François-Xavier Coudert

Marie Jeffroy

Alain H Fuchs

Anne Boutin
email: anne.boutin@lcp.u-psud.fr

Caroline Mellot-Draznieks
email: c.mellot-draznieks@ucl.ac.uk

Thermodynamics of guest-induced structural transitions in hybrid organic-inorganic frameworks

We provide a general thermodynamic framework for the understanding of guest-induced structural transitions in hybrid organic-inorganic materials. The method is based on the analysis of experimental adsorption isotherms. It allows the determination of the free energy differences between host structures involved in guestinduced transitions, especially hard to obtain experimentally. We discuss the general case of adsorption in flexible materials and show how a few key quantities, such as pore volumes and adsorption affinities, entirely determine the phenomenology of adsorption, including the occurrence of structural transitions. Based on adsorption thermodynamics, we then propose a taxonomy of guest-induced structural phase transitions and the corresponding isotherms. In particular, we derive generic conditions for observing a double structural transition upon adsorption, often resulting in a two-step isotherm. Finally, we show the wide applicability and the robustness of the model through three case studies of topical hybrid organic-inorganic frameworks: the hysteretic hydrogen adsorption in Co(1,4-benzenedipyrazolate), the guest-dependent gate-opening in Cu(4,4bipyridine)(2,5-dihydroxybenzoate) 2 and the CO 2 -induced "breathing" of hybrid material MIL-53.

Introduction

Organic-inorganic framework materials display an extremely large range of crystal structures and host-guest properties, ranging from coordination polymers, porous metal-organic frameworks (MOFs), and extended inorganic hybrids. [1][2][3][4][5][6] This makes them an important class of materials with potentially major impact in adsorption/separation technologies of strategic gas (H 2 , CO 2 or CH 4 ). [7][8][9][10] The combination of tunable porosity, the functionalization of the internal surface together with the structural flexibility of the host opens the way to an extremely rich host-guest chemistry, putting this class of materials in a unique position.

The distinctive chemistry that results from the flexibility of hybrid frameworks may be set against that of zeolites, that are characterized by a relatively limited framework flexibility and a high thermal stability due to the strength of the metal-oxygen bonds (Si-O bonds are among the strongest covalent bonds known), allowing their permanent porosity upon adsorption-desorption processes. By contrast, hybrid materials involve significantly weaker bonds (coordinative bonds, π-π staking, hydrogens bonds...) that are responsible for their intrinsic structural flexibility. Thus, one fascinating aspect of hybrid frameworks is the ability of a subclass of structures to behave in a remarkable guest-responsive fashion. [11][12][13][14] As recently classified by Kitagawa et al., 13,14 such hybrid frameworks exhibit a variety of guest-induced structural phase transitions upon gas adsorption and desorption.

They are typically reported to possess a bistable behavior controlled by an external stimulus such as gas pressure.

It is striking that a substantial number of experimental adsorption data collected on guest-responsive hybrid frameworks exhibit S-shape or step-wise adsorption isotherms, frequently assorted with hysteresis loops, this for a surprisingly large variety of polar and non polar sorbates (CO 2 , CH 4 , methanol, ethylene...). [15][16][17][18][19][20][21][22][23] They include a recent example of hysteretic H 2 multi-step isotherms 24 that do not conform to any of the IUPAC isotherm types. 25 Despite the richness of the experimental data at hand on these systems, the fundamental thermodynamics that underlie these particularly complex {host-guest} systems is far from being understood today and its phenomenology remains to be developed. In this respect, we may put forward two strictly distinct situations.

For a number of hybrid frameworks, the S-shape of adsorption isotherms may be clearly attributed to the very large pore size of the material and the occurrence of strong sorbate-sorbate interactions reminiscent of the bulk, as shown in the case of CO 2 adsorption in a series of MOFs. 26,27 In these situations, the shape of the isotherm does not arise from a phase transition of the host but from the details of the structural arrangement of the adsorbed phase, similarly to what occurs for a number of {zeolite, guest} systems (e.g. {silicalite-1/heptane} 28 and {AlPO 4 -5/methane} 29 ). The phenomenology of such systems has been extensively studied and is appropriately described in the frame of the widely used Grand Canonical ensemble. We will not consider them further in this paper.

For another subclass of hybrids, which is the primary focus of this article, the presence of one or more steps in the isotherms has been attributed to guest-induced structural transitions of the host material. In this case, the phenomenology of gas adsorption is deemed to be far more complex, as shown experimentally by the great variety of isotherms profiles, hysteresis loops and types of phase transitions (classified as amorphous-to-crystal and crystal-to-crystal transformations). 13 For example, among the most eye-catching guest-induced phase transitions of hybrids are the so-called "gate-opening" and "breathing" phenomena, attracting much attention due to their potential applications in sensing, gas separation or low pressure gas storage. 23,30,31 "Gate-opening" typically involves an abrupt structural transition between a non-porous state and a porous crystalline host that is induced by gas adsorption. It is characterized by highly guest-dependent "gate-opening" and "gate-closing" pressures (on the adsorption and desorption branches respectively). 31 The "breathing" phenomenon includes two recent examples, MIL-53 32 and MIL-88 33 transition metal terephtalates, that exhibit massive guest-induced crystal-to-crystal transformations. While MIL-88 exhibits a gradual cell expansion up to 200 % upon fluid adsorption, 34 an abrupt structural transition is observed in MIL-53 upon H 2 O 32 or CO 2 adsorption, 35 with a transition resulting in a ∼38 % cell variation.

Despite a very rich experimental corpus of data, there is a dearth of systematic understanding of the thermodynamics of guest-responsive hybrid frameworks. Key questions remains to be answered such as, for example: (i) predicting the occurrence or absence of a guest-induced transition for a given {host, gas} system and predicting the resulting shape of the adsorption isotherm; (ii) elucidating the thermodynamics that underlie the guest-induced transition itself, including the relative stabilities of the two states; or (iii) identifying the key factors that drive the strong guest-dependence of the gate-opening processes.

One of the reasons that have hindered the systematic understanding of guest-responsive hybrid frameworks until now is that the thermodynamics of the two concomitant processes (host-guest interactions and the host structural transition) cannot be easily deconvoluted experimentally, even though it is generally agreed that the interplay between the host-guest interactions and the energetic cost of the structural transition governs the guestinduced structural transitions. The current approach in the field relies on the characterisation of the energetics of the adsorption, typically using calorimetric measurements which however do not distinguish the host and hostguest contributions. Alternatively, forcefield-based and DFT single point energy calculations 36,37 have been used to elucidate the relative energetic contributions pertaining to the host and to the guest adsorption, although leaving the thermodynamic picture incomplete.

A complete description of the thermodynamics of guest-responsive frameworks can be obtained directly by carrying simulations of the fully flexible solid in presence of adsorbate, in the so-called osmotic ensemble. However, this method puts a demanding constraint on the forcefield development, as they are required to describe the full energy landscape of the host material, including the structural transition itself. Moreover, it is frequent with hybrid materials that the low crystallinity of phases involved in the transition hinders their structural determination from powder data, therefore eliminating the possibility of simulating their adsorption isotherms.

The present paper aims at rationalizing the thermodynamics of adsorption in flexible frameworks when guest-induced structural transitions of the host are involved. We first develop a thermodynamic description of the process of adsorption in flexible structures, and then devise a method to calculate the difference in free energy between the different structures involved in guest-induced structural transitions. This method relies on adsorption isotherms, which can be obtained experimentally. We then use the description developed to discuss the general case of type I adsorption in flexible materials, and we show that the thermodynamics of this process depends only on a few key quantities (pore volumes, adsorption affinities). This allows us to develop a taxonomy of the different behaviors that can be observed upon adsorption, and the resulting types of isotherm.

Finally, we show the wide applicability and the robustness of our free energy calculation method by applying it to three topical guest-responsive hybrid materials: we study the hysteretic hydrogen adsorption in Co(1,4benzenedipyrazolate), the guest-dependent gate-opening in Cu(4,4 -bipyridine)(2,5-dihydroxybenzoate) 2 and the CO 2 -induced "breathing" of hybrid material MIL-53.

Thermodynamic potential in the osmotic ensemble Expression of the thermodynamic potential of host-guest systems

We consider here the general process of adsorption of a fluid in a nanoporous material, where the host framework undergoes structural phase transitions induced by the adsorption of the fluid. It has been shown that this process is most appropriately described in the osmotic statistical ensemble, [38][39][40][41][42] where the control parameters are the number of molecules of the host framework N host , the chemical potential of the adsorbed fluid µ ads , the mechanical constraint σ exerted on the system (which is simply here the external pressure P) and the temperature T .

The osmotic ensemble in this formulation 39 is an extension of the grand canonical ensemble that accounts for the presence of a flexible host material with variable unit cell. Its thermodynamic potential Ω os and configurational partition function Z os are the following: 43 Ω os = U -T Sµ ads N ads + PV

(1)

Z os = ∑ V ∑ N ads ∑ q exp [-βU(q) + β µ ads N ads -β PV ] (2)
where q denotes the positions of the atoms of the system (host and adsorbate) and β = 1/kT (k being the Boltzmann constant).

In order to study the relative stability of different phases of the host structure upon adsorption, our method relies on the factorization of the expressions above in two contributions, one characterizing the framework structures themselves (i.e. their free energy), the other describing the fluid adsorption in each solid phase involved in the transition. The summation over the positions q of atoms in the system is thus split into a double summation over the coordinates of the material, q host , and coordinates of the atoms of the adsorbate, q ads . The total energy is the sum of the energy of the host, the energy of the adsorbed fluid and the host-adsorbate interactions:

U(q) = U host (q host ) +U ads (q ads ) +U host-ads (q ads , q host )

As our interest lies in the determination of adsorption-induced structural transitions of the host material, we make the assumption that while the fluid adsorption may favor one phase or another, each single structure is only marginally changed by the adsorption of fluid. This is translated into a mean field approximation by writing that the host-adsorbate interactions depend only on the average positions of the atoms (and unit cell vectors) of the material in a given phase i: U host-ads (q ads , q host ) U host-ads (q ads ; q host i ). Using this separation of variables, the configurational partition function can be written as a sum of configurational partition functions for each phase i of the solid:

Z os = ∑ i Z (i) os .
The configurational partition function of any given phase i of the material is then given by:

Z (i) os = ∑ V ∈i ∑ q host ∈i exp [-βU host (q host ) -β PV ] × ∑ N ads ∑ q ads exp [-βU ads (q ads ) -βU host-ads (q ads ; q host i ) + β µ ads N ads ] (4)
It is apparent that the first term is the configurational partition function (restricted to phase i) of the isolated host structure in the (N host , P, T ) ensemble, Z (i) host . The second term can be recognized as the grand canonical configurational partition function Z (i) GC of the fluid in the external field created by the host framework considered as rigid. Thus, for each host phase (i), Z (i) os = Z (i) host ×Z (i) GC , can be written in terms of the thermodynamic potentials for each ensemble:

Ω (i) os = G (i) host + Ω (i) ,
where Ω is the grand canonical potential for the adsorbate and G host is the free enthalpy of the host material.

To express the grand canonical potential as a function of quantities directly available from experiments or simulations, we calculate it from its derivatives, using the fundamental relation involving the chemical potential µ ads : [44][45][46] ∂ Ω ∂ µ ads V,T = -N ads (5) Integrating this equation and taking into account that Ω(P = 0) = 0, the grand canonical potential Ω can be rewritten as a function of pressure P rather than chemical potential µ ads , as follows:

Ω(P) = - P 0 N ads (p) ∂ µ ∂ p T,N dp (6)
which can be simplified further by introducing the molar volume of the pure fluid, V m = (∂ µ/∂ P) T,N :

Ω(P) = - P 0 N ads (p)V m (p)dp (7)
The free enthalpy of the host can also be written as a function of the free energy of the host at zero pressure,

F host : G host = G host (P = 0) + PV = F host + PV .
Finally, this leads to a complete expression of the osmotic potential which involves only three key parameters:

the free energy of the solid phase, the adsorption isotherm of fluid inside that phase (N ads (T, P)) and the molar volume of the pure fluid as a function of pressure:

Ω os (T, P) = F host (T ) + PV - P 0 N ads (T, p)V m (T, p)dp (8)

Prediction of the free energy of phases involved in structural transitions

We now have an expression for the thermodynamic osmotic potential as a function of gas pressure for each solid phase. The comparison of Ω os for each phase of the host allows us to determine the relative stability of the phases, the number of structural transitions that will take place and the pressure at which they occur. This requires the prior determination of the three quantities in Equation 8. The relative free energies of the empty solid phases (F (i) host ) are especially difficult to evaluate both experimentally and by simulation methods. While relative energies between solid phases can be experimentally measured, for example by differential scanning calorimetry or the measurement of dissolution enthalpies, 47 or calculated using first-principles methods, the determination of relative free energies is a much harder task.

By contrast, the pressures at which guest-induced transitions happen are relatively easy to measure experimentally, either because they show up as steps on the adsorption isotherm, or by calorimetry (structural transitions often result in a sudden change in heat of adsorption) or even by in situ X-ray diffraction studies at different gas loadings. For that reason, we propose here a method to calculate the relative free energies of the empty solid phases (F (i) host ) from the readily available experimental quantities that are the sorption isotherms and the phase transition pressure, using Equation 8.

The quantities involved in Equation 8 and that we need to calculate free energy differences are now the transition pressures, the adsorption isotherms for each isolated phase (N (i) ads (P)), the molar volume of the pure adsorbate as a function of pressure (V m (P)) and the unit cell volume of each phase (V i ). That last two are rather straightforward to obtain: V i is known from crystal structures determined by X-ray diffraction and V m (P) can be found tabulated for a large number of gases or alternatively approximated by the relation for an ideal gas,

V m (P) = RT /P. The pressures at which structural transitions occur may be determined from the experimental adsorption and desorption isotherms of the material or other methods described above. However, the "virtual" rigid-host isotherms N (i) ads (P) related to each phase are harder to obtain experimentally: they correspond to each of the solid phases assuming the absence of transition over the whole range of pressures, so that only parts of these rigid-host isotherms are observed experimentally, as shown in Figure 1. To overcome this issue, we propose here to fit the distinct parts of a stepped isotherm to obtain full "rigid-host" isotherms needed for each of the phases.

They correspond to the dashed lines in Figure 1. Alternatively, it is possible to use isotherms calculated from Grand-Canonical Monte Carlo simulations in each phase, with the host structure being considered rigid. 40 Let us illustrate here the method on an isotherm possessing a single step that corresponds to a guest-induced structural transition (Figure 1) between two phases labeled 1 and 2, assuming 1 and 2 have been structurally characterized and have known unit cell volumes, V 1 and V 2 . From the step in the experimental isotherm, the pressure P eq of the structural transition can be determined and the two parts of the isotherm (below and above the step) are fitted by Langmuir isotherms 48 (dashed lines on Figure 1). From the fitted isotherms N (i) ads (P), we can now plot (Fig. 1, lower panel) the two functions f 1 (P) and f 2 (P) defined as:

f i (P) = Ω (i) os (P) -F (i) host = PV i - P 0 N (i) ads (p)V m (p)dp (9)
At the transition pressure, the thermodynamic equilibrium between the two phases Ω (1) os (P eq ) = Ω (2) os (P eq ) and thus ∆F host = f 1 (P eq )f 2 (P eq ).

It is clear that this method is straightforward to extend to cases where more than two solid phases come into play, simply by applying the above construction to each structural transition. It is also worth noting that, if sorption isotherms are available for different temperatures, both energy and entropy differences, ∆U host and ∆S host , can be extracted from the free energies: ∆F host (T ) = ∆U host -T ∆S host .

Taxonomy of guest-induced transitions in flexible frameworks

In this section, we apply the equations described above to the case of a nanoporous, flexible material that has two different metastable phases, and for which fluid adsorption follows type I isotherms in each structure. This case, including the possibility of one structure having no microporosity at all, has frequently been observed for hybrid organic-inorganic frameworks. We will show that this case yields a straightforward analytical expression for ∆Ω os (P) and enables us to propose a phenomenology of guest-induced structural transitions induced by adsorption and a classification of the resulting isotherms into distinct categories.

Let us consider that, for each phase i of the material, the adsorption happens in the gas phase (considered ideal) and follows a type I isotherm. 25 In order to keep the analytical expressions simple and to include only a few key physical quantities, the Langmuir equation is used to describe the gas adsorption:

N (i) ads = K i P 1 + K i P N i max ( 10
)
where K i is the Henry constant for adsorption, which measures the adsorption affinity, and N i max is the number of adsorbed gas molecules at the plateau of the isotherm. Plugging Equation 10 into Equation 8, and taking for V m the expression of the ideal gas, the osmotic potential of phase i can be written as follows:

Ω (i) os = F (i) host + PV i - P 0 K i 1 + K i P N i max dP = F (i) host + PV i -N i max RT ln 1 + K i P N i max ( 11
)
The guest-induced structural transitions of the host material are dictated by the osmotic potentials Ω (i) os of the different solid phases as a function of pressure. We discuss in this section the conditions of occurrence of guest-induced transitions. We consider two structures of a single host material, labeled 1 and 2 in such a way that, in the absence of adsorbate, structure 1 is more stable than the structure 2 (i.e., ∆F host = F (2) host -F (1) host is positive). The osmotic potential difference between these structures, as a function of pressure, is expressed as:

∆Ω os = ∆F host + P∆V -RT N (2) max ln 1 + K 2 P N (2) max -N (1) max ln 1 + K 1 P N (1) max ( 12
)
All the terms in this equation have a clear physical meaning and the full thermodynamic behavior can be discussed from there. To shorten the discussion, we now proceed to simplifying Equation 12. In all the cases presented in this article, gas adsorption happens in a range of pressures for which the term P∆V in the expression above is of limited importance. To simplify the analytical formulas and the discussion that follows, P∆V will thus be neglected it in the rest of this section. Moreover, we replace the saturation values of the isotherms, N i max , with the accessible porous volume of the material in phase i, V (i) p , by writing N i max = ρV (i) p , with ρ the density of the adsorbed gas at high pressure. 49 Equation 12 can then be rewritten as follows:

∆Ω os (P) = ∆F host -RT ρ V (2) p ln 1 +

K 2 P
ρV (2) p -V (1) p ln 1 + We now study the evolution of ∆Ω os (P) given by Equation 13and in particular the solutions of the equation ∆Ω os (P) = 0 (i.e., the structural transitions). As demonstrated in Appendix A, the behavior of the system and the occurrence of guest-induced phase transitions are entirely determined by five key parameters characteristic of the system: the difference in free energy between the empty host structures, ∆F host , the pore volumes, V (1) p and V (2) p , and the Henry constants, K 1 and K 2 . We can identify four distinct cases, synthesized in Figure 2:

K 1 P ρV (1) p (13)
• Case a, V (2) p > V (1) p : Under this condition, whatever the values of K 1 , K 2 and ∆F host , only one structural transition is observed. Structure 1 will be more favorable at low pressure, and as pressure increases, the larger accessible pore volume of structure 2 will make it more and more favorable, leading to a structural transition upon adsorption. 50 This transition leads to a one-step isotherm (see Fig. 2, column a). This case is quite common and is actually observed in many of the so-called "third generation coordination polymers" 13 that have attracted a lot of interest in the recent years. In the next section, we will show that our method can usefully be applied to two materials exhibiting crystal-to-crystal transformations of this type, including the example of [Cu(4,4 -bipy)(dhbc) 2 ], and can apply to "gating processes", 31 where one of the structures has no microporosity at all.

• Case b, V (1) p > V (2) p and K 1 > K 2 : Under these conditions, we can predict that no structural transition will be observed. This stems from the fact that all factors favor structure 1: the empty structure has lower free energy, it has higher adsorption affinity and higher pore volume (Fig. 2, column b).

• Cases c and c , V (1) p > V (2) p and K 2 > K 1 : Under these conditions, two different behaviors must be dis-tinguished. At low pressure, structure 1 is favored. At high pressure, it is also favored because it has a larger pore volume. In between, however, there might exist a regime where structure 2 is thermodynamically favored because of its larger adsorption affinity. Whether this regime occurs or not depends on the balance between the terms in Equation 13. If it happens, the system will undergo two successive structural transitions upon adsorption (from phase 1 to phase 2, then back to phase 1).

The condition for the occurrence of the double transition can be most easily expressed as an upper bound on the value of ∆F host :

∆F host ρRT < (V 2 -V 1 ) ln K 2 V 1 -K 1 V 2 V 1 -V 2 +V 1 ln K 2 -V 2 ln K 1 (14)
Thus, if ∆F host is small enough (or, seen the other way, if ∆K is large enough), there will be a domain of stability for structure 2, and two transitions will be observed upon adsorption. This leads to an adsorption isotherm with two successive steps, as shown in third column of Fig. 2 (case c). On the other hand, if ∆F host is too large (or ∆K too small), there will be no domain of stability for structure 2, and structure 1 will be the most stable phase in the entire pressure range (see last column of Fig. 2, case c ), leading to the absence of structural transition. We will show later that the occurrence of a double guest-induced structural transition was indeed suggested experimentally in the case of CO 2 adsorption in MIL-53. 35,51 The cases considered above correspond to guest-induced transitions between two distinct host structures.

More generally, this taxonomy can be extended to more complex cases where more than two host structures are involved.

Case studies of guest-induced transitions of hybrid frameworks

In this section, the above method is used to investigate the thermodynamics of three distinct cases of interest.

The first one relates to the study of a recently discovered H 2 -induced phase transition in a cobalt-based hybrid material. The second aims at studying the topical case of gate-opening processes. Finally, we will show how our method may be valuably applied to elucidate the thermodynamics of the rather complex case of the "breathing" of MOFs upon gas adsorption.

Structural phase transition in Co(1,4-benzenedipyrazolate) induced by H 2 adsorption

An interesting H 2 stepped isotherm was reported recently by Choi et al., in the case of the Co(BDP) (BDP = 1,4-benzenedipyrazolate). 24 The area of hydrogen storage in hybrid materials is a particularly challenging and topical one, 7,52 and any improvements in terms of H 2 storage capacity and performances are considered highly valuable. Until now, most hybrid frameworks had exhibited a traditional type I reversible H 2 adsorption isotherm, that allowed a ranking of the performances of materials on the sole basis of the weight adsorption capacity. By contrast, in the case of Co(BDP), the particular eye-catching feature is a stepped isotherm with a wide hysteresis, 24 allowing the adsorption of H 2 at high pressure and its storage at lower pressure, 53,54 a highly desirable feature for practical applications such as transportation. The structural transition in Co(BDP) is reported to be induced by H 2 adsorption and therefore highlights that subtle energetic and thermodynamic effects are at play, having in mind that H 2 is associated with weak host-guest interactions in hybrid materials.

This case caught our attention in the context of this work. Experimental adsorption and desorption isotherms of H 2 in Co(BDP) were reported at 50 K, 65 K, 77 K and 87 K, showing various complex multistepped isotherms. 24 In this study, we have selected the isotherm at 77 K (Fig. 3b) because of its well-defined single-step isotherm, which was interpreted as resulting from a A1→A2 structural transition. We apply here the method presented above to this experimental isotherm in order to calculate the free energy difference between A1 and A2. The adsorption isotherm is fitted to a Langmuir equation in the 0-15 bar region to provide the "rigid-host" isotherm of phase A1; the desorption isotherm is fitted similarly in the 15-40 bar region to obtain the "rigid-host" isotherm of phase A2. Using the experimental isotherm, we estimate the thermodynamic transition to occur at P eq ≈ 15 bar, considering that the isotherm deviates from the Langmuir fits at 14 bar and 17 bar, for the desorption and adsorption branches respectively.

Applying Equation 9 and neglecting the PV i terms, 55 we find that the difference in free energy between A1 and A2 is of 3.3 kJ/mol (±0.2 kJ/mol). This result is in very good agreement and much more accurate than the estimation put forth by Choi et al. that the energy of the structure change process lies in the 2-8 kJ/mol range.

It is interesting to note that this value of 3.3 kJ/mol, associated with an isotherm measured at 77 K, is significantly larger than the thermal energy kT (∼ 5 times larger). Following the reasoning of

Gate-opening transition in a flexible coordination polymer

Gate-opening in flexible hybrid frameworks occurs when a material exhibit a structural transition from nonporous to porous structure at a specific pressure, and has been reported in a number of compounds. 12,23,30,31 These materials are expected to find applications as sensors or switches, as well as gas separation. We focus here on the case of a particular flexible coordination polymer, Cu(4,4 -bipy)(dhbc) and N 2 ) at 298 K. 31 The crystal-to-crystal structural transition occurs between two phases hereafter labeled B1 and B2, whose structures have not been solved. B1, the dehydrated material, shows no microporosity and, upon gas adsorption, its channels open up at a given (guest-dependent) gate-opening pressure to yield the fully open B2. The framework structure of B2 is the same as that of Bs, and is shown in Fig. 4a. It is composed of interdigitated two-dimension motifs (Fig. 4a, top) formed by copper (II) ions linked by bipyridine and dihydroxybenzoate linkers, and stacked due to π-π interactions between parallel dhbc ligands. Their interdigitation creates unidimensional channels along the a axis, with a 8 Å diameter.

The adsorption isotherms of various small molecules (N 2 , CH 4 and O 2 ; see Figure 4b) in structure B1 at 298 K present common features: little to no adsorption in the lower pressure region, followed by a abrupt increase attributed to the B1→B2 transition. Following Kaneko 30 and Kitagawa, 31 we call here "gate-opening" pressure the point of the isotherm at which the structural transition happens during adsorption. The desorption isotherms, reversely, show an abrupt drop starting at a pressure that we will call the "gate-closing" pressure, where the B2→B1 transition takes place. Each gate-closing pressure is lower than the corresponding gate-opening pressure, and the isotherms all exhibit hystereses. Only the isotherm for CO 2 is different in that transition happens at such a low pressure that no hysteresis was detected with the experimental setup.

We apply our method to each set of experimental isotherm (including CO 2 ) and calculate the difference in free energy between the two phases B1 and B2. As structure B1 is not porous, only the adsorption isotherm of structure B2 is needed for the calculation. We use a Langmuir-type fit of the experimental desorption isotherms in the region of pressure higher than the gate-closing pressure. The fits, shown as dashed line for each isotherm in Fig. 4b, are all quite satisfactory. However, because the pressure of the thermodynamic structural transition can only be bracketed by the gate-opening and gate-closing pressures, the method does not lead to a single value of ∆F host but to a range of free energy difference between structures.

Table 1 presents the values of free energy obtained from the isotherms of each gas. In each case, we find that the nonporous structure is more stable than the open one by 4 kJ/mol (±0.5 kJ/mol) at 298 K. It is striking that, using different gas adsorption isotherms exhibiting different gate-opening and gate-closing pressure ranges, we obtain such narrow and consistent ranges for the value of ∆F host . This excellent agreement between all the values independently obtained validates our approach and is a strong sign of the robustness and reliability of the method presented in this article. Indeed, in the absence of structural characterization of the transition, we show that a unique mechanism is at play behind the seemingly different features of all four isotherms. "Breathing" phenomenon in the 3D hybrid framework MIL-53

The metal-organic framework MIL-53 has attracted a lot of attention due to the massive flexibility it exhibits, including structural characterization, 32,35 adsorption of strategic gases H 2 , CO 2 and CH 4 51,56 and simulations. 36,37,57 The MIL-53 framework topology is formed of unidimensional chains of corner-sharing MO 4 (OH) 2 octahedra (M=Al 3+ , Cr 3+ ) linked by 1,4-benzenedicarboxylate (BDC) ligands, which results in linear lozengeshaped channels large enough to accommodate small guest molecules. This structure may oscillate between two distinct states, a large pore form (lp) and a narrow pore form (np), upon adsorption and desorption of gases;

there is a ∼ 38% difference in cell volume between these two forms. Both structures are depicted in Figure 5a.

The adsorption isotherm of CO 2 in MIL-53 at 304 K exhibit a step at approximately 6 bar (upper panel of Figure 5b), demonstrated to emanate from a structural transition upon adsorption from the CO 2 -loaded np form to the CO 2 -loaded lp one. 35 Moreover, as the most favorable guest-free MIL-53 is the large pore form at room temperature, another transition, lp→np, is expected at low pressure. Although it was not seen in the original adsorption isotherm, 51 this low pressure transition was very recently confirmed by a combined simulation and microcalorimetry study. 36 Recently, a lot of effort has been put into understanding the energetics of these two guest-induced structural transitions, by Density Functional Theory 37 and forcefield-based calculations using rigid 57 or flexible 36 MIL-53

structures. Here, we bring the discussion one step further by giving valuable insight on the thermodynamics of this guest-induced structural transition using our method and the simple model developed above, yielding quantitative information on the relative stability of the two MIL-53 structures, which are virtually impenetrable experimentally.

The upper pannel of Figure 5b shows the experimental adsorption isotherm of CO 2 in MIL-53 (Al). The step between 5 and 9 bar is a signature of the transition between the two phases of MIL-53 (Al), separating the part of the isotherm that corresponds to the np structure (P < 5 bar) and the part where the structure is fully open (P > 9 bar). Both parts were satisfactorily fitted by Langmuir isotherms (Fig. 5b) and Henry constants were extracted (they are the slopes of the Langmuir fits at P = 0). The latter show clearly that the adsorption affinity of CO 2 is much higher in the np structure than in the lp one (3.5 times higher; K np 9.0 × 10 -5 mol kg -1 Pa -1 and K lp 2.6 × 10 -5 mol kg -1 Pa -1 ), in line with the energetics published so far. 36,37,51 Indeed, with such features, MIL-53 exactly fits in case c of our taxonomy: the lp form is more stable in the absence of adsorbate and has a larger pore volume, but has a smaller affinity for CO 2 than np. Thus, according to our model, MIL-53 is expected to exhibit a double structural transition upon gas adsorption if the affinity is high enough: when empty and at very low pressure, the lp structure is intrinsically more stable; when pressure increases, the high affinity of the np structure leads to a lp→np transition; upon further rise of gas pressure, the larger pore volume of the open structure makes it favorable again and a np→lp transition should be observed. Our simple model predicts that, for this type I adsorption of gas in MIL-53 (Al, Cr), there is either no structural transition or two of them, depending on the nature of the gas. In the case of CO 2 , the experimental observation of one transition leads to the conclusion that another one must exist, at lower pressure, which was indeed recently evidenced. 36 By contrast, CH 4 has a much smaller adsorption affinity in MIL-53 and appears not to induce any structural transition, 51 in agreement with our discussion above. These results highlight the correctness of the model and its ability to predict the occurence or the absence of structural transitions for a given adsorbate, on the sole basis of its respective affinities for the host structures.

To quantify the effects discussed above, we have calculated the thermodynamic osmotic potential, Ω os (P), of the lp and np phases of MIL-53 (Al) as a function of CO 2 pressure using the Eq. 8 and the fitted isotherms described above. From the experimental isotherm, we assume the equilibrium pressure for the np→lp transition to be around 5 bar. We then find a difference in free energy between the two structures of ∆F lp-np 2.5 kJ/mol per unit cell. A similar value was obtained for the chromium-containing form of MIL-53. The osmotic potential profiles (Fig. 5b, lower panel) clearly confirm the existence of two structural transitions upon adsorption and predict the low pressure lp→np transition to happen at 0.3 bar. It is noteworthy that this predicted value of the transition pressure is in very good agreement with the experimental value of 0.25 bar found very recently by microcalorimetry, 36 which is once again indicative of the predictive quality of the method exposed here. It also helps explaining why this low pressure transition was not visible on the original adsorption isotherm.

Conclusions

This article focuses on the specific class of hybrid materials exhibiting guest-induced structural transitions upon gas adsorption. Studies in this rapidly growing field mainly revolve around the structural characterization of these systems, the determination of their adsorption properties and the microscopic understanding of the coupling between the framework and adsorbate, e.g. by atomistic simulations. Here, we expose a general thermodynamic framework for gas adsorption in flexible hybrid materials and put forward a taxonomy of guest-induced structural transitions. This allows us to predict the occurence or the absence of transitions of the host on the sole basis of a few key parameters: pore volumes, adsorption affinities and relative free energies of the framework structures involved in the transitions. Conversely, we show that these relative free energies can be extracted from experimental stepped isotherms, therefore quantifying the energetics of the phase transitions, which are especially difficult to access experimentally, however vital to the fundamental understanding of these systems. The robustness of our thermodynamic approach is illustrated through the analysis of three distinct cases of guest-induced transitions: the hysteretic H 2 adsorption in a cobalt-based 3D framework, the gate-opening process in an interdigitated coordination polymer and the guest-dependent "breathing" phenomenon of MIL-53. We show that the free energy differences between the host structures involved in these guest-induced transitions fall into the 2-6 kJ/mol range. To our knowledge, this is the first general thermodynamic framework successfully rationalizing the variety of behaviors reported for this class of materials. We believe this work should prove very useful to explore new guest-induced transitions and further complement the current experimental and simulation approaches in the field.

Table of Contents Graphic

Figure 1 :

 1 Figure 1: Schematic representation of the determination of free energy difference ∆F host from an experimental adsorption isotherm (upper panel) by calculation of f i (P), the pressure-dependent part of the thermodynamic potential of the osmotic ensemble (lower panel).

Figure 2 :

 2 Figure 2: The four possible cases of Langmuir-type adsorption in materials with two metastable phases 1 and 2. Top panels show the adsorption isotherms, middle panels depict the step-free isotherms corresponding to each phase, and bottom panels plot the osmotic potential of both phases. Green arrows indicate guest-induced structural transitions.

Figure 3 :

 3 Figure 3: (a) 2 × 2 × 1 supercell of structure As, Co(BDP) • 2 DEF • H 2 O, viewed along the c axis. A2 is believed to have the same framework structure as As. (b) Adsorption and desorption isotherms, respectively in filled circles and empty squares, of H 2 in Co(BDP) at 77 K.24 The blue dashed line is the fit of the upper part (P ≥ 15 bar) of the desorption isotherm. The red dashed line is the fit of the lower part (P ≤ 15 bar) of the adsorption isotherm.

Figure 4 :

 4 Figure 4: (a) 2 × 2 supercell of coordination polymer [Cu(4,4 -bipy)(dhbc) 2 ]• H 2 O: top, along the a axis; bottom, along the c axis. (b) Adsorption and desorption isotherms, respectively in filled circles and empty triangles, of various small molecules in Cu(4,4 -bipy)(dhbc) 2 at 298 K, as found in Ref. 31: CO 2 (green, upper panel), O 2 (red, upper panel), CH 4 (blue, lower

Figure 5 :

 5 Figure 5: (a) Narrow-pore (np) and large-pore (lp) forms of the MIL-53 structure, viewed along the axis of the unidimensional channels. (b) Upper panel: experimental adsorption isotherm of CO 2 in MIL-53 (Al) at 304 K, 51 as well as Langmuir fits of the 0-5 bar and 9-30 bar regions (in red and blue, respectively). Lower panel: osmotic thermodynamic potential as a function of CO 2 pressure for the lp (in blue) and np (in red) structures of MIL-53 (Al), calculated from the Langmuir fits.

  Neglecting entropic effects at 77 K, this equation allows us to propose a heat of adsorption of ∼6.5 kJ/mol for H 2 in Co(BDP). This heat of adsorption is rather in the lower region of the range typically observed, from 5 to 11 kJ/mol.52

Choi et al., we can then use the calculated free energy difference between host phases to estimate the heat of adsorption of H 2 in Co(BDP), ∆H ads , as ∆H ads = ∆H host + ∆ f H. The latter term, ∆ f H, is the formation enthalpy of a Co(BDP):H 2 clathrate complex and was estimated by Choi et al. to be 3.2 kJ/mol (±0.3 kJ/mol).

  2 • H 2 O (4,4 -bipy = 4,4bipyridine; dhbc = 2,5-dihydroxybenzoate), whose as-synthesized structure Bs has been solved and which is known to exhibit a guest-induced structural transition upon adsorption of a large variety of gases (CO 2 , O 2 , CH 4

  4,4 -bipy)(dhbc) 2 at 298 K, as found in Ref. 31: CO 2 (green, upper panel), O 2 (red, upper panel), CH 4 (blue, lower panel) and N 2 (black, lower panel). The dashed lines are the fits of the upper part of each desorption isotherm (at pressure higher than the gate-closing pressure) by a Langmuir equation.

	Adsorbate Gate-opening Gate-closing Calculated ∆F host
	N 2	30 bar	49 bar	3.3 -4.5 kJ/mol
	CH 4	7 bar	12 bar	3.6 -5.1 kJ/mol
	O 2	25 bar	37 bar	3.4 -4.3 kJ/mol
	CO 2	< 2 bar	< 2 bar	< 6 kJ/mol
	Table 1: Gate-opening and gate-closing pressures extracted from adsorption and desorption isotherms of various molecules
	in [Cu(4,4 -bipy)(dhbc) 2 ]• H 2 O, as well as free energy difference (at 298 K) between the open and closed structures of the
	material calculated from each isotherm; see text for details.

Acknowledgments

This work was supported by the EPSRC (Advanced Research Fellowship awarded to CMD) and the FP-6 -European funding STREP "DeSanns" (SES6-CT-2005-020133) and "SURMOF" (NMP4-CT-2006-032109).

Supporting Information Available

Calculation details for the derivation of the taxonomy (Appendix A).

Supplementary information for:

Thermodynamics of guest-induced structural transitions in hybrid organic-inorganic frameworks

Appendix A: Calculation details

Starting with the expression of ∆Ω os (P) in Equation 13, we want to study the sign of ∆Ω os (P) and the solutions of the equation ∆Ω os (P) = 0. To do so, we first compute the derivative of ∆Ω os (P) and we can show that:

The sign of the derivative is thus the same as that of -K 1 K 2 ∆V p P + ρV (1) p V (2) p ∆K . Four different cases need to be considered, depending on the sign of both ∆K and ∆V p :

• If ∆V p > 0 and ∆K > 0, the derivative will always we always be negative, and ∆Ω os (P) is strictly decreasing. As ∆Ω os (P = 0) = ∆F host > 0, the equation ∆Ω os (P) = 0 has one unique solution.

• If ∆V p > 0 and ∆K < 0, the derivative will be positive around P = 0 and will become negative at higher pressure. ∆Ω os (P) has a positive value at P = 0, is first increasing, then decreasing; the equation ∆Ω os (P) = 0 will once again have one solution (and only one).

• If ∆V p < 0 and ∆K < 0, the derivative will always be positive and ∆Ω os (P) is increasing: at its value in P = 0 is positive, the equation ∆Ω os (P) = 0 will have no solution.

• If ∆V p < 0 and ∆K > 0, the derivative will be negative at small P and positive at larger P. Starting with a positive value at P = 0, ∆Ω os (P) will thus first decrease and then increase. Depending on the value it reaches at its minimum, the equation ∆Ω os (P) = 0 will have zero, one (in the limiting case) or two solutions.

It is worth noting that the first two cases can be joined into a single case: if ∆V p > 0, the equation ∆Ω os (P) = 0 has one unique solution.


Christelle Raffaitin
email: christelle.raffaitin@isped.u-bordeaux2.fr

MD Henri Gin

PhD Jean-Philippe Empana

PhD Catherine Helmer

PhD Claudine Berr

PhD Christophe Tzourio

PhD Florence Portet

MD Jean-François Dartigues

PhD Annick Alpérovitch

Pascale Barberger-Gateau

PhD Pascale Barberger- Gateau

Metabolic syndrome and risk for incident Alzheimer's disease or vascular dementia: the Three

Keywords: metabolic syndrome, hypertriglyceridemia, diabetes, Alzheimer disease, vascular dementia, aged, epidemiology, longitudinal study

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

OBJECTIVE

Associations between metabolic syndrome (MetS), and its individual components, with risk of incident dementia, and its different subtypes, are inconsistent.

RESEARCH DESIGN AND METHODS

The 7087 community-dwelling subjects aged 65 yr and over were recruited from the French Three-City (3C) cohort. Hazard ratios of incident dementia over four years -and its subtypes (vascular dementia and Alzheimer's disease (AD)) -associated with MetS (defined according to the NCEP ATP III Criteria) -and its individual components (hypertension, large waist circumference, high triglycerides, low HDL-cholesterol, elevated fasting glycemia) -were estimated in separate Cox proportional hazard models.

RESULTS

MetS was present in 15.8% of the study participants. The presence of MetS increased the risk of incident vascular dementia but not AD over four years independently of socio-demographic characteristics and Apolipoprotein E ε4 allele. High triglycerides level was the only component of MetS that was significantly associated with the incidence of all-cause dementia (HR = 1.45 [1.05 -2.00]; p = 0.02) and vascular dementia (HR = 2.27 [1.16 -4.42]; p = 0.02), even after adjustment on Apolipoprotein E genotype. Diabetes, but not impaired fasting glycemia, was significantly associated with all-cause dementia (HR = 1.58 [1.05 -2.38]; p = 0.03) and vascular dementia (HR = 2.53 [1.15 -5.66]; p = 0.03).

CONCLUSIONS

The observed relation between high triglycerides, diabetes and vascular dementia emphasizes the need for detection and treatment of vascular risk factors in older persons in order to prevent the likelihood of clinical dementia.

Dementia is the most severe form of pathological brain aging, defined by the coexistence of memory disorders and deficit in at least one other cognitive function (impairment of abstract thinking, impaired judgment, other disturbances of higher cortical function such as aphasia, apraxia, agnosia, or constructional difficulty, personality change), with a significant impact on activities of daily living [START_REF] Ritchie | The dementias[END_REF]. Eighteen percent of people over 75 yr have dementia. Its most frequent form is Alzheimer's disease (AD) (about 50-60% of cases,(1)), vascular dementia accounting for most of the other causes [START_REF] Ritchie | The dementias[END_REF]. The underlying pathophysiological mechanismsand hence risk factors -of AD and vascular dementia remain uncertain [START_REF] Gorelick | Risk factors for vascular dementia and Alzheimer disease[END_REF] and no etiologic treatment is available yet. Nevertheless, several potentially modifiable risk factors of dementia have been identified. Hypertension, especially in mid-life, increases the likelihood of developing dementia [START_REF] Qiu | The age-dependent relation of blood pressure to cognitive function and dementia[END_REF]. Although less studied, obesity (4) and dyslipidemia ( 5) are also being recognized as possible modifiable risk factors of dementia. Some studies have shown that diabetes increases the risk of developing dementia [START_REF] Biessels | Risk of dementia in diabetes mellitus: a systematic review[END_REF].

These factors usually coexist under the heading of metabolic syndrome (MetS) which is a cluster of five cardiovascular risk factors including hypertension, abdominal obesity, dyslipidemia: elevated triglycerides, low HDL-cholesterol and elevated fasting glycemia (impaired fasting glucose or diabetes) [START_REF] Reaven | Role of insulin resistance in human disease[END_REF].

MetS is associated with increased risk of cardiovascular disease [START_REF] Lakka | The metabolic syndrome and total and cardiovascular disease mortality in middleaged men[END_REF]. If MetS were also associated with increased risk of developing dementia, the screening and management of its components might offer avenues for prevention of cardiovascular disease and dementia as well. However, the association between MetS, or its individual components, and dementia has received little attention [START_REF] Kalmijn | Metabolic cardiovascular syndrome and risk of dementia in Japanese-American elderly men. The Honolulu-Asia aging study[END_REF][START_REF] Yaffe | The metabolic syndrome, inflammation, and risk of cognitive decline[END_REF][START_REF] Yaffe | Metabolic syndrome and cognitive decline in elderly Latinos: findings from the Sacramento Area Latino Study of Aging study[END_REF][START_REF] Komulainen | Metabolic syndrome and cognitive function: a population-based follow-up study in elderly women[END_REF][START_REF] Dik | Contribution of Metabolic Syndrome Components to Cognition in Older Persons[END_REF][START_REF] Muller | Metabolic syndrome and dementia risk in a multiethnic elderly cohort[END_REF]. Three longitudinal studies showed that MetS as a whole is related to higher risk of cognitive decline, but they did not examine the association with the risk of dementia [START_REF] Yaffe | The metabolic syndrome, inflammation, and risk of cognitive decline[END_REF][START_REF] Yaffe | Metabolic syndrome and cognitive decline in elderly Latinos: findings from the Sacramento Area Latino Study of Aging study[END_REF][START_REF] Komulainen | Metabolic syndrome and cognitive function: a population-based follow-up study in elderly women[END_REF].

While a cross-sectional study (13) found a higher prevalence of dementia in women with MetS, the single published prospective study [START_REF] Muller | Metabolic syndrome and dementia risk in a multiethnic elderly cohort[END_REF] showed no association between MetS and incident dementia. On the other hand, only some components of the MeS -and not MetS itself -might be associated with an increased risk of incident dementia. In particular, several prospective studies evidenced that presence of diabetes was associated with increased incidence of AD and vascular dementia [START_REF] Biessels | Risk of dementia in diabetes mellitus: a systematic review[END_REF][START_REF] Muller | Metabolic syndrome and dementia risk in a multiethnic elderly cohort[END_REF]. However, the role of hyperglycemia in the absence of diabetes (impaired fasting glucose) needs to be well-defined [START_REF] Messier | Impact of impaired glucose tolerance and type 2 diabetes on cognitive aging[END_REF].

We therefore aimed to estimate the prospective association between MetS, or its individual components, with the risk of incident dementia -and the potential differences according to the subtype of dementia: AD or vascular dementia -in a large prospective epidemiological study conducted in community-dwelling older persons. We focused on the component "elevated fasting glycemia" to study the specific role of diabetes versus impaired fasting glucose.

RESEARCH DESIGN AND METHODS

Study population

The subjects included in the present study were a sub-sample of the Three-City (3C) study, a large French multicenter prospective epidemiologic study of vascular risk factors for dementia.

The study protocol has been described previously [START_REF]Vascular factors and risk of dementia: design of the Three-City Study and baseline characteristics of the study population[END_REF] and was approved by the Consultative Committee for the Protection of Persons participating in Biomedical Research (CCPPRB) of the University Hospital of Kremlin-Bicêtre (Paris, France). All participants gave their written informed consent. Briefly, non-institutionalized persons aged 65 yr and older were selected from electoral rolls of three cities and their suburbs: Bordeaux in the southwest, Dijon in the northeast, and Montpellier in the southeast of France, and then invited to participate in the study. Sample size was estimated to achieve a sufficient number of health events over four years. At baseline in 1999-2000, the sample included 9,294 subjects (3,718 men and 5,576 women): 2,104 in Bordeaux, 4,931 in Dijon, and 2,259 in Montpellier.

For the present study of incident dementia (screening procedure described below), we excluded 215 participants already demented at baseline (1999)(2000). We also excluded 1341 subjects because of incomplete data for defining MetS (623 persons with no blood sample and 971 with no waist circumference measurement). This gave a baseline sample of 7738 subjects. At the second round of the study (2001)(2002), 6964 (90.0%) were examined again, 134 were deceased, and 640 refused to participate or were lost to follow-up. At the third round (2003)(2004), 6262 were examined (84.9% of the survivors after the second round), the cumulated number of deaths was 358, and 1118 refused to participate or were lost to follow-up. Finally, 7087 (91.6%) participants had at least one follow-up examination over the four years and formed the sample of the present study.

Diagnosis of dementia

A three-step procedure was used to diagnose cases of dementia [START_REF]Vascular factors and risk of dementia: design of the Three-City Study and baseline characteristics of the study population[END_REF]. First, screening was based on a thorough neuropsychological examination by trained psychologists. This examination included a battery of cognitive tests covering memory, attention, language and visuo-spatial abilities. Data on activities of daily living, severity of cognitive disorders and, where possible, magnetic resonance images (MRI) or CT-Scan were collected [START_REF]Vascular factors and risk of dementia: design of the Three-City Study and baseline characteristics of the study population[END_REF]. Second, the participants who were suspected of having dementia on the basis of their neuropsychological performance (in particular Mini Mental Status Examination, Benton Visual Retention Test and Isaac's Set Test administered uniformly in all three study centres at each follow-up ( 16)) were examined by a neurologist in the three study centres [START_REF]Vascular factors and risk of dementia: design of the Three-City Study and baseline characteristics of the study population[END_REF]. Finally, all suspected dementia cases were analyzed by a common independent committee of neurologists according to the criteria of the Diagnostic and Statistical Manual of Mental Disorders, fourth edition (DSM-IV) [START_REF]Vascular factors and risk of dementia: design of the Three-City Study and baseline characteristics of the study population[END_REF]. This committee of neurologists reviewed by teleconference all potential cases of dementia of the three study centres to obtain a consensus on its diagnosis and etiology based on all existing information. With regard to the different subtypes of dementia, we considered the two most frequent causes of dementia as determined by the committee: AD according to NINCDS-ADRDA criteria ( 16) and vascular dementia based on history of vascular disease, Hachinski score [START_REF]Vascular factors and risk of dementia: design of the Three-City Study and baseline characteristics of the study population[END_REF] and MRI whenever possible.

Definition of MetS

Baseline MetS was defined according to the National Cholesterol Education Program Adult

Treatment Panel III Criteria (NCEP ATP III), which requires the presence of three or more alterations among the following cardiometabolic parameters (being dichotomized as present or absent): 1) elevated systolic (> 130 mmHg) or diastolic blood pressure (> 85 mmHg) or use of antihypertensive medication, 2) large waist circumference (> 88 cm in women and > 102 cm in men), 3) elevated triglycerides (≥ 150 mg/dL), 4) low HDL-cholesterol (men < 40 and women < 50 mg/dL), and 5) elevated fasting glycemia (≥ 110 mg/dL) or non-fasting glycemia ≥ 200 mg/dl or antidiabetic medication [START_REF] Grundy | Diagnosis and management of the metabolic syndrome: an American Heart Association/National Heart, Lung, and Blood Institute Scientific Statement[END_REF].

In order to better understand the specific role of diabetes versus impaired fasting glucose, we made further analyses distinguishing subjects with diabetes (fasting glycemia ≥ 126 mg/dl or non-fasting glycemia ≥ 200 mg/dl or antidiabetic medication) (definition of WHO, 1999) (23), subjects with impaired fasting glucose (fasting glycemia ≥ 110 mg/dl without diabetes) and those with normal fasting glycemia (< 110 mg/dl) without diabetes (category of reference, subjects with no glucose regulation disorder).

Assessment of MetS components

Blood pressure was measured twice in a sitting position using a digital electronic tensiometer, and the average was used in the statistical analyses. Height, weight, and waist circumference were measured in lightly-dressed subjects. Waist circumference was measured between the lower rib margin and the iliac crest following a normal expiration. All pharmacological treatments taken during the month preceding the time of the interview were recorded. Medical prescriptions and, when feasible, the medications themselves were checked by the interviewer.

Blood samples were collected (EDTA plasma samples stored at -80°C) and centralized measurements of fasting blood glucose, HDL-cholesterol and triglycerides were performed. As we had to include only Type 2 diabetes patients in the definition of MetS component [START_REF] Grundy | Diagnosis and management of the metabolic syndrome: an American Heart Association/National Heart, Lung, and Blood Institute Scientific Statement[END_REF], we excluded patients who were considered to have Type 1 diabetes based on the definition of WHO, 1999 [START_REF] Gabir | Plasma glucose and prediction of microvascular disease and mortality: evaluation of 1997 American Diabetes Association and 1999 World Health Organization criteria for diagnosis of diabetes[END_REF] and who were treated by insulin alone (n = 26).

Potential confounders and effect modifiers

Socio-demographic information recorded at baseline during a face-to-face interview included age, gender, and educational level (in two classes: subjects who received primary or no education vs the others (6 years of schooling or more), 10 missing data). Determination of Apolipoprotein E (ApoE) genotype was carried out at the Lille Genopole located in Lille, France (http://www.genopole-lille.fr/spip/). All these characteristics are potential risk factors of dementia [START_REF] Ritchie | The dementias[END_REF] and might be associated with some MetS components, thus acting as potential confounders.

Statistical analysis

Baseline characteristics of the subjects were compared according to the presence of MetS using Student and Pearson Chi square tests for continuous and categorical variables respectively.

Associations between MetS, or each of its components, with incident dementia over four years were estimated using separate proportional hazards models with delayed entry and age as time scale [START_REF] Lamarca | Left-truncated data with age as time scale: an alternative for survival analysis in the elderly population[END_REF]. This model allowed estimation of the strength of the association between MetS and each of its components with age at onset of dementia, taking censoring by death or loss-tofollow-up into account, over the four years of follow-up. All MetS components were then included in a single model to evaluate their own independent effect on the risk of dementia.

Models tested associations with all-cause dementia and then its different subtypes (Alzheimer's disease / vascular dementia). Models were adjusted for socio-demographic characteristics: age (already taken into account by the model), gender, education level, and study centre. We tested the potential interactions between each component of the MetS and ApoE genotype (presence of at least one vs no ε4 allele) on the risk of incident dementia. If no significant interaction was detected, analysis was further adjusted on ApoE genotype. We also tested the interaction between age and waist circumference. All statistical tests were two-sided and a p value of less than 0.05 was considered statistically significant, except for the interaction terms, for which a level of significance of 0.10 was used in order to better identify biologically meaningful associations.

Statistical analyses were performed with the SAS Statistical package release 9.1 (SAS Institute Inc., Cary, NC, USA).

RESULTS

General baseline characteristics

The study sample consisted of 4323 women and 2764 men who had at least one re-examination over four years and for whom all MetS components were measured at baseline. The 651 subjects with no follow-up did not differ significantly from the 7087 subjects having at least one reexamination in terms of MetS components (p = 0.07 for hypertension, p = 0.12 for high waist circumference, p = 0.22 for high triglycerides, p = 0.10 for elevated fasting glycemia; data not shown), except for low HDL-cholesterol (p = 0.001) which was more frequent in subjects with no follow-up (data not shown).

The baseline characteristics of the whole cohort, and according to the presence of MetS, are reported in Table 1. Mean age was 73.4 (4.9) yr. According to the NCEP-ATP III Criteria, MetS was present in 15.8% of the study participants. Subjects with MetS were more often men, slightly older and less educated than those without MetS. There were no differences in prevalence of ApoE ε4 allele between the two groups. High blood pressure was the most frequent MetS component, followed by high waist circumference, high triglycerides, low HDLcholesterol, and elevated fasting glycemia. About two thirds of participants with so defined hyperglycemia had diabetes.

Associations between MetS at baseline, its components, and incident dementia during follow-up

During the four years of follow-up, 208 incident cases of all-cause dementia were validated, i.e. 0.84 incident dementia cases per 100 person-years (95% CI: 0.72 -0.95). This incidence was 1.13 (0.79 -1.46) per 100 person-years in subjects with MetS (44 cases) and 0.78 (0.66 -0.90) in the others (164 cases). The incidence of AD (134 cases, 64% of dementia cases) was 0.54 (0.45 -0.63) per 100 person-years in the whole cohort, 0.51 (0.29 -0.74) in subjects with MetS (20 cases) and 0.55 (0.45 -0.65) in the others (114 cases). The incidence of vascular dementia (40 cases, 19.2% of dementia cases) was 0.16 (0.11 -0.21) per 100 person-years in the whole cohort, 0.33 (0.15 -0.51) in subjects with MetS (13 cases) and 0.13 (0.08 -0.18) in the others (27 cases).

Subjects with MetS at baseline had a non-significant increased risk of all-cause dementia over four years (Table 2). Table 2

also reports the association between each component of MetS and

the risk of all-cause dementia over the four years of follow-up in separate models. High triglycerides level at baseline was the only component significantly associated with all-cause dementia, the risk being increased by 45%. This association remained significant and virtually unchanged after additional adjustment on ApoE genotype (HR = 1.47 [1.06 -2.03]; p = 0.02).

Regarding the link between MetS and the risk of dementia sub-types, no association was found between MetS at baseline and the risk of AD over four years (Table 2). None of the MetS components was associated with a significantly increased risk of AD. On the contrary, a high waist circumference was associated with a decreased risk of AD (Table 2). This association remained unchanged after additional adjustment on ApoE genotype (HR = 0.62 [0.41 -0.92]; p = 0.02).

Subjects with MetS at baseline had a significantly increased risk of vascular dementia over four years (Table 2) even after additional adjustment on ApoE4 genotype (HR = 2.44 [1.25 -4.77]; p = 0.01). High triglycerides level was the only component significantly associated with vascular dementia. This significant association persisted after additional adjustment on ApoE genotype (HR = 2.36 [1.21 -4.61]; p = 0.02). Elevated fasting glycemia at baseline according to the NCEP ATP III criteria was not significantly associated with increased risk of incident dementia. However, when this variable was split into two dummy variables the presence of diabetes was significantly associated with increased risk of all-cause and vascular dementia, not AD, while impaired fasting glucose was not associated with any type of dementia (Table 2).

There was no interaction between age and waist circumference. (p = 0.44 in the model with allcause dementia, p = 0.76 in the model with AD and p = 0.45 in the model with vascular dementia). We also found no interaction between age and elevated fasting glycemia for the risk of all-cause dementia (p = 0.78), AD (p = 0.24) or vascular dementia (p = 0.23) (data not shown).

The results remained unchanged when all five MetS components were entered simultaneously as explanatory variables in a single model (Table 2 bottom, single model). In this analysis, the results for each MetS component were similar to those obtained with separate regressions with each component. Hazard ratios and significance levels were only slightly attenuated because of some collinearity among MetS components. Similarly, diabetes but not impaired fasting glucose, remained associated with risk of all-cause and vascular dementia when adjusting for the other MetS components.

CONCLUSIONS

In this prospective study of French non-institutionalized elderly subjects aged 65 yr and over, the presence of MetS (NCEP ATP III criteria) at baseline was associated with an increased risk of incident vascular dementia but not all-cause dementia and AD over four years, independently of socio-demographic characteristics and ApoE ε4 allele. High triglycerides level at baseline was the only MetS component that was significantly associated with all-cause dementia and vascular dementia, even after adjustment on ApoE genotype. On the other hand, a high waist circumference was associated with a decreased risk of AD. Diabetes, but not impaired fasting glucose, was significantly associated with all-cause dementia and vascular dementia.

Our findings suggest that the concept of MetS may not give any additive value in predicting the development of dementia compared to its components considered separately. Indeed, when we tested the contribution of each MetS component by putting them together into a single model, the results were similar to those obtained with one regression for each component, suggesting that the association of each component with dementia risk was independent. Moreover, we found no statistically significant association between the number of MetS components and the risk of incident dementia (data not shown). This result is in agreement with the study of Yaffe et al [START_REF] Yaffe | The metabolic syndrome, inflammation, and risk of cognitive decline[END_REF] where the number of MetS components did not affect the risk of cognitive decline. Accordingly, the concept of MetS is controversial and the MetS would be "a set of statistical associations believed to carry an excess of cardiovascular risk" [START_REF] Ferrannini | Is insulin resistance the cause of the metabolic syndrome?[END_REF] and then of vascular dementia.

The association between high triglycerides level at baseline and risk of incident dementia is an original finding. Few studies have focused on hypertriglyceridemia and its relations to dementia.

A transversal study made known the potential relations between elevated triglyceride levels and poorer cognitive function in patients with diabetes [START_REF] Perlmuter | Triglyceride levels affect cognitive function in noninsulin-dependent diabetics[END_REF]. A similar association was found in a longitudinal study with a long-term follow-up: in the Honolulu-Asia Aging Study, an 1 SD increase in triglycerides levels during mid-life increased the risk of dementia 25 years later [START_REF] Kalmijn | Metabolic cardiovascular syndrome and risk of dementia in Japanese-American elderly men. The Honolulu-Asia aging study[END_REF].

In our study, high triglycerides level was only associated with incident vascular dementia (even if there were only 40 cases) but not with AD. Whether moderate hypertriglyceridemia is an independent risk factor for cardiovascular disease remains a debated question but a metanalysis of thousands of patients followed up for more than 10 yr showed that a triglycerides elevation of 1 mmol/L increased the risk of cardiovascular disease, independently of HDL-cholesterol [START_REF] Hokanson | Plasma triglyceride level is a risk factor for cardiovascular disease independent of high-density lipoprotein cholesterol level: a meta-analysis of populationbased prospective studies[END_REF].

However, the precise mechanisms, especially in the brain, by which hypertriglyceridemia might increase dementia risk still have to be elucidated. Few studies have examined the relations between HDL-cholesterol and incidence of dementia, most focusing on total cholesterol or LDLcholesterol with conflicting results [START_REF] Rosendorff | Cardiovascular risk factors for Alzheimer's disease[END_REF].

The fact that diabetes itself, and not impaired fasting glucose, was associated with all-cause and vascular dementia but not AD, reinforces the hypothesis that vascular risk factors are involved.

Vascular pathophysiological mechanisms are more involved in diabetes than in impaired fasting glucose. Indeed, "diabetes" is defined by risk of microvascular complications (18) (as occurrence of microanevrysms).

In opposition with our results, higher waist circumference was found to be associated with higher risk of AD but only in persons younger than 76 years [START_REF] Luchsinger | Measures of adiposity and dementia risk in elderly persons[END_REF]. Conversely, some studies found that patients might lose weight prior to AD diagnosis and this has been interpreted as a consequence of the disease rather than a risk factor. So, the inverse association between adiposity, measured by waist circumference, and AD could be explained by weight loss due to pre-clinical disease in our participants.

In contrast to other studies (3), we found no association between hypertension and risk of developing dementia. This could be due to the high frequency of hypertension in the whole 3C cohort. Therefore, the relationship between hypertension at middle age and risk of dementia [START_REF] Kivipelto | Midlife vascular risk factors and Alzheimer's disease in later life: longitudinal, population based study[END_REF] may no longer hold at older ages.

Moreover, in our study, the duration of the presence of the various components of MetS was not known and may have influenced the risk of dementia. Indeed, risk factors which are already present at mid-life might have a stronger impact on dementia risk, e.g. hypertension [START_REF] Kivipelto | Midlife vascular risk factors and Alzheimer's disease in later life: longitudinal, population based study[END_REF]. Results from several studies indicate that relations between these factors and disease become increasingly complex with advancing age [START_REF] Luchsinger | Measures of adiposity and dementia risk in elderly persons[END_REF]. Findings may therefore be different when the risk factors are measured at mid-life, in particular for hypertension and low HDL-cholesterol but also for BMI [START_REF] Kivipelto | Midlife vascular risk factors and Alzheimer's disease in later life: longitudinal, population based study[END_REF]. Another explanation to the lack of association in our study may lie in the fact that the thresholds currently used for defining MetS might not be suitable for an older population, as suggested by the very high prevalence of hypertension found in our sample of rather healthy participants.

Nevertheless, several limitations may affect the interpretation of our results. Selective survival might explain some paradoxical results in the oldest old, in whom MetS was found to be associated with slower cognitive decline. The association between MetS and dementia might indeed be underestimated by censoring by death since subjects with MetS are more likely to die from cardiovascular disease before developing dementia. It would be interesting, in a future study, to explore the causes of death in patients with MetS.

Similarly, the 651 subjects with no follow-up did not differ significantly from the 7087 subjects having at least one re-examination in terms of MetS components, except for HDL-cholesterol which could be explained by a worse nutritional status linked to higher cumulative mortality rates (data not shown).

Our results emphasize the need for early detection and treatment of hypertriglyceridemia and diabetes in older persons in order to delay the onset of clinical dementia. However, the concept of MetS as a whole does not seem to have any added value regarding dementia risk in this population. The putative interactions between single metabolic risk factors -high triglycerides level and diabetes -and dementia need to be further studied.

  BP: blood pressure, TG: triglycerides, HDL-chol: HDL-cholesterol, Gly: glycemia. High BP: SBP > 130 mm Hg or DBP > 85 mm Hg or medication; SBP: systolic blood pressure, DBP: diastolic blood pressure High Waist: Waist circumference > 88 cm in women or > 102 cm in men High TG: TG ≥ 150 mg/dl; TG: triglycerides Low HDL-chol: HDL-chol < 50 mg/dl in women or < 40 mg/dl in men; HDL-chol: HDL-cholesterol High Gly: Fasting glycemia ≥ 110 mg/dl or non-fasting glycemia ≥ 200 mg/dl or antidiabetic medication IFG: Impaired fasting glucose: fasting glycemia ≥ 110 mg/dl without diabetes Diabetes: fasting glycemia ≥ 126 mg/dl or non-fasting glycemia ≥ 200 mg/dl or antidiabetic medication

Table 1 . Baseline characteristics according to the presence of metabolic syndrome (MetS) and associations between the different components of the MetS. The Three City Study, France, 1999-2004. N = 7087.

 1

		Total	MetS	No MetS	p		Components of the MetS
		Cohort				High BP	High Waist	High TG	Low HDL-chol	High Gly
		N = 7087	N = 1121	N = 5966		N = 6064	N = 2049	N = 1247	N = 741	N = 816
	Age, yr, mean (SD)	73.4 (4.9)	73.8 (4.8)	73.4 (4.9)	0.05	73.7 (4.9)	74.0 (4.9)	73.3 (4.9)	73.6 (4.9)	73.2 (4.7)
	Men (%)	39.0	41.6	38.5	0.05	40.8	32.3	45.9	35.4	39.0
	Education level:	23.6	29.1	22.6	<0.0001	24.4	28.8	25.9	29.9	23.6
	no or primary (%)
	ApoE ε4 (%)	20.4	21.0	20.3	0.60	20.3	19.6	20.4	23.5	20.4
	High BP (%)	85.6	98.2	83.2	<0.0001	na	91.6	90.0	88.3	95.2
	High Waist (%)	28.9	78.5	19.6	<0.0001	31.0	na	44.2	50.4	50.2
	High TG (%)	17.6	66.4	8.4	<0.0001	18.5	26.9	na	50.5	32.0
	Low HDL-chol (%)	10.5	47.6	3.5	<0.0001	10.8	18.3	30.2	na	22.8
	High Gly (%)	11.5	47.6	4.7	<0.0001	12.9	20.0	20.9	25.0	na
	IFG without diabetes	3.9	15.3	1.8	<0.0001	4.3	6.9	6.1	7.0	34.2
	Diabetes	7.6	32.3	2.9	<0.0001	8.6	13.1	14.8	18.0	65.8

Table 2 . Association between the metabolic syndrome (MetS) at baseline, or each of its components -in separate models and then in a single model -and risk for incident dementia over four years. The Three-City study, France, 1999-2004. Models are adjusted on age, gender, educational level and city center: N = 7077, without missing data.

 2 Regressions analyzed by proportional hazards models with delayed entry and age as time scale. Results expressed as hazard ratios and 95% CI.

	All-cause dementia	p	Alzheimer's disease	p	Vascular dementia	p
	N = 7077		N = 7077		N = 7077
	Hazard ratio (95% CI)		Hazard ratio (95% CI)		Hazard ratio (95% CI)

ACKNOWLEDGMENTS

We thank the members of the independent committee of neurologists who reviewed the potential cases of dementia: F. Pasquier, F. Portet, S. Auriacombe, M. Poncet, O. Rouaud. We thank S. Jarman and R. Cooke for their revisions of the english manuscript.

Funding/Support

The 3C study is conducted under a partnership agreement between the Institut National de la Santé et de la Recherche Médicale (INSERM), the Institut de Santé Publique et Développement of the Victor Segalen Bordeaux 2 University, and Sanofi-Aventis. The Fondation pour la Recherche Médicale funded the preparation and initiation of the study. The 3C Study is also supported by the Caisse Nationale Maladie des Travailleurs Salariés, Direction Générale de la Santé, Mutuelle Générale de l'Education Nationale, Institut de la Longévité, Regional Councils of Aquitaine and Bourgogne, Fondation de France, and Ministry of Research -INSERM Programme "Cohortes et collections de données biologiques." The Lille Genopole was supported by an unconditional grant from Eisai. C. Raffaitin received funding from the Federation Hospitalière de France for her Master training.

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