V Langlois

C T Nguyen

F Detrez

‡ J Guilleminot

C Perrot

Permeability of polydisperse solid foams

he e'et of polydispersity on fom permeility is investigted y numeril simultionsF pom strutures re (rst generted y vguerre tesselltions vi the xeper softwreD nd relxed to minE imize the surfe energy vi the urfe ivolver softwreF he )uid )ow nd permeility re then lulted y mens of poreEnetwork simultionsD y onsidering either fully openEell foms or foms with rndomlyEseleted losed windowsF hi'erent on(gurtions of window perture re usedD inluding identil window perture sizeD identil window perture rtioD or rndom window perture rtioF he min results re otined for the se of foms hving identil nd uniform window perture rtiosF por suh foms nd t onstnt men pore sizeD fom permeility is found to strongly inrese with the polydispersity degreeF he numeril results re employed to disuss the vlidity of the men pressure (eld ssumption used to lulte the fom permeilityD the e'et of smll poresD nd the de(nition of n equivlent uelvin fom sizeF e show tht s long s the )ututions of the window perture rtio remin lowD fom permeility n e estimted y using the men pressure (eld hypothesisF he wek e'et of smll pores on permeility is relted to their smll ontriution to the overll )uid volume frtionF pinllyD vrious estimtions of the equivlent uelvin fom size sed on poreEsize distriution re proposedF NOTICE: his is the uthors9 version of pper pulished in hysil eview i vnglois et lF@PHPPAD ermeE ility of polydisperse solid fomsD hysF evF iD IHSD HISIHID doiX IHFIIHQGhyseviFIHSFHISIHIF is possible. Therefore, these materials can be used in various applications, such as heat-exchanger, catalyser, lter, and acoustic insulator [1]. For such applications, the permeability is a key parameter to control and to ensure the eciency of foam material. For example, in the context of sound absorption of monodisperse open-cell * vincent.langlois@univ-eiel.fr † cong-truc.nguyen@univ-eiel.fr ‡ fabrice.detrez@univ-eiel.fr § johann.guilleminot@duke.edu ¶ camille.perrot@univ-eiel.fr pigure IF wirosopy imges reveling the mirostruture of solid fomX @A ligments @or lteu ordersD strutsA nd verE tiesD @A open or losed windows @toner prtiles re spred over the fom smple to highlight the memrne surfesAF foam, it was shown that modeling transport properties such as permeability is a key step to nd the optimal microstructures dened by the pore size and the porosity [2,3]. However, foams can have various degrees of polydispersity [4,5] and/or various membrane contents [6,7]. Therefore, polydispersity and membrane content can be taken into account to search for new optimal foam microstructures.

Dierent works have focused on the eects of morphological parameterspore or strut diameter, porosity, specic surface area, window aperture size, etc. on the permeability of monodisperse open-cell foams [1,8,9]. The eects of partially or fully closed membranes separating neighbor pores on foam permeability have recently been investigated [10].

Dierent approaches for modeling foam permeability can be found in the literature.

A rst approach is based on the so-called Carman-Kozeny equation [11,12]:

K = C[φ 3 /(1 -φ) 2 ]d 2 c
where d c is the inverse of the specic surface area a c (pore surface/solid volume), φ is the porosity, and C is a dimensionless constant depending on the microstructure geometry. This equation was introduced to estimate the permeability of granular material. In the case of foam permeability, some alternative choices for the characteristic micro-structure length d c have been proposed in the literature, such as the pore size or the hydraulic radius d h = 4φ/a c (see Ref. [9] for a recent review). As C depends on the microstructure geometry, these approaches cannot give a full analytical formulation and require numerical calculations to elucidate the relationship between C and the microstructure geometry.

A second approach, especially relevant for foams having membranes [10] or for porous media having small constrictions connecting large pores [1,13], was suggested by Despois and Mortensen [1]. In this approach, the uid ow passing through constrictions is supposed to be governed by the dierence of uid pressures between interconnected pores. The relationship between uid ow and pressure drop used by Despois and Mortensen is due to Sampson [14]. For monodisperse foam, the foam permeability can be fully derived by considering the symmetries of the periodic unit cell in the case of ordered foams [10],

or by considering a mean pore in the case of disordered foams [13]. This approach gives a very accurate estimation of foam permeability in a large range of constriction sizes. Moreover, this approach is similar to the porenetwork approach introduced by Fatt [15] to study the permeability of a network of tubes. The pore-network approach is also very useful to study the permeability of percolating porous media [16] and can be easily implemented in numerical calculations with low computational costs compared to other numerical methods (nite element, nite volume, boundary element). If the permeability of monodisperse and ordered foam has been extensively studied, then the permeability of polydisperse and disordered foams has received much less attention to date.

Due to the natural polydispersity of soil, the eects of polydispersity on the permeability of granular material have been studied in the past. Dierent equations for permeability can be found in the literature. For example, the Hazen formula [17], cited in numerous geotechnical textbooks [1820], links the permeability K to an eective size D 10 : K = C Hazen D 2 10 where C Hazen is an empirical constant, and D 10 is the grain size for which the mass of particles smaller than D 10 is equal to ten percent of the total mass of particles. However, Hazen formula is not very accurate to predict the permeability of granular material [21], and formulations derived from the Carman-Kozeny formula give better predictions. Carrier [21] proposed the expression d e = ( i x i /d i ) -1 for the eective grain size to be used in the Carman-Kozeny formula instead of d c ; where x i is the mass fraction of particles having a size d i . These formulas suggest that the sizes, D 10 and d e , may be interpreted as the grain size of an equivalent monodisperse granular media for the permeability. With a similar approach, formulas for the permeability of polydisperse foams involving the pore size of an equivalent monodisperse foam could perhaps be derived.

In this paper, pore-network simulations are used to study the permeability of random polydisperse foams with various membrane contents in the case of high porosity (corresponding to the dry foam limit in liquid foam, i.e, ligaments are lines, vertices are points and porosity φ is equal to 1). In the rst section, we detail the porenetwork method used for the calculation of permeability and pressure uctuations within a single pore are not described in detail. In the case of circular aperture of radius r o,i = d o,i /2, the volume uid ow rate q i passing through the window aperture i depends on the pressure dierence (P i2 -P i1 ) and on a local uid ow conductance G i , as established by Sampson [14]:

q i2→i1 = G i (P i2 -P i1 ) , with G i = r 3 o,i 3µ , (1)
pigure PF @A plow pttern nd @A its equivlent poreE network shemeF @A he(nition of geometril prmeter inE volved in permeility lultionF he plne rossEsetion A of re A divides the smple into two prtsF sf two poresD ind i+D ssoited with window re loted from either side of the rossEsetionD then the window is loted on the rough surfeF where µ is the dynamic viscosity of the uid.

The pore-network model describes the foam pore-space as a network of uid ow conductances [Fig.

2(b)].

Only periodic networks are considered. The total number of elements within the elementary structure will be denoted N p for pores and N w for windows. Assuming that the ow of the incompressible uid is steady, the sum of currents owing towards or away from the pore n is equal to zero (similarly to Kirchho 's junction rule):

m G mn (P m -P n ) = 0, where G mn is equal to the local conductance associated with the window shared by the pores n and m (and equal to 0 if pores are not connected).

A macroscopic pressure gradient is imposed by applying a macroscopic pressure dierence ∆P macro between upper and lower boundaries. When the pores are connected from top to bottom of the sample, the pressure in each pore can be calculated from the following matrix form equation:

[G][P ] = [F ], (2)
where

[P ] is the pressure vector for all pores;

[G] is the matrix of local conductances:n G mn on the diagonal and G mn elsewhere;

[F ] is a forcing vector containing zeros except for the components associated with the pores located on the upper and lower limits of the sample (the upper pores have lower pores as neighbors by periodicity and vice versa). For a boundary pore m, F m = n,lower G mn ∆P macro if m corresponds to an upper pore, while F m =n,upper G mn ∆P macro if the pore belongs to the lower boundary.

From the knowledge of pore pressures, the global ow rate Q z is equal to the sum of the local ow rates q i passing through the windows covering the rough surface associated with a cross-section A(z) of area A [see Fig. 2(a) for a denition of the rough surface] :

Q z = N wA i=1 q i = N wA i=1 G i ∆P i , (3)
where ∆P i = P i+ -P i-[where i + and i -refers to the pores associated with the window i with i + located above the cross-section and i -below, see Fig. 2(c)].

As the uid is considered incompressible, the global ow rate Q z is the same for any rough surface, or any continuous surface of windows joining lateral faces and respecting the lateral periodicity. In our calculation, we use the upper limit of the samples: the pore i -is an upper pore and the pore i + is a lower pore rebuilt by periodicity.

Finally, the permeability K z is calculated as follows :

K z = µ Q z A 1 λ , (4)
where λ = ∆P H and H is the length of the sample in the direction of the macroscopic pressure gradient (equal to the spatial period in periodic structures). Note that for isotropic porous media:

K x = K y = K z .
From Eqs.( 2)-( 4), it appears that the macroscopic permeability K depends on local conductances G i through the determination of the pressure eld [Eq. ( 2)] and the global ow rate Q z [Eq. ( 3)]. It should be noticed that the eect of local conductances on the global ow rate is direct, whereas the eect on the pressure eld is of the second order compared to that of the macroscopic pressure dierence ∆P macro . For example, consider the case where the same coecient α is applied to all local conductances (that is, G i,1 = αG i,0 ). From Eq. ( 2) and the expression of the forcing vector [F ], it appears that the local pressures are not modied (P n,1 = P n,0 ), while the coecient α applies to both the total ow rate and the permeability (K 1 = αK 0 ).

The pore-network model with Sampson's law has been successfully validated by comparing its predictions to numerical calculations on Kelvin structures and to experimental measurements performed on monodisperse foams [10,13,22]. Pore-network models are usually considered to describe the conductivity through a network of tubes. In such a case, the local conductance is given by the Poiseuille law [23,24]: G tube = πR 4 t,i /(8µl t,i ), where R t,i and l t,i are the inner radii and the length of the tube i, respectively. In such a model, the distance between the connected pores corresponds to the length of the tubes. The pressure drop is uniformly distributed over the length of the tube, whereas in foams, the pressure drop is located near the membrane aperture. In a foam structure, the distances between pores and the positions of pores are not explicitly present in the porenetwork model. The input geometrical parameters of the pore-network model are the sizes of the membrane apertures and the structure of the pore network. This structure is specically dened by knowing: (i) the indices of the pores associated with each window ; (ii) if a window is inside the periodic structure or located on its boundary ;

(iii) which pore is inside or outside the periodic structure for a boundary window. Note that the connectivity of the pores in the network can be represented by a square matrix, called the adjacency matrix and denoted by a ij , which is dened as follows [25]: a ij = 1 if the pores, i and j, are neighbors, and 0 otherwise.

In Appendix A, we present two alternative methods to calculate the permeability. The rst approach follows from the denition of the permeability [Eq. ( 4)], while the second is based on energetic consideration. Moreover, we introduce two macroscopic parameters on which permeability can depend, namely the volume density of pores, given by ρ p = N p /V , and the volume density of windows, ρ w = N w /V with V = AH, the volume of the elementary structure. Note that the inverse of the cube root of the pore density is a good estimator of the mean distance between pores δ p . These methods lead to the following expressions for the permeability:

K av = µρ w λ G∆P ∆z w , (5)
K e = µρ w λ 2 G|∆P | 2 w , (6)
where w = 1 Nw i∈{w} is the average over the windows.

Our pore network simulations show that Eqs. (4), ( 5) and

(6) lead to the same values of permeability. Because a term ∆z i = z i+ -z i-appears in Eq. ( 5), the positions of the pores seem to have a direct eect on permeability. This result, however, is in contradiction with the previous discussion and the form of Eq. ( 6). To check that permeability does not depend on the exact positions of the pore centroids (with a xed structure of the pore network), we performed calculations by adding a random term to the position of the pores, and obtained results identical to the ones without the perturbations. However, it is important to note that the spatial distribution of pores leading to ∆z i , aects the permeability by its eect on the structure of the pore network.

For any window i, the pressure drop ∆P i is proportional to ∆P macro . Moreover, increasing the number of pores (or the pore density ρ p ) by a factor 2 3 will increase by a factor 2 the number of pressure drops when the uid ows through the pore network. Therefore, the pressure drop can be written as ∆P i = λρ -1/3 p ∆ Pi . Similarly to the pressure eld P , the normalized pressure P depends on the structure of the network and the normalized local conductances G i / G w = r 3 o,i / r 3 o w . However, unlike P , P does not depend on the size of the elementary structure (while its size is larger than the size of the representative elementary volume). Combining these results with Eq. ( 6) yields to the following expression for the permeability:

K = ρ w ρ 2/3 p µG w F, (7)
where F = r 3 o ∆ P and illustrated in Fig. 4 for our relaxed foam samples).

In this work, each step is based on the use of a specic powerful and free software: Neper software for the generation of periodic and isotropic Laguerre structures [30] and Surface Evolver for the relaxation calculation [31].

a b

pigure QF @A Qh periodi vguerre mirostruture performed y xeper @pth ID C d = 0.95AD @A the sme mirostruture fter relxtion lulted y urfe ivolverF

In the rest of this section, we rst introduce some common parameters to Laguerre structures and to the relaxed ones. Then, we give some details concerning both calculation steps and present some results related to structural parameters involved in pore-network model.

Microstructural parameters

We dene two microstructural parameters characterizing the pore size and the pore shape (compared to a sphere). The pore size is dened by a sphere equivalent diameter d p , which represents the diameter of a sphere having a volume equal to the pore volume V p . The pore shape is measured by the sphericity Ψ, which is dened as the ratio of the surface area of a sphere having the same volume as the pore to the surface area S p of the pore: Ψ = (36π) 1/3 V 2/3 p /S p . The pore sphericity takes a maximal value equal to one for a spherical pore. For example, the sphericity is equal to 0.67 for a regular tetrahedron, 0.81 for a cube and 0.91 for a Kelvin structure (truncated octahedra). Moreover, we dene the face size (or window size) as the diameter d w of a circle having an area equal to the face area S w (see Fig. 1): d w = 2r w = (4S w /π) 0.5 .

Note that the face surface area takes into account the curvature of the membrane in the relaxed structures.

From the pore-size distribution, dierent pore size averages can be dened based on the weighting of the mean.

The calculation of these averages follows the general formula:

d m,n = p d m p p d n p 1/(m-n) . (8)
In this paper, we will consider three dierent pore di- To measure the polydispersity degree, we used the normalized standard deviation of pore sizes C d dened as:

C d = σ dp d 1,0 , (9)
where σ dp is the number-weighted standard deviation of pore sizes.

Similarly, the pore sphericity is characterized by the number-weighted mean Ψ and the standard deviation σ Ψ value (or the pore sphericity variation coecient

C Ψ = σ Ψ / Ψ ).

Polydisperse Laguerre structures

Neper software uses the distributions of pore size d p and sphericity Ψ as inputs (i.e., the probability distributions and the associated hyperparameterstypically a mean and a standard deviationmust be specied).

Concerning the probability distribution, we use lognormal distributions for the pore size and the variable (1 -Ψ), so that highly polydisperse microstructures can be obtained. Dierent Laguerre structures were generated by changing the polydispersity parameter C d with the mean pore size d p = d 1,0 set to 1.0.

We have considered two ways to modify Ψ and C Ψ as the polydispersity parameter C d changes: (1) in the rst case (path 1), we considered known structures of foams,

(2) in the second case (path 2), we consider Laguerre microstructures maximizing Ψ for each polydispersity parameter C d . In the rst case (path 1), we relied on known structures of foam samples: monodisperse foams and a highly polydisperse polyurethane foam sample (characterized by x-ray microtomography, see Ref. pigure SF indow size distriutions for vguerre nd relxed mirostruturesX @A vguerre pth ID @A vguerre pth PD nd @A relxed pth IF xote tht dotted retngles re dded to grphs in the viinity of their origin to highlight tht the vguerre pth I ondition genertes polyhedr with smll fes unlike the other onditionsF

The structure of monodisperse foam samples can be approximated by ordered monodisperse structures such as the Kelvin structure and that of 34].

For these two ordered monodisperse structures, we have For each conguration of input parameters (C d , Ψ , C Ψ ), ve samples were generated with at least 1000 pores to ensure that the averages of the parameters of interest over the ve samples did not depend on the number of pores (see details in Ref. [5]). The range of variation of the polydispersity degree C d is 0.09 to 0.95 for path 1 and 0.11 to 0.78 for path 2.

C d = 0, Ψ ≈ 0.

Relaxed microstructures

The Surface Evolver is used to relax the Laguerre structures, i.e., to minimize the surface area (surface energy, see Appendix B for details) and to satisfy Plateau's laws.

Therefore, a signicant fraction of small faces (or short edges) shrink to zero area (or zero length).

All edges and faces smaller than 10 -3 d 1,0 for edges and 10 -4 d 2 1,0 for faces were removed to produce topological transitions favorable to surface-area minimization. Consequently, the relaxation process modies the windows size distribution as shown in Note that the cases, r o cst and t o cst, were previously considered for the permeability calculation of ordered Kelvin foam in Refs. [10,22]. In such foam structures, symmetries of the periodic unit cell simplies the problem resolution and analytical formulas can be derived (in the dry foam limit):

K = 4 3 π 1/3 µG 0 d 1,0 , for r o cst , (10)
K = c K t 3 o d 2 1,0 , for t o cst , (11)
where c K ≈ 0.0283. pigure VF ermeility lulted y poreEnetwork method forX @A onstnt perture size @ro stAD @A onstnt perture rtio @to stAF sn pnel @AD the solid line is Nv F the eects of the evolution of the macroscopic (or mean) parameters (ρ p , ρ w , N v , G w ). The permeability K depends on the whole eects, whereas F depends only on the former (1). Figure 9 pigure IHF ermeility evolution ginst ∆to/ to for pth P relxed struture nd the se of rndom perture rtio @to rndomAF he sme trends re oserved with the pth I ondition nd the unrelxed struturesF xote tht to = 0.5 nd the stndrd devition of to is ∆to/3 G is inappropriate to calculate the foam permeability is perfectly known for monodisperse and ordered foam as shown in Ref. [10]. In this case, the foam permeability is deduced from an eective local conductance which can be estimated by a specic averaging method given by Kirkpatrick [16] and not by the arithmetic average. However, this averaging method, eective for topologically ordered structures (N v constant), is not appropriate for polydis- perse structures.

t 3 o d 3 w = t 3 o d 3 w = t o 3 1 + (∆t o / t o ) 2 d 3 w . If

B. Eect of closed windows

In this section, we present results concerning the effect of non-perforated membranes that close windows and prevent ow between neighboring pores from occurring. A open porosity Rop @de(ned s the rtio of the volume of the pores elonging to the perolting pore network to the volume of ll the poresAD nd dimensionE less permeility K(xo)/K(1) for @A ro st nd @A to stF sn pnel @AD the dshed line orresponds to the se ro st with low polydispersity nd highlights the e'et of the lol ondutne dispersion on the urve shpeF snset in pnel @A shows the peroltion threshold lulted from iqF @IQA @full lineA nd the one estimted from poreEnetwork resultsD αp,PN ginst the prmeter Nv / N 2 v -Nv F endD the dshed line is 0.865 X + 0.033 with X = the sissF merical structure are duplicated once in each direction before performing our pore-network simulations. Calculations were repeated on 80 independent realizations and averaged.

As shown in Ref. . This eect of polydispersity on the percolation threshold could be interpreted as the eect of large pores producing shortcuts in the random walk through the pore-network. Shortcuts then reduce the number of steps required to cross the sample. We have calculated the minimal number of pressure drops (or the minimal number of steps from pore to pore) required to connect each pore to its one-period-translated counterpart. This calculation is performed by using the property of the adjacency matrix (the term ij of the kth power of a ij is non zero if a path having k steps links pore i to pore j). Fig. 12 shows the mean number of drops n drop between pores and their translated counterparts. For a low polydispersity degree, the mean number of pressure drops n drop is between the mean number obtained with a Kelvin structure (n drop /N 1/3 p = 1/2 1/3 ≈ 0.79) and that obtained with a Weaire Phelan structure (n drop /N 1/3 p = 1). When the polydispersity degree increases, the number of steps to join two points separated by a period decreases. Rigorously, this parameter applies for open-cell structures but the cluster of pores that forms during the percolation process can benet from the existence of shortcuts (to progress more quickly). Now, we turn our interest into the estimation of the percolation threshold. For ordered networks (3D lattices) for which N v is constant, Galam and Mauger [39] found an approximate formula linking

x p to N v : x p,GM = 0.754 2 3 (N v -1) -0.9346
. This expression predicts that the percolation threshold decreases as N v increases, but our results [Figs. 6 and 11(a)] show a contrary trend between the number of neighboring pores N v and x p . In tree-like networks where loops can be neglected [25,40,41] (i.e. there is an unique path linking any two nodes or pores), the percolation threshold is given by:

x p,t = N v N 2 v -N v = 1 N v 1 + C 2 Nv -1 . (12)
When applied to ordered networks (such as the structure of an ordered monodisperse foam), Eq. ( 12) yields

x p,t = (N v -1)

-1

and leads to underestimate the percolation threshold (x p,GM /x p,t = 1.101 [N v -1] 0.0654 > 1).

Note that for ordered networks, the percolation threshold can be written as: x p,GM = 0.754 3 2 x p,t 0.9346

. Extend this relationship to the case of polydisperse structures leads to:

x p ≈ 0.754 3 2

N v N 2 v -N v 0.9346 . ( 13
)
Inset of Fig. 11(a) shows that this approximation underestimates the percolation thresholds when the polydispersity degree increases. A linear relationship between x p and the structural parameter N v / N 2 v -N v could be more appropriate. However, it is worth mentioning that our simulations are performed on structures that are too small to determine an accurate value of the percolation threshold [42].

Other expressions for the percolation threshold have been suggested in literature [41,43]. For example, the percolation threshold can be approximated by the largest eigenvalue of the adjacency matrix or a matrix combining the adjacency matrix and the degree matrix (which is a diagonal matrix whose elements are equal to the degree of the nodes, i.e. the number of neighboring pores). In some circumstances [41], these estimators can be more accurate than the naive estimator x p,t . Alternatively, in fractured porous media, percolation threshold expressions can be derived from the concept of excluded volume [44]. In such porous media, it is shown that the number of fractures per excluded volume controls the percolation [45].

In the case r o cst, the permeability K exhibits a linear dependence on the fraction of open windows x ow (except for x ow close to the percolation threshold x p ). This result is in good agreement with the behavior of ordered lattice networks.

Indeed, in ordered lattice networks, if the fraction of open windows x ow is not too close the percolation threshold (x ow > 3/N v ), the permeability K (x ow ) is well predicted by the eective medium approximation (EMA) [16]: K (x ow ) /K (1) = (x ow -x ) / (1 -x ) with x = 2/N v . However, this relationship predicts an eect of the pore-network structure through the term x . But, our pore-network simulations show that the relationship between K and x ow is almost independent on the polydispersity degree C d . This result could be interpreted by invoking two phenomena having opposite eects on permeability. (1) The rst one involves the eect of disorder on the percolation threshold.

As the percolation threshold decreases when the polydispersity degree increases, it can be expected, at fraction of open windows constant, that the permeability is all the higher as the polydispersity degree is high. (2) However, it is observed in ordered structure that a decrease of the connectivity degree reduces permeability. Therefore, the decrease of N v when the polydispersity degree increases should also contribute to reduce the permeability.

Concerning the case t o cst, the permeability depends not only on x ow but also on the polydisperse C d as shown on Fig. 11(c). In ordered foams, the dispersion of local conductances induces a reduction of ratios K(x ow )/K(1), and the eect is more pronounced when the dispersion is important [10]. This eect is shown on Fig. 11(c

IV. DISCUSSIONS

In this section, we discuss various points regarding the permeability of fully open-cell polydisperse foams. First, the use of the mean pressure eld assumption to calculate the foam permeability is discussed. Then, we specically address two main questions associated with the case t o cst: how do small pores contribute to permeability? Is it possible to dene a Kelvin foam equivalent size from the pore-size distribution, i.e. to determine the pore size of a foam having a Kelvin structure and exhibiting the same permeability as the original polydisperse foam?

A. Mean pressure eld approximation

As shown in Sec. II A, the permeability can be calculated from expressions depending on the pressure dierence ∆P i associated with each window [see Eqs. (A2) and (A6)]. Consequently, the accuracy of the permeability prediction depends on the accuracy of the pore pressure prediction. The most simple way to estimate the pore pressure is to consider the mean pressure eld approximation for which the pressure of a pore located to the position z is equal to λz + cst. We introduce the random uctuations P * i of pressure around the mean pressure eld and write P i = λz + P * i + cst. Therefore, the pressure drop ∆P i = P i+ -P i-associated with the window i is equal to: ∆P i = λ∆z i + ∆P * i where

∆z i = z i+ -z i-> 0 [see Fig. 2(c)].
K ρ w µ = G|∆z| 2 w + G ∆P * ∆z λ w (14) K ρ w µ = G|∆z| 2 w + 2 G ∆P * ∆z λ + |∆P * | 2 λ 2 w . ( 15
)
The rst term G (∆z) 2 w in the right-hand side appears in both equations and leads to the following expression of the permeability K mf with the mean pressure eld approximation:

K mf = N v πd 3 3,0 r 3 o (∆z) 2 w . ( 16
)
Note that ∆z i is related to the inter-pore distance δ p,i and the angle θ i between the inter-pore direction and the direction of the macroscopic pressure gradient [Fig.

2(c)]

: ∆z i = δ p,i cos (θ i ). pigure IRF gomprison etween poreEnetwork @xA simE ultions on relxed strutures nd permeility lultions from men (eld hypothesis iqF @ITAX @A onstnt perture size ro stD @A onstnt perture rtio to stF cos 2 (θ) w = 1 3 . Therefore, we have that r 3 o (∆z)

2 w = 1 3 r 3 o δ 2
p w . Moreover, Eq. ( 16) applies to highly porous foams (dry foam limit), i.e. φ = 1. In the case where the porosity φ is less than 1, the values of permeability given by Eq. ( 16) must be multiplied by φ.

Figure 14 shows that the permeabilities estimated with the mean eld approximation [Eq. ( 16] are quite accurate for the cases r o cst and t o cst. The relative deviation (K mf /K PN ) -1 between the mean eld prediction K mf and the pore-network calculation K PN is less than 12% on the variation range of C d used in the simulations on relaxed structures. In the case t o random, Fig. 15 shows that the values of permeability calculated with the mean eld assumption evolve

as t 3 o A = t o 3 1 + (∆t o / t o ) 2
, contrary to porenetwork results (Fig. 10), and become less accurate as the variation ∆t o of the aperture ratio or the polydispersity degree C d increases. Specic expressions for K mf can be calculated for each pigure ISF gomprison etween poreEnetwork simultions nd permeility lultions from men (eld hypothesis in the se to rndom nd relxed pth PX @A C d = 0.11D @A C d = 0.78F snset grphs orrespond to the reltive preE dition error with the men (eld hypothesisF hshed lines re lulted with K mf (∆to) = K mf (0) 1 + (∆to/ to ) 2 while rosses orrespond to the lultions performed with the dt ssoited with the rndomly drwn on(gurtions used for the pore network simultionsF case considered in the simulations:

K mf = 3 π N v µG 0 d 3 3,0 |∆z| 2 w , for r o cst , (17)
K mf = N v πd 3 3,0 r 3 w |∆z| 2 w t 3 o , for t o cst , (18)
K mf = N v πd 3 3,0 r 3 w |∆z| 2 w t 3 o w , for t o random . ( 19
)
The permeability expressions [Eqs. ( 17), ( 18) and ( 19 We now consider the modeling of the corrective term E error of the mean eld approximation. In Eqs. ( 14) and ( 15), the second terms in the right-hand sides are two different expressions for the corrective term to be applied to the mean pressure eld approximation. Requiring these expressions to be equal implies that the corrective term must be negative:

1 λ G∆P * ∆z w = - 1 λ 2 G (∆P * ) 2 w < 0. (20)
Consequently, the permeability calculated with the mean pressure eld approximation is an upper bound for the permeability (as conrmed by our pore-network simulations). This result was shown previously in Ref. Durand et al. [24].

Finally, the error in the permeability calculation with the mean pressure eld approximation can be written as follows:

K -K mf = -Nv π d 2 3,0 E error , (21)
with

E error = r 3 o (∆P * /λ) 2 w d 5 3,0 .
In the cases t o cst and t o variable, the aperture sizes r o and the uctuating term ∆P * are not independent random variables, and consequently, r

3 o |∆P * | 2 w = r 3 o w |∆P * | 2 w + cov r 3 o , |∆P * | 2 . Figure
∆P * λd 3,0 2 w ≈ 0.118BC + 4.101AC -0.207AB -0.578C 2 + 0.012B 2 -8.656A 2 -0.159C + 0.034B -0.159A. (22)
Figure 18 compares the values of R error found by porenetwork simulations to the values estimated by using Eq. ( 22). The coecient of determination R 2 is equal to 0.999.

Finally, an improved version of the permeability formula based on the mean pressure eld assumption can be derived from Eqs. ( 16) and ( 22): pigure IVF gomprison etween Rerror lulted y poreE network simultions @to st nd to rndomA for ll fom relxed strutures @ pth I nd pth P A to vlues luE lted with the pproximtion given y iqF @PPAF assumption of constant aperture rate ( t o cst). In polydisperse structures, small pores are numerous but do not occupy a large volume. This common result can be illustrated by using the cumulative pore-size distribution with various weightings. These cumulative pore-size distributions can be dened as follows:

K = N v πd 3 3,0 r 3 o (∆z)
F n (d) = i where dp,i≤d d n p,i N p,t i=1 d n p,i , (24)
where N p,t is the total number of pore.

By considering n = 0, we obtain the proportion in number of pores having a size smaller than d (numberweighted distribution). For n = 3, we obtain the volume fraction of all the pores smaller than d (volume-weighted distribution). The dierent cumulative pore-size distributions are almost identical in structures having a low polydispersity degree, as shown in Fig. 19(a). On the contrary, for structures having a high polydispersity degree, the volume-weighted distribution is wider than the number-weighted distribution [Fig. 19(b)]. As previously pointed and illustrated in Fig. 19(b), small pores are numerous but do not occupy a large volume in structures having a high polydispersity degree. For example, 61% of small pores in number ll only 4% of the pore space when C d = 0.78.

To highlight the eect of small pores on the permeability of open cell foam samples, pore network simulations were performed on structures for which pores smaller than a threshold size d cp are considered to be completely closed (i.e. their windows are closed or all their window conductances are set to be equal to zero). As expected, the permeability values decreases while the threshold size d cp increases [Fig. 20]. Note that the permeability tends to zero before all the pores are closed. This is brought about by the vanishing of percolation (site percolation) which occurs when the closed pores number is enough to stop the ow of uid. Now, we focus more specically on the case where the volume fraction of small closed pores is low (and d cp < d 1,0 ). Figs. 20 and19 show that a decrease of the permeability value equal to 5% is obtained by closing: 2% of pores in volume (or 5% in number) for structures having a low polydispersity degree (C d = 0.11), and 4% of pores in volume (or 63% in number) for structures having a high polydispersity degree (C d = 0.78). Clearly, the volume fraction of small pores is the key parameter to evaluate the eect of small pores on foam permeability. This result is in good agreement with a model of electric conductivity in a suspension of spheroids due to Fricke [46] and used to model the permeability of fractured rock by Ref. [47]. In this model, the eective permeability of a porous media of permeability K m containing randomly oriented spheroidal inclusions of permeability κ K m is equal to

K = K m (1 -βφ) , ( 25
)
where φ is the volume fraction of inclusions, and β is a coecient depending on the ratio of permeabilities κ and the shape of inclusions (β = 1.5 for spherical inclusion with zero permeability, κ = 0).

C. Equivalent Kelvin foam

In this section, we restrict the analysis to the case where the aperture ratio is constant ( t o cst). For such a polydisperse foam, we dene an equivalent monodisperse/Kelvin pore size d eq , which is dened as the pore size of a foam sample having a Kelvin structure and exhibiting the same permeability and aperture ratio t o as the initial polydisperse foam sample. In the following, we aim to estimate d eq from the pore-size distribution of the polydisperse foam sample.

The equivalent Kelvin pore size can be calculated from our pore-network simulation results [Fig. 8(b)] and from Eq. 11:

d eq = K PN / c K t 3 o 0.5 . ( 26
)
Dierent sizes calculated from pore-size distribution were tested to estimate the equivalent Kelvin pore sizes:

(i) d m,n (d 1,0 , d 3,0 and d 3,2 ), (ii) two sizes dened from the cumulative pore size functions (F 2 and F 3 ), (iii) a size dened from the mean volume of pores larger than a threshold size dened from the volume-weighted distribution F 3 .

As the ratio K/d 2 1,0 depends on C d (Fig. 8), the number-weighted mean diameter d 1,0 is not relevant to estimate d eq . Indeed, the use of d 1,0 as equivalent Kelvin pore size would lead to underestimate the permeability to an order of magnitude for very polydisperse structures.

We also checked that the mean diameter d 3,0 is not suitable (not shown here). Sauter diameter d 3,2 seems more 1.00 0.83 pigure PHF i'et of losing smll pores in the se to st @pth P relxed strutures nd C d = 0.11 or 0.78AF he windows of pores hving pore size less thn dcp re losedF appropriate as it would lead to overestimate the permeability of about 30%, (see Fig. 22).

For granular material, Hazen uses the grain size for which the mass of grains smaller than D 10 is equal to 10% of the mass of all grains: F 3 (D 10 ) = 10% (by assuming that all grains have the same density). Similarly, 10 -1 5 10 -4 5 10 -3 5 10 -2 10 -2 10 -3 26) and the cumulative pore size functions, the values of can be calculated for each case, n = 2 or 3. Figure 23 shows that no unique value of can be found for F 2 and F 3 . We note however that the values of F 2 (d eq ) and F 3 (d eq ) remain in a limited range of variations. Therefore, we chose two values in order to dene two pore sizes from the cumulative functions, F 2 and F 3 , giving an estimate of the equivalent Kelvin pore size: d eq,2 is the size for which F 2 (d eq,2 ) = 0.65 and d eq,3 is the size for which F 3 (d eq,3 ) = 0.43.

1/3 . ( 27
)
where d is the size for which F 3 (d ) = . i.e. d = F -1 3 ( ). The introduction of this pore size function is motivated by the fact that the contribution of small pores to permeability is low.

The function D ( ) depends on the cumulative functions, F 1 and F 3 , as follows:

D ( ) = d 3,0 1 - 1 -F 0 F -1 3 ( ) 1/3 . ( 28
)
From Eqs. ( 26) and ( 28), the values of = D -1 (d eq ) can be calculated (D -1 is the inverse function of D). As for the others attempts, Fig. 23 shows that no unique value of can be found for D -1 (d eq ). However, a value of ≈ 0.14 is in agreement with most of the simulated data for moderately high to high polydispersity degree and can be used to dene another approximate of the equivalent pore size : d eq,1 = D (0.14).

Figure 22 compares the simulated data to the permeability values calculated by using Eq. ( 11) with d 1,0 set to be equal to d eq,i . From which it becomes apparent that the most accurate denitions to estimate d eq are d eq,1 = D (0.14) and d eq,2 = F -1 2 (0.65).

Importantly, we note that the equivalent Kelvin pore Finally, while the following results may be general in character for polydisperse open-cell foams, they will have to be conrmed on other realistic microstructures:

(1) the correction of the permeability calculations based on the mean pressure eld assumption [Eq. ( 22)], (2) the empirical estimate of the equivalent Kelvin size from pore-size distributions (d eq,1 and d eq,2 ) when a constant aperture ratio t o constant is assumed.

pigure PRF urfe free energy density E plotted ginst the polydispersity prmeter p nd ompred to the theory due to urynik et lF PUF E is sled y σ/ Vp 24 shows that the energies in our relaxed foam samples follow the same relationship.

These energies remain slightly higher than their theoretical values, which suggests that our relaxation pro- pigure PVF i'et of polydispersity degree C d on ovrine etweenX @A pore size nd spheriityD @A pore size nd numE er of neighoring poresD @A lol ondutne nd the vE erge of the pore sizesD dp1 nd dp2D ssoited with eh window @mking sense in the se of onstnt perture rE tioAF xote tht the error rs gives the ovrine etween the lol ondutne nd the mximl @or minimlA pore size max(dp1, dp2) @or min(dp1, dp2)AF he rndom vriles used for the lultions of ovrine re normlized or dimenE sionlessF cess is not as advanced as the one used by Kraynik and co-authors. However, regarding the computational costs to go further in the relaxation process, we consider our relaxed microstructures to be quite satisfactory for the purpose of our work (i.e. a deeper convergence of the relaxed microstructures would not signicantly modify the relationship between the permeability and C d , see Sec.

III A).

Number of neighboring pores

Figure 25 shows that the evolution of the distribution of N v with respect to the polydispersity degree is similar for both the Laguerre and relaxed structures: the distribution becomes wider and the number of neighbors associated with the maximal value decreases when the polydispersity degree increases. Similar results were found in

Ref.

[27].

Local conductances

Figure 26 shows the evolution of two parameters linked to the local conductances in the case of constant aperture ratio t o cst (as

µG i = 1 3 t 3 o r 3 o,i ).

Covariances between microstructural parameters

Figure 28 shows the evolution of dierent covariances between microstructural parameters. In Fig. 28(a), we

show that the sphericity is strongly correlated to the pore size in Laguerre structures, and not in relaxed structures.

These results were reported in Ref.

  is made of membranes (also called lms for liquid foams), ligaments or Plateau's borders (junction of three membranes), and vertices or nodes (junction of four ligaments). Solid foam microstructures inherit some properties of the original liquid foam microstructure (Fig. 1): in particular, the poresize distribution and pore position in the solid foam are similar to the bubble size distribution and bubble position in the liquid foam (if coalescence and ripening stay limited). Whereas membranes are necessary closed in liquid foam, they can be open or totally absent in solid foam [Fig. 1(b)], hence allowing for the foam pores to be connected through windows [the term window will refer to the surface where a membrane is located before its possible opening, Fig. 1(b)]. As soon as the fraction of open membrane exceeds a certain amount, called the percolation threshold, a uid ow through the solid foam

  and the procedure used to generate numerical foam microstructures. Results concerning polydispersity degree and morphological parameters are presented in this section. The following section presents the results of porenetwork simulations performed on both fully open cell foam samples and foam structure having a fraction of closed membranes. The last section is devoted to the discussion of various points regarding the permeability of open cell foam samples: (i) the estimation of the permeability by assuming a mean pressure eld, (ii) the eect of small pores and (iii) the estimate of the size of the equivalent monodisperse foam for the permeability, i.e. the pore size of a monodisperse Kelvin foam having the same permeability as the polydisperse foam sample. II. METHODS A. Pore-network model for permeability Because most of the pressure drop is located in the vicinity of the membrane aperture, a simplied calculation of the uid ow based on the pore-network hypothesis is possible [10]. In this framework, a value of uid pressure is associated with each pore and the ow between two connected pores is based on a local uid ow conductance [Figs. 2(a) and (b)]. Therefore, in the pore-network model, the details of local velocity

  the structure of the network and the normalized local conductances G i / G w = r 3 o,i / r 3 o w . A similar reasoning can be used to derive the same kind of dimensionless function with Eq. (5), because ∆z i /H is sensitive to the same size eects than the inverse of the number of pressure drops. B. Numerical microstructure of random polydisperse foams The process used to generate 3D periodic structures of random polydisperse foam is inspired by the work of Kraynik and co-authors [2629]. It involves two steps: (1) lling space with Laguerre polyhedra and (2) relaxing the Laguerre structures to satisfy Plateau's laws and minimize surface area. The pore geometry evolves during the relaxation calculation: starting from polyhedra with at faces and straight ligaments [Fig. 3(a)] to converge to pores with curved faces and ligaments [Fig. 3(b)]. Note that the pore-size distribution remains unchanged during the relaxation process, while the sphericity is deeply affected by the curvatures of interfaces, leading to a mean sphericity ( Ψ ≈ 0.91) independent on the polydispersity degree (as shown previously by Kraynik et al. [27]

  ameter averages: the number-weighted mean diameter d 1,0 = d p ( . denotes the number average), the volume/surface mean (Sauter) diameter d 3,2 = d 3 p / d 2 p , and the diameter of the sphere having a volume equal to the mean pore volume V p , d 3,0 = d 3 p 1/3 . It is worth mentioning that the size d 3,0 is related to the pore density ρ p through the relationship d 3,0 = π 6φ ρ p -1/3(with φ = 1 in the dry foam limit), and that d 3,0 /d 1,0 ≈ spheriity Ψ ginst polydispersity prmE eter C d F irror rs orrespond to stndrd devitionsF he point monodisperse orresponds to the spheriity of uelvin or eireEheln strutureF he point rystlline mteril orresponds to the spheriity prmeters found in polyrysE tlline mterils s reported in efF QPF 1.0119C 2 d -0.0107C d + 1 when the pore size follows a log-normal distribution.

  [5]).

  91 and C Ψ = 0. Similar values for Ψ are reported for random monodisperse foam in Ref.[35] (note that the sphericity Ψ corresponds to the inverse of the parameter β used in Ref.[35]). The microstructure of our highly polydisperse polyurethane foam sample is characterized by Ψ = 0.77, σ Ψ = 0.07, and C d = 0.79. Therefore, for the rst case, the pore sphericity ( Ψ and C Ψ ) for dierent values of C d is linearly interpolated between the pore sphericity of monodisperse foam (C d ≈ 0, Ψ = 0.91, C Ψ = 0) and the pore sphericity of the real polydisperse PU foam sample having (C d = 0.79, Ψ = 0.77, C Ψ = 0.09). The evolution of sphericity parameters are plotted against C d for the simulations used to generate Laguerre tessellations in Fig. 4. Note that this path of pore sphericity evolution with C d encompasses a case observed on polycrystals for which the size distribution of crystals is governed by the minimization of the interfacial energy [32, 36]. Unfortunately, Laguerre tessellation is known to produce polyhedra having very small faces [2628, 30]. Fig.5(a) shows the distribution of face sizes d w and conrms that the number of very small faces rises as polydispersity increases.(2) A second path was explored to limit the generation of polyhedra having small faces. The strategy is based on the observation that Laguerre microstructures maximizing Ψ for each polydispersity parameter C d do not exhibit the above limitation[Fig.5(b)].

Fig. 5 :

 5 the large amount of small faces observed in a Laguerre structure [Fig. 5(a)] is not present in the relaxed structure [Fig. 5(c)]. The relaxation process is time consuming, and its duration depends on the structure of the initial Laguerre samples: to few hours for low C d up to few weeks for high C d (with Surface Evolver running on eight processors). Moreover, the microstructure does not evolve linearly with the computation time: numerous topological transitions occur in the beginning of the calculation, and as the microstructure relaxes, an additional topological transition requires more computation time. Structural parameters involved in pore-network model The pore-network model emphasizes dierent structural eects on permeability, namely (1) eects of the pore network structure and (2) local conductance eects. Topological eects can be partly estimated via the number of neighboring pores N v . It is an important parameter in the context of uid ow through foam samples, particularly when they contain closed membranes [10]. The mean number of neighboring pores N v is an overall measure of the connectivity degree of pores and is useful to monitor the evolution of the pore network structure. The evolution of the distribution of N v with the polydispersity degree is given in Appendix B, but global information can be found by using the mean number of neighbors N v and the normalized standard deviation C Nv of the number of neighbors. The density of windows ρ w and the density of pores ρ p are both aected by the polydispersity and are related to the average number of neighbors N v : ρ w = ρ p N v 2.Figure 6 shows the evolution of N v and C Nv with the polydispersity degree, and compares our results to experimental results found in literature and numerical results determined by Kraynik et al. [27] from their simulations of random soap froth. As a general trend, the average number of neighbors N v decreases as the polydispersity C d increases [Fig. 6(a)] ; and the average number of neighbors N v decreases with the relaxation process. This eect is more pronounced in the case of path 1 which has a large amount of small faces at the beginning of the relaxation process and consequently starts far from the relaxed state. Note that our results dier from Ref. [27] here. The observed discrepancies may be explained by the values for the cut-o thresholds in edge length and face area chosen for the relaxation process. In our simulations, we used very low threshold values, while Kraynik et al. chose larger values that lead to the removal of an important amount of connections between pores. The dierences between numerical results and the fact that they do not reproduce experimental data can be understood by considering that there is no unique relationship between N v and C d : C d essentially depends on the distribution of the pore size and N v depends on the way the pores are arranged in space (and on the curvature of the interfaces). Concerning the normalized standard deviation C Nv , which is a measure of the topological disorder [27] , Figure 6(b) shows that it increases with polydispersity. Moreover, our numerical results are quite similar to experimental results found in literature [5, 29]. Concerning the second structural eect, the local conductance depends on the window aperture size, which is always less than or equal to the window size: G w = r 3 o /3µ ≤ r 3 w /3µ = d 3 w /24µ. Since window aperture is left undened at this stage, we focus on the eect of polydispersity on d 3 w below. As shown in Fig. 7, d 3 w /d 3 1,0 signicantly increases together with the polydispersity degree. The relationships between d 3 w /d 3 1,0 and C d are similar for all foam structures except for Laguerre structures path 1. At a constant mean pore size d 1,0 , we nally get as a consequence that the mean local conductance must signicantly increase with the foam polydispersity (if the aperture size r o evolves as d w ). In fact, the evolution of r 3 w w with C d is similar to that of d 3 3,0 since the ratio r 3 w w /d 3 3,0 is almost constant in the whole range of C d : r 3 w w /d 3 3,0 = 0.0244±0.010 for relaxed structures, and 0.0265 ± 0.025 for Laguerre structures. Additional results concerning foam microstructure are given in Appendix B.

  foam for both cases [Eqs. (10) and (11)]. The case of constant aperture size r o cst gives a decreasing trend of permeability with the polydisperse degree C d (at d 1,0 and G o constant), while the permeability increases strongly for the case of constant aperture ratio t o (at d 1,0 constant). We recall that the local conductances are constant in the case of constant aperture size while they increase with the polydispersity degree in the case of constant aperture ratio. The increase of permeability with polydispersity has been shown by Skibinski et al. [37] with permeability measurements on model open-cell foams without membrane. Besides, the fact that increasing the variation coecient of window sizes increases the permeability was also found in Westho et al. [38]. As shown in Sec. II B, dierent morphological parameters ( N v , G w , ρ p , ρ w , and C Nv ) evolve as the polydispersity degree changes. To distinguish their various contributions to the permeability, we introduced previously (Sec. II A) the dimensionless function F = Kρ 2/3 p (ρ w µG w ). This function allows us to separate the eects of (1) both the relative uctuations of local conductances and the structure evolution from (2) w ) ginst C d see iqF @UAF @A F ginst Nv / N 2 v -Nv in the se of onstnt perture size @ro stAF he solid line is 105X 2 -21.6X + 1.51 with X the

  (a) shows the evolution of F as a function of the polydispersity degree C d . With this representation, the normalized permeability F increases with the polydisersity degree C d for both cases r o cst and t o cst. The function F is especially interesting in the case of constant aperture size ( r o cst), because it is expected that the function depends only on parameters describing the pore-network structure (such as N v or C Nv ). In the next section, we will consider the parameter N v / N 2 v -N v to build an approximate expression for the percolation threshold. In Fig. 9(b), we show that we obtain a master curve F N v / N 2 v -N v with all data obtained for the case of constant aperture size ( r o cst). The eect of a random variation of t o on the permeability is shown in Fig. 10 for structures having various polydispersity degree C d . For all polydispersity de-

  the permeability were simply deduced from mean local conductance, then one would expect the permeability to increase as 1 + (∆t o / t o ) 2 and regardless of polydispersity degree. Figure10shows that the increase is less important than the one predicted by this simple expression. Actually, the fact that the mean local conductance

For

  each foam sample, a fraction of windows is closed with a random spatial distribution over the foam sample. The remaining windows are open either with the same aperture size r o cst or with the same aperture ratio t o cst. The fraction of open windows is denoted x ow . To reduce the eects of the sample size, our nu-pigure IIF i'et of open window frtion xow nd polydisE persity degree C d onX @

  [10] for ordered monodisperse structures, foams containing both open and closed windows can be subjected to percolation phenomena. Fig. 11(a) shows that the pore space is closed (zero open poros-numer of drops n drop etween pores nd their trnslted ounterprt ginst C d F xote tht n drop is normlized y N 1/3 p to orret the size e'etF ity) until the fraction of open windows exceeds a value x p , called the percolation threshold. For the foam structures considered in this work, the percolation threshold is around 0.1 and slightly decreases as the polydispersity degree increases (see Fig. 11(a) and the corresponding inset)

  ) by comparing the both curves t o cst and r o cst at low C d . This eect, observed for low polydispersity, is enhanced when the polydispersity degree increases because the dispersion of local conductances increases in the same time.

  Figure 13(a) shows the evolution of the distribution of ∆P * i /(λd 3,0 ) for various polydispersity degree C d . It is seen that the dispersion increases with C d and that values uctuate around 0. Note that the average ∆P * w is dierent from 0 as the polydispersity degree increases [Fig. 13(b)]. In addition, the squared uctuating term |∆P * | 2 w [involved in the corrective term E error , see Eq. (20)], increases as the polydispersity degree increases [Fig. 13(c)].Using ∆P i = λ∆z i + ∆P * i , we can rewrite Eqs. (5) )ututions ∆P * /(λd3,0)X @A distriutionD @A men vlue ∆P * w ginst C d D @A |∆P * | 2 w ginst C d F and (6) as:

  )] deduced from the mean eld hypothesis give simple analytical expressions revealing the inuence of morphological parameters. For example, in the case of a constant aperture ratio t o cst, it appears that the permeability depends on r 3 w |∆z| 2 w = r 3 w w |∆z| 2 w + cov r 3 w , |∆z| 2 where cov (., .) is the covariance of random variables associated with windows. The random variables r 3 w and ∆z are not independent and their covariance is strictly positive, as shown in Fig. 16. Therefore, the increase of permeability with C d shown in Fig. 8(b) s funtion of C d in the se to stF is the cumulative eect of the evolution of the product r 3 w w |∆z| 2 w and the covariance cov r 3 w |∆z| 2 for the main part when C d > 0.65.

  17 shows the dierent contributions to the error term E error for the case t o cst. The main contribution to the error comes from the product r 3 o w |∆P * | 2 w , and the contribution from the covariance increases as the polydispersity degree increases. The mean eld hypothesis assuming a decorrelation between the pressure eld and the porenetwork structure is no longer possible when the degree of polydispersity is high and/or the uctuations of aperture ratios are important. In this case, nding a strong correction to the mean eld predictions involving disorder parameters (such as C d , C Nv or [d 3,0 /d 2,0 ] -1) and permeility preditions with men (eld pproximtion ginst C d for the se to stF xote tht pigsF IT nd IU give omplementry inE formtion leding to the reltive error (K -K mf )/K mf = -r 3 w |∆P * /λ| 2 w r 3 w |∆z| 2 w F dimensionless parameter (such N v ) is of interest. To this end, and with the aim of estimating the disorder induced by the variation of the local conductance over the rough surface, we introduce a specic disorder parameter dened as: C G = σ G / G w , where σ G is the standard deviation of the local conductance. Assuming the validity of the Sampson's law, C G is equal to the normalized standard deviation of the window aperture size raised to the power of 3: C G = σ r 3 o / r 3 o w . Hereafter, we use A=[d 3,0 /d 2,0 ] -1, B = C G and C = C Nv , to build an approximate formula for d 3 3,0 E error / r 3o w that we will de- note by R error :

  error . (23) B. Eect of small pores In this section, we investigate the eect of small pores on the permeability of open-cell foam samples under the

Fig. 21

 21 Fig.21shows that our results obtained with structures made of discrete pores follow the relationship given by Fricke's equation with a specic value of β for each polydispersity degree.

  on normlized pore size dp/d1,0 of umultive poreEsize distriutions Fn(d) with n = 0, 2 or 3 for pth P vguerre struturesX @A C d = 0.11D @A C d = 0

  C d F xote tht the de(nition of depends on the wy of lulting the orresponding size deq,iX = D -1 (deq) for deq,1D F2 (deq) for deq,2D F3 (deq) for deq,3F Finally, we dene a mean pore size function D ( ), as the size of an average pore of volume equal to the mean volume of the pores having d i > d : D ( ) = i, where di>d d 3 i i, where di>d 1

  size does not correspond to a specic pore belonging to the set of all pores and which could be taken as representative of all pores. For example, for a path 2 polydisperse sample with C d = 0.79, the pores having a size approximately equal to the equivalent Kelvin pore size have on average about 30 neighboring pores. This is much more than the 14 neighbors expected for a Kelvin structure.V. CONCLUSIONTo study the eects of polydispersity on solid foam permeability, we performed dierent numerical simulations to generate foam structures (Laguerre tessellations via Neper software, followed by relaxation calculations to minimize the surface area and satisfy Plateau's laws via Surface Evolver software), and to calculate the uid ow through foam with pore-network simulations by considering either open-cell foam (no closed window) or foam with some closed windows (randomly chosen). Concerning foam microstructures, it appears that dierent structures sharing the same pore size dispersion (as a Laguerre structure and its relaxed version) can have dierent morphologies (in a topological sense). Despite the variety of microstructures generated, certain general conclusions can be drawn concerning the permeability of polydisperse open-cell foam: (1) in the case of constant aperture size ( r o constant), we show that the permeability can be calculated from parameters derived from a dimensional analysis and a dimensionless function F depending on N v / N 2 v -N v (see Fig. 9); (2) in the case of constant aperture ratio t o constant (i.e. the apertures of windows connecting neighboring pores increase as the associated pore sizes), permeability at constant mean pore size increases as the polydispersity degree increases and in a similar way for all foam microstructures (as Laguerre structures and their relaxed versions) ; (3) as long as the dispersion of the window aperture rate remains low, the assumption of mean pressure eld is accurate enough to calculate the permeability [Eqs. (18) and (19)] from key microstructural parameters, d 3,0 , t o , r w , N v and ∆z (however, their characterization requires 3d tomography) ; (4) in highly polydisperse structures, small pores do not contribute signicantly to the global permeability (due to their low volume fraction). Concerning polydisperse foams containing closed windows, the percolation threshold x p decreases with the polydispersity degree while the mean number of neighbors decreases; and the parameter N v / N 2 v -N v was helpful for establishing an explicit link to the microstructure. Moreover, in the case of constant aperture size ( r o constant), the relationship between the permeability and the fraction of open windows is very insensitive to the polydispersity degree itself [Fig. 11(b)]. We interpret this observation as the result of two opposite mechanisms resulting from polydispersity: the decrease of percolation threshold x p accompanied by a lower mean number of neighbors N v . Further study on percolation of polydisperse foams will be necessary to build a complete analytical model as done for monodisperse and ordered foams. For this work, the extension of the Kirkpatrick's model due to Klimenko et al. [48] will be considered.

  Figure27shows the evolution of two parameters linked to the distance between neighbor pores. It appears that the mean inter-pore distance δ p and the size d 3,0 are almost equal[Fig. 27(a)]. This result is expected as

  [27]. The zero covariance in relaxed structures is due to the curvatures of the interfaces leading to a constant sphericity. In Fig.28(b), we show that the covariance between the pore size and its number of neighboring pores increases as the polydispersity degree increases (Large pores have more neighbors).And, Fig. 28(c) shows that the covariance between local conductance and the average of the pore sizes, d p1 and d p2 , associated with each window increases as the polydispersity degree increases (Large windows and high conductances are associated with large pores).I tFEpF hespois nd eF wortensenD ermeility of openE pore miroellulr mterilsD et mterili 53D IQVI @PHHSAF P pF ghevillotte nd gF errotD i'et of the threeE dimensionl mirostruture on the sound sorption of fomsX e prmetri studyD he tournl of the eoustiE l oiety of emeri 142D IIQH @PHIUAF Q F vngloisD eF uddmiD yF itoisD nd gF errotD eoustis of monodisperse openEell fomX en experiE mentl nd numeril prmetri studyD he tournl of the eoustil oiety of emeri 148D IUTU @PHPHAF R wF rongD qF fonnetD rF un vuuD nd gF errotD viner elsti properties derivtion from mirostrutures representtive of trnsport prmetersD he tournl of the eoustil oiety of emeri 135D QIUP @PHIRAF S gF F xguyenD Acoustic foams with pore size distributions and controlled interconnections: Structures-propertiesfabrication relationshipsD hd thesisD niversité risEist upD prne @PHPIAF T yF houtresD xF etllD nd uF hongD i'et of the miE rostruture losed pore ontent on the ousti ehvior of polyurethne fomsD tournl of epplied hysis 110D HTRWHI @PHIIAF U F rF rinhD F vngloisD tF quilleminotD gF errotD F uhidsD nd yF itoisD uning memrne ontent of sound soring ellulr fomsX pritionD experimenE tl evidene nd multisle numeril simultionsD wteE rils 8 hesign 162D QRS @PHIWAF V hF idourdD wF vroixD gF ruuD nd pF vukD ressure drop modeling on solid fomX tteEofEthe rt orreltionD ghemil ingineering tournl 144D PWW @PHHVAF W F uumr nd pF opinD rediting pressure drop in openEell foms y dopting porhheimer numerD snterE ntionl tournl of wultiphse plow 94D IPQ @PHIUAF IH F vngloisD F rF rinhD gF vussoD gF errotD F ghteuD F uhidsD nd yF itoisD ermeility of solid fomX i'et of pore onnetionsD hysil eview i 97D HSQIII @PHIVAF II tF uozenyD eer kpillre veitung des ssers im foE denD oyl edemy of ieneD iennD roF glss s 136D PUI @IWPUAF IP F gF grmnD pluid )ow through grnulr edsD rnsF snstn ghemF ingF 15D ISH @IWQUAF IQ yF itoisD eF uddmiD nd F vngloisD ermeility of monodisperse solid fomsD rnsport in orous wedi 134D TQS @PHPHAF IR F mpsonD xiiF on stokes9s urrent funtionD hilosophE il rnstions of the oyl oiety of vondonF@eFAF IS sF pttD The Network Model in the Study of Porous MediaD hFhF thesisD niversity of outhern gliforni @IWSSAF IT F uirkptrikD eroltion nd ondutionD eviews of modern physis 45D SUR @IWUQAF IU eF rzenD ome physil properties of snd nd grvel with speil referene to their use in (ltrtionD PRth ennD epFD wssF tte ford of relthD fostonD IWVQ @IWVQAF IV uF erzghiD F ekD nd qF wesriD Soil mechanics in engineering practice, 3rd ed.D john wiley edF @tohn ileyD IWWTA pF SIHF IW fF hsD Principles of Geotechnical EngineeringD l enginee edF @genggeD PHHWA pF TTTF PH F roltzD F uovsD nd F hehnD Introduction to Geotechnical Engineering, 2nd EditionD person edF @ersonD PHIIA ppF I!VSQF PI F hF grrierD qoodyeD rzenY helloD uozenyEgrmnD tournl of qeotehnil nd qeoenvironmentl ingineerE ing 129D IHSR @PHHQAF PP F vngloisD F rF rinhD nd gF errotD iletril onE dutivity nd tortuosity of solid fomX i'et of pore onE netionsD hysil eview i 100D HIQIIS @PHIWAF PQ hF vF tohnsonD tF uoplikD nd F hshenD heory of dynmi permeility nd tortuosity in )uidEsturted porous mediD tournl of pluid wehnis 176D QUW @IWVUAF PR wF hurndD tFEpF doD nd hF eireD wximum eleE tril ondutivity of network of uniform wiresX the vemlih lw s n upper oundD roF F oF vondonF erF e wthF hysF ingF iF 460D IPTW @PHHRAF PS wF xewmnD Networks: An Introduction.D oxford edF @yxE ford nivF ressD PHIHA ppF I!VSQF PT eF wF urynikD hF eF eineltD nd pF vn wolD truture of rndom monodisperse fomD hysil eview i 67D HQIRHQ @PHHQAF PU eF wF urynikD hF eF eineltD nd pF vn wolD truE ture of rndom fomD hysil eview vetters 93D PHVQHI @PHHRAF PV eF wF urynikD hF eF eineltD nd pF vn wolD truture of rndom idisperse fomD golloids urfes e hysiE ohemF ingF espF 263D II @PHHSAF PW sF ehioD gF edenhD uF hlditzD nd eF wF urynikD smproved models of solid foms sed on sop frothD gomputtionl wterils iene 120D TH @PHITAF QH F uey nd vF enversdeD yptiml polyhedrl desripE tion of Qh polyrystlsX method nd pplition to stE tistil nd synhrotron xEry di'rtion dtD gomputer wethods in epplied wehnis nd ingineering 330D QHV @PHIVAF QI uF frkkeD he surfe evolverD ixperimentl mthemtE is 1D IRI @IWWPAF QP eF pettlD F erzD gF urillD nd F hmidtD rmetri representtion of Qh grin ensemles in polyrystlline mirostruturesD tournl of ttistil hysis 154D WIQ @PHIRAF QQ hF eire nd F helnD e ounterEexmple to uelvin9s onjeture on miniml surfesD hilosF wgF vettF 69D IHU @IWWRAF QR sF gnttD F gohenEedddD pF ilisD pF qrnerD F röhE lerD yF itoisD pF ouyerD eF intEtlmesD F pltmnD nd F goxD Foams: Structure and Dynamics. @yxford niversity ressD snorportedD PHIQA pF PVVF QS eF wF urynikD pom strutureX from sop froth to solid fomsD w ulletin 28D PUS @PHHQAF QT hF owenhorstD eF vewisD nd qF pnosD hreeE dimensionl nlysis of grin topology nd interfe urE vture in βEtitnium lloyD et wterili 58D SSII @PHIHAF QU tF kiinskiD uF gwiekD F uowlkowskiD fF ysokiD F ejrznowskiD nd uF uurzydlowskiD he in)uene of pore size vrition on the pressure drop in openEell fomsD wterils 8 hesign 87D TSH @PHISAF QV hF estho'D tF kiinskiD yF ediv yD fF ysokiD F eE jrznowskiD nd F hmidtD snvestigtion of the reltionE ship etween morphology nd permeility for openEell foms using virtul mterils testingD wterils 8 hesign 147D I @PHIVAF QW F qlm nd eF wugerD niversl formuls for peroE ltion thresholdsD hysil eview i 53D PIUU @IWWTAF RH F gohenD uF irezD hF enEevrhmD nd F rvlinD eE siliene of the snternet to ndom frekdownsD hysF evF vettF 85D RTPT @PHHHAF RI pF dihiD rediting peroltion thresholds in netE worksD hysF evF i 91D HIHVHI@A @PHISAF RP rF vF prishD tF wF rmmersleyD nd hF tF eF elshD wonte grlo istimtes of eroltion roilities for rious vttiesD hysF evF 126D WRW @IWTPAF RQ wF viD FEF viuD vF vüD wFEfF ruD F uD nd FEgF hngD eroltion on omplex networksX heory nd pplitionD hysF epF 907D I @PHPIAF RR F edlerD tF hovertD nd F wourzenkoD Fract. Porous MediaD olF WUVHIWWTTT @PHIQA ppF I!IVRF RS F F wourzenkoD tFEpF hovertD nd F wF edlerD wroE sopi permeility of threeEdimensionl frture netE works with powerElw size distriutionD hysF evF i 69D HTTQHU @PHHRAF RT rF prikeD e wthemtil retment of the iletri gondutivity nd gpity of hisperse ystems sF he iletri gondutivity of uspension of romogeneous pheroidsD hysF evF 24D SUS @IWPRAF RU eF iigoD F vngD eF lusznyD nd F immermnD snlusionEfsed i'etive wedium wodels for the erE meility of Qh prtured ok wssD rnspF orous wedi 113D IQU @PHITAF RV hF eF ulimenkoD uF roomnD nd eF F ulimenkoD ivlE uting trnsport in irregulr pore networksD hysF evF i 86D HIIIIP @PHIPAF RW F en(eldD F peneD nd F huinkerD e generlized form of ellegen9s theoremD siii rnsF giruit heory 17D QHP @IWUHAF

  Three cases are specically considered for the permeability calculations by pore-network simulations: (1) all windows are open with an identical aperture size r o proportional to the mean pore size d 1,0 : r o,i = r o regardless of the window i and the polydispersity degree C d (we arbitrarily chose r o = 3 1/3 d 1,0 which does not matter if the values of permeability are normalized by µG 0 /d 1,0 ); (2) all windows have an identical aperture ratio t o (t o,i = t o whatever the window i and whatever the polydispersity degree C d ) where t o = S o /S w = r o /r w with S o the area of the window aperture, S w the total area of the window, and r w = d w /2 the equivalent window radius; (3) the aperture ratios of windows t o,i are drawn randomly, according to a uniform law on the interval [0.5 -∆t o , 0.5 + ∆t o ], where ∆t o varies between 0.01 and 0.49. Consequently, t o = 0.5 and the standard deviation of t o is equal to ∆t o /3 0.5 . In the third case, calculations were repeated on 40 independent realizations and averaged.The rst case r o cst focuses on topological disorder because, at constant mean pore size d 1,0 , all local conductances G i where equal to G o = 1 3µ r 3

	15
	14
	13
	12
	11
	10
	9
	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
	1.2
	1.0
	0.8
	0.6
	0.4
	0.2
	0.0
	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
	1.6
	1.4
	1.2
	1.0
	0.8
	0.6
	0.4
	0.2
	0.0
	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
	pigure UF	xormlized men of the ued window size
	d 3 w /d 3 1,0 ginst C d F
	III. PORE-NETWORK RESULTS

pigure TF xumer of neighoring pores NvX @A men vlue Nv nd @A normlized stndrd devition CN v F

A. Permeability of polydisperse open-cell foam

In this section, we pay attention to fully open-cell polydisperse foams (i.e., the membranes separating neighbor-ing pores in Laguerre or relaxed structure have holes).

o , regardless of the window and the polydispersity degree C d . This case is addressed more for theoretical reasons than for practical purposes, because it would be correspond to cases where the aperture sizes would be lower than the smallest pores (leading to very low permeability material in highly polydisperse structures). The second case t o cst combines both topological disorder and window size uctuations (two disorder sources which are correlated since coming from the foam morphology). As the window size uctuates, the local conductances G i are dierent for each window: G i = t 3 o r 3 w,i /(3µ). This case is of practical interest because there are some experimental evidences suggesting that real foams tend to satisfy this hypothesis

[7]

.

The third case t o random introduces a random variable to the aperture rate as expected for real foams. It emphasizes the eect of the window size uctuations on local conductances: G i = t 3 o,i r 3 w,i /(3µ) where t o,i and r w,i are independent random variables.

  0.5 in our poreE network lultionsF C d vries from HFII to HFUVF grees, the permeability increases with ∆t o / t o , although the eect is more pronounced when the polydispersity is low. As for the case t o cst, the increase of permeability with ∆t o / t o is an eect of the increase of the mean local conductance. Indeed, as t o is randomly chosen, the mean local conductance G is proportional to

  For random isotropic structure, δ p,i and θ i are independent random variables and

	4.2
	4.0
	3.8
	3.6
	3.4
	3.2
	3.0
	2.8
	2.6
	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
	0.45
	0.40
	0.35
	0.30
	0.25
	0.20
	0.15
	0.10
	0.05
	0.00
	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0

  pigure PIF i'et of losing smll pores in the se to st nd in the dilute regime φcp = F3(dcp)1F oints orreE spond to dt otined with pth P relxed strutures nd vrious polydispersity degree C d from HFII to HFUV @eh olor orresponds to vlue of C d AF pull line is plotted y ssuming spheril inlusion with zero permeility nd φ = φcp in iqF @PSAF snset grph orresponds to vlues of β djusted for tht the preditions of iqF @PSA (t to our dtF

							2.2
							2.0
							1.8
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							1.0
								0.0	0.2	0.4	0.6	0.8
	1.4
	1.3
	1.2
	1.1
	1.0
	0.9
	0.8
	0.0	0.1	0.2	0.3	0.4	0.5	0.6		0.7	0.8	0.9	1.0
	pigure PPF tio of the vlues KPN of permeility found y
	poreEnetwork simultions to vlues of permeility of uelvin
	fom hving pore sizes equl to d3,2D deq,1D deq,2 or deq,3 for
	vrious C d nd ll fom strutures @pth ID pth PD pth
	I relxedD pth P relxedAF
	for each cumulative function (F 2 and F 3 ), we checked if it
	is possible to nd a unique value	for which F n (d ) =
	and d = d eq for all foam samples with n = 2 or 3. From
	Eq. (

  1/3 D where σ is the surfe tension nd Vp is the verge volume of the poresF

	0.25
	0.20
	0.15
	0.10
	0.05
	0.00
	0	5	10	15	20	25	30
	0.25
	0.20
	0.15
	0.10
	0.05
	0.00
		5	10	15	20	25	30
	pigure PSF histriution of the numer of neighoring pores
	Nv for A vguerre strutures nd A relxed strutures @pth
	IAF he sme trends re oserved with the pth P ondiE
	tionF xote tht the distriutions for fom smples hving
	high polydispersity degree re not fully plottedX the numer
	of neighors Nv of the lrgest pores of fom smples hving
	high polydispersity degree is higher thn one hundredF
	of polydisperse foam structures: foam with a continuous

pore-size distribution in Ref.

[27]

, and random bidisperse foam in Ref.

[28]

. Figure

Appendix A: Alternative permeability calculations

In this appendix, we present the derivation of two alternative expressions for permeability. One calculation is based on the denition of permeability, and the second approach involves energetic considerations.

In Eq. ( 4), the permeability is derived from the global ow rate Q z calculated from a sum of local ow rates passing through any cross-section, A(z). Consequently, we can write:

where

and i∈A(z) and i∈{w} are respectively the sum over all windows covering the rough surface and the sum over all the windows contained in the whole structure. Note that a window i is active in the integral calculation as long as the position of the cross-section is between z i-and z i+ .

From this calculation, we derive the rst alternative expression for permeability:

where V = AH is the volume of the elementary structure.

Another expression for permeability can be derived from energetic considerations. This expression is easily understood by considering the analogy between uid ow through the pore-network and the electrical current transport through a network of wires. Here, ∆P i can be seen as the equivalent of the tension U i , q i as the analogous of the current I i , G i can be viewed as the electrical conductance G e,i , and K/µ is the equivalent of the electrical conductivity σ e . In a network of electrical resistances, the dissipated energy by unit of time P can be expressed as a function of the equivalent conductance of the network G e,e and the power dissipated locally [24]:

where U m = λ e H is the tension applied between the boundaries of the network.

The equivalent conductance of the network G e,e is related to the eective conductivity of the network as follows:

When expressed in terms of the eective uid ow conductance G e and permeability K, Eq. (A4) leads to:

The equivalence between Eqs. (A2) and (A6) requires that:

In the context of electrical current transport, the above equation is equivalent to: