Molecular Dynamics Phenomena of Water in the Metalorganic

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Molecular Dynamics Phenomena of Water in the Metalorganic Framework MIL-100(Al), as Revealed by Pulsed Field Gradient NMR and Atomistic Simulation Tobias Splith, Evangelia Pantatosaki, Panagiotis D. Kolokathis, Dominik Fröhlich, Kang Zhang, Gerrit Fueldner, Christian Chmelik, Jianwen Jiang, Stefan K. Henninger, Frank Stallmach, and George K. Papadopoulos J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b06240 • Publication Date (Web): 03 Aug 2017 Downloaded from http://pubs.acs.org on August 3, 2017

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The Journal of Physical Chemistry

Molecular Dynamics Phenomena of Water in the Metalorganic Framework MIL-100(Al), as Revealed by Pulsed Field Gradient NMR and Atomistic Simulation

Tobias Splith†,a, Evangelia Pantatosaki†,b, Panagiotis D. Kolokathisb, Dominik Fröhlichc, Kang Zhangd, Gerrit Füldnerc, Christian Chmelika, Jianwen Jiangd, Stefan K. Henningerc, Frank Stallmach*,a, and George K. Papadopoulos*,b,e

a

Faculty of Physics and Earth Sciences, Leipzig University, Linnéstr. 5, D-04103

Leipzig, Germany b

School of Chemical Engineering, National Technical University of Athens, 15780

Athens, Greece c

Fraunhofer Institute for Solar Energy Systems ISE, 79110 Freiburg, Germany

d

Department of Chemical and Biomolecular Engineering, National University of

Singapore, 117576, Singapore e

Institute for Medical Engineering and Science, Massachusetts Institute of

Technology, Cambridge, MA 02139, USA

†

Authors equally contributed. Authors to whom correspondence should be addressed: Frank Stallmach ([email protected]) George K. Papadopoulos ([email protected])

*

ACS Paragon Plus Environment

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ABSTRACT Measured, via pulsed field gradient (PFG) NMR, and computed molecular dynamics (MD) were utilized for the study of the phase equilibrium and kinetics of water sorbed in a bed of MIL-100(Al) crystallites. The computations rely on our recent methodology for modeling water equilibria and dynamics in the Fe-homologue MIL100 crystal; in that sense, the particular NMR technique serves also as a validation tool of the previous simulation work which is adapted to the current system. In addition, a computational scheme for assigning partial charges on the host framework atoms was devised; it involves density functional theory (DFT) combined with electronegativity equalization method (EEM) calculations. The derived this way electronegativity, hardness and gamma parameters for the specific MIL-100(Al) atoms, can be used in EEM calculations of other aluminum metalorganic frameworks (MOF) bearing similar atom types. The thermodynamics predictions obtained via MD, comprising equilibria, enthalpies, adsorbate probability densities, and host’s terminal species effects, were compared with data from the real system’s phase equilibria measured in this work. The intra-crystalline self-diffusivity of the sorbed water was extracted by means of the spin echo curves obtained by PFG NMR for various guest loadings as a function of observation time, and, a theoretical short-time expansion of the diffusion coefficient of random walkers, assuming spherical particles under reflecting boundary conditions following Mitra et al. The experimental activation energies for diffusion confirmed previous, in MIL-100(Fe), and current modeling results, with respect to the adsorbed water dynamics and singlet probability density distribution.


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I. INTRODUCTION The success of metalorganic frameworks (MOF), also known as porous coordination polymers, is based on the feasibility of tailoring selected molecular blocks together, thus resulting in well-defined pore network morphologies. Further chemical modification of their linking units via computer aided energy minimization techniques and combinatorial synthesis, may lead to the creation of a whole series of homologues with varying organic linkers and metal centers.1,2 The choice of aluminum as the MOF’s metal center, has proved to provide stable porous materials, yet its light weight furnishes low density sorbents, that along with its natural abundance allow for low-cost production on industrial basis.3 In addition, aluminum’s environmentally benign nature and recyclability do promote Al-frameworks for use in ecofriendly applications.

The target sorbent of the presented work is the mesoporous MIL-100(Al),4 synthesized and characterized for this study by the Fraunhofer Institute for Solar Energy Systems. A variety of experimental studies have considered aluminum-based MOFs, and particularly MIL-100(Al), as potential candidates: for gas5-7 and energy8-11 storage applications, for removing sulfur and nitrogen heterocyclic compounds from fuels,12 and toxic biological pollutants from water,13 and for fresh water production.14 Separations of anti-inflammatory drugs from water and urine samples have also been reported.15

The performance of these materials for the aforementioned applications is vastly influenced by the transport rates of guest sorbates inside the pores of the host matrix. An advanced experimental technique for studying mass transport phenomena in such


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systems is the PFG NMR. The particular NMR technique can probe the guest mobility in an assemblage (bed) of porous crystallites; the latter form of substrate, constitutes a commonly employed design for many adsorption engineering applications.16,17 It is noticeable that the particular technique has been the first attempt for measuring diffusivity in MOFs,18 by the group of Leipzig University.

The hybrid nature of the MIL inorganic-organic frameworks, results in a significant structural and chemical complexity bearing a pronounced electron charge distribution that constitutes a challenging subject for the computer modelling of guest molecular dynamics phenomena therein. The diffusivity of several sorbates in MOFs has been explored over the last years by means of atomistic simulations;19-25 in MILs particularly, diffusion studies reported so far pertain to MIL-53,26 MIL-47,27 and MIL-100.28 Moreover, previous studies combining physical and computer experiments, have allowed exploration of the behavior of the digitized system vis–à– vis the real one by considering various force-field parameters “options” for the host framework.21-23, 25

Primary objective of this study is to probe the dynamics phenomena of the water sorbed phase within MIL-100(Al) crystals focusing on energy conversion and storage applications. The computation of the host partial charges on the framework atoms proceeded through a computation scheme combining: DFT calculations carried out at the National University of Singapore, with EEM, conducted in the National Technical University of Athens the way the method was employed in the MIL-100(Fe) along with the molecular dynamics simulations. 28


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II. EXPERIMENTAL AND MODELING DETAILS Material synthesis. The MIL-100(Al) was synthesized from aluminum nitrate nonahydrate (1.80 g, 4.80 mmol) and trimesic acid (1.01 g, 4.80 mmol), similar to reported methods. The educts were each dissolved in 10 ml distilled water. After vigorous stirring for 10 min both solutions were combined and stirred for about 3 hours at room temperature. Acetic acid (100%, 0.2 ml) was added, and the reaction mixture was stirred for 5 min. The synthesis was carried out in a microwave oven (ETHOS - MLS GmbH) at 210 °C for 30 min, with a heating rate of 20 °C/min. After hydrothermal treatment the reaction mixture was cooled by means of an ice bath for half an hour and centrifuged to collect a yellow solid at 10,500 rpm for 15 min. Subsequently, the solid was washed in methanol and centrifuged off at 10,500 rpm for 15 min. This procedure was performed twice. The washed MIL-100(Al) was dried at 90 °C under normal pressure and then at 120 °C in a vacuum oven to activate it.

Powder X-Ray diffractograms were acquired on a Bruker D8 Advance with DaVinci using Cu-Kα radiation and a step size of 0.02° with 1.0 s/step. The surface area and pore volume values of 1166 m2/g, and 0.49 cm3/g respectively were evaluated by means of N2 isotherm at 77 K using a Quantachrome Nova apparatus. The H2O adsorption isotherms were measured using a Quantachrome Hydrosorb.

PFG NMR. The mobility of water sorbed phase in the MIL was studied by means of PFG NMR experiments.29,30 Also, the water longitudinal relaxation time, T1, and the transverse relaxation time, T2, were measured via inversion recovery and CPMG (Carr-Purcell-Meiboom-Gill) pulse sequences, respectively. The in-house built FEGRIS NT NMR spectrometer operating at the 1H-resonance frequency of 125 MHz


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was used for these experiments. For the PFG NMR diffusion studies this spectrometer is equipped with a z-gradient system30,31 capable of delivering gradient pulses up to 37 Tm-1.

Due to short transverse relaxation times of water in MIL-100(Al) (approx. 1.0 ms at 125 MHz), the stimulated spin echo PFG NMR sequence30,32 was utilized. In this sequence the signal is decaying with transverse relaxation for time intervals between the first and the second π/2 RF pulse; and between the third π/2 RF pulse and the spin echo. Both time intervals were of the same length, t2. The strong gradients achieved through the FEGRIS setup allowed for short t2 values of around 0.6 ms, thus achieving significant attenuation of the spin echo signal. During these two time frames, the pulsed field gradients of amplitude, G, and width, δ, were applied to encode the molecular motion. By increasing the amplitude G, the magnitude M of the echo was attenuated, and eventually from the spin echo attenuation curve the timedepended effective diffusivity of guest water, D=D( ), was calculated through the equation,

,

0,

(1)

where Δ is the observation time of the experiment, namely, the time elapsed from the beginning of the first, up to beginning of the second gradient pulse; the b-value was calculated from the NMR pulse sequence parameters, whereas the gyromagnetic ratio, γ, is given by the relationship

(2)


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The observation time was varied from an upper limit, set by the longitudinal relaxation of approx. 100 ms for MIL-100(Al), up to a lower limit of 5 ms, determined by the internal time limits of the stimulated echo PFG NMR sequence, and the requirement for observing significant attenuation of the spin echo signal. The guest diffusivity was measured at four loadings within a bed of 120 mg MIL100(Al) as follows: three samples were first water-loaded in a closed vessel (exsiccator) via the vapor phase of oversaturated solutions of LiCl, CH3COOK and MgCl2, resulting respectively in the relative humidities of 11 %, 22 % and 32 %;

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finally, a forth sample was loaded in the presence of the vapor phase of pure water. By virtue of our measured adsorption isotherm the obtained loadings can be estimated so that to correspond to 0.12 g/g, 0.17 g/g, 0.32 g/g and 0.60 g/g, respectively. All samples remained under equilibrium at their relative humidity environment for five days and they were sealed inside vessels. Temperature dependent NMR diffusivity measurements up to 335 K were also carried out for the water loadings of 0.17 g/g, 0.32 g/g and 0.60 g/g, and relaxation times were measured for all loadings and the temperatures of 298 K, 310 K, 323 K and 335 K.

MIL-100(Al) model. The building block of MIL-100(Al) forms super tetrahedral units consisting of a trimer of aluminum octahedra sharing a common vertex and chelated by 1,3,5-benzenetricarboxylate bridging ligands. The entire unit cell of MIL100(Al) was digitized by means of powder X-ray diffraction data4 to obtain the fractional coordinates and the atomic occupancy probabilities; then, implementation of the symmetry operations of the Fd-3m (No. 227 in Ref. 34) crystallographic space group led to the final reproduction of the unit cell atomic coordinates.


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The diffraction data provide the terminal crystallographic positions (oxygen atoms) of the metal octahedral trimers, giving no information on the actual distribution of the OH- groups and structural H2O at these locations, however. The experimental ratios for the anion terminal groups and the structural waters over the metal atoms are 1:3 and 2:3, respectively.35 Following the experimental data, we distributed the hydroxyl oxygens uniformly among each of the three terminal positions in all trimers; the remaining positions were occupied by water oxygens. In a sequent step hydrogen atoms were assigned to the oxygen atoms.

The resulted cubic unit cell contains 13,056 framework atoms, namely, Al816O272(OH)272(H2O)544(C9H3O6)544, forming sixteen small and eight large mesoporous cavities of effective diameter approx. 2.5 nm and 2.9 nm respectively which construct a pore network of MTN topology. The physical entrances to the small cavity (pentagon dodecahedron) form pentagonal apertures with effective diameter of 0.52 nm; the large cavity (hexakaidecahedron) is delimited by pentagonal and hexagonal window apertures with effective diameters of 0.52 nm and 0.88 nm respectively.

Energetics. Given that the kinetic diameter of water is nearly twice smaller than the narrowest window aperture in the MIL-100 structure, and also in view of previous studies where adsorbate molecules experience tight fitting in pores,21,23 we opted to the reasonable decision of a rigid unit cell model. Moreover, experiments have shown that the MIL-100 framework does not exhibit guest-induced “breathing” effects contrary to other MILs.36 The guest-MIL dispersion interactions were described by the Lennard-Jones potential, utilizing the Dreiding force field,37 based on results of


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previous computations in MOFs20,21,23 (parameters are reported in Table SI in the SI). The guest-guest water interactions were described by the SPC/E model,38 which includes partial charges to all atoms with the oxygen acting as the molecule’s sole Lennard-Jones center. For the size and strength parameters of the OH- groups and the structural waters in the MIL we tested values from both Dreiding and SPC/E model resulting in trivial computational differences.

For the calculation of the partial charges on the sorbent framework atoms we followed a hybrid scheme combining DFT39 in small atomic clusters of the host framework, and EEM40 based on the principle of electronegativity equalization.41 In particular the methodology entails: (i) DFT calculation of atomic charges on a MIL cluster cleaved from the cell and terminated by methyl groups to become neutral; (ii) inverse EEM whereby the input parameters, electronegativity, hardness and gamma factor, for each atom type in the cluster were obtained in view of the atomic charges calculated by DFT; (iii) implementation of EEM on the entire MIL unit cell utilizing the previous parameters to obtain partial charges on each framework atom.

Molecular dynamics.

In order to enrich the statistics of the modeling part we

digitized the entire large MIL-100(Al) unit cell instead of the primitive cell. To overcome the additional computational burden of such a CPU demanding system, the energy and force values at each step were being interpolated on the fly during the runs through 3-D Hermitian interpolation in a pre-tabulated grid, bearing 300×300×300 nodes of 0.024 nm spacing and spanning the entire MIL unit cell. It must be noted that the explicit tabulation procedure of all nodes was parallelized, otherwise extremely high amounts of CPU times would be required;28 the latter procedure proved to be of


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considerable importance for the computation of electrostatic interactions via the Ewald summation. Finally, the above procedure was embodied into the LAMMPS open source code42 for the integration of the equations of the molecular motion for NG number of water molecules in the host MIL volume, VH, in the canonical (NG,VH,T) ensemble following the logic of a recent work,28 up to the time length of 40 ns, through the velocity Verlet algorithm based on the Nosé method.43 The water model was kept rigid by constraining its bonds and angle through the SHAKE algorithm.44

The loading of the sorbent was achieved by inserting uniformly water molecules in the unit cell cavities up to a certain value. Then, successive energy minimizations were performed prior to the actual run through sequential MD cycles, increasing conservatively the time step up to the final value of 1 fs which eventually was used in the numerical integration of equations of motion.

The unknown pressure, P, at the prescribed T and NG was calculated by means of the bulk water phase described by the pressure-explicit Peng–Robinson equation of state, being in equilibrium with the guest sorbed phase, both under uniform pressure, temperature and fugacity. The latter quantity was calculated by Widom averaging.28 The finally obtained equation28 relating fugacity and its conjugate pressure was solved numerically to provide P values for each MD simulation run.

Subsequently, by post-processing the produced dynamics trajectories through a Widom-like averaging procedure the thermodynamics of the system water-MIL (equilibria, enthalpies) was obtained. The adsorbate probability density distributions


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were also calculated, and the self-diffusivity of the sorbates was extracted through standard mean squared displacement time plots of the water oxygen.

III. RESULTS AND DISCUSSION A. Computer modeling The structural and chemical complexity of the hybrid MIL structures creates an amphiphilic interior of a hydrophilic region in the vicinity of the metal octahedra trimers (metal atoms bound to the linkers’ oxygen atoms) having terminal species (OH- groups and structural waters in the MIL) and local hydrophobic areas, formed by the non-polar backbone of the organic linker (Figure S1a). The DFT calculations (details in SI) on the selected MIL cluster (Figure S1b) were carried out by means of the DMol3 package;45 the partitioning of the electron density to obtain the partial charges on each atom was performed using the electrostatic potential (ESP) scheme.46,47 Then an inverse EEM procedure was employed on the same cluster to derive parameters for each atom type (Table S2), thereby partial charges for the 13,056 framework atoms of the entire unit cell were eventually obtained via direct EEM (Figure S2 and Table S2 in SI). It must be noted that the derived EEM parameters for the specific MIL-100(Al) atom types (Figure S1a) can be used in EEM computations for other aluminum MOFs comprising similar atom types in their pore network; as an example the reader may refer to a recent work on the aluminum MOF CAU-10-H.8

In Figure 1 experimental, and computed water isotherms obtained via MD, are compared with experiments found elsewhere.


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Figure 1. Computed (triangles) and experimental (rhombi) amount of sorbed water per MIL-100(Al) mass, [g g-1], at 300 K; measurements found elsewhere (open8 and filled48 circles, and squares14) at 298 K are also shown. The inset presents comparison of computed and measured isotherms of this work with the sorbed amount reduced per MIL-100(Al) surface, [g m-2].

The observed deviations between the measured isotherms may be attributed mainly to the variety of structural characteristics of the synthesized MIL-100(Al) material as presented in Table 1. The observed differences in specific surface areas and pore volumes may be due to the presence of additional hydrophobic sites, e.g. amounts of unremoved trimesic acid during the washing procedure that inhibit complete filling of the pores. Also, the existence of an impurity phase of MIL-96(Al), occurring during the MIL-100(Al) synthesis4,49,50 of a specific surface area varying from 216 m2g-1, 49 to 530 m2 g-1,51 and pore volume of 0.07 cm3g-1.49


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Table 1. Specific surface area and pore volume values reported for MIL-100(Al). specific surface area (m2 g‐1) 1056 1200 1166 1342 1786 2152

pore volume (cm3 g‐1) 0.33 0.40 0.49 0.65 ‐‐‐‐ 0.82

Ref. 49 7 this work

8 14 4

The digitized material of this study was based on the experimental work of Ref. 4 (Table 1) because to the best of our knowledge this reference reports the only source of XRD refined crystallographic data (CIF file). The nearly 2:1 over-prediction with respect to the measured values, therefore, is due to the nearly 2:1 ratios of the surface areas and pore volumes of the digitized material vs. the synthesized material of this work (Table 1, cf. 2,152 m2g-1 vs. 1,166 m2g-1, and 0.8 cm3g-1 vs. 0.49 cm3g-1). Thus, the deviations almost disappear when the sorbed amount is expressed per unit area, as seen in the inset of Fig. 1.

It is noteworthy, that force-field still remains an issue, because convenient mixing of generic force fields, or even re-parameterization of them with respect to experiments, are not attempted in this study in contrast to several works found elsewhere.

It is known that temperature, heating time, modulator and solvent type, as well as dwelling time, affect the particle size, polydispersity and crystallinity of the final synthesized material;50 in particular, the crystal size increases as the dwelling time of hydrothermal treatment increases. Because also long dwelling times favor presence of


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the MIL-96(Al) phase,49 the demand for producing large and at the same time pure MIL-100(Al) crystals is subject to antagonistic effects.

In Fig. 2 the powder XRD data for two reactions producing MIL-100(Al) for different dwelling times are shown; all the other reaction conditions are identical.

Figure 2. Powder XRD data for two reaction times of the synthesized MIL-100(Al).

The characteristic reflexes for MIL-96(Al), appearing at 9.08 ° and 7.65 °, become higher as the reaction time increases, whereas the peaks of MIL-100(Al) remain nearly unchanged. For this reason we compromised on reaction conditions of 30 minutes.


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Figure 3. Calculated (triangles), and measured (circles) sorbed water isosteric heat, qst ≡ –ΔsH, at 300 K in MIL-100(Al); the value of liquid water enthalpy of evaporation52 at 300 K is shown (line).

The calculation of the water differential enthalpy of sorption, ΔsH, predicted from MD under given equilibrium pressure, the fugacity of the guest phase, fG, for the given loading, NG, and temperature, β = 1/RT, by means of a Widom type “ghost   ln f G  water” perturbation to calculate the partial derivative,   , is detailed in Ref.    N G 28, and the results appear in Fig. 3. The computed isosteric heat, –ΔsH, being in compliance with the preceding sorption analysis, are in qualitative agreement with the measured data. The slight rise of isosteric heat after the density of 0.5 g g-1 may be connected to a weak small-to-large-cavity transition of water that was found marked at the iron MIL homologue, also causing a small step in the simulated28 and measured isotherms.8,53,54

The computed guest singlet probability density distribution (probability of finding a particle in a certain volume element within host’s interior), was calculated within the entire MIL-100(Al) unit cell. In particular, in Fig. 4 the contours depict density profiles, being related to the inverse exponential of free energy profiles,9,55 throughout


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the plane, z = 18 Å; the latter was selected among several planes so that to achieve optimum visualization results. In the figure an essentially synchronous filling is observed for both small and large cavity systems at low loadings. Subsequently, filling of the small pores proceeds first, and the occupancy of the large pore system evolves further until saturation is attained. Interestingly, this gradual filling of S- and L-cavity systems was found to be more pronounced in the isostructural MIL-100(Fe) giving rise to the aforementioned small-to-large-cavity transition and verifying a previous hypothesis based on experimental findings.54

Figure 4. Computed singlet probability density profiles of water [molec./Å3], (lnvalues), depicted as contours on a plane (see text) spanning the small (S) and large (L) cavities of the MIL-100(Al) unit cell, for two loadings: 0.24 g g-1 (top) and 0.54 g g-1 (bottom). Color code: low (blue) to high (red) probability densities.


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It should be noted that utilizing the entire large cubic unit cell (a = 7.1687 nm) instead of its primitive unit, enabled sampling from a statistically much richer system consisting of 16 small and 8 large cavities.

As shown in Fig. 4 the density value corresponding to the interior of the filled cavities is in agreement with the density of liquid water at ambient conditions indicating water condensation inside the mesoporous MIL cavities, and the guest-guest potential energy is approaching the potential energy of the liquid water close to saturation (Fig. S3). At low loadings the guest-host potential energy is lower than the guest-guest energy up to the filling of the small cavities at 0.5 g g-1; as pressure increases further the guest-guest interactions dominate, mainly because of the hydrogen bonds with the already adsorbed guest molecules (Fig. S3). These interactions are eventually responsible for the stability of the water sorbed phase reducing the total potential energy below the value of the liquid water as denoted by the same figure.

Elaboration of the water molecular trajectories through the Einstein equation, led to self-diffusivities at various loadings within the host lattice as presented in Fig. 10, where is shown that guest mobility grows, starting from the lowest guest occupancies and gradually approaches the computed self-diffusivity of liquid water nearby saturation, supporting the findings of the preceding sorbed phase thermodynamics. It is obvious that the presence of aluminum results in lower self-diffusivity values compared to the iron homologue28 especially at low loadings, thus reflecting the stronger potential field exerted to guests by the MIL-100(Al) host.


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Effect of Terminal species. The computed singlet probability density distribution contours of Fig. 4 reveal primary sorption sites at the pentagonal windows whereon the terminal species of the framework, OH- and structural water, form strong hydrogen bonds with the guest water molecules. Formation of hydrogen bonds with two neighboring terminal species simultaneously was proved to be possible when the guest water is located at the pentagonal windows only (and not at the hexagonal windows) due to the different relative size and geometry of the two window types (Figure S4). Also, among pentagonal windows with various OH- group distributions (Figure S5), minimum energy is obtained when a guest water forms hydrogen bonds with two neighboring OH- groups (OH−OH pair). This is depicted in Fig. 5 in the form of the correlation of the water content of each S-cavity with the number of its OH−OH pairs. We chose two low loadings so that to eliminate any competitive guest−guest interaction from adjacent cavities. The plots of Fig. 5 indicate that the sorbate population increases as the number of OH−OH pairs per S-cavity increases; hence the presence of hydroxyl pairs gives rise to potent primary adsorption sites located at the pentagonal windows.


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Figure 5. Water amount inside each S-cavity (totally sixteen) in the MIL-100(Al) unit cell, as a function of the number of OH-OH pairs (illustrated on the top) for two water loadings: 0.10 g g-1 (middle); 0.16 g g-1 (bottom); least squares regression lines are shown.

Similar study on the number of OH- groups, structural waters (SW), as well as SW−SW and OH−SW pairs, showed zero correlation with the cavity content (see Figure S6).

B. PFG NMR The water mobility in the MIL-100(Al) was measured at the four loadings mentioned above for several observation times. The resulting spin echo attenuation curves of


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Fig. 6 depend on observation time and show a multi exponential decay. This indicates that the water molecules in the MIL-100(Al) bed experience different environments, such as the inter-particle vapor phase and the intra-crystalline sorbed phase, with the guest water molecules being able to exchange between them on the time scale of the PFG NMR measurements. For self-diffusion in beds of porous crystallites, this situation was first considered by Kärger et al.56 who developed a corresponding twosite exchange model for diffusion processes and later the so called fast NMR tracer desorption technique.57 In the following, we adapt these ideas to the mesoporous MIL-100(Al) host.

Figure 6. Spin echo attenuation curves of water in MIL-100(Al) at 298 K for the loading of 0.60 g g-1, measured for multiple observation times; the lines represent mono-exponential fits of Eq. (1) to the slower decay of data thereby the time evolution of the effective intra-crystalline diffusivities and the relative amounts of molecules remaining in the intra-crystalline space during the observation time were evaluated (see text). In general, the fast decay at low b-values in Fig. 6 indicates the exchange of the adsorbed water molecules with the vapor phase in the inter-crystalline space. The slower decay observed when higher b-values are reached is predominantly influenced


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by diffusion in the intra-crystalline space. By invoking the above mentioned models56,57 for the description of guest diffusivity in beds of porous materials, we fitted Eq. (1) to the slower decaying parts of the spin echo attenuations at high b-values (lines in Fig. 6). These single-exponential fits yield the time-dependency of the intra-crystalline effective self-diffusion coefficient, D(Δ), as analyzed below, and the PFG NMR signal intensity, M(b=0, Δ). The latter quantity is proportional to the number of water molecules remaining in the MIL crystallite during the observation time Δ.

It is realized that the population of water molecules which are able to exchange with the inter-crystalline space, increases progressively with increasing observation time; thus, diffusivity becomes higher in the bed as denoted by the fast decaying part of the spin echo attenuations. Consequently, the experimentally determined timedependency of M(b=0, Δ) can be used to evaluate a mean intra-crystalline life time,

according to the relationship

0,

(3)

The value M0 therefore, corresponds to the NMR signal intensity of the water equilibrium loading in the MIL. Figure 7 shows the signal intensities M(b = 0, Δ) as a function of observation time at multiple loadings and their evaluations (fittings) by Eq. 3. This evaluation procedure was applied to all measured data to yield the mean intra-crystalline life time, of water in the MIL, as well as the corresponding M0 values (see Table 2).


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Figure 7. PFG NMR signal intensity dependency on the observation times measured for various water loadings at 298 K in the MIL-100(Al); lines represent fits of Eq. (3).

It must be noted that in order to compare the water equilibrium loading measured by the adsorption isotherms with the obtained M0 values, the latter quantity had to be corrected in relation to the transverse and longitudinal NMR relaxation effects by means of the relaxation term of Eq. (1), and the independently determined relaxation times T1 and T2 (see Table S3) extracted from the inversion recovery and the CPMG techniques. Figure 8 shows plots of the M0 values against the measured water equilibrium loading. The correlation between these values is satisfactory as shown by their linear fit. This indicates that the PFG NMR technique and the proposed data analysis of the slower decaying parts of the spin echo attenuation curves are in fact representative of the water dynamics in the intra-crystalline space of the MIL100(Al).


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Figure 8. Correlation of the PFG NMR signal intensity, M0, with the loading measured from the sorption isotherm of water in MIL-100(Al) at 298 K; the line represents linear fit.

The observed time-dependence of the intra-crystalline effective self-diffusion coefficient D(Δ) of water in MIL-100(Al) is plotted in Fig. 9. The diffusion coefficients tend to slightly decrease with increasing diffusion time. This is best understood by using the concept of partially restricted diffusion at the external MIL100(Al) crystallite boundaries. It is based on the short-time approximation model of Mitra et al,58 where the influence of particle boundaries and their geometry on the measured diffusion coefficient is discussed via time-expansions of the Laplace transformed diffusion equation. Since our data analysis procedure (single exponential fits in Fig. 6) only takes into account water molecules that stay inside the MIL crystallites over the entire observation time, the case of reflective boundary conditions applies leading to the expression58

1

(4)


23


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where r represents the crystallite radius (assuming spherical crystals) and Ds is the (unrestricted) intra-crystalline self-diffusion coefficient of molecules, namely, not being subject to the influence of the particle boundaries. It is the quantity that molecular dynamics simulation computes by elaborating the water trajectories.

Figure 9 presents the measured time-dependent effective (restricted by the particle boundaries) diffusivity of water in the bed of particles as a function of the observation time at various loadings. The fit of Eq. (4) to experimental data provides the pure (unrestricted) intra-crystalline diffusivity, Ds, and the MIL-100(Al) average crystal radius, r. The corresponding results for all loadings and temperatures employed are given in Table 2.

Figure 9. Intra-crystalline diffusivity, D(Δ), of water in MIL-100(Al) as a function of the observation time, measured via PFG NMR (see Fig. 6) for various loadings at 298 K; lines represent fits of Eq. (4) thereby the pure intra-crystalline diffusivity, Ds, was evaluated.


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The estimated mean crystal radius, r, is in a reasonable agreement with the average size of 2 μm to 3 μm obtained from SEM micro-imaging (Figure S7). At the lowest loading the diffusivity does not depend significantly on observation times. Obviously, as a consequence of the low diffusion coefficient at this loading, the majority of the water molecules do not encounter the crystal surface during the applied observation time and no reduction of the diffusion coefficient due to surface phenomena could be measured. At this low guest concentration region therefore, it is reasonable to assume D(Δ) ≈ Ds, being in agreement with the simulated MD value at this regime.

The higher error-bars are explainable on the basis of the lower signal-to-noise ratio (less water, low signal); yet, because of the slower diffusion, one needs to reach higher b-values to attenuate the PFG NMR signal further in order to get accurate diffusion coefficients; nevertheless, since the gradients need to be applied at the t2 intervals the b-values are limited by the transverse relaxation time.

Table 2. NMR signal intensity, M0, mean residence time, τ, intra-crystalline selfdiffusivity, Ds, of water sorbed in the MIL-100(Al) crystals at various loadings, and the temperatures shown in Fig. 11, as they resulted from fittings of Eqs (3) and (4). a/gg

-1

T/K

M0 a.u.

τ/s

2 -1

DS / m s

0.6

298

566 ± 18 0.62 ± 0.19 1.58E-10 ± 1.1E-11

0.6

310

480 ± 12 0.24 ± 0.03 1.92E-10 ± 3.9E-12

0.6

323

359 ± 7

0.6

335

341 ± 17 0.09 ± 0.01 2.50E-10 ± 1.7E-11

0.32

298

345 ± 9

0.52 ± 0.18 1.41E-10 ± 1.0E-11

0.32

310

458 ± 7

0.36 ± 0.09 1.69E-10 ± 7.6E-12

0.32

323

268 ± 11 0.40 ± 0.14 1.66E-10 ± 2.2E-11

0.32

335

206 ± 9

0.12 ± 0.02 1.65E-10 ± 3.5E-11

0.17

298

124 ± 5

0.34 ± 0.10 2.82E-11 ± 5.0E-12

0.17

310

127 ± 9

0.14 ± 0.03 2.87E-11 ± 1.6E-12

0.17

323

117 ± 13 0.08 ± 0.02 2.84E-11 ± 4.9E-12

0.17

335

111 ± 10 0.06 ± 0.01 3.14E-11 ± 6.4E-12

0.12

298

53 ± 5

0.13 ± 0.01 2.40E-10 ± 5.6E-12

0.25 ± 0.14 5.47E-12 ± 1.7E-12


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Comparisons between the intra-crystalline self-diffusivities as extracted by the PFG NMR data via the above described procedure, and the MD computations are presented in Fig. 10.

Figure 10. Intra-crystalline water self-diffusivity as a function of loading in MIL100(Al), extracted from PFG NMR (blue squares), and predicted by MD simulations (red diamonds); simulation data from the Fe homologue28 (triangles) are also present for comparison; measured59 (dashed line), and computed using the SPC/E model (solid line) liquid water self-diffusivities at 300 K are also shown.

Figure 11 shows the temperature dependence of self-diffusivity at various water loadings as obtained by PFG NMR. Table 3 presents the activation energies, EA, as estimated by the regression of the Arrhenius formula. It is observed that all activation energies are smaller than the isosteric heat (enthalpy of desorption) shown in Fig. 3; therefore, they must constitute low energy barriers separating adjacent adsorption sites within the crystals’ framework. This finding validates further our applied elaboration toward yielding the intra-crystalline self-diffusivity.


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Figure 11. Variation of the extracted self-diffusivities from PFG NMR as a function of temperature at various loadings, and regression lines after fitting Arrhenius equation.

Table 3. Activation energies of the diffusion process derived from regression of the Arrhenius equation to the plots shown in Fig. 11.

As seen in the table, the activation energy of the intra-crystalline diffusion increases as the host framework is progressively populated with water adsorbate molecules. This can be readily confirmed by Fig. 11, where it is seen that Ds becomes almost insensitive to temperature at low water loadings. This finding is in line with the conclusion drawn from the predicted guest higher probability density profile (low free energy profiles) around the apertures of both S- and L-cavity systems at low loadings as illustrated in Fig. 4; hence diffusion proceeds mainly through a rather homogeneously dispersed adsorbed layer, which is translated to a slow process of low activation energy too.


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As the adsorbate concentration increases further, the predicted guest probability density profiles indicate progressive filling of the L-cavity connecting windows, thereby water molecules start surmounting the higher free energy barriers at the cavity centers (lower probability density profile), therefore contributing further to the overall sorbate flow in a sense of a diffusion process of higher activation energy.

CONCLUSIONS It is presented an a posteriori experimental verification of a computational methodology developed lately, comprising molecular dynamics modeling of the equilibria and dynamics phenomena of water sorbed in the MIL-100(Al) host framework, first employed in a previous study of water sorbed in the iron homologue of MIL-100.28 Application of PFG NMR at that study, nonetheless, proved to be impossible because the

1

H-relaxation times of water in the MIL-100(Fe) are

significantly shorter (T1 < 1 ms) than the ones in its aluminum homologue of this work.

A computational scheme was devised toward assigning partial charges on the host framework atoms that combines: firstly, DFT calculations in a representative finite framework cluster cleaved from the MIL-100(Al) unit cell, with an inverse EEM set of calculations to derive electronegativity, hardness, and gamma parameters for each atom type; and secondly, EEM calculations based on the principle of electronegativity equalization of the entire host crystal to obtain partial charges on the MIL-100(Al) atoms. It is noteworthy that the derived EEM parameters for the specific atom types, can be used in EEM calculations for other aluminum MOFs comprising similar atom


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types in their framework. Subsequently, MD thermodynamics predictions for various sorbate concentrations, as well as differential enthalpies of adsorption were compared with data from the real system’s equilibria measured in this work.

PFG NMR signal intensities extrapolated to zero time after having been corrected in relation to the transverse and longitudinal NMR relaxation effects, were linearly correlated with the measured equilibrium loadings of water in the MIL-100(Al). This finding indicates that the particular NMR technique and the proposed data analysis of the slower decaying parts of the spin echo attenuation curves are representative of the water dynamics phenomena in the MIL’s intra-crystalline space.

The PFG NMR experiment measured an effective observation time-dependent diffusion coefficient, representing the guests’ motion within the intra-crystalline mesopore network up to times they have either reflected or left the surrounding surface barriers in various beds of MIL-100(Al) crystallites. To extract the pure intracrystalline self-diffusivity from the measured spin echo attenuation curves, a theoretical expression of random walkers’ diffusivity as a function of the NMR observation time was utilized, assuming spherical interfaces, and reflecting boundary conditions for the solution of the diffusion equation following Mitra et al. The intracrystalline self-diffusivities computed via MD for a series of water concentrations were found in reasonable agreement with the ones extracted from PFG NMR taking into consideration the convenient hypothesis of spherical crystallites.


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The measured activation energies for diffusion were in agreement with the predicted trend in the evolution of the water adsorbed phase probability density profile (being analogous to free energy profile) with increasing loading, verifying also the progressive filling of the small and large pore systems first proposed in the MIL100(Fe).28

ASSOCIATED CONTENT Supporting Information. Details of the electrostatic calculations, terminal group distributions, MD, and PFG NMR supplementary data (PDF). This material is available free of charge via the Internet at http://pubs.acs.org.

ACKNOWLEDGMENTS The German Federal Ministry of Education and Research (BMBF) and the Greek General Secretariat for Research and Technology (GSRT) are kindly acknowledged for financial support under the bilateral project “WASSERMOD”. GKP thanks the GRNET High Performance Computing Services for the CPU time provided on ARIS computer nodes.

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Molecular Dynamics Phenomena of Water in the Metalorganic

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