Theoretical investigation of water sorption in perfluorinated sulfocationic membranes

Colloids and Surfaces A: Physicochemical and Engineering Aspects 158 (1999) 385 – 397 www.elsevier.nl/locate/colsurfa

Theoretical investigation of water sorption in perfluorinated sulfocationic membranes Y.K. Tovbin *, N.F. Vasyatkin Karpo6 Institute of Physical Chemistry, ul. Vorontso6o Pole 10, Moscow, 103064, Russia Received 30 August 1997; accepted 24 March 1999

Abstract A water sorption process in the perfluorinated sulfocationic membranes (PSM) is of interest from the point of view of study of formation of monolayer and multilayer fillings in strongly nonuniform environments and influence of a character of membrane fillings on the diffusion characteristics of their permeability. In the practice tasks the interest to the PSM for the experimental and theoretical investigation is due to unique properties (high chemical and thermal firmness, high permissible density of current). At this work the model of membrane structure is proposed. The interparticle potentials created on the quantum-chemical MNDO calculations are used in the PSM-cation-water system’s potential energy calculation. Influence of the water content of a membrane on character of distribution of molecules of a water is investigated by method of atom – atomic potentials. On analysing of water locations at different water contents in membrane, a lattice structure for water distribution in pore space was created. With lattice-gas model a macroscopic characteristics such as the isotherm and heat of sorption and the coefficient of diffusion of water molecules are calculated. The results are comparing with experimental data and the results of the molecular dynamics simulation method. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Water; Sorption; Isotherm; Perfluorinated sulfocationic membranes; Membrane structure; Diffusion of water molecules; MNDO approximation; Atom–atomic potential; Lattice – gas model

1. Introduction High chemical stability and heat resistance, as well as a high value of limiting current density [1,2] characterize perfluorinated sulfocationic membranes (PSMs). Because of their unique properties, these membranes are used widely in the electrolytic preparation of sodium hydroxide

* Corresponding author. Fax: +7-95-975-2450.

and chlorine. Perfluorinated sulfocationic membranes are synthesized from the copolymer of tetrafluoroethylene and perfluoro-3,6-dioxa5methyl-8-sulfonylfluorideoctene-1-[CF2CFOCF2CF–(CF3)OCF2CF2SO2F]. As a result of the subsequent hydrolysis in NaOH, SO2F groups in the side chains are transformed to produce ionogenic SO− 3 groups, which are known to control transport properties of the membranes. It has been established that the membrane properties are to a large degree determined by the molecular

0927-7757/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 9 9 ) 0 0 1 7 7 - 6

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structure of the amorphous part of the PSM with the structural formula

where n =6 – 13. A water sorption process in the PSM is of interest from the point of view for study of monolayer formation and multilayer fillings in strongly non-uniform environments and influence of a character of membrane fillings on the diffusion characteristics of their permeability. Below, we will consider the structural and energy aspects of the molecule of water transport in PSM. We suggest a three-dimensional local structural model of the amorphous part of the membrane and consider model estimations of the activation energy of lithium cation migration for low moisture contents, as well as influence of the water content of a membrane on character of distribution of water molecules and a macroscopic characteristics such as the isotherm and heat of sorption and the coefficient of diffusion for water molecules. To obtain a more detailed description of the mechanism for molecules transport in PSM, a wide range of experimental methods was used [3]. The most detailed information on the cation transport is given by the NMR technique (see for example [4,5]). Transport characteristics and selectivity of PSM are known to depend strongly on water content as well as on the nature of the cations. However, a molecular interpretation of the experimental data does not always enable one to make unambiguous conclusions. So, the NMR data cannot be interpreted unambiguously and used directly to characterize membrane permeability. In this connection, theoretical calculations of the membrane properties seem to be important. Our analysis begins with an estimation of the + energy characteristics of the SO− system. The 3 Li simplest approximation is the atom-atom approximation [6]. This approximation was widely used for calculations of structural and thermodynamic characteristics of polymer systems [7], as well as to describe the transfer of cations of alkali metals in protein canals by means of the method of molecular dynamics [8 – 13]. In this case, the po-

tential energy of cations accounts for both the Lennard–Jones and Coulomb contributions. However, as it was shown in Ref. [14] the application of this approximation is not justified in calculations of the energy characteristics of the SO− 3 Li+ system. Because of large variations in the charges of the lithium cation and the atoms in the SO− 3 group with distances between them, the po+ system can not be tential energy of the SO− 3 Li considered additive. In this connection, one of goals of this work was to construct the potential of the interaction of the lithium cation and the ionogenic sulfo group of PSM to be used in theoretical calculations of macroscopic characteristics. In the case of non-additive potentials, difficulties of calculations with help of molecular dynamics and Monte Carlo techniques increase significantly. So in order to obtain an isotherm of sorption as well as sorption heat and a coefficient of diffusion of water molecules the lattice –gas model [15,16] was used. Our paper consists of next sections: 2–a model structure of PSM; 3–potential of ion Li+ with the side chain of PSM, 4–energetics of water shell around of the ion Li+ in PSM, and Section 5 contents calculations with a help of lattice–gas model [15,16] of a macroscopic characteristics of water sorption in PSM: an isotherm and heat of sorption and the coefficient of diffusion of water molecules.

2. Model structure of PSM The initial cluster channel model suggested by Gierke [17–19] as it was proven by neutron and X-ray scattering experiments, the Mo¨ssbauer spectroscopy data, etc. [20–25], is incorrect. This made it necessary to formulate a three-dimensional structure model. The another structure model for the amorphous PSM part [3] is based on an analogy with the comb-shaped polymers [26]. It is a model of a layer structure with side chains directed towards each other. A schematic two-dimensional image of the channel model [3] is insufficient for relating the permeability properties of a membrane to the elementary process of cation migration. We should, therefore, construct

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a three-dimensional structural model of the amorphous PSM part. Consideration of all of the experimental data mentioned above allows us to suggest the following simplest local structural model of the amorphous PSM part (Fig. 1). The membrane consists of pores formed by closely packed polymer chains. Pore walls are characterized by a bilayer packing. The width of pore increases with moisture content: L1 =30–50 A, [20]. The lower value corresponds to a dry membrane and agrees with the estimate made for the layer packing of comb-shaped polymers: twice the fluorocarbon chain diameter ( 5 A, ) plus twice the side chain length (10 A, ). We remind that a dry membrane at 25°C contains : 1.7 water molecules per one SO− 3 group [27], which is about 2.5% of the maximum amount of the sorbed water. According to [18,21,28,29], the linear dimensions of pore width are estimated as L2 =100–170 A, and L3 =50 – 100 A, . Chain bends, their entanglement, and fastening in the crystal phase making up 3 – 8% of the total matrix volume disturb the regular packing of chains forming the pore ‘walls’ and make the chains relatively rigid (i.e. stabilize the structure). Such intermediate regions surrounds the space of pore. They may be roughly considered as an ensemble of cylindrical pores (the number of chains is not sufficient to produce dense structures) which are built by bent fragments of polymer chains. The characteristic diameter of these pores in the given model is L5 =15 – 25 A, , which agrees with the known estimates [18,30,31]; the

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length of the intermediate regions (cylindrical pores) is of the order of magnitude of L4  20 A, [31]. Side chains in cylindrical pores may be deformed. In a pore the average distance between side chains in the ‘free’ state is approximately  8 }9 A, [21]. On the one hand, the suggested model complements the two-dimensional scheme [3] and extends it to the third dimension. The existence of the intermediate region L4 prevents the channel walls from being ‘suspended’ in free space [3] and brings the model closer to the ideas introduced in Refs. [17–19]. On the other hand, Fig. 1 may be interpreted as the replacement of spherical associates [17–19] by periodic bilayer structures [20,26], which is confirmed by recent experiments. However, it should be noted that the problem of the long-range spatial organization of pores remains open.

3. Potential of ion Li+ with the side chain of PSM To construct a three-dimensional potential describing a nonisotropic potential surface in the vicinity of the sulfo group, we considered the following fragment of the side chain.

Fig. 1. The structural model of the amorphous PSM part. L1 =30…50 A, , L2 =100…170 A, , L3 =50…100 A, , L4 =20 A, , L5 =15…25 A, .

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Table 1 Numbers and angles (valence and dihedral) of the rays for constructing the Li–SO3 potential N

1

2

3

4

5

6

7

8

9

10

11

12

13

a 8

69 125

101 125

141 125

69 45

101 45

141 45

69 −45

101 −45

141 −45

69 −100

101 −100

141 −100

180 –

The same fragment was used earlier [14] to determine the energy and electron characteristics of localization of H+ ions at different hydrations of the ionogenic sulfo group by using semiempirical quantum-chemical methods within the MNDO and CNDO/BW approximations and similar characteristics of lithium ion in the absence of water (the MNDO approximation). Our calculations [14] showed that as the Li – O distance increases from 1.5 to 3 A, the charge on the Li+ ion changes from 0.5 to 0.7 e−, and the negative charge is localized (70 – 80%) on the CF2 –SO− 3 fragment. The above fragment enables one to calculate the energy of removal of the cation from the sulfo group within an accuracy of 0.1%. We used the MNDO approximation (calculations performed with the AMPAC program) with the optimization of the location of atoms of the + SO− relative to the 3 group. The position of Li sulfur atom was fixed and specified by a set of coordinates R, a, 8. Here, R is the length of the Li–S bond, a is the valence angle of C9 –S12 –Li, and 8 is the dihedral angle formed by the C6 –C9 – S12 and C9 – S12 – Li planes. Removal of the cation was assumed to occur along chosen directions (‘rays’) specified by fixed angles a and 8. The value of R was varied from 2.0 to 5.0 A, in 0.3 A, increments; angle a was 69, 101, 141, 180°; angle 8 was − 100, − 45, 45,125°. To construct a threedimensional potential, we analyzed cation removal along the rays described by the combinations of angles a and 8 (see Table 1, where N is the ray number). Note that the optimization procedure involves displacements of the atoms of the side chain from their initial positions. Therefore, coordinates R, a, 8 were measured with respect to the coordinates of sulfur atoms in a fully optimized SO3Li group (the shift of the sulfur atom did not exceed 0.2 A, ). The energy of interaction between the lithium and

the side fragment was estimated as DE = ES – Li − ES, where ES – Li is the energy of formation of a cluster with the lithium cation, ES is the energy of this cluster at an infinite distance from Li+ (this corresponds to R= 6 A, ). All the data on DE, the charges on all atoms of the cluster, and the displacements of the atoms of the SO3 group relative to C9 were collected in a three-dimensional array that was used as the basis for the approximate calculations of the corresponding values by using cubic spline interpolation [32]. Fig. 2 shows the total potential energy of the system U=ELJ + ECoul (for example for ray N= 13) is calculated by different methods (the Lennard–Jones contribution is shown by the dashed line). The result of the quantum–chemical calculation of ECoul is shown with the curve a. The case of the standard atom–atom approximation [6,7] the total energy was calculated as the sum of atom–atomic potentials in the Lennard–Jones and Coulomb potentials for respective particles

Fig. 2. The potential Li – SO3: (a) constructed on the base of quantum chemical calculations; (b) atom – atomic potential. The dotted lines are the Lennard – Jones parts.

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Table 2 The parameters of atom–atom potentials Li

SO3

Conditions

A,B,C

O

Li q = +1 [35]

A B C

−25.30 4483.2 332.156

−97.28 27834.2 −332.156

−445.20 19200.0 −225.813

−248.2 1800.0 112.9065

SO3 q = −1 [33]

A B C

−97.28 27834.2 −332.156

−361.0 160963.3 332.156

−282.0 282923 225.813

−126.29 27312.8 −112.9065

O q = −0.68 [34]

A B C

−445.2 19200 −225.813

−282.0 282923 225.813

−200.0 410000 153.5168

−3760.0 9700.0 −76.7584

H q =0.34 [34]

A B C

−248.20 1800.0 112.9065

−3760.0 9700.0 −76.7584

−40.0 3800.0 38.3792

−126.29 27312.8 −112.9065

H

(1)

approximation. It is assumed that each sulfo group has its ‘own’ cation in the most favorable energy state, i.e. the system is neutral (Fig. 3). The top of Fig. 3 shows the projections of SO3 groups located at the lower boundary of the pore onto

In the formula (1) the charge on lithium cation is + 1, and the charges on the atoms of the PSM fragment correspond to R “ , whereas the real distance between the Li+ and PSM side chain can be arbitrary. This charges on the atoms of the PSM fragment have been obtained by MNDO approximation also. For curve b, the Lennard– Jones’s contribution is higher because the geometry of the sulfo group is fixed. The differences between the above curves are most pronounced at R = 2 – 3.5 A, . The positions of the minima on curves a and b virtually coincide, however, the corresponding minimal values are different. Thus, we see that the application of the atom–atom approximation for the calculation of potential surfaces in the systems containing charged ions and functional groups may result in considerable errors in the values of energy, so that the problem of the accuracy of describing the system energy assumes primary importance. The obtained potential was used to calculate the energy profile describing cation migration in amorphous regions of dry PSM. The energy profiles were compared with the results of the calculations performed within the atom–atom

Fig. 3. The potential profile of the migration of Li ion in the pore between two SO3 groups. (a) calculated with quantum chemical potential; (b) with atom – atomic approximation.

with the parameters listed in Table 2 (A is measured in kcal/mol A, 6; B, in kcal/mol A, 12, and C, in kcal/mol A, ): Uij = Aij /R 6ij +Bij /R 12 ij +Cij /Rij

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the xOz plane. The distance between the groups is 8.91 A, . This value correlates with data [3]. The minimum energy positions of lithium cations near SO3 groups are shown by crosses. The trajectory of the cation was chosen to pass as close as possible to the minimum energy positions without intersecting the Lennard – Jones spheres of the side chains. In the calculations, we used a trajectory parallel to the x-axis passing at a distance z=12.5 A, from the lower boundary of the pore. The y coordinate of the trajectory coincided with the coordinates corresponding to the optimal positions. We took into account the interactions of Li+ with all the side chains located at pore boundaries as well as with the pore boundaries themselves (a potential of the ‘3 – 9’ type, which can be obtained by averaging the atom–atom potential over a half-space). We considered the potential of interaction of the lithium cation with the nearest sulfo group using two approximations. At large distances, the potentials coincide and they are described within a standard atom–atom approximation. In the system all sulfo groups are symmetric relative to section passing through the central SO3 group. However, all these groups are oriented along the x-axis, and the symmetry is violated (potential curves a and b are not symmetric relative to section passing throw the minimum). Potential curves a and b markedly differ near the minima associated with the sulfo groups, but they virtually coincide between the sulfo groups. The difference in the values of the minima is related to the peculiarities of potentials a and b on Fig. 2. Curve a has two local minima corresponding to the minimum energy near the neighbouring sulfo groups. The oscillations of curve a is related to rearrangements in the geometry of the SO3 group. However, one may see that, in the region when the displacement x =10 – 13 A, , the atom-atom curve with constant charges b differs from curve a. This difference can be explained as follows: for curve b, the charge on the cation bonded to the right-hand sulfo-group is +1, whereas, for curve a, the cation carries a fractional charge corresponding to the position of minimum energy.

4. Energetics of water shells around of the ion Li+ in PSM Water molecules influence essentially on the transport of ions. The matrix PSM, in which there are the processes of carry of cations and molecules of water, was simulated as the pore, the walls of which are formed by the polymers– (CF2 –CF2)-with periodic side branches, directed towards the other to other. In pore there are also cations of the lithium, number of which equally to number of sulfo groups in the pore, and molecules of water, the number of which on one sulfo group defines water content of a membrane. The elementary cell contained eight side chains (4 from above and 4 bottom-up). To avoid influence of boundary effects periodic boundary conditions were applied. The walls of the pore were considered as particles with parameters, appropriate to atom F. The calculations were carried by two Li+ –SO− potentials: quantum–chemical and 3 atom–atomic. For water, we used a simple rigid three-point modification of the SPC model [34] which agrees quite well with the experimental density and the data on the heat of vaporization. This model rather well describes hydrogen bonds. The energetics of a Li–(H2O)n system in pore of PSM was investigated for n= 1,…,10. The structures according the minimum of the potential energy of a system were found with optimization DFP method [36]. It have been found that for small number of molecules of water, the main type of interaction for water molecules is the interaction with cation. At increase of water content, the interaction between molecules of water is increased. The complete energy of system PSM– Li–(H2O)n, where n =1–10, can be written as U= USI + UIW + USW + UWW + UWall,

(2)

where USI is the energy of interaction of a side chain with the cation, UIW is the energy of interaction for cation with molecules of water, USW is the energy of interaction for a side chain with molecules of water, UWW is the energy of interaction for molecules of water among themselves, UWall is the energy of interaction for cation and molecules of water with pore wall. The contributions in the formula (2) was presented as a sum of

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391

Fig. 4. The energy of the successive attachment water molecules to the lithium hydrate shell in the pore. 1-atom-atomic potential, 2-calculated with quantum chemical potential, 3-calculated macroscopic heat of adsorption, 4-experimental heat of adsorption [39].

Lennard–Jones and Coulomb potentials (as in Eq. (1)) at exception USI, where instead of Coulomb tabulated potential, constructed on the basis of the quantum chemical calculations, was used. The interaction of particles with walls was presented as the potential of the ‘3–9’ type [37]. For water – water interaction we have UWW =Aij /R nij +Bij /R 12 ij +Cij /Rij, where n=6 for O…O, H…H, and n = 10 for O…H. The Lennard – Jones’s potentials for remaining interactions were obtained with the use of the Lorentz – Berthelot combination rules [35]. The interaction with the main polymer chain was ignored because of its small contribution to the process (B 0.01%). The special attention was given to selection of parameters of interparticle potentials for cation Li with water. The adjustment of the parameters was carried out under experimental meanings

for change of the energy of sequential connection of molecules of water to Li ion [38]. The Lennard–Jones’s parameters for interaction of lithium with atoms O and H were selected so that the value and the distances up to minima of potential functions Li–(H2O)n coincide to appropriate ones, received from experiments on measurement of heats of formation of microclusters in vacuum with n from 1 up to 4. A procedure of optimization by the DFP method was used. The received Lennard–Jones’s parameters are resulted in Table 2. Through procedure DFP search of a minimum of potential function of system PSMcation Li–(H2O)n, where n= 0–10 was made. For each meaning of n a set of minima for Un were found. The most deeper minimum was used as so an energy of sequential connection for molecules of water was determined as DEn = Un − Un − 1

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The results of accounts are resulted on Fig. 4. The atom–atomic approximation curve (without the charge distributions) and experimental data on heats of water sorption in PSM [39] are presented for comparison. The first three molecules of water, engaging in first hydrated shell of lithium, are strongest connected with sulfo group. Further increase of the number of water molecules results in removal them from cation. The analysis of the geometry of the water clusters has shown that the main interaction for water molecules is their interaction with cation. The curves are qualitatively coincide. Most differences are observed for the first three molecules and besides the curve calculated by quantum–chemical potential is nearer to the experimental curve. The calculations have been shown that the first rigidly localized hydrate shell consisting of the three water molecules strongly connected with cation exists around the cation for all numbers n. As increasing n, the first hydrate shell displaces few off a sulfo group saving nevertheless their geometry (Table 3). Others water molecules form next hydrate shells. They are located between the neighbouring side chains and can form a common water layer when hydrate shells from different SO3 groups overlap and thus create the conditions for cation migration on the water layer along the pore. On analysing of water locations at different water contents, a lattice structure for water distribution in pore space was constructed. At distances more than the second hydrated coordination shell of the cation, we used a lattice structure which coincides with the model offered in work [40,41]. This lattice structure was used for calculations of macroscopic characteristics of system under investigation. Table 3 The distances of water molecules in the first hydrate shell of Li near the SO3 group

Li 1-wat. 2-wat. 3-wat.

1-wat.

2-wat.

3-wat.

O in SO3

1.87 – 2.87 2.85

1.85 2.87 – 3.59

1.85 2.85 3.59 –

3.17 2.89 3.07 4.69

5. Lattice–gas model calculations In order to calculate some macroscopic characteristics for water sorption in PSM we have used the lattice–gas model [15,16,40–42]. In this model a volume of porous space of PSM is divided into an elementary volumes (sites) of size V equaling a size of water molecule. We were assumes that different orientations of water molecule were considered as different kinds of particles. Each site of the lattice is a sorption center, and can be occupied by a particle of any kind i (including a vacancy), 15 i5 s, s is the number of components. The state of occupancy of each site f 15 f5 N, N is the number of sites in repeated fragment of porous space), will be characterized by the variable g if, with g if = 1 if the site f contains a particle of kind i and g if = 0 if it contains a particle of any other kind. These variables obey the following equations g 1f + …+ g sf = 1, g ifg fj = Dijg if, where Dij is the Kronecker symbol, which states that any site can be occupied by any particle. The type of the site numbered f will be characterized by the parameter, h qf assumed to be known and unvarying during the sorption process (non reconstructing lattice structure), with 15 q5t, where t is the number of types of site potential with an interaction radius R, where R has any given value. Distances will be expressed as the number of coordination shells (c.s.). The number of sites in the rth c.s. of a type-q site numbered f will be denoted by zf (r) [or by zq (r)], with 1 5 r5R. The interaction parameter of the species i and j on site of types q and p numbered f and g at a distance r will be denoted by o ijfg (r) [or by o ijqp (r), since the number of the site and its type are uniquely correlated by the parameters h qf ]. The total energy of the sorption system in the grand canonical ensemble is described as follows [15,16,42]: N

s

H= % % h if,



h if

f=1 i=1

= 6 iq −



s 1 R % % % o ijqp(r)g gj rh pgr g ifh qf , 2 r = 1 gr j = 1

= − b − 1Ln(Y iq), = F iqb exp(bQ iq)/F oi .

Y iq = A iqPi,

6 if

A iq (3)

Y.K. To6bin, N.F. Vasyatkin / Colloids and Surfaces A: Physicochem. Eng. Aspects 158 (1999) 385–397

where the subscript gr numbers the sites around of site f in the rth c.s., A iiq is the sorption coefficient of particle i on the site q; F iq and F oi are the partition of the sorbed particle i on site q and of the molecules i in the gas phase; (b = (kBT) − 1; Pi is the partial pressure. Q iq is the energy of the bond between the particle i and polymer chains in the site q. We limit the discussion to direct correlations between the interacting particles, the closed system of equations for u if (the probability of finding i particle in the site numbered f ), and u ijfg (r) (the probability of finding i and j particles in sites f and g at a distance r) becomes [15,16,42] A A (1− u A f )Y q = u f

R

5

r = 1 g  zf(r)

5 Sfg (r),

Sfg (r)

A = [1 +u AA fg (r)xqp (r)/u f ],

u AA fg (r)

xqp (r) =exp [−bo AA qp (r)] − 1, =2u u /[dfg (r) + bfg (r)], A f

A gr

dfg (r)

A =1+ xqp (r) (1 −u A f −u g ),

={[dfg (r)] +4xqp (r)u u } , 2

A + u A6 fg (r) = u f ,

=u 6g,

A f

A 1/2 g

bfg (r) u AA fg (r)

66 u 6A fg (r) + u fg (r)

6 uA f +u f =1.

(5)

This system of Eqs. (4) and (5) offers a detailed description of the distribution of particles upon different sites. It has been solved numerically. Once the equilibrium distribution of molecules is established the isosteric heat of sorption can be computed by the thermodynamic definition [37] Q(u)= −(d ln P/db)u.

(6)

The diffusion coefficient was derived [43,44] considering the direct and opposite jumps of a molecules from the f-site to the g-site at a distance r, 1 5 r5 r*, where r* is a maximum length of a jump. The transition state theory enables one to estimate the direct and opposite jump rates, and a difference between these rates gives the diffusion coefficient. It expressed as follows r*

N

r=1

f=1

D(u)= % r 2 % z*f (r)

%

g  zf*(r)

Vfg (r)Tfg (r),

where z*f (r) is the number of jumps paths from site f over distance r in the direction of diffusion, a function Tfg (r) is describe a changing in populations of different lattice sites with variation of total coverage u, Vfg (r) is the rate of an elementary jump between sites f and g spaced r apart R

(7)

pr

Vfg (r)= Kfg (r)u A6 5 5 S A6 fg (r) 5 fgh (vr r), r = 1 vr = 1 h

S A6 fgh(vr r) 6i u Ai fh (r1)u gh(r2) exp{b[o*fh (r1)+ o*gh (r2) A 6 i u f u gu h i=A n

= %



−ofh (r1)]}, =

(4)

393

Tfg (r)

n

d af (1−uf )P ln , du (1−tfg (r))Sfg (r)

= K 0fg(r) exp [− bEfg (r)],

Kfg (r) (8)

K0fg (r) is the preexponential factor in jump rate constant Kfg (r) of a molecule between sites f and g at distance r, Efg (r) is the jump activation energy, ofg (r) and o*fg (r) are the energy parameters of interaction of a molecule in site f in the ground and transition states, respectively, with an adjacent molecules in ground state at a distance r in site g. Here i =A is the sorbed molecule of water and 6 is the vacancy. Index h is the number of site with orientation vr at distances r1 and r2 from central sites f and g, respectively; 15 vr 5 pr, pr is the total number of orientation of neighbouring sites in the common coordination sphere around the central sites f and g. There are two types of lattice parameters in the theory: (1) the lateral interaction parameters between sorbate–sorbate o; and (2) the parameters of interaction of the sorbed particle with the polymer chains (sorbate–sorbent) Qq and Aq. In order to compute this lattice parameters we have been used the potential of interactions discussed above. Let us sphere around the central sites f and g. There are two types of lattice parameters in the theory: (1) the lateral interaction parameters between sorbate–sorbate o; and (2) the parameters of interaction of the sorbed particle with the polymer chains (sorbate–sorbent) Qq and Aq. In order to compute this lattice parameters we have

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Fig. 5. The local coefficients of molecule-wall bond energy Qq and Henry constant Aq.

been used the potential of interactions discussed above. Let us use the expressions derived in Ref. [45] for the calculation of the lattice parameters Qq, Aq and o: q= =

&& && &&&& &&&& Vq

Vq

Vq

Vq

=−

U(r) exp[ − bU(r)]dVq dVq /Aq, exp[ − bU(r)]dVq dVq,

Aq

o

E(r) exp[ − bE(r)]dVq dVq dVp /

E(r) exp[ − bE(r)]dVq dVp dVq dVp. (9)

where E(r) is the potential of the interaction of sorbate–sorbate, Vq is the volume of the site with the number q in porous space of the sorbent, Vq is the ‘volume’ of orientation space of water molecule (dV = da sinh dh dg, a, h, g are the angles of orientations); U(r) is the potential of interaction of sorbate – sorbent, derived by the sum of all atom–atomic potentials for interactions of atoms in water with ions in solid inside the sphere

with the radius R. It is assumed that in the atom–atomic approximation the meanings of the parameter o do not depend on the value of the sorbate–sorbent interaction. We have been restricted by next four orientation between two neighbouring water molecules:  ’ ,  ’ ,    and  ¡. For these orientations the values o have been obtained 3.6, − 3.6, − 4.3 and 4.3 kcal/mol, respectively The calculated values of the local values of the heat and constant Henry for the system PSM– H2O are shown on the Fig. 5. It is convenient to use for the non-uniform lattices the distribution ` ])) characterizing the function f(Q) (or f(Ln[A portion of the lattice sites with the energy of the binding in interval from Q to Q+dQ (analogously for Ln[A]). (The calculation of the distribution functions was carried out with the precision till 0.5 kJ/mol). The using of the logarithmic scale for the values A is caused by their changing up to 5–6 orders. The obtained lattice parameters are used to calculate isotherm and heat of sorption of water molecules. Heat sorption is presented by curve 3 on Fig. 4. It demonstrates rather well coincide at

Y.K. To6bin, N.F. Vasyatkin / Colloids and Surfaces A: Physicochem. Eng. Aspects 158 (1999) 385–397

middle and completely coverages and qualitative agreement at low coverages. Isotherm sorption is shown on Fig. 6. Since we have not a meaning of the full volume of membrane porous space we adjusted this meaning with the calculated u and experimental DP (for water pressure) for dry and full filling states of membrane. Fig. 6 demonstrate agreement between experimental [27] and calculated curves. For the sake of simplicity of the lattice–gas model calculation of concentration dependence of diffusion coefficient, we assume that R = 2, r*= 1, o*fg (r) =ofg (r)/2 and K 0fg =1. Fig. 7 shows the plot D(u). At low coverages a molecule transport is negligible. With growing of coverage a molecule transport have be rather intensive. Such step-wise dependence D(u) at u B0.15 agree with experimental data [3] and with results of molecular dynamics simulations [46,47]. The simulation confirms that when n B5 molecules of water on one sulfo group, they are localizing around cation and their mobility is small. Increasing water content in membrane gives growing transport both for water molecules and cations.

Fig. 6. Isotherm of adsorption of the water in PSM: desorption branch (1) and adsorption branch (2) [27], 3-calculated curve.

395

Fig. 7. The calculated diffusion coefficient of water in PSM.

6. Conclusion At this work some macroscopic characteristics of the water sorption and diffusion in the pores of PSM are estimated by the theoretical methods. The pore presents a split-like structure formed by closely packed polymers with side chains including SO3 groups and directed one to other. The potential, which in the sum of the additive Lennard–Jones’s potential and non-additive electrostatic contribution calculated according to the quantum–chemical semiempirical MNDO method, was constructed. The obtained potential accounts for changes in the geometry of the SO3 group and the redistribution of electron density between the atoms of the side chain of PSM induced by the migration of the lithium cation relative to the sulfo group. These factors strongly affect the energetics of ion transfer in charged pores and they must be taken into account. This potential was used in theoretical calculations for the migration activation energy of water and cations as well as for isotherm sorption of water in PSMs. A localization of water molecules in hydrate shells of cation Li in the pore at the different + water contents n around SO− 3 Li –(H2O)n (where n changes from 1 to 10) was studied by the minimizing of the potential energy of the system. At small water contents, isolated clusters form around cations. When water contents increase, neighbouring water clusters can overlap. They form crossing water layer connecting water molecules of opposite chains or between chains

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from each side of the pore space. And at last, multilayer filling of the inner part of pore in a membrane occurs further. In particular, the nearest to the cation three molecules of a water will form a rigid structure – a hydrate environment, the properties of which essentially differ from properties of the other molecules of a water. On analysing of water locations at different water contents, a lattice structure for water distribution in pore space was created. With help of the lattice – gas model, some macroscopic characteristics such as the isotherm and heat of sorption, and the coefficient of diffusion of water molecules have been calculated. The temperature and concentration dependencies of calculated values are investigated. The results are comparing with experimental data and the results of the molecular dynamics simulation method.

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