10828
J. Phys. Chem. B 2002, 106, 10828-10833
Transport Properties of Porous PVDF Membranes E. Quartarone,* P. Mustarelli, and A. Magistris Department of Physical Chemistry and INFM, UniVersity of PaVia, Via Taramelli 16, I-27100 PaVia, Italy ReceiVed: October 26, 2001; In Final Form: March 27, 2002
Polymer electrolytes based on polyvinyilidene fluoride are widely studied for their applications in lithium batteries and proton fuel cells. Semicrystalline, porous membranes are conveniently obtained by phase inversion methods. In this paper, we report on the preparation of membranes with porosity ranging from ∼7% to ∼85% and morphology which depends on the preparation conditions. The contributions to the overall electrolyte uptake given by pores filling up and swelling of the polymer amorphous phase are separated. The ionic conductivity of the membranes is discussed in terms of tortuosity and percolation through a continuous medium, that can be approximated by a fractal Bethe lattice.
1. Introduction The success of PVDF-based gel electrolytes is now widely confirmed by the recent developments in Li-ion polymer batteries. During these last two years, in fact, many batteries companies from Japan and the United States have begun the marketing of PVDF-based devices for applications in computers and portable phones and for near future applications in electric vehicles1. The reasons that make these gel-electrolytes particularly exciting are well-known and accepted.2-4 They are multicomponent systems, in which poly(vinilydene fluoride) uptakes large amounts of nonaqueous liquid electrolyte, thus reaching liquidlike conductivities at room temperature, as well as exhibiting good mechanical properties. A critical point in the optimization of these gel electrolytes is the preparation procedure. The conventional casting method4,5 and the Bellcore technology2 have been the most followed approaches to coat free-standing films of PVDF copolymers, but both of them have limitations, namely, difficulty in large scale applications (first technique) and a complex plasticizer extraction step (the second one). Recently, phase separation has received some attention as an alternative procedure for preparing PVDF films which are able to absorb and retain a large amount of liquid electrolyte.6-8 It is a well-known method in the field of the ultra and microfiltration processing,9 because highly porous membranes with controlled and planned morphology can be obtained. When this technique is performed via immersion precipitation, it is possible to prepare PVDF membranes with different porosities and morphologies, ranging from “sponge” like to “finger” like, simply by modulating experimental parameters such as the nature and the ratio of the solvents and nonsolvents, the concentration, and the crystallinity of the polymer.10 The morphology of the membrane plays a fundamental role in the transport of mass and ions through the porous texture and, therefore, in the permeability and conductivity behavior. However, if this kind of gel electrolytes has been widely investigated from a technological point of view, a lack of knowledge exists about their physical chemistry. Some NMR studies on cast PVDF-based electrolytes have showed that they are multiphase systems, in which a crystalline region coexists * To whom correspondence should be addressed. E-mail: ELIANA@ chifis.unipv.it. Fax: +39 0382 507575.
with a swollen gel phase with liquid pockets. A very low level of chemical interactions has been observed between nonaqueous electrolyte and polymer, which acts like a quasi-inert host if the solution content is higher than, roughly, 50 vol %.11,12 On the other hand, when substantial quantities of polymer are present, the interaction in the swollen phase becomes relevant.11 In the case of the microporous PVDF membranes, where the swollen gel is a minority phase compared with the coexisting liquid phase in the pores, few data are now available about the flow phenomena. On the other hand, the transport of mass and ions are open questions in porous media.13 In our previous paper,14 we reported some preliminary results about the conductivity of PVDF membranes with different morphologies and porosities ranging from 55 and 85%, activated by the solution EC/DEC/LiPF6 1.0 M, pure and with the addition of TEGDME as a third component. We showed that nearly all of the pore structure is accessible to the liquid and that it is possible to separate the swelling contribution from the total electrolyte uptake. In this paper, we present a structural and electrical study on several PVDF homopolymer and copolymer membranes for applications in lithium batteries. A wide porosity range (085%) and different morphologies, from a honeycomb to a fingerlike texture, and from skinned to unskinned types, have been explored. Our aim was to highlight the influence of the morphology on the total uptake of liquid electrolyte, in terms of tortuosity and pore connectivity. Particular attention has been then focused on the behavior of the effective conductivity of these membranes versus the actual electrolyte uptake and porosity. In particular, an attempt to describe the ionic transport process in these gels has been preliminarily discussed by means of the continuum percolation theory approach. 2. Experimental Details 2.1. The Raw Materials. The porous membranes were prepared starting both from PVDF homopolymer (Elf Atochem) and from P(VdF-5 mol % HFP) copolymer (Ausimont). Table 1 reports the crystallinity, Χc, calculated from the DSC data by attributing a melting enthalpy of 105 J/g to the totally crystalline PVDF.15 The degree of crystallinity of the compact homopolymer is 49%. A crystallinity increase of ∼10% with respect to
10.1021/jp0139843 CCC: $22.00 © 2002 American Chemical Society Published on Web 09/28/2002
Transport Properties of Porous PVDF Membranes
J. Phys. Chem. B, Vol. 106, No. 42, 2002 10829
TABLE 1: Crystallinity Degree and Morphology of Porous Membranes Based on PVDF Homopolymer and Copolymer membrane homo-COMPACT homo-HYL-7 homo-SOL-8 homo-SOL-4 homo-HYL-4 homo-PF-100 (DMF)a homo-PF-900 (TEP)b homo-PF-901 (TEP) homo-PF-419 (NMP)c homo-PF-421 (NMP) homo-PF-413 (NMP) homo-PF-423 (NMP) homo-PF-424 (NMP) copo-COMPACT copo-COP-AU 3 copo-COP-AU 5 copo-COP-AU 6 copo-COP-AU 9
morphology spongelike spongelike spongelike spongelike fingerlike spongelike spongelike fingerlike fingerlike fingerlike fingerlike fingerlike
Χc (%) 49.0 48.0 57.0 55.5 50.0 59.3 63.0
60.0 29
spongelike spongelike spongelike spongelike
a DMF, N,N-dimethylformamide. b TEP, Triethyl phosphate. c NMP, N-methyl-2-pyrrolidone.
the compact sample is observed in porous matrixes; this is a possible consequence of the precipitation process during the phase separation. The membranes were activated in a drybox by immersion in the liquid electrolyte, consisting of a 1.0 M solution of LiPF6 in a 1:1 w/w EC-DEC mixture, provided by Merck (Selectipur, H2O content 98%), using a Brabender internal mixer. Films having a thickness of about 100 µm were prepared by compression moulding at 200 °C and then quenched in a large water bath within few seconds. Plasticizer extraction was finally carried out in a large excess of a nonsolvent for PVDF, and the resulting membranes were dried by exposing in air. For more details see ref 16. 2.3. Characterization. The thermal properties of the membranes were studied by a modulated differential scanning calorimeter (MDSC 2910, TA Instruments, U.S.A.). The measurements were carried out at 5 °C/min with a modulation period of 40 s and a modulation amplitude of 0.5 °C/min. The microstructure of dried PVDF membranes was observed by means of scanning electron microscopy (Leica Stereoscan
Figure 1. SEM micrographs of two highly porous (∼75 vol %) PVDF homopolymer membranes with a spongelike (a) and a fingerlike texture (b).
440) and atomic force microscopy (Autoprobe CP, Park Scientific Instruments). Ionic conductivity was measured at room temperature by impedance spectroscopy, using a frequency response analyzer (Solartron 1255), connected to an electrochemical interface (Solartron 1287), over the frequency range 10 Hz-1 MHz at an AC amplitude of 100 mV. The density values were determined with a pycnometer, using water as the reference liquid, also estimated from the sample geometry, and used to calculate the volume % porosity as discussed below. 3. Results and Discussions 3.1. Morphology and Structure of the PVDF Membranes. Figure 1 (parts a and b) shows the cross-sectional SEM micrographs of two highly porous (∼75 vol %) homopolymer membranes, cast from 20 wt % PVDF solutions in triethlyl phosphate (TEP; a) and N-methyl-2-pyrrolidone (NMP; b), respectively. The microstructure of the two samples is strongly different. The film cast from TEP (a) shows a homogeneous honeycomb structure, in which the size of the alveoli reaches diameters up to ∼5 µm. The membrane cast from NMP exhibits a highly inhomogeneous fingerlike structure, in which channels of different sizes are separated by layers of discrete polymer globules. These cavities extend through almost the whole film thickness, and they reach diameters up to 30 µm. As already stated in the Introduction, different morphologies of the membranes may be tailored by modulating experimental factors such as the polymer concentration in the cast solution, the nature of the solvent, and mostly the precipitation rate.9,10
10830 J. Phys. Chem. B, Vol. 106, No. 42, 2002
Quartarone et al. TABLE 2: Some Physicochemical Properties of Porous Membranes Based on PVDF Homopolymer and Copolymer Membranes membrane homo-COMPACT homo-HYL-7 homo-SOL-8 homo-SOL-4 homo-HYL-4 homo-PF-100 homo-PF-900 homo-PF-901 homo-PF-419 homo-PF-421 homo-PF-413 homo-PF-423 homo-PF-424 copo-COMPACT copo-COP-AU 3 copo-COP-AU 5 copo-COP-AU 6 copo-COP-AU 9
Φ 0.00 0.23 0.30 0.38 0.46 0.55 0.64 0.71 0.74 0.78 0.81 0.84 0.85 0.07 0.15 0.20 0.29
dm (g/cm3) 1.820 1.402 1.270 1.125 0.970 0.650 0.529 0.496 0.470 0.402 0.342 0.286 0.264 1.760 1.637 1.496 1.408 1.250
Tot 0.18 0.33 0.39 0.45 0.54 0.64 0.73 0.77 0.81 0.80 0.85 0.86 0.88 0.45 0.47 0.51 0.55 0.57
Swel 0.18 0.10 0.09 0.07 0.08 0.14 0.09 0.06 0.07 0.02 0.05 0.02 0.03 0.45 0.40 0.36 0.34 0.28
σ25 °C (S/cm)
τ
2.20 × 10 2.20 × 10-5 2.40 × 10-5 1.45 × 10-4 2.00 × 10-4 4.30 × 10-4 4.80 × 10-4 1.00 × 10-3 1.77 × 10-3 1.63 × 10-3 1.36 × 10-3 1.46 × 10-3 1.06 × 10-3
75.48 90.25 18.92 16.61 9.23 9.63 5.13 3.02 3.45 4.30 4.15 5.79
2.28 × 10-4 3.40 × 10-4 3.90 × 10-4 4.60 × 10-4
2.22 3.18 3.70 4.55
-8
The porosity of the membranes, Φ, is conveniently expressed in terms of the volume occupied by the pores in the matrix, by the following relationship:
Φ)1-
Figure 2. AFM images of a spongelike membrane. (a) The upper surface (skin); (b) the bottom surface (sublayer).
The surfaces of the membranes are generally asymmetric, as the effect of the preparation method, both for what concerns the thickness and the porosity. Thickness differences are well evident in the fingerlike sample (1b): the upper surface is a thin mesoporous layer (called skin), whereas the lower one is a thick macroporous sublayer (∼20-30 µm) with a globular structure. Concerning the surface porosity, Figure 2 reports the noncontact AFM images of the upper (part a) and bottom surfaces (part b) of the spongelike membrane shown in Figure 1a. Very similar images have been obtained for the fingerlike sample. The upper surface (a) shows pores with dimensions of just 1-2 µm, whereas the globular sublayer (b) is less dense with pores reaching diameters larger than 10 µm. The asymmetric morphology of the host membrane, which is generally related to the kinetics of precipitation in water, does not seem to influence the electrochemical behavior of the gel electrolytes, contrary to what happens in the filtration processes where the selective structure of the skin is the main parameter affecting the characteristics of the membrane itself. In fact, no relevant differences in the gel anodic limit voltages were observed by changing the orientation of the skin.17
dm dc
(1)
where dc and dm are the densities of the compact and porous film, respectively. Table 2 reports the apparent densities and the Φ values, ranging from 0 to 0.85, of all of the analyzed membranes. The porosity seems to be strictly dependent on the preparation method and on the nature of the matrix, because of a different compatibility between the homopolymer and copolymer and the casting solvents.9 In the case of the homopolymer, in fact, the phase separation technique via immersion precipitation gives rise to highly porous membranes with Φ between 0.55 and 0.85, depending on the degree of solubility of PVDF in the casting solvent (see Table 1). To obtain homopolymer and copolymer samples with Φ < 0.5, the Bellcore-like procedure was adopted, varying the nature of the plasticizer extraction solvents. 3.2. Membrane Porosity and Uptake of Liquid Electrolyte. Figure 3 reports the behavior of the uptake of liquid electrolyte, , in PVDF homopolymer and copolymer membranes with different porosity. The liquid uptake, , may be conveniently defined by the following equation:
)
(mw - md)dc (mw - md)dc + mddel
(2)
where mw and md are the weights of the wet and dried membranes and dc and del are the densities of the compact membrane and the electrolyte solution, respectively. Filled circles and triangles show the values of the PVDF homopolymer and copolymer membranes respectively, prepared as described in the experimental part; the open circles and triangles represent PVDF and P(VDF-HFP) membranes studied by Michot et al.,7 which we report for comparison. The solid line represents the total liquid uptake which one would obtain if all and only the pores were accessible to the electrolyte. The measured liquid uptake, , strictly depends on the pore volume inside the membranes, and this dependence is roughly linear over the whole explored porosity range. However, all of
Transport Properties of Porous PVDF Membranes
Figure 3. Behavior of total electrolyte uptake versus the pore volume of homopolymer (filled circles) and copolymer (filled triangles) based membranes. The open circles and triangles refer to PVDF and P(VdFHFP) data reported by Michot et al.7 The dotted and dashed lines show the linear fits performed over the experimental values of pure PVDF and PVDF-HFP membranes, respectively. The solid line represents the total uptake one should obtain if all and only the pore volume is accessible.
Figure 4. Liquid absorption normalized to the sample porosity, /φ, vs log t in the case of a spongelike membrane (open squares) and of a fingerlike one (filled squares).
the experimental values lie above the solid line. This is clearly due to a swelling contribution to the overall liquid absorption that decreases with the enhancement of Φ and then with the decrease of the amorphous phase of the compact polymer fraction in the membrane. The dotted line is a linear fit performed over the experimental values of pure PVDF membranes. The intercept provides the swelling degree of an ideally totally compact matrix, which is in good agreement with the experimental value reported on the graph. This value is clearly higher for the copolymers, where the amorphous fraction is larger. These results suggest that the overall electrolyte uptake in PVDF porous membranes is governed by two processes: (1) the filling up of the voids, which are completely accessible, and (2) the swelling of the polymer compact amorphous phase. The morphology of the porous texture seems to affect scarcely the overall liquid uptake, whereas it seems to influence the uptake rate. Figure 4 shows the uptake normalized to the sample porosity against the logarithm of time for two samples with different microstructure: the open and filled squares represent a spongelike texture (porosity 71%) and a fingerlike one (porosity 74%), respectively. In both cases, the absorption of liquid is very fast, and the bulk of the uptake (>70%), which corresponds to the pores filling up, is reached in less than 1
J. Phys. Chem. B, Vol. 106, No. 42, 2002 10831
Figure 5. Behavior of the membrane tortuosity factor, τ, versus the porosity of the homopolymer (circles) and the copolymer (triangles) films.
min. In the honeycomb membrane, however, the uptake is roughly a one-step process, whereas in the fingerlike film, two steps are clearly observed. This fact is likely due to the differences in the pore dimensions. In the spongelike film, in fact, the pores are relatively small (∼5 µm) and “good” spatial continuity exists between the pore structure and the swollen phase (see Figure 1a). Therefore, it is reasonable to imagine that the filling up of the swollen phase will take place in a short time or, even contemporarily, after the filling up of the pores. In contrast, the cavities of the fingerlike membrane are very large (>30 µm; see Figure 1b), and less contact surface is available with the swollen amorphous phase. This picture has been confirmed by the equivalence of the uptake amount of solvent during the second step and the amount of the swelling of the fingerlike membrane reported in Table 2. Preliminary NMR results have shown the existence of one and two 7Li selfdiffusion rates in spongelike and fingerlike membranes, respectively.18 3.3. Membrane Tortuosity. Tortuosity, τ, is a long-range property of a porous medium, which qualitatively describes the average pore connectivity of the solid. Although there are many definitions of tortuosity, with the simplest based on geometrical considerations, it is usual to define τ by electrical conductivity measurements. This parameter is widely used to describe the ionic transport by providing information on the effect of the pore blockage.19 Figure 5 shows the behavior of the membrane tortuosity factor, τ, as a function of the porosity of homopolymer (circles) and copolymer (triangles) films:
τ)
σ0 Φ σb
(3)
where Φ is the membrane porosity and σ0 and σb are the conductivities of the free liquid electrolyte and of the activated polymer one, respectively. A tortuosity factor τ ) 1, therefore, describes an ideal porous body with cylindrical and parallel pores, whereas values of τ > 1 refer to more or less hindered systems. The figure shows substantially different behaviors for the copolymer and homopolymer membranes. In the first case, the tortuosity increases monotonically, and the observed values do remain relatively low. This can be rationalized by recalling that in the P(VDF-HFP) membranes the amorphous swollen gel phases are much more larger than in the PVDF ones. This, in turn, leads to a much more important contribution of the swollen phase to the conduction process (from which the tortuosity is evaluated). Because in these copolymers the swollen
10832 J. Phys. Chem. B, Vol. 106, No. 42, 2002
Figure 6. Plot of σe versus the total electrolyte uptake for the porous homopolymer (filled circles) and copolymer samples (filled triangles). The open circles and triangles are referred to the PVDF and P(VdFHFP) films reported by Michot et al.7
phase is dominant for the conduction,18 therefore, the tortuosity decreases with the extent of the swelling (see Table 2). In contrast, the homopolymer membranes are characterized by a rapidly decreasing behavior that changes its direction for Φ > 0.75. To account for this behavior, we recall that these membranes are characterized by higher crystallinity and lower compatibility towards the electrolyte, which further reduces the amount of swollen phase. Here, the tortuosity behavior chiefly accounts for the remarkable changes in conductivity (see Table 2). For Φ < 0.35, the membranes are characterized by a spongelike morphology and show high τ values (between 70 and 90), probably because the pores are not well connected and the ionic transport is hindered by the pore blockage. Above ∼35% of porosity, the tortuosity abruptly decreases, because the ionic transport is now supported by a better pore connectivity, which further increases when the membranes change their morphology from spongelike to fingerlike texture (Φ < 0.7). The final increase of τ from ∼3 to ∼6 is likely related to skin effects. 3.4. Ionic Conductivity. Figure 6 reports the behavior of the effective conductivity, σe ) σb/σ0, against the electrolyte uptake for PVDF (filled circles) and PVDF-HFP (filled triangles) based electrolytes. For both cases, the conductivity values are in good agreement with the data reported elsewhere by Michot and co-workers7 (see caption). For a given electrolyte content, the copolymer-based gels generally show higher conductivities than the homopolymer membranes. On the other hand, the homopolymer-based technology reaches higher values of and, therefore, higher conductivities. The behavior of the effective conductivity reported in Figure 6 suggests the existence of a threshold for low uptakes ( < 0.3), above which the logarithm of conductivity seems to increase quasilinearly with the electrolyte uptake. It is wellknown that the diffusion of liquids in porous media is conveniently described by the percolation theory.20 Although the theory is well established for hard porous media (e.g., rocks), when the porous structure is deformable by reactions or physicochemical interactions with the fluid, the physical description is further complicated by the changes in morphology.21 This is indeed the case of gel electrolytes based on porous PVDF membranes, in which, mainly for low Φ, the swelling contribution of the amorphous phase cannot be neglected. In these materials, the percolation may be conveniently discussed in terms of total volume fraction of the liquid electrolyte in the membrane.22 At the percolation threshold, the conductivity
Quartarone et al.
Figure 7. Ln plot of σe vs ( - c) for the electrolytes based on porous homopolymer membranes (filled circles of Figure 6). The solid line is a nonlinear best fit (see text).
behavior is generally described by the following scaling power law:
σe() ) k( - c)µ
(4)
where c is the critical uptake required for ions to percolate, µ is a critical exponent which depends on the system dimensionality, and k is an adjustable parameter which, in our case, takes into account the contributions to the conductivity coming from the amorphous swollen polymer gel phase. In the following, we will adopt this equation to fit our data for homopolymers (filled circles of Figure 6). Figure 7 shows the plot of ln σe vs ( - c) for the gel electrolytes. The continuous line is a nonlinear multiparametric best fit performed with a commercial tool (Mathcad, Levenberg-Marquardt algorithm). We obtained the following parameters: c ) 0.158 ( 0.007, µ ) 3.24 ( 0.3, k ) 0.68 ( 0.3, and r2 ) 0.99. The value of c is in excellent agreement with the one expected for percolating continua with dimensionality d ) 3,23 whereas the large uncertainty observed on k is likely related to the fact that the amount of swollen gel phase is varying across the large porosity range we are examining. Our result for µ, finally, requires a detailed comment, chiefly because in percolating continua its values are not as well defined as in regular lattices. The µ value expected for percolation in a regular d ) 3 lattice is 2.0, whereas 3.0 has been exactly calculated for an ideal Bethe lattice,21 that is a network of branching structures with no closed loops. Whereas transport in percolating continua can be quite different from that in discrete networks,20 Jerauld et al.24 showed that, as long as the average coordinations of a regular network and a random one are about the same, many transport properties are, for most practical purposes, identical. Under this assumption, we can infer that the value we obtained for µ describes ionic transport in a Bethe network. Percolation on a Bethe lattice is used to rationalize the gelation theory developed by Flory about the branching of small molecules in a polymerization process which leads to a gel.20,25 We can adopt the same approach in our case by supposing that the electrolyte formation takes place in a “fractal fashion”, i.e., starting from occupation of the pores and then developing through progressive physicochemical interactions of the solution with the membrane amorphous layers around the pores. In fact, the liquid spans the whole porous system during the imbibition process, during which there is also swelling of the polymer. By means of NMR diffusion measurements, this second
Transport Properties of Porous PVDF Membranes phenomenon has been demonstrated to be relevant in determining the transport properties of these electrolytes.18 Further NMR studies are in progress in order to investigate the dynamics of membrane permeation and gel formation. 4. Conclusions In this paper, we discussed the fabrication and some physicochemical properties of porous PVDF membranes. By means of phase inversion methods, it was possible to obtain films with porosity up to ∼85%. Two different morphologies, spongelike and fingerlike, can be obtained by changing the experimental parameters (chiefly the solvent). The bulk of the electrolyte solution is absorbed by filling of the pores, whereas a fraction ranging from ∼5 to ∼15 vol %, depending on the residual amorphous polymer phase, results from swelling. The ionic conductivity obtained with a standard electrolyte solution (EC-DEC-LiPF6 1 M) reaches 1.77 × 10-3 ohm-1 cm-1 at RT. The conduction mechanism may be rationalized as a percolation process through a Bethe lattice. Acknowledgment. The authors thank Prof. P. Piaggio and Dr. A. Bottino for the interesting discussions of some aspects of this work, concerning the phase inversion technique and the resulting morphology of the membranes. Furthermore, the authors are particularly grateful to Dr. Bottino, who provided some porous PVDF membranes and the relative SEM micrographs, reported in Figure 1. References and Notes (1) Osaka, T. Electrochem. Soc. Interface 1999, 8, 9. (2) Gozdz, A. S.; Schumutz, C.; Tarascon, J. M.; Warren, P. C. U.S. Patent No. 5,296,318, 1995. (3) Tarascon, J.-M.; Gozdz, A. S.; Schmutz, C.; Shokoohi, F.; Warren, P. C. Solid State Ionics 1996, 86-88, 49.
J. Phys. Chem. B, Vol. 106, No. 42, 2002 10833 (4) Jiang, Z.; Carrol, B.; Abraham, K. M. Electrochim. Acta 1997, 42, 2667. (5) Quartarone, E.; Brusa, M.; Mustarelli, P.; Tomasi, C.; Magistris, A. Electrochim. Acta 1998, 44, 677. (6) Boudin, F.; Andrieu, X.; Jehoulet, C.; Olsen, I. I. J. Power Sources 1999, 81-82, 804. (7) Michot, T.; Nishimoto, A.; Watanabe, M. Electrochim. Acta 2000, 45, 1347. (8) Bodin, F.; Lenhof, C.; Carillon, G.; Olsen, I. I. Paper No. 276 presented at 10th International Meeting on Lithium Batteries, “Lithium 2000”; Como, Italy, May 28-June 2, 2000. (9) Strathmann, H.; Kock, K. Desalination 1977, 21, 241. (10) Bottino, A.; Camera-Rota, G.; Capannelli, G.; Munari, S. J. Membr. Sci. 1991, 57, 1. (11) Capiglia, C.; Saito, Y.; Kataoka, H.; Kodama, T.; Quartarone, E.; Mustarelli, P. Solid State Ionics 2000, 131, 291. (12) Mustarelli, P.; Quartarone, E.; Capiglia, C.; Tomasi, C.; Ferloni, P.; Magistris, A. J. Chem. Phys. 1999, 111, 3761. (13) Sahimi, M.; Gavalas, G.; Tsotsis, T. T. Chem. Eng. Sci. 1990, 45, 1443. (14) Magistris, A.; Mustarelli, P.; Quartarone, E.; Piaggio, P.; Bottino, A. J. Power Sources 2001, 97-98, 65. (15) Wunderlich, B. Macromolecular Physics 3; Academic Press: 1980; p 50. (16) Arcella, V.; Sanguineti, A.; Quartarone, E.; Mustarelli, P. J. Power Sources 1999, 81-82, 790. (17) Magistris, A.; Mustarelli, P.; Quartarone, E.; Piaggio, P.; Bottino, A. Electrochim. Acta 2001, 46, 1635. (18) Kataoka, H.; Saito, Y.; Sakai, T.; Quartarone, E.; Mustarelli, P. J. Phys. Chem. B 2000, 104, 11460. (19) Micka, K.; Svata`, M. J. Power Sources 1978, 3, 167. (20) Sahimi, M. Applications of Percolation Theory; Taylor & Francis: London, 1994. (21) Stauffer, D.; Aharony, A. Introduction to Percolation Theory, 2nd ed.; Taylor & Francis: London, 1994. (22) Hsu, W.; Barkey, J. R.; Meakin, P. Macromolecules 1980, 13, 198. (23) Scher, H.; Zallen, R. J. Chem. Phys. 1970, 53, 3759. (24) Jerauld, G. R.; Scriven, L. E.; Davis, H. T. J. Phys. C 1984, 17, 3429. (25) Flory, P. J. J. Am. Chem. Soc. 1941, 63, 3083.
