Anionic Polymerization - American Chemical Society

11 Dynamics of Ionic Processes in Low Polar Media A N D R E PERSOONS

and J A C Q U E S

EVERAERT

Downloaded by PENNSYLVANIA STATE UNIV on March 16, 2013 | http://pubs.acs.org Publication Date: November 30, 1981 | doi: 10.1021/bk-1981-0166.ch011

Department of Chemistry, University of Leuven, Leuven, Belgium

A thorough discussion i s given of the f i e l d modulation technique, a new stationary relaxation method based on electric f i e l d perturbation of ionic equilibria. Concomitantly the theory of electric f i e l d effect in ionic systems i s reviewed especially stressing their importance for conductance phenomena in low polar solutions. Some examples of conductance relaxation measurements i n a solution of tetraalkylammonium-salts are given. The results confirm convincingly the applicability of the sphere-in-continuum model as a basic model for ionic interactions: a complete treatment of ionic processes can be given from the diffusion of ions i n a continuous medium. Conductance relaxation i s also shown to be c r i t i c a l l y dependent upon aggregation equilibria affecting non-conducting (ion-pairs) as well as ionic species. The relaxation behavior in the presence of quadrupoles (ion-pair dimers) and t r i p l e ions i s thoroughly analyzed. The experimental results show the potential of the f i e l d modulation techniques as a method for the investigation of ionization processes, independent of conductance measurements.

The thorough understanding of elementary i o n i c i n t e r a c t i o n s i s a necessary p r e r e q u i s i t e f o r a s u c c e s s f u l mechanistic d e s c r i p t i o n of any chemical process i n v o l v l i n g i o n i c s p e c i e s . Conductance techniques are pre-eminently used to i n v e s t i g a t e i o n i z a t i o n phenomena. The data obtained y i e l d , w i t h i n the framework o f i o n i c s o l u t i o n theory, d e t a i l e d i n f o r m a t i o n on the d i s t r i b u t i o n between conducting and non-conducting e n t i t i e s the exact nature of which cannot always u n e q u i v o c a l l y assessed. Although t h i s d i s t r i b u t i o n between " f r e e " charges and i o n i p h o r e s "

0097-6156/81/0166-0153$05.75/0 ©

1981 American Chemical Society

In Anionic Polymerization; McGrath, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.

154

ANIONIC POLYMERIZATION


i s a primary c o n s t r a i n t the c e n t r a l problem i n any a n a l y s i s of i o n i z a t i o n mechanisms i s the k i n e t i c study of the i n t e r c o n v e r s i o n processes between the d i f f e r e n t species; f o r such a k i n e t i c I n v e s t i g a t i o n to be complete a l l the elementary processes should be analyzed f o r t h e i r e n e r g e t i c and dynamic p r o p e r t i e s . Since the elementary steps i n i o n i c a s s o c i a t i o n - d i s s o c i a t i o n processes a r e u s u a l l y v e r y f a s t - to the l i m i t of d i f f u s i o n - c o n t r o l l e d r e a c t i o n s - t h e i r k i n e t i c i n v e s t i g a t i o n became only f e a s i b l e with the advent of f a s t r e a c t i o n techniques, mainly chemical r e l a x a t i o n spectrometrie techniques. In chemical r e l a x a t i o n spectrometry the i n f o r m a t i o n on the k i n e t i c parameters of f a s t processes i s obtained from an a n a l y s i s of the response of a chemical system upon a f a s t p e r t u r b a t i o n i n an i n t e n s i v e thermodynamic v a r i a b l e d e f i n i n g the e q u i l i b r i u m of the system. To a t t a i n an acceptable accuracy the response of the system upon the p e r t u r b a t i o n should be of s u f f i c i e n t amplitude, a requirement which i s g e n e r a l l y met only f o r chemical systems where products and r e a c t a n t s are evenly d i s t r i b u t e d over the e q u i l i b r i u m s t a t e . T h i s requirement e x p l a i n s the r e l a t i v e abundance of chemical r e l a x a t i o n s t u d i e s on i o n i c processes i n aqueous, or aqueous-like media as compared to such s t u d i e s i n low p o l a r s o l v e n t s . In the former media conducting s p e c i e s are u s u a l l y present i n a p p r e c i a b l e amounts while i n the l a t t e r i o n i z a t i o n e q u i l i b r i a are almost completely s h i f t e d towards non-conducting s p e c i e s , which i s due to the long range of coulombic i n t e r a c t i o n s a t low p o l a r i t y . However a multitude of o r g a n i c r e a c t i o n s are c a r r i e d out i n s o l v e n t s of low p o l a r i t y and i t i s t h e r e f o r e of the utmost importance to have a d e t a i l e d i n s i g h t i n i o n i c processes i n these media i n order to assess u n e q u i v o c a l l y the i n t e r v e n t i o n and r o l e of i o n i c species i n o r g a n i c r e a c t i o n mechanisms. To i n v e s t i g a t e the dynamic p r o p e r t i e s of i o n i c species present a t very low c o n c e n t r a t i o n i n a p o l a r s o l v e n t s the f i e l d modulation method i s p a r t i c u l a r l y s u i t e d (l-JD · In t h i s r e l a x a t i o n technique, developed some years ago In our l a b o r a t o r y , an i o n i c e q u i l i b r i u m i s perturbed w i t h an e l e c t r i c f i e l d and the response of the system measured from the conductance p r o p e r t i e s of the sample s o l u t i o n . Since both p e r t u r b a t i o n and d e t e c t i o n are c a r r i e d out i n the frequency domain t h i s s t a t i o n a r y r e l a x a t i o n technique has a very good s e n s i t i v i t y and accuracy which i s due to the averaging inherent i n r e p e t i t i v e measurements. In t h i s paper we review the f i e l d modulation method e s p e c i a l l y to i n d i c a t e the a p p l i c a b i l i t y f o r the i n v e s t i g a t i o n of o r g a n i c r e a c t i o n mechanisms where i o n i c s p e c i e s may i n t e r v e n e . The p o t e n t i a l of t h i s r e l a t i v e l y simple technique w i l l be shown from r e s u l t s obtained the the i n v e s t i g a t i o n of i o n i z a t i o n processes i n low p o l a r media. We w i l l g i v e f i r s t an i n t r o d u c t o r y summary of some aspects of e l e c t r i c f i e l d e f f e c t s i n i o n i c e q u i l i b r i a s i n c e a knowledge of t h i s t o p i c i s necessary


11.

PERSOONS AND EVERAERT

Ionic Processes in Low

Polar Media

155

p r e r e q u i s i t e f o r a proper understanding of the f i e l d modulation


E l e c t r i c F i e l d E f f e c t s i n Ionic E q u i l i b r i a The b a s i c assumption i n conductance measurements i s the independence of the sample r e s i s t a n c e on e l e c t r i c f i e l d s t r e n g t h . However a d e v i a t i o n from the l i n e a r r e l a t i o n between c u r r e n t d e n s i t y and f i e l d s t r e n g t h w i l l be observed i f any f i e l d e f f e c t on the m o b i l i t y and/or the number of f r e e c h a r g e - c a r r i e r s i s present. At high f i e l d strengths a conductance increase i s observed both i n s o l u t i o n of strong and weak e l e c t r o l y t e s . The phenomena were discovered by M. Wien (6-8) and are known as the f i r s t and the second Wien e f f e c t , r e s p e c t i v e l y . The f i r s t Wien e f f e c t Is completely explained as an increase i n i o n i c m o b i l i t y which i s a consequency of the i n a b i l i t y of the f a s t moving ions to b u i l d up an i o n i c atmosphere ( 8 ) . T h i s m o b i l i t y increase may a l s o be observed i n s o l u t i o n of weak e l e c t r o l y t e s but s i n c e the second Wien e f f e c t i s a much more pronounced e f f e c t we must invoke another explanation, i . e . an increase i n f r e e c h a r g e - c a r r i e r s . The second Wien e f f e c t i s t h e r e f o r e a s h i f t i n i o n i c e q u i l i b r i u m towards f r e e ions upon the a p p l i c a t i o n of an e l e c t r i c f i e l d and i s t h e r e f o r e a l s o known as the F i e l d D i s s o c i a t i o n E f f e c t (FDE). Only the smallness of the f i e l d d i s s o c i a t i o n e f f e c t safeguards the use of conductance techniques f o r the study of i o n i z a t i o n equilibria. The energy d i s s i p a t i o n of a system c o n t a i n i n g f r e e charges subjected to e l e c t r i c f i e l d s i s w e l l known but t h i s i n d i c a t e s a non-equilibrium s i t u a t i o n and as a r e s u l t a thermodyanmic d e s c r i p t i o n of the FDE i s impossible. W i t h i n the framework of i n t e r i o n i c a t t r a c t i o n theory Onsager was able to d e r i v e the e f f e c t of an e l e c t r i c f i e l d on the i o n i c d i s s o c i a t i o n from the t r a n s p o r t p r o p e r t i e s of the ions i n the combined coulomb and e x t e r n a l f i e l d s (9.). I t i s not improper to mention here the n o t o r i o u s mathematical d i f f i c u l t y of Onsager's paper on the second Wien e f f e c t . A conceptual d i f f i c u l t y i n the t h e o r e t i c a l d e s c r i p t i o n of i o n i z a t i o n e q u i l i b r i a i s the d i s t i n c t i o n between " f r e e " and "bound" ions (molecules or ionophores). Somewhat a r b i t r a r i l y Onsager adopted Bjerrum's convention w r i t i n g the d i s t r i b u t i o n between f r e e and bound ions as an a s s o c i a t i o n - d i s s o c i a t i o n equilibrium: k

A+b"

d

> A+ + B" K = k /k [1] . r Where ANB i s the s t a t e of two bound i o n s , i . e . two o p p o s i t e l y charged ions at d i s t a n c e s from each other where the e l e c t r o s t a t i c i n t e r a c t i o n of the two ions exceeds t h e i r thermal i n t e r a c t i o n w i t h the surrounding medium. T h i s p i c t u r e introduces the Bjerrum d

d

r

k



156


i o n - p a i r concept together with the Bjerrum d i s t a n c e , i . e . the c r i t i c a l d i s t a n c e a t which the mutual e l e c t r o s t a t i c i n t e r a c t i o n equals the thermal energy. The p r e v a i l i n g c o n f u s i o n about these concepts i s mainly due to the s t r i c t d i s t i n c t i o n between f r e e and n o t - f r e e ions w h i l e conductance d i s c r i m i n a t e s between species c o n t r i b u t i n g and n o t - c o n t r i b u t i n g to conductance. An important q u e s t i o n i s t h e r e f o r e i f two o p p o s i t e l y charged ions at the Bjerrum-distance, i . e . "bound" to each other, can s t i l l c o n t r i b u t e to the conductance. Since i n low p o l a r media the Bjerrum-distance i s r a t h e r l a r g e (125 A) i n benzene a t room temperature) i t i s very p l a u s i b l e that such c o n f i g u r a t i o n s c o n t r i b u t e to the conductance e.g. by an apparent i n c r e a s e i n m o b i l i t y of a " f r e e " i o n passing by. Nevertheless i t should be noted that, e s p e c i a l l y i n low p o l a r s o l v e n t s , the overwhelming c o n t r i b u t i o n ot he i o n - p a i r s t a t e i s from c o n f i g u r a t i o n s with the two ions very c l o s e to c o n t a c t . A p p l y i n g the laws of Brownian motion to the d i s t r i b u t i o n of f r e e ions and i o n - p a i r s i n the presence of a e x t e r n a l e l e c t r i c f i e l d Onsager c a l c u l a t e d the r a t e constants of recombination and d i s s o c i a t i o n f o r e q u i l i b r i u m [1] as: k

r

- 8ïïqkT (ω+ + w )

[2]

and k

d

» 8nqkT (ω+ + ω )

K (E=0) F(2pq)

[2b]

d

Here q i s the Bjerrum-distance (= -e+e-/2DkT), k the Boltzmann constance, ω the mechanical m o b i l i t y of an i o n K ( E O ) the e q u i l i b r i u m constant i n the absence of e l e c t r i c f i e l d and D and d i e l e c t r i c constant of the medium. From these expressions we see that the s h i f t i n d i s s o c i a t i o n constant upon the a p p l i c a t i o n of an e l e c t r i c f i e l d i s g i v e n by: e

d

K ( E ) / K ( E - 0 ) = F(23q) d

[3]

d

F(2$q) i s somewhat complex f u n c t i o n which can however be w r i t t e n i n a c l a r i f y i n g s e r i e s expansion: E(23q) » 1 + 23q + l / 3 ( 2 3 q )

2

+

l/18(23q)

The argument 2£q Is the r a t i o of two d i s t a n c e s , the d i s t a n c e q and the d i s t a n c e (2β)""* g i v e n by:

le+Wf - e-ω-Ι 2B -

|E| kT(ui+ + ω )

1:1 =

e

3

+ ...

[4]

Bjerrum-

0

|E|

[5]

kT

The dependence on the a b s o l u t e value of the f i e l d s t r e n g t h i s a noteworthy f e a t u r e of Onsager s theory; of course symmetry c o n s i d e r a t i o n s Impose always a dependence of the FDE on an even 1


11.


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Media

157

f u n c t i o n of f i e l d s t r e n g t h . From the s p e c i f i c a t i o n f o r a 1:1 e l e c t r o l y t e i t i s seen that ( 2 β ) " i s the d i s t a n c e a t which two o p p o s i t e l y charged ions form a d i p o l e , "a Bjerrum d i p o l e " w i t h an e l e c t r o s t a t i c i n t e r a c t i o n energy kT when o r i e n t e d p a r a l l e l to the field. For a u n i v a l e n t e l e c t r o l y t e and r e t a i n i n g only the l i n e a r terra i n [ 4 ] the dependence of the d i s s o c i a t i o n constant upon e l e c t r i c f i e l d s t r e n g t h can formally be w r i t t e n as a Van 1 H o f f type equation: 3 dlnK e = [6] d|E| 2Dk T 1

1


d

0

2

2

Eq. [6] c l e a r l y shows that the FDE i s most pronounced i n low p o l a r media a t low temperatures. A numerical c a l c u l a t i o n shows t h a t the i n c r e a s e i n d i s s o c i a t i o n constant i s a few percent w i t h r e l a t i v e l y low f i e l d strengths i n those media, e.g. 5% i n benzene a t room temperature, about 2% i n t e t r a h y d r o f u r a n -100C with a f i e l d of 1 k V . c n r . As the e q u i l i b r i u m constant i s mostly d e r i v e d from conductance data the s h i f t i n i o n i z a t i o n with f i e l d can a l s o be determined from the measurement of the conductance i n c r e a s e upon a p p l i c a t i o n of a high e l e c t r i c f i e l d . Conductance i n c r e a s e and f i e l d d i s s o c i a t i o n a r e g e n e r a l l y r e l a t e d as: 1

1 - α

AK

AG

d

[7]

e

2 - α

K

d

G

Eq. [7] shows that the r e l a t i v e change i n conductance, AG/G, i s maximal a t low degree o f i o n i c d i s s o c i a t i o n a, a c o n d i t i o n

The

F i e l d Modulation Technique

From the foregoing d i s c u s s i o n o f e l e c t r i c f i e l d e f f e c t s i n i o n i c e q u l i b r i a i t i s c l e a r that a s o l u t i o n o f a weak e l e c t r o l y t e shows a n o n - l i n e a r behavior i n conductance ( o r r e s i s t a n c e ) a t h i g h f i e l d s t r e n g t h s . With an i n t e r d i s c i p l i n a r y look a t the f i e l d of e l e c t r o n i c s we note that such n o n l i n e a r i t i e s a r e a t the h e a r t of a l l modern e l e c t r o n i c c i r c u i t s and d e v i c e s . We t h e r e f o r e can use a s o l u t i o n of a weak e l e c t r o l y t e s u b j e c t s t o high e l e c t r i c f i e l d s as an e l e c t r o n i c device, which i s the b a s i c i d e a of the F i e l d Modulation Tecnnique, the g e n e r a l p r i n c i p l e s we w i l l d i s c u s s now. I f a sample s o l u t i o n of a weak e l e c t r o l y t e i s subjected to an a l t e r n a t i n g e l e c t r i c f i l e d of high amplitude the sample-conductance G becomes a p e r i o d i c f u n c t i o n o f time which f o r m a l l y can be w r i t t e n as:


ANIONIC

158

POLYMERIZATION

G « G + AG s i n u> t Due to the dependence on the absolute value of the f i e l d s t r e n g t h (Eq. 6) a s o l u t i o n o f a weak e l e c t r o l y t e has a l s o r e c t i f y i n g p r o p e r t i e s f o r high a l t e r n a t i n g f i e l d s and ω i s t h e r e f o r e not the frequency o f the high a l t e r n a t i n g f i e l d : cos 2n tot S i n c e I s i n u>t| - 2/ΐΐ + 4/ϊΐ·Σ 1 - 4 η p

ρ

2

ω w i l l be twice t h i s frequency and the higher harmonics w i l l a l s o be present* I f t h i s conductance i s measured w i t h an a l t e r n a t i n g , low amplitude s i g n a l V , o f frequency % (from the same or another p a i r of e l e c t r o d e s i n the sample c e l l ) we see i n the c u r r e n t frequency components which a r e not present i n the s i g n a l sources:


ρ

m

I " G.V

m

» (G + AG s i n ω Ο V s i n u) t ρ

M

M

[8a]

or I - GV sin M

1 u>t + -V AG c o s i i D p - a t M

-

1 - V AG cos(u> + in)t M

p

[8b]

These new frequency components a r i s e from a mixing between the h i g h - f i e l d p e r t u r b a t i o n ( ω ) and the measuring s i g n a l ( ω ) . We hence conclude that an e l e c t r o l y t i c r e s i s t a n c e can be used as a s i g n a l mixer i f one of the s i g n a l s has high amplitude. From eq. 8b i t i s a l s o seen that the amplitude of the mixed s i g n a l s i s a unique f u n c t i o n of the conductance i n c r e a s e due to the high e l e c t r i c f i e l d and t h e r f o r e a d i r e c t measure of the amplitude of the FDE. T h i s i s the b a s i c p r i n c i p l e of the f i e l d modulation technique. So f a r we have considered the conductance i n c r e a s e as i n s t a n t a n e o u s l y upon the a p p l i c a t i o n of the high f i e l d . Since the conductance i n c r e a s e i s due to an i n c r e a s e i n number of i o n i c s p e c i e s generated i n i o n i z a t i o n processes i t i s evident that the conductance i n c r e a s e should have a f i n i t e r a t e determined by the r a t e of i o n i c e q u i l i b r a t i o n . T h i s introduces a phase l a g between the high a l t e r n a t i n g f i e l d and the sample conductance i n c r e a s e a t high frequencies and a l s o a smaller amplitude of the conductance i n c r e a s e . The conductance i n c r e a s e shows t h e r e f o r e a d i s p e r s i o n w i t h frequency c h a r a c t e r i z e d by a time constant. In the subsequent d i s c u s s i o n we w i l l see how t h i s time constant i s d i r e c t l y r e l a t e d to the r e l a x a t i o n time(s) o f i o n i z a t i o n e q u a l i b r i a present i n s o l u t i o n . ρ

Before proceeding to a complete c a l c u l a t i o n r e l a t i n g the conductance i n c r e a s e to the amplitude of the f i e l d d i s s o c i a t i o n e f f e c t and i t s temporal behavior we w i l l give a short d e s c r i p t i o n of the a c t u a l c i r c u i t used f o r the f i e l d modulation technique ( F i g u r e 1) w i t h r e f e r e n c e to the previous d i s c u s s i o n .



11.


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159

A sample c e l l c o n t a i n i n g three e l e c t r o d e s , p r e f e r a b l y i n a symmetric arrangement i s used as a mixer f o r two s i g n a l s d e r i v e d from two s i g n a l generators. One s i g n a l i s a square wave of r e l a t i v e l y high amplitude which i s f u r t h e r i n c r e a s e d by the use of a step-up transformer. The other s i g n a l i s d i r e c t l y a p p l i e d through an i s o l a t i o n transformer to the e l e c t r o d e s of the sample cell. The a p p l i c a t i o n of the square wave f i e l d r e s u l t s i n a modulation, hence the name f i e l d modulation, o f the sample conductance a t the same frequency as the square wave s i g n a l . S i n c e the measuring s i g n a l i s locked i n phase and frequency on the square wave s i g n a l the f r e q u e n c i e s ω and (eq. 8b) a r e equal r e s u l t i n g i n a d.c. component i n tne mixed s i g n a l c u r r e n t . T h i s d.c. c u r r e n t i s used to charge a c a p a c i t o r from which the amplitude can be measured as a d.c. v o l t a g e over the c a p a c i t o r . T h i s way of measuring the d.c. c u r r e n t has the advantage t h a t , a f t e r completely charging the c a p a c i t o r , no d.c. c u r r e n t flows i n the sample a v o i d i n g thus p o l a r i z a t i o n e f f e c t s . Moreover, a l l s i g n a l s i n the c i r c u i t being a.c. s i g n a l s , a d.c. s i g n a l can be measured w i t h an extreme accuracy. ρ

For obvious experimental reasons the square wave s i g n a l i s not a pure square wave but i s an envelope of a high frequency s i g n a l which i s square wave modulated. R e t a i n i n g only t h i s square-wave envelope of the h i g h - v o l t a g e , high frequency p e r t u r b i n g f i e l d , a complete mathematical a n a l y s i s of the e l c t r o n i c c i r c u i t g i v e s f o r the d.c. v o l t a g e measured over the c a p i c i t o r cj|(2): V v

M

AG

d. . c

Π

G

1

V

; — ; 1 + ω τ PRF 2

2

d # c

.

m a x

; — ;

1 + ω τ PRF ζ

f9]

1

Where ω ρ p i s the frequency of the square-wave-envelope o r , e q u i v a l e n t l y , the frequency o f the measuring s i g n a l ( ω ) V max i s the d.c. s i g n a l measured a t low f r e q u e n c i e s where the time-dependent behavior of the conductance Increase i s completely i n phase w i t h the high f i e l d s i g n a l , τ i s the time constant f o r the conductance i n c r e a s e upon the a p p l i c a t i o n o f the high f i e l d and I t i n t i m a t e l y r e l a t e d to the chemical r e l a x a t i o n of the i o n i z a t i o n processes present i n the sample s o l u t i o n . Eq. 9 i s d e r i v e d assuming a small p e r t u r b a t i o n , a necessary assumption to s o l v e the n o n l i n e a r d i f f e r e n t i a l equations d e s c r i b i n g the c i r c u i t , and i s recognized as a c l a s s i c a l d i s p e r s i o n equation. From the g e n e r a l d i s c u s s i o n of the use of n o n l i n e a r p r o p e r t i e s to i n v e s t i g a t e dynamic phenomena i t i s c l e a r that the f i e l d modulation techniques I s but one example of a broader c l a s s of methods f o r the study of f a s t processes. A drawback, p a r t i c u l a r t o e l e c t r i c f i e l d modulation, i s the p r o h i b i t i v e heat d i s s i p a t i o n i n conducting systems. However, any f o r c i n g parameter imposing a conductance modulation c o u l d be used i n p r i n c i p l e as, f o r example, In the study of the dynamics of photoconduc t i v e phenomena· d # C e


160



R e s u l t s and D i s c u s s i o n The f i e l d modulation technique has been s u c c e s s f u l l y a p p l i e d to a broad v a r i e t y of systems showing a f i e l d dependent conductance. S i n c e a l a r g e amount of conductance data i s a v a i l a b l e f o r low p o l a r s o l u t i o n s of tetraalkylammonium s a l t s these systems were s t u d i e d most thoroughly. In the subsequent d i s c u s s i o n we w i l l r e l y mainly on the r e s u l t s obtained f o r these systems but the g e n e r a l c o n c l u s i o n s apply d i r e c t l y to other systems, e.g. f l u o r e n y l - s a l t s i n e t h e r e a l s o l v e n t s , metal complexes i n benzene, e t c . Conductance data c l e a r l y show that low p o l a r i t y s o l u t i o n s of tetraalkylammonium s a l t s behave r a t h e r simply a t low s a l t c o n c e n t r a t i o n and the main p r o p e r t i e s can be completely d e s c r i b e d by the simple e q u i l i b r i u m [1] between an i o n - p a r i and the f r e e i o n s . F o r such an e q u i l i b r i u m the chemical r e l a x a t i o n time can be w r i t t e d as: 1

τ""

-

kd+k (C +C ) r

A

B

88

(electroneutrality) - k +2k C d

r

A

[10a]

chem and f o r low degrees of d i s s o c i a t i o n , the u s u a l s i t u a t i o n i n low p o l a r i t y s o l u t i o n s , t h i s may be approximated by ( w i t h C the t o t a l s a l t concentration): Q

τ-l

- 2(k k )l/2 l / 2 ο d

r

C

IyM* d

(Μ*

k r

[10b]

ο

The r e l a x a t i o n time measured from the conductance modulation Is of course the same as t h i s chemical r e l a x a t i o n time s i n c e the i o n - p a i r d i s s o c i a t i o n e q u i l i b r i u m i s the only charge generating process present i n s o l u t i o n . These c o n c l u s i o n s are completely borne out by the experimental r e s u l t s : whenever the conductance data I n d i c a t e only the presence of the simple e q u i l i b r i u m [1] the r e c i p r o c a l r e l a x a t i o n time measured f o r the conductance modulation i s a l i n e a r f u n c t i o n of the square root of the t o t a l c o n c e n t r a t i o n as can be seen from F i g u r e 2. F o r these cases i t becomes p o s s i b l e , from a knowledge of the d i s s o c i a t i o n constant K and r e l a x a t i o n time, to determine through eq. 10b the r a t e constants f o r i o n - p a i r d i s s o c i a t i o n and f o r i o n i c recombination. These data, some of which are presented i n T a b l e I , a r e extremely v a l u a b l e to I n v e s t i g a t e the a p p l i c a b i l i t y and v a l i d i t y of the sphere-in-continuum model, which i s b a s i c to a l l e l e c t r o l y t e s o l u t i o n t h e o r i e s . In t h i s conceptual model Ions are viewed as (hard) spheres subjected to Brownian motion i n a continuous medium. An i o n - p a i r i s d e f i n e d as two o p p o s i t e l y charged ions w i t h i n a g i v e n d i s t a n c e a which i s the most c r i t i c a l parameter of d


11.


Ionic Processes in Low Polar

161

Media

v


d.c

Figure 1.

Schematic circuit for the field modulation technique.

Generator 1 is a square wave (0.01-30 kHz) modulated high frequency (0.1-1 MHz) high voltage (0.1-ak V ) signal generator driving a step-up (1:10) ferrite core trans former. Generator 2 is a low-voltage (1-20 V ) sinusoidal oscillator synchronously locked on the square wave signal modulating the power oscillator 1. The sample cell contains, in a very symmetric arrangement, three electrodes, two of which (E and E ) are connected to the high-voltage transformer output terminals. Electrode E is, over capacitor C (10 pF-22 nF), grounded for a.c. signals. The spacing between the Η ν electrodes and the center electrode E is of the order of 0.2-1 mm, depending on the particular cell design. The center tap of the high-voltage transformer is connected over a 1:1 isolation transformer to oscillator 2. pp

pp

t

3

u

3

Figure 2. Reciprocal conductance relax ation times as a function of the square root of TBAP concentration in media and concentration range where conductance indicates a preponderance of the simple ions. TEA Ρ in diphenyl ether (D = 3,55) at 318 Κ (Ο). TBAP in benzene-chlorobenzene (60:40 v/v; D = 3, 55) at 298 Κ (A). TBAP in benzene-chlorobenzene (60:40 v/v; D = 3, 64) at 298 Κ and 350 at m (2). TBAP in benzene-chlorobenzene (60:40 v/v; D = 3, 36) at 323 Κ (0).


2

162


TABLE I Rate Parameters f o r the D i s s o c i a t i o n of Tetrabutylammonium S a l t s (TBA-salts) i n Low P o l a r Media

Experimental* 1 2

K /1(T M

a/A**


d

k r / K W

1

Calculated

k s "

1. TBA-picrate i n benzene-chlorobenzene

1

V I O

1

W

1

k /s" d

mixtures a t 298 Κ (15,16)

3.22

0.363

5.92

3.01

0.109

3.01

0.098

3.55

3.83

5.84

2.61

1.00

2.70

0.97

5.78

2.64

6.45

2.44

6.56

3.87

24.4

2. TBA-bromide i n benzene-nitrobenzene

3.54 3.85

2.44 10.3

mixtures a t 298 Κ (17)

5.76

3.23

0.79

2.95

0.70

5.58

4.92

5.07

2.71

2.8

0.75

1.37

3. TBA-picrate i n diphenylether a t 318 Κ (3)

3.55

18.1

5.82

0.82

1.48

* f o r a l l experimental data the experimental e r r o r i s u s u a l l y l e s s than 4% and nowhere exceeds 7% c a l c u l a t e d from c o r r e c t e d values f o r K obtained from l n K v s . 1/D l i n e a r p l o t s i n case 1, otherwise d i r e c t l y from K . d

d

d


11.



Media

163

the model. T h i s d i s t a n c e should be considered as the boundary were f r e e ions cease to e x i s t but i s o f t e n taken as the d i s t a n c e of c l o s e s t approach, i . e . the sum of the i o n i c r a d i i . From t h i s model the recombination and d i s s o c i a t i o n rate constants may be c a l c u l a t e d as (10.11):

4ΠΝ e k

r

(

β

2

D a +

D

g ) / [ i - exp(-e /DkTa)]

3


[11a]

0

10 DkT and 3 e

k

d

2 o 2

=

( D + D ) / [exp(e /DkTa) - 1] ** A

DkTa

3

B

[12a]

Where 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 species 1 i n the continuous medium w i t h d i e l e c t r i c constant D and v i s c o s i t y η· With reasonably good assumption of s p h e r i c a l ions of equal r a d i i , and using the S t o k e s - E i n s t e i n r e l a t i o n f o r the d i f f u s i o n c o e f f i c i e n t s these r e l a t i o n s reduce to ( s i n c e i n low p o l a r media e DkTa < 1 ) : ο 2

8 χ 10" K

r

=

2

3

N E 0

1

[lib]

0

3

Dna

and 2 2e kd

0

s

II

2 e

1 — Dna

0

exp 4

-

[12b] DkTa

These equations allow the c a l c u l a t i o n of the r a t e constants, w i t h i n the assumptions made, from a knowledge of the d i s t a n c e a. Some of these r a t e constants c a l c u l a t e d w i t h an aj-value d e r i v e d from conductance data (12) a r e g i v e n i n T a b l e I ; the agreement between experimental and c a l c u l a t e d values i s a good i n d i c a t i o n t h a t the a s s o c i a t i o n processes of ions i n low p o l a r media can be d e s c r i b e d as simple d i f f u s i o n - c o n t r o l l e d processes. The d i s s o c i a t i o n process can a l s o be d e s c r i b e d as the d i f f u s i o n c o n t r o l l e d s e p a r a t i o n of two spheres bound i n an e l e c t r o s t a t i c i n t e r a c t i o n ; we w i l l c a l l t h i s a d i f f u s i o n - c o n t r o l l e d d i s s o c i a t i o n ( U ) · However, i n view of the u n d e r l y i n g assumptions made i n the d e r i v a t i o n of eqns. 11 and 12 and the u n c e r t a i n t y of the meaning of the d i s t a n c e a, i t i s necessary t o c o r r o b o r a t e t h i s important c o n c l u s i o n from other experimental data. Whatever the exact form of eqn. 11 and 12 and exact meaning of £ i t i s c l e a r that the temperature-dependence o f d i f f u s i o n - c o n t r o l l e d recombination and d i s s o c i a t i o n ( a p a r t from the e l e c t r o s t a t i c work o f charge-separation) shoud be mainly


164


governed by the temperature-dependence of the v i s c o s i t y . T h i s i s shown i n T a b l e I I where i t i s seen that Δ* Δ* - ΔΗ « ΔΗ viscosity. The mechanistic importance of the study of the temperature dependence of the r a t e constants i s very c l e a r l y i n d i c a t e d i n the study of the dynamics of f l u o r e n y l l i t h l u m i n d i e t h y l ether (5,13). Although the values of the r a t e constants f i t t e d r a t h e r w e l l w i t h the v a l u e s expected f o r a d i f f u s i o n a l a s s o c a t i o n - d i s s o c i a t i o n process a t one temperature, the d i s s o c i a t i o n r a t e constant decreased w i t h i n c r e a s i n g temperature. T h i s i s a d e f i n i t e proof f o r the i n t e r v e n t i o n of a s o l v e n t separated i o n - p a i r i n the d i s s o c i a t i o n pathway. Since the design of the measuring c e l l f o r f i e l d modulation s t u d i e s Is not very c r i t i c a l i t was r e l a t i v e l y easy to study a l s o the pressure-dependence of i o n - p a i r d i s s o c i a t i o n and i o n i c recombination (14). Here a l s o the values (Table I I ) f o r the a c t i v a t i o n volumes show the e s s e n t i a l l y d i f f u s i o n c o n t r o l l e d aspects of the a s s o c i a t i o n - d i s s o c i a t i o n phenomena, s i n c e the c a l c u l a t e d v a l u e s , e s s e n t i a l l y the pressure dependence of the v i s c o s i t y , and the e x p e r i m e n t a l l y determined values agree r a t h e r well. At i n c r e a s i n g s a l t c o n c e n t r a t i o n s the conductance of a l l s o l u t i o n s i n v e s t i g a t e d shows the emergence of other e q u i l i b r i a i n v o l v i n g the ions and i o n - p a i r s : t r i p l e i o n s , quadrupoles . . . become s t a b l e s p e c i e s . Althought one expects from chemical r e l a x a t i o n theory the appearance of more r e l a x a t i o n processes, experimentally only one r e l a x a t i o n process, whatever the c o n c e n t r a t i o n of s a l t , Is observed. T h i s odd behavior i s i n t i m a t e l y r e l a t e d to the nature of e q u i l i b r i u m p e r t u r b a t i o n and o b s e r v a t i o n of the response i n the f i e l d modulation method. Indeed the a p p l i c a t i o n of a high e l e c t r i c f i e l d upon a weak e l e c t r o l y t e s o l u t i o n r e s u l t s not so much In a p e r t u r b a t i o n of the equilibrium between i o n - p a i r s and f r e e ions but r a t h e r of the e q u i l i b r i u m between a non-conducting s t a t e and the conducting s p e c i e s . T h i s i o n i z a t i o n mode i s slow compared to other processes i n each s t a t e s i n c e t h i s mode i n v o l v e s charge-separation w h i l e the aggregation modes(s) only i n v o l v e s i o n - d i p o l e and/or d i p o l e - d i p o l e I n t e r a c t i o n s . Such a pathway f o r i o n - f o r m a t i o n i n low p o l a r media i s c o n v i n c i n g l y i n d i c a t e d from measurements of the FDE o f tetraalkylammonlum s a l t s In benzene s o l u t i o n ; the FDE i s completely d e s c r i b e d by Onsager s theory although conductance measurements c l e a r l y show the presence of a v a r i e t y of i o n i c and aggregated species ( 1 8 ) . 38


r

1

Taking the presence of t r i p l e Ions and quadrupoles Into account the r e a c t i o n scheme f o r i o n i z a t i o n becomes very complex compared to the simple i o n - p a i r d i s s o c a t i o n [ 1 ] :


11.



165

Media

TABLE I I A c t i v a t i o n Parameters f o r the I o n i z a t i o n o f Tetrabutylammonium P i c r a t e (TBAP) i n Benzene-Chlorobenzene Mixtures (15,16)


D

298K

A

H

A/kj.mol-

d

1

AV

3

A */cm mol""

d

1

d

1. Experimental 3.22

31.6

7.2±2.0

38.8

-51.8

19.6

-32.3

3.55

27.4

6.8±1.4

34.4

-48.5

18

-30.8

3.87

27.4

4.5±2.6

32.7

50.8

20.5

-30.6

2. C a l c u l a t e d

x

+ >

18.91

-33.9

32.9

8.9

40.9

-53.0

29.8

8.7

37.7

-49.6

18.4

-32.0

26.1

8.6

33.9

-48.6

18.0

-30.4

Physico-chemical parameters of the mixtures D 98K

n/mP -(dD/dT)/K (dD/dP)/atm

3.22

5.45

5.07xl0~

3.55

6.61

6.63x10-

3.87

6.78

8.24x10'

2

v

,-3

-1

.1

1

(dlnn/dT" )/K (dlnn/dP/atra"

,.5 23.7x10"

1.19x10°

7.37x10-

,.5 27.1x10"

1.17x10°

7.17x10'

,-5 30.6x10"

1.15x10°

6.98x10'

4

° The e r r o r s on these small values a r e r e l a t i v e l y l a r g e w h i l e the e r r o r s f o r a l l other experimental values a r e t y p i c a l l y 3-4%. +

f o r an exact agreement the temperature dependence of a shoulg be accounted f o r ; dlna/dT i s [-0.34, -0.04 and"0.16] χ 10" f o r the three mixtures s t u d i e d . dlnad/dP i s always found to be n e g l i g i b l e .


166

ANIONIC

Β

+

A

+

s

2AB

Β

+

ABA

+

AB.

s

ABA

POLYMERIZATION

+

BAB


[13]

BAB

+

ABA

+

(A i s the c a t i o n A and Β the anion B" as i n eq. 1; charges a r e omitted here f o r c l a r i t y ) . Such a complex scheme i s however e q u i v a l e n t , from the viewpoint of i o n i z a t i o n , with a simple e q u i l i b r i u m between a non-conducting s t a t e ( i o n - p a i r s and quadrupoles) and a conducting s t a t e ( i o n s and t r i p l e i o n s ) . With the simple assumption o f f a s t e q u l i b r i a w i t h i n each s t a t e (the " h o r i z o n t a l " e q u i l i b r i a i n scheme 13) and noting that u s u a l l y the c o n c e n t r a t i o n o f a l l species i s n e g l i g i b l e compared to the i o n - p a i r c o n c e n t r a t i o n C the r e c i p r o c a l r e l a x a t i o n time a s s o c i a t e d w i t h the slow i o n i z a t i o n mode can be c a l c u l a t e d as: 0

τ"

1

=

K

d

+ + ( k + k )4C /K D

D

0

1 + 9/16.k K (4C /K )

s

F

s

0

2

s

1 + 4C /K 0

K

1/2 1/2 + 1/2 - 1/2 C (1+C /K ) (1+C /K ) d

0

0

3

0

3

k kR kR |k +2| — + — 1 C +3 1

r

l+2Co/K3+2C /K +3Co/K3K3 0

3

T

0

*3

S

2

Co

[14]

K3K3

The d e r i v a t i o n o f t h i s equation from general r e l a x a t i o n theory, w i t h i n the assumptions s t a t e d , i s s t r a i g h t f o r w a r d but tedious and w i l l be published elsewhere. At f i r s t s i g h t equation 14 i s so i n v o l v e d that i t may be everything but u s e f u l f o r any p r a c t i c a l purpose. F o r t u n a t e l y , depending on the ( r e l a t i v e ) s t a b i l i t y o f the d i f f e r e n t aggregated species and on the t o t a l c o n c e n t r a t i o n , d i f f e r e n t l i m i t i n g formes of eq. 14 a r e very i l l u m i n a t i n g . F i r s t , and g r a t i f y i n g , a t low c o n c e n t r a t i o n and/or low s t a b i l i t y o f the aggregated species eq. 14 reduces to:


11.



r-1

k

d

Media

1/2 1/2 + 2K K C r

d

167

[10]

c

which i s indeed the expression obtained f o r the simple i o n - p a i r d i s s o c i a t i o n . Moreover, as long as the c o n c e n t r a t i o n of quadrupoles i s n e g l i g i b l e compared to the i o n - p a i r s the f i r s t term i n eq. 14, which d e s c r i b e s the d i s s o c i a t i o n , reduces always to k o r depending on the r e l a t i v e magnitude of k , Kjy and kp t o a concentration-dependent erm (which i s always small compared to the recombination term)· Focusing on the recombination term i n eq. 14 we see that t h i s term reduces to: 1


d

d

1/2 3/2

Kd 2k

[14a]

T

K

K

3 3 J at high i o n - p a i r c o n c e n t r a t i o n and/or high s t a b i l i t y of the t r i p l e ions s i n c e then 0 / Κ 3 * < 1 and k « k ± « k assumed to be d i f f u s i o n - c o n t r o l l e d recombination r a t e constants. T h i s i s i n t u i t i v e l y c l e a r s i n c e a t these c o n d i t i o n s the main i o n i c recombination process w i l l be between the a n i o n i c and c a t i o n i c t r i p l e i o n s , the c o n c e n t r a t i o n of which depends on the 3/2 power of the t o t a l i o n - p a i r c o n c e n t r a t i o n . Reasoning along the same l i n e s we note that f o r the case of u n i l a t e r a l t r i p l e i o n formation (e.g. K — » ) a t high i o n - p a i r c o n c e n t r a t i o n , o r f o r the c o n d i t i o n K > C > K "", the recombination term would be g i v e n as: ο

r

+

3

+

3

0

3

Kd

1/2

! [14b]

2kR

κ 3

A l i n e a r c o n c e n t r a t i o n dependence of the r e c i p r o c a l r e l a x a t i o n time upon t o t a l i o n - p a i r c o n c e n t r a t i o n would t h e r e f o r e p o i n t to a main recombination process between a simple i o n and a t r i p l e i o n . C o n s i d e r i n g these d i f f e r e n t l i m i t i n g forms of the recombination term an important t e n t a t i v e c o n c l u s i o n emerges: the c o n c e n t r a t i o n dependence of the r e c i p r o c a l r e l a x a t i o n time i s a d i r e c t measure of the main i o n i c recombination process and y i e l d s t h e r e f o r e i n f o r m a t i o n on the i o n i c species present i n s o l u t i o n . A l i n e a r dependence on t o t a l i o n - p a i r c o n c e n t r a t i o n would t h e r e f o r e i n d i c a t e u n i l a t e r a l t r i p l e i o n formation or, i f both kinds of t r i p l e ions a r e present as i n d i c a t e d by conductance, a s u f f i c i e n t d i f f e r e n c e i n t h e i r s t a b i l i t y . At t h i s p o i n t i t should be noted that the u s u a l method of Fuoss and Draus (19) t o determine s t a b i l i t y constants f o r t r i p l e ions assumes an equal s t a b i l i t y . In many i n v e s t i g a t i o n s t h i s assumption i s o f t e n


ANIONIC


168

POLYMERIZATION

i m p l i c i t l y accepted even with c o n t r a - i n d i c a t i o n s such as a d i f f e r e n c e i n s i z e of the i o n s . E x p e r i m e n t a l l y a l l the l i m i t i n g cases of the recombination terra are encountered i n a l l systems i n v e s t i g a t e d up to now. In s o l u t i o n s of tetraalkylammonium s a l t s In benzene the r e c i p r o c a l r e l a x a t i o n time i s even dependent on the square of t o t a l c o n c e n t r a t i o n (but here the quadrupoles may be comparable In c o n c e n t r a t i o n w i t h i o n - p a i r s ) which would i n d i c a t e d a preponderance of q u i n t u p l e and t r i p l e ions i n the i o n i c recombination. However a very p u z z l i n g reature of the experimental r e s u l t s , as can be seen d i r e c t l y i n F i g u r e 3, i s the r e l a t i v e l y important i n t e r c e p t which i s always measured f o r systems where the r e c i p r o c a l r e l a x a t i o n time has a higher than square-root dependence on c o n c e n t r a t i o n . T h i s i s i n c o n t r a d i c t i o n w i t h the d i s s o c i a t i o n term of the general eq. 14 which always gives a v a n i s h i n g l y small monomolecular constant f o r systems where the aggregation i n t o quadrupoles i s small. K i s of the order 1-10-% f o r tetrabutylammonium s a l t s i n low p o l a r s o l u t i o n s , the s m a l l e r values a p p l y i n g to benzene-solution (20) and somewhat depending on s i z e and form of the i o n s ; the extent of the d i m e r i z a t i o n of i o n - p a i r s with p i c r a t e i s u s u a l l y small as i n d i c a t e d from conductance. The r e l a x a t i o n behavior i n d i c a t e s a s i t u a t i o n as i f each aggregated form d i s s o c i a t e s i n t o the corresponding ions independently of the other aggrgated s p e c i e s present, indeed i f t h i s would be the case the i n t e r c e p t (which has the dimensions of a r e c i p r o c a l time) c o u l d be e x p l a i n e d as the sum of the d i s s o c i a i t o n r a t e constants of a l l monomolecular d i s s o c i a t i o n processes present. T h i s would be so i f the d i f f e r e n t aggregated s p e c i e s were not i n e q u i l i b r i u m or, which i s the same, would e q u i l i b r a t e more slowly than the i o n i z a t i o n processes. However t h i s assumption i s untenable as confirmed by some measurements on the r e l a x a t i o n of ion-pair-quadrupole e q u i l i b r i a ( 2 1 ) . I t Is t h e r e f o r e not a t r i v i a l problem to d e f i n e which r e l a t i v e l y f a s t monomolecular process present In s o l u t i o n g i v e s r i s e to the i n t e r c e p t . A s a t i s f a c t o r y e x p l a n a t i o n c o u l d only be g i v e n from a d e t a i l e d numerical c a l c u l a t i o n of the extent of p e r t u r b a t i o n of the i o n i z a t i o n mode upon a p p l i c a t i o n of the electric field. The c o n c e n t r a t i o n of i o n i c s p e c i e s , which i s d i r e c t l y p r o p o r t i o n a l to the conductance, i s very small i n these low p o l a r media and s i n c e the r e l a t i v e change i n conductance i s of the order of 1% w i t h f i e l d the a b s o l u t e change i n i o n i c c o n c e n t r a t i o n i s exceedingly s m a l l . As a r e s u l t the change i n c o n c e n t r a t i o n of the aggregated s p e c i e s vanishes almost completely. Otherwise s t a t e d , the aggregation e q u i l i b r i a are not perturbed by the a p p l i c a t i o n of the e l e c t r i c f i e l d which f o r m a l l y , i f we c o n s i d e r the aggregation mode(s), uncouples these equlibria! T h i s means that the d i s s o c i a t i o n term i n eq. 14 should be r e p l a c e d by the sum of a l l monomolecular r a t e constants p e r t a i n i n g to the d i s s o c i a t i o n of a l l aggregates present In s





10 C 6

J

1

o

3 / 2

/M

Media

169

3 / 2

I

2

10*C /M o

Figure 3. Dependence of the reciprocal conductance relaxation time on TBAsalt concentration in media and concentration range where conductance indicates an important fraction of triple ions. TBA-bromide in benzene-nitrobenzene (3, 22 vol %; D — 2, 90) at 298 Κ (Ο). TBA-picrate in benzene-chlorobenzene (16 vol %;D = 2, 78) at 298 Κ (0).


170

ANIONIC

POLYMERIZATION

s o l u t i o n , i . e . the i n t e r a c e p t should be g i v e n by k

Anionic Polymerization - American Chemical Society

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