Mechanisms of Proton Conductance in Polymer Electrolyte Membranes
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3646 J. Phys. Chem. B 2001, 105, 3646-3662
Mechanisms of Proton Conductance in Polymer Electrolyte Membranes
M. Eikerling,†,‡ A. A. Kornyshev,*,† A. M. Kuznetsov,§ J. Ulstrup,| and S. Walbran†
Institute for Materials and Processes in Energy Systems, Research Center “Jülich” GmbH,
D-52425 Jülich, Germany, The A. N. Frumkin Institute for Electrochemistry, Russian Academy of Sciences,
117071 Moscow, Russia, and Department of Chemistry, The Technical UniVersity of Denmark,
DK-2800 Lyngby, Denmark
ReceiVed: September 5, 2000
We provide a phenomenological description of proton conductance in polymer electrolyte membranes, based
on contemporary views of proton transfer processes in condensed media and a model for heterogeneous
polymer electrolyte membrane structure. The description combines the proton transfer events in a single pore
with the total pore-network performance and, thereby, relates structural and kinetic characteristics of the
membrane. The theory addresses specific experimentally studied issues such as the effect of the density of
proton localization sites (equivalent weight) of the membrane material and the water content of the pores.
The effect of the average distance between the sulfonate groups, which changes during membrane swelling,
is analyzed in particular, and the factors which determine the temperature dependence of the macroscopic
membrane conductance are disclosed. Numerical estimates of the specific membrane conductivity obtained
from the theory agree very well with typical experimental data, thereby confirming the appropriateness of the
theoretical concepts. Moreover, the versatility of the models offers a useful and transparent frame for combining
the analysis of both experimental data and the results of molecular dynamics simulations.
I. Introduction which is induced by configuration fluctuations of donor and
acceptor water molecules; this mechanism is discussed further
The levels of research and development of new proton below and is the basis for the model of proton conduction
conductors as well as those currently available have increased addressed in this paper. The main reason for the facile proton
dramatically over the past decade.1-6 This activity has been transport in aqueous systems is, however, simple. No matter
driven by the booming demand for fuel cell technology, how many “excess” protons there are in the system, all of them
particularly for vehicles and portable applications.7 These impose become indistinguishable in the “sea” of the background protons
rather strict requirements on proton conductors, such as stable of water. There are thus no “fixed” and “free” protons; they
performance in the temperature range between 80 and 200 °C are interchangeable. A proton, mobile a moment ago, takes a
with a conductivity not lower than 0.1 S cm-2, chemical,
rest to become a part of a water molecule. The privilege to run
mechanical, and thermal stability, and impermeability to gases,
is then relayed to another proton, which was a part of this
methanol, and charge carriers other than protons.
molecule or of other molecules clustered with the first one. It
Some of these demands are met by the best currently available is the relay character typical of aqueous hydrogen-bonded
polymer electrolyte membranes such as Nafion, Aciplex, systems which makes proton transport fast.
Flemion, and Dow membranes. These materials are essentially
While providing an environment for facile long-range proton
strong polymer acids. Exposed to water, they dissociate upon
transfer, aqueous-based membranes are also fraught with several
hydration into the immobile ionomer (anionic) groups residing
disadvantages. One is the permeability to methanol,8 which
on side chains of the polymer and free mobile protons in the
aqueous solution. Thus, the free protons move through the makes it hard to use them in direct methanol fuel cells. They
hydrogen-bonded network of water molecules inside the poly- are also, obviously, unstable at elevated temperatures (J100
mer. °C), where electrocatalysis could otherwise run fast enough to
consume all methanol at the anode so that the problem of
Compared to solid state proton conductors, the major
methanol permeation would simply not arise. A third shortcom-
advantage of aqueous polymer electrolyte conductors is the high
ing is the electroosmotic effect: the transport of water induced
conductivity of 0.01-0.1 S cm-2 available at moderate tem-
by the classical migration of hydrated H3O+ ions. This leads to
peratures. This is due to the high concentration of protons in
dehydration of the anodic side of the membrane, and thus limits
water-filled pores and the high proton mobility. The mobility
the current.9 Last, but not least, is the prohibitive aspect of high
is facilitated by a “collective” character of the proton transfer,
cost for the best-performing polymer electrolyte membranes,
such as Nafion and its derivatives.
* To whom correspondence should be addressed. E-mail: a.kornyshev@
fz-juelich.de. Progress in “dry” solid state proton conductors that offer high
† Research Center “Jülich” GmbH.
conductance without the listed deficiencies is, however, far from
‡ Present address: Materials Science Division, Los Alamos National
outclassing aqueous polymer electrolyte membranes in hydrogen
Laboratory, P.O. Box 1663, MS D429, Los Alamos, NM 87545.
§ Russian Academy of Sciences. energy technology. Theoretical and experimental efforts toward
| The Technical University of Denmark. better understanding of membrane operation are therefore very
10.1021/jp003182s CCC: $20.00 © 2001 American Chemical Society
Published on Web 04/07/2001Proton Conductance in Polymer Electrolyte Membranes J. Phys. Chem. B, Vol. 105, No. 17, 2001 3647
much needed. This is especially true because such efforts are Equipped with such a theory, we discuss the available
likely to shed new light on structural and chemical modifications experimental data on membrane conductance and make some
which could improve the membrane performance. conclusions about the optimal membrane architecture.
The structure of the polymer electrolyte is very complex, and
there is still no ultimate view of its detailed spatial organization. II. Mechanisms of Proton Transport in Water
There is also no accepted picture of transport mechanisms. There The motion of excess protons in water as a free proton, as a
are thus several ways to approach theoretically the structure and proton dressed in a polaron cloud, as a classical H3O+ ion, or
function of polymer electrolyte membranes. One way is to by mechanisms rooted in the Grotthus mechanism of structural
undertake molecular dynamic simulation studies, based on diffusion is treated in a voluminous literature.15,16 New gates
interatomic potentials;10 in the context of polymer electrolyte toward detailed understanding of proton conduction at the
membranes, such attempts are still in their infancy. Another molecular level appear to have been opened by recent ab initio
way is to investigate idealized “physical” models, such as quantum chemical and classical molecular dynamics simula-
random or correlated network models, to compare rigorous tions.15,17-19 The views emerging from these studies are also
solutions with experimental data, and in the case of success to the basis for the investigation in this report, but we first discuss
explore them in the context of new predictions. There are, shortly a few general notions related to the chemical identity
finally, ways based on phenomenological theories.11 Possibly of the proton-conducting species in aqueous solutions and the
having less predictive power, they could, however, provide nature of the fundamental physical event of proton transfer
transparent frameworks for the interpretation of computer between dynamic proton carriers in aqueous solutions.
simulations, experimental data, and model calculations.
Due to the high rate of proton exchange in the hydrogen-
Among the features of the membrane functions, which are bonded system, the excess proton is accommodated equally
first to be explained, are the proton conductivity dependence among the other water protons. There is therefore no free excess
on temperature, equivalent weight (the number of grams of dry proton in water. Three main options were discussed in the
polymer per mole of anionic groups, in the following abbreviated literature.20,21 First, the excess proton can be a part of an H3O+
as “EW”), and water uptake. Each of these observable correla- ion, in which all three protons are equivalent. Second, it can
tions undoubtedly reflects several physical effects. These cannot reside in a Zundel20 H5O2+ cation complex with a proton
necessarily be separated, but, framed by analytical theory, they between two water molecules, binding them in a cluster. Third,
can be illuminated and qualitatively assessed. it can be present as an Eigen21 H9O4+ cluster, which consists
With such views, we provide in this report a semiphenom- of H3O+ and three strongly bound H2O molecules, each attached
enological, theoretical description of proton conduction in to one of the H3O+ protons. One can, of course, imagine other
polymer electrolyte membranes. We focus on membranes of intermediate structural arrangements; the overall weight of the
the Nafion family, but the approach may be applied to a wider H5O2+ and H9O4+ clusters seems to be, however, dominant. In
range of membranes. The description rests on models which any case, all the clusters are short-lived in water, and their life
are necessary but drastic simplifications of real membrane and disappearance are crucial parts of the proton transport
architecture and performance. In a different sense, the descrip- dynamics. The generic structural diffusion pathways of the three
tion, although crude, is comprehensive, as it addresses in species are as follows.
principle all membrane components and available conduction A. The “Simplistic” Picture. In this view, the excess proton
features. Furthermore, it can be straightforwardly augmented hops from the H3O+ donor site to any neighboring water
or updated, as warranted by new structural or functional data, acceptor molecule, leaving a neutral water molecule behind. The
or simulation results. proton hop is preceded by suitable configurational polarization
The description covers the following three equally important fluctuations in water. The fluctuations appear as if water were
elements of membrane proton conduction. (1) Proton transfer polarized by the neighboring acceptor molecule, with the excess
between localized ionic states or water molecules is considered proton bound to the same extent as if it resided on the donor
as a fundamental chemical rate process.12 This part rests on the molecule. Such fluctuations determine the transition state. For
contemporary theory of proton transfer in condensed media, but the proton transfer (in the following abbreviated as PT) to occur,
the chemical identity of the donor and acceptor sites and the the electronic interaction between the donor and acceptor which
single-pore environment are crucial and specific to the mem- triggers the transfer must, moreover, be strong enough.
brane system. (2) In the next step, the proton transfer rate The latter is only possible when the donor and acceptor
constant is converted to the proton conductance in a single- acquire the proper coordination, resembling an H5O2+ cluster.
pore confinement. The key issue here is the calculation of the At the distance between the two oxygen atoms close to those
proton distribution function in the pore. In combination with of the equilibrium ground state of the H5O2+ cluster in the gas
the rate constant, it gives the single-pore conductivity. In the phase (2.4 Å), the interaction between the H3O+ and H2O
models, this also involves a crucial distinction between the moieties is so strong that there is no barrier to proton motion
proton conduction in the bulk of the pore and along the inner along the O-H-O coordinate and the proton is delocalized
pore surface. The resulting single-pore conductivity yields between the two water molecules.16,17 The equilibrium O-O
interesting observations regarding the effects of temperature, distance may be slightly longer in the condensed phase, still
pore dimensions (swelling), and density of proton localization leaving a very small barrier, and a shallow double well may
sites (equivalent weight) on the proton conductivity. (3) The emerge.
third part is the construction of the total pore network. This The proton is displaced from the shallow double well by
part involves a statistical approach to both fluctuations in the donor and acceptor molecular fluctuations out of the H5O2+
dimensions of individual pores and statistical combination of configuration. This makes the donor molecule “H2O-looking”
their conductances into the performance of the whole net- and the acceptor molecule “H3O+-looking”. The H5O2+ cluster
work.13,14 This part bridges the single-pore proton transfer is, thus, destroyed by such a fluctuation and the H3O+ generated
events and the observable macroscopic membrane proton at a new site, displacing the excess charge over roughly the
conductivity. diameter of a water molecule. Overall, this is a collective3648 J. Phys. Chem. B, Vol. 105, No. 17, 2001 Eikerling et al.
mechanism for propagation of the “proton dislocation”, in which However, near the surface it may be very different. In the bulk,
many particles are involved, rather than consecutive single- all proton transfer takes place between symmetrical states, so
proton hopping. The pathway for proton translocation is along while there is environmental nuclear reorganization associated
any of the three protons of the initial H3O+ ion, and the with the transfer, in the absence of macroscopic driving forces
efficiency of such proton exchange exceeds by several-fold that the reaction Gibbs free energy or driving force is zero. At the
of the classical diffusion of H3O+ ion.22,23 surface, proton transfer cannot proceed directly between SO3-
B. Zundel’s and Eigen’s Picture. Comprehensive infrared groups because the average distance between these groups is
spectral data have framed a view within which the H3O+ ion is far too large (∼7-12 Å).27 PT is, therefore, invariably to
a less stable moelcular entity than the H5O2+ complex. H5O2+ intermediate sites different from the SO3- groups. These sites
would then be the basic state of the proton, and the H5O2+ can hardly be any chemical species other than water, and in
structural defect would move together with its charge in water. principle, surface proton conductivity involves an additional
Again, this does not involve classical defect diffusion as a whole, driving force contribution to the activation Gibbs free energy
but structural diffusion based on PT dominates and gives the due to this asymmetry.
high proton transport mobility. A water molecule, next to such The precise state of the intermediate protonated water is not
a cluster, fluctuates into a configuration where it forms a new known, but the following considerations are an approach toward
H5O2+ together with the nearest molecule of the original H5O2+. characterization. First of all, SO3H is a strong acid. Although
This also requires orientational adjustment and deformation of dissociation involves an additional driving force contribution,
both adjacent and remote water molecules in the original H5O2+ this is not likely to be energetically a large quantity. Second,
cluster. Subsequent fluctuations move remote water molecules PT from SO3H to water in a direction perpendicular to the pore
out of the initial H5O2+ configuration, shifting the charge by walls is thermodynamically unfavorable, at least for narrow
one water molecule. This motion may proceed along any of pores, due to the cooperative electrostatic attraction of the proton
the four external protons of the H5O2+ cluster. both from the SO3- donor group and from several nearby SO3-
Fluctuational preorganization followed by virtually barrierless groups. PT along the pore wall toward a neighboring SO3- group
PT carries over to mechanisms based on the H9O4+ cluster. This is here much more feasible. Third, the state of the intermediate
is followed by decoupling of one of the peripheral molecules H2O and H3O+ is then likely to be engaged in the hydrogen
in the cluster, shifting the center of the cluster in the direction bond network, broadly similar to the hydrogen bond network
of the incoming acceptor water molecule. The process runs along in which H3O+, H5O2+, and H9O4+ are engaged in bulk water.
one of the six peripheral protons. An SO3- interspacing of 7-8 Å accommodates a single H3O+
Common to the three simplistic views is the fact that H+- in a configuration suitable for facile consecutive PT steps subject
nH2O clusters self-assemble spontaneously up to a certain size. only to minor environmental configurational fluctuations and
If a cluster were disrupted by the configurational fluctuations preorganization.
of the same molecule the fluctuational motion of which has
created the cluster, there would be no charge transfer. If, IV. Parametrization of the Proton Mobility in a Pore
however, the disruption comes from one of those molecules, A. Bulk Transition Probability and Mobility. Common to
which were a part of the initial cluster, the charge is shifted. In the PT event that triggers structural diffusion in bulk water and
the class of models discussed above, the activation free energy at the pore surface is a strong interaction between the proton
of cluster motion is of the same order of magnitude as that for donor and acceptor. Ab initio electronic structure calculations
breaking a hydrogen bond. and quantum path integral calculations, thus, concur with an
Recent Kohn-Sham density functional simulations for mo- extremely shallow or even vanishing barrier, once the transition
lecular wires24 have shown that H3O+ and H5O2+ clusters are configuration, preorganized for PT, is prepared. Proton tunneling
the dominant defects in water. Ab initio calculations in multi- is, therefore, only indirectly reflected in the PT transition
molecular water drops17 demonstrated the presence of all types probability in determining the residual barrier height for proton
of clusters in more or less equal amounts. The particular cluster transfer. This is equivalent with the totally adiabatic limit known
size may be important for the proton conductivity of water in from molecular proton transfer theory.12 The transition prob-
a confined geometry. Squeezing water inside a pore or embed- ability in this limit takes the form
( )
ding charged side chains may also change the distribution
balance in the contribution of clusters of different size, since Eb
ωb b +
their condensed state energies are similar. In other words, the 4
spatial constraints on the pathways via clusters of different size
Wb ) exp - (1)
2π θ
may affect the proton mobility. In our further construction, the
size of the clusters will only affect certain parameters of the Hereafter, the subscript b refers to the parameters of the bulk
phenomenological theory. transfer. Eb is the reorganization Gibbs free energy of all the
modes that contribute to the Franck-Condon factor. These are
III. Proton Conductivity and Mobility in a Nafion Pore local translational, librational, and hydrogen bond deformational
motions, but also bulk collective inertial polarization modes.
To a certain extent, concepts and models of proton conductiv- b is a term which parametrizes the energy needed for the
ity in bulk water carry over to proton conductivity in Nafion configurational preorganization of the solvated PT complex
and other polymer electrolyte membranes. A water-saturated, before the proton can move from the initial to the final state on
proton-conducting polymer electrolyte membrane behaves like the adiabatic potential energy surface. ωb is the effective
an acid. Swollen in water, the SO3H groups of its side chains frequency of all the environmental modes,28 and θ()kBT) is
dissociate into SO3- anions and H+ cations. The SO3- groups the thermal energy, kB being Boltzmann’s constant and T the
remain attached to the pore walls, whereas the protons are temperature. Equation 1 represents a broad, parametrized PT
donated to water. rate constant form. It can be extended as warranted by
The proton mobility in the central region of a water-filled straightforward interfacing with contemporary proton transfer
pore is expected to be close to the mobility in bulk water.25,26 theory, in particular as follows.Proton Conductance in Polymer Electrolyte Membranes J. Phys. Chem. B, Vol. 105, No. 17, 2001 3649
(1) All environmental nuclear modes are regarded as linear The mobility can be estimated as28
and “classical”, i.e., low-frequency. High-frequency nuclear
tunneling effects, e.g., in the librational models (≈800 cm-1), ksVs eas2Ws
anharmonicity, and mode mixing between the reactant and µs ) (5)
product states can, however, be incorporated as warranted. 2 θ
(2) Equation 1 interpolates a variety of different specific
where ks is no longer 1/3. It can, generally, be expected that
reaction pathways, such as those based on the simplistic views
ksVs/2 < kbVb/2 and as ∼ ab.
and the more precise composite pathway rooted in the Marx-
C. Mobility in the Pore. A Two-State Model. Separation of
Parinello scheme and similar schemes.
the overall proton conductivity into surface and bulk mobility
The proton mobility can be estimated as28
mechanisms constitutes an idealization. A more comprehensive
notion would be the mobility as a function of the distance from
kbVb eab2Wb the pore walls. The mobility for a cylindrical pore model, µ(r),
µb ) (2) with r being the radial coordinate, is thus
2 θ
Here, ab is the effective distance of the net charge transfer; this ea2(r)W(r)
quantity is on the order of the “effective” diameter of a water µ(r) ) h(r) (6)
θ
molecule (∼3 Å), but can be larger for cooperative two-proton
transfer in larger clusters (6-9 Å). The geometrical factor kb where
≈1/3 for “homogeneous and isotropic” bulk conductivity. Vb is
the coordination number of the transfer. It would have a value k(r)V(r)
of 3 for the simplistic mechanism, 4 for the Zundel mechanism, h(r) ) (7)
and 6 for the Eigen mechanism of structural diffusion. As none 2
of these models uniquely determines proton transfer pathways,
where k(r) and V(r) are the radius-dependent geometrical factor
Vb can be assigned an effective value between 3 and 6.
and coordination number of PT, respectively. It is convenient
Altogether, thus
and presently necessary, however, to use a two-state model.
Surface mobility (index s) is assumed to prevail in boundary
1 k bV b layers with a thickness δ of ≈3 Å, which is roughly the thickness
e e1 (3)
2 2 of a water monolayer, while bulk mobility (index b) prevails in
the remaining pore volume. The overall pore mobility to which
B. Surface Transition Probability and Mobility. The experimental data could be referred is thus
surface process can in principle be represented by a form
analogous to eq 1 eas2Ws
( )
µ(r) ) hs Θ(r - R + δ)Θ(R - r) +
θ
∆G Es
ωs s + + eab2Wb
2 4 hb Θ(R - δ - r)Θ(r) (8)
Ws ) exp - (4) θ
2π θ
where the subscript s stands for the surface. As noted before, a where R is the pore radius and Θ is the step function; i.e., Θ(x)
driving force term ∆G has been added in the form of the free ) 1 when x > 0 and zero otherwise.
energy of an elementary endothermic PT step from the region
near an SO3- group to an intermediate position. s is equivalent V. Proton “Cloud” in a Pore
to b in eq 1 and represents the total environmental preorgani- As we have discussed, a thermalized excess proton in water
zation preparing the proton donor and acceptor for PT. Es is exists within an H+-nH2O cluster: H3O+, H5O2+, and H9O4+.
the nuclear reorganization Gibbs free energy, equivalent to Eb The number n fluctuates with time; i.e., the cluster dynamically
in eq 1, accompanying PT from the initial prepared state to the changes with spontaneous breakage and recovery of hydogen
final, similarly prepared state. While the two forms (eqs 1 and bonds in the clusters and between the clusters and the remaining
4) are thus quite analogous, the values of the parameters are of water. At a given time, several protons in a given volume can
course, in general, different. In addition, although palatable occupy different states with close but still different energies.
models for the surface conductivity can be constructed im- These states, however, are never fixed; they fluctuate and
mediately, the specific operational models and their independent transform one into another. Therefore, a lattice gas model for a
substantiation are much more elusive than models for bulk distribution of charged particles over a limited set of sites
proton conductivity, for which, for example, comprehensive characterized by different energies29 really may not be a better
molecular dynamics simulation studies are available (see approximation than a simple Poisson-Boltzmann mean-field
above). theory; dynamics averages out the difference between different
When the distance between neighboring SO3- groups in- hydrated excess proton states. There are exact solutions of this
creases, new pathways of PT in the surface layer via an problem for a flat or cylindrical pore. We consider here
increasing number of intermediate water molecules are opened. specifically the cylindrical pore case, as recent neutron scattering
The specific expression for the rate constant of PT between two data (V. Gordelyi, unpublished results) have provided new
groups then changes. These modifications lead, however, merely experimental evidence in favor of the predominance of the
to an enhancement of the already predicted tendency of cylindrical pore type in swollen Nafion. Nevertheless, we are
increasing activation energy, or decreasing rate constant, upon aware of the other pore geometries, lamellae and spherical
increasing the SO3- distance. This is addressed in detail in the micelles, that have been promoted on the basis of experimental
Appendix. data30,31 as well.3650 J. Phys. Chem. B, Vol. 105, No. 17, 2001 Eikerling et al.
We consider a cylindrical pore of length Lp and radius R with The R-dependent variable
a negatively charged wall and positive countercharges distributed
(R -R δ)
within it. The charge on the walls is characterized by the surface 2
∆) (17)
charge density Σ.
e Λe has been introduced for further notational convenience. ∆ varies
Σ)- )- (9)
l2 2πLpR between the limits 0 (only surface) and 1 (only bulk).
The proton concentration in the surface layer is determined
where l is the “average separation” between neighboring fixed by
SO3- groups. In the second equation on the right-hand side of
eq 9, Λ is the total number of SO3- groups (and protons) in the F0
[ ]
pore, which is assumed to be conserved. The quantity Λe/Lp is, Fs ) (18)
thus, an invariant of the system, giving the charge per unit length (κbR)2 2
1- ∆
of the pore. 8
The pore is filled with water with a dielectric constant s in
the surface layer and b in the bulk. b and s are assumed to be In the limit of large R, F0 is given by
constant within the bulk and surface regions, respectively, but
this assumption could be relaxed by using a nonlocal dielectric 2 Λe
F0 ) (19)
approach or other means. Within the surface layer, a constant, 1 + 2λ πR2L
B
i.e., independent of the radial coordinate r, proton density Fs is,
moreover, assumed VI. Conductance of a Pore
) s and F(r) ) Fs for R - δ < r < R (10) The conductance of a single cylindrical pore of length Lp and
radius R is given by
The proton distribution F(r) in the bulk of the pore is obtained
from the Poisson-Boltzmann equation
Gp )
2π
Lp
∫0Rr dr F(r) µ(r) (20)
1 d dψh
ú( )
ú dú dú
) -(κbR)2 exp(-ψ
h) (11)
Integration using eq 8 gives
[ x ]
where ú ) r/R, ψ
h ) eψ/θ, where ψ is the electrostatic potential,
and κb ) (eF0/0bθ)1/2 ) (4πF0L B/e)1/2. The Bjerrum length 2
L B has been introduced for later notational convenience µb ∆ + λ 1 - 1 + ∆(1 - ∆)
λ
Gp )
[ x ]
G0p
µ0b
1 2
e2 1- 1 + ∆(1 - ∆)
∆-
{ }
LB) (12) 2 λ
4π0bθ µs 1 1-∆
1+ 1+ ( )
[ x ]
(21)
L B will be used in the following as a characteristic length scale µb 2λ 1 2
∆- 1- 1 + ∆(1 - ∆)
in the system to simplify the formalism. The proton distribution 2 λ
for 0 < r < R - δ is given in analytical form by
Here, we introduced a reference conductance
F0
[ ]
F(r) ) (13) Λe 0
(κbr)2 2
G0p ) µb (22)
1- Lp2
8
where the proton concentration at the center of the pore, F(r)0) representing the ideal conductance that a single cylindrical pore
) F0, must be determined from the condition would obtain, if all protons in it were moving with bulk mobility
µ0b at the reference temperature T0, determined via eqs 1 and 2.
-2πRΣ ) 2π ∫0Rr dr F(r) (14) The actual bulk mobility at temperature T is determined by the
ratio
[ )]
Equation 14 states that the number of protons in a slice of the
(
cylinder with unit thickness is equal to the number fixed of µb T 4b + Eb 1 1
) exp - - 0 (23)
SO3- charges on the circumferential surface of that slice. After µ0b T 0 4k B T T
integration and determination of F0, the following relation is
obtained In the limit ∆ f 0 (surface dominating), eq 21 simplifies to
x∆λ (2 + ∆λ ) - 2λ { ) ]} (24)
λ µb µs
1+ - 1 1 1 2
F0 ) 2
2e 1 ∆
1 + 2λ
(15)
Gp ≈ G0p
µ0b
∆+
µb
1+[ (
2λ
-1∆-
2λ
1+
2λ )
∆ (
πR L B ∆
where the parameter λ is and in the limit ∆ f 1 (bulk dominating) to
λ)
Lp
ΛL B
(16) Gp ≈ G0p
µb
µ0b {
1-2 1+ ( 1 δ
)
2λ R
µs
+2 1+
µb
1 δ
2λ R ( ) } (25)Proton Conductance in Polymer Electrolyte Membranes J. Phys. Chem. B, Vol. 105, No. 17, 2001 3651
In eq 21, the parameters that determine the balance between
surface and bulk contributions are µs, µb, and λ. Additional,
implicit R dependencies of Gp arise due to the effect of the finite
pore size on µs (cf. section VII) and b. The latter may appear
to be due to the spatial dielectric dispersion in confined systems
or due to effects of the hydrophobic walls on the water structure.
A simple parametrization of this effect is given by
[ ( )]R -1
b(R) ) free
b 1 + (E - 1) exp - (26)
Λc
where E is the ratio of the water dielectric constant of free bulk
water, free
b , to the short-range dielectric constant and Λc (≈3
Å) is the structural correlation length in water.32-35
VII. Conductivity of a Single, Regularly Swelling Pore Figure 1. Activation energy of proton transfer along the pore surface,
Ea ) s + ∆G/2 + Es/4, as a function of pore radius for a regular pore,
in which the separation between SO3- groups varies according to eq
Equations 1, 4, and 8 combined with the proton carrier 27 with l ) 7 (case I) or l ) 12 Å (case II) at R ) 5 Å.
distribution functions in eqs 13, 15, and 16 leading to eq 21
constitute the basis for the investigation of the single-pore swelling is, thus, 2-fold. Probably, the impact of l will determine
conductance as a function of the pore radius R. This kind of the overall tendency, which will then be manifested as a
expression also has a direct bearing on experimental observa- considerable overall increase of the surface activation free
tions where such relations are available. In this section, we energy upon pore growth. One should however note the
present model calculations for a single, ideal, cylindrical pore competing effects which inhibit a precise a priori prediction of
with a fixed number Λ of SO3- groups that are regularly the net tendency in the surface activation free energy. It is even
distributed on the pore walls. In section IX, this is converted to harder to predict the dominating variation of the pre-exponential
the overall observable conductivity of a membrane. factor.
When R is increased, the average separation l between SO3-
groups will increase according to Instead, we have varied the parameters of the surface mobility
to investigate the interplay between the different effects upon
swelling. Cases where the surface contribution is insignificant
x
2πLp are, further, compared to cases when it gives an important
l) R (27)
Λ contribution. In practical terms, these different effects may be
controlled by variation of the equivalent weight of the mem-
Equation 27, together with results that we obtained in a specific brane, defined as the mass of dry polymer material per mole of
study of the activation free energy in a single pore in ref 36, ion exchange sites (SO3- groups). Within the model calculations,
gives an increase in the calculated surface activation free energy presented here, a smaller equivalent weight, i.e., a larger
upon pore growth, i.e., increasing l. The reaction free energy concentration of SO3- groups, is reflected in a smaller l and
contribution to the overall activation free energy in the surface therefore in smaller activation free energies of surface conduc-
layer increases with l due to attenuated assitance of neighboring tion.
SO3- groups in the SO3H deprotonation process. In the
Electrostatic calculations on activation free energies in pores
following, we will denote pores in which eq 27 is valid as
of polymer electrolyte membranes have been reported previously
regular pores. It is expedient at first to discuss separately bulk
in ref 36. They form the basis for the subsequent analysis. The
and surface contributions.
activation free energy of the bulk mobility is taken to be 0.12
The mechanism underlying the bulk mobility is expected to
eV, determined from experimental Arrhenius plots,37 disregard-
be weakly affected by the variation of R. The pore size effect
ing formal entropy effects. The pre-exponential factor is taken
on this contribution is, therefore, manifested mainly through
to be a fixed, adjustable number. These two quantitites are fixed
the proton distribution, which is explicitly taken into account
in all calculations. Figure 1 shows the activation free energy of
in eq 21 via the inclusion of eq 13.
the surface mobility as a function of R for distinct values of the
The mechanism underlying the surface contribution may,
however, change drastically with R. In a dry membrane, the appropriate parameters. The upper curve corresponds to separa-
water structure close to the surface is rigid, with a large number tions between SO3- groups with a reference value l of 12 Å
of directed bonds. The dielectric constant is low due to the high with an R of 5 Å and variation with R according to eq 27. The
concentration of SO3- groups and the proximity of the activation free energies are considerably lower and less de-
hydrophobic polymer walls. On one hand, this water structure pendent on R for the lower curve when l ) 7 Å and R ) 5 Å.
produces a tendency toward a large activation free energy. On These features reflect mutual SO3- assistance on the horizontal
the other hand, the smaller the pores are, the smaller the average proton transfer at short SO3- separations. Pre-exponential factors
separation between neighboring SO3- groups is. This gives an of the surface mobility are considered to be roughly equal or
opposite tendency toward a reduced activation free energy. slightly larger (maximum of 1 order of magnitude) than for the
In a swollen pore, the water structure at the surface may bulk mobility. For the explicit calculations, presented in the
become softer. This effect creates a tendency toward a dimin- following, equal pre-exponential factors will be used for both
ished reaction free energy contribution to the surface activation mechanisms.
free energy. However, the increasing separation l between SO3- All subsequently presented conductance data will be scaled
groups upon swelling creates the opposite tendency toward an relative to the parameter G0p, defined in eq 22. The corre-
enhanced reaction free energy contribution. The impact of pore sponding reference mobility µ0b is specified at temperature T0.3652 J. Phys. Chem. B, Vol. 105, No. 17, 2001 Eikerling et al.
Using the definitions in eqs 1, 2, 9, and 22, G0p is given by
( )
Eb
k V ω e2
a 2 b +
b b b R b 1 4
G0p ) exp -
2kB Lp l2 T0 kBT 0
revealing important dependencies on pore geometry (R and Lp),
microscopic distances (ab and l), and T0. We fix the reference
temperature T0 at 373 K and use, for an order-of-magnitude
estimate, the following parameter values: kbVb/2 ) 1 (cf. eq
3), ωb ) 5 × 1014 s-1 (cf. refs 28 and 38), ab ) 3 Å, Lp ) 50
Å, b + Eb/4 ) 0.12e eV. Due to the assumption of charge
conservation, i.e., conservation of Λ (cf. eq 9), within a single
pore upon swelling, the ratio R/l2 is independent of R. Two cases
with different surface densities of fixed charges will be
considered: (I) l ) 7 Å at R ) 5 Å (Figures 2 and 4) and (II)
l ) 12 Å at R ) 5 Å (Figure 3). With these values, we obtain
the estimates
G0p ≈ 5.5 × 10-7 S (I) and 1.8 × 10-7 S (II)
Single-pore Arrhenius plots [log10(GpT/G0pT0) vs 1000/T]
and the R dependence of Gp (normalized to G0p) correspond-
ing to different parametrizations of the surface mobility are
shown in Figures 2-4. Figure 2, obtained with the lower curve
of Figure 1 (case I), represents a large surface mobility with a
small pore size. However, the surface mobility decreases rapidly
upon pore growth due to the increasing activation free energy.
The total conductance plotted as a function of R at different
temperatures in panels b (273 K) and c (373 K) has, interest-
ingly, a pronounced minimum at R ≈ 8 Å. When the activation
free energy of the surface mobility exceeds ≈0.24 eV, corre-
sponding to radii of >10 Å, the surface mobility becomes
insignificant.
Figure 3, obtained with the upper curve of Figure 1 (case
II), represents the case of a small surface contribution due to
large SO3- separation. The surface contributes insignificantly
to the overall conductance (cf. panels b and c of Figure 3). The
conductance increases monotonically with pore size in this case.
In contrast to the correlations shown in Figures 2 and 3, which
were calculated with the R-dependent surface activation energies
of Figure 1, the correlations depicted in Figure 4 have been
calculated with the activation energy of the surface contribution
fixed. In this case, the conductance increases monotonically with
increasing R, since the dependence of the surface mobility on
R is much weaker than in the previous cases.
In the various plots of Gp(R) at 373 K, the conductance at Figure 2. High surface mobility in small pores. Arrhenius plots
the largest depicted radii amounts to about 75% of the ideal {log10[GpT/(G0pT0)] vs 1000/T} of single-pore conductivity at various
radii (a) and dependence of pore conductivity (normalized to G0p) on
conductance G0p. radius at 273 (b) and 373 K (c), obtained from eq 21. The activation
energy of the bulk mobility was fixed as 0.12 eV. For the surface
VIII. Discussion of the Single-Pore Results activation energy, the R-dependent parametrization with l ) 7 Å at R
Our calculations show that there are definitely two contribu- ) 5 Å (cf. Figure 1) was used (case I, for which the numerical estimate
gives G0p ≈ 5.5 × 10-7 S). The pre-exponential factor was assumed to
tions to the proton conductance of a single pore: one contribu-
be the same as for both mobility mechanisms. The conductivity is high
tion dominates in the middle of the pore and proceeds in a at small R due to the excellent surface mobility. With increasing R
manner similar to that of bulk water, and the other contribution values, the surface mobility decreases rapidly (see panels b and c),
dominates near the pore surface along the array of SO3- groups. whereas the bulk contribution increases due to the increasing fraction
The surface mobility is usually considerably smaller and has of bulk protons.
a higher activation free energy than bulk mobility. This may,
however, be partially compensated by the accumulation of The families of Arrhenius relations at various R values in
protons in the surface region. Squeezing the pore to smaller Figures 2-4 look rather different and depend on whether the
dimensions will diminish the size of the bulklike portion of surface contribution is significant. In a regularly swelling pore,
water, and practically all the proton transport will then be in which l increases with R according to eq 27, the potential
surface-like in sufficiently narrow pores. wells in which the surface protons are trapped become deeperProton Conductance in Polymer Electrolyte Membranes J. Phys. Chem. B, Vol. 105, No. 17, 2001 3653
Figure 3. Low surface mobility in small pores. The same set of plots Figure 4. R-independent surface mobility. The same set of plots as in
as in Figure 2. For the surface activation energy, the R-dependent Figure 2 with density of SO3- groups corresponding to l ) 7 Å at R )
parametrization with l ) 12 Å at R ) 5 Å (cf. Figure 1) was used 5 Å. For the surface activation energy, a fixed value 0.24 eV was used.
(case II, G0p ≈ 1.8 × 10-7 S). Overall, due to the high activation
energy of the surface mobility, the conductivity is mainly determined protons, a pronounced minimum in conductance at intermediate
by the bulk transport. The surface mobility contributes considerably pore sizes appears. Overall, the conductance is higher due to a
only in the range of very small R. higher proton concentration, as indicated by the corresponding
value of G0p.
with increasing R.36 Surface protons therefore become im- A regular pore with a high surface density of SO3- groups
mobilized at the SO3- groups upon swelling, whereas could perform equally well at low and high water contents (cf.
the fraction of highly mobile bulk protons increases continu- Figure 2). Due to the larger value of G0p, this kind of pore will
ously. be superior in conductance compared to the kind of pore used
Generally, a pore with a high surface density of SO3- groups to generate Figure 3. A low degree of swelling could then be
and, thus, a higher proton concentration performs better. Due sufficient for good overall performance, characterized by a good
to a considerable surface contribution, the conductance can be conductance. Moreover, due to the prevalence of the surface
excellent even at small pore sizes. Since, however, the surface mobility, permeation of methanol and water would probably
mobility rapidly decreases with increasing R, whereas the bulk be suppressed under these conditions. These properties of the
conductance increases due to an increasing fraction of bulk single pore would be directly related to characteristics of the3654 J. Phys. Chem. B, Vol. 105, No. 17, 2001 Eikerling et al.
whole membrane, if the appropriate type of pores would considered (parallel model). Another step of complication is to
dominate in the percolating network of membrane pores. allow nonuniform tube cross sections, i.e., to virtually cut the
The presence of a considerable fraction of irregular pores with tube into n cylindrical segments of equal lengths lp ()Lp/n) with
large SO3- group separations could, however, hamper the overall a distribution in segment diameters (serial model).44 In calcula-
performance of the membrane. In this case, a bundle of parallel, tions of transport properties (conductivity and permeability)
equally performing regular pores is not a good model of the across the membrane, the fact that on average only one-third
membrane. In particular, in the dry membrane, there may be a of all the tubes should be pointing in each of the three spatial
significant fraction of poorly conducting irregular pores, the directions must be taken into account39 (provided that the
heterogeneities of which would require separate attention. membrane is isotropic).
As demonstrated above, the EW and the density of SO3- Finally, the pore space model, which should provide the
groups per unit pore length may affect not only the proton closest approach to reality, consists of a three-dimensional
concentration but also the competition between the surface and system of interconnected tubular segments of various lengths
bulk contributions, with a considerable effect on the membrane and diameters (network model). Effective medium theory (the
performance. A lower EW may, thus, give a higher surface abbreviation EMT will be used hereafter) could be used as a
density of SO3- groups at the pore surfaces, resulting in an good approximation for a calculation of the conductivity of such
enhanced surface contribution. an interconnected network.11,45,46
Parallel and serial type models require the incorporation of a
IX. Basic Equations for Conductance of a Membrane tortuousity factor τ2 {the tortuousity factor τ, as defined by von
Karman in 1937, is the square of the ratio of the hypothetical
The overall membrane conductance is determined by the tortuous capillary length, Lp, to the thickness of the porous
interplay between the charge distribution inside pores, the membrane in the direction of macroscopic flow, L [τ ) (Lp/
mobility mechanisms, and the membrane porous structure. The L)2]39} to account for a possibly curved form of the channels.
effect of pore structure will be discussed in this section. Two The non-interconnected, one-dimensional model variants (serial
major impacts of the pore structure variations on electrical type models) tend to underestimate the transport properties.39
conductance can be distinguished: the variation of the single- Furthermore, they are very sensitive to details of the pore size
pore cross-sectional area, available for proton conduction; and distribution for the smallest pore sizes due to the “bottleneck
the orientation and topology of conducting pores in the network. effect”: a single poorly conducting segment can limit the overall
Protons are released to the enclosed bulk from the interior pore transport, since in one dimension it cannot be bypassed.
surface. Therefore, the fixed surface density of SO3- groups Our calculations of the membrane conductance will be based
directly influences the concentration of protons in the bulk. on serial and network type models. In the serial model,44 the
There is, thus, no intrinsic, isotropic bulk conductivity of the pore space is formed by parallel tortuous channels, each of
pore-filling medium (water). The latter would be observed if which is assumed to consist of elementary cylindrical segments
pores were filled with a bulk electrolyte solution at a fixed with distributed radii according to a given pore size distribution.
concentration. This principal difference hampers the straight- This pore size distribution evolves with the water content. An
forward transfer of results of well-known studies on conductivity extension to a three-dimensional network with interconnected
in porous media13,39,40 to polymer electrolyte membranes. As elementary segments will be discussed in terms of the effective
we will see below, even simple models lead to relations between medium theory.46
conductivity and porosity more complex than Archie’s law.41 A. Integral Relations. We begin with an outline of how the
A polymer membrane differs from a rigid porous body, such pore size distribution and its evolution are related to the water
as a sandstone; there is no fixed, pre-existing porosity in it. In content and to the membrane conductance in the serial and
the membrane, the porous structure is formed upon the uptake network type models.
of water. Water is the pore former, not only a pore filler. The The mass of dry membrane material and the total number of
pore network can, thus, be envisaged as an “evolving organism” parallel tubes N are assumed to be constant upon swelling.
that reorganizes continuously during swelling. Furthermore, the effective tube length Lp along the thickness
In the past, there have been extensive studies about the of the membrane is assumed to be fixed. The water content can
evolution of pore geometry in this network.11,27,30,42,43 However, then be expressed in terms of the volume or weight fraction
no single accepted view of membrane structure has emerged.
Probably, no simple geometry will give an appropriate descrip- Vw Mw
tion. Instead, it is most likely that the water fractions can be wV ) or wM ) (28)
Vp + V w Mp + M w
considered to form tortuous pathways with ramified boundaries.
Nonetheless, this highly interconnected porous structure may
where Mw and Vw are the mass and the volume of the water
be subdivided, at least for model purposes, into elementary
fraction in the membrane, respectively, and Mp and Vp are the
segments resembling lamellae, cylinders, or spheres. According
mass and the volume of the dry polymer material, respectively.
to recently obtained experimental data,42 cylindrical pores prevail
Due to the fixed tube length Lp, the total water volume is given
in the swollen membrane. Upon dehydration, the number of
by
pore constrictions increases, possibly leaving structures that look
like networks of spherical pores connected by narrow cylindrical
Vw ) Nπ⟨R2⟩Lp, Mw ) VwFw (29)
channels.
The simplest possible calculation of the sample conductance
where Fw is the density of water. The mean square radius of
as a function of water content would rest on a model of parallel
pores (cylindrical segments) is given by
identical straight tubes of length Lp with a uniform cylindrical
cross section.39 Within such an approach, the single pore
properties completely carry over to the membrane (simplistic ⟨R2⟩ ) ∫0∞dRR2 g(R) (30)
model). As a next step toward a more sophisticated description,
a distribution of diameters of parallel straight tubes can be where g(R) is the “pore size distribution” function, abbreviatedProton Conductance in Polymer Electrolyte Membranes J. Phys. Chem. B, Vol. 105, No. 17, 2001 3655
as psd hereafter. Each element of the linear chain has a Equations 30 and 31 establish the relation between the psd and
probability density g(R) to have the radius R. The shape of the w. The psd acquires new shapes with the variation of w. Note
psd and its evolution upon water uptake determine the water that the definitions used within the calculation of w are
volume fraction. The general expression of the water content independent of the system dimensionality and the specific model
is, thus, given by variant. By means of eqs 32-34, GM is related to the psd.
To calculate GM(w), we need to specify Gp(R) and g(R). Gp(R)
π⟨R2⟩ was calculated in section VI. The procedure of the generation
w) (31) of a suitable psd which continuously evolves with w is described
A + π⟨R2⟩
in the next subsection.
Here, the effective “area” A is a parameter of the dry membrane, B. Model of Pore Space Formation during Swelling and
a normalized cross section of the polymer part. It is defined Deswelling. The requirements for the model of pore space
differently, subject to whether the water content is given in terms formation that we will introduce here are as follows. The psd
of the volume fraction [AV ) Vp/(NLp)] or the mass fraction should closely resemble experimentally observed psds of
[AM ) Mp/(NLpFw) ) AVFp/Fw]. polymer electrolyte membranes such as Nafion and its relatives.
In the serial model of cylindrical segments, identical in length, The data in ref 49 are well-approximated by a log-normal psd,50
the membrane conductance is given by39 a typical psd of ultrafiltration membranes.51 Moreover, the model
should allow the straightforward incorporation of single pore
N e information, i.e., Gp(R).
GM ) G (32)
3n p Particularly convenient in this respect is the so-called bond
shrinkage/expansion model,14 constructed subject to the fol-
where Gep, an effective conductance of a segment, is defined by lowing algorithm. Consider a single straight cylindrical tube
with a uniform cross section. This single tube of length Lp is
Gep ) { ∫0∞dR[Gp(R)]-1 g(R)}-1 (33) now subdivided into n cylindrical segments of length lp
()Lp/n). At each step of the iterative procedure, choose
The factor 3 in the denominator is due to the fact that only randomly a single segment i and multiply its radius by a fixed
one-third of all pores point in the spatial direction of macro- factor t,
scopic flow.
Ri f tRi
In the three-dimensional, multiply interconnected random
resistance network, the system of Kirchhoff equations for current Depending on t, the water fraction will shrink (t < 1) or ex-
densities through all nodes and for potential drops between the pand (t > 1). This model resembles a regular, continuous
nodes must be solved. This complex task can be handled exactly (de-)swelling process. It is straightforward and simple, and at
only by means of numerical simulations. An approach toward any stage of the swelling procedure, it allows us to relate w,
this problem, developed by Kirkpatrick,45 is based on EMT. g(R), and GM, using Gp(R) as an input. For the serial pore model,
Here, the random resistance network is replaced with an analytical expressions (or at least integral relationships) for w
effective network in which the elementary conductances Gp(R), and GM can be obtained. Moreover, the procedure can be
distributed according to the psd g(R), are replaced with identical extended to three-dimensional interconnected networks.
effective conductances Gep. The value of the latter is deter- It will be shown in section IX.E that the psd evolving out of
mined self-consistenly from the integral equation46,47 the initial configuration after M elementary procedures is a log-
normal distribution. The parameters which control its shape are
e
∞[Gp - Gp(R)]g(R) dR
∫0
the shrinkage/expansion parameter t and the number M. The
)0 (34) closer t is to 1, the narrower the psd is. In the limit t f 1, the
Gp(R) + (z/2 - 1)Gep
psd will approach a δ function-like distribution, equivalent to
where z is the network coordination number (the number of the simplistic pore space model.
bonds meeting at a node), and z ) 6 for a cubic lattice [a refined The model provides various options. If, for example, t ) 0,
approximation suggested by Bernasconi and Wiesmann48 in- segments completely vanish at a single step, the usual bond-
corporates correlations among water-filled bonds meeting at a percolation problem with finite conductance threshold at w >
node; however, it leads to an equation equivalent to eq 34 if 0 will be recovered.53 Otherwise, at t > 0, the system will have
one replaces z/2 - 1 with a dimension-dependent parameter, the conduction threshold at w ) 0. Further options can be
and in complex disordered media with ill-defined coordination introduced by additional constraints. For instance, one may fix
numbers, this parameter can be treated as a fitting parameter an initial monodisperse psd for the dry system and then allow
which reproduces the correct percolation behavior11]. Equation each segment to swell exactly one time by an equal amount.
This resembles the two-state model, where pores are either dry
34 establishes the link between the effective conductance Gep
with poor residual conductivity or swollen to the equilibrium
of an individual cylindrical segment and the structural charac-
radius in the water-filled state with good conductivity. The latter
teristics of the pore network, i.e., the psd and z. The overall
problem was studied in detail in ref 11.
effective network conductance GM is proportional to Gep. For
C. Analytical Expressions. The initial distribution of the
the serial model, corresponding to z ) 2, this proportionality
cross sections of the cylindrical segments of length lp is given
factor is given by N/3n (cf. eq 32), where the term in the
by
denominator accounts for the tortuousity of the tubes, since n
stands for the ratio of actual tube length Lp to the membrane Ai,0 ) πRi,02, i ) 1, ..., n
thickness. For the network model, the proportionality factor
depends on the interconnetivity and tortuousity of the pore space where the first index i denotes the number of the element in
in a somewhat more complex way. the ensemble and the second index denotes the number M of
Equations 30-34 form the general basis on which the iterations performed on this particular segment. The initial
calculation of relations between GM and w is performed. distribution corresponds to M ) 0. Since lp is fixed, the ensemble3656 J. Phys. Chem. B, Vol. 105, No. 17, 2001 Eikerling et al.
average of Ai,M will determine the mean water volume in a tube the pores were filled with a homogeneous bulk electrolyte of
fixed, R-independent conductivity, then µ ) 2 and GM(w) ∝
Vw ) Lp⟨Ai,M⟩n ) Lpπ⟨Ri,M2⟩p (35) w(1/t2). These proportionality relations indicate that one should
expect to find an effect of hysteresis between swelling (t > 1)
where the terms in broken brackets denote an ensemble average and drying (t < 1) processes in GM(w). t > 1 produces a
over all segments. tendency toward convex relations GM(w) (with an exponent ê
After M statistically independent elementary shrinking or < 1), whereas t < 1 gives concave forms (ê > 1). For t ≈ 1,
expanding processes have been performed, a new distribution corresponding to a very narrow, δ function-like psd, one would
of average cross sections, A h i,M, will emerge. The probability expect a simple proportionality GM ∝ w and a small hysteresis.
for any particular segment i to shrink or expand exactly m times Membrane with Residual Surface ConductiVity. Suppose now
(0 e m e M) is given by the binomial distribution an initial distribution
M! 1 m n - 1 M-m Ri,0 ) δ
P(m) )
(M - m)!m! n ()(n ) (36)
corresponding to a residual water content
The average cross section of that segment is then given by
( )
πδ2
M
t2 - 1 M wres )
A ∑
h i,M ) Ai,0
m)0
t2m P(m) ) Ai,0 1 +
n
(37) A + πδ2
and consider the membrane swelling process t > 1. In the limit
and the new average cross section of the one-dimensional of small extra water contents, i.e., w close to wres and Ri,M close
network is to δ for all segments, Gp(R) is given by eq 24 which can be
( )
written in the form
t2 - 1 M
[ ( )]
π⟨R2⟩n ) ⟨A
h i,M⟩n ) ⟨Ai,0⟩n 1 + (38) 2 1 µb 1
n Gp(R) ) Gres
p 1 + k2 1 - + 2m , k2 ) + - 1 (44)
t m
t µs 2λ
This expression will also be valid in three dimensions.
The resistance of the serial array of conductances in one tube Here, Gres 2
p ()(Λe/lp )µs) is the residual conductance due to the
of length Lp is obtained from surface contribution. After insertion of Gp(R) into eq 39,
M straightforward algebraic analysis gives the following ap-
R tube ) n⟨ ∑ [Gp(tmRi)]-1P(m)⟩n (39) proximation
{ [ ( )
m)0
N res t-1 w
GM(w) ≈ G 1 + k2 2 -1 -
Equation 38, inserted into eq 31, and eq 39 establish the relation 3n p t (t + 1) wres
]}
between w and Gs [)N/(3R tube)]. This relation is determined
1 t4 - 2t - 1
by the initial psd, i.e., by the set Ri,0, by Gp(R), by the parameter (w - 1)2 (45)
t, and by the number of processes M. 2 t4(t + 1)2
D. Analytical Solutions. Power Law Dependence. There are
analytical expressions for Gs(w), if Gp(R) follows a power law which is the law expected at small w f wres. This describes a
linear dependence GM(w) for w f wres. The maximum slope of
Gp(R) ) k1Rµ the linear term is obtained for t ) 1.62. For t close to 1, the
quadratic term will become dominating. The behavior of GM(w)
with a constant k1. The summation in eq 39 then gives for the in the limit of small water contents can, thus, provide informa-
single tube resistance tion about the pore space evolution and about the character of
the psd. Figure 5 depicts, how the parameter t of pore space
R tube )
n
k1 (
⟨Ri,0-µ⟩n 1 +
t-µ + 1
n )
M
(40)
evolution affects GM(w) in the limit w f wres. The closer t is to
1, the more concave is GM(w). t ≈ 1.5 gives a more or less
linear relation over the plotted range.
For t < 1, in the thermodynamic limit (n f ∞) and at small w, E. PSD Continuum Limit. Here, we present expressions for
one finds after elementary algebra a power law the psd in the continuum limit. We consider a monodisperse
initial configuration with Ri,0 ) R0 for all pores. In the limit n
GM(w) ∝ gwê (41) and M f ∞, the binomial probability distribution P(m), eq 36,
approaches a Gaussian distribution for M/n . 1
[ ]
with the exponent
1 (m - mj )2 M
1 t2 - t2-µ P(m) ) exp - j)
,m (46)
ê) 2 2 (42) x2πm
j j
2m n
t t -1
Note that, in the limit of small w, i.e., for M/n ) O (1), the
and the proportionality constant continuuum limit corresponds to a Poisson distribution
( )
N A ê
g) k (43) jm
m
3n 1 πR 2 P(m) ) j)
exp(-m (47)
i,0 m!
Since the exponent is determined by the parameters t and µ, This limit, however, was already studied by means of the
it is, thereby, affected by the shape of the psd. If, in particular, analytical expressions in the previous subsection. In the furtherYou can also read
