.

Introduction Zeolites are porous aluminosilicates, which are applied as molecular sieves and host materials for active sites in heterogeneous catalysis, where two important applications are hydrocarbon conversion 1,2 and selective reduction of NOx to N2 . 3 As the active sites in zeolites are wellseparated, these kinds of systems can promote high selectivity. 4–10 Adsorption and reaction rates are determined by the free energy profile, and considerable efforts have been made to elucidate the enthaply and entropy of molecules in zeolites. 5,11–18 As the enthalpy of adsorption in many cases is low, it is central to make estimates of the entropy, especially when formulating microkinetic models for reactions. With regards to the entropy of molecules located in zeolite channels, two limiting cases can be identified: The molecule can either be (i) chemically bound to one site or (ii) weakly bound with retained translational and rotational degrees of freedom. In the case of a chemically bound species, it is reasonable to invoke the harmonic approximation, where all degrees of freedom are treated as vibrations. An improvement over this approach is to include anharmonic contributions, 12,19,20 or explicit molecular dynamics simulations of the energy changes. 10,18 For weakly bound species, the 2


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situation is less clear. It is obvious that translations and rotations are hindered by the presence of the framework. For example, for a gas of N2 at 1 mbar and 200◦ C, the mean-free path is ca 0.2 mm, which is 106 times the cage diameter in chabazite. The rotations are also perturbed as N2 only undergoes about 1.2 rotations in the gas-phase when diffusing across a chabazite cage length. To estimate the entropy resulting from the hindered molecular motion, heuristic estimates have been suggested, which assume that one 11 or two 21 translational degree of freedom are lost upon adsorption. Experimentally, it has been measured that closed-shell molecules lose roughly 1/3 of the gas-phase entropy when adsorbing on single crystal surfaces. 22 In addition to the molecular entropy, there is a macroscopic configurational entropy related to occupation of different zeolite unit-cells. Thus, to fully understand and quantify catalytic reactions in confined systems, there is a need for a systematic method to evaluate molecular and configurational entropy taking the full potential energy surface into account. In the present work, we introduce Monte Carlo Complete Potential Energy Sampling (MCPES), which is a computational method for entropy calculations of weakly bound molecules in zeolites. The method is intended for cases where the harmonic approximation is not suitable. The approach is similar to Complete Potential Energy Sampling 23 (CPES), which we introduced recently for entropy estimations of adsorbates on extended surfaces. An approach similar to the CPES method including zero point energy correction was discussed in Ref. 24 On single-crystal surfaces, the high symmetry of the potential energy surface allows for a simple evaluation of the adsorbate partition function. However, due to the lower symmetry in zeolite channels, the current method involves Monte Carlo integration of the semi-classical partition function including rotational degrees of freedom. We apply the method to N2 and CH4 in Chabazite (CHA) and MFI silicalites, and find that the molecules lose between 1/3 and 1/2 of the entropy when entering the zeolite. By providing quantitative data for the entropy of molecules in zeolites, the MCPES method provides a possibility for more accurate descriptions of catalytic reactions in confined systems.

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Method Density Functional Theory Electronic structure calculations were performed using Density Functional Theory 25,26 (DFT) with VASP 27–29 and the Atomistic Simulation Environment 30 (ASE). A plane-wave basis was used with a kinetic cutoff of 450 eV, and the BEEF-vdW functional 31 was applied to describe exchange and correlation effects. Core-valence electron interactions were described in the Projector Augmented Wave 32 (PAW) scheme. The number of valence electrons treated explicitly were: Si(4), O(6), N(5), C(4), and H(1). Gas-phase CH4 and N2 were treated in a (22 ˚ A × 22 ˚ A × 22 ˚ A) cell, and in singlet spin-states. Ionic relaxations were performed in ASE using the BFGS-linesearch algorithm until all forces were lower than 0.05 eV/˚ A. Vibrational energies were calculated with ASE in the harmonic approximation with two-point finite differences using a displacement of 0.01 ˚ A.

Molecular Dynamics Simulations Born-Oppenheimer ab-initio molecular dynamics (AIMD) simulations were performed in the NVT ensemble. As for the static calculations, the AIMD was performed with the BEEFvdW functional. The simulation temperature was controlled to be 473 K by a Nosé-Hoover thermostat, 33,34 and a time step of 1 fs was applied. The simulations were performed for 50 ps where the first 10 ps were used to equilibrate the system.

Entropy Calculations Entropies were calculated using partition functions. In the canonical ensemble, the relation between the entropy and the partition function (Z) is: S = kB lnZ + kB T

4

∂ln Z ∂T

, V,N


(1)

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where T is the temperature, V is the volume, N is the number of particles, and kB is Boltzmann’s constant. For independent degrees of freedom, the partition function is a product over the individual degrees of freedom:

Z=

Y

Zd

(2)

d

Gas-phase molecules were treated in the ideal gas approximation, where translations, rotations, and vibrations are assumed to be independent. Three translational degrees of freedom were included according to the partition function:

Z

Trans

(V ) = V

2πmkB T h2

3/2 (3)

kB T is the volume of the gas, p is the pressure, and m is the molecular mass. p Rotations of gas-phase molecules were treated as free rigid-rotors, 35 where the partition

where V =

function for non-linear molecules is:

Rot Znon-linear

1 = σ

8π 2 kB T h2

3/2 p

πIA IB IC ,

(4)

here σ is the symmetry factor of the molecule, and IX is the moment of inertia around principal axis X. For linear molecules, this becomes 35

Rot Zlinear

1 = σ

8π 2 kB T h2

I.

(5)

The remaining degrees of freedom are vibrations, which were calculated by the quantum harmonic oscillator partition function:

Z Vib

~ωi Y exp − 2kB T , = ~ωi i 1 − exp − kB T 5


(6)


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where ~ωi is the energy of mode number i. The entropy of molecules entering a zeolite channel is restricted as the translations and rotations are hindered. To account for this, the following semi-classical partition functions are evaluated by Monte Carlo integration (See Supporting Information for derivation):

Z`CPES

=Z

Vib

Z Z`Rot

Trans

(V )

Z

V

exp

−U (r, Φ) kB T

dr dΦ,

(7)

where ` represents linear or non-linear molecules, U is the potential energy of the adsorbate relative to the gas-phase, r is the molecular center of mass position, and Φ = (θ/π, φ/2π, ψ/2π) is a generalized coordinate in the interval [0,1] describing molecular rotations, see Supporting Information. The integrals are evaluated over all molecular configurations in the zeolite unit-cell. Thus, the integral is a weighting factor that reflects the potential energy of the zeolite-molecule interactions. The Monte Carlo integration of equation (7) was realized by generating a large number of molecular configurations, where the value of the integral in equation (7) was estimated using the mean-value theorem of integration:

Z`CPES

≈Z

Vib

Z`Rot Z Trans (Vuc )

N −U (ri , Φi ) 1 X exp , N i=1 kB T

(8)

where ` is linear or non-linear, i enumerates the molecular configurations (positions and rotations), N is the number of considered configurations, and Vuc is the volume of a unitcell. In the case of free translation and rotations [U (ri , Φi ) = 0], equation (8) reduces to the expression for an ideal gas, where the rotations and translations are decoupled and free. The Monte Carlo integration was performed by generating configurations using three random uniform numbers u1,2,3 ∈ [0, 1[, which define the center of mass position r, of the molecule in the zeolite: r=

X

ui ai

i

6


(9)

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Where ai are the unit-cell vectors. Secondly, rotations were taken into account by generating random uniformly distributed Euler-angles for each non-degenerate rotation around the center of mass. As phase-space integrals run over generalized coordinates and conjugate momenta, 36 this simple procedure ensures a uniform sampling. In the present study, bondlengths of the molecule were kept at the gas-phase value, and the zeolite structure was not relaxed. This is a reasonable approximation as interactions with the zeolite are weak. The fixed-framework approximation can be omitted for cases with stronger adsorbate-zeolite interactions. Configurations generated with atom centers closer than 0.2 ˚ A were not evaluated, and instead the energy was considered to be infinite. The presented method is intended for weakly bound molecules that have a low concentration in the zeolite. This is not a limitation as strongly bound molecules likely are well-described in the harmonic approximation. 23 The error in the evaluation of the partition 1 function decreases as √ , which makes the partition function converge rather slowly with N N . However, as the entropy depends logarithmically on Z, the convergence to a reasonable value of the free energy is rapid with respect to N . We note that less symmetric potential energy surfaces likely require a larger number of included configurations for convergence. The applied method is semi-classical, however, using an integral to calculate the partition function of translations and rotations instead of a sum over states is a reasonable approximation. The related error of this procedure is estimated using a particle in a box with length 5˚ A, where the difference between integration and summation is less than 1 meV at 1000 K. The use of a semi-classical partition function neglects the zero point energy in the entropy. However, this contribution to the free energy is negligible for translational and rotational degrees of freedom at relevant temperatures. 24 In addition to the molecular entropy, the system has a configurational entropy (S cfg ) arising from the fact that molecules can occupy different cages. The configurational entropy is derived in the mean-field approximation by considering a system with Nuc unit-cells and an adsorbate concentration θ. Here we consider full coverage to be one molecule per unit

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cell. The binomial coefficient Ω describes the number of indistinct ways to distribute the θNuc molecules: 37 Ω=

Nuc ! (Nuc − Nuc θ)!(Nuc θ)!

(10)

The logarithm of the partition function Z cfg is found using Stirling’s approximation: lnZ cfg = ln Nuc θ

1−θ θ

−

ln (1 − θ) θ

(11)

∂lnZ cfg The configurational entropy per molecule can be found assuming that kB T is negli∂T gible: S cfg ln (1 − θ) 1−θ = kB ln − . (12) Nuc θ θ θ Figure 1 shows the configurational entropy calculated using (12) for the entire system per unit-cell (top) and for one molecule (bottom). The highest configurational entropy for the entire system is obtained at θ = 1/2, which is owing to this state having the highest multiplicity in similarity with adsorption on extended surfaces. For an individual molecule, the highest configurational entropy is reached for small concentrations, which reflects the fact that the molecule is free to occupy a large number of unit-cells. The individual configurational entropy diverges in the limit θ → 0. This occurs as the entropy per adsorbate is ill-defined in this limit. To evaluate the concentration entering the configurational entropy, we take as an example N2 in the gas-phase in equilibrium with N2 adsorbed in the zeolite:

N2 (g) + ∗

↔

N∗2

(13)

Solving the kinetic equations in the mean-field approximation at steady-state yields the

8


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Langmuir isotherm: K 1 + K ∆S N2 −∆E N2 exp K = exp kB T kB

θ N2 =

(14) (15)

Where θN2 is the average concentration of N2 in the zeolite and K is the equilibrium constant. ∆E N2 is the adsorption energy of N2 in the zeolite, and ∆S N2 is the entropy difference between N2 in the zeolite and the gas phase. It should be noted that the entropy of N2 in the gas phase depends on the pressure and temperature. Although the Langmuir model for adsorption is an approximation, higher level models are not necessary as small differences in concentration do not significantly change the configurational entropy.

Results and Discussion Prior to modeling the entropy, we analyze the active degrees of freedom by performing MD simulations of N2 in CHA. Figure 2 (a) reveals that vibrational degrees of freedom are active as the average potential energy fluctuates with sub-ps periods, corresponding to the N2 stretch vibration. Figures 2 (b-d) show active rotations as the orientation of the vector between the two atoms in N2 is seen to oscillate considerably during the simulation. Counting the number of full oscillations, the molecule undergoes only 1-2 full rotations around each axis. Figure 2 (e) shows the trajectory projected on two-dimensions. The molecule visits a large fraction of the cage during the 50 ps, where the shortest observed N-framework distance is 2.4 ˚ A. The relatively straight translation across the cage suggests that the potential energy landscape for diffusion is flat. The MD simulation indicates that vibration, rotation, and translation must be included in the entropy model. The translations and rotations are coupled and occur at quite similar time-scales, whereas the vibrations are significantly faster. In fact, the N-N vibration is identical to the calculated gas-phase value of 2433 cm−1 (see

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Figure 1: Mean-field configurational entropy per unit-cell for the total system (top) and for one individual molecule (bottom).

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Relative Energy (eV)

(a)

(b) 𝐜𝐨𝐬 𝜶

𝒓

𝐜𝐨𝐬 𝜷

(c)

(d)

𝐜𝐨𝐬 𝜸

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2

(e)

4

6

Time (ps)

8

10

Figure 2: (a) The potential energy of N2 in the 12 T site CHA unit cell along a AIMD trajectory of 40 ps. Energies are relative to the average potential energy of the entire AIMD (dashed blue line). (b-d) Cosine of the angle between the vector along the N-N bond and the x, y, and z axis, respectively. (e) Projection of the trajectory onto the xy-plane. Temperature: 473 K. 11 ACS Paragon Plus Environment


also SI). Moreover, we note that the molecule hits the zeolite framework frequently. These observations again demonstrate the weak adsorbate-framework interactions.

Figure 3: Entropy change with respect to the gas-phase as a function of configurations included in the Monte Carlo integration. Configurational entropy is excluded. Temperature: 473 K and pressures: 1 mbar. The MCPES method requires a reasonable number of energy evaluations to calculate the entropy of molecules in a zeolite cage. Figure 3 shows the convergence of the Monte Carlo integration for CH4 and N2 in CHA and MFI silicalites. The entropic contribution to the free energy is converged to within 0.05 eV already at 50 Monte Carlo steps. At the considered conditions (T = 473 K and p = 1 mbar), the calculated gas-phase entropy amounts to 1.49 eV and 1.26 eV for N2 and CH4 , respectively. There are abrupt jumps in the entropy at a lower number of included configurations. These are attributed to lowenergy molecular configurations far away from the framework or low energy points close to the wall. Longer convergence tests are shown for N2 in the SI. It should be noted that other systems may require more configurations for convergence, and local relaxations for stronger 12


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adsorbate-framework interactions. The presented method relies on adequately sampling the potential energy surface in the zeolite cell. Thus, it is instructive to consider the sensitivity. If one new energy minimum is found that lies 0.5 eV lower than the present minimum, the entropy differences of Figure 3 would vary by about 0.1 eV. However, finding one single configuration that lies 0.5 eV lower is improbable considering the physisorped nature of the considered molecules, which leads to a smooth energy landscape that depend mainly on the distance between the molecule and the wall of the cage. The calculated molecular entropy in the zeolites and in the gas-phase are shown as functions of temperature at 1 mbar pressure in Figure 4. The entropies in the gas-phase are in the range 0.5-2.75 eV. The corresponding range for the molecules in the zeolite is 0.4-1.5 eV. The entropy is lower inside the zeolite, owing to more restricted molecular motion. In the zeolite, N2 has a lower entropy than CH4 , and CHA results in a lower entropy than MFI. The difference between MFI and CHA is related to steric effects as CHA has smaller pores than MFI. The ratio of the entropy inside the zeolite and gas-phase is fairly constant. N2 loses roughly 1/2 of its entropy upon entering the zeolite, whereas CH4 loses about 1/3. Thereby, the heuristic estimate that molecules lose 1/3-2/3 of the entropy upon entering the zeolite 11,21 can serve as a fair initial approximation. Figure 5 shows the gas-phase entropy and entropy inside the zeolites as functions of pressure at 473 K. The entropy inside the zeolite does not depend on gas-phase pressure in this low concentration regime, and is only shown for comparison. The gas-phase entropies decrease with increasing pressure, as pressure and volume are inversely proportional in the ideal gas equation of state. The gas-phase entropy of N2 ranges between 1.3-1.5 eV over the investigated pressure range, and the corresponding values for CH4 are 1.09-1.25 eV. As observed for the temperature variations, N2 loses roughly 1/2 of its entropy when entering the zeolite from the gas-phase, and CH4 loses about 1/3. The configurational entropy and equilibrium concentration depend on the Gibbs free

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Figure 4: Entropy contribution to the free energy of the gas-phase molecule (top) and inside the zeolite excluding configurational entropy (bottom) as functions of temperature. Pressure: 1 mbar.

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Figure 5: Entropy contribution to the free energy of the gas-phase molecule (top) and inside the zeolite excluding configurational entropy (bottom) as functions of pressure. Temperature: 473 K. energy differences. The adsorption energies of the molecules in the zeolite channel were calculated by local relaxations. For CHA, N2 and CH4 bind with 0.22 eV and 0.25 eV, respectively. In MFI, the corresponding binding energies are 0.32 eV and 0.38 eV. The calculated MFI values are in fair agreement with previously reported calorimetric measurements of isosteric heats of adsorption 38 (N2 : 0.18 eV, CH4 : 0.22 eV). The equilibrium concentrations and corresponding configurational entropies are shown as functions of temperature in Figure 6, where the concentrations are estimated using (14). The concentrations are maximal at low temperatures and approach zero in the limit of high temperatures. This can be understood by the larger entropy of the molecules in the gas-phase, which results in desorption at elevated temperatures. The CHA zeolite leads to low concentrations in general, primarily owing to the low binding energies as compared to MFI. Similarly, N2 is present in lower concentrations than CH4 . That the molecules are present in low concentrations is interesting as it results in a high configurational entropy. For the considered cases, the 15



Figure 6: Top: Equilibrium concentrations as functions of temperature. Bottom: Configurational entropy per molecule. Pressures: 1 mbar.

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configurational entropy varies between 0-1.2 eV and grows with temperature. The configurational entropy is high at elevated temperatures, which to some degree compensates for the loss of molecular entropy when entering the zeolite framework. A similar compensation effect showing that entropy loss is correlated with the heat of adsorption has been observed experimentally. 39,40 Thus, the entropy is quite sensitive to the potential energy landscape. The presented results demonstrate that entropic differences between molecules in zeolites and the gas-phase are considerable. As the binding energies of the considered molecules are relatively low, the entropy determines to a large extent the equilibrium concentration. Moreover, the binding energies and entropic losses are of a similar value, which suggests that entropy losses disfavor N2 and CH4 adsorption in the zeolite. In terms of catalytic reaction kinetics, the predicted low concentrations might be important as the reactants need to be simultaneously in close proximity to the active site.

Conclusions We have presented the Monte Carlo Complete Potential Energy Sampling (MCPES) for calculating molecular entropy in zeolites. The method uses Monte Carlo integration of the semi-classical partition function, which results in a low computational cost. Using N2 and CH4 as prototype cases, we find that the entropic losses range between 1/3 and 1/2 of the gas-phase entropy upon adsorption. In the present example, only a few hundreds of single point calculations are necessary as opposed to molecular dynamics simulations, where the potential energy surface needs to be sampled at different temperatures using long trajectories. The method is directly applicable to more complex systems including ion-exchanged zeolites. With the presented method, computational studies of catalysis in zeolites can include adsorbate entropy at feasible computational costs, which allows for an increased understanding of chemical reactions in confined systems.

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Acknowledgement Financial support is acknowledged from Chalmers Area of Advance Nanoscience and Nanotechnology and the Swedish Research Council (2016-05234). The calculations were performed at PDC (Stockholm) and C3SE (Göteborg) via a SNIC grant. The Competence Centre for Catalysis (KCK) is hosted by Chalmers University of Technology and is financially supported by the Swedish Energy Agency and the member companies AB Volvo, ECAPS AB, Haldor Topsøe A/S, Scania CV AB, Volvo Car Corporation AB, and Wärtsilä Finland Oy.

Supporting Information Available Derivation of the semiclassical partition function integrals. Analysis of N2 vibrational spectrum from MD simulation. Movie showing an MD simulation. Extended convergence test of N2 . Configurations as ASE trajectory files and energies used to perform the Monte Carlo integration.

This material is available free of charge via the Internet at http:

//pubs.acs.org/.

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Monte Carlo Potential Energy Sampling for Molecular Entropy in

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