V Langlois
email: vincent.langlois@u-pem.fr

V H Trinh

C Perrot

Electrical conductivity and tortuosity of solid foam: eect of pore connections

Numerical and analytical methods at both micro-and mesoscales are used to study how the electrical resistivity and the high frequency tortuosity of solid foam are modied by the presence of membranes that partially or totally close the cell windows connecting neighbor pores. Finite element method (FEM) simulations are performed on two pores connected by a single-holed membrane and on well-ordered Kelvin foam. For two pores connected by a single-holed membrane, we show that the equation for pore access resistance obtained by Sahu and Zwolak (Phys. Rev. E 98, 012404, 2018) can predict, after a few modications, the electrical resistivity at the membrane scale for a large range of membrane apertures. In the second part, considering these analytical results, we build a pore-network model by using two kinds of conductances at the pore scale -inter-pore conductance and intra-pore conductance. Local inter-pore resistances govern foam electrical conductivity at small membrane aperture size, but when the membrane aperture has the same order of magnitude as the pore size, the intra-pore resistances are no longer negligible. An important success of this pore-network model is that it can be used to study the eects of percolation on the foam electrical conductivity by using pore-network simulations on larger samples containing a few thousands of pores and having dierent proportions of closed membrane randomly distributed over the sample.

The tortuosity is found to be drastically larger than one in foam containing membranes with small apertures or a signicant fraction of closed membranes.

I. INTRODUCTION

When an acoustic wave propagates through a viscous uid-saturated porous media having a motionless skeleton (rigid frame), the relative displacement between uid and frame generates uid shear at the surface of the pore walls. This results in a viscous dissipation and an attenuation of the sound. Due to its diusive propagation, any vorticity generated at the pore walls decays to zero as one moves away from the pore wall into the bulk of the pore [1]. Therefore, viscous dissipation occurs in a boundary layer located at the surface of the pore and of thickness δ. This length δ, called viscous skin depth, is frequency dependent: δ ≈ µ/(ρ f f ) where µ and ρ f are respectively the dynamic viscosity and density of the uid, and f is the frequency. In the limit of high frequencies, the viscous skin depth becomes much smaller than any characteristic pore size, and the uid tends to behave as an ideal uid having no viscous eect (except in the thin boundary layer). More generally, the dynamic behavior of uid-saturated porous media is due to a balance between the power developed by the uid pressure gradient and both the viscous dissipative power and the inertial power developed in the uid [2,3]. The viscous dissipative power dominates the inertial power at low frequency and inversely at high frequency. The visco-intertial frequency f v dening the transition between low-and high-frequency behaviour, corresponds to the frequency for which the viscous dissipated and kinetic (inertial) powers are equal [3]. Likewise, the distinction between the low-and high-frequency behaviour depends also on whether the viscous skin depth is large or small compared to a characteristic pore size [1]. An estimate of f v follows from this phenomenological description, and is given by f v = µ/(Kα ∞ ρ f ); where K is the permeability, a parameter associated with the low frequency behavior; α ∞ is the tortuosity, a parameter associated with the high-frequency behavior. In the high-frequency limit, the eective density ρ e only depends on the tortuosity, ρ e = α ∞ ρ f . The tortuosity is a key parameter describing dispersion of microscopic velocities with respect to the average value of microscopic velocities (Eq. A4). Because at high frequency, the potential ow is formally identical to electric conduction assuming that the porous solid is insulating [4], the tortuosity can be experimentally estimated by electrical conductivity measurement or calculated for specic pore geometries by solving the electrical boundary value problem (Appendix A, Sec. Inertial ow, Eq. A3). Several properties of α ∞ are important to mention.

(1) α ∞ is a scale invariant parameter (i.e. remaining unchanged if the sample is uniformly dilated or contracted by a scale factor) and is greater than or equal to one. (2) α ∞ is dependent on the shape of the solid matrix, for example it accounts for both sinuosity and convergingdiverging pore geometry. (3) In the theory proposed by Biot, α ∞ is a "structure constant" dened as a purely geometric parameter, independent of material parameters, which relates an inertial uid-solid coupling density ρ a to the density of the uid lling the pore space ρ f , ρ a = (α ∞ -1)ρ f [2,4,5]. This coupling term is related to the force applied to a uid to prevent its displacement when the solid is accelerated. (4) At high frequency, the eective sound speed is proportional to 1/ √ α ∞ , meaning that the eective distance of travel for the wave between two points is increased by √ α ∞ due to the tortuous path [6].

Foam is a dispersion of gas in a liquid or solid matrix. Its structure consists of membranes (also called lms for liquid foams), ligaments or Plateau's borders (junction of three membranes) and vertices or nodes (junction of four ligaments). Whereas closed membranes are necessary to ensure the mechanical stability of liquid foam [START_REF] Cantat | Foams: structure and dynamics[END_REF], they can be open or totally absent in solid foam allowing for the foam cells (pores) to be connected through windows. The closed membranes have drastic eects on transport phenomena, such as uid ow or electric current: open windows participate in the transport through the material, whereas closed windows stop it. Therefore the fraction of closed windows is crucial for several applications (ltering or sound absorption). The eects of membrane on acoustical properties are now the subject of active research [3,811]. It has been shown that closed membranes can increase the eective density of foam [START_REF] Gaulon | [END_REF] and therefore, in agreement with the work of Jonhson et al. [1], the foam tortuosity α ∞ [10]. Moreover, closed membranes imply some percolation eects at the mesoscale which can drastically reduce the foam permeability K as shown in [9].

Addressing the percolation issue of the electric conductivity of porous media requires numerical simulations with large samples involving a few thousand pores [12]. As the size of samples increases, the computational costs for the major numerical methods (nite element, nite volume and boundary element methods) become prohibitive, so that alternative approaches are preferable. Such methods involve determining the ow behavior at the local scale (i.e. a throat between two linked pores) by numerical simulations or analytical solutions (e.g. Hagen-Poiseuille equation), then pore-network simulations at the mesoscopic scale are performed [13,14]. This intermediate scale between micro-and macro-scales is necessary to capture collective eects such as percolation behavior. Such a method was used by Johnson et al. to study the electrical conductivity of porous media. In their simple model, the pores are connected by tubes for which the electrical resistance is perfectly known. The electrical conductivity of the porous media is determined by calculating the equivalent resistance of the tube network. Moreover, it was recently shown that the porenetwork method can successfully predict the eects of closed membranes on foam permeability, and can be used to validate a model of eective permeability [9]. As the problem of foam electrical conductivity and the one of foam permeability share some similarities [15], it is appealing to check if the method used to estimate the effective permeability of foam could be transposed to compute the eective electric conductivity and the tortuosity of foam with membrane.

In this paper, we use a multi-scale approach to study the electrical conductivity of solid foam with various window congurations. Firstly, we deal with the case of two pores separated by a thin membrane having a circular hole. We calculate both the electrical conductance and the uid ow conductance by using FEM. Then, we consider a more realistic pore geometry by using a Kelvin partition of space. We conduct FEM simulations on periodic unit cells (PUC) containing two interconnected pores with various open membrane fractions. The two previous steps are used to build a pore-network model able to reproduce the FEM results. Then, mesoscopic eects induced by the structure of the pore network are studied by pore-network simulations on large (16×16×32 pores) networks of electrical resistances.

II. FLUID PERMEABILITY AND ELECTRICAL CONDUCTION IN POROUS MEDIA

In this section, we recall the denition of the macroscopic parameters involved in this paper: permeability for the issue of Newtonian uid ow through the pore space, and electrical conductivity for the issue of electrical current through a conducting uid lling the pore space.

Permeability K can be determined by measuring the rate of uid ow Q through a surface of porous media when the latter is subjected to a pressure dierence ∆P sp between its opposite faces [15,[START_REF] Adler | Porous media : geometry and transports[END_REF]:

K = [µQH sp ] / [A sp ∆P sp ] (1)
where µ is the dynamic viscosity of the uid, H sp the thickness of the sample and A sp its cross-section area.

Eective electrical conductivity σ e is determined by measuring the total current I passing through a surface of porous media when this one is subjected to an electrical potential dierence ∆V sp applied across the system:

σ e = [IH sp ] / [A sp ∆V sp ] (2)
As shown in [4], the high frequency tortuosity α ∞ is related to the electrical conductivity as follows: where φ is the porosity, σ f the electrical conductivity of the uid lling the pore space, F the formation factor (like α ∞ , a scale invariant property).

φ α ∞ = σ e σ f = 1 F (3)
The macroscopic behavior of porous media regarding the transport of viscous uid and electrical current, results from the cumulative eects of specic local mechanisms. In the case of foam permeability [9], the dominating local mechanism of viscous uid ow corresponds to the ow passing through a single membrane aperture. Hereafter, we reexamine local mechanisms governing electrical conduction properties in order to elaborate on the simplest description that provides a reliable estimate of the tortuosity α ∞ in micro-macro models of partially closed foam.

III. TWO PORES CONNECTED BY A SINGLE PERFORATED MEMBRANE

In this section, we consider the simplest model of foam: two pores connected by a thin membrane containing a circular hole of radius r o (Fig. 1). Periodic conditions are applied on the lateral faces of the pore space. By using FEM simulation, we determine the eective uid ow conductance G f l (dened as the ratio of the uid ow to the pressure dierence applied between the opposite faces of the pore space volume) and the eective electrical conductance G e (dened as the ratio of the total current to the electrical potential dierence applied between the opposite faces of the pore space volume). A commercial code, Comsol Multiphysics v5.2, was used ("creeping ow" and "electrostatic" modules). Details of the boundary value problems solved for each fundamental case are given in appendix A.

Figure 2 shows that the uid ow conductance is strongly dependent on the membrane aperture and quite insensitive to the aspect ratio H/D b of the pore space. This result shows that the viscous losses are essentially dominated by a mechanism acting in the vicinity of the membrane aperture. Indeed, when the membrane disappears (2r o /D b -→ 1), the viscous loss disappears and the uid ow conductance diverges (Fig. 2b). Moreover, as shown in [9], the uid ow conductance can be estimated by a Sampson law [START_REF] Sampson | [END_REF]:

G f l = r 3 o 3µ (4)
Figures 3 and4 show that the electrical conductance increases when the membrane aperture size increases. For low membrane aperture sizes, the electrical conductance exhibits a linear dependence with the aperture size (inset graph of Fig. 3b). Note that, when the membrane thickness is taken into account, this linear dependence is no longer observed at very low membrane aperture sizes. If the electrical conductivity and the permeability follow the same trend as previously described, they have a dierent behavior when the membrane aperture reaches the lateral side of the pore space. Indeed, unlike permeability, the electrical conductance tends to a nite value equal to D 2 b σ f /H where σ f is the electrical conductivity of the uid. This result is fundamental because it reveals that the whole pore space, and not only a part located in the vicinity of the membrane aperture, contributes to the electrical conductivity.

In the last century, the eect of constrictions on electric conductance (Fig. 5) was studied by several authors coming from dierent scientic communities. Hall [18] showed that the electrical resistance R e (the inverse of the electrical conductance G e ) due to a circular aperture in an isolating membrane immersed in a conducting uid having an innite size (electrodes are located at innity) is given by 1/R e = 2σ f r o . Earlier, in the context of electrical conduction through two conducting bodies in contact, several authors [2022] showed that electrical resistance for low area of contact between bodies is given by 1/R e = 2σ f r c where r c is the contact radius. Later, Rosenfeld and Timsit [19] found the exact solution for the electrical resistance of a long constricted cylinder (cylinder length constriction radius = membrane aperture radius): R e = (H/σ f A c ) + R s where A c is the crosssectional surface of the tube (A c = πD 2 b /4 where D b is the cylinder diameter). The rst term H/σ f A c corresponds to the resistance of the cylinder without constriction, and the second term R s is the spreading resistance due to the constriction eect. As the exact solution found by Rosenfeld and Timsit implies an innite sum of terms, the authors gave a compact expression for the spreading resistance R s . The total resistance is then given by:

R e = H A c σ f + R s = H A c σ f + 1 2σ f r o P r 0 D b (5)
where

P (x) = 1 + C 1 x + C 2 x 2 + C 3 x 3 + C 4 x 4 with C 1 = -2.83162, C 2 = 0.25288, C 3 = 1.22088, C 4 = 3.19968.
Note that the polynomial P is built with the constraints, P (1/2) = 0 and P (0) = 1 with the aim of recovering the asymptotic solutions (r o → D b /2 corresponds to a nite uniform cylinder and r o → 0 corresponds to an innite constricted cylinder).

Recently, Sahu and Zwolak [23] have studied the resistance of two interconnected pores having cylindrical or rectangular shape and nite size with various aspect ratios H/D b . The authors break down the pore volume into several parts, each contributing to the electrical conductance of the pore space (Fig. 5). The rst subpart, only observed for long cylinders (H > D b ), is located far from the membrane aperture where the current streamlines are almost parallel to the external electrical eld as in a uniform conductor. The third subpart is located in the vicinity of the membrane aperture where the current streamlines converge to the aperture. The second subpart is a transition region between the rst and the third, where the current streamlines change their behavior. The fourth subpart corresponds to the small channel inside the membrane thickness itself. The rst and fourth subparts can be approximated by uniform conductors having a known resistance (R

I = (H -L 2 ) /σ f A with L 2 ≈ D b /2 and R IV = h m /πσ f r 2
o respectively). For the third subpart, Sahu and Zwolak use an approximation based on the Taylor expansion of the analytical solution for the access resistance with a spheroidal electrode:

R III = 1 2σ f ro 1 -2r0 πL1 + O r0 L1 3
where

L 1 ≈ D b /2
is the size of the spheroid region (Fig. 5). For the transition region, the authors consider a contribution proportional to R II = 1/σ f D b . Therefore, by considering a series connection of the sub-part resistances and after some calculations, the total resistance R e is given by:

R e = H Ψσ f D 2 b + 1 σ f 1 2r o - f l ΨD b + h m πr 2 o ( 6
)
where Ψ and f l are coecients depending on the geometric details of the cell: Ψ = 1 and f l ≈ 1.2 for rectangular pores, or Ψ = π/4 and f l = 1.0 for cylindrical pores [24]. Note that the authors determined the coecients f l by adjusting the equation's predictions to their numerical simulation results.

The term in brackets in Eq. 6 corresponds to the spreading resistance R s as dened by Rosenfeld and Timsit. Similarly to the case of long constricted cylinder, the spreading resistance of a nite constricted cylinder should be equal to zero when the membrane aperture is equal to the cross-section radius or, more precisely, when the surface of the membrane aperture is equal to the cross-section surface (2r o = D b for cylindrical pores, and √ πr 0 = D b for rectangular pores). However, the equation given by Sahu and Zwolak do not satisfy this constraint since f l = Ψ for cylindrical pores and D b = H. Therefore by comparison with FEM results, we show in Fig. 3a that Eq. 6 cannot predict the electric resistance for high membrane aperture size (2r 0 → D b ).

Because we aim at generating an expression enabling us to recover bulk and membrane eects for the whole range of membrane aperture sizes (and not only for tiny holes corresponding to graphene nanopores) with great accuracy, a modication of Eq. 6 is further proposed in order to improve the transition region mixing the access and bulk like behavior:

R e = H Ψσ f D 2 b + 1 σ f 1 2r o 1 + C 1 r o D b + C 3 r o D b 3 + h m πr 2 o (7) with C 1 = -2.89 and C 3 = -8 1 + 1 2 C 1 for cylindri- cal pores; C 1 = -2.52 and C 3 = -π 1.5 1 + 1 √ π C 1 for rectangular pores.
Note that C 1 is determined by tting the equation's predictions to our numerical simulation results ; and C 3 is chosen with the aim of recovering the resistance without membrane R e σ f = H (by neglecting the membrane thickness term). The relative dierence between FEM simulations results and resistances calculated by using Eq. 7 does not exceed 1% as long as the aspect ratios H/D b are superior to 0.75 and the aperture size r o is inferior to 0.9D b . Figure 3a shows that Eq.7 predicts FEM results for a wide range of membrane aperture size with a better accuracy than Eq. 6. Moreover, Figure 3b compares the prediction of Eq. 7 to the solution given by Rosenfeld and Timsit (Eq. 5 ). Both equations have identical values of resistance except when the membrane aperture size is close to the radius of the cylinder. In this case, predictions of Rosenfeld and Timsit equation are slightly better even for short cylinders. However, for very short cylinders (H/D b ≤ 0.25), all equations fail in their predictions. Finally, in Fig. 4, we compare both the predictions given by Sahu and Zwolak (Eq. 6) and provided by Eq. 7 to FEM results on rectangular pores. As observed for the case of cylindrical pores, Eq. 7 predicts FEM results for a wide range of membrane aperture size with a better accuracy than the Sahu and Zwolak expression.

Before moving to the next section, we briey discuss the golden aspect ratio introduced by Sahu and Zwolak [23,24]. For this special aspect ratio (H/D b ) * , nite size eects should be eliminated in numerical simulations (i. cancel each other out. The golden aspect ratio is then given by

(H/D b ) * = -C 1 Ψ/2
, and is equal to 1.26 for rectangular pores, and 1.13 for cylindrical pores (or 1.11 if the Rosenfeld and Timsit equation is used for the spreading resistance). As these values of golden aspect ratios are in the range of aspect ratios where Eqs. 5 and 7 give accurate predictions, our calculation is justied. Note that Sahu and Zwolak [23] nd 1.2 for rectangular pores, and 1.07 for cylindrical pores from FEM simulations and molecular dynamics simulations. Finally, nite eects are cancelled as long as the higher terms in the expansion of the spreading resistance are negligible compared to one. From Rosenfeld and Timsit equation's (Eq. 5), this condition is veried with a relative error inferior to 1% if 2r o < 0.28D b .

IV. ORDERED KELVIN FOAM

In this section we describe the properties of high porosity ordered foam exemplied by a Kelvin structure. As shown in Fig. 6, a periodic unit cell (PUC) of characteristic size D b is used to represent the pore structure in foam samples [10,25]. The number of pores, N p , contained within the unit cell is equal to 2. The cell is based on the Kelvin partition and is a 14-sided polyhedron (8 hexagonal and 6 square faces) corresponding to windows shared with the N v = 14 neighbors. The cell skeleton is made of idealized ligaments having length L = D b /(2 √ 2) and an equilateral triangular cross section of edge length r = 0.58D b (1 -φ) 0.521 , where φ is the gas volume fraction. The thickness of the membranes is taken equal to h m /L = 0.001, such that the eect of the channel inside the membrane thickness can be neglected (because In perfect agreement with the previous results of ref. [9], Figure 8 shows that the permeabilities of the simulated PUCs follow the power-law given in Sec. III (Eq.4). Moreover, because most of the pressure drop is located in the vicinity of the membrane aperture, a simplied calculation of the permeability based on the pore-network hypothesis is possible. In this framework, a value of uid pressure is associated with each pore and the ow between two connected pores is based on the Sampson uid ow conductance (Fig. 9a&b). The foam permeabil- Note that the horizontal median plane is a plane symmetry.

ity calculation based on a pore-network model needs to compute the amount of ow passing through the median plane of the PUC, as shown in Fig. 10. Due to the symmetries of the PUC congurations, the pressure difference associated to the ow passing through the square windows is twice the one associated to the hexagonal windows. Therefore it is straightforward to show that the permeability of the studied PUC is given by:

K = µ D b n sq G f l,sq + 1 2 n hex G f l,hex (8)
where G f l,sq and G f l,hex are the uid ow conductance associated with each window; n sq and n hex are the numbers of open square and hexagonal windows. Note that from the FEM simulations, G f l,sq = G f l,hex = r 3 o 3µ . It can be checked that this formula has the ability to predict the whole FEM results as shown by Fig. 8, whichever way the membranes are distributed throughout the PUC.

We further focus on the calculation of foam electrical conduction. As has been shown for the case of two interconnected pores, we also demonstrate here that the electrical conductivity increases with the membrane aperture size and the degree of interconnections between pores (see Fig. 11). For small membrane apertures, the electrical conductivity exhibits a linear dependence on the aperture size. As noticed for the case of two interconnected pores, this behavior also reveals that the membrane aperture governs most of the electrical resistivity for small apertures. In that case, the electrical conduction is dominated by a mechanism acting at the scale of the membrane aperture just as observed for the permeability. Based on this . Fig. 11a shows that this approach allows us to estimate the electrical conductivity for small aperture, 2r o /D b < 0.1. By contrast, for higher membrane aperture sizes, the contribution of the pore volume is no longer negligible.

We now focus on the elaboration of a pore-network model able to describe most of the electrical conduction eects in foam. As the whole pore volume must be considered in the electrical conduction issue, the general scheme used for permeability, which only accounts for inter-pore conductances (Fig. 9b), does not apply in this case. Additional intra-pore conductances must be introduced in the network as shown in Fig. 9c. Compared to the pore-network model for permeability which uses only one node for each cell, we introduce 14 nodes per cell in the pore-network model for electrical conduction. The nodes are associated with the dierent windows of the cell, and are linked to the neighbor windows' nodes sharing the same cell by the intra-pore conductance, and to a node of a neighbor cell by the inter-pore conductance if the membrane separating the cells is opened. Moreover, the neighborhoods of windows are not equivalent: the hexagonal windows within a cell have two kinds of neighbor windows, squares and hexagons, whereas the square windows have only hexagons as possible neighbors. Consequently, we introduce two intra-pore conductances: G i,sh which links square to hexagon, and G i,hh which links hexagon to hexagon. For the inter-pore conductance G ep , we consider the term due to the membrane aperture in Eq. 7 with appropriate values of C 1 and C 3 : Figure 10. Fluid ow passing through the K0 conguration of the PUC (Fig. 7a). Color lines correspond to the active bond in the pore-network model for foam permeability (Fig. 9b).

1 Gep = 1 2σ f ro 1 + C 1 ro di + C 3 ro di
for square windows and 3 1.5 /4π 0.5 for hexagonal windows. We do not take into account the term involving the thickness of membrane as its eect is negligible in the range of membrane aperture sizes and thicknesses that we investigate (r o h m ). The coecients, C 1 and C 3 , and the intra-pore conductances are chosen to best t the results of FEM simulations. As shown in Fig. 11a, a good agreement between both numerical methods is achieved with: G i,sh /D b σ f = 0.16, G i,hh /D b σ f = 0.30, C 1 = -2.53 and C 3 = 1.32. Finally, from a practical point of view, it is possible to estimate the electrical conductivity for the reference conguration K 0 with a good accuracy (error < 2% with φ = 0.995) by using the following equation:

σ e,K0 = 4σ f D b 2r o + h m D b πr 2 o -1 1 + 2.7 r o D b + 5.1 r o D b 2 -1 (9
) Fig. 11b shows the tortuosities of each PUC conguration for various membrane aperture sizes. When the membrane aperture size is high, the tortuosity is closed to 1. However, when the membrane aperture size or the number of open windows decreases, the tortuosity increases. Tortuosity can be signicantly larger than one in foam containing membranes with small aperture. Therefore this model predicts a signicant increase of the dynamic eective density of the eective uid in the high frequency inertial regime. To reach the inertial regime, the frequency has to be higher than the viscoinertial characteristic frequency f v . Fig. 12 shows the visco-inertial characteristic frequency f v as a function of the membrane aperture size for each conguration.

For a fully open foam K 0 , the membrane aperture size for which the tortuosity is higher than 10 is equal to r o /D b ≈ 0.0125. For foam having millimetric pores and lled with air, the corresponding visco-inertial characteristic frequency f v is approximately equal to 100kHz which is higher than the audible frequency range. However, for foam containing a fraction of closed membrane as in the conguration K 3 and with D b ≈ 2mm, the visco-inertial characteristic frequency can reach the audible frequency range (f v ≈5000Hz).

As the porosity is a parameter which is involved in the relationship between the tortuosity and the electrical conductivity, we evaluated the eect of porosity on each parameter by performing FEM simulations on K0 congurations for various values of φ. As shown on Fig. 13, the dependence of the electrical conductivity on φ is low when compared to the dependence of the tortuosity on φ. This is due to the fact that the eective electrical conductivity σ e is only depending on r 0 when r 0 → 0 (bulk eects are vanishing, Eq. 6) whereas the tortuosity involves a factor φ to account for the scaling from the uid phase to the overall volume of the porous aggregate (Eq. 3).

V. ELECTRICAL CONDUCTION IN FOAM WITH RANDOMLY DISTRIBUTED CLOSED MEMBRANES

Eects of pore network features on electrical conductivity are studied on lattices having a structure similar to Kelvin's structure: each pore is surrounded by v = 14 neighbor pores. The samples have a size L 2 H sp = 16 3 (D b units) and contain 8192 pores. As we are interested lines are plotted by using Eq. 8 taking into account only interpore electrical conductances (Fig. 9b). Full lines are plotted by using pore-network calculations taking into account both intra-and inter-pore electrical conductances (Fig. 9c). Due to the symmetries of the PUC congurations, analytical expressions for eective conductivity can be found for each conguration (see appendix B).

in the eect of closed membranes on electrical conduction, we close a part of the windows by randomly cancelling some electrical bonds between nodes connecting two pores. The values of intra-pore conductances, G and G i,hh , found previously on Kelvin having φ = 0.995 are used in this pore-network calculation. Boundary eects are avoided by resorting to periodic conditions imposed in the directions perpendicular to the macroscopic gradient. Details of the calculations of electrical conductivity are given in appendix B.

Fig. 14 shows the electrical conductivity calculated with simulations performed on large samples having random positions of closed windows and various open window fractions x ow having identical membrane aperture sizes. Below a critical concentration x ow < x p ≈ 0.1 ≈ 1.5/N v , the network of interconnected pores does not connect the top of the sample to the bottom. Therefore the open porosity (porosity of the percolating pore space) is null and no electrical conduction through the foam is possible. In the range [0.1, 0.2] of open window fraction, on the one hand the fraction of open porosity increases from 0 to 0.93, and on the other hand, the electrical conductance gently rises. For x ow > 0.2, the electrical conduction gradually increases as the fraction of open windows rises. For low aperture size, the electrical conduction exhibits a quasi-ane dependence on the open window fraction x ow similar to the one observed in [9] for permeability. As we previously saw, the uid ow conductance corresponds to an inter-pore conductance, Therefore as the inter-pore resistance dominates the electrical conduction at low membrane apertures, a similar behavior between electric conductivity and per- open pore space. Moreover, σe(1) = σe,K 0 can be calculated by using Eq.9.

meability was expected for low membrane apertures. In that case, electrical conductivity and tortuosity can be estimated by using the eective model for foam permeability of Langlois et al. [9] (see appendix C).

VI. CONCLUSION

In order to study the eects of both the fraction of open windows and their aperture sizes on both electric conductivity and high frequency tortuosity of foam, we performed simulations at dierent scales: 1) FEM simulations computing the ow of electric current at the pore scale and on ordered Kelvin foam; and at the higher scale, 2) pore-network simulations capturing the physics at the lower scale without describing the entire velocity eld. Eects of the membrane aperture size and the open window fraction on the permeability are recalled to highlight the dierence between permeability and electrical conductivity. The FEM simulations at pore scale were useful to identify the mechanisms of electric conduction through a single pore. The analysis reported in this paper shows that, for low membrane aperture size, the foam electrical conductivity can be described by a mechanism acting at the scale of the membrane aperture. However, when the membrane aperture size is comparable to the pore size, another mechanism acting on the whole pore volume must be considered. By considering these results, we build a pore-network model at the pore scale by using two kinds of conductance: inter-pore conductance and intra-pore conductance. By using appropriate conductances, we show that pore-network model can reproduce the FEM results performed on ordered Kelvin foam having dierent congurations of closed membranes. Pore-network simulations on large samples having random position of closed windows show that percolation occurs when the fraction of open windows is close to x p = 1.5/N v (≈0.11 for Kelvin foam having N v = 14) and that the fraction of closed windows can signicantly increase high frequency tortuosity.

An experimental conrmation of this numerical study would require the production of well-controlled monodisperse foam having a known (controlled or measured) fraction of closed membranes. Emulsion and foam templating techniques can be used to produce tailor-made porous polymers, see for instance Ref. [26] and references therein. In particular, experimental results were reported [27] to generate porous polyurethane by a sphere templating method as an attempt to control the interconnection size for a given bead diameter (bead sintering). We are not aware of any experimental setup able to produce well-dened (geometrically) polymeric structures having closed and open pores of dierent ratios. Milliuidic techniques were, however, employed to generate monodisperse foams with various membrane contents [10]; (geometrically) characterized according to the proportions of closed and open windows and aperture size. While tortuosity estimates were reported using acoustical measurements [28] by Trinh et al. [10], the acoustical method failed to provide tortuosity results when the membrane content was too high. This suggests that the acoustical method is not appropriate for the characterization of foam having a large proportion of fully closed membranes and/or very low membrane aperture sizes due to the low amplitude of signal transmitted through the sample at the third microphone and the high values of the viscoinertial frequency (i.e. the inertial regime is out of the audible frequency range). Moreover, in all treatments of acoustical characterization it is assumed that the uid motion is completely decoupled from the solid motion. This is certainly not true if the tortuosity is much larger than one. For example, in the case of porous media with soft and/or lightweight frame, sound propagation through uid can induce frame resonance at specic frequencies as observed in [START_REF] Gaulon | [END_REF]. It should be possible in the future to see a comparison between the predictions of the current model and the results from direct experiments using alternative techniques, such as (i) electrical conductivity measurements [4,29], (ii) superuid acoustics (He II) taking advantage of the fact that the acoustics of He II can become identical with the acoustics of an ideal uid [29,30], or (iii) ultrasonic measurements of velocity dispersion in porous media saturated by various uids [31].

We conclude this section with a summary of this main results:

(1) A unied set of viscous uid ow and electrical conduction calculations have been carried out on a family of idealized three-dimensional model foams with emphasis on membrane eects.

(2) We rst studied ordered cellular foams, for which excellent analytical estimates of the permeability K and tortuosity α ∞ were provided. More particularly, the structure of the equations predicting K and α ∞ is determined based on phenomenological considerations and the relevant coecients tted against numerical results. Moreover, the radius of a perforated membrane r 0 is the most important geometrical parameter governing the viscous uid ow and electrical conduction properties. This is especially true when r 0 /D b → 0.

(3) Estimates of α ∞ were derived based on a bulk and a membrane eect contribution, the later mechanisms dominating the overall eective properties of the ow of the electric current when r 0 /D b → 0.

(4) The relation K/D 2 b ∝ (r 0 /D b ) 3 based on the Sampson law appears to be valid over the entire range of membrane aperture with dierent congurations of closed membranes in Kelvin cells.

(5) These analytical results then enabled the use of pore-network simulations to study the behavior of the tortuosity, including percolation eects in foams containing large number of cells (∼ 16 3 ) having randomly distributed closed membranes. It is quite interesting to see that this to a drastic increase of the tortuosity α ∞ with the closed window fraction, 1 -x ow .

(6) Finally, we examine the relationship between permeability K and tortuosity α ∞ when both the aperture size r 0 /D b and the membrane thickness h m are very small:

K/D 2 b ∝ (r 0 /D b ) 3 and α ∞ ∝ φ/ (r 0 /D b ). v = 0 on ∂Ω (A1b) v and p are Ω -periodic (A1c)
where ∇p m is the macroscopic pressure gradient acting as a source term. Symbols v and p are the velocity and pressure of the uid, respectively. It can be shown that the local eld of the static viscous permeability is obtained from the local velocity eld as v = -k 0 /η∇p m .

The static viscous permeability k 0 is calculated by the standard denitions below,

k 0 = φ k 0 , (A2)
Inertial ow: At the high frequency range with ω large enough, the viscous boundary layer becomes negligible and the uid tends to behave as perfect, since it has no viscosity. Consequently, the perfect incompressible uid formally behaves according to the electrical conduction problem [START_REF] Avellaneda | [END_REF]:

∇.E = 0 with E = -∇ϕ + e, in Ω f (A3a) E.n = 0, in ∂Ω f (A3b) ϕ is Ω -periodic (A3c)
where e is a given macroscopic electric eld, E the solution of the boundary problem having -∇ϕ as a uctuating part, and n is unit normal to the boundary of the pore region. The high frequency tortuosity α ∞ is calculated,

α ∞ = E 2 E 2 (A4)
Appendix B: Pore-network simulations

The general principle of calculation of the electric conductivity of a tube network as a model of porous media can be found in Johnson et al 1987. Details about the use of pore-network simulation for the calculation of foam permeability, can be found in [9]. Hereafter, we give some details about the pore-network simulations for the case of foam electrical conduction.

In the pore-network model, the details of electrical potential uctuations within a single pore are not described in detail. To take into account the current displacement within a pore, the inner volume of a pore is decomposed into 14 parts which are each associated with a node in the electrical pore-network. The nodes are linked by intrapore conductance for two neighbor nodes located within the same pore or by inter-pore conductance for two pores located in two dierent pores linked by an open window. The rst step is to build the conductance network.

We consider, for each node, a unique value of potential without calculating its uctuations inside the pore. At the local scale, the current J j→i from node j to node i is governed by the dierential electrical potential dierence between the nodes ∆V ij = V j -V i :

J j→i = G ij ∆V ij where the coecient G ij is the conductance between the nodes i and j.

At steady state, the sum of current coming from neighbor nodes is equal to zero, leading to: Nv j=1 G ij (V j -V i ) = 0. To generate a current displacement through the sample, a potential dierence is imposed between the top and bottom faces of the sample (V top = ∆V sp , V bot = 0). By considering these boundary conditions, the previous equation can take a matrix form:

GV i = S i (B1)
where V i is a vector containing the electrical potential of nodes (except that the nodes located on top and bottom faces are excluded); G is the matrix dened from local conductances (j G ij along diagonal and G ij elsewhere) and S i is a vector containing zeros except for nodes located on the top face of the sample as neighbors where S i =jtop G ijtop ∆V sp .

When the pore network is linked from top to bottom of the sample, and by considering only the nodes located in the interconnected pores, Ḡ can be inverted and the electrical potential in each node can be calculated from Eq.B1. Finally, the macroscopic current I and the eective conductivity σ e can be calculated as follows:

I = i bot jv J jv→i bot = i bot ,jvi G i bot jvi ∆V i bot ,jvi (B2)
σ e = IH sp /L 2 ∆V sp (B3)

For simple networks, it is possible to derive analytical expressions for the eective electrical conductivity. Due to the symmetries of the PUC congurations (Fig. 7), their equivalent conductance pore-networks can be simplied as shown in Fig. 15. From these equivalent conductance pore-networks, a calculation of the equivalent conductance can be performed leading to the eective electrical conductivity given in Tab. I.

Appendix C: Eective medium model for electric conductance

Here we present an eective medium model for electric conductivity in the case of small membrane aperture size (2r 0 /D b < 0.1). This model is based on the eective medium model of foam permeability detailed in [9]. ≈ n 2

The mean local conductance Ḡ is iteratively calculated from [14,[START_REF] Adler | Porous media : geometry and transports[END_REF]: where the coecient n w depends on the structure of the porous medium (Tab. II). Note that when the fraction of open windows x ow is close to the percolation threshold, it is necessary to modify these equations to consider the structure of the openpore space as shown in [9].

1 Ḡ + n Ḡ = i x i G i + n Ḡ (

Figure 1 .

 1 Figure 1. Two pores connected by a single perforated membrane. Depending on the physical problem under study, the boundary conditions are dierent. For the uid ow case, a pressure drop is applied between the top and the bottom faces, a no slip condition is applied over the membrane and a slip condition is applied on the lateral faces. For the electrical conductivity case, an electrical potential dierence is applied between the top and bottom faces, and the electrical eld is parallel to all the others faces (membrane and lateral faces).

Figure 2 .

 2 Figure 2. Fluid ow conductance G f l of two inter-connected pores as a function of the membrane aperture size for various aspect ratio H/D b . Dots are the FEM results. Full line is plotted by using Sampson's equation.

Figure 3 .

 3 Figure 3. Cylindrical pores: electrical conductance Ge = 1/Re of two inter-connected pores normalized by the electrical conductance without membrane (D 2 b σ f /H) as a function of the membrane aperture size for various aspect ratios H/D b . (a) Comparison between FEM results (dots), predictions of Eq. 6 from Sahu & Zwolak (dashed lines) and Eq. 7 (full lines). (b) Comparison between FEM results (dots), predictions of Eq. 5 from Rosenfeld and Timsit (dashed lines) and Eq. 7 (full lines). The same data are used in graph (b)and in its inset graph.

Figure 4 .

 4 Figure 4. Rectangular pores: normalized electrical conductance Ge = 1/Re of two inter-connected pores as a function of the membrane aperture size for various aspect ratios H/D b .Dots are the FEM results. Dashed and full lines are plotted by using, respectively, Eq. 6 and Eq. 7.

  e. for any L or D b such as H/D b = (H/D b ) * , R e = lim {L/D b }→∞ (R e )). According to Eqs. 5 or 7, it is possible to nd a particular aspect ratio for which the rst term of the spreading resistance expansion, 1 2ro C 1 ro D b , and the bulk resistance H ΨD 2 b

Figure 5 .

 5 Figure 5. Shape of current streamlines in a pore associated to four regions: I. bulk like, II. transition region, III. access region and IV. membrane thickness channel. Note that the rst region is observed only if the pore is long enough (H D b ).

Figure 6 .

 6 Figure 6. Periodic unit cell based on the Kelvin structure (φ = 0.995). Depending on the physical problem considered, the boundary conditions are dierent.

Figure 7 .

 7 Figure 7. Top view from the median plane of the tested congurations showing the positions of the closed membranes.

Figure 8 .

 8 Figure 8. Permeability K of PUC having dierent congurations of closed membranes for various membrane apertures. Dots are the FEM results. Full and dashed lines are plotted by using Eq.8.

3 where d i = 1 Figure 9 .

 319 Figure 9. (a) Flow pattern and (b) its equivalent pore-network scheme. (c) Pore-network scheme to solve the electrical conduction problem by using two kinds of conductances: interpore conductance and intra-pore conductances. (d) Intra-pore conductances for square and hexagonal windows of a Kelvin cell.

Figure 11 .

 11 Figure 11. (a) Electrical conductivity σe and (b) tortuosity α∞of the PUCs as a function of the membrane aperture size for various congurations of closed membranes (K0, K1, K2, K3) with φ = 0.995. The same data are used to the inset graph of (a). Dots correspond to the FEM results. Dashed

Figure 12 .Figure 13 .

 1213 Figure 12. Visco-inertial characteristic frequency fv of Ki conguration as a function of the aperture size calculated from FEM results: on the left abscissa, the frequency is normalized, and on the right abscissa, the frequency corresponds to the case of foam lled by air with D b = 1mm and φ ≈ 1.

4 {Figure 14 .

 414 Figure 14. Pore-network simulations on random Kelvin foam with various fractions of open windows and identical membrane aperture size (φ = 0.995): (a) electrical conductance σe and fraction of open porosity φo/φ as a function of open window fraction, xow, and ratio σe(1)/σe(xow) as a function of the open window fraction xow and (b) ratio σe(1)φo/σe(xow)φ as a function of the open window fraction xow. Note that (b) leads to the ratio α∞ (xow) /α∞ (1) in considering for the pore space volume to use in the denition of the tortuosity, the

Figure 15 . 1 andK1 4D - 1 b- 1 b[ 1 K3 D - 1 b G -1 eh + 3 2 (

 15111112 Figure 15. Conductance pore-networks associated to each PUC conguration.

1 .

 1 C1) with x i the fraction of local inter-pore electrical conductance G i and n = Nv 2 -The macroscopic eective conductivity σ e is then deduced from the mean local conductance Ḡ, σ e = n w Ḡ/D b (C2)

Table II .

 II Coecients nw for various lattices and weakly disordered foam. Note that for SC, BCC or Kelvin lattices, nw

	is isotropic.
	Structure SC BCC (φ 0.88) BCC (Kelvin) disordered
	Nv	6	8	14	2(n + 1)
	nw	1	2	4

Appendix A: calculation of transport parameters

In this section, we briey introduce the boundary value problem used for computing the permeability and the electric conductivity.

Viscous ow: The low Reynolds number ow of an incompressible Newtonian uid is governed by the usual Stokes equations in the uid phase [START_REF] Auriault | Homogenization of coupled phenomena in heterogenous media[END_REF]:


E Stefanutti

L E Bove

‡ G Lelong

M A Ricci

A K Soper

F Bruni

Ice crystallization observed in highly supercooled confined water †

We investigate the state of water confined in the cylindrical pores of MCM-41 type mesoporous silica, with pore diameters of 2.8 nm and 4.5 nm, over the temperature range 160 -290 K by combining small angle neutron scattering and wide angle diffraction. This allows us to observe simultaneously the intermolecular correlations in the local water structure (which shows up in a main water peak around Q = 1.7 Å -1 ), the two-dimensional hexagonal arrangement of water cylinders in the silica matrix (which gives rise to a pronounced Bragg peak around Q = 0.2 Å -1 ), and the so-called Porod scattering at smaller Q, which arises from larger scale interfacial scattering within the material. In the literature, the temperature evolution of the intensity of this Bragg peak has been interpreted as the signature of a density minimum in confined water at approximately 210 K. Here we show that, under the conditions of our experiment, a fraction of freezable water coexists with a layer of non-freezable water within the pore volume. The temperature dependence of all three aspects of our data, as well as comparing the data from two different pore sizes, leads us to conclude that the observed variation in the intensity of the main Bragg diffraction peak is actually caused by a liquid to ice transition in the freezable fraction of confined water.

Introduction

The density maximum of liquid water at 277 K 1 is one of the best known water anomalies as it has a direct impact on our daily life and on our ecosystem. Water, indeed, as few other anomalous liquids, displays a maximum density in the liquid phase, related to a negative thermal expansion coefficient, and implying that the liquid expands upon cooling or shrinks upon heating. As an example, this anomalous expansion plays an important role by only freezing the upper layer in lakes and rivers, leaving life in the bottom layers free to develop. Hydrogen bonding between water molecules and their local tetrahedral arrangement, which characterizes both the liquid and the several crystalline ice phases, are responsible at a microscopic level for this anomaly. The presence of a density maximum suggests the possible existence of a * ; E-mail: fabio.bruni@uniroma3.it † Electronic Supplementary Information (ESI) available:

[details of any supplementary information available should be included here].

See DOI: 10.1039/cXCP00000x/ ‡ Present address: Dipartimento di Fisica, Sapienza Università di Roma, P-le A. Moro 2, 00185 Roma, Italy density minimum in the supercooled region [START_REF] Angell | Water and its anomalies in perspective: tetrahedral liquids with and without liquid?liquid phase transitions[END_REF] , because the density of water decreases rapidly below the temperature of the density maximum and its value can not go below that of ice.

The quest for this density minimum [START_REF] Poole | Density minimum and liquid-liquid phase transition[END_REF][START_REF] Paschek | How the Liquid-Liquid Transition Affects Hydrophobic Hydration in Deeply Supercooled Water[END_REF][START_REF] Xu | Relation between the Widom line and the dynamic crossover in systems with a liquid-liquid phase transition[END_REF] , as well as for the observed thermodynamic anomalies in the supercooled region, has prompted the formulation of different hypotheses to describe undercooled water. Among other proposals, of particular interest is the idea that, at low enough temperatures, there exist two metastable liquid phases, namely low-density (LDL) and high density (HDL) water, whose domain of existence is believed to be separated by a first-order transition line ending in a second critical point [START_REF] Xu | Relation between the Widom line and the dynamic crossover in systems with a liquid-liquid phase transition[END_REF] .

The hypothesis for a liquid-liquid transition in supercooled water has been tested in a number of computer simulation studies. Both the predicted critical temperature and pressure vary quite widely between the different models [START_REF] Poole | Phase beahviour ofmetastable water[END_REF][START_REF] Stanley | Cooperative molecular motions in water: The liquid-liquid critical point hypothesis[END_REF][START_REF] Buldyrev | A system with multiple liquid-liquid critical points[END_REF][START_REF] Abascal | Widom line and the liquid-liquid critical point for the TIP4P/2005 water model[END_REF][START_REF] Gallo | Water: A Tale of Two Liquids[END_REF] . Mean field models which have a second critical point have also been quite successful at predicting thermodynamic properties of water, but give a different prediction for the location of the second critical point compared to the computer simulation results [START_REF] Franzese | Liquid-liquid critical point in a Hamiltonian model for water: analytic solution[END_REF][START_REF] Holten | Thermodynamics of Liquid?Liquid Criticality in Supercooled Water in a Mean-Field Approximation[END_REF] . The mW model of water is particularly interesting in this regard, since it predicts quite successfully several water anomalies, including a density maximum at T > 0 and a density minimum in the supercooled region, but at low enough temperature the ice nucleation time becomes faster than the liquid relaxation time, therefore no liquid-liquid transition is observed [START_REF] Moore | Growing correlation length in supercooled water[END_REF][START_REF] Moore | Structural Transformation in Supercooled Water Controls the Crystallization Rate of Ice[END_REF] . Under the assumption that there is a second critical point in water, the extension of the phase boundary into the single phase region of the P -T plane, the so-called Widom line [START_REF] Franzese | The Widom line of super-cooled water[END_REF] , typically intersects the temperature axis at around 210 -230 K at ambient pressure. Within this scenario, this latter temperature should be close to that of the water density minimum.

Experimental evidence for the presence of a density minimum is hard to obtain. This is because, when cooling the supercooled liquid, the onset of homogeneous crystal nucleation occurs at a temperature, T H , ∼ 235 K. Attempts to explore the supercooled region from below, i.e. by melting the amorphous phases, are hindered by the spontaneous crystallization of ice at ∼150 K. These two temperatures delimit the so-called no man's land, a region of the phase diagram where probing bulk liquid water appears impossible. However recent X-ray laser experiments appear to have penetrated a few degrees K into no-man's land for water [START_REF] Sellberg | Ultrafast X-ray probing of water structure below the homogeneous ice nucleation temperature[END_REF][START_REF] Kim | Maxima in the thermodynamic response and correlation functions of deeply supercooled water[END_REF] . Using femto-second X-ray pulses and the huge intensity available at such facilities, it is possible to capture the structure of individual micron-sized droplets immediately before they crystallize below T H . The authors of that work claim they have seen a peak in the isothermal compressibility, at ∼ 229 K, which, if it were true, could correspond to the crossing of the purported Widom line, although other explanations of the data cannot be discounted [START_REF] Caupin | Comment on "Maxima in the thermodynamic response and correlation functions of deeply supercooled water[END_REF] .

In the past decade or so, in order to circumvent such hurdles, a series of experiments have been performed on water confined in nanoporous materials. Upon confinement, the melting temperature of water is lowered and apparently crystallization is only observed, by means of Differential Scanning Calorimetry (DSC), for pore diameters larger than ∼ 3.0 nm [START_REF] Kittaka | Enthalpy and interfacial free energy changes of water capillary condensed in mesoporous silica, MCM-41 and SBA-15[END_REF][START_REF] Oguni | Calorimetric Study of Water's Glass Transition in Nanoscale Confinement, Suggesting a Value of 210 K for Bulk Water[END_REF] . For water confined in smaller pores the suppression of crystallization would, allow confined water to remain in the deeply supercooled state below the homogeneous crystallization temperature of bulk water. This approach has been used to derive hints of the possible existence of a liquid-liquid phase transition in the no man's land of bulk water, provided that confined water can be assumed to be a realistic model for bulk water [START_REF] Ricci | Similarities" between confined and supercooled water[END_REF][START_REF] Bruni | How safe is to safely enter in the water no-man's land?[END_REF] . Whatever the nature of the surface of the porous matrix (polar, hydrophilic or hydrophobic), the restriction of the accessible space induced by confinement has indeed important structural, dynamical and thermodynamical consequences. Despite many experimental and computational studies, the effects of a specific nanoconfinement on the properties of water are still largely unknown.

The first experimental claim for evidence of a density minimum in confined water was obtained with small angle neutron scattering (SANS) on water confined in the cylindrical pores of MCM-41-S-15 with nominal pore diameter of 1.5 ± 0.1 nm, arranged in a 2D hexagonal lattice [START_REF] Liu | Observation of the density minimum in deeply supercooled confined water[END_REF] . The rationale behind this experiment was to consider the intensity of the (10) Bragg peak, which arises from the 2D hexagonal array of unidirectional pores, as proportional to the difference between the scattering density of the confined liquid and that of the confining substrate, under the assumption of a uniform water density distribution across the diameter of the pore. This assumption plays a crucial role in the interpretation of the data, as we shall see in the following. This first study was followed by other experiments and simulations, which either confirmed the finding or criticized it. Briefly, supporting evidence has been provided by similar small angle neutron or x-ray scattering (SAXS) studies [START_REF] Zhang | Density hysteresis of heavy water confined in a nanoporous silica matrix[END_REF][START_REF] Kamitakahara | Temperature dependence of structure and density for D 2 O confined in MCM-41-S[END_REF][START_REF] Liu | Density and anomalous thermal expansion of deeply cooled water confined in mesoporous silica investigated by synchrotron X-ray diffraction[END_REF] . In addition, crystallization at 220 K has been observed for water confined in MCM-41 with larger pore size (2.4 nm) [START_REF] Liu | Density and anomalous thermal expansion of deeply cooled water confined in mesoporous silica investigated by synchrotron X-ray diffraction[END_REF] . The apparent lack of a liquid water density minimum in this latter case was considered as an overall validation of the DSC data and of the method based on the intensity of the (10) Bragg peak as a probe of water density under confinement. An overview of these findings, discussing both the density minimum and the hypothesized liquid-liquid critical point, can be found in reference [START_REF] Bertrand | Deeply-cooled water under strong confinement: neutron scattering investigations and the liquid?liquid critical point hypothesis[END_REF] . Conversely, atomistic simulations of wide angle scattering data on water confined in MCM material with nominal pore diameter of 1.5 ± 0.1 nm pointed out that the water density profile across the pore is not flat and can change with temperature [START_REF] Mancinelli | Controversial Evidence on the Point of Minimum Density in Deeply Supercooled Confined Water[END_REF] . Other simulations of similar systems also reported a non-uniform density profile, which included the idea that there is a layer of "bound" water near the pore surface different from bulk water [START_REF] Gallo | Dynamic Crossover in Supercooled Confined Water: Understanding Bulk Properties through Confinement[END_REF][START_REF] Lerbret | Water Confined in Cylindrical Pores: A Molecular Dynamics Study[END_REF] . It must be also recalled that a temperature dependent density, in turn, would lead to a change in Bragg peak intensity, even when the overall density across the pore stays constant [START_REF] Soper | Density minimum in supercooled confined water[END_REF][START_REF] Soper | Density profile of water confined in cylindrical pores in MCM-41 silica[END_REF] . SAXS experiments on water confined in MCM-41-S material with pore size ranging from 2.0 to 4.4 nm were used to demonstrate that the scattering data cannot be described as resulting from a homogeneous density distribution of water within the pores [START_REF] Erko | Density minimum of confined water at low temperatures: a combined study by small-angle scattering of X-rays and neutrons[END_REF] . Interestingly, in the same study [START_REF] Erko | Density minimum of confined water at low temperatures: a combined study by small-angle scattering of X-rays and neutrons[END_REF] , the authors reported the presence of a water fraction in the center of the pore with different properties compared to that at the pore wall, and identified a density minimum similar to that found by Liu et al. [START_REF] Liu | Observation of the density minimum in deeply supercooled confined water[END_REF] . However, at odds with [START_REF] Liu | Density and anomalous thermal expansion of deeply cooled water confined in mesoporous silica investigated by synchrotron X-ray diffraction[END_REF] , density minima were apparently observed [START_REF] Erko | Density minimum of confined water at low temperatures: a combined study by small-angle scattering of X-rays and neutrons[END_REF] for all investigated samples, even when water was confined in MCM with the largest pore size.

NMR studies revealed that confined liquid water exhibits a high-temperature crossover of rotational motion at 220â ĹŠ230 K, which coincides with an emergence of solid water, where, most probably, the liquid and solid fractions can be identified with interfacial and internal waters, respectively [START_REF] Sattig | Dynamic Crossovers and Stepwise Solidification of Confined Water: A 2 H NMR Study[END_REF][START_REF] Sattig | NMR studies on the temperature-dependent dynamics of confined water[END_REF] .

Notably, the temperature of the density minimum was found to depend on the pore size, shifting to lower temperatures as the pore diameter decreases [START_REF] Erko | Density minimum of confined water at low temperatures: a combined study by small-angle scattering of X-rays and neutrons[END_REF] . This last result casts some doubt on the use of confined water as a tool to probe the existence of a density minimum in bulk water, because such a minimum should be an intrinsic characteristic of water, not depending on the size of the confining matrix. Very recently, in a series of measurements on the velocity of sound in stretched water (negative pressure) in micron-sized occlusions, a line of compressibility maxima (corresponding approximately to the Widom line) has been identified which proceeds to lower temperature as the pressure is raised [START_REF] Holten | Compressibility Anomalies in Stretched Water and Their Interplay with Density Anomalies[END_REF] . The temperature at which this line would cross the ambient pressure line is unclear at this stage, but, assuming it does continue into the positive pressure region of the water phase diagram, its observation is at least consistent with the liquid-liquid transition hypothesis, though it does not rule out other interpretations either. Hence, as it currently stands, it would be fair to state that the question of the presence of a density minimum in deeply supercooled water and the associated crossing of the Widom line, along with a liquid-liquid transition at lower temperatures, remains unresolved.

The aim of the present study is to shed light on the above issues, exploiting the possibility offered by the NIMROD diffractometer [START_REF] Bowron | NIMROD: The Near and InterMediate Range Order Diffractometer of the ISIS second target station[END_REF] , at the ISIS pulsed neutron source, to investigate a wide Q range, extending from 10 -2 to 30 Å -1 , thus combining traditional SANS and wide angle diffraction in a single experiment. This is of particular importance here, because it means it is possible to look at the peak (around Q = 0.2 Å -1 ) resulting from the (10) plane of the 2D hexagonal arrangement of the water cylinders in the silica matrix and at the main water peak (around Q = 1.7 Å -1 ) at the same time.

As a consequence, we can, simultaneously, tackle the question of the purported density minimum, check the phase of the confined water, and observe the scattering in the very low Q region. This latter Q region provides information on the density fluctuations, that are expected when approaching the liquid-liquid transition temperature.

Materials and Methods

We report neutron diffraction (ND) experiments performed on water confined in MCM-41/C10 (pore size 2.8 nm) and in MCM-41/C18 (pore size 4.5 nm) as a function of temperature, in the range 160 -290 K. Details on the synthesis procedure are described elsewhere [START_REF] Lelong | Effect of surfactant concentration on the Morphology and texture of MCM-41 materials[END_REF] . The smallest pore size was chosen because no water freezing has been observed with differential scanning calorimetry (DSC) in MCM-41 with pore diameter ≤ 3 nm 39,40 . To better exploit NIMROD potentialities and to distinguish features of the dry MCM material from those due to confined water, three samples were investigated for each MCM, namely a dry MCM matrix, one filled with D 2 O, and one filled with a "zero-water" mixture made up of 92% H 2 O and 8% D 2 O. The neutron scattering length of this mixture is such that the average coherent scattering length of the water molecule as a whole is zero, so that in the event that the water is randomly distributed and orientated across the pore there should be no coherent water scattering, which in turn would mean the scattering signal from this sample should be the same as the dry MCM-41. MCM samples were carefully hydrated up to 90% of the total pore volume available, to rule out the possibility of observing contributions to the scattering pattern from water molecules that left the pore volume due to expansion on cooling, either due to the presence of a density minimum or due to crystallization.

Experimental Data and Discussion

Figure 1 shows the raw scattering data collected at T = 284 K for the MCM-41/C10 samples (similar data have been collected for the larger pore size MCM-41/C18). This figure illustrates several interesting features. First, the steep rise at very low Q values due to interfacial (Porod) scattering. Second, the peak around Q = 0.2 -1 , due to the (10) plane of the 2D hexagonal arrangement of water cylinders in the silica matrix. Third, the water peak around Q = 1.7 -1 , which is better visible in the figure insert. Also to be noted is that the zero-water sample is not identical to the dry MCM signal, suggesting a non-uniform distribution of water across the pore. Finally, we note that, at this temperature, the main water diffraction peak is at a markedly lower Q value than is found in bulk water, suggesting that the confined water is on average at lower density than bulk water. Following the same procedure described in [START_REF] Liu | Observation of the density minimum in deeply supercooled confined water[END_REF] , we investigated the temperature dependence of the intensity of the Bragg peak around Q = 0.2 Å -1 , treating it, as in [START_REF] Liu | Observation of the density minimum in deeply supercooled confined water[END_REF] , as a direct measure of the average mass density of water in the pores.

The first step of the procedure requires a subtraction of a power law decay from each diffraction pattern, in order to remove the contribution from interfacial scattering. Details on this procedure and on the data analysis methods adopted to investigate the temperature dependence of the Bragg peak are described in the Supplementary Information.

Figure 2 shows the temperature dependence of the Bragg peak after background subtraction, along with the fits obtained with eq. 4 in the Supplementary Information. We remark the very good agreement between the model and the experimental data. The mass density of D 2 O, extracted as a fit parameter, is plotted as a function of temperature in Figure 3A, and compared to the trend reported in [START_REF] Liu | Observation of the density minimum in deeply supercooled confined water[END_REF] . Even though the absolute values of the mass density of confined D 2 O may not be reliable, yet the adopted normalization does not affect its temperature dependence.

The measured apparent D 2 O density shows a minimum at about 210 K, albeit less marked compared to previous data [START_REF] Liu | Observation of the density minimum in deeply supercooled confined water[END_REF][START_REF] Liu | Density measurement of 1-D confined water by small angle neutron scattering method: pore size and hydration level dependences[END_REF] . However, it should be noted that the accuracy of the density values extracted from the fitting procedure strongly depends on the background subtraction. Figure 3B shows the temperature dependence of the thermal expansion coefficient α P = -ρ -1 m (∂ ρ m /∂ T ) P , calculated from the data in Fig. 3A. If we compare our results (red markers) with those obtained in [START_REF] Liu | Observation of the density minimum in deeply supercooled confined water[END_REF] (black markers), we can notice a good overall agreement. The temperature of the density minimum corresponds to the temperature where α P is equal to zero, and α P is expected to exhibit a maximum crossing the Widom line Fig. 2 Color On-line. Analysis of the Bragg peak intensity with the model given by eq. 4 in the Supplementary Information. Experimental neutron scattering intensity (open symbols) obtained with the D 2 O-hydrated MCM-41/C10 sample and fits (solid lines) according to eq. 4 in the Supplementary Information. To make the figure clearer, only some selected temperatures have been shown. However, the same good agreement between the model and the experimental data is observed at all the investigated temperatures.

above the critical point [START_REF] Liu | Density and anomalous thermal expansion of deeply cooled water confined in mesoporous silica investigated by synchrotron X-ray diffraction[END_REF][START_REF] Liu | Density of hy-drophobically confined deeply cooled water investigated by small angle X-ray scattering[END_REF] ; this implies that both data sets suggest a density minimum in the temperature range 210-215 K, and a crossing of the Widom line around 240 K at ambient pressure.

As discussed above, if below a second critical point supercooled water is regarded as a mixture of two states of the same liquid, differing in density [START_REF] Poole | Density minimum and liquid-liquid phase transition[END_REF][START_REF] Bertrand | Deeply-cooled water under strong confinement: neutron scattering investigations and the liquid?liquid critical point hypothesis[END_REF][START_REF] Soper | Structures of High-Density and Low-Density Water[END_REF][START_REF] Debenedetti | Condensed matter: One substance, two liquids?[END_REF][START_REF] Palmer | Metastable liquid-liquid transition in a molecular model of water[END_REF][START_REF] Wang | Liquid-Liquid Phase Transition and Its Phase Diagram in Deeply-Cooled Heavy Water Confined in a Nanoporous Silica Matrix[END_REF][START_REF] Wang | Evidence of the existence of the high-density and low-density phases in deeply-cooled confined heavy water under high pressures[END_REF] , then density fluctuations above that critical point will reach a maximum close to the Widom line. In principle, this should, in turn, give rise to enhanced scattering intensity at low Q near the transition temperature [START_REF] Hansen | Theory of Simple Liquids[END_REF] . In order to make this point clearer, we have performed a quantitative, model free, analysis of the Porod region, looking at the scattered intensity at selected Q values as a function of temperature. The results (Fig. 4) show a continuous decrease of the scattered intensity with increasing Q. For Q values below 0.059 Å -1 the scattered intensity shows no (or very weak) temperature dependence, suggesting the absence of density fluctuations and, in turn, of two coexisting liquid states. Such a result is remarkable as it clearly provides evidence that the Widom line is not crossed upon cooling confined D 2 O at ambient pressure. This result is at odds with a recent report, based on extremely fast X-ray diffraction in supercooled micron-sized water droplets [START_REF] Kim | Maxima in the thermodynamic response and correlation functions of deeply supercooled water[END_REF] . However, even in this latter work the rise in scattering at low Q is very modest in practice. Interestingly, the drop in intensity with temperature (below T = 240 K and for Q ≥ 0.059 Å -1 ) indicates a reduction in compressibility, as it would be expected for a solidification transition.

At this stage, we appear to have two conflicting pieces of evidence: an apparent density minimum at about 210 K, along with an associated maximum in the thermal expansion coefficient at about 240 K, on the one hand, and the absence of obviously enhanced density fluctuations at low Q at all investigated tempera- m (∂ ρ m /∂ T ) P , as a function of temperature. α P has been calculated with the ρ m data shown on Fig. 3A; those indicated with red symbols have been obtained from our experimental data, while those indicated with black symbols have been calculated using data reported in [START_REF] Liu | Observation of the density minimum in deeply supercooled confined water[END_REF] . Lines connecting symbols are guides for eyes.

tures on the other. To reconcile these two observations, we looked at the short-range correlation of confined D 2 O. In particular, we investigated in detail the temperature dependence of the main water peak around Q = 1.7 Å -1 (see Fig. 1, Fig. 1 insert, and Fig. 5), the rationale being that any modifications of the diffraction profile must come from changes in the state of confined water [START_REF] Morishige | Freezing and melting of water in a single cylindrical pore: The pore-size dependence of freezing and melting behavior[END_REF] .

Figure 5 shows the scattering data in two Q regions, namely in the low Q region of the (10) Bragg peak, and in the region of the main water diffraction peak (1 ≤ Q ≤ 6 Å -1 , right hand graphs), for both MCM-41/C10 (top row) and MCM-41/C18 (bottom row). Starting from the left, plot (A), we notice that the temperature dependence of the (10) Bragg peak in the 2.8 nm pores of MCM-41/C10, already discussed above and shown in detail in Figure 2, goes along with a change in position and reduction in width of the main water peak on cooling below 240 K, as shown in plot (B). The observation of a peak narrowing strongly sug- gests crystallization or solidification of at least a fraction of confined water (more on this issue below). At temperatures below 210 K further changes are relatively small. The situation is similar for the larger pores of MCM-41/C18, plots (C) and (D). Also in this case, the temperature dependent decrease of intensity of the Bragg peak is accompanied by a sharp change in the position and width of the main water diffraction peak between T = 240 K and T = 230 K, with minor changes below 230 K. Once the substantial background from Porod scattering at very low Q has been removed (see Supplementary Information for a detailed description of this procedure), for both MCM materials the intensity of the (10) peak drops as the temperature is lowered: by a factor of ∼ 50% for 2.8 nm pores, and ∼ 33% for 4.5 nm pores. The drop in intensity levels out at the temperature where crystallization or solidification reaches completion in both cases (230 K for 4.5 nm pores, 210 K for 2.8 nm pores); notably this is the temperature at which a density minimum in the confined water has been reported.

T (K) Q = 0.016 -1 Q = 0.020 -1 Q = 0.022 -1 Q = 0.025 -1 Q = 0.031 -1 Q = 0.040 -1 Q = 0.059 -1 Q = 0.083 -1 Q = 0.115 -1
The comparison between MCM-41/C10 and MCM-41/C18 regarding the temperature evolution of the water main diffraction peak, points to a more sudden change with temperature and a sharper main peak at the lowest temperature for water confined in MCM-41/C18. These observations can be accounted for by considering the larger constraints imposed by confinement of water in MCM-41/C10 and the substantial pore size broadening in this latter case. However, the narrowing and shift of the main peak is unequivocal for both pore sizes.

Remarkably, the main diffraction peak of water confined in MCM-41/C10 no longer appears as a single peak below 240 K, but rather looks like the convolution of substructures, located approximately between 1.5 and 1.9 Å -1 . These substructures were not present in the liquid phase, and can be assigned to the formation of stacking faults ice [START_REF] Hansen | Formation and annealing of cubic ice: I. Modelling of stacking faults[END_REF] albeit with a change in intensities due to the likely partially oriented nature of the growing crystallites. The observation of stacking faults ice I formation when water crystallizes at low temperature is a well known phenomenon observed when water freezes from vapor deposited nanodroplets [START_REF] Malkin | Stacking Disorder in Ice I[END_REF] , or from salty solutions [START_REF] Ludl | Structural characterization of eutectic aqueous NaCl solutions under variable temperature and pressure conditions[END_REF] , as well as in other environments favoring deep undercooling [START_REF] Hansen | Formation and annealing of cubic ice: I. Modelling of stacking faults[END_REF] . Interestingly, the intensity of these small features does not appear to grow as the temperature is lowered, suggesting they do not arise from water extruding from the pores as proposed by [START_REF] Kamitakahara | Temperature dependence of structure and density for D 2 O confined in MCM-41-S[END_REF] , but may arise from a small amount of water in larger interstices in the MCM structure. However, the assignment of the freezable fraction of water to a homogeneous ice structure is not straightforward. We report in Figure 6 a comparison of a diffractogram of stacking faults ice I, as shown in Figure 5 of ref. [START_REF] Hansen | Formation and annealing of cubic ice: I. Modelling of stacking faults[END_REF] , and the diffraction pattern of D 2 O confined in MCM-41/C10 and MCM-41/C18 at 209 K. We can easily recognize stacking disordered ice I main peaks at 1.6, 1.7, 2.8 and 4.3 Å -1 . The lack of some -minor -reflections in the black curve in Fig. 6 (D2O in MCM-41/C10) with respect to the reference diffractogram can be attributed to the non perfectly powdered nature of the ice growing in the nanopores. Moreover, stacking sequences can be significantly different depending on the thermal history of the ice sample. The comparison here with stacking faults ice I obtained by crystallization of recovered ice V is therefore only indicative. The difference in stacking sequences (see ref. [START_REF] Hansen | Formation and annealing of cubic ice: I. Modelling of stacking faults[END_REF] ) can also explain why ice I structure ob- [START_REF] Hansen | Formation and annealing of cubic ice: I. Modelling of stacking faults[END_REF] , blue line) with the experimental differential cross-section of D 2 O confined in MCM-41/C10 (pore size 2.8 nm, black line) and in MCM-41/C18 matrix (pore size 4.5 nm, red line). The diffraction patterns have been arbitrarily shifted along the vertical axis in order to have the same offset with respect to the origin of the reference frame.

served in the larger pores of MCM-41/C18 shows a slightly different diffraction signature, which is indeed more similar to the one observed for stacking faults ice I obtained on warming up fast quenched amorphous salty solutions above their glass transition temperature [START_REF] Ludl | Structural characterization of eutectic aqueous NaCl solutions under variable temperature and pressure conditions[END_REF] . This observation suggests that the effect on water structure imposed by the template (namely, the pore surface) depends on pore size, being larger for the smaller pores of MCM-41/C10. In turn, this would imply that a uniform ice structure, starting from a disordered template, can be found only after a number of stacked disordered ice layers is present in the pore volume, in agreement with recent simulation studies [START_REF] Leoni | Effects of confinement between attractive and repulsive walls on the thermodynamics of an anomalous fluid[END_REF] .

At this point, there is strong evidence that confined D 2 O undergoes a crystallization transition around 240 K, and, consequently, the maximum of the thermal expansion coefficient appearing at 240 K (Fig. 3B) is not related to the crossing of the Widom line but rather to the onset of a crystallization event. Based on the uncertainties associated with the measurement of the pore size [START_REF] Jahnert | Melting and freezing of water in cylindrical silica nanopores[END_REF] , being sensibly dependent on the particular method adopted to determine it, one may argue that a pore diameter of 2.8 nm, as in MCM-41/C10, is too close to the reported value of 3 nm representing the largest size below which no water freezing can indeed be observed by DSC [START_REF] Jahnert | Melting and freezing of water in cylindrical silica nanopores[END_REF] . In addition, it has been shown that the freezing temperature of water can be lowered by about 60 K when confined in MCM-41 with pore diameter ∼ 3 nm 40 , a freezing point depression clearly not consistent with our observations. All these issues can be rationalized as follows. The confinement-induced shift of the melting temperature, ∆T p = T 0 -T p , where T 0 is the freezing temperature of bulk water and T p the freezing temperature of water confined, in a series of MCM-41 material, was studied using DSC [START_REF] Findenegg | Freezing and Melting of Water Confined in Silica Nanopores[END_REF] . It was found that ∆T p can be related to the pore radius R by a modified form of the Gibbs-Thomson equation:

∆T p = C GT R -λ (1)
where C GT is a constant value [START_REF] Findenegg | Freezing and Melting of Water Confined in Silica Nanopores[END_REF] , and the parameter λ takes into account the presence of a layer of non-freezable water at the pore walls. In the Supplementary Information the procedure to obtain the thickness of λ based on our diffraction data is described in detail.

We have estimated λ = R -(C GT /∆T p ): for R = 1.4 nm, we obtain λ = 0.38 nm, that is very reasonable and in agreement with literature data for this sort of porous matrix, implying between 1 and 2 mono-layers of non-freezable water at the pore wall [START_REF] Jahnert | Melting and freezing of water in cylindrical silica nanopores[END_REF][START_REF] Schreiber | Melting and freezing of water in ordered mesoporous silica materials[END_REF] . These results suggest a coexistence between freezable water, in the center of the pore, and non-freezable water at the pore wall below 240 K. A rough estimate of the fraction of non-freezable water, φ , corresponding to a layer with thickness λ = 0.38 nm gives φ = 1 -(1λ /R) 2 = 47%. This outcome has several consequences. First, we can safely assume that changes in shape and position of the D 2 O peak around Q = 1.7 Å -1 can be interpreted as a freezing phase transition of a not negligible fraction of water confined in MCM-41/C10 matrix. This, in turn, implies that the density of water inside the pores cannot be considered as constant across the pore, and therefore, the approach based on the Bragg peak intensity to determine the homogeneous density, ρ m , of confined water has no ground. This observation rules out the existence of a minimum density of confined water as determined by SANS technique. The temperature dependence of ρ m reported in Fig. 3A should be then interpreted as resulting from a temperature-dependent rearrangement of a constant number of water molecules in the available pore volume, possibly due to increasing hydrogen-bonding upon cooling.

Conclusions

In conclusion, we have found that a non negligible fraction of water confined in MCM-41/C10 (pore diameter 2.8 nm) crystallizes at 240 K as a stacking faults ice I at the center of the pore. This fraction coexists with about one or two layers of non-freezable water at the pore wall. As a result, variation of water density across the pore cannot be considered as unchanging with temperature, and previous work, [START_REF] Liu | Observation of the density minimum in deeply supercooled confined water[END_REF][START_REF] Zhang | Density hysteresis of heavy water confined in a nanoporous silica matrix[END_REF][START_REF] Kamitakahara | Temperature dependence of structure and density for D 2 O confined in MCM-41-S[END_REF][START_REF] Liu | Density measurement of 1-D confined water by small angle neutron scattering method: pore size and hydration level dependences[END_REF] , which appears to be based on this assumption, needs to be revisited. This conclusion neither implies that a density minimum in supercooled water does not exist, nor that there is no liquid-liquid transition to explain the water anomalies, but rather demonstrates that the claim of a density minimum in confined liquid water at 210 K, as determined looking only at the temperature dependence of the intensity of the (10) Bragg peak of the 2D hexagonal internal structure of the porous matrix, is incorrect. Moreover, extension of the investigated Q region both to lower and to higher Q values, allows us on the one hand to rule out the presence of a maximum in density fluctuations, expected if a Widom line were being crossed in confined water, and, on the other hand, to identify a crystallization or other solidification event starting at about 240 K.

a

  Dipartimento di Scienze, Università degli Studi Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy b Sorbonne Université, Muséum National d?Histoire Naturelle, UMR CNRS 7590, IRD, Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie, IMPMC, 4 Place Jussieu, 75005 Paris, France c ISIS Department, UKRI-STFC Rutherford Appleton Laboratory, Harwell Campus, Didcot, OX11 0QX, U.K.

Fig. 1

 1 Fig.1Color On-line. Raw Differential Cross Section (DCS) data of MCM-41/C10 sample (pore size 2.8 nm), hydrated with heavy water (D 2 O) or with the zero-water mixture (92% H 2 O and 8% D 2 O) compared to data obtained for the dry matrix. The zero-water mixture composition was specifically calculated to suppress the signal from water inside the pores. The peak around Q = 0.2 Å -1 is due to the (10) plane of the 2D hexagonal arrangement of water cylinders in the silica matrix. The peak around Q = 1.7 Å -1 , better visible in the figure insert, is ascribed to water.

Fig. 3

 3 Fig. 3 Color On-line. (A) Comparison between the temperature dependence of the mass density of D 2 calculated in the present work (red symbols) with those reported in 24 (black symbols). (B) Thermal expansion coefficient, α P = -ρ -1m (∂ ρ m /∂ T ) P , as a function of temperature. α P has been calculated with the ρ m data shown on Fig.3A; those indicated with red symbols have been obtained from our experimental data, while those indicated with black symbols have been calculated using data reported in[START_REF] Liu | Observation of the density minimum in deeply supercooled confined water[END_REF] . Lines connecting symbols are guides for eyes.

Fig. 4

 4 Fig. 4 Color On-line. Temperature dependence of the Porod region of DCS obtained with D 2 O confined in MCM-41/C10 sample at several Qvalues. For each Q value the DCS data have been normalized to their value at T =280 K. The figure shows that below Q = 0.59 Å -1 the scattered intensity does not depend strongly on temperature.

Fig. 5

 5 Fig. 5 Color On-line. Comparison between the differential cross sections of D 2 O in MCM-41/C10 and MCM-41/C18 in the low Q-region, plots (A) and (C), and in the range of the main water diffraction peak, 1 ≤ Q ≤ 6 Å -1 , plots (B) and (D). Details on the procedure adopted to remove the substantial background from Porod scattering at very low Q, particularly evident in plot (C), are given in the Supplementary Information.

Fig. 6

 6 Fig.6 Color On-line. Comparison between the diffractogram of stacking fault ice I (data taken from ref.[START_REF] Hansen | Formation and annealing of cubic ice: I. Modelling of stacking faults[END_REF] , blue line) with the experimental differential cross-section of D 2 O confined in MCM-41/C10 (pore size 2.8 nm, black line) and in MCM-41/C18 matrix (pore size 4.5 nm, red line). The diffraction patterns have been arbitrarily shifted along the vertical axis in order to have the same offset with respect to the origin of the reference frame.

Acknowledgments

This work was supported within the CNR-CCLRC Agreement No. 01/9001 concerning collaboration in scientific research at the spallation neutron source ISIS. The financial support of the Consiglio Nazionale delle Ricerche and of the ANR Blanc International program PACS under reference no. ANR-13-IS04-0006-01in this research is hereby acknowledged.


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