Thermal Transport in Interpenetrated Metal–Organic Frameworks

Article pubs.acs.org/cm

Cite This: Chem. Mater. 2018, 30, 2281−2286

Thermal Transport in Interpenetrated Metal−Organic Frameworks Kutay B. Sezginel, Patrick A. Asinger, Hasan Babaei,* and Christopher E. Wilmer Department of Chemical and Petroleum Engineering, University of Pittsburgh, 3700 O’Hara Street, Pittsburgh, Pennsylvania 15261, United States S Supporting Information *

ABSTRACT: In practice, the usefulness of metal−organic frameworks (MOFs) for many gas storage applications depends on their ability to rapidly dissipate the heat generated during the exothermic adsorption process. MOFs can be precisely designed to have a wide variety of architectures which can allow tuning their thermal conductivity. In this work, we use molecular dynamics simulations to investigate the effect of interpenetration on the thermal conductivity of MOFs. We find that the addition of a parallel thermal transport pathway yields a thermal conductivity nearly the sum of the two constituent frameworks. This relationship holds for a variety of interpenetrating MOFs with different atomic masses and a wide range of interframework interaction parameters. We show that both the strength and range of interactions between constituent frameworks play a significant role in framework mobility as well as framework coupling which can result in deviation from this relationship. We propose a simple model to predict thermal conductivity of the interpenetrated framework based on thermal conductivities of individual frameworks and an interframework interaction parameter which can account for this deviation.

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INTRODUCTION Metal−organic frameworks (MOFs) show promise in a variety of gas adsorption applications such as gas storage,1−5 gas separation,6,7 sensing,8,9 and catalysis10−12 due to their high porosity and internal surface area. A wide array of organic ligands and metal centers are available for the synthesis of MOFs, providing the opportunity for properties by design when the underlying principles are well-understood.13 While the focus for structure−property relationships in MOFs has expectedly been gas-uptake capacity and selectivity, limited insight exists on the thermal transport characteristics of MOFs.14−24 Since the adsorption capacities of MOFs are reduced at higher temperatures, the exothermic process of gas adsorption can limit the rate at which tank-filling, separations, or other processes can be performed.25 Thus, investigating design principles for thermal transport in MOFs is critical for allowing these materials to reach their full potential for gas storage and separations applications.4 MOFs with particularly high void fractions are often capable of having interpenetrated26 or interwoven27 structures. Mostly, interpenetration is observed within the same framework (homo-interpenetration). However, on rare occasions interpenetration of different frameworks (hetero-interpenetration) has been observed as well. Even though interpenetration decreases the available space within the framework and increases heat of adsorption,28 it has been shown to enhance framework stability29 and increase gas adsorption selectivity.30 Moreover, interpenetrated MOFs have sometimes shown enhanced gas adsorption capacity for small gases such as H2 and N2 by the introduction of additional adsorption sites.31,32 The formation, or not, of interpenetrated networks can often be controlled by either rational geometric design of the structure © 2018 American Chemical Society

or manipulation of experimental conditions. Deliberate modification of organic ligands,33 using infinite secondary building units,34,35 and choosing solvents with appropriate molecular size36 have been used as geometrical restraints to control interpenetration whereas manipulating reactant concentration and reaction temperature were used to control interpenetration by regulating reaction kinetics.37 Additionally, reversible interpenetration control was achieved by ligand removal and addition,38 and anisotropic entanglement was realized by partial interpenetration.39 Using these experimental methods, advanced control of final architecture can be achieved by controlling degree of interpenetration,40,41 reaching up to 54-fold interpenetration,42 as well as modifying interpenetration distance to design minimally interpenetrating (interweaving) MOFs.27 It was also shown that computational methods can also be used to rationally design interpenetrated structures.43,44 Recently, we developed an algorithm to design heterointerpenetrated MOFs with which we were able to discover 18 novel hetero-interpenetrating hypothetical structures.44 As we will show in this Article, the advanced control over MOF interpenetration demonstrated by others can be leveraged as a valuable way to tune thermal transport behavior. Interpenetrating frameworks provide parallel pathways for thermal transport. The effect of thermal coupling between frameworks, however, has so far been uncertain. In the present work, we use molecular dynamics (MD) simulations to investigate thermal transport in a variety of doubly interpenetrated MOFs. To isolate the effects of interpenetration from other structural parameters, a simple cubic idealized Received: November 30, 2017 Published: January 16, 2018 2281

DOI: 10.1021/acs.chemmater.7b05015 Chem. Mater. 2018, 30, 2281−2286

Article

Chemistry of Materials

Figure 1. (a) Idealized porous crystal (8 × 8 × 8 cubic unit cells). (b) Bonding arrangement for a single unit cell using Morse potentials (red bonds are modeled more strongly than blue bonds). (c) Doubly interpenetrated unit cell with entangled frameworks depicted as red and blue (initial frameworks in each simulation are 5 Å apart in each dimension). (d) Interpenetrated idealized porous crystal (8 × 8 × 8 cubic unit cells). See the Supporting Information for more details about the structures and simulation details. ligands (see IRMOF-13 vs IRMOF-9 in Figure 2) that correspond to interpenetrated frameworks that are “locked in” or “free”, respectively.

framework was considered. Effects of pore structure on thermal conductivity have been reported elsewhere.18 We find interpenetration yields a system thermal conductivity approximately equal to the sum of the two independent frameworks, which can be used as a “rule of thumb” for design purposes. To account for deviations from this general rule, we present equations that can be used as guidelines for predicting thermal conductivity in doubly interpenetrated MOF structures.

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METHODS

We used an idealized cubic MOF structure inspired by IRMOF-1 to better understand parameters that affect thermal transport, as we had done in prior studies17,18 (see Figure 1a). The idealized interpenetrated MOF allows us to study the effect of framework interactions parametrically which would not be possible in a real interpenetrated MOF. We use two-body bonded interactions between atoms within each of the interpenetrated structures, modeled using the Morse potential (see Figure 1b).45 The force field parameters were chosen so that the thermal conductivity of the simple cubic structure with a pore size of 1 nm was of the same order as a typical MOF (∼1 W/(m K)). The interpenetrated structure is generated by creating a copy of the idealized structure and translating it 5 Å in each dimension which corresponds to the maximal distance between frameworks (see Figure 1c). We also calculated the thermal conductivity for IRMOF-1 and its doubly interpenetrated MOF using UFF46 parameters which were shown to provide accurate values for predicting bulk modulus.47 All thermal conductivity predictions were done using the Green−Kubo approach and equilibrium MD simulations (LAMMPS48). The force field parameters for structures and MD simulation details are provided in the Supporting Information, with sample LAMMPS48 input files provided in therMOF GitHub repository: https://github.com/ kbsezginel/thermof. The nonbonded interactions between interpenetrating frameworks were modeled by a Lennard-Jones potential:

⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ VLJ = 4ε⎢⎜ ⎟ − ⎜ ⎟ ⎥ ⎝r⎠ ⎦ ⎣⎝ r ⎠

Figure 2. Different interpenetration configurations based on cubic IRMOF series:49 (a) metal center distribution, (b) BPDC linker (IRMOF-9), and PDC linker (IRMOF-13) representing less and more bulky linkers, respectively. Color scheme is as follows: Zn (blue), O (red), C (gray), H (white).

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RESULTS AND DISCUSSION Framework Interaction Dependency of Constituent Position and Thermal Conductivity. The effect of interframework interactions on thermal conductivity and framework mobility was investigated by calculating thermal conductivity and mean squared displacement (MSD) for a wide range of ε and σ values (see Figure 3). On average, the thermal conductivity of the interpenetrated framework was found to be around 1.6 W/(m K) which is approximately double that of the single framework (0.82 W/(m K)). Consequently, this led us to explore the idea that the thermal conductivity of the interpenetrating frameworks can be predicted by a simple linear combination rule: kIP = k1 + k 2

(2)

where kIP, k1, and k2 are the thermal conductivities of the interpenetrating framework, single framework 1, and single framework 2, respectively. In order to understand the conditions under which this relationship holds, percentage error for predicting thermal conductivity using eq 2 was calculated (Figure 3b). As seen in Figure 3a,b the thermal conductivity is relatively constant in the region 0.01 < ε < 0.2 kcal/mol and 1 < σ < 4.5 Å. However, increasing ε above 0.1 kcal/mol decreases thermal conductivity for σ values lower than 4.5 Å as apparent from the dark blue region in Figure 3a. This low thermal conductivity region described by 1.5 ≤ σ ≤ 4.5 Å and 0.1 ≤ ε ≤ 1.0 corresponds to increased coupling of the frameworks, which likely leads to higher phonon scattering rates introduced by the interactions between two frameworks.

(1)

where ε is the depth of the potential well, and σ is the distance at which the interatomic potential is zero. A range of ε values between 0.01 and 1 kcal/mol and σ values between 1 and 6 Å were chosen to systematically understand the effects of interframework interactions on thermal transport. Then, an ε value of 0.105 kcal/mol (corresponding to a carbon atom in UFF46) and σ values of 3.5 and 4.5 Å were selected to represent different interpenetration configurations for further analysis. The σ influences framework mobility where a value of 3.5 Å allows interpenetrating frameworks to “freely” translate relative to each by 2 Å in each direction, while a σ value of 4.5 Å models interpenetrated frameworks that are “locked in” and whose distance from each other remains constant. These configurations can be imagined as, for example, isoreticular MOFs with more or less bulky 2282

DOI: 10.1021/acs.chemmater.7b05015 Chem. Mater. 2018, 30, 2281−2286

Article

Chemistry of Materials

Figure 3. Effect of framework interaction on thermal conductivity and framework mobility in terms of force field parameters. For each plot, each bin represents a simulation with corresponding ε and σ values. The black bins on the upper right-hand side are for simulations where high repulsive forces caused frameworks to collapse. (a) Thermal conductivity (W/(m K)). (b) Thermal conductivity prediction error (%) for interpenetrated framework using the relationship in eq 2. (c) Coupling constant (γ). (d) Thermal conductivity prediction error (%) for interpenetrated framework using the relationship in eq 3. (e) Mean squared displacement, MSD (Å2).

Figure 4. Effect of interframework interaction on framework distances during simulation. Interframework interactions are modeled using LennardJones potential. (a) σ, 3.5 Å, and (b) σ, 4.5 Å, with varying ε values in kcal/mol shown in the figure. Grid size, 6 Å, and bin size, 1 Å. (c) ε, 0.105 kcal/mol, and varying σ values shown in the figure. Grid size, 12 Å, and bin size, 2 Å. (d) Framework distances are measured according to middle red atom with reference to the corner atoms of the surrounding blue framework. A 5 Å × 5 Å × 5 Å cube shown as dashed box in 2 dimensions. The yellow−red bins represent the position of the other framework. (e) Time spent in given location are given in percentage. Color scale is limited to 50% of the simulation time for clarity. All results are for x and y directions from a single run. Methods used for calculating relative distance are given in section 3c of the Supporting Information.

thermal conductivity for the interpenetrated IRMOF-1 was found to be 0.68 W/(m K) with a standard deviation of 0.04 W/(m K) over 10 runs. The single-framework thermal conductivity was found to be 0.34 W/(m K) again with a standard deviation of 0.04 W/(m K). According to eq 2 the thermal conductivity of the interpenetrated framework is predicted as 0.68 W/(m K), which is in excellent agreement with the simulated value and suggests no significant interframework coupling. Framework mobility is also significantly affected by the force field parameters as evident from the MSD shown in Figure 3e. In addition to MSD, the framework mobility was investigated by calculating the relative distance between reference atoms selected from interpenetrating frameworks (see Figure 4). The relative distances of one corner atom of the second framework (Figure 4d shown in red) to the 8 corner atoms of the surrounding first framework (Figure 4d shown in blue) were calculated from MD trajectories. The resulting 2D histograms given in Figure 4 show where the framework spends time during the simulation. The effect of ε is shown in Figure 4a for σ 3.5 Å and Figure 4b for σ 4.5 Å, and the effect of σ is shown in Figure 4c. Generally, σ is inversely related to the total area the frameworks can translate relative to each other whereas ε is directly related with the amount of time frameworks spend in a

We then modified eq 2 by adding a coupling constant (γ) that depends on the interframework interactions as follows: kIP = (k1 + k 2)(1 − γ )

(3)

We defined individual coupling constants for σ and ε as γ = γεγσ and derived equations to represent the coupling seen for the region 1.5 ≤ σ ≤ 4.5 Å and 0.1 ≤ ε ≤ 1.0. Derivation of the equations for coupling constants are given in section 3b of the Supporting Information. As seen in Figure 3c the coupling constant was modeled to increase linearly with ε and change quadratically with σ between 0 and 0.25. Including the coupling constant in eq 3 decreases the average error for the whole interaction range from 13.9% to 7.8% (Figure 3d). We limited the region of the coupling constant because we believe the deviations seen from eq 2 outside this region are not related to framework coupling. It can be seen in Figure 3a that increasing σ above 4.5 Å increases the system thermal conductivity especially at higher ε values. This increase is a result of stronger interactions between frameworks providing better thermal transport pathways. As for σ < 1.5 Å, we observe very high framework mobility (Figure 3e) which is attained to unphysically low repulsive forces causing frameworks to slide through each other. As a proof of concept, we calculated thermal conductivity for single and interpenetrated IRMOF-1. The 2283

DOI: 10.1021/acs.chemmater.7b05015 Chem. Mater. 2018, 30, 2281−2286

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Figure 5. Effect of atomic mass on thermal conductivity. (a) Single MOF. (b) Free interpenetration (σ: 3.5 Å). Dashed lines are equations shown in the figure. (c) Parity plot for predicted (eq 1) and simulated thermal conductivities for “free” and “locked” interpenetrated MOFs for different masses. For interpenetrated structures, atomic mass of the second framework (M2) is modified whereas first framework atoms are always chosen as Ar (M1 ∼ 40 Da). Error bars represent one standard deviation of uncertainty.

acoustic phonons in the lattice scales as 1/√M whereas the mean free path and specific heat do not change considerably with mass. We then investigated interpenetrated frameworks, where the atomic mass of one of the frameworks was changed and the other kept constant. We found the same relationship to hold for both “free” and “locked” interpenetration (Figure 5b). Moreover, thermal conductivity values for the interpenetrated framework are approximately the sum of the thermal conductivity of individual frameworks. For instance, thermal conductivity of individual frameworks made of atoms that have masses of either 39.95 Da (Ar) or 83.80 Da (Kr) have thermal conductivities of 0.83 W/(m K) and 0.57 W/(m K), respectively. Summing their individual thermal conductivities gives 1.40 W/(m K), which is between 1.49 and 1.34 W/(m K) given by the interpenetrated frameworks having σ values of 4.5 and 3.5 Å, respectively. As shown in Figure 5c, the predicted thermal conductivities (using eq 2) for different masses match closely with simulated thermal conductivity values. This suggests that a linear combination of single-framework thermal conductivities is a good model for the conductivities of interpenetrated frameworks, even when their corresponding masses are different.

given configuration. Consequently, MSD is generally higher at lower ε and σ values. Above σ value of 4 Å, the frameworks are tightly locked; therefore any change in force field parameters do not reflect directly on the MSD. Below 4 Å, however, MSD increases with decreasing σ as seen in Figure 3e. For lower σ values, there are multiple potential wells in between the frameworks. As a result, frameworks reside in different potential wells during each run and can jump between wells during the simulation (see Figure 4a). With increasing ε, the energy barrier to jump between potential wells increases, and frameworks become more closely coupled. Depending on the initial velocities, frameworks are trapped (for the duration of the simulation) in a position relative to each other and stay there during the simulation (Figure 4c). For low ε values, however, the frameworks easily move relative to each other (Figure 4a). Given an infinite simulation time, the average distance between frameworks should correspond to optimal distance since, by symmetry, each corner would be visited equally many times resulting in complete sampling of the configuration space. For low σ values, the frameworks have increased mobility as the amount of space they can translate with respect to each other with the same energy penalty is increased. This is also complemented by the MSD (Figure 3e) where the displacement generally increases toward lower σ values. As seen in Figure 3e, between σ values of 1 and 2 Å the displacement is quite high, and between 2 and 4 Å the displacement is moderate, decreasing as σ goes higher. For σ ≥ 2 Å increasing ε slightly decreases MSD since fewer potential wells are visited during a simulation. As the MSD is calculated in reference to the initial position of the frameworks, simulations with higher ε values result in frameworks spending more time on a single potential well distant from their initial position. For lower ε values, however, as the frameworks travel between potential wells they get closer to their initial position, consequently decreasing the MSD. Again, with better sampling of the configuration space these differences would be minimal. The high MSD for σ < 2 Å is due to frameworks sliding through each other since interatomic distances within a framework are 3.3 Å. In this region increasing ε decreases the MSD as the interaction strength is increased. However, these extraordinary framework interactions are not expected to be observed in real MOFs. Mass Dependency of Thermal Conductivity. Initially thermal conductivities for a single framework were investigated for different atomic masses between 4 and 200 Da (i.e., changing the mass of every atom in the framework, see Figure 5). As seen in Figure 5a, thermal conductivity as a function of atomic mass shows a 1/√M relation. As previously shown by others,50 this is because frequency and group velocity of

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CONCLUSION

The work presented here shows that, based on MD simulations, doubly interpenetrated MOFs will have thermal conductivity approximately the sum of the constituent frameworks. We show that the relation holds for both cubic idealized MOFs and IRMOF-1 as presented here. For rare cases where frameworks are not maximally interpenetrating, and interframework attraction is high, the effect of coupling might result in lower thermal conductivity than the sum of individual frameworks. This suggests that the nature of interpenetration is quite important for thermal conductivity. We show that the decrease in thermal conductivity can be approximated by calculating a coupling factor using force field parameters (σ and ε) that define interframework interactions. For interactions described by 0.01 < ε < 0.1 kcal/mol and 1 < σ < 4.5 Å, the effect of coupling was found to be minimal. We believe these results provide important insights into the gas adsorption application of interpenetrated MOFs especially for small gases such as H2. The inclusion of additional frameworks could increase storage capacity of the gas by increasing adsorption sites and provide additional thermal transport pathways which allows the generated heat to dissipate more quickly. 2284

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.chemmater.7b05015. Generation of idealized MOF structures, force field parameters for idealized MOFs and IRMOF-1, molecular dynamics simulation details, construction of framework coupling model, and framework mobility calculations (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Kutay B. Sezginel: 0000-0003-3060-2945 Hasan Babaei: 0000-0002-3291-0290 Christopher E. Wilmer: 0000-0002-7440-5727 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS C.E.W., H.B., and K.B.S. gratefully acknowledge the Donors of the American Chemical Society Petroleum Research Fund for support of this research. They also acknowledge both the Swanson School of Engineering and the Center for Simulation and Modeling (SAM) at the University of Pittsburgh for early financial support, and for providing computational resources, respectively.

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REFERENCES

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DOI: 10.1021/acs.chemmater.7b05015 Chem. Mater. 2018, 30, 2281−2286