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. 2025 Jan 25;16(5):1213–1220. doi: 10.1021/acs.jpclett.4c03070

Combining Brillouin Light Scattering Spectroscopy and Machine-Learned Interatomic Potentials to Probe Mechanical Properties of Metal-Organic Frameworks

Florian P Lindner †,‡,*, Nina Strasser †, Martin Schultze ‡, Sandro Wieser §, Christian Slugovc ∥, Kareem Elsayad ⊥, Kristie J Koski #, Egbert Zojer †, Caterina Czibula ∇,*
PMCID: PMC11808784  PMID: 39862191

Abstract

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The mechanical properties of metal-organic frameworks (MOFs) are of high fundamental and practical relevance. A particularly intriguing technique for determining anisotropic elastic tensors is Brillouin scattering, which so far has rarely been used for highly complex materials like MOFs. In the present contribution, we apply this technique to study a newly synthesized MOF-type material, referred to as GUT2. The experiments are combined with state-of-the-art simulations of elastic properties and phonon bands, which are based on machine-learning force fields and dispersion-corrected density functional theory. This provides a comprehensive understanding of the experimental signals, which can be correlated to the longitudinal and transverse sound velocities of the material. Notably, the combination of the insights from simulations and experiments allows the determination of approximate values for the components of the elastic tensor of the studied material even when dealing with comparably small single crystals, which limit the range of accessible experimental data.

Almost 30 years after their discovery,1,2 research interest in metal-organic frameworks (MOFs) is still growing steadily. The versatile combination of inorganic and organic building blocks, intrinsic to MOFs allows the formation of microporous, complex, and yet still crystalline structures.3 A major reason for the lasting research interest in these materials is the plethora of their possible applications in various fields including gas storage, gas separation, catalysis, sensorics, and energy storage.4−7 Aside from their functional properties, for virtually any of the envisaged applications, the mechanical characteristics of the used MOFs are relevant. While for a gas storage application the available pore volume determines the hypothetical performance, for a real-world implementation one also needs to consider how easily the MOF is deformed by mechanical forces during loading or deloading cycles of the pores.8,9 High mechanical forces exerted on MOF crystals during operation could lead to a loss of structural integrity and in extreme cases even to the onset of amorphization.10−12 It is thus evident that structural deformations under mechanical stress, which are typically quantified by engineering constants like Young modulus, E, or the Poisson’s ratio, ν, will crucially affect the functional properties of MOFs. Hence, for many applications, a sound understanding of the mechanical properties of the MOFs is highly relevant. In view of their undeniable importance, also a number of theoretical studies predicting mechanical properties MOFs exist.13−19 Nevertheless, experimental data especially at the single crystal level, to verify theoretical predictions, for example, the elastic constants Cij of MOFs, is scarce: Besides employing pressure-dependent (powder) X-ray diffraction (PXRD),20−24 elastic properties of MOFs were measured by nanoindentation or atomic force microscopy.25−28 The latter approach is, however, prone to potential errors, when the anisotropy of the probed samples and nonunidirectional stress fields generated by the indenter tips are not correctly accounted for.9,29−31 Measurements of bulk moduli in pressure PXRD experiments tend to give a better agreement with theoretical predictions32 but are often restricted to applying hydrostatic pressure, e.g., when using diamond-anvil cells.33

In this letter, we demonstrate that Brillouin spectroscopy in combination with atomistic simulations is a promising, non-invasive experimental method to investigate the anisotropic mechanical properties of MOF single crystals in a contactless manner. It relies on analyzing light scattered from thermally excited acoustic phonons causing density fluctuations in the probed sample.34−36 This provides access to the sound velocity tensor, from which the full elastic tensor, Cij, of the studied material can be derived.37−40 In the context of framework materials, Brillouin spectroscopy has, for example, been applied to study the mechanical properties of single crystals of the prototypical zeolitic imidazolate framework ZIF-841,42 and, recently, also of perovskite-like dense MOFs.43

Here, we conducted Brillouin scattering experiments on a single crystal of a newly designed zinc(II) MOF/coordination polymer, with chemical formula C14H18N4O4Zn–H2O. In the following, it will be referred to as GUT2 to be consistent with the literature. Its synthesis and structural, thermal, and vibrational properties are described in Kodolitsch et al.44 The material features common characteristics of MOFs,45 consisting of metal-ion derived secondary building units connected by organic linkers and displaying a non-negligible porosity. It crystallizes in the orthorhombic space group Pcca and contains 8 Zn2+ ions in the primitive unit cell. Its density is calculated to be 1.502 g/cm3. The unit cell and crystal structure were determined by single crystal XRD (sc-XRD) measurements, which yield (room temperature) lattice constants of: a = 15.1861(12) Å, b = 15.0082(13) Å and c = 15.0568(13) Å.44 The crystal structure of the investigated MOF is visualized in Figure 1. It shows zinc ions tetrahedrally coordinated with two imidazole nitrogen atoms and two carboxylate groups. Two zinc ions are linked by two bridging ligands (2-methyl-imidazol) pointing in the same direction. Overall, the coordination of each zinc ion to two imidazole-nitrogen atoms and to two carboxylate ligands results in the formation of chains of coordination polymers. These are connected via hydrogen bonds between the carboxylate groups. The interchain-bonding is reinforced by H-bonds involving water molecules in well-defined positions. The water molecules block the pore channels along the b-axis, see Figure 1, while the pore channels along the a- and c-axes remain open. Further details are provided in Kodolitsch et al.44

Figure 1.

Figure 1

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Crystal structure of the studied zinc(II) coordination polymer GUT2 viewed from different directions. Pores blocked by adsorbed water molecules are shaded in yellow, whereas open pore channels along a- and c-directions are shaded in blue. Zinc coordination polyhedra are shaded in gray. The primitive unit cell is indicated by thin black lines.

To measure the single crystal elastic constants of GUT2, an as-synthesized plate-like GUT2 crystal picked from the mother solution, with an extent of roughly 0.5 mm was isolated and fixed by double sided adhesive tape onto a stainless-steel plate containing a 5 mm hole in its center, as shown in Figure 2. Subsequently, the stainless-steel plate holding the MOF crystal was attached to an optic rotation mount (Newport RSP-1T) to allow precise adjustment of the sample rotation angle θ, as defined in Figure 2. Then the Brillouin scattering experiments were conducted in a forward symmetric (90a) scattering geometry.39,46 A schematic of this scattering geometry is shown in Figure 2. For details regarding the experimental setup, see Supporting Information. The wave vector q of the thermally activated phonons participating in the Brillouin scattering process, is always within the scattering plane, defined by the wave vectors ks and ki of the scattered and incident laser light (532 nm) within the sample, see Figure 2. Notice that this coplanarity is not necessarily preserved when considering ks and ki outside the sample due to the refraction of the laser light at the sample surfaces. The collected scattered light was analyzed using a six-pass tandem Fabry–Perot interferometer (JRS Scientific Instruments, TFP-1). Starting from an arbitrary reference position, for which the tilt angle was defined to be 0°, the sample was rotated in steps of 10° up to a tilt angle θ = 90°. At tilt angles >90°, the sample quality did not allow the detection of a high-quality Brillouin spectrum. The Brillouin shifts of the recorded spectra were obtained by fitting Lorentzian functions to the observable quasilongitudinal (QL) and quasitransversal (QT) peaks.

Figure 2.

Figure 2

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Forward symmetric (90a) Brillouin scattering geometry and single crystal MOF mounted on a metallic sample holder (top left). The optical paths of the incident (532 nm) and scattered laser light are shown in dark and light green, respectively. Vectors ki and ks denote the wave vectors of incident- and scattered laser light. The wave vector of the thermally activated phonon involved in the scattering process is denoted by q. Refraction of the light at the surfaces of the sample is accounted for in the schematic sketch. By rotation of the sample around the axis indicated in purple (by an angle θ), directionally dependent sound velocities can be determined.

A key challenge encountered in the studies of GUT2 is that the available MOF single crystals reached sizes of only a few hundred μm. This makes handling the investigated specimens during experiments difficult. Here, a tight integration of the experiments with state-of-the-art atomistic simulations of the elastic properties of GUT2 provides the necessary complementary information: on the one hand, the simulations ease the interpretation of the experimental results, and on the other hand, they also justify approximations concerning the crystal symmetry, which will be made below. As described in the Supporting Information, a direct calculation of the elastic tensor elements Cij using density functional theory with a fully converged plane-wave basis set is hardly possible due to the complexity of the MOF material. Moreover, approximate approaches like the clamped-ion method47 implemented in the internal routines of, e.g., the VASP code48,49 is numerically not stable here (see the Supporting Information). Thus, we resorted to the use of machine-learned potentials of the moment-tensor (MTP)50 type, parametrized following the procedure described by Wieser et al.51 The latter builds on using data calculated at the density functional theory (DFT) level generated during VASP active learning52 to parametrize MTPs utilizing the MLIP package.53 The DFT reference data were calculated using the PBE functional,54,55 combined with Grimme’s D3 dispersion correction56 with Becke–Johnson damping57 (DFT-PBE), and the obtained potentials reach essentially DFT-PBE accuracy.50 They are, however, many orders of magnitude faster than the parent DFT-PBE approach. Moreover, as we outline in the Supporting Information, the procedure of acquiring MTPs requires fewer single point DFT-PBE calculations than a “direct” simulation of the elastic tensor using ab initio calculations. The availability of the MTPs allows a full relaxation of atomic positions in the strained cells and, in combination with the phonopy package,58,59 the simulation of phonon band structures employing converged GUT2 supercells. To confirm the appropriateness of the force-field calculated elastic constants, D3 dispersion-corrected56,57 DFT-PBE calculations with a potentially less complete, but numerically more efficient basis set were performed employing the CRYSTAL23 package60,61 (see the Supporting Information). A comprehensive description of the applied simulation methodology can be found in the Supporting Information. While the simulations performed with the machine-learned potential in the following will be referred to as MTP results, the CRYSTAL23 results will be denoted DFT-PBE results. A typical spectrum observed in our Brillouin experiments is shown in Figure 3a. It displays two distinct peaks on either side of the elastic peak at 0 GHz. The two sets of peaks are shifted by ∼±5 and ∼±10 GHz. Their nature can be identified based on the calculated low-frequency phonon band structure of GUT2, displayed in Figure 3b. It reveals two nearly degenerate acoustic bands at lower frequencies and a single higher-energy acoustic band. Analyzing the degree of longitudinality of the bands (for details see Supporting Information) shows that the higher frequency band is of primarily longitudinal character, while the lower two bands are primarily transverse in nature. These results allow us to identify the peaks shifted by ∼±10 GHz as quasilongitudinal (QL) and the ones shifted by ∼±5 GHz as quasitransverse (QT). The near degeneracy of the transverse bands also explains, why in the measured Brillouin spectra depicted in Figure 3a, only one QT peak is resolved.

Figure 3.

Figure 3

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(a) Typical Brillouin spectrum of GUT2 with quasilongitudinal (QL) and quasitransversal (QT) peaks, as observed during experiments (a). (b) MTP-calculated low-frequency phonon band structure of GUT2. The bands in (b) are colored according to their acoustic character.62

In the present case of forward scattering geometry, the interfaces of the sample and environment are parallel to each other and both are at the same angle with respect to incident and scattered wave vectors ki and ks. In this case, the velocities of the acoustic phonons, ν, that were involved in the scattering process, are directly related to the observed frequency shifts, ΩB, and the wavelength, λ, (532 nm) of the incident laser light via:39,46,63

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Using eq 1, a distribution of sound velocities for different tilt angles θ, i.e., along different directions within the investigated MOF single crystal, are obtained. These are shown in Figure 4.

Figure 4.

Figure 4

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(a) Direction-dependent sound velocities in GUT2 derived from Brillouin experiments rotating the sample about the axis shown in Figure 1. The data points are drawn with an ad hoc uncertainty bar of 10%.

The data reveal that the transverse sound velocities in the GUT2 crystal are within 1.4 and 1.9 km/s, while the longitudinal sound velocities vary between 3.5 and 4.3 km/s (see also Table 1). Importantly, the measured sound velocities agree rather well with the calculated ones, even though the spread of sound velocities is somewhat larger in the experiments (Table 1). The situation in GUT2 is reminiscent of the observations for ZIF-8 by Tan et al.41 There, however, smaller sound velocities between 1.0 km/s–1.2 km/s for transverse and 3.1 km/s–3.2 km/s for longitudinal waves were observed in similar measurements.36 As will be detailed below, this suggests that GUT2 is stiffer than ZIF-8. It is also interesting to compare the results for GUT2 with the sound velocity measured by phonon acoustic spectroscopy for the closed pore phase of zinc-based flexible MOF DUT-1(Zn). As in this case ν amounted to only 0.8 km/s,64 one can conclude that even in its closed pore phase DUT-1(Zn) is significantly less stiff than both ZIF-8 and GUT2.

Table 1. Measured and Calculated Speeds of Sound, Elastic ConstantsCij and Hill Averaged Mechanical Properties for GUT2, Extracted from Brillouin Spectroscopy, and Calculated Using the Moment Tensor Potential and DFT-PBEa.

    Brillouin spectroscopy MTP DFT-PBE
long. sound velocity (min/max) [km/s] 3.49 ± 0.35/4.31 ± 0.43 3.77/4.20 3.54/3.93
trans. sound velocity (min/max) [km/s] 1.38 ± 0.14/1.86 ± 0.19 1.77/1.92 1.74/1.99
            
cubic approximation C11 [GPa] 18.4 ± 3.7 23.6 21.4
  C12 [GPa] 12.7 ± 2.5 13.0 11.8
  C44 [GPa] 5.2 ± 1.1 5.7 4.9
  bulk modulus K [GPa] 14.6 ± 2.9 16.5 15.0
  Young’s modulus E [GPa] 11.2 ± 2.3 14.9 13.1
  shear modulus G [GPa] 4.1 ± 0.8 5.5 4.8
  Poisson’s ratio [1] 0.372 ± 0.004 0.349 0.355
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a

The elastic constants are reported within the cubic approximation. Reported error bars are calculated assuming an ad hoc ±10% estimate for the uncertainties of the measured sound velocities.

The fully quantitative determination of the components of a material’s elastic tensor, Cij, from measured sound velocities along specific crystallographic directions requires the inversion of the so-called Christoffel equation.65,66 The latter describe the dispersion relation for plane sound waves traveling through a crystalline solid and relate direction-dependent sound velocities to a material’s elastic constants Cij.67 Despite considerable efforts, for the available single crystals the set of experimental data points is rather limited. Thus, to gain further insights, it is inevitable to introduce certain approximations. Here, we pursued a dual approach: On the one hand, we extracted elastic constants assuming a higher (cubic) crystal symmetry. This is motivated by the observation that the calculated elastic tensor is reasonably close to cubic, for both the MTP and DFT-PBE approaches, as is shown in the Supporting Information. On the other hand, from the experiments we estimated limits to certain elements of the elastic tensor for the actual orthorhombic symmetry and compare these estimates to the calculated values.

Within the cubic approximation, the calculated tensor elements Cij (listed in Table 1) are obtained by averaging over the respective elements of the orthorhombic elastic tensors listed in the Supporting Information. Regarding the experiments, estimates for the elastic constants can be made based on the measured direction dependent values of the sound velocities for the longitudinal and transversal acoustic modes, using the so-called “envelope method”.42 Originally introduced in high-pressure Brillouin experiments,68,69 the “envelope method” allows deriving an estimate of the elastic constants of a crystalline sample with unknown orientation. It assumes that the measured direction-dependent sound velocities depicted in Figure 4 form an envelope to the maximum and minimum acoustic velocities, which, in the cubic case, are straightforwardly related to the elements of the elastic tensor. The mathematical details of this approach for the case of GUT2 are described in the Supporting Information. Notably, when applying the “envelope method” to determine the elastic tensor of the actually cubic ZIF-8 crystal,42 a very good agreement with the values obtained by a full inversion of Christoffel relations41 was obtained. Assuming that the measured minimum quasilongitudinal sound velocity (3.5 km/s) and the minima and maxima of the transverse sound velocity from Figure 4 (1.9 and 1.4 km/s, respectively) are good estimates for the true extremal velocities, one then obtains the following estimates for the independent elements of the elastic tensor of GUT2 in the cubic approximation: C11 = 18.4 GPa, C44 = 5.2 GPa, and C12 = 12.7 GPa. As shown in Table 1, the values for C44 and C12 agree very well with both types of simulations. The value of C11 extracted from the experiments is somewhat smaller than the calculated value, but the deviation is still acceptable.

To estimate, how well the sound velocity distributions are sampled in the experiments (c.f., Figure 4), we used the “christoffel” python package67 in combination with the MTP- and DFT-calculated elastic tensors to solve the Christoffel dispersion relation in forward direction. The obtained simulated velocity distributions based on the MTP elastic tensor for the QL and QT modes are shown in Figure 5 (with the DFT-PBE results in the Supporting Information). The calculated velocity distributions reveal that there is a reasonable correspondence between the ranges of the calculated and measured sound velocities, especially considering the experimental errors and the differences in the values obtained with the two simulation approaches (Table 1). In fact, the observation that the experimental spread in sound velocities is even larger than the calculated one, supports the assumption that the true range of sound velocities is suitably probed in the experiments.

Figure 5.

Figure 5

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Directional dependence of the MTP-based sound velocity distributions plotted on a unit sphere. (a) Slow quasitransversal mode vQT2, (b) fast quasitransversal mode vQT1 and (c) quasilongitudinal mode vQL. The similar DFT-PBE-calculated distributions are contained in the Supporting Information.

Notably, the directions of the maxima and minima of the sound velocities are consistent with the assumptions made in the “envelope method” for a subset of the crystallographic directions (see discussion in the Supporting Information). We also derived equations, which for the orthorhombic symmetry allow us to at least determine limits to the values of the components of the elastic tensor from the measured minimum and maximum sound velocities. Applying these relations (for details, see Supporting Information), we find that the first three diagonal elements of the elastic tensor (C11, C22, and C33) should be in the range between 18.5 and 29.7 GPa, while C44, C55, and C66 must be smaller than 5.2 GPa, which is consistent with the results presented above for the cubic approximation. For the set of diagonal elements related to compressive strain (C11, C22, and C33), the conditions derived from the experiments are fulfilled for both the MTP and the DFT-PBE simulated elastic tensor (with values of 21.4 (19.8), 26.6 (22.8), and 22.8 (21.4) GPa in the MTP (DFT-PBE) simulations). For the shear-related components (C44, C55, and C66) only the DFT-PBE results are strictly below the experimentally set limit, while the MTP results are slightly higher (albeit by at most 0.6 GPa with values of 5.4 (4.4) GPa, 5.7 (5.0) GPa, and 5.8 (5.2) GPa in the MTP (DFT-PBE) cases; see also the Supporting Information).

From the elements of the elastic tensor, so-called engineering constants can be obtained. They are typically used to quantify the mechanical robustness of a material in technical applications and are derived from the components of the elastic tensor using averaging schemes.70,71 Averaged engineering constants like Young’s, bulk or shear moduli are typically scalar values that account for the fact that in applications, one often has to deal with polycrystalline samples,72 which can be assumed to behave in good approximation as isotropic.63 Using the ELATE package,73 we calculated the Hill-averaged71 Young’s modulus E, shear modulus G, bulk modulus K, and Poisson’s ratio ν for GUT2. This was done using the Brillouin scattering elastic constants, as well as the values from the MTP and DFT-PBE simulations in the cubic approximation (see Table 1). Again, the engineering constants derived from the simulations and from the experiments are in good agreement. The deviations are slightly larger between experiments and the MTP simulations, which we attribute to the larger (∼20%) deviation between the experimental and the MTP-calculated values of C11. In passing, we mention that the values obtained for the averaged engineering constants when using the simulated, full orthorhombic elastic tensor are essentially identical to the results from the cubic approximation (see the Supporting Information).

To put the obtained results into perspective, we compare the elastic constants of GUT2 to those obtained in the past for two prototypical cubic systems, ZIF-8 and MOF-5: comparing the values reported by Tan et al.41 for ZIF-8 (C11 = 9.5 GPa, C12 = 6.9 GPa, and C44 = 0.9 GPa) with the elastic constants of GUT2, one sees that ZIF-8 displays a significantly larger structural flexibility. The most striking difference is that the shear constant C44 of GUT2 is almost 6 times as large as that reported for ZIF-8. The above trends prevail for the engineering constants: Tan et al. report a Hill average for the shear modulus of ∼1.1 GPa for ZIF-8, which suggests an almost 4 times lower resistance against shear stresses in ZIF-8 than in GUT2. Furthermore, GUT2 with its bulk modulus of 14.6 GPa and a Young’s modulus of 11.2 GPa is almost twice as resistant against hydrostatic compression and uniaxial loading as ZIF-8, for which values of 7.7 and 3.1 GPa have been reported, respectively.41

For MOF-5 (as the prototypical example of an isoreticular MOF), to the best of our knowledge, no experimental values of the elastic constants are available. Nevertheless, it is interesting to compare our results for GUT2 to the available theoretical predictions for MOF-5: based on GGA level DFT calculations, Bahr et al.74 report the elastic constants for MOF-5 to be C11 = 27.8 GPa, C12 = 10.6 GPa, and C44 = 3.6 GPa. This means that for this classical isoreticular MOF rather similar elastic constants as in GUT2 are to be expected.

In summary, we conducted state-of-the-art Brillouin scattering experiments for a newly synthesized MOF and deduced its mechanical properties from the determined single crystal elastic constants Cij. These were obtained from measuring direction-dependent sound velocities. Using a machine-learned moment tensor potential (MTP) as well as within DFT-PBE, we were able to also simulate the elements of the MOF’s elastic tensor. In doing so, we also demonstrate that the MTP approach overall requires less expensive DFT-PBE steps then a direct ab initio calculation, and thus is in fact a cheaper alternative. Morevover, as we demonstrate here by using the trained MTPs to calculate phonon band structures, which crucially help in interpreting the measured Brillouin spectra, the MTP approach is way more flexible than standard ab initio procedures, which basically only give one quantity at a time. The main challenge faced in the evaluation of the experimental data was that the latter is limited due to the comparably small single crystals which often displayed a significant surface roughness, making the collection of Brillouin spectra difficult. To overcome this issue, the crystal symmetry in the data evaluation was approximated as cubic and, as an alternative approach, estimates for certain tensor elements of the true, orthorhombic system were performed. Despite these challenges, an overall very good agreement between the elastic constants derived from the experiments and from the simulations was obtained. This also testifies to the predictive power of the numerically extremely efficient machine-learned potential, which is applied here for the first time to directly simulate the elastic tensor elements of a newly synthesized MOF. Thus, in future studies this type of potential will be used in combination with molecular dynamics simulations to include the effects of temperature in the prediction of the mechanical properties of MOFs. This will even better link simulations and experiments, as the latter are typically performed at elevated temperatures. The presented approach provides an avenue for obtaining the elastic constants of MOFs in a reliable and efficient manner, complementing established methods like pressure-dependent powder X-ray diffraction or Raman spectroscopy. Moreover, as a next step, Brillouin scattering could potentially be used to probe viscous properties of MOFs in order to investigate how they behave under variable (time-dependent) loadings.

Acknowledgments

C.C. acknowledges the Hertha Firnberg program of the Austrian Science Fund (FWF) for funding. This research was funded in part by the Austrian Science Fund (FWF) [T 1314-N], grant DOI: 10.55776/T1314. C.C. further acknowledges the program Unconventional Research [grant no. PN 38] of the Styrian Government (Land Steiermark) for funding. For open access purposes, the author has applied a CC BY public copyright license to any author accepted manuscript version arising from this submission. F.P.L. was funded by the Graz University of Technology through the Lead Project Porous Materials @ Work for Sustainability (PMWSP03). Computational results have been obtained using the Vienna Scientific Cluster, VSC-5.

Data Availability Statement

All experimental data and the corresponding analysis are available at https://doi.org/10.3217/fvahy-htj04.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpclett.4c03070.

  • Experimental details; theoretical methodology; simulation-assisted data analysis (PDF)

The authors declare no competing financial interest.

Supplementary Material

jz4c03070_si_001.pdf (814KB, pdf)

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

jz4c03070_si_001.pdf (814KB, pdf)

Data Availability Statement

All experimental data and the corresponding analysis are available at https://doi.org/10.3217/fvahy-htj04.

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