Guest-Induced Gate Opening and Breathing Phenomena in Soft

FEATURE ARTICLE pubs.acs.org/JPCC

Guest-Induced Gate Opening and Breathing Phenomena in Soft Porous Crystals: Building Thermodynamically Consistent Isotherms Marc Pera-Titus* and David Farrusseng Universit
e de Lyon, Institut de Recherches sur la Catalyse et l’Environnement de Lyon (IRCELYON), UMR 5256 CNRS - Universit
e Lyon 1, 2, Av. A. Einstein, 69626 Villeurbanne Cedex, France ABSTRACT: Metal
organic frameworks (MOFs) have emerged as a novel and fascinating family of porous materials offering promising perspectives for designing tailor-made adsorbents and catalysts. Particularly, great attention has been paid to flexible nanoporous MOFs (or Soft Porous Crystals) displaying structural phase transitions promoted by external stimuli. This is the case of guest-induced structural transitions upon adsorption being at the origin of the striking gate opening and breathing phenomena. We present here a short overview of the formulation of thermodynamic isotherms for describing S-shaped adsorption/desorption curves typical of such phenomena, as well as recent thermodynamic methods for estimating phase transition energies. These methods might be valuable for the rational design of gate opening/breathing MOFs.

1. INTRODUCTION The past decade has seen the emergence of metal
organic frameworks (MOFs), and more extensively hybrid inorganic
organic coordination polymers, as a fascinating new class of nanoporous materials with many potential applications in gas storage and separation. Metal
organic frameworks are crystalline materials synthesized by self-assembly of organic ligands (linkers) and metal clusters, creating highly regular and flexible porous frameworks with different topologies, defined pores, and chemical functionalities that can be tuned by modifying the metal group or the organic ligand.1
3 MOF materials can also be subjected to postsynthetic modification,4
6 whereby both the metal unit and the ligand can undergo heterogeneous chemical transformations while keeping the overall crystalline topology of the material. These features combined with the large versatility of porous frameworks make MOFs a practically inexorable source of creativity for adsorbent and catalyst design. Unlike zeolites and microporous (alumino)silicates, characterized by relatively rigid frameworks ascribed to strong Si
O covalent bonds, MOF materials display an inherent structural flexibility due to their weaker bonds (e.g., Π
Π stacking, hydrogen bonds, and van der Waals interaction). These interactions are at the origin of structural transformations promoted by external stimuli (e.g., temperature, hydrostatic pressure, presence of a sorbate) not observed in purely inorganic materials. If the structure is rigid, these transformations can be inexistent or moderate as in the case of ZIF-8, where partial reorientation of the imidazolate linkers can only be attained at very high hydrostatic pressures (up to 14 700 bar)7 and/or by low-pressure N2 adsorption at 77.4 K,8 increasing the accessible pore volume and the size of sixring windows. Related phenomena are the intraframework dynamic rotation of bridging ligands upon adsorption/desorption of r 2011 American Chemical Society

guest molecules without large-scale geometrical changes,9,10 as well as the shearing of pore cages in IRMOF-1 with the temperature providing negative thermal expansion effects.11 In contrast, when dealing with highly flexible frameworks, the materials can display either a gradual (elastic) swelling driven by guest accommodation as for the MIL-88/pyridine system12 or a rich variety of abrupt but reversible phase transitions between metastable phases involving either amorphous-to-crystal or crystalto-crystal transformations.13,14 The term “Soft Porous Crystals” (SPCs) has been recently coined to account for such materials.9 The above stated structural changes in SPCs translate into anomalous adsorption isotherm patterns (either in the presence of polar or nonpolar sorbates) such as the paradigmatic “gate opening” and “breathing” phenomena. Gate opening phenomena are characterized by a large hysteresis loop between the adsorption and desorption curves due to an abrupt transition between a nonporous and a porous crystalline phase promoted by guest accommodation. On the other hand, breathing phenomena are characterized by an abrupt expansion of the unit cell upon adsorption (the unit cell parameters can suffer variations >5 Å) due to guest-induced crystal-to-crystal transformations. A typical example of breathing phenomena can be found in the MIL53(Cr,Al,Fe) family, exhibiting an abrupt phase transition upon water15 and CO216 adsorption (and to a lesser extent upon adsorption of C1
C9 alkanes17) resulting in an ca. 38% unit cell expansion (MIL-53(Cr)-CO2). This transition is accompanied by a prominent inflection in the CO2 isotherm (S-shaped) at about 6 bar and 304 K.18 This particular sorption behavior Received: October 23, 2011 Revised: December 14, 2011 Published: December 14, 2011 1638

dx.doi.org/10.1021/jp210174h | J. Phys. Chem. C 2012, 116, 1638–1649

The Journal of Physical Chemistry C opposes to the typical Type I or even Type V19 patterns in the presence of strong sorbate
sorbate interactions displayed by rigid MOFs. F
erey and Serre18 have recently published a topical review establishing some rules for breathing in SPCs. On a molecular perspective, bistability is promoted by strong sorbate
sorbate interactions in the framework, acting as a backbone for an onset of symmetric interactions, most often with the interplay of bridging OH groups. This interaction set can be tuned through either a proper functionalization of the ligands before solvothermal synthesis, as in the case of the synthesis of amino-MIL-53(Al)20 with different amine loadings, or through postsynthetic functionalization.6 The potential technological impact of MOFs makes the development of suitable isotherm models imperative for describing gas/vapor adsorption. By now, most of the studies dealing with adsorption properties of MOFs have been aimed at providing a materials screening and ranking for target separations. The reason for this is based not only on the novelty of the field, but also on the international competition for discovering and patenting novel structures. Notwithstanding this fact, a number of authoritative Monte Carlo, DFT, and molecular dynamics simulation studies have been reported, elucidating the main interactions and adsorption mechanisms in MOFs (see for instance refs 21
29). A question that emerges in these studies is how to model framework flexibility to guide the design of SPCs. As a matter of fact, most of the reported simulation studies rely on the transposition of well-known concepts developed for robust zeolites and carbons to hybrid porous materials, where the framework is regarded as “frozen”. While this approach is reasonable for rigid MOFs, some doubts have been raised in the case of SPCs since most of these models take explicitly into account neither framework swelling nor bistability upon gas/vapor adsorption. Moreover, although other methods based on simulations of the fully flexible solids in the presence of the sorbate (i.e., osmotic ensemble) have been considered, their practical implementation appears to be intricate and dissuasive. In view of the challenges posed by molecular simulations, it seems necessary to develop analytical models describing properly gas adsorption in flexible MOFs. At first glance, this task is not straightforward since classical adsorption models address rigid materials with negligible guest-promoted strains. This shortcoming combined with the extremely large specific surfaces offered by some MOFs poses obvious questions on the compatibility of the BET theory for measuring specific surfaces30,31 and the classical Dubinin
Radushkevic (DR) and Dubinin
Astakhov (DA) isotherms. This latter point will be discussed in detail below. Thermodynamics, and most specifically solution thermodynamics, offers a suitable framework for modeling gas adsorption in flexible MOFs. Thermodynamics has been applied with success for estimating phase transition energies between metastable states in SPCs driven by sorbate-induced stresses using restricted osmotic ensembles.32
34 In previous studies, our group has also developed a quantitative methodology for inferring complex sorption patterns through the formulation of “thermodynamic isotherms”.35,36 This approach relies first on the transformation of the bare q vs P isotherm into a thermodynamic equivalent that can be fitted to a “universal” model including a reduced set of affinity and energy heterogeneity parameters. Subsequently, the fitted thermodynamic isotherm can be subjected to antitransformation, allowing the reconstruction of the q vs P isotherm. This approach enables a quantitative characterization of the flexibility of MOF materials with small structural differences. This feature

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paper is intended to provide a global picture on the formulation of such isotherms for modeling S-shaped curves in SPCs, stressing how they can help interpret gate opening and breathing phenomena. Several case examples will be presented showing the potentials of such formulation.

2. INTEGRAL FREE ENERGY OF ADSORPTION RELATIVE TO SATURATION AS A DESCRIPTOR OF THE ADSORPTION PROCESS Before beginning our mathematical analysis, let us recall first the general guidelines and hypotheses considered in the derivation of thermodynamic isotherms. Our starting point is the solution thermodynamics formalism proposed by Myers37 for modeling gas adsorption in porous materials. In this formalism, a porous system is regarded as consisting of three phases: the gas/ vapor phase (g), the solid phase(s) or “solvent” (s), and the sorbate phase or “solute” (a). The sorbate has ideally no volume, i.e. Va = 0. This phase, together with the solid phase, constitutes the “condensed phase” (c). For such a system, the specific free energy of the sorbate reads G̅ a ¼ F̅ a ¼ U̅ a
T S̅ a ¼ μq þ ̅Φ

ð1Þ

where U is the internal energy, S the entropy, μ the chemical potential, and Φ the surface potential, which equals the difference between the surface potential of the condensed phase and the chemical potential of the adsorbent without loading, i.e., μC
μS. The specific free energy of the sorbate includes two contributions: (1) the free energy of q moles adsorbed per kilogram of adsorbent at equilibrium with the gas phase, namely, with the same chemical potential, and (2) the surface potential ̅ , which depends on the sorbate
of the condensed phase, Φ adsorbent interaction and tends to zero in the absence of sorbate. The differentiation of eq 1 at constant temperature yields δGa = μδq and δΦ = δμ (Gibbs
Duhem equation), relating the surface potential to the chemical potential of the solid/ sorbate system. Considering δΦ = RTδ ln(P) for ideal gases (otherwise the pressure has to be substituted by a fugacity), eq 2 can be obtained ̅Φ ¼
RT

Z P 0

qδ lnðPÞ ¼
qM

Z P 0

θδ lnðPÞ

ð2Þ

̅ /qM. where θ = q/qM, qM = q(P°), and Φ = Φ The application of the fundamental equation of energy for the condensed phase at isothermal conditions combined with eq 1 provides an expression for the integral free energy of the sorbate, ΔGa, which is defined as the difference between the free energy of the sorbate and that of the same sorbate loading at saturation pressure ΔG̅ a ¼ qðμ
μo Þ þ ̅Φ

ð3Þ

From eq 3, a dimensionless integral free energy of adsorption relative to saturation for the sorbate,
Ψ/RT, can be defined as follows (see ref 38 for its derivation)


Ψ 1 ¼ jΔG̅ a ðqM Þ
ΔG̅ a ðqÞj RT RTqM ¼


1639

½ΦðP°Þ
ΦðPÞ ½μo
μ
θ RT RT

ð4Þ

dx.doi.org/10.1021/jp210174h |J. Phys. Chem. C 2012, 116, 1638–1649

The Journal of Physical Chemistry C In the derivation of eq 4, we have implicitly assumed that the chemical potential of the sorbate equals that of the gas/vapor phase. Note that
Ψ/RT is defined positive and reaches a maximum value at P f 0 (or θ f 0). The first term on the right-hand side of

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eq 4 collects the information about the sorbate/sorbent interaction, while the second term accounts for the compression of the gas/vapor phase. Moreover, with any lack of physical significance, the chemical potential term in eq 4 includes, if any, the elastic energy stored by the adsorbent in the presence of sorbate-induced stresses. The integral free energy of adsorption relative to saturation can be rewritten as a function of a Kiselev integral through integration of the differential free energy of adsorption


Ψ 1 Z 1 a ¼
Δg δθ ¼ RT RT θ ̅

Z 1 θ

½
lnðΠÞδθ

ð5Þ

Equation 5 provides a valuable state variable for describing gas/ vapor adsorption in micro- and mesoporous solids. If the functional dependence of
Ψ/RT with Π is known, eq 5 defines the equation of an isotherm. The next chapter is devoted to the formulation of such an isotherm.

Figure 1. Representation of the thermodynamic isotherm in double logarithmic axes relating the integral free energy of adsorption relative to saturation (
Ψ/RT) and the inverse of the dimensionless chemical potential (Z), as defined by eq 10.

3. FORMULATION OF THERMODYNAMIC ISOTHERMS: APPLICATION TO “RIGID” SOLIDS In two inspiring studies, Neimark39,40 proposed a thermodynamic equation relying on the Kiselev integral defined in eq 5 for the determination of the surface area of an adsorbed film, S, as a function of the sorbate loading relying on a 2D formalism for

Figure 2. Top: adsorption isotherms of N2 at 77.4 K on zeolite NaY, MOF-5, and CAU-1. Bottom: corresponding representations of the thermodynamic isotherm (eq 12). The dashed and straight lines correspond to the fittings of eqs 12 and 13, respectively, for the low and high relative pressures domains. Images adapted from ref 43. 1640

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Figure 4. Calculation strategy for the determination of the relevant parameters in the set of eqs 11
16, showing the input and output data.

Figure 3. Evolution of the slope of CO2 adsorption isotherms computed using eq 17 as a function of the relative pressure. Image adapted from ref 43.

gas/vapor adsorption SðΠÞ ¼

RT Z qmax ½
lnðΠÞδq γ q

ð6Þ

where γ is the surface tension of the sorbate. A first insight into eq 6 reveals that this equation can be rewritten in the form of eq 5 through the equality ψ = γS/RTqM. The surface area of an adsorbed film can be related to the pore radius for micro-/mesoporous fractal solids. In this case, taking the pore radius as a “stick” of size r, the surface area of an adsorbed film, S, can be described by the following expression lnðSÞ ¼ const1 þ ðDS
2ÞlnðrÞ

ð7Þ

In eq 7, DS represents the surface fractal dimension of the solid (2 e DS e 3) as a global measure of structural and surface irregularities remaining invariant over a certain degree of resolution (selfsimilitude). For highly porous systems, the surface fractal dimension does not reflect the structure of basic objects such as pores or clusters but their distribution.41 Equation 7 can be expressed as a function of the relative pressure through the Kelvin equation assuming a capillary condensation regime lnðSÞ ¼ const2 þ ðDS
2Þln½
lnðΠÞ

Ψ ¼ RT

Z 1 θ

½
lnðΠÞδθ ¼ C1 Z
m




ð9Þ

where C1 is a constant; Z = 1/
ln(Π) = μ/RT; and m = DS
2 for a fractal solid with supermicropores and mesopores. Equation 9 can be extended to purely microporous solids by reinterpreting the physical meaning of parameter m in eq 9. Parameter m can be linked to the degree of energy heterogeneity (or energy distribution) of the solid/sorbate system. The energy heterogeneity can be ascribed in its turn to the degree of

Z 1

Ψ ¼ RT

θ


lnðΠÞδθ ¼

G° G° ¼ 1 þ kZm 1 þ λ

ð10Þ

where G° =
Φ(P°)/RT; Z = 1/
ln(Π); and k is an affinity parameter providing information about the sorbate/sorbent interaction. For convenience, we have defined variable λ as λ = kZm1. Equation 10 relates
Ψ/RT with Π in such a way that
Ψ/RT f G° at Π f 0 and
Ψ/RT µ Z
m at higher Π values (see Figure 1). Parameter G° can be regared as the total dimensionless free energy dissipated during the adsorption process. In real practice, this energy can be computed from an experimental isotherm by the following expression G° ¼

ð8Þ

Jaroniec42 has shown that the thermodynamic method proposed by Neimark is compatible with the classical FHH isotherm in the capillary condensation regime (slope DS
3) for pore sizes >8 Å. Combining eqs 6 and 8, a potential equation or “scaling law” can be proposed relating the free energy of adsorption relative to saturation and the inverse of the chemical potential of the gas/vapor


confinement and packing of sorbate molecules in the zeolite cavities. Equation 9 can be further refined by incorporating a bound at low pressures. Indeed, at sufficiently low relative pressures, no adsorption occurs. This means that, at such conditions, the function
Ψ/RT should tend to a plateau value corresponding to the maximum surface potential available for adsorption, i.e.,
Φ(P°)/RT (see eq 4). Although such observation was already provided by Neimark,40 eq 8 was not corrected to incorporate a low-pressure bound. A scaling law isotherm that fulfills this requirement can be defined by eq 10

Z 1 0


lnðΠÞδθ ¼

Z 1
1 0

Z δθ

ð11Þ

For gas physisorption in zeolites and rigid MOFs (Type I), the thermodynamic representation of adsorption isotherms (
Ψ/RT vs Z) reflects most often the presence of two linear trends in double logarithmic axes with slopes m1 and m2. Figure 2 plots some examples of typical trends obtained for N2 adsorption at 77.4 K in a NaY zeolite as well as in rigid MOF-5 and CAU-1 materials.43 Similar trends can also be obtained for CO2 adsorption (Type I) and water adsorption/intrusion (Type V) in MFI zeolites.44 These experimental trends can be well covered by the following modified thermodynamic isotherm equation


Ψ ¼ RT

Z 1 θ


lnðΠÞδθ ¼

G° λ1 λ2 1 þ λ1 þ λ2

ð12Þ

where λ1 = k1Zm1 and λ2 = k2Zm2, being m1 > m2 and k1 . k2 in the case of zeolites. 1641

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Figure 5. Adsorption/desorption isotherms and corresponding representation of the thermodynamic isotherm for the silicalite-1/benzene system at 273 K (on top) and for the ZIF-7/CO2 system at 303 K (on bottom). Images adapted from refs 35 and 36.

One outstanding property ascribed to eq 12 is that it incorporates two different m parameters accounting for the energy heterogeneity of the solid. Parameter m1, obtained in the very beginning of the adsorption process, can be intrinsically linked to the energy heterogeneity of the sorbate/sorbent interaction or even to sorbate confining effects in zeolites.44 In contrast, parameter m2, obtained at higher loadings (most often θ > 0.1), is also expected to incorporate information about sorbate/sorbate interactions. Moreover, parameter m2 can be linked to the characteristic α exponent of the DA isotherm at Π > 0.01 in a simple way (α = m2
1)38 and to the surface factal dimension for self-similar solids, where the fundamental relation DS = m2 + 2 applies. A q vs P isotherm equation can be further obtained by numerical integration of the derivative of eq 12 2

3

δθ δ6 ¼
Z 6 δZ δZ4

7 G° 7 λ1 λ2 5 1 þ λ1 þ λ2

ð13Þ

(the value Zβ establishes the beginning of existence of the second zone) ! Z m1 þ 1 ð14Þ for 1=λ1 e 1 θ ¼ G°k1 1 þ k1 Zm1

θ ¼ 1



G° m1 Z
ðm1
1Þ k1 ðm1
1Þ

G° m2 Z
ðm2
1Þ for Zβ , Z < 1 k2 ðm2
1Þ

ð15Þ

We have already shown that eq 14, accomplished at relative pressures 0.1, being formally equivalent to a McLaurin development of the DA isotherm (see ref 45 for further details) θ ¼ 1


Equation 13 allows the derivation of two analytical expressions describing gas adsorption in two limiting pressure zones 1642

G° m2 Z
ðm2
1Þ k2 ðm2
1Þ

ð16Þ

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Figure 6. From top to bottom: Experimental and reconstructured CO2 adsorption/desorption isotherms at 303 K, thermodynamic isotherm representations with the corresponding fittings, and difference of integral free energies between the adsorption and desorption curves for ZIF-7 and MIL-53(Cr). The filled and empty symbols correspond, respectively, to the adsorption and desorption branches. Images adapted from ref 36.

Henry’s linear region at low pressures can be obtained by derivation of eq 12  δθ  1 δθ G°m1 λ1 ¼ ¼ H ¼  δΠ Π ln2 ðΠÞ δZ Π ln2 ðΠÞ ð1 þ λ1 Þ2 θf0 ð17Þ

One of the characteristics of gas/vapor adsorption in microporous rigid solids is the presence of sharp adsorption trends at low pressures (usually