On Condensation and Evaporation Mechanisms in DisorderedPorous MaterialsFabien Bonnet, Mathieu Melich, Laurent Puech,† Jean-Christian Angles d’Auriac,and Pierre-Etienne Wolf*

Univ. Grenoble Alpes, CNRS, Institut Neel, 38000 Grenoble, France

*S Supporting Information

ABSTRACT: Sorption isotherm measurement is a standardmethod for characterizing porous materials. However, suchisotherms are generally hysteretic, differing between con-densation and evaporation. Quantitative measurement of porediameter distributions requires proper identification of themechanisms at play, a topic which has been and remains thesubject of intensive studies. In this paper, we compare high-precision measurements of condensation and evaporation ofhelium in Vycor, a prototypical disordered porous glass, to amodel incorporating mechanisms on the single pore levelthrough a semimacroscopic description and collective effectsthrough lattice simulations. Our experiment determines boththe average of the fluid density through volumetric measure-ments and its spatial fluctuations through light scattering. We show that the model consistently accounts for the temperaturedependence of the isotherm shape and of the optical signal over a wide temperature range as well as for the existence ofthermally activated relaxation effects. This demonstrates that the evaporation mechanism evolves from pure invasion percolationfrom the sample’s surfaces at the lowest temperature to percolation from bulk cavitated sites at larger temperatures. The modelalso shows that the experimental lack of optical signals during condensation does not imply that condensation is unaffected bynetwork effects. In fact, these effects are strong enough to make most pores to fill at their equilibrium pressure, a situationdeeply contrasting the behavior for isolated pores. This implies that, for disordered porous materials, the classical Barrett−Joyner−Halenda approach, when applied to the condensation branch using an extended version of the Kelvin equation, shouldproperly measure the true pore diameter distribution. Our experimental results support this conclusion.

1. INTRODUCTION

Barrett−Joyner−Halenda (BJH) method1 is routinely used todetermine the pore diameter distribution of mesoporousmaterials from the measurement of sorption isotherms. Itrelies on the fact that, for a cylindrical pore of radius R and awetting fluid, the vapor pressure corresponding to phaseequilibrium is shifted below the saturated vapor pressure, theshift increasing when the radius decreases. For wide pores(typically larger than 100 nm), this shift can be understood interms of capillarity through Kelvin’s law. For narrower pores,the situation is more complex: the interaction with thesubstrate should be taken into account, and below a fewnanometers, the atomistic structure of the fluid and thesubstrate should also be considered. Beyond these effects, themain difficulty to apply the BJH scheme is the existence of anhysteresis between condensation and evaporation below acritical temperature Tch, smaller than the bulk criticaltemperature Tc. Below Tch, a central question is then toidentify which of the condensation and evaporation branches,if any, corresponds to a phase equilibrium.Thanks to the combination of experiments and analytical or

numerical theoretical approaches, the situation is today well-

understood for single cylindrical or slit pores or regular systemsof pores (for reviews, see refs2,3). For cylindrical pores, studiesusing local4 or nonlocal5 density functional theories haveshown that, above a diameter of about several nanometers, amacroscopic capillary description of the condensation andevaporation phenomena is reasonably accurate. For a perfectlywetting substrate and a cylindrical pore, condensation firstproceeds by the adsorption of a film on the inner surface of thepore because of substrate attraction. This film becomesmetastable with respect to the fully filled state above theequilibrium pressure, corresponding to its coexistence with aliquid region, and fully unstable above a spinodal-likepressure.6 In the absence of thermal activation, condensationin a single pore open at both ends then proceeds through thespinodal instability, whereas evaporation takes place atequilibrium by recession of the meniscus. In this situation,the equilibrium branch to be considered in the BJH theory isexpected to be the evaporation one.

Received: December 24, 2018Revised: March 4, 2019Published: March 13, 2019

Article

pubs.acs.org/LangmuirCite This: Langmuir XXXX, XXX, XXX−XXX

© XXXX American Chemical Society A DOI: 10.1021/acs.langmuir.8b04275Langmuir XXXX, XXX, XXX−XXX

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However, coupling between different pores modifies thissimple picture. For a wide pore (cavity) surrounded bynarrower pores (constrictions), condensation in the con-strictions can trigger condensation of the cavity below itsspinodal pressure (advanced condensation). Inversely, theenergy cost for creating a liquid−vapor interface blocks theevaporation inside the cavity at its equilibrium pressure.Evaporation is delayed until thermal activation allows thecreation of such an interface (cavitation) or the constrictionsthemselves empty, creating a meniscus at the cavity entrance,which can then recess. Both possibilities have beendemonstrated in patterned silicon wafers7,8 and in orderedporous materials consisting of regular networks of cavitiesconnected by narrow constrictions.9−11

The case of disordered porous materials is more complex. Ifcavitation is negligible, evaporation should proceed by acollective invasion percolation mechanism from the surfa-ces.12−14 Such a mechanism has been experimentallydemonstrated by room temperature light-scattering experi-ments on hexane in Vycor, a prototypical disordered glassconsisting of interconnected cylindrical pores.15 In theopposite case, it has been predicted that evaporation couldalso proceed by invasion from the cavitated pores.16 In bothcases, the pore diameter distribution cannot be simplyretrieved from the evaporation branch of the isotherm.Experimentally, based on an analysis of the evolution of the

isotherm shape with temperature,17 a temperature-drivencrossover between pure percolation and cavitation has beenreported for evaporation of nitrogen in Vycor. However, latermeasurements using helium as a fluid and light scattering as aprobe have suggested that the temperature dependence of thescattered intensity was a better marker of a crossover than thatof the isotherm shape.18

On the theoretical side, local mean field density functionaltheory (LMF-DFT) has been successfully applied to describecondensation and evaporation in disordered porous materials19

such as silica aerogels and Vycor. In the first case, the predictedbehavior20 accounts well for later measurements.21 Calcu-lations on Vycor22 based on a realistic description of the actualporous structure of Vycor remarkably reproduce not only theexperimental isotherm shape and its temperature dependencebut also the observed shape of scanning curves. Furthermore,Monte Carlo simulations based on the same description doevidence the occurrence of cavitation for specific simulationparameters.23

Despite this impressive success, two reasons justify toexplore an alternative theoretical description. First, the LMF-DFT approach assumes a known microstructure. Inferring poresizes for an arbitrary porous material would thus have a veryhigh computational cost. Second, at least in its present state,LMF-DFT does not provide a general physical picture of theinterplay between basic mechanisms. In this paper, we developsuch a picture based on a multiscale approach, benchmark itagainst experimental measurements, and discuss how its resultscan be used to obtain pore diameter distributions fromsorption isotherm measurements.

2. METHODS2.1. Experimental Section. We summarize here our exper-

imental methods. Details can be found in the Supporting Informationand in refs.24,18 Our sample is a 14 mm diameter, 4 mm thick disk ofVycor 7930 contained in an annular copper cell closed by twosapphire windows. The cell was regulated between 3 K and Tc ≈ 5.2 K

in an optical cryostat.18 Condensation and evaporation of helium werecontrolled by sweeping the temperature of a gas reservoir connectedto the cell.25 The cell pressure, the confined fluid density, and thecorresponding liquid fraction were obtained as described in theSupporting Information. The high accuracy of these measurementswas central to study the shape and closure of the hysteresis loop indetail.

Isotherms were measured for about 20 temperatures ranging from3 K up to 5.22 K, slightly above Tc = 5.195 K. We typically scan thecondensation or evaporation branch of the hysteresis loop in 10−20 hbelow 4 K and several hours near the closure temperature of thehysteresis loop, Tch ≈ 4.65 K. We checked such flowrates to be smallenough to avoid thermal gradients within the sample because of theheat released (respectively absorbed) by the condensation (respec-tively evaporation) process. Relaxation effects due to thermalactivation were studied by stopping the flowrate at various fillingfractions and measuring the pressure relaxation at a fixed reservoirtemperature over several hours.

Sorption isotherms only give access to the fluid density averagedover the Vycor sample. Spatially resolved information was obtained bysimultaneous light scattering measurements. The sample wasilluminated with a thin laser sheet at a wavelength of 632 nm under45° incidence with respect to the common disk and windows axis,with polarization perpendicular to the scattering plane. Theilluminated slice was imaged using two CCD cameras at oppositeobservation angles (45° or 135° from the incident direction). In ourregime of single scattering, the signal of each CCD pixel measures thelight scattered in the direction of observation by the correspondingimaged region and probes the static spatial fluctuation of the fluiddensity at the considered point on a spatial scale set by theobservation angle.

2.2. Modelization. 2.2.1. Description of the Model. To modelizethe effect of connections between pores, we combine a realisticcalculation of the condensation and evaporation processes within asingle cylindrical pore26 with a lattice description of the pore network.This approach compromises between those used by Guyer andMcCall,27 to discuss evaporation by percolation in three-dimensional(3D) disordered porous networks, and by Puibasset28 to modelcondensation and evaporation in a single corrugated cylindrical pore.Our treatment of the single pore is more realistic than in Guyer andMcCall’s approach, as it includes, in particular, the key effect ofthermal activation. On the other hand, based on the assumption that,locally, the fluid obeys the bulk equation of state, it is less exact thanthe grand canonical Monte Carlo approach used by Puibasset.However, the gain in simplicity allows to modelize a complex 3Dnetwork. Our approach, while similar in spirit, differs from that usedin ref 16 by two important aspects. On the single pore level, it takesinto account the size dependence of the homogeneous cavitation,18 acrucial fact not recognized by the authors of ref 16. On the networklevel, we consider a cubic rather than a Bethe lattice. Although thisdoes not allow for analytical calculations, it provides a more realisticdescription of an actual porous material.

Our single pore model predicts three temperature-dependentcharacteristic pressures for a cylindrical pore of radius R. Thesepressures are (i) Peq(R), the pressure corresponding to the liquid-filmcoexistence within the pore, (ii) Pact(R), the pressure at which thepore fills by thermally activated nucleation of a liquid bridge, and (iii)Pcav(R), the cavitation pressure below which the pore empties bythermally activated nucleation of a vapor bubble. These pressuresmust be understood as the vapor pressure of the reservoir (far awayfrom the pore walls). They are obtained from the correspondingfugacities through the equation of state of helium. The latter arepredicted by the model from the fluid properties at temperature T, thepore radius, the van der Waals attraction between the fluid and thesubstrate, and the barrier height which can be overcome on theexperimental time scale. For simplicity, we will speak below ofpressure P rather than fugacity; but it should be remembered that, forall theoretical curves, P is actually a fugacity, with P = 1 correspondingto the saturated vapor pressure Psat. Accordingly, comparisons of the

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model and the experiment will be made by converting theexperimental pressures to fugacities.Figure 1a shows the radius dependence of these pressures for three

temperatures computed for a van der Waals interaction of 1200 K Å3

when a barrier 38 kBT high can be overcome. Below a critical radiusRc(T) and a pressure P3, the three pressures coincide, correspondingto a reversible condensation−evaporation process. Above Rc(T),which increases with temperature, the three pressures differ. For largeenough pores, Pcav(R) approaches the bulk cavitation pressurePcav(∞), whereas Pact(R) and Peq(R) tend to unity. Note thatPact(R) and Pcav(R) behave quite differently when thermal activation isinhibited by decreasing the temperature. While cavitation issuppressed, condensation below the saturated vapor pressure remainspossible through the spinodal instability of the adsorbed film.6,26 Thisdifference will turn out to be responsible for the different opticalbehavior between evaporation and condensation.For the same pore within a network, the actual evaporation and

condensation mechanisms depend on the surrounding pores. Thecoupling mechanisms described in the introduction can be translatedinto four evolution rules. The two first ones apply to whatever thestate of the pore’s neighbor, whereas coupling comes into play for thetwo last ones.

• (i) For P ≤ Pcav(R), the pore is empty (but for the adsorbedfilm).

• (ii) For P ≥ Pact(R), the pore is filled.• (iii) For Peq(R) < P < Pact(R), the pore fills and only if it has at

least a filled neighbor.• (iv) For Pcav(R) < P < Peq(R), the pore empties if and only if it

has at least an empty neighbor.

The two first rules give rise to hysteresis if T is smaller than thehysteresis closure temperature Tch, such that the critical radius Rc(Tch)is the radius of the largest pore. Above Tch, Pcav = Peq = Pact for allpores, and the sample’s configuration is simply determined by rules(i) and (ii). Below Tch, the four rules allow to determine the systemstate for any given pressure history. Our model is thus described, asmany out-of-equilibrium models, by a cellular automaton rather thanby a hamiltonian. In so far as the barrier height that can be overcomeis fixed, Pcav(R) and Pact(R) do not depend on time so that our modelhas no relaxation dynamics.These rules can be applied to any geometry of the pore network.

To be specific, we consider a cubic lattice of lattice constant a andlinear size L = Na with N = 512. a is a free parameter, which isexpected to be of the order of the experimental correlation length of

the Vycor pore network. Each site of the lattice is occupied by thehexapod unit shown in Figure 1b. The pores of the hexapod have aradius randomly drawn at each site according to a Gaussiandistribution. The pore radii are thus independent, identicallydistributed (iid) variables. The length of each pore of the hexapodbeing equal to a/2, the volume of an hexapod unit scales as R2. Forthe sake of simplicity, we will ignore the adsorbed film and the liquidcompressibility, so that each hexapod can only be in two states, empty(0) or filled (1). Contact with the vapor phase at the sample’s surfacesis simulated by adding an extra layer of sites around the sample andfixing their state to 0. Alternatively, we can suppress the influence ofthe surface by using periodic boundary conditions. We can also probethe roles of cavitation and pore coupling by comparing the fullsolution to those, respectively, obtained by forcing Pcav to zero ordisregarding rules (iii) and (iv).

For the chosen geometry, the neighboring pores are directlycoupled (Figure 1b). In contrast, former studies13 have consideredlarge cavities (sites) separated by narrow windows (bonds).Evaporation in the absence of thermal activation is then a bondpercolation problem instead of a site percolation problem in ourmodel. Beyond this difference, our model is oversimplified withrespect to real Vycor. First, pores of different radii only branch axiallyand not laterally. Second, for our cubic lattice, a pore has sixneighbors, a larger number than suggested from transmission electronmicroscopy pictures.29 Third, neighboring pores have uncorrelateddiameters. These simplifications will affect the percolation threshold,so that we cannot hope for a detailed quantitative agreement betweenthe model and the experiment. However, as we will show, ourdescription allows to capture the qualitative features of theexperimental results, in particular, the temperature evolution of theoptical signal.

2.2.2. Obtaining the Fluid Configuration. To predict thecondensed fraction and the optical signal for a given pressure historyand temperature, we first have to determine the configuration for thetarget pressure. With that aim, we can use two equivalent numericalmethods. The first one, used in the present work, is described in theSupporting Information (§:S31). Here, we describe an alternatemethod, based on the concept of Bernoulli clusters, as this concept isuseful to discuss our numerical results. We consider only the case of amonotonous pressure history; but, as described in the SupportingInformation, the method can be generalized to an arbitrary one. Themethod starts by marking, at the target pressure of interest P, the sitespotentially filled for condensation (respectively empty for evapo-ration), that is, the sites of radius R such that Peq(R) < P (respectivelyPeq(R) > P). We then group these sites into maximally connectedclusters (the so-called Bernoulli clusters) by using a fast algorithm.30

According to the coupling rules (iii) or (iv) above, all sites within thegiven cluster are in the same state. Among these potential clusters, theeffectively filled clusters for condensation are those containing at leasta site with Pact(R) < P. According to rule (ii), this site (denominatedgerm below) will be filled (state 1), thus triggering the filling of thefull cluster. For evaporation, the effectively empty clusters are thosecontaining a vapor site (state 0). Such sites are either the boundarysites or the cavitated sites (Pcav(R) > P). Correspondingly, we havetwo kinds of germs and two selection rules instead of one forcondensation. Once the relevant germs at pressure P are identified, wefind the corresponding selected clusters and hence the configuration.In the remaining of this paper, clusters of sites potentially filled (forcondensation) or empty (for evaporation) will be referred to aspotential clusters, whereas the clusters effectively filled (respectivelyempty) will be referred to as selected clusters. For both condensationand evaporation, some clusters can percolate throughout the samplewhen the fraction of sites potentially filled or empty reaches 31%,which is the percolation threshold for site percolation on a 3D cubiclattice. Introducing Rperc, the radius such that 31% of the pores have alarger radius, we define the percolation pressure for evaporation, Pperc= Peq(Rperc).

2.2.3. Computation of Isotherms. From the configuration, weobtain the total amount of the liquid as a function of the externalpressure, that is, the isotherm. In our lattice model, the volume of a

Figure 1. (a) Three single pore characteristic reduced pressures(actually fugacities) Pcav, Peq, and Pact. P = 1 corresponds to thesaturated vapor pressure Psat. Their dependence on the pore radius,shown for three temperatures, is computed from ref 26 and used as aninput for the coupled model. The bullet corresponds to thetemperature-dependent critical radius Rc(T) and the triple pointpressure P3. (b) Modelization of the pore network by a cubic lattice ofsites occupied by hexapod building blocks. The picture shows twoadjacent building blocks meeting at the middle of the bond.

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pore is proportional to R2, but other distributions could be chosen toevaluate the impact of the actual geometry for a fixed connectivity. Asexplained above, we neglect the film contribution and the effect ofliquid compressibility. These approximations affect the isothermshape little, so that qualitative, if not quantitative, comparison to theexperimentally measured isotherms will be possible.2.2.4. Computation of the Structure Factor. To compare our

model to our light-scattering measurements, we also compute thestructure factor of the local liquid density. Within our lattice model,this quantity, to which the scattered intensity is proportional, isobtained from the fluid configuration as

∑= ıF aS q n qn( ) ( )exp( )n

2

x y z, , (1)

Here, n is the site of integer coordinates (nx, ny, nz) (nx, ny, nz runbetween 0 and N − 1), q = (qx, qy, qz) is the transfer wave vectordefined by the directions of incidence and observation and the lightwavelength, a is the lattice constant, and F(n) is the scatteringamplitude at site n.31

We approximate F(n) as

α δ α= [ − + ]F a a CR CRn( ) ( )n n n2 2

Si2

He (2)

where Rn is the site radius, CaRn2 is the associated void volume, δn = 0

(respectively δn = 1) if the site is empty (respectively filled), and αSiand αHe are the silica and helium polarizabilities, respectively, relatedto the corresponding dielectric constants ϵ through the Clausius−Mosotti equation, α = ϵ −

ϵ +12. Since ϵ = 1.057 for helium and 2.4 for

silica, we use αSi ≈ 1 and αHe ≈ 0.05. For non-zero q, S(q) reduces to

∑ α δ α∝ − ıR aS q qn( ) ( )exp( )n

n n2

Si He

2

(3)

Because Rn is randomly distributed, this expression, in contrast tothat corresponding to the simpler choice Fn = δn, accounts for thefinite scattering by the bare silica network (δn = 0). It also accountsfor the experimentally observed reduction, due to index matching, ofthe scattered intensity for the filled sample (δn = 1).For computation, we restrict q to the reciprocal lattice, thus

allowing the use of the discrete Fourier transform. We thus write

π=q 2 QNa, where Qx, Qy, and Qz are integers between 0 and N − 1.

The modulus Q of Q then corresponds to the scattering angle θthrough Q = 2Na sin(θ/2)/λ. S(Q), now independent on a, isevaluated using the fast Fourier transform algorithm provided at www.fftw.org.The result strongly fluctuates as a function of Q, corresponding to

the so-called speckle phenomenon. In an actual structure factormeasurement, these fluctuations are washed out by using a detectorcovering many speckle spots around a given scattering direction.Therefore, we average S(Q) over the direction of Q to keep only themodulus dependence. To this aim, we sum the computed structurefactor over all Q vectors located in a shell of width 3 centered on

= +Q Q( 1/2) 3 , where Q is an integer. The final isotropizedoptical signal S(Q)32 , that we will compare to the experiment is thereal piecewise constant function

∑=S QN Q

S Q( )1( )

( )i

2

where the sum runs over the shell and N(Q) is the number of pointsin the shell.In the case where the states of the sites are not correlated, for

example above Tch where Pcav = Pact = Peq for all sites, the δns are iidrandom variables, so that ⟨S(q ≠ 0)⟩ = ⟨F(n)2⟩ − ⟨F(n)⟩2.Because αSi ≫ αHe, this expression is dominated by the silica signal

and hardly depends on the liquid fraction. Obtaining a signal largerthan for bare silica is only possible if the site occupancy is spatially

correlated. As we will show, this requires not only pore coupling butalso a selection process of the resulting clusters.

3. RESULTS AND DISCUSSION3.1. Experimental Results. 3.1.1. Isotherm Measure-

ments. Figure 2 shows the temperature evolution of isotherms

from around 3.4 K to above Tc. Below approximately 4.4 K, thehysteresis loop is triangular, the evaporation branch presentinga sharp kink followed by a steep section as the pressure isdecreased. Such a H2-type behavior has been previouslyreported in Vycor for a large variety of fluids (e.g., argon,33,34

nitrogen,17,34,35 hexane,15 and water36) and often discussed asan evidence for the invasion percolation evaporationmechanism.12−14 Above 4.4 K, the loop becomes lesstriangular, and at a flowrate of 0.3 STPcc/min, closes around4.65 K (see Figure S2 for a zoom). This closure temperatureTch is about 10% lower than the bulk critical temperature Tc. Itis larger than previously observed by Brewer and Champeney37

for helium in Vycor, who found no hysteresis at 4.2 K. Thisdifference could be due to a smaller resolution in their case.38

Loop closure below Tc has been previously reported inVycor for other fluids (nitrogen,17 carbon dioxide39) or otherporous glasses.40,41 However, to our knowledge, our experi-ment is the first one to follow the loop closure with such a highaccuracy. In particular, we clearly show that the evaporationisotherm becomes less steep as temperature increases up toTch, in contrast to the previous conclusion of Morishige fornitrogen in Vycor.17 Above Tch, the isotherm is fully reversible,with no sharp kink but an elbow which becomes fainter as thetemperature increases up to Tc. On the basis of the theoreticalcalculations for a distribution of independent pores,26 weinterpret this elbow as the pressure where all pores are fullyfilled with the liquid.Two mechanisms can account for the closure of the

hysteresis loop at Tch < Tc. The first one is a shift of thecritical temperature because of confinement,42 which wouldsuppress the surface energy and hence the energy barrier. Thesecond one is thermal activation, which would allow toovercome the energy barrier at equilibrium, as first proposedby Machin.41 Similar to Burgess et al.,39 we can extract fromour isotherm measurements a “coexistence curve” representing,as a function of temperature, the fluid density at the lower and

Figure 2. Selected condensation−evaporation isotherms of helium inVycor from 3.4 K up to above Tc, showing the average density of theconfined fluid as a function of P/Psat, the pressure relative to thesaturated pressure. The hysteresis loop changes in shape and extent asthe temperature is increased. It closes between 4.6 and 4.7 K, wellbelow the bulk critical temperature.

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upper ends of the hysteresis loop. As discussed in theSupporting Information, this analysis shows that the hysteresisloop is nonuniversal, unlike expected for a confinement-induced shift of Tc.

42 In contrast, our measurements areconsistent with a thermally activated closure mechanism. First,according to our single pore modelization (Figure 1a), aclosure temperature at Tch ≈ 4.65 K corresponds to a maximalpore radius of the order of 7 nm. As we will find below, thisvalue agrees with the measured pore diameter distribution(Figure 10). Second, the thermal activation scenario is directlysupported by the relaxation measurements we now describe.3.1.2. Relaxation Effects. Relaxation effects along the

hysteresis loop were studied, as described in the methodssection. When the reservoir temperature sweep is stopped atsome fixed set point within the hysteresis loop, we observeslow relaxation of the pressure. On the evaporation branch, thepressure increases corresponding to the evaporation of the celltoward the reservoir and reversely on the condensation branch.Figure 3 shows such measurements for a temperature of

4.34 K. The relaxation is obvious on the evaporation branch,whereas it is hardly visible on the condensation branch.a Theinset of Figure 3 shows that the relaxation time scale is longerthan several hours for both branches. This long time scale andthe fact that relaxation is only observed within the hysteresisloop show that it cannot be due to the transport of mass orheat within the sample.As discussed in the Supporting Information, relaxations are

observed for temperatures ranging from 3.16 K up to Tch andare consistent with thermally activated condensation andevaporation for cylindrical pores with a diameter distribution.In the temperature region where the loop closes, relaxationmakes Tch dependent on the waiting time (see Figure S3 of theSupporting Information). This strengthens the above inter-pretation of the hysteresis closure in terms of a thermallyactivated mechanism.We note that long time scale relaxations for condensation

and/or evaporation in Vycor have also been reported usingultrasonic or nuclear magnetic resonance (NMR) measure-ments, respectively.15,43 The relaxing variables being different,we cannot determine whether these relaxations are related ornot. NMR relaxations were interpreted as the signature of athermally activated reorganization of the adsorbate. In ourcase, the temperature and pressure dependence of therelaxation times is consistent with the idea that this

reorganization takes place on a local scale, on the level ofthe single pore.

3.1.3. Optical Signal. The evolution of the isotherm shapewith temperature can be qualitatively explained by assumingthat condensation and evaporation take place independently ineach pore.18 This is surprising as Vycor’s pores are highlyconnected, and coupling effects are to be expected. As we nowshow, light-scattering measurements do reveal such couplingeffects during evaporation.Figure 4 shows the images of the Vycor sample during

evaporation at 3 K. In the initial state (image 1), the sample is

filled and the light is only scattered by the small-scale silicaheterogeneities. Right at the kink in the isotherm, the samplegets slightly brighter. As evaporation proceeds further, thebrightness increases nearly uniformly, reaching a maximum atan average liquid fraction Φ around 0.9. At the lower end ofthe hysteresis loop (image 5), the image is slightly brighterthan for the filled sample. The difference reflects the indexmatching by helium.At its maximum, the image brightness is about twice that for

the filled sample. Because the dielectric constant of helium ismuch smaller than that of silica, this implies that the vapordistribution is correlated over distances much larger than thesilica correlation length. Such long-range correlations are inagreement with the fact that, at the maximum, the scattering isslightly forward peaked, being larger for 45° than for 135°.When the temperature increases, the brightness enhancementgradually decreases, and scattering becomes isotropic, con-sistent with a decrease of the correlation range. This isquantified in Figure 5a, which shows how the scatteredintensity, averaged over the rectangle shown in Figure 4,evolves with the liquid fraction from 3 to 3.75 K.44 In contrast,in the same temperature range, the scattered intensity duringcondensation continuously decreases from the empty Vycorvalue to the index matched value in the liquid state. Thisdifference between condensation and evaporation disappearsfor larger temperatures, that is, at 4.34 K and beyond. In thistemperature range, the dependence on fraction of the scatteredintensity is the same for condensation and evaporation, eventhough the isotherm is hysteretic below 4.6 K.This temperature evolution of the scattered intensity is a

result of cavitation. Because at all temperatures, the pressure

Figure 3. Relaxations of pressure and fluid density within thehysteresis loop at 4.34 K recorded after stopping the condensation orthe evaporation. The inset shows the time dependence of the pressurerelaxation.

Figure 4. Successive (false color) images of a Vycor slice duringevaporation at 3 K. The slice is imaged at 45° from the illuminatinglight sheet. The positions of the different images on the evaporationisotherm are indicated on the graph. Averaging the brightness over thewhite rectangle gives the optical signal shown in Figure 5a.

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Peq(R) monotonously increases with radius R, the sequence ofpotential clusters during evaporation does not depend on thetemperature. If evaporation would only proceed by vaporpercolation from the sample’s surfaces,13 the selection rule willalso be temperature-independent, and hence the successiveconfigurations and the optical signal would be identical for anytemperature, in contrast to our observations.Light-scattering measurements were also performed during

the above-described relaxation experiments. At the loweststudied temperature (3.16 K), we found that, close to the kinkof the evaporation isotherm, the optical signal grows over thesame long time period during which the pressure relaxes (seeSupporting Information). This shows that this signal iscavitation driven. However, evaporation by cavitation alonecannot produce an optical signal, in contrast to what wasapparently assumed in the LMF-DFT study of evaporation inVycor.23 Indeed, for pure cavitation, the vapor sites would notbe spatially correlated. As explained in the modelizationmethods section, this would result in a negligible dependenceof the optical signal on the liquid fraction. Hence, ourexperiment demonstrates that both pore coupling andcavitation should be taken into account to properly describeevaporation in Vycor at all temperatures. In the following, weshow that our model incorporating both effects indeedaccounts for all our experimental observations.3.2. Analysis. 3.2.1. Comparison of Experimental Results

to Simulations. Figure 6 compares the experimentallymeasured isotherms to those computed within the model.

For the experimental curves, the pressure scale of Figure 2 isconverted to fugacity using the bulk equation of state ofhelium. The theoretical curves have been computed using aGaussian distribution truncated to positive values, with therespective mean and width, 3 and 1.4 nm, chosen toapproximately reproduce the characteristic pressures forcondensation and evaporation, and the hysteresis closuretemperature Tch ≈ 4.6 K. Figure 6 shows that our modelproperly reproduces the triangular (H2) shape of the hysteresisloop for the three temperatures shown.45

The optical signal predicted by the model, for the samedistribution as above, is plotted in Figure 5 for Q = 5 and 12.These values correspond to the experimental angles of 45° and135° when choosing a lattice constant a = 15 nm, reasonablyclose to 30 nm, the measured correlation length of the Vycorstructure.29 Comparison of Figure 5a,b shows that the modelremarkably accounts for the very different behavior of theoptical signal during condensation and evaporation. Startingwith condensation, it predicts the signal to be very weak,temperature- and angle-independent, and decreasing withfraction from its empty value down its lower filled value.This is in agreement with the experiment. Next, forevaporation, the model reproduces the four main featuresobserved in the experiment: (i) for the largest temperatures(4.34 K and above), the identity of the signal to that observedfor condensation; (ii) for smaller temperatures, the gradualemergence of a peak of intensity for fractions around 0.8−0.9;(iii) the observation of an anisotropy at the lowest temper-atures; and (iv) the maximal enhancement of the signal, about

Figure 5. Comparison of the (a) measured and (b) predicted opticalsignals for condensation (C) and evaporation (E) between 3 and4.34 K. The signals are plotted vs the liquid fraction Φ for twoscattering angles (θ = 45° and θ = 135° corresponding to Q = 5 and12 for the model). θ = 135° data are shown only when they differfrom the θ = 45° data.

Figure 6. Comparison of (a) the selected experimental isotherms to(b) the model. For both (a,b), the liquid fraction Φ is plotted as afunction of fugacity. For the theoretical isotherms, the horizontalarrows indicate Pperc and the vertical arrows indicate Pcav

max.

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twice the empty Vycor contribution at the lowest temperature(this specific value being fixed by the ratio αHe/αSi). Beyondthe difference in the fraction scale (due to the filmcontribution), the quantitative differences probably reflectthe approximations made by the model.We conclude that our approach captures the important

ingredients of the phenomena at play. We now discuss theunderlying physics.3.2.2. Analysis of the Model’s Predictions. From the model,

we can construct the P−T diagram shown in Figure 8, whichpredicts the nature of the evaporation mechanism dependingon the temperature and pressure. This diagram (which isdetailed in the Supporting Information) allows to define acrossover temperature Tco such that, below Tco, evaporation isessentially governed by percolation from the surfaces, andabove, by percolation from the cavitated pores. Tco is definedby the intersection of two lines, Pcav

max, the cavitation pressure ofthe largest pores (for our distribution Pcav

max ≈ Pcav(∞)), andPperc, the percolation pressure where (potential) clusterscontact to form a larger cluster percolating throughout thesample.For the parameters used above, Tco = 3.35 K. Figure 7a,b

shows the hysteresis loops computed for two temperatures,respectively, below and above Tco, 3 and 3.7 K.Below Tco, evaporation above Pcav

max is fully controlled by thesurfaces. Only the potential clusters contacting these surfacesempty as the pressure is decreased. The liquid fraction starts

decreasing approximately at the equilibrium pressure of thelargest pore,Peq

max, as the clusters contacting the sample’ssurfaces grow in size, due to the increase of the correlationlength for percolation, ξ. ξ being finite, their localizedcontribution however vanishes in the limit of an infinitelylarge sample. This corresponds to the dark pink region inFigure 8. In contrast, when the pressure reaches Pperc, ξ

diverges, and a cluster percolates through the sample. Forsmaller P values, the number of sites in the percolating clusterrapidly grows and the fraction sharply drops (pink region inFigure 8). Decreasing the pressure further, P reaches Pcav

max andcavitation comes into play, decreasing the liquid fraction withrespect to the case of pure invasion from the surfaces (inset ofFigure 7a and white region in Figure 8). For P < Pcav

max, the insetof Figure 7a shows that using periodic boundary conditionsand hence suppressing invasion from the surfaces negligiblyaffects the liquid fraction. This is due to the fact that the largestcluster contacting the surfaces contains a macroscopic fractionof sites, hence one of the very first cavitated sites. Finally,below the pressure P3 of the triple point in Figure 1, allremaining pores empty reversibly at their equilibrium pressure(blue region in Figure 8).Above Tco, Pperc < Pcav

max. Except close to the sample’s surfaces,evaporation is driven by cavitation (white region in Figure 8).In contrast to the situation below Tco, isotherms are thenidentical for periodic and nonperiodic conditions (Figure 7b).Coupling effects are nevertheless important, as cavitation of agiven site selects the whole cluster to which it belongs. Thisexplains why the evaporation branch of Figure 7b is steeperthan for the uncoupled case. The closer Pcav

max is to Pperc, thelarger the typical extent of the potential clusters and the largerthe decrease of the liquid fraction with respect to theuncoupled case.Comparison of Figure 7a,b shows that the change of

evaporation mechanism at Tco has no obvious signature interms of the isotherm shape, which is of the H2-type bothabove and below Tco. In contrast, Figure 9 shows that thischange has a distinct optical signature. Below Tco, the opticalsignal only depends on the temperature below Pcav

max(T), whencavitation sets in. Above this pressure, a temperature-independent peak occurs below the percolation pressure.The peak is angle-dependent: it shifts toward smaller fractionsand pressures and decreases in amplitude as Q is increased(compare the curves at Q = 5 and 12 at 3.2 K in Figure 5b). In

Figure 7. Combined effect of pore coupling and thermal activation onthe isotherm shape computed from our model at two symmetrictemperatures with respect to the percolation-cavitation crossover at3.35 K: (a) at 3 K, the evaporation is controlled by invasion from thesurfaces down to Pcav

max; (b) at 3.7 K, the evaporation is controlled byinvasion from the cavitated sites. In both cases, the condensationisotherm coincides with the equilibrium prediction.

Figure 8. P−T diagram showing the different mechanisms ofevaporation above and below Tco for a 5123 lattice (see text). Theline PmaxO gives the pressure of the maximal optical signal for Q = 5.

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contrast, above Tco, the peak occurs above the percolationpressure. Its position and amplitude are sensitive to thetemperature. The peak amplitude is reduced with respect tothe low temperature case, and its dependence on Q is weakerthan below Tco (compare the curves at 3.2 and 3.4 K in Figure5b).As detailed in the Supporting Information, these different

features can be traced to the light-scattering properties ofinvasion percolation. Below Tco and above Pcav

max(T), the peak iscontrolled by the percolation cluster. Above Tco, thecomparison of the signals in the non-periodic and periodiccases show that it is mainly due to the clusters selected fromthe cavitated germs. Starting from large liquid fractions, theoptical signal grows because of the increase of the number andextent of these clusters, approximately up to the point wherethe average distance between germs equals the typical clusterextent. Beyond this point, the vapor density becomes morehomogeneous and the signal decreases.Comparing Figure 5a and Figure 9, we see that the predicted

saturation of the optical signal below 3.35 K is notexperimentally observed. At 3.16 K, the signal is significantlysmaller than at 3 K, and the anisotropy is smaller. This pointsto the fact that, for our experimental system, Tco lies below3.16 K (and possibly below 3 K). This is consistent with theobservation of relaxation effects at the isotherm kink at thelatter temperature (see Figure S8 in the SupportingInformation). This discrepancy can be explained if the modelof a cubic lattice overestimates the coordination number of theactual network. 3D networks with a lower coordinationnumber (3 or 4) have larger percolation thresholds, withcritical fractions ranging between 40 and 60% instead of 31%.Such topologies would decrease the crossover temperature Tcobelow 3.35 K and would better account for our experimentalresults.Let us now turn to condensation, for which coupling has a

dramatic effect. Figure 7a,b indeed shows that, at alltemperatures, the condensation isotherm markedly differsfrom that in the uncoupled case and nearly coincides with that

computed assuming condensation to take place at equilibrium.The computed optical signal precisely coincides with thatcomputed in the absence of any coupling, that is, for poresfilling in the order of their diameters (Figure 9). This lack ofoptical signal follows from the fact that pores condense atequilibrium in the order of their diameter up to P3, the lowerclosure pressure of the hysteresis loop. Because the sites areuncorrelated, the scattered intensity then corresponds to arandom distribution of filled pores and is very small. At T = 3K, Figure 1 shows that P3 corresponds to a radius of 2.5 nm.For our distribution, more than 50% of the sites are then filledat P3. This explains why the percolation of the filled sites at alower fraction (≈31%) has no optical signature. Above P3, thesites already filled at P3 act as germs for correlatedcondensation. Their density being very large, most largerpores condense at equilibrium, with no optical counterpart,because of the destructive interferences between selectedclusters (see the Supporting Information).The large difference between condensation and evaporation

below Tco reflects that, for the chosen distribution, the germdensity at the percolation point is large for condensation andvery small for evaporation. Here, this results from the fact thatthe smallest pores can condense at equilibrium, whereas thesymmetrical process is forbidden by pore-blocking. Thisdifference is however more general and follows from the factthat the spinodal instability for condensation has no counter-part for evaporation. Even for a distribution such that noisolated pore can fill at equilibrium, the smallest pores fill dueto the spinodal instability and act as germs allowing some otherpores to condense at equilibrium because of the coupling. Atthe percolation point (for condensation), the density of suchgerms is very large, so that the optical signal is much smallerthan for evaporation. This accounts for the weak or null opticalsignature of condensation of, respectively, argon33 or hexane15

and helium,18 in Vycor.b Correlatively, because of this largedensity of germs, we show in the Supporting Information that agood approximation of the pore diameter distribution can beretrieved from the condensation branch by assuming the poresto fill at equilibrium (see Figure S11).

3.2.3. Determination of the Pore Volume Distribution. ForVycor, this major prediction of our model is supported by ourexperimental observation that, on the condensation branch, therelaxation is much smaller than on the evaporation branch (seethe Supporting Information for a detailed discussion). Wetherefore determine the pore size distribution for Vycor fromour condensation branches from 3.16 K up to Tch using theBJH procedure to account for the contribution of the adsorbedfilm to the density change along the isotherm. The single poreproperties (equilibrium pressure and average density at a givenpressure) were obtained from our full single pore model.26

Figure 10 shows the inferred distribution, again taking a vander Waals interaction strength of 1200 K Å3. Quite remarkably,it is identical over the full temperature range, which validatesour assumption that condensation takes place at equilibrium.The distribution is asymmetrical, being flatter on the lowradius side than on the large radius one. It peaks around 5 nmand extends up to radii of 8 nm. The resulting expected closuretemperature, computed using our single pore model26 is about4.75 K when a barrier of 40 kBT can be overcome. This is closeto the measured Tch ≈ 4.625 K. We therefore conclude that thepore size distribution derived from our multiscale approach isglobally consistent.

Figure 9. Dependence of the theoretical optical signal on theevaporation process (for Q = 5). The temperature-independent signalfor condensation coincides with that computed without couplingeffects. Below the crossover temperature Tco = 3.35 K and above thepressure Pcav

max (indicated by arrows), the signal for evaporation istemperature-independent and corresponds to percolation invasionfrom the sample boundaries. Above Tco, the temperature-dependentsignal corresponds to invasion from the cavitated sites. For eachtemperature, the liquid fraction at which percolation takes place isindicated by a dashed line. Continuous lines extending up toPcav

max (T) give the optical signal in the periodic case.

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This conclusion might seem at odds with our aboveassumption of a mean pore diameter of 3 nm. However, westress that we determine the pore volume distribution ratherthan the pore diameter distribution. For the assumed Gaussiandistribution, the pore volume distribution would peak between4 and 4.2 nm if we assume that the average volume of a pore ofradius R scales as R2 or R3. This is closer to the observed 5 nm.Moreover, as discussed in the modelization part, ourdescription of Vycor is oversimplified. We cannot thus expectthe theoretical distribution accounting for the experimentaltemperature evolution to quantitatively agree with the actualone.Using the above determined experimental pore volume

distribution and assuming the pore volume to scale as thesquare of its radius, we can deduce the pore diameterdistribution. The percolation threshold of 31% then corre-sponds to a value of Rperc between 3 and 4 nm, comparable toRperc ≈ 3.7 nm for the assumed Gaussian distribution. Thepredicted Tco is then around 3.35 K, whereas we observe thescattered intensity to increase between 3.16 and 3 K. However,as above, this discrepancy can be explained by our simplifieddescription of Vycor. A model with a lower coordinationnumber would increase the 31% threshold. This would pushthe percolation pressure to lower values and hence decreasethe predicted crossover temperature (see Figure 8).

4. CONCLUSIONSIn this paper, we presented an extensive experimental study ofthe condensation and evaporation of helium in a widetemperature range below the bulk critical temperature Tc.We also developed a lattice model including the effect ofthermal activation on the single pore level and the collectiveeffects resulting from the coupling between neighboring pores.This model allows to predict the isotherms and structure factorfor any pressure history, including arbitrary scanning curves.Our main experimental findings are the demonstration ofactivated relaxation, which, in agreement with the Machin’stheory,41 explains the closure of the hysteresis loop below Tcand the determination of the fluid spatial correlations duringcondensation and evaporation. These findings are consistentlyexplained by the model, demonstrating that the evaporationmechanism smoothly evolves from pure invasion percolationfrom the sample’s surfaces at the lowest temperature to

percolation from bulk cavitated sites at larger temperatures.Since a key ingredient of this crossover is the dependence ofthe cavitation pressure on the pore radius, the agreement oftheory and experiment indirectly validates this prediction ofour single pore description. Our collective model alsoreproduces the experimental lack of optical signal duringcondensation, a fact already recognized in several experimentsbut never explained. Using percolation concepts (selection ofBernoulli clusters), we explain this lack of optical signature andshow that it does not contradict our model’s prediction thatcondensation, like evaporation, is a collective process. Animportant feature of this prediction is that, under a wide rangeof conditions, condensation of pores should proceed atequilibrium. This leads us to suggest that, for disorderedporous materials, the classical BJH approach, when applied tothe condensation branch using a properly extended version ofthe Kelvin equation, should give a good approximation of thepore size distribution in complex porous materials. Thisconclusion is supported by the application of this scheme toour experimental results, which yields the same pore volumedistribution in a wide temperature range. To our knowledge,such an agreement had never been reported before. It wouldbe highly valuable to further test this conclusion using otherfluids (e.g., nitrogen) in Vycor as well as other materials.

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.lang-muir.8b04275.

Details on the experimental and modeling methods, alarger set of relaxation data, and detailed analysis of themodelization results (PDF)

■ AUTHOR INFORMATIONCorresponding Author*E-mail: pierre-etienne.wolf@neel.cnrs.fr.ORCIDPierre-Etienne Wolf: 0000-0001-8633-7824NotesThe authors declare no competing financial interest.†Deceased 2008, Dec 2nd.

■ ACKNOWLEDGMENTSWe acknowledge ANR for funding the experimental work andM.M.’s postdoctoral stay (ANR-06-BLAN-0098).

■ ADDITIONAL NOTESaAs discussed in Section S4.3, the small relaxation on thecondensation branch is consistent with the idea that because ofpore coupling, condensation takes place close to equilibrium.bApplying our single pore model (with the parameters given inref 26) to argon, and taking into account that the Vycor sampleused in ref 33 has a larger average pore diameter than ours, weestimate that, for argon at 87 K, the germ density at thecondensation percolation point is below 5%, to be compared tomore than 20% for helium at 3 K. We believe that thisdifference, combined with the larger index of refraction ofargon compared to helium, explains why a small decrease ofoptical transmission during condensation is observed in ref 33.

Figure 10. Pore volume distribution inferred from a BJH-like analysisof the condensation branches of the experimental isotherms, assumingcondensation to take place at equilibrium. Isotherm 3.16 K (f) ismeasured using the flowmeter described in the SupportingInformation (§S2.2) instead of the temperature-controlled reservoir.

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■ REFERENCES(1) Barrett, E. P.; Joyner, L. G.; Halenda, P. P. The Determination ofPore Volume and Area Distributions in Porous Substances .1.Computations from Nitrogen Isotherms. J. Am. Chem. Soc. 1951, 73,373−380.(2) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; Sliwinska-Bartkowiak, M. Phase Separation in Confined Systems. Rep. Prog.Phys. 1999, 62, 1573−1659.(3) Monson, P. A. Understanding Adsorption/Desorption Hyste-resis for Fluids in Mesoporous Materials Using Simple MolecularModels and Classical Density Functional Theory. MicroporousMesoporous Mater. 2012, 160, 47−66.(4) Evans, R. Fluids Adsorbed in Narrow Pores: Phase Equilibriaand Structure. J. Phys.: Condens. Matter 1990, 2, 8989−9007.(5) Neimark, A. V.; Ravikovitch, P. I.; Vishnyakov, A. BridgingScales from Molecular Simulations to Classical Thermodynamics:Density Functional Theory of Capillary Condensation in Nanopores.J. Phys.: Condens. Matter 2003, 15, 347−365.(6) Saam, W. F.; Cole, M. W. Excitations and Thermodynamics forLiquid-Helium Films. Phys. Rev. B: Solid State 1975, 11, 1086−1105.(7) Wallacher, D.; Kunzner, N.; Kovalev, D.; Knorr, N.; Knorr, K.Capillary Condensation in Linear Mesopores of Different Shape. Phys.Rev. Lett. 2004, 92, 195704.(8) Grosman, A.; Ortega, C. Cavitation in Metastable FluidsConfined to Linear Mesopores. Langmuir 2011, 27, 2364−2374.(9) Ravikovitch, P. I.; Neimark, A. V. Experimental Confirmation ofDifferent Mechanisms of Evaporation from Ink-Bottle Type Pores:Equilibrium, Pore Blocking, and Cavitation. Langmuir 2002, 18,9830−9837.(10) Morishige, K.; Tateishi, N. Adsorption Hysteresis in Ink-BottlePore. J. Chem. Phys. 2003, 119, 2301−2306.(11) Rasmussen, C. J.; Vishnyakov, A.; Thommes, M.; Smarsly, B.M.; Kleitz, F.; Neimark, A. V. Cavitation in Metastable LiquidNitrogen Confined to Nanoscale Pores. Langmuir 2010, 26, 10147−10157.(12) Mason, G. A Model of Adsorption Desorption Hysteresis inwhich Hysteresis is Primarily Developed by the Interconnections in aNetwork of Pores. Proc. R. Soc. A 1983, 390, 47−72.(13) Mason, G. Determination of the Pore-Size Distributions andPore-Space Interconnectivity of Vycor Porous Glass from Adsorption-Desorption Hysteresis Capillary Condensation Isotherms. Proc. R. Soc.A 1988, 415, 453−486.(14) Parlar, M.; Yortsos, Y. Percolation Theory of VaporAdsorption-Desorption Processes in Porous Materials. J. ColloidInterface Sci. 1988, 124, 162−176.(15) Page, J. H.; Liu, J.; Abeles, B.; Herbolzheimer, E.; Deckman, H.W.; Weitz, D. A. Adsorption and Desorption of a Wetting Fluid inVycor Studied by Acoustic and Optical Techniques. Phys. Rev. E: Stat.Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1995, 52, 2763−2777.(16) Parlar, M.; Yortsos, Y. C. Nucleation and Pore GeometryEffects in Capillary Desorption Processes in Porous Media. J. ColloidInterface Sci. 1989, 132, 425−443.(17) Morishige, K. Hysteresis Critical Point of Nitrogen in PorousGlass: Occurrence of Sample Spanning Transition in CapillaryCondensation. Langmuir 2009, 25, 6221−6226.(18) Bonnet, F.; Melich, M.; Puech, L.; Wolf, P. E. Light ScatteringStudy of Collective Effects during Evaporation and Condensation in aDisordered Porous Material. Europhys. Lett. 2013, 101, 16010.(19) Kierlik, E.; Monson, P. A.; Rosinberg, M. L.; Sarkisov, L.;Tarjus, G. Capillary Condensation in Disordered Porous Materials:Hysteresis Versus Equilibrium Behavior. Phys. Rev. Lett. 2001, 87,055701.(20) Detcheverry, F.; Kierlik, E.; Rosinberg, M. L.; Tarjus, G.Mechanisms for Gas Adsorption and Desorption in Silica Aerogels:The Effect of Temperature. Langmuir 2004, 20, 8006−8014.(21) Aubry, G. J.; Bonnet, F.; Melich, M.; Guyon, L.; Spathis, P.;Despetis, F.; Wolf, P.-E. Condensation of Helium in Aerogel andAthermal Dynamics of the Random-Field Ising Model. Phys. Rev. Lett.2014, 113, 085301.

(22) Woo, H.-J.; Sarkisov, L.; Monson, P. A. Mean-Field Theory ofFluid Adsorption in a Porous Glass. Langmuir 2001, 17, 7472−7475.(23) Woo, H.-J.; Porcheron, F.; Monson, P. A. Modeling Desorptionof Fluids from Disordered Mesoporous Materials. Langmuir 2004, 20,4743−4747.(24) Bonnet, F. Etude de la Condensation et de l’Evaporation del’Helium dans les Milieux Poreux : Effets du Confinement et duDesordre. Ph.D. Thesis, Grenoble University, 2009.(25) Cross, B.; Puech, L.; Wolf, P. E. A Temperature-ControlledDevice for Volumetric Measurements of Helium Adsorption inPorous Media. J. Low Temp. Phys. 2007, 148, 903.(26) Bonnet, F.; Wolf, P.-E. Thermally Activated Condensation andEvaporation in Cylindrical Pores. J. Phys. Chem. C 2019, 123, 1335−1347.(27) Guyer, R. A.; McCall, K. R. Capillary Condensation, InvasionPercolation, Hysteresis, and Discrete Memory. Phys. Rev. B: Condens.Matter Mater. Phys. 1996, 54, 18−21.(28) Puibasset, J. Monte-Carlo Multiscale Simulation Study ofArgon Adsorption/Desorption Hysteresis in Mesoporous Heteroge-neous Tubular Pores like MCM-41 or Oxidized Porous Silicon.Langmuir 2009, 25, 903−911.(29) Levitz, P.; Ehret, G.; Sinha, S. K.; Drake, J. M. Porous VycorGlass - the Microstructure as Probed by Electron-Microscopy, DirectEnergy-Transfer, Small-Angle Scattering, and Molecular Adsorption. J.Chem. Phys. 1991, 95, 6151−6161.(30) Hoshen, J.; Kopelman, R. Percolation and Cluster Distribution.I. Cluster Multiple Labeling Technique and Critical ConcentrationAlgorithm. Phys. Rev. B: Solid State 1976, 14, 3438−3445.(31) Here, we ignore the vectorial nature of the scattered field. Thisis justified in the Rayleigh regime if the incident polarization isperpendicular to the scattering plane. Both conditions are fulfilled inour experiments.(32) This procedure assumes the configuration to be statisticallyisotropic. While this is true for the simulation, this might not be so inan actual scattering experiment where the structure factor is measuredlocally for a given incident direction. In this case, comparison to themodel requires to average over different configurations rather thanover the full shell.(33) Soprunyuk, V. P.; Wallacher, D.; Huber, P.; Knorr, K.; Kityk, A.Freezing and Melting of Ar in Mesopores Studied by OpticalTransmission. Phys. Rev. B: Condens. Matter Mater. Phys. 2003, 67,144105.(34) Thommes, M.; Smarsly, B.; Groenewolt, M.; Ravikovitch, P. I.;Neimark, A. V. Adsorption Hysteresis of Nitrogen and Argon in PoreNetworks and Characterization of Novel Micro- and MesoporousSilicas. Langmuir 2006, 22, 756−764.(35) Brown, A. J. The Thermodynamic and Hysteresis ofAdsorption. Ph.D. Thesis, School of Chemistry, University of Bristol,U.K., 1963.(36) Quinn, H. W.; McIntosh, R. The Hysteresis Loop inAdsorption Isotherms on Porous Vycor Glass and AssociatedDimensional Changes of the Adsorbent II. Can. J. Chem. 1957, 35,745−756.(37) Brewer, D. F.; Champeney, D. C. Sorption of Helium andNitrogen on Vycor Porous Glass. Proc. Phys. Soc., London 1962, 79,855−868.(38) A difference in the pore size distributions between our twosamples, which could also explain a different Tch, is not likely as the 77K nitrogen sorption isotherm reported in ref 37 is quite similar toours.(39) Burgess, C. G. V.; Everett, D. H.; Nuttall, S. AdsorptionHysteresis in Porous Materials. Pure Appl. Chem. 1989, 61, 1845−1852.(40) Thommes, M.; Findenegg, G. H. Pore Condensation andCritical-Point Shift of a Fluid in Controlled-Pore Glass. Langmuir1994, 10, 4270−4277.(41) Machin, W. D. Properties of Three Capillary Fluids in CriticalRegion. Langmuir 1999, 15, 169−173.

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(42) Ball, P. C.; Evans, R. Temperature Dependence of GasAdsorption on a Mesoporous Solid: Capillary Criticality andHysteresis. Langmuir 1989, 5, 714−723.(43) Valiullin, R.; Naumov, S.; Galvosas, P.; Karger, J.; Woo, H.-J.;Porcheron, F.; Monson, P. A. Exploration of Molecular Dynamicsduring Transient Sorption of Fluids in Mesoporous Materials. Nature2006, 443, 965−968.(44) As shown by Figure 5a, the optical signal at θ = 45° and 3 Kshows strong fluctuations. These fluctuations are uniform over theilluminated slice. They likely result from the fluctuations of thecondensed mass due to the small oscillations of the room temperatureinduced by the air-conditioning system.(45) The larger fraction span for theory is explained by the fact that,for the empty pores, the present model neglects the adsorbed film.The more precise model developed in ref 26 for a single pore showsthat this film significantly contributes to the liquid fraction.

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Deciphering the Mechanisms of Condensation

and Evaporation in a Complex Porous Material:

Supporting Information

Fabien Bonnet,† Mathieu Melich,† Laurent Puech,†,‡ Jean-Christian Anglès

d’Auriac,† and Pierre-Etienne Wolf∗,†

† Université Grenoble Alpes, CNRS, Institut Néel, F-38042 Grenoble, France

‡Deceased 2008, Dec 2nd

E-mail: pierre-etienne.wolf@neel.cnrs.fr

Contents

S1 Outline S3

S2 Experimental Methods S3

S2.1 Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S3

S2.2 Isotherms measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S3

S2.2.1 Determination of the pressure . . . . . . . . . . . . . . . . . . . . . . S4

S2.2.2 Determination of the fluid density . . . . . . . . . . . . . . . . . . . . S5

S3 Modelisation S6

S3.1 Obtaining the fluid configuration . . . . . . . . . . . . . . . . . . . . . . . . S6

S3.2 Generalization to scanning curves . . . . . . . . . . . . . . . . . . . . . . . . S7

S1

S4 Experimental Results S7

S4.1 Loop closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S7

S4.2 Phase diagram for the confined fluid . . . . . . . . . . . . . . . . . . . . . . S9

S4.3 Relaxations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S10

S4.3.1 Relaxation measurements . . . . . . . . . . . . . . . . . . . . . . . . S10

S4.3.2 Analysis of relaxation measurements . . . . . . . . . . . . . . . . . . S12

S4.3.3 Optical signal during relaxation . . . . . . . . . . . . . . . . . . . . . S14

S5 Analysis of the modelisation results S16

S5.1 Numerical phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . S16

S5.2 Analysis of the theoretical optical signal . . . . . . . . . . . . . . . . . . . . S17

S6 Influence of the pore diameter distribution on the process of condensation

at equilibrium S20

References S21

S2

S1 Outline

This Supporting Information gives details on our experimental and modeling methods. We

also present a larger set of relaxation data than in the paper, and show that their main

features are consistent with our model of thermal activation in a single pore.? Details are

also provided on the analysis of the modelisation results.

S2 Experimental Methods

S2.1 Sample

Our Vycor sample was purchased from Advanced Glass and Ceramics (Holden MA). Its

porosity estimated from its weight using the standart silica density (2.2 g/cm3) is 32± 2%,

to be compared to the value 28% given by Corning. The corresponding pore volume is 0.23

cm3±7%. A sample from the same batch was characterized by nitrogen sorption at 77 K

using a commercial set-up (Micromeritics ASAP 2010). Figure S1 compares the measured

loop to those previously reported for other Vycor samples. Excepting the data from Ref. 1,

the measurements point to the fact that Vycor 7930 has been as produced with two different

mean pore sizes, our sample falling in the large pores category. Analysis of our data using

the set-up software yields a mean pore radius of 2.8 or 3.5 nm depending on whether the

desorption or adsorption data were used.

S2.2 Isotherms measurements

Isotherms measurements were performed in two steps. First, the full circuit (including the

reservoir, the connecting lines and valves, and the cell) was filled with a known mass of helium

using a calibrated flowmeter. Second, this circuit was sealed and the cell content was changed

by sweeping the reservoir temperature. Given the dynamics of the reservoir temperature

(80-200 K, or even 300 K for the lowest sample’s temperatures where the isotherm loop is

S3

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1Li

quid

Fra

ctio

n

P/Psat

HuberGruner

ThommesMorishige

Present study

Figure S1: Characterization of our sample by nitrogen adsorption isotherms at 77 K. Themeasured hysteresis loop is compared to previously reported nitrogen isotherms for Vycor7930 at 77 K.1–4 Here, the volume liquid fraction is obtained by dividing the measureddensity by its value at Psat. The same H2-type loop is found in all cases, but the strongdependence of fraction observed by Morishige in the liquid phase1 is distinct from the othermeasurements. Comparison of the other four isotherms suggests that at least two kinds ofVycor 7930 were manufactured.

largest), we could follow the hysteresis loop over the full range of cell temperatures by using

only two different helium loads, minimizing systematic errors. We typically used a flowrate

of 0.1-0.2 STPcc/min (the corresponding sweeping rate of the reservoir temperature was 2 to

5 K/hour, depending on the cell temperature). At low temperatures, several isotherms were

performed using a 5 or 10 times larger flowrate to check that the thermal gradients within

the sample, or the pressure gradients due to the fluid viscosity, have a negligible influence

on the hysteresis loop.

S2.2.1 Determination of the pressure

The cell pressure was measured with an accuracy of ±0.01 mbar by a Digiquartz pressure

transducer (Paroscientific 245A) connected to the cell through a separate line. Two factors

influence the pressure stability. First, along the steepest portion of the isotherm, a change

of the cell temperature shifts the pressure by an amount comparable to the change in the

saturated vapor pressure, typically 1-2 mb/mK in the temperature range studied. The cell

S4

temperature was regulated within a fraction of a mK below 4.2 K, and several tens of µK

above, the difference reflecting two different modes of operation of the cryostat.5 This is

translated in a pressure stability of a fraction of a mbar below 4.2 K, and one order of

magnitude better above. Second, the pressure was found sensitive to the temperature of the

experimental room. Although this temperature is controlled to within one degree Celsius,

the remaining oscillations make the pressure oscillate with an amplitude of several tens of

mbar. This effect has a negligible influence on the isotherms but is visible when studying

the long time behaviour of the relaxations (see Figure S5). Overall, the pressure stability

was better than 0.5 mbar, which is more than two orders magnitude below the pressure

difference between the hysteresis loop and the saturated pressure, typically of order 100 mb

at all temperatures.

S2.2.2 Determination of the fluid density

In a standard way, the confined fluid density was found by subtracting from the total mass

contained in the closed circuit the mass of the gas contained in the reservoir and the filling

lines (known from temperature and pressure measurements), and the vapor mass inside the

cell (deduced from the cell dead volume and the equation of state for bulk helium6,7). The

precision on the helium mass inside the Vycor sample is better than 1 cm3 STP, corresponding

to a systematic error of 0.8 g/l on the average density. From the liquid density ρ at a pressure

P and a temperature T , we compute the average liquid fraction Φ as

Φ =ρ− ρg(P, T )

ρcl(P, T )− ρg(P, T )(S1)

where ρg(P, T ) is the gas density at P and T and ρcl(P, T ) is the density of the confined

liquid at the (reservoir) pressure P . This density is larger than the bulk liquid density, due to

the silica attraction, and varies linearly with pressure up to Psat, due to the compressibility

of liquid helium.? ρcl(P, T ) is then found by linearly extrapolating down to P the density

S5

measured on the linear portion of the evaporation branch. Note that equation S1 is only

an approximation as, due to the spatial dependence of the attractive potential, the average

liquid density is larger for a partly filled pore than for a fully filled one.

S3 Modelisation

S3.1 Obtaining the fluid configuration

In this paragraph, we describe the algorithm used for the simulations. It produces exactly

the same results as the one described in the text.

For a given temperature, we first compute and sort the three characteristic pressures

for the whole pore distribution. This yields 3V pressures where V = N3 is the number

of sites, (N being the lattice linear size, 512 in the present study). For a given pressure

history starting either from a filled (P = 1) or empty (P = 0) configuration, we then

compute the configuration at each successive stage (i.e. at each characteristic pressure),

according to the four rules explained in the main text. For an evaporation phase, when the

current characteristic pressure is a cavitation pressure, the corresponding pore is emptied,

while if it is a equilibrium pressure, it empties only if at least one of its neighbors is empty.

Symetrically, during a condensation phase, when the characteristic pressure is an activation

pressure, the pore fills while, when it is an equilibrium pressure, it fills only if it has at

least one already filled neighbor. In both cases, when a pore changes state, it triggers an

avalanche of all pores connected to it having an equilibrium pressure smaller or larger than

the present characteristic pressure, depending on whether the pressure is currently increasing

or decreasing.

In the paper, we restrict ourselves to a monotonous pressure variation, corresponding to

the major hysteresis loop. Then, instead of considering all intermediate pressures, one can

go directly to the target pressure, then loop over the pores to determine their new state.

Note that, in this case, it is possible to map the problem of evaporation with cavitation to

S6

standard percolation by replacing the two relevant pressures by one single effective pressure

Peff . For the case of evaporation, this pressure is obtained by repeatedly replacing Pcav by

min (max (P vcav, Pcav) , Peq) and Peq by max

(min

(Pcav,max

(P veq

)), Pcav

), the superscript ν

refering to the neighboring site of the current site, until the two pressures become equal for

each site. At a given pressure P , the clusters of sites such that P < Peff are precisely the

selected clusters of the original problem.

S3.2 Generalization to scanning curves

Even though we do not use this possibility in the present work, it is important to stress

that scanning curves can be readily computed within our approach. Indeed, the algorithm

above described can be used for any pressure variation, monotonous or not. The version

described in the paper can also be generalized to the case of a non monotonous pressure

variation. On that purpose, one simply adds new selection rules. Namely, for evaporation, a

potential cluster can also be selected if it contains in the current configuration a vapor site

(a site which has been emptied, but not filled again, in the previous history). Similarly, for

condensation, a potential cluster can be selected if it contains a condensed site.

For any pressure variations, the vapor clusters for evaporation and the liquid clusters for

condensation are then subsets of the potential clusters. This shows that the influence of any

complicated pressure variation is only in the selection of the potential clusters, not in the

nature of the clusters themselves.

S4 Experimental Results

S4.1 Loop closure

Figure S2 shows how the hysteresis loop closes for a flowrate corresponding to scanning the

loop within two hours. The closure temperature is then between 4.65 K and 4.725 K. A

lower closure temperature is obtained when stopping the flowrate at several points along the

S7

125

130

135

0.90 0.94 0.98ρ

(g/L

)

P/Psat

4.34 K4.5 K4.6 K

4.625 K4.65 K

4.675 K4.7 K

4.725 K

Figure S2: Details of the loop closure for isotherms in the range 4.34 K-4.725 K.

loop and letting the pressure relax at a constant total mass.

111

112

113

114

0.94 0.95 0.96

T= 4.625 K

ρ (g

/L)

P/Psat

0.03 STPcc/minevaporation

condensation

Figure S3: Closure of the hysteresis loop at 4.625 K. The continuous curve corresponds to aconstant flowrate of ± 0.03 ccSTP/min. Discrete points are obtained by waiting two hoursat different pressures along the hysteresis loop.

As shown by Figure S3, the relaxed hysteresis loop is then nearly closed, within our

resolution, at 4.625 K. This dependance of the closure temperature with the time scale of

measurements clearly points to a thermally activated closure mechanism.

S8

3.0

4.0

5.0

0 50 100

(a)

ρ (g/L)

T(K

)

He bulkHe confined

0.6

0.8

1

0.5 0.8 1.1

(a)

(b)

ρ/ρc

T /

Tc

XeCO2He

Figure S4: (a) Helium average density at the lower and upper ends of the hysteresis loop inVycor. The data are compared to the bulk coexistence curve of helium. (b) Comparison ofthe same helium data - in reduced units- to those previously reported for xenon and carbondioxyde.8

S4.2 Phase diagram for the confined fluid

Similarly to Burgess et al,9 we can extract from our isotherms measurements a ”coexistence

curve” representing, as a function of temperature, the fluid density at the lower and upper

closure points of the hysteresis loop. Figure S4a shows that this ”coexistence curve” is much

narrower than for the bulk phase. The larger density on the ”vapor” side reflects the large

contribution of the film at the lower end of the hysteresis loop. The increased density of the

dense phase with respect to the bulk can be quantitatively accounted for by the attractive

interaction with the silica combined with the large compressibility of liquid helium.5? By

definition, the two branches merge at Tch, the hysteresis closure temperature.

Figure S4b compares our coexistence curve to those reported by Burgess et al. for xenon

and carbon dioxyde confined in Vycor, when plotted as a function of reduced density and

pressures. While the agreement between xenon and carbon dioxyde data has been considered

by Burgess et al. in terms of an universal phase diagram for fluids confined in Vycor, the

discrepancy with the helium data (much larger than in the bulk case, where it results from

the large De Boer parameter of helium) shows that the loop closure is non universal. This

S9

suggests that it does not reflect a shift of the critical temperature due to confinement.

S4.3 Relaxations

S4.3.1 Relaxation measurements

0.94

0.96

0.98

1

0.92 0.94 0.96

1

2

34

5

Liqu

id fr

actio

n Φ

P/Psat

0.2 STPcc/minfixed points

transient

10-2

10-1

1

0 1 2

∆P (

mb)

time (hours)

12345

Figure S5: Relaxations along the evaporation branch at 4.5 K. (a): in P − Φ coordinates.Hysteresis loop obtained at a flow rate of ± 0.2 STPcc/minute and transient and relaxedpoints following successive stops along the hysteresis loop. (b): for pressure as a function oftime. The pressure oscillations reflect the small oscillations of the room temperature. Seetext for details.

We have measured pressure relaxations on the evaporation branch between 3.16 K and

4.625 K. Figures S3 and S5 shows how the relaxation amplitude and rate depend on the

position on this branch at 4.625 K and 4.5 K. The rate increases when decreasing the fugacity

S10

from the upper closure point of the hysteresis loop to the lower closure point. The amplitude

first increases, then decreases.

10-2

10-1

1

0 1 2

∆P (

mb)

time (hours)

T=3.16 KT=4.34 K

T=4.5 KT=4.625 K

10-1

1

0 5 10 15 20 25

∆P (

mb)

time (hours)

T=3.16 KT=4.34 K

Figure S6: Relaxations close to the elbow/kink of the evaporation branch for different tem-peratures on a timescale of (a): 2 hours ; (b): 15 hours.

For a given position along the loop, relaxation also depends on temperature. Figure S6a

shows measurements close to the elbow over a two hours time range for T ≥4.34 K. At each

temperature, the baseline is determined by fitting to an exponential decay. The relaxation

rate measured is then of order 1 h−1 for all temperatures. Smaller rates are obtained when

monitoring the relaxation over longer times. This is illustrated by Figure S6b, which shows

relaxations over up to 36 h measured close to the elbow at 3.16 K and 4.34 K (see respectively

Figure S8a and Figure 3 of the paper for the position within the loop). At 4.34 K, slightly

S11

below the elbow, the decay rate decreases from about 1 h−1 at short times to 0.1 h−1 at long

times. At 3.16 K, slightly above the elbow, the departure from a single exponential is much

weaker, the decay rate decreasing from 0.2 to 0.08 h−1.

0

20

40

60

4 5 6 7 8

E∗ /(

k BT

)

R (nm)

y- y1 y2 yk y3 y+

Figure S7: Radius dependence of the barrier energy for activated condensation (left of max-imum) or evaporation (right of maximum), computed according to the single pore modelof Ref. ? for T=4.5 K. The different curves correspond to different fugacities y alongthe hysteresis loop. y− and y+ correspond to the fugacities at the lower and upper ends ofthe hysteresis loop for a pore radius distribution extending up to 7 nm, when a barrier of40kBT can be overcome on the time scale of the continuous measurements. Between y− andy+, the different curves allow to analyze how relaxation on longer time scales depends onintermediate y’s values, and account for the behaviour observed in Figures S5 and S6.

S4.3.2 Analysis of relaxation measurements

We now show that the above features are qualitatively consistent with the behaviour expected

for thermally activated condensation/evaporation in independent cylindrical pores with dis-

tributed radii. Figure S7 shows the radius dependence of the energy barrier E∗(R, y) at

4.34 K, computed following Ref. ? for a van der Waals constant of 1200 KÅ3, and different

fugacities y. For a given fugacity, the barrier is maximal at the corresponding equilibrium

radius. For smaller radii, the barrier corresponds to activated condensation, and, for larger

radii, to cavitation. The relaxation time to equilibrium for a pore of given radius R scales as

exp(−E∗(R, y)/kBT ), which determines which pores are in their equilibrium state depending

S12

on the waiting time at the fugacity y. For illustrative purpose, we assume that these pores

correspond to E∗(R, y) < 40kBT when the hysteresis loop is described at a typical flow rate

of 0.3 STP cc/minute. When the flowrate is stopped, the longer one waits, the larger is the

barrier which can be overcome, and the larger the number of pores which can relax. Due

to the exponential dependence of the relaxation time on E∗, an increase of the acccessible

barrier of 5 kBT approximately corresponds to a one hundred fold increase of the waiting

time, comparable to our largest observation times (10-20 hours). Within this time, the pores

which relax are those such that 40 kBT < E∗(R, y) < 45 kBT . In Figure S7, the lowest

fugacity y− corresponds to the lower closure point of the hysteresis loop for an accessible

barrier of 40 kBT (i.e. y− = yeq(Rc(40 kBT )), where Rc(40 kBT ) is the critical radius for

40 kBT ). For the same barrier, the largest fugacity y+ corresponds to the upper closure

point of the hysteresis loop when the pore radius distribution extends up to Rmax=7 nm (i.e.

y+ = yact(Rmax, 40kBT )). Relaxation can only be observed between these extreme values:

• (i) For y− < y < yeq(Rc(45kBT )), the maximal barrier increases, as well as the range

of R’s of relaxing pores (curve y1 in Figure S7). The longest relaxation time then

increases, as well as the total amplitude of relaxation. This behaviour is consistent

with the curves 2-6 in Figure. S5b.

• (ii) For larger y and up to yk = ycav(Rmax, 40kBT ), the fugacity at the elbow (or

kink) of the evaporation isotherm, relaxing pores have barrier energies covering the

full interval between 40 and 45 kBT (curves y2 and yk in Figure S7). This corresponds

to a multiexponential relaxation, with a longest relaxation time set by the observation

time window. Close below the kink, one thus expect the long relaxation time to be

constant and independent on fugacity or temperature. This accounts for the long time

behaviour measured in Figure S6b. The range of R’s of relaxing pores is approximately

constant for condensation. For cavitation, it increases, but the corresponding weight

decreases for a reasonable pore radius distribution (mean at or below 5 nm, width of

order 1 nm). This explains why the relaxation amplitude decreases close to yk (curve

S13

1 in Figure. S5b).

• (iii) Above the kink fugacity yk, the relaxation weight still decreases. For cavitation,

the smaller barrier energy now increases, corresponding to the suppression of the short

time scales and of the multiexponential character (curve y3 in Figure S7). For y >

ycav(Rmax, 45kBT ), no relaxation due to cavitation can be observed. In this range, the

relaxation weight for condensation still decreases to eventually vanish at y+.

Points (i) and (ii) account for the observed behaviour at 4.5 K along the loop (Figure S5).

Point (iii) explains why the weight of fast relaxation right above the kink at 3.16 K is smaller

than below the kink at 4.34 K (Figure S6b).

Around yk, the radius interval over which pores cavitate is much larger than the interval

for activated condensation. This accounts for the larger weight of cavitation, with respect

to condensation, observed in Figure 3. However, for a gaussian distribution centered around

4 or 5 nm, a large part of the difference in the radius interval is compensated for by the

smaller weight of large pores. We then typically expect the weight of cavitation to be at

most twice that of condensation. The observed difference is more than a factor of 10 (see

Figure 3). We believe that this results from the coupling between pores, which, as discussed

in the paper, makes that pores condense close to their equilibrium pressure.

S4.3.3 Optical signal during relaxation

In the paper, we discuss the optical signature of evaporation below 4.34 K when the hysteresis

loop is followed at a constant flow rate. Figure S8 shows the optical counterpart of the

pressure relaxation at 3.16 K (Figure S6b). A slow increase of the optical signal is observed.

This increase does not depend on the position within the sample (close to or far from the

surface). This homogeneous behaviour implies that the optical signal is due to clusters

connected to cavitated pores rather than to surfaces. Thus, at 3.16 K, evaporation is still

driven by invasion from cavitated germs. This conclusion is also consistent with the fact

S14

0.6

0.7

0.8

0.9

1

0.7 0.8 0.9 1

frac

tion

Φ

y

3.16 Krelaxation 3.16 K

1

1.1

1.2

0 4 8

228

232

Opt

ical

Sig

nal

P (

mb)

t (h)

center 45surface 45center 135

surface 135Pressure

Figure S8: Optical signature of the relaxation at 3.16 K. (a) Position within the loop ;(b) time dependence of the optical signal and pressure. The different optical signals aremeasured at the center and close to the surface of the sample for scattering angles of 45◦

and 135◦.

S15

that the maximal optical signal is smaller at 3.16 K than at 3 K.

S5 Analysis of the modelisation results

S5.1 Numerical phase diagram

2.5

3

3.5

4

4.5

0.4 0.5 0.6 0.7 0.8 0.9 1

TCO

T

P

Pcavmax

Pperc

P3

PmaxO (Q~

=5)

surfaces

percolatio

n

cavitation

reversible

Figure S9: Reproduction of Figure 8 of the paper. (P -T ) diagram showing the differentmechanisms of evaporation above and below Tco for a 5123 lattice (see text). The line PmaxO

gives the pressure of maximal optical signal for Q=5.

Figure S9 reproduces Figure 8 of the paper. It summarizes how the evaporation mech-

anismes depend on temperature and pressure. Four lines subdivide the (P ,T ) plane in four

regions. The first line is Pmaxcav (T ) obtained for the pore of largest radius. Since Pcav saturates

with increasing R, this line does not depend on the size L. The second line is P3(T ), the

largest pressure such that Pcav, Peq and Pact are equal. Both lines intercept at the tempera-

ture Tco. The third line is the largest pressure below which the vapor percolates. In principle,

it depends on the realization and on the size L. However, for L = 512, it is extremely little

sensitive to these, and nearly coincides with Pperc(T ). The fourth line gives the pressure

PmaxO at the peak of optical signal, for Q=5. Only this line cannot be computing without

ressorting to simulations. This line is only drawn when the peak corresponds to the largest

maximum of the optical signal. Beyond T =3.75 K, the peak is only a secondary maximum,

S16

the main maximum being identical to that observed for condensation, and the theoretical

optical signal coincides with that for condensation, except in the narrow peak region.

In the darker pink region (T < Tco, Pperc < P ), the liquid fraction φ is 1 in the thermo-

dynamical limit. Indeed the correlation being finite, the contribution to the liquid fraction

of the boundary clusters is of order 1/L . In this limit, this region corresponds to a filled

sample. In the blue region (P < P3(T )) , the pores empty (and fill) at equilibrium. In the

white region (P3 < P < Pmaxcav ), evaporation is hysteretic and sensitive to cavitation. In the

light pink region (T < Tco, Pmaxcav < P < Pperc) no cavitation can take place, and evaporation

is driven by invasion from the sample’s surfaces. Below Tco, the maximum of the optical

signal coincides (on the scale of the figure) with the percolation line (blue). Above Tco, it

takes place close to the cavitation line (green).

S5.2 Analysis of the theoretical optical signal

1

1.5

2

0.2 0.4 0.6 0.8 1

Q~

=5

I/IS

i02

Φ

no cavitationT= 3.0 K

Per-T= 3.2 KT= 3.3 K

Per-T= 3.3 KT= 3.4 K

Per-T= 3.4 KT= 3.5 K

Per-T= 3.5 KT= 3.7 K

Per-T= 3.7 KT= 3.8 K

Per-T= 3.8 Kcondensation

Figure S10: Reproduction of Figure 9 of the paper. Dependence of the theoretical opticalsignal on the evaporation process (for Q = 5). The temperature independent signal forcondensation coincides with that computed without coupling effects. Below the crossovertemperature Tco=3.35 K, and above the pressure Pmax

cav (indicated by arrows), the signal forevaporation is temperature independent and corresponds to percolation invasion from thesample boundaries. Above Tco, the temperature dependent signal corresponds to invasionfrom the cavitated sites. For each temperature, the liquid fraction at which percolation takesplace is indicated by a dashed line.

S17

In this section, we detail the analysis of Figure 9 of the paper, which we reproduce here

for convenience (Figure S10).

Below Tco, and above Pmaxcav (T ), the optical signal does not depend on temperature. As

discussed in §, this reflects the temperature independence of the selection criterion for pure

invasion percolation from the surfaces. When P approaches the percolation pressure Pperc

from above, the signal increases, due to the divergence of ξ and the growth of clusters

connected to the surface. At and below Pperc, we find that the signal is dominated by the

percolating cluster. In the Q −→0 limit, this signal peaks at Pperc, resulting from the fact

that the percolation cluster is fractal at all scales at the percolation pressure, and only up

to a finite ξ below. For an infinite sample, the vapor fraction at the peak would then be

vanishingly small. In our case, it is finite due to the combination of two effects. First, the

finite size of the numerical sample, and, second, the fact that, for a finite Q, the peak occurs

below the percolation pressure (see Figure 9). Indeed, because the scattered intensity probes

the vapor heterogeneities at scales smaller than 1/q, its maximum corresponds to ξq ≈ 1,

i.e. to a finite ξ, as we will detail in another publication. The corresponding vapor fraction

is of order ξdf−3 ∝ Q3−df , where df is the fractal dimension of the percolating cluster (2.54

for the cubic lattice), and the scattered intensity of order Q3−2df ≈ Q−2. This explains

why increasing Q shifts the peak towards smaller pressures and decreases its amplitude

(compare the curves at Q=5 and 12 at 3.2 K in Figure 5b). Note that Figure S10 does not

contradict the identity of the P = Pperc and P = PmaxO lines in Figure S9. Because the liquid

fraction abruptly drops below P = Pperc, the difference of fraction between the peak and the

percolation point in Figure S10 translates to a tiny difference in pressures, not visible on the

scale of Figure S9.

Again below Tco, but now below Pmaxcav (T ), the observed drop of the scattered intensity

results from the fact that cavitation selects clusters which, although not directly contact-

ing the percolating cluster, reduce the fractal range of the vapor distribution by filling its

large scale voids, hence decrease the optical signal. Note that, in this regime, the optical

S18

signal coincides with that computed for periodic boundary conditions. As in the case of the

isotherm, this results from the fact that the large clusters connected to the surfaces also con-

tain cavitated sites. Decreasing the pressure further, the optical signal finally merges with

that computed in the absence of coupling when entering the blue region in Figure 8. Indeed,

below P3, the only vapor sites correspond to non hysteretic sites, insensitive to coupling.

Above Tco, both the peak position and height are sensitive to temperature. In this regime,

the optical signal is due to the clusters selected from the cavitated germs rather than to the

largest cluster only. As the percolation pressure is approached from above, the number of

such germs increases and the typical cluster extent, of order ξ, increases. As long as the

average distance between clusters remains larger than their typical extent, their scattered

intensities add up, and the optical signal increases. Beyond this point, correlations between

clusters induce destructive interferences, which decrease the optical signal. The maximum of

the optical signal thus approximately corresponds to the point where ξ equals the distance

between germs.

Increasing the temperature increases the density of germs. Hence, at the peak, ξ and

the typical germ extent decrease when the temperature increases. Correspondingly, the

vapor fraction (∝ ξdf−3) increases, while the scattered intensity (∝ ξ2df−3) decreases. The

vapor distribution being homogeneous at scales larger than the distance between germs, the

scattered intensity becomes Q independent for ξq << 1.

These considerations allow to explain the features observed in Figures 5 and 9 for T > Tco.

For a very large sample, we would expect the liquid fraction at the peak to continuously

increase as the temperature decreases to Tco. The comparison of the 3.3 K (below Tco) and

3.4 K (above Tco) does not fit into this picture. This is a finite size effect, resulting from

the fact that below Tco, the non percolating clusters connecting the boundaries significantly

contribute to the liquid fraction.

S19

S6 Influence of the pore diameter distribution on the

process of condensation at equilibrium

1 3 5 R (nm)

T=3 K flip at Pact

(a)

N(R

)

1 3 5 R (nm)

T=3 K flip at Peq

(b)

N(R

)1 3 5 R (nm)

T=3 K flip at Peq

truncated

(c)

N(R

)

1 3 5 R (nm)

T=4 K flip at Peq

truncated

(d)

N(R

)

Figure S11: Testing the equilibrium nature of the condensation branch. Pore radius distri-bution inferred from isotherms computed within the coupled model, and analyzed withoutcoupling. The red line is the actual distribution. (a) and (b) at 3 K for a gaussian distri-bution of mean 4 nm and width 0.8 nm, truncated below 0 nm, assuming pores to fill (a)at their activated pressure or (b) at their equilibrium pressure. (c) and (d) at 3 K and 4 Kfor the same distribution truncated below 3 nm, assuming pores to fill at their equilibriumpressure.

In this section, we test the influence of the pore diameter distribution on our paper’s

conclusion that, due to coupling, most pores condense at equilibrium. To this aim, we

compare in Figure S11 the pore distributions obtained by analyzing the isotherms computed

in the full model for two distributions. The first one is a gaussian (truncated to positive

radii) with larger average radius and smaller standard deviation than in the paper, namely an

average radius of 4 nm (instead of 3 nm) and a standard deviation 0.8 nm (instead of 1.4 nm).

These parameters were chosen so that the proportion of pores filled at P3 is 20%, instead

of more than 50% in the paper. The second distribution is the same one, now truncated at

3 nm, so as to ensure that no pore fills reversibly. For each distribution, the isotherms were

computed as explained in the paper (the selected clusters are those containing at least one

pore above its activated condensation pressure).

S20

For the first distribution, Figure S11 shows the inferred distributions from the computed

isotherm at 3 K, assuming that the pores fill at (a) their activated condensation pressure,

or (b) their equilibrium pressure. The first assumption underestimates the pore’s diameters,

since pores fill more easily than without coupling. In contrast, the inferred distribution

in the second case correctly reproduces the actual distribution, not only for low diameter

small pores (which indeed fill at equilibrium), but also for other pores. A small deviation is

observed for diameters between the critical diameter and the maximum of the distribution.

For larger diameters, this deviation gets smaller as the fraction increases and the probability

to have at least a neighbour already filled at the equilibrium pressure of a given pore increases.

For the second distribution, no pore is narrow enough to fill at its equilibrium pres-

sure. However, in contrast to cavitation, condensation remains possible without thermal

activation thanks to the spinodal instability. The first pores then fill at their own activated

condensation pressure Pact. Once a large enough proportion of such pores is filled , the larger

ones mostly fill at equilibrium. This explains why the distribution inferred at 3 K assuming

condensation to occur at equilibrium is still reasonably accurate (Figure S11c). As shown

by Figure S11d, increasing the temperature improves the agreement. We thus expect the

hypothesis of condensation at equilibrium to hold for a broad range of distributions shapes.

Hence, for disordered porous materials, the application of BJH method to the condensa-

tion branch of sorption isotherms should yield a good approximation of the pore diameter

distribution.

References

(1) Morishige, K. Hysteresis Critical Point of Nitrogen in Porous Glass: Occurrence of Sam-

ple Spanning Transition in Capillary Condensation. Langmuir 2009, 25, 6221–6226.

(2) Huber, T. E.; Tsou, H. L. Temperature-Dependent Adsorption of Nitrogen on Porous

Vycor Glass. Phys. Rev. B 1998, 57, 4991.

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(3) Thommes, M.; Smarsly, B.; Groenewolt, M.; Ravikovitch, P. I.; Neimark, A. V. Adsorp-

tion Hysteresis of Nitrogen and Argon in Pore Networks and Characterization of Novel

Micro- and Mesoporous Silicas. Langmuir 2006, 22, 756–764.

(4) Simon, G. Rheology and Dynamics of Simple and Complex Liquids in Mesoporous Ma-

trices. Ph.D. thesis, Universität des Saarlandes, 2010.

(5) Bonnet, F. Étude de la Condensation et de l’Évaporation de l’Hélium dans les Milieux

Poreux : Effets du Confinement et du Désordre. Ph.D. thesis, Grenoble University, 2009.

(6) McCarty, R.; Arp, V. A New Wide Range Equation of State for Helium. Adv. Cryog.

Eng. 1990, 35, 1465–1475.

(7) Kierstead, H. PVT Surface of 4He Near Its Critical Point. Phys. Rev. A 1973, 7, 242–251.

(8) Burgess, C. G. V.; Everett, D. H.; Nuttall, S. Adsorption of Carbon Dioxide and Xenon

by Porous Glass over a Wide Range of Temperature and Pressure-Applicability of the

Langmuir Case VI Equation. Langmuir 1990, 6, 1734–1738.

(9) Burgess, C.; Everett, D.; Nuttall, S. Adsorption Hysteresis in Porous Materials. Pure

Appl. Chem 1989, 61, 1845–1852.

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