Metal-Organic Framework Breathing in Electric Field: A Theoretical

Article Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

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Metal−Organic Framework Breathing in the Electric Field: A Theoretical Study Andrei L. Kolesnikov,*,† Yury A. Budkov,*,‡,§ Jens Möllmer,† Michael G. Kiselev,§ and Roger Gläser† †

Institut für Nichtklassische Chemie e.V., Permoserstr. 15, 04318 Leipzig, Germany Tikhonov Moscow Institute of Electronics and Mathematics, National Research University Higher School of Economics, Tallinskaya 34, 123458 Moscow, Russia § G. A. Krestov Institute of Solution Chemistry of the Russian Academy of Sciences, Akademicheskaia 1, 153045 Ivanovo, Russia

J. Phys. Chem. C Downloaded from pubs.acs.org by UNIV OF LOUISIANA AT LAFAYETTE on 04/18/19. For personal use only.

‡

S Supporting Information *

ABSTRACT: In this manuscript, we study the electrically induced breathing of a metal−organic framework (MOF) within a 2D lattice model. The Helmholtz free energy of the MOF in the electric field consists of two parts: the electrostatic energy of the dielectric body in the external electric field and the elastic energy of the framework. The first contribution is calculated from the first principles of statistical mechanics with an account of MOF symmetry. By minimizing the obtained free energy and solving the resulting system of equations, we obtain the local electric field and the parameter of the unit cell (angle α). The paper also studies the cross-sectional area of the unit cell and the polarization as functions of the external electric field. We obtain the hysteresis in the region of the structural transition of the framework. Our results are in qualitative agreement with the literature data of the molecular dynamics simulation of MIL-53(Cr).

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INTRODUCTION Stimuli-responsive materials attract the attention of researchers because of their numerous applications and remarkable properties.1,2 Among external stimuli, the most frequently used are adsorption/absorption,3−6 temperature,3,7 electric and magnetic fields,8−10 mechanical stimuli,11−13 light.10 There are numerous types of materials that respond to certain environmental changes. Among them are polymers14−20,20−24 (and as a consequence derived materials and composites containing polymers), liquid crystals, ionic liquids, solid adsorbents. It is well known that solid materials can deform during gas adsorption with typical magnitudes of volume strain less than one percent.25,26 However, there are groups of materials, for example, aerogels, whose deformation can exceed tens of percent.27 Another interesting class of materials with an enhanced flexible response is soft metal−organic frameworks (MOFs). MOFs are inorganic−organic hybrids, with a lattice structure consisting of metal ions bound by ligands.28 Although the mechanism of flexibility is different from that of polymers and depends on the type of MOF,3−6,29,30 they react to the similar stimuli; among the most frequently used is gas adsorption. Because of the MOF’s unique properties, there are a lot of potential applications, for example, gas separation,31−34 gas/heat storage,35−39 sensing,40−42 controllable capture and release.43−45 The unusual properties of MOFs, especially gate-opening MOFs, attract the attention of a lot of researchers and motivate them to find new possibilities to control material properties contact-free. The recent literature shows intensive work in the field of stimuli-responsive MOFs, and here we will discuss some of these publications. Sievers et al.46 studied the dependence of © XXXX American Chemical Society

MOF mesoparticles on water vapor pressure. The remarkable feature is that its crystalline structure consists of a twodimensional coordination polymer, packed in parallel sheets, with organized clusters of water molecules lying between the sheets and bridging them via a dense H-bond network.46 Authors observed that the particles respond with shrinking to the decrease of vapor pressure. Li et al.47 presented dual stimuli-responsive MOF (magnetic PCN-250), which shows responses to both magnetic induction and ultraviolet light. Authors demonstrated that the combination of both triggers results in high CO2 desorption at 1 bar. Huang et al.48 investigated the structural behavior of the gallium-based MOF having the MIL-53 topology. The dry material can switch between a narrow-pore phase and a large-pore phase by means of a temperature increase; also, the hysteresis accompanies the structural transition. Baimpos et al.49 studied real-time deformation of HKUST-1 crystals caused by humidity adsorption and desorption. Also, the authors observed the nonmonotonic structural changes during the initial hydration of crystals. Namasani and Yazaydin,50 using density functional theory and molecular dynamics (MD), studied the behavior of MOFs in an external electric field. In particular, they showed the possibility to rotate the organic linkers with permanent dipole moment by changing the strength of the electric field. Tam and Yazaydin8 demonstrated the potential mechanism of electric field-controlled molecular gates. They are complex molecules with permanent dipole anchored on the host MOF Received: December 3, 2018 Revised: March 27, 2019 Published: March 28, 2019 A

DOI: 10.1021/acs.jpcc.8b11630 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

field. Clearly, the list is not comprehensive, and highly sensitive to the structure of the MOF. As far as both induced dipoles are located in the close vicinity of an ion, for the sake of simplicity, we will put them in the same position, namely, in the center of an ion. Thus, nodes can be treated as the positions of the induced local dipoles. Now, let us consider the model of MOF as a 2D lattice with N junctions (see Figure 1). Each node represents the metal ion and the gray connections are the linkers. Also, induced dipole moment can be decomposed into two contributionsan orientational part from the permanent dipole moments and an instantaneous induced dipole moment from the molecular polarization. We assume, for simplicity, that both permanent and instantaneous induced dipole moments have only two possible opposite orientations along which the homogeneous constant electric field E is applied, as it is shown in Figure 1. The total Helmholtz free energy of MOF can be written as a sum of two terms

and rotating by the changing of the direction of the electric field. Recently, it has been shown that an external electric field has a great potential in separation processes,51,52 as the authors showed an increase in the separation factor for the mixture C3H6/C3H8. Moreover, by means of MD simulations, Ghoufi and co-authors9 showed the reversible structural transition of MIL-53(Cr) by applying an external electric field. In an empty host material, the first-order transition with hysteresis was observed at 1−2 V/nm. It should be noted that electric field values are far above the breakdown electric field of air. Schmid proposed in his work53 a possible molecular explanation of the electric field-induced phase transition. The mechanism is based on the mutual dipole−dipole interactions while the external electric field induces the average dipole moment of the interacting groups. In addition, the author proposed several possible mechanisms of polarization. To the best of our knowledge, there are no statistical models describing the effect of an external electric field on the MOFs. Thus, in this manuscript, we propose a model describing the phase transition of a solid matrix induced by dipole−dipole interactions under an external electric field. As a starting point, we use the physical model, proposed in the mentioned work,53 that is, we consider a rhombic lattice 2D structure (see Figure 1) with induced dipoles in the nodes of the lattice. In this approximation, each node corresponds to the metal ion, whereas the organic linkers are treated with a fixed length a.

F = Fm + Fel

(1)

where the first term on the right hand side is the elastic free energy of the 2D matrix, which can be expressed as a function of angle (α) and the second term is the electrostatic free energy of the lattice. In order to obtain the elastic free energy, we expand it in the power-law series around equilibrium angle (α0) at zero value of the external electric field (the derivation is presented in the Supporting Information: Appendix II) Fm(α) = Fm,0 + κ0(α − α0) + κ1(α − α0)2 + κ2(α − α0)3 + κ3(α − α0)4

(2)

where Fm,0 is the free energy of MOF at α0 and the κn (n = 1, 2, 3) are the adjusted coefficients which describe the elastic behavior of the particular MOF. The first constant κ0 is related to the initial stress in the MOF and will be discussed at the end of this section. The elastic coefficients κ1, κ2, and κ3 were obtained, in order to qualitatively reproduce the structural transition behavior like in the MIL-53, namely, the jump from one phase [large pore (lp)-phase] to the other [narrow pore (np)-phase]. Naturally, the proposed elastic free energy does not describe the complex two-stage transitions, which were obtained in the experiment,54 but at the same time contains all the necessary information to describe the structural transition, driven by dipole−dipole interactions (see below). In order to calculate the electrostatic contribution to the total free energy of MOF, we write the Hamiltonian of electrostatic interactions ÉÑ N Ä ÅÅÅ ξ 2 ÑÑ i Å Å Hel = ∑ ÅÅ − pEsi − Eξi ÑÑÑÑ ÅÇ 2γ ÑÑÖ i=1 Å N N (ξ + psi)(ξj + psj) 1 ∑∑ i (1 − 3 cos2 θij) + 8πε0 i = 1 j = 1 rij 3

Figure 1. Illustration of the 2D MOF matrix, where E is the external electric field, θ is the sharp angle between the field direction and the dipole−dipole radius vector, a is the length of the linker. Each node represents the metal ion and the gray connections are the linkers. Red arrows are the induced dipoles on the two arbitrary chosen nodes, other dipoles are not shown for the simplicity reasons. The α is the expansion angle in the elastic Helmholtz free energy.

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THEORY In the manuscript, we will follow the idea that the MOF transition is induced by mutual dipole−dipole interactions.53 Accordingly, the system is polarized by an external electric field and carry some polarization density. The possible polarization mechanisms are proposed in ref 53. The author supposes that the external electric field could reorient the O−H groups, which carry local dipole moments. Also, as a positively charged metal ion is surrounded by negatively charged atoms, a strong electric field could deform the configuration and thus induce a dipole moment. The result of both mechanisms is the appearance of the local dipole moment induced by the electric

(3)

where p is the permanent dipole moment, θij is the angle between two dipoles, ϵ0 is the vacuum permittivity, si = ±1 are the numbers specifying the projections of the permanent dipoles on the electric field direction, ξi is the instantaneous induced dipole moments, and γ is the molecular polarizability of nodes. The first sum in the eq 3 takes into account the selfenergy of the instantaneous induced dipoles and the interaction between both permanent and induced dipoles with the external electric field. The second contribution takes B


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The Journal of Physical Chemistry C into account dipole−dipole interactions between both types of dipoles. Now, in accordance with the standard variation method, the Hamiltonian (eq 3) can be rewritten in terms of Hamiltonian H0 of the reference system and the perturbation Hamiltonian ΔH Hel = H0 + ΔH ÑÉÑ ÅÅ 2 N Ä Ñ ÅÅ ξi Å − p ,si − ,ξi ÑÑÑÑ H0 = ∑ ÅÅ ÑÑÑ ÅÅÅ 2γ Ö i=1 Ç

χ=

(4)

N N (ξ + psi)(ξj + psj) 1 ∑∑ i (1 − 3 cos2 θij) 3 8πε0 i = 1 j = 1 rij

where we have introduced variational parameter , , which physically describes the electrostatic mean field in the lattice. We would like to stress that we chose a lattice of noninteracting dipoles under “external” electric field , as a reference system. Thus, the partition function Z055,56 of the reference system can be easily calculated, which yields ···

s1=±1 s2 =±1

sN =±1

∫−∞ ∏ dξi e−βH

0

1+

= exp[−βF0] (7)

where β = 1/kBT, kB is the Boltzmann constant, T is the temperature. The reference Helmholtz free energy is ÅÄÅ ÑÉ Å γ ,2 ÑÑÑ 1 ÑÑ F0 = −kBTN ÅÅÅÅln(e β ,p + e−β ,p) + ln(2πkBTγ ) + ÅÅÇ 2 2kBT ÑÑÑÖ

κ0 = −

(8)

From the Bogolyubov inequality one can obtain the upper limit for the free energy (9)

i=1

i

e βp , − e − βp , = tanh(βp ,) e βp , + e − βp ,

(10)

(11)

and of induced dipole ξi is ∞

⟨ξi⟩ =

2

∫−∞ dξξ e−βξ /2γ + β ,ξ 2

2πkBTγ eγβ , /2

= γ, (12)

Now, the electrostatic free energy can be written in the final form Fel = F0 + N (, − E)η(,) +

Nη2(,)χ 2ϵ0

(18)

NUMERICAL RESULTS AND DISCUSSIONS Here, we use the following set of dimensionless parameters: p̃ = p/(a3ϵ0kBT)1/2, χ̃ = χa3, γ̃ = γ/ϵ0a3, Ẽ = E(ϵ0a3)1/2(kBT)−1/2, Ã = A/a2. Let us discuss the behavior of the matrix unit cell area and the polarization density as a function of an external electric field. At first, we discuss the influence of the external electric field on the cross-sectional area (A) of the MOF. The crosssectional area is a function of the angle α, so that we will discuss the changes in the context of the angle. Initially, the lattice exists in the state with the angle α0, we call this state lp. After the application of the external electric field, the angle changes from its initial value to a larger one (that in turn causes the decrease of A). Further field alteration generates abrupt change in α from lp to np. Figure 2 shows three possible types of transitions occurring in the 2D MOF. The np-transition and lp-transition correspond to the situation when the matrix enters a stable state from metastable states corresponding to the narrow pore and large pore, respectively. The eq-transition corresponds to the equilibrium transition, that is, to the intersection point of the free energies of lp- and np-phase. In this case, the energy barrier between lp- and np-

The average value of dipole parameter si is S = ⟨si⟩ =

η2(, 0) dχ (α) |α = α0 2ϵ0 dα

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∞ N

∫−∞ ∏ dξi exp[−βH0](. . )

(17)

where , 0 is the solution to eq 16 with E = 0 and α = α0. Minimization of the total free energy with respect to α will allow us to obtain the comprehensive description of the system of interest.

where the symbol ⟨..⟩ denotes the average over statistics of the reference system 1 ⟨(. . )⟩ = ∑ Z0 {s }

(γ + βp2 )χ ϵ0

which corresponds to the linear response theory limit. Parameter κ0 can be obtained from the condition of free energy minimum at α = α0. As it follows from the derivation in Appendix II (see Supporting Information), the reference state can be a state with initial stress. In the context of present consideration, the initial stress is not zero in the absence of the external field (E = 0), if the MOF at α = α0 has nonzero spontaneous polarization, that is, MOF is in a ferroelectric state.57 Following the study given in ref 9 we consider only that case. Thus, κ0 is obtained from the extreme condition

i=1

Fel ≤ F0 + ⟨ΔH ⟩

E

,=

∞ N

∑

(15)

It is instructive to point out that in the case of small permanent dipole moment p and weak electric fields E, eq 16 can be transformed into

(6)

∑ ∑

(14)

has been introduced as the averaged total dipole moment of the node. Minimizing the total free energy (eq 13) with respect to , , we obtain the self-consistent field equation χ E = , + η(, ) ϵ0 (16)

i=1

Z0 =

j(≠ i) = 1

1 (1 − 3 cos2 θij) rij 3

η(,) = γ , + pS(,) (5)

∑ [psi(, − E) + ξi(, − E)] +

N

∑

The parameter χ describes the dipole−dipole interactions in the lattice. It can be treated as dipole analogue to the Madelung constant in an ionic solid. Also, the auxiliary function

N

ΔH =

1 4π

(13)

where in the thermodynamic limit N → ∞ in the last term we have neglected the boundary corrections; χ is determined by the relation C


Article


obtained the hysteresis on the unit cell volume of the empty MIL-53(Cr) as a function of an external electric field. The polarization density obtained in our model demonstrates similar to MD behavior at a low strength of the external electric field but does not contain a pronounced maximum in the region of structural transition. The latter may be the results of the model assumptions and simplifications, for example, the 2D dipole model and not accounting for the real structure of the polarized unit. The polarization process in this model is due to two effects: orientation polarizability of permanent dipoles and polarizability of induced dipoles. While the latter has no saturation, the permanent contribution can achieve it, when all permanent dipoles oriented along the external electric field. Thus, in the case of only permanent dipoles, the MOF phase transition can occur only at the values of the external electric field below the value corresponding to the saturation of the polarization density. The further increase of the field will not increase the contribution of the dipole−dipole interactions to the Helmholtz free energy and, thus, the system can be trapped in the large phase. On the other hand, if the host has only induced a polarization mechanism, the phase transition should occur, but at a higher value of the electric field compared to the case when both contributions are included. As mentioned in the Introduction, the structural transition occurs at the electric field of several GV/m. These values exceed the breakdown limit in the air; thus, we can offer two possible situations when the field effect can be significant. The first one, the MOF, is already under other external stimuli and the electric field can shift the structural transition. The second one, the material, is significantly softer, then MIL-53(Cr) and there is no need for such high electric field strength to provoke the structural transition. Worth noting, that despite the “2Dassumption” the model should be applicable to the real materials with layered-like structures separated by the distances, which is much bigger than the length of the linker in the plane. Also, the described theoretical formalism is, in general, applicable to 3D structures. However, that will require additional information about the material structure and accurate estimation of the stiffness constants. Such an analysis is beyond the scope of the present paper, while our main goal was to make a simple qualitative model of the electric fieldinduced structural transition of a flexible MOF.

Figure 2. Cross-sectional area of the unit cell as a function of the external electric field. The np-transition and lp-transition correspond to the situation when the matrix enters a stable state from metastable states corresponding to the narrow-pore and large-pore, respectively. The eq-transition corresponds to the equilibrium transition, that is, to the intersection point of the free energies of lp- and np-phase. The nonstable branch corresponds to the maximum on the Helmholtz free energy. The insertions demonstrate the free energy as a function of angle (α) at three different strength values of the external electric field (Ẽ = 0, 2.75, and 4.5).

phases at the equilibrium transition point is very low. It is a possible sign that the obtained transition could be shifted or suppressed by other external stimuli, for example, gas adsorption or mechanical stress. Figure 3 shows the polarization density (the induced dipole moment of the dielectric divided by the number of nodes) as a

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CONCLUSIONS Herein, we report a simplified mathematical description of the structural transition of a MOF in an electric field based on a 2D lattice model. We have obtained the hysteresis on the unit cell cross-sectional area and on the polarization density as a function of the electric field. Despite significant simplifications, our results are in qualitative agreement with the MD simulation of the similar phase transition in empty MIL53(Cr).9 Also, it is worth noting that our results confirm the qualitative picture proposed by Schmid.53 The availability of this model can provide the basis for future description and prediction of electric field-induced phase transitions not only of MOFs but also of other materials.

Figure 3. Polarization density, defined as the total dipole moment of dielectric media divided per number of nodes, as a function of an external electric field. The insertion presents the magnified region corresponding to the structural transition in the matrix.

function of an external electric field. There is an inflection point at Ẽ ≈ 1, which corresponds to the saturation of orientation polarization. The following increase in polarization corresponds to the mechanism, based on molecular polarizability of the matrix nodes. The insertion shows the magnified region with hysteresis of the polarization curve. As in the case with the unit cell area, the three jumps correspond to two metastable transitions and one equilibrium transition. The results from Figure 2 are in qualitative agreement with the MD simulation made by Ghoufi,9 where the authors

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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.jpcc.8b11630. D


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Detailed calculations of χ parameter and one derivation of elastic free energy (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (A.L.K.). *E-mail: [email protected] (Y.A.B.). ORCID

Andrei L. Kolesnikov: 0000-0003-3429-1478 Roger Gläser: 0000-0002-8134-4280 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The reported study was funded by the RFBR according to research project no. 18-31-20015.

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Metal-Organic Framework Breathing in Electric Field: A Theoretical

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