Perfluorinated Ionomer Membranes - American Chemical Society

13

The

Cluster-Network

Model

of

Ion

Clustering

in

Perfluorosulfonated Membranes

1

Downloaded by UNIV OF MISSOURI COLUMBIA on June 17, 2013 | http://pubs.acs.org Publication Date: February 4, 1982 | doi: 10.1021/bk-1982-0180.ch013

T. D. GIERKE and W. Y. HSU Ε. I du Pont de Nemours & Co., Inc., Central Research and Development Depart ment, Experimental Station, Wilmington, DE 19898

A model for ionic clustering in "Nafion" (registered trade mark of Ε. I. du Pont de Nemours and Co.) perfluorinated mem branes is proposed. This "cluster-network" model suggests that the solvent and ion exchange sites phase separate from the fluorocarbon matrix into inverted micellar structures which are con nected by short narrow channels. This model is used to describe ion transport and hydroxyl rejection in "Nafion" membrane pro ducts. We also demonstrate that transport processes occurring in "Nafion" are well described by percolation theory. The solvent and ion exchange sites in "Nafion" perfluori nated membranes phase separate from the fluorocarbon matrix to form clusters (1-5). This ionic clustering w i l l not only affect the mechanical properties of the polymer (1), but should also have a direct effect on the transport properties across these membranes (2). In addition the exchange sites in the resin are strongly acidic and the polymer is extremely hydrophilic. Com bined with the polymer's exceptional thermal and chemical stabi lity, these properties make "Nafion" membranes particularly suit able for a variety of applications. These include applications as membrane separators in several electrochemical processes (6-9), as a superacid catalyst in organic syntheses (10-12), and as a membrane electrode (13). The principal application of "Nafion" currently is as a mem brane separator in c h l o r - a l k a l i c e l l s , shown schematically in Figure 1. In this process water is decomposed in the cathode compartment to produce caustic and hydrogen, while saturated brine is fed to the anode compartment where the chloride ion is reduced to chlorine gas. The role of the membrane is to separate the two compartments, allow the facile transport of sodium ions from the anode to cathode compartments, and to restrict the flux of hydroxyl ions across the membrane. In the classical picture of ion exchange membranes (14) where the ion exchange sites are 1

Current address: Parkersburg, WV 26101.

0097-6156/82/0180-0283$06.25/0 © 1982 American Chemical Society In Perfluorinated Ionomer Membranes; Eisenberg, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1982.


284

PERFLUORINATED

IONOMER

M E M B R A N E S

homogeneously d i s t r i b u t e d , the membrane w i l l f u l f i l l t h i s r o l e only i f the i n t e r n a l m o l a l c o n c e n t r a t i o n of i o n exchange s i t e s g r e a t l y exceeds the c o n c e n t r a t i o n of hydroxide i o n i n the cathode compartment. T h i s c o n d i t i o n i s not s a t i s f i e d . The i n t e r n a l conc e n t r a t i o n of exchange s i t e s i s g e n e r a l l y between 5 and 15 m o l a l , w h i l e the c o n c e n t r a t i o n of the c a u s t i c produced i n the cathode compartment t y p i c a l l y exceeds 15 m o l a l . Indeed we have observed t h a t t r a c e s of calcium i n the b r i n e tend to p r e c i p i t a t e at the anode s u r f a c e , which suggests that the membrane i s v e r y b a s i c throughout i t s t h i c k n e s s . However, i n these same membranes, over 90% of the current may be c a r r i e d by the sodium i o n s . I t appears that the hydroxide i o n can get i n t o the membrane, but i t s motion i s h i g h l y r e s t r i c t e d i n r e l a t i o n to the f l u x of sodium i o n s . I t seems l i k e l y that i o n i c c l u s t e r i n g plays some r o l e i n r e l a t i o n to t h i s behavior. In t h i s work we propose a model f o r i o n i c c l u s t e r i n g , which we have c a l l e d the cluster-network model ( 2 ) , to account f o r h y d r o x y l r e j e c t i o n i n "Nafion" p e r f l u o r i n a t e d membranes. I n deve l o p i n g t h i s model we have been guided by two requirements: 1. the model should be c o n s i s t e n t w i t h the a v a i l a b l e data on the microscopic s t r u c t u r e of the polymer (1-5) 2. the model should be cast i n a form which can be used to d e s c r i b e i o n t r a n s p o r t . In pursuing the second c o n d i t i o n , we make s e v e r a l approximations i n our model which are c l e a r l y i d e a l i z a t i o n s . We f e e l that these assumptions are reasonable and are j u s t i f i e d , i n p a r t , by the u t i l i t y and success of the model which r e s u l t s . In the next s e c t i o n we w i l l present the data and arguments on which the c l u s t e r - n e t w o r k model i s based. We w i l l a l s o d i s cuss the e f f e c t s of e q u i v a l e n t weight, i o n form, and water content on the dimensions and composition of the c l u s t e r s . I n the t h i r d s e c t i o n we w i l l present a formalism, which f o l l o w s from the c l u s t e r - n e t w o r k model, based on a b s o l u t e r e a c t i o n r a t e theory (2) and h y d r o x y l r e j e c t i o n i n "Nafion" p e r f l u o r i n a t e d membranes. F i n a l l y we w i l l o u t l i n e the concepts of p e r c o l a t i o n theory and demonstrate that i o n t r a n s p o r t trough " N a f i o n " i s w e l l described by p e r c o l a t i o n . The C l u s t e r on Network Model. P r e v i o u s s m a l l angle x-ray (1-5) and neutron (4) s c a t t e r i n g experiments c l e a r l y i n d i c a t e that i o n i c c l u s t e r i n g i s present i n "Nafion". However, d e t a i l s of the arrangement of matter i n these c l u s t e r s cannot be obtained from these r e s u l t s alone. For hydrocarbon ionomers s e v e r a l d i f f e r e n t i n t e r p r e t a t i o n s have been advanced as the cause of the SAXS maximum. These i n c l u d e a model of s p h e r i c a l c l u s t e r s on a p a r a c r y s t a l l i n e l a t t i c e proposed by Cooper et a l . (16), the s h e l l core model of Macknight et a l . (17) and more r e c e n t l y a l a m e l l a r model (18). At present there i s no consensus about which of these models best d e s c r i b e s c l u s t e r i n g i n hydrocarbon ionomers. In c o n s i d e r i n g these v a r i o u s models, we concluded that the s h e l l core model was h i g h l y u n l i k e l y because of the unfavorable e l e c t r o -

In Perfluorinated Ionomer Membranes; Eisenberg, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1982.


13.

GIERKE AND

HSU

Ion Clustering

Model

285

s t a t i c e n e r g e t i c s of having i s o l a t e d i o n d i p o l e p a i r s imbedded i n a f l u o r o c a r b o n medium which possesses a very low d i e l e c t r i c cons t a n t (19). Of the remaining geometries which could l e a d to the observed SAXS r e f l e c t i o n , one could i n c l u d e c l u s t e r i n g i n t o spheres, c y l i n d e r s , or sheets. Transmission e l e c t r o n m i c r o s c o p i c s t u d i e s i n d i c a t e that the s p h e r i c a l c l u s t e r morphology i s the most l i k e l y geometry of the three geometries j u s t mentioned (2,20,21). M i c r o graphs from s t a i n e d , ultramicrotomed s e c t i o n s always r e v e a l metall i c c l u s t e r s approximately c i r c u l a r i n shape w i t h diameters of 3-10 nm. An example i s shown i n F i g u r e 2. No evidence to support an e x t e n s i v e c y l i n d r i c a l or s h e e t - l i k e morphology, which would g i v e n o n c i r c u l a r p r o j e c t i o n s i n the micrographs, has been observed by us (21). These r e s u l t s do not exclude the p o s s i b i l i t y t h a t the c l u s t e r s are a s p h e r i c a l . I n t h i s respect our observations are not i n c o n s i s t e n t w i t h the model of Roche et a l . (18), provided the aspect r a t i o of the l a m e l l a r s t r u c t u r e s i s not too l a r g e . The model of i o n i c c l u s t e r i n g we b e l i e v e to be most l i k e l y at present i s that of an approximately s p h e r i c a l , i n v e r t e d m i c e l l a r s t r u c t u r e . I n t h i s model the absorbed water phase separates i n t o approximately s p h e r i c a l domains, and the i o n exchange s i t e s are found near the i n t e r f a c e , probably imbedded i n t o the water phase. Such a s t r u c t u r e s a t i s f i e s the strong tendency f o r the s u l f o n i c a c i d s i t e s to be hydrated, and at the same time t h i s s t r u c t u r e w i l l minimize unfavorable i n t e r a c t i o n s between water and the fluorocarbon matrix. The e f f e c t of deformation and o r i e n t a t i o n on t h i s c l u s t e r i n g can be understood by r e c a l l i n g that the SAXS peak, a s s o c i a t e d w i t h c l u s t e r i n g , i s normal to the molecular c h a i n a x i s 03,5). In a WAXD study of t h i s f i b e r Starkweather has suggested the c r y s t a l l i n e p o r t i o n of the polymer e x i s t s as a f r i n g e d m i c e l l e w i t h a pseudo-hexagonal l a t t i c e (22). Even i n the n o n - c r y s t a l l i n e port i o n of the polymer one might expect short range order of t h i s nature. We b e l i e v e i o n c l u s t e r s form on e i t h e r s i d e of t h i s f r i n g e d m i c e l l a r s t r u c t u r e , and thus the average d i s t a n c e between c l u s t e r s i s h e a v i l y weighted towards a v a l u e equal to the sum of the t h i c k n e s s of the f l u o r o c a r b o n f r i n g e d m i c e l l e p l u s the c l u s t e r diameter. T h i s could p o t e n t i a l l y account f o r our a b i l i t y to d e t e c t the SAXS r e f l e c t i o n a t low water contents. N a t u r a l l y any deformation which o r i e n t s the polymer chains w i l l r e s u l t i n t h i s most probable c l u s t e r s e p a r a t i o n being observed normal to the d i r e c t i o n of deformation, which i s i n agreement w i t h o b s e r v a t i o n C3, 5). We b e l i e v e t h i s model i s i n s u b s t a n t i a l agreement w i t h experimental o b s e r v a t i o n ; however, f u r t h e r experimentation w i l l be r e q u i r e d before the complete d e t a i l s of i o n i c c l u s t e r i n g i n p e r f l u o r i n a t e d ionomers are d e f i n e d . I f the i o n i c c l u s t e r s are approximately s p h e r i c a l i n shape, then t h e i r average s i z e may be estimated from s o l v e n t a b s o r p t i o n s t u d i e s (6) u s i n g s t r a i g h t - f o r w a r d geometric arguments. Suppose, f o r the sake of c a l c u l a t i o n o n l y , that the c l u s t e r s were d i s t r i -


286

PERFLUORINATED IONOMER M E M B R A N E S

t H '

t

;ci OVERFLOW (15-25% NaCl)

2

2

NAFION PERFLUORINATED MEMBRANE

NaOH (10-40%)

+

Na H 0-

2.


2

Cathode

Anode OH

NaCL

Cl

-

H0

-

2

'A

Figure 1. Nafion perfluorinated membrane as a membrane separator in a typical chlor-alkali cell.

Figure 2. Transmission electron micrograph from 1200 EW cross section, stained by counter-diffusion of Ag+ and S ions across film. +2

n


13.

GIERKE AND

Ion Clustering

HSU

Model

287

buted on a simple cubic l a t t i c e w i t h an average l a t t i c e constant, d, given by the observed Bragg r e f l e c t i o n . We a l s o know that the change i n volume, AV, which occurs during the s w e l l i n g of polymer w i t h water i s described by the e m p i r i c a l r e l a t i o n (6)


AV = p Am/p p w

(1)

3 3 where AV i s the f r a c t i o n a l volume change i n (cm /cm of d r y p o l y mer), m i s the f r a c t i o n a l weight gain i n (g/g d r y polymer), p i s the d e n s i t y of d r y polymer and p i s the d e n s i t y of water. The values of Am are e i t h e r taken from reference 6 or are measured. For a simple cubic l a t t i c e the volume of a c l u s t e r , V , w i l l be given by v

c =

i1+ . - rAV l? d

3

+ N pVp

(2)

where N i s the number of i o n exchange s i t e s i n a c l u s t e r and V the volume of an exchange s i t e . N can be obtained d i r e c t l y fr8m the d e n s i t y and equivalent weight,^EW, of the polymer N

= [N p /EW(1 + AV]d A p A

P

3

(3)

i s Avogadro's number, and the term i n s i d e the b r a c k e t s , [-] represents the number of exchange s i t e s i n a cubic centimeter of swollen polymer. The c l u s t e r diameter, d , i s , of course, given b y

d

= [6V /TT] c

c

±

1/3 /

(4)

J

F i n a l l y , the number of water molecules i n our average c l u s t e r i s given by w t P Am/18(l AV)]d (5) N

=

N

+

A

We emphasize that only estimates of c l u s t e r dimension are o b t a i n ed by using Equations (1.-5) . Yet, r e c o g n i z i n g t h i s l i m i t a t i o n one may be able to o b t a i n a d d i t i o n a l i n s i d e f o r the s t r u c t u r e of i o n i c c l u s t e r s and how t h i s s t r u c t u r e v a r i e s w i t h equivalent weight, c a t i o n form, water content, and other f a c t o r s . Table 1 shows the r e s u l t s of such c l u s t e r morphology c a l c u l a t i o n s f o r polymers w i t h _ ^ f f e ^ e n t equivalent weights. The value f o r V was taken as 68x10 cm and corresponds to an e f f e c t i v e r a d i u s f o r the i o n exchange s i t e s of 0.25 nm. This i s the same order of magnitude assumed by Hora et a l . (23) . As the equival e n t weight i n c r e a s e s , the c l u s t e r diameter, i o n exchange s i t e s per c l u s t e r , and water per exchange s i t e decrease. Qualitatively, these trends may be understood by r e c o g n i z i n g that the c r y s t a l l i n i t y and polymer s t i f f n e s s i n c r e a s e w i t h i n c r e a s i n g EW. At higher EW s, i t w i l l thus r e q u i r e more energy to hydrate each exchange s i t e and t o have the exchange s i t e s aggregate. Table 2 shows the r e s u l t s f o r c l u s t e r morphology c a l c u l a t i o n s f o r 1200 EW polymer n e u t r a l i z e d w i t h v a r i o u s c a t i o n s . As T


In Perfluorinated Ionomer Membranes; Eisenberg, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1982. 78.2 49.8 37.5 17.2 13.3

37.5 23.8 17.8 8.1 6.3

2.093

2.103

2.113

2.135

2.144

971

1100

1200

1600

1790

Am

96

1944

4.22 43

266

4.07

4.55 52

370

4.78

5.09

5.12

d , nm

73

86 3

84

95

2102

1200

Cluster

H„0/Cluster I

Bragg Spacing

Fixed Charge/

% Volume G a i n 100 Av 87.3

100

Gain

42.0

g/cc

%

2.088

P,

Mass

944

EW

Density

Polymer

Table I. Results of Cluster Morphology Calculations According to Equations 1-5 for Polymers with Difference Equivalent Weights in Na Ion Form. Samples Conditioned by Boiling 1 h in 0.2% N a O H .


Cluster

d

, nm

2.74

3.03

3.88

4.31

4.97

5.09

c

Diameter

to oo oo


+

Cs

a

+

+

Rb

K

Taken

+

Na

from

Reference

2.304

k.221

2.141

2.113

2.078

+

L i

1.

5.9

8.1

8.7

21.0

29.7

33.6

2.075

+

H

%

Mass G a i n 100 Am

Polymer Density P. * / c c

Cation Form

%

13.6

17.9

18.7

44.3

61.7

69.7

470

560

520

1120

1430

1690

Volume G a i n H^O/Cluster 100 A v 2

120

103

89

80

72

76

Fixed Charge/ Cluster

2

4.90

4.78

4.61

4.78

4.82

4.98

f

Bragg Spacing d nm

Table II. Results of Cluster Morphology Calculations According to Equations 1-5 for 1200 E W P o l ymers in Different Cation forms. Samples Conditioned by Boiling 1 h in H 0 .


3.50

3.56

3.45

4.21

4.49

4.74

Cluster Diameter d c , nm

oo

3

§ X

w w >


290

PERFLUORINATED IONOMER

M E M B R A N E S

the c a t i o n weight i n c r e a s e s , the c l u s t e r diameter and water per exchange s i t e decrease, but the number of exchange s i t e s per c l u s t e r i n c r e a s e s . C l e a r l y , the h y d r o p h i l i c i t y of the exchange s i t e i s lower w i t h the h e a v i e r c a t i o n s which i s c o n s i s t e n t w i t h the o b s e r v a t i o n that the heavier c a t i o n s are more t i g h t l y bound to the exchange s i t e (24, 25). One e x p l a n a t i o n f o r the i n c r e a s e i n number of exchange s i t e s per c l u s t e r f o r the h e a v i e r c a t i o n s might be r e l a t e d to the balance of energy of e l a s t i c deformation on one hand and h y d r a t i o n and i o n aggregation on the other (26). As the h y d r a t i o n of the i n d i v i d u a l exchange s i t e s decreases w i t h heavier c a t i o n s , the e l a s t i c s t r a i n of the f l u o r o c a r b o n m a t r i x a s s o c i a t e d w i t h h y d r a t i o n w i l l a l s o decrease. This w i l l make i t p o s s i b l e f o r a d d i t i o n a l c l u s t e r i n g to occur, w i t h an a s s o c i a t e d i n c r e a s e i n e l a s t i c s t r a i n , u n t i l thermodynamic e q u i l i b r i u m i s achieved w i t h the e x t e r n a l s o l v e n t . Table 3 shows c l u s t e r morphology c a l c u l a t i o n s f o r 1200 EW polymer w i t h d i f f e r e n t i n t e r n a l water content. The r e s u l t s i n t h i s t a b l e provide some i n s i g h t i n t o the growth of c l u s t e r s . As the polymer absorbs more water, the c l u s t e r diameter, exchange s i t e s per c l u s t e r , and waters per exchange s i t e i n c r e a s e . F i g ure 3 shows more c l e a r l y the v a r i a t i o n of c l u s t e r diameter and exchange s i t e per c l u s t e r w i t h water content. As noted e a r l i e r , c l u s t e r s do e x i s t i n the dry polymer, and i n t h i s sample they are about ^1.9 nm i n diameter and c o n t a i n ^26 i o n exchange s i t e s . The i n c r e a s e i n the number of exchange s i t e s per c l u s t e r w i t h i n c r e a s i n g water content i s noteworthy because i t suggests that c l u s t e r growth does not merely occur by an expansion of the dehydrated c l u s t e r . Rather the growth of c l u s t e r appears to occur by combination of t h i s expansion and a continuous r e o r g a n i z a t i o n of exchange s i t e s so there are a c t u a l l y fewer c l u s t e r s i n the f u l l y hydrated sample. The type of r e o r g a n i z a t i o n v i s u a l i z e d i s shown s c h e m a t i c a l l y i n F i g u r e 4. This f i g u r e s i l l u s t r a t e s on dehydration how the exchange s i t e s from two c l u s t e r s (#6-10) could be r e d i s t r i b u t e d to form a t h i r d new c l u s t e r without a s i g n i f i c a n t t r a n s l a t i o n of polymer chains. Of course the i n c e n t i v e f o r o b t a i n i n g a b e t t e r understanding of i o n i c c l u s t e r i n g i n "Nafion" i s to determine the r e l a t i o n s h i p between i o n c l u s t e r i n g and mass t r a n s p o r t . With t h i s i n mind we have measured the h y d r a u l i c p e r m e a b i l i t y and d i f f u s i o n c o e f f i c i ent of water through membranes of d i f f e r e n t e q u i v a l e n t weights. These data are l i s t e d i n Table 4. These t r a n s p o r t measurements were used to estimate the average s i z e of the s t r u c t u r a l f e a t u r e c o n t r o l l i n g t r a n s p o r t , or the e f f e c t i v e pore diameter, D . The d i f f u s i o n data were analyzed according to equation (6) o? (7) (27). 2

( D / D ° ) = (A/Ao) = (u/4d ) (D - a ) WW p w T T

2

(6)

or



27.2

12.9

2.075

13.3 11.4

6.3 5.4

2.075

2.078

Li

+

Li

H

18.2

8.6

2.078

+

Li

+

25.3

12.0

2.078

+

H

32.3

15.3

2.078

Li

+

39.1

18.5

41.0

2.078

19.4

%

149

3.52

41

2.5?

2.52

3.46 39

233

2.97

53

288

3.38

4.13

3.90

57

456

3.70

4.38

3.27

59

505

3.94

4.08

Cluster Diameter dc, nm

4.55

4.66

Bragg Spacing d.nm

4.07

67

72

76

Fixed Charge/ Cluster

687

886

984

Volume Gain 100 Av H.O/Cluster

+

2.075

% Mass Gain 100 Am

Li

+

Polymer Density p. R / C C

+

H

Cation Form

2

Table III. Results of Cluster Morphology Calculations According to Equations 1-5 for 1200 E W Polymers with Different H 0 Content.



292


M E M B R A N E S

10 20 WATER CONTENT, g H 0 / 1 0 0 g DRY POLYMER 2

Figure 3. The variation of cluster diameter (O) and ion exchange sites (A) per cluster with water content in 1200 EW polymer.


Ion Clustering

Model


GIERKE A N D H S U

Figure 4. Representation of redistribution of ion exchange sites that occurs on dehydration of polymer.


In Perfluorinated Ionomer Membranes; Eisenberg, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1982. cm

-

2.4

x

10"

D°

b. ) 5

2

0.7

0.16

1600

cm /sec

(28)

0.9

0.34

1500

From Reference

1.0

0.42

1400

a. )

1.2

Dia.

0.69

nm

Pore

1200

sec

2

a

1.6

6

Coef.

1.28

D ,10" w

Diff.

1100

EW

b i10n "

z

1

3

3

Perm.

0.2

-0.7

1.1

1.2

Dia.

1.3

nm

Pore

2.1

dyne-sec

cm

Hyd.

Table I V . Effective Pore Diameters Derived from H O Transport Measurements.


13.

GIERKE AND

D

= o

295

Ion Clustering Model

HSU

2

+ [4d (D /D° ) / i r ]

1 / 2

(7)

where D i s the measured d i f f u s i o n c o e f f i c i e n t (28), D° i s the d i f f u s i o n c o e f f i c i e n t of water i n water, a i s the diameter o f a water molecule (taken as 0.35 nm) and d i s the d i s t a n c e between c l u s t e r s which i s obtained from the SAXS data 03, J5). The hydraul i c p e r m e a b i l i t y were analyzed according t o the expression (27).


D

2

p

= (128 t n L d / 7 r ) p

1 / 4

(8)

where t i s the t h i c k n e s s of the membrane, n i s the v i s c o s i t y of water, and L i s the h y d r a u l i c p e r m e a b i l i t y c o e f f i c i e n t . The results for e f f e c t i v e pore diameter a r e a l s o l i s t e d i n Table 4. Note that the two experiments r e s u l t i n values which a r e s e l f consistent. Combining the r e s u l t s of these water t r a n s p o r t experiments w i t h the i n v e r t e d m i c e l l a r s t r u c t u r e proposed f o r the c l u s t e r s , we a r r i v e a t the c l u s t e r - n e t w o r k model shown i n F i g u r e 5. I n t h i s model the c l u s t e r s a r e connected by short narrow channels whose dimensions a r e d e r i v e d from the water t r a n s p o r t measurements. The c l u s t e r s e p a r a t i o n (5.0 nm) i s c o n s i s t e n t w i t h the SAXS experiments, and the c l u s t e r diameters (4.0 nm) a r e c o n s i s t e n t w i t h the r e s u l t s given i n Tables 1-3. The s i g n i f i c a n c e of the c r o s s hatched area w i l l be explained s h o r t l y . As we w i l l demonstrate, t h i s model of i o n i c c l u s t e r i n g i s very u s e f u l i n d e s c r i b i n g i o n t r a n s p o r t i n "Nafion". Absolute Reaction Rate Formalism. Given the s t r u c t u r e shown i n F i g u r e 5, we can e x p l a i n the o b s e r v a t i o n described above of h i g h current e f f i c i e n c y i n membranes which are b a s i c throughout t h e i r t h i c k n e s s . Because of r e p u l s i v e e l e c t r o s t a t i c i n t e r a c t i o n s , hydroxide ions w i l l be excluded from the s u r f a c e of the c l u s t e r s and connecting channels by the polymeric f i x e d charges which a r e assumed, i n the model, t o be l o c a t e d i n these r e g i o n s . From the theory of the e l e c t r i c double l a y e r (29,30), we know that the e f f e c t i v e range of these i n t e r a c t i o n s w i l l be about 0.5 nm a t the concentrations that e x i s t i n s i d e the membrane. This r e g i o n i s r e presented by the cross-hatched area i n Figure 5. I n a l a r g e port i o n of the c l u s t e r , the hydroxide i o n w i l l be e f f e c t i v e l y s h i e l d ed from these i n t e r a c t i o n s by sodium i o n s , and by Boltzmann s t a t i s t i c s the hydroxide c o n c e n t r a t i o n i n the i n t e r i o r of the c l u s t e r w i l l be s i m i l a r t o the e x t e r n a l c o n c e n t r a t i o n . This would exp l a i n why the membrane i s b a s i c throughout i t s t h i c k n e s s when i n a c h l o r - a l k a l i c e l l . However, f o r a h y d r o x y l i o n to migrate from one c l u s t e r t o the next, i t would have t o overcome a f a i r l y l a r g e e l e c t r o s t a t i c b a r r i e r i n the channel, which a p o s i t i v e i o n l i k e Na w i l l not experience. I t i s t h i s b a r r i e r which would account f o r the high c u r r e n t e f f i c i e n c y . These q u a l i t a t i v e concepts may be cast i n t o a q u a n t i t a t i v e formalism u s i n g absolute r e a c t i o n r a t e theory (31). As the hydro-



296


M E M B R A N E S

5.0 nm

Figure 5. Cluster-network model for Nafion perfluorinated membranes. The polymeric ions and absorbed electrolyte phase separate from the fluorocarbon backbone into approximately spherical clusters connected by short, narrow channels. The polymeric charges are most likely embedded in the solution near the interface between the electrolyte and fluorocarbon backbone. This configuration minimizes both the hydrophobic interaction of water with the backbone and the electrostatic repulsion of proximate sulfonate groups. The dimensions shown were deduced from experiments. The shaded areas around the interface and inside a channel are the double layer regions from which the hydroxyl ions are excluded electrostatically.


13.

GIERKE AND

Ion Clustering

HSU

297

Model

x y l i o n migrates through the membrane, i n a c h l o r - a l k a l i c e l l i t w i l l encounter an o s c i l l a t i n g p o t e n t i a l which i s low i n the c l u s t e r and high i n the channel. This s i t u a t i o n i s shown s c h m a t i c a l l y i n F i g u r e 6. The o v e r a l l p o t e n t i a l g r a d i e n t , A, i s provided by the v o l t a g e drop across the membrane, and the b a r r i e r h e i g h t , a, contains both a term due t o the geometric r e s t r i c t i o n , $* and a term due t o the e l e c t r o s t a t i c r e p u l s i o n , >. For i o n i c speci e s M w i t h charge q (M), we may w r i t e


a(M) = 3(M) - q(M)

(9)

«f»

A, a, B, and are a l l expressed i n reduced u n i t s ( u n i t s of k T). I n F i g u r e 6, d corresponds t o the e f f e c t i v e Bragg spacing deduced from the SAXS experiments. Using absolute r e a c t i o n r a t e theory (31) an expression f o r the r a t i o of the hydroxide i o n f l u x t o sodium i o n f l u x may be derived: fi

-[J(OH)/JCHa)] -gGg-

§g§

exp [-2.0 ] (10)

In equation (10), J(M) i s the f l u x of species M, C(0H,n) i s the c o n c e n t r a t i o n of hydroxide i o n i n the cathode compartment, C(Na,0) i s the c o n c e n t r a t i o n of sodium ions i n the anode compartment, and y(M) i s d e f i n e d y(M) = k (M)

exp [-B(M)]

Q

(11)

where k (M) i s the i n t r i n s i c r a t e of t r a n s p o r t of species M. The experimental quanity i s the current e f f i c i e n c y , CE, which i s d e f i n e d by, CE = 1.0/(1.0-J(OH)/J(Na))

(12)

In equation (10) there are two unknown q u a n t i t i e s : the r a t i o of m o b i l i t i e s ,^C(OH)/jtt(Na) , and the e l e c t r o s t a t i c c o n t r i b u t i o n t o the b a r r i e r , >. A v a l u e f o r can be estimated by assuming that the channel i s a c y l i n d r i c a l c a p i l l a r y of diameter, D , w i t h a uniform charge d e n s i t y a t the s u r f a c e , a^, g i v i n g . P

a

c

= [dp /(2EW(l+AV))] [•£ ( ^ p

2

) ]

1 7 3

(13)

The r a d i a l p o t e n t i a l d i s t r i b u t i o n i n s i d e the c a p i l l a r y , (f)(r), i s then obtained by s o l v i n g the Poisson-Boltzmann equation f o r c y l i n d r i c a l symmetry (30). The r e s u l t i n g p o t e n t i a l depends on a s i n g l e a d j u s t a b l e constant which i s f i x e d by the boundary c o n d i t i o n on the p o t e n t i a l which r e l a t e s the p o t e n t i a l gradient a t r=l/2D t o the s u r f a c e charge density,(p . Then we d e f i n e P 1/2D / n 1/2D «J» = J

Perfluorinated Ionomer Membranes - American Chemical Society

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