Drit tes Physikalisches Inst i tut , Universi tgt G6tt ingen, BiirgerstraBe 42-44. D-37073 Grt t ing en. Germ any
Abstract--An account is given on dielectric relaxa tion spectroscopy as a dom ain of current interest inl iquid s tate physics . Based on results mainly from this laboratory, dielectric propert ies of l iquids atmicrowave frequencies are summarized an d the underlying mo lecular mechanisms are discussed. Part icularat tent ion is paid to the unique behaviour of aqueous systems.
1. INTRODUCTION: DIELECTRIC RELAXATION
SPECTROSCOPY OF LIQUIDS
L i q u i d s a r e c h a r a c t e r i z e d b y a s h o r t - r a n g e m o l e c u l a r
o r d e r w h i c h r a p i d l y v a r i e s i n t i m e . T o m o n i t o r t h i s
o r d e r a n d i t s th e r m a l f l u c t u a t i o n s d i e l e c t r i c s p e c -
t r o s c o p y u t i li z e s e l e c t ri c a l c h a r g e d i s t r i b u t i o n s a s
n a t u r a l l y p r e s e n t m o l e c u l a r m a r k s ( B r o w n , 1 95 6;
F r6 h l i ch , 1 9 5 8 ; Dan i e l , 1 9 6 7 ; H i l l et al . , 1969 ; Dav ies ,
1972, 1975, 1977; B6 t tche r , 1973 ; B6 t tch er and Bo r-
d e w i j k , 1 9 7 8 ; G r a n t et al . , 1978 ; Scai fe , 1989) . Par-t i c u l a r l y su i t a b l e l a b e l s a r e p e r m a n e n t e l e c t r ic d i p o l e
m o m e n t s ~ . H e n c e s p e c i a l e m p h a s i s i s u s u a l l y d i -
r e c t ed t o w a r d t h e s t u d y o f d i p o l a r l i q ui d s. H o w e v e r ,
e x a m p l e s a r e g iv e n b e l o w t o s h o w t h a t n o n p o l a r
l i q u id s m a y b e a l s o o f i n te r e s t. A m o n g t h e v a r i e ty o f
d i p o l a r l i q u id s w a t e r a n d a q u e o u s m i x t u r es a r e m o s t
i m p o r t a n t w i t h r e s p e c t t o a p p l i c a t i o n s in m e d i c i n e ,
b i o l o g y , c h e m i c a l e n g i n e e r i n g , a n d e n v i r o n m e n t a l
s c i en ce a s we l l (F ran k s , 1 9 72 , 1 9 7 3 a , b , 1 9 7 5 a , b ,
q u e n t l y , m u c h a t t e n t i o n h a s b e e n p a i d ( a n d w i l l b e
g i v e n i n t h i s r e v i e w ) t o a q u e o u s s y s t e m s .B a s i c a l l y , d i e l e c t r i c s p e c t r o s c o p y a i m s a t t h e p r e -
c i se k n o w l e d g e o f t w o q u a n t i t i e s , ( i) t h e r . m . s , v a l u e
p = (p 2 ( t ) ) , / 2 (1 )
o f t h e n o i s e s i g n a l o f th e e l e c t r i c p o l a r i z a t i o n P a t
t h e r m a l e q u i l i b r iu m ( ( P ( t ) ) = 0 ), a n d ( ii ) t h e n o r -
m a l i z e d a u t o c o r r e l a t i o n f u n c t i o n
( P ( t ) - P (O ) )qb(t ) = (2)
(P (O) • P (O) )
o f th i s si g n a l . ~ ( t ) i s a l s o c a l le d d i e l e c t r i c d e c a y
f u n c t i o n . A t t h e r m a l e q u i l ib r i u m , h o w e v e r , t h e
n o i s e r e s u l t i n g f r o m t h e f l u c t u a t i n g p o l a r i z a t i o n o f
t h e l i q u id is m a s k e d b y t h e n o i s e f r o m t h e m e a s u r i n g
a p p a r a t u s i t s e lf . I t is th u s i m p o s s i b l e t o m o n i t o r t h e
f l u c t u a t i n g p o l a r i z a t i o n o f t h e s a m p l e w i t h s u f f i c i e n t
a c c u r a c y i f e q u i l i b r iu m r e m a i n s c o m p l e t e l y u n d i s-
t u r b e d . T o i n c r e a s e t h e s e n s i ti v i t y i n t h e m e a s u r e -
549
m e n t s t h e s a m p l e l i q u i d is u s u a l l y e x p o s e d t o a
m o n o c h r o m a t i c e l e c t r ic f i e ld E ( v ) o f l o w f ie l d
s t r e n g t h ( P - E 0 < < k T ; E 0 , a m p l i t u d e ) . V a r y i n g t h e
f r e q u e n c y v t h e p o l a r i z a t i o n P ( v ) i s m e a s u r e d a s a
f u n c t i o n o f v a n d t h e d i e l e c t r ic p r o p e r t i e s o f t h e l i q u i d
a r e e x p r e s s e d b y t h e c o m p l e x ( e l e c tr i c ) p e r m i t t i v i t y
d e f i n e d b y
E (v ) = ¢ ' (v ) = iE (V) = 1 P ( v )eo Ev ~ + 1 . (3 )
H e r e i n , e0 d e n o t e s t h e e l e c t r i c f ie l d c o n s t a n t . T h e
r e a l p a r t e ' ( v ) o f t h e p e r m i t t i v i t y r e p r e s e n t s t h e
i n p h a s e c o m p o n e n t o f t h e p o l a r i z a t i o n w h i l e t h e
( n e g a t i v e ) i m a g i n a r y p a r t E ( V ) r e p r e s e n t s t h e c o n t r i -
b u t i o n s t o P ( v ) w i t h a ~ / 2 p h a s e s h i f t w i t h r e s p e c t t o
E ( v) . H e n c e u s e o f a c o m p l e x p e r m i t t i v i t y a ll o w s t o
t a k e i n t o a c c o u n t p o s s i b l e p h a s e l a g s b e t w e e n t h e
r e s p o n d i n g p o l a r i z a t i o n a n d t h e e x c i t i n g s i n u s o i d a l
e l e c t ri c f i e ld w h i c h m a y r e s u l t f r o m m o l e c u l a r i n t e r -
a c t i o n s p r e v e n t i n g P ( v ) f r o m i n s t a n t a n e o u s l y f o l l o w -
i n g E ( v ) . A p h a s e l a g b e t w e e n P ( v ) a n d E ( v ) m e a n s
t h a t e l e c t r ic e n e r g y i s d i s s i p a t e d a s h e a t w i t h i n t h es a m p l e l i q u id . I f d i s p l a y e d a s a f u n c t i o n o f v th e
n e g a t iv e im a g i n a r y p a r t e t h e r e fo r e l o o k s l ik e a n
a b s o r p t i o n c u r v e ( F i g . 1 ). A s p r e d i c t e d b y t h e f l u c t u -
a t i o n / d i s s ip a t i o n t h e o r e m t h e r e a l p a r t e ' ( v ) s h o w s
d i s p e r s i o n c h a r a c t e r i s t i c s ( F i g . 1 ).
L e t u s b r i e f ly r e t u r n t o t h e t i m e d o m a i n . I n t e r -
m o l e c u l a r f o r c e s a r e a l s o r e f l e c te d b y t h e n o i s e s i g n a l
o f th e p o l a r i z a t i o n a n d t h u s b y a n o n - t r i v i a l d i e l e c t r ic
d e c a y f u n c t i o n . F o l l o w i n g li n e a r s y s te m s t h e o r y t h e
c o m p l e x p e r m i t t i v i ty e ( v) a n d t h e a u t o c o r r e l a t i o n
f u n c t io n ~ ( t ) o f t h e p o l a r i z a t i o n n oi s e s ig n a l a r e
r e l a te d a c c o r d i n g t o t h e L a p l a c e t r a n s f o r m
1 P s f ~ [ d q b ( t ) l e , ~ V t d t+ le (v) = E0 E~ J0 dt
= [E (0 ) - 11 ~ - e - '~ 'd t + 1 (4 )
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550 UDOKAATZE
8 0
4
2
~
t c 1 . } _ 7 _ _ -0 I ~ t I
4 L i t i i
10 I 1
1 2 ~ r
0 I I I I I
2 / , 7 1 0 2 0 G H z t O 7 0
V
Fig. 1 . Real par t e ' ( v ) and nega t i ve im ag ina ry pa r t e ( v ) o fthe com plex perrni t t iv i ty plot ted versus f requency v forwater a t 25 °C (Kaatze , 1989a). The ful l curves are grap hs o fthe D ebye - type r e l axa ti on func t i on de f ined by equa t ion (6 )wi th the fol lowing va lues for the parameters: E oo)= 5.2 ,
e (0) - 78.36, z = 8.27 ps.
w h e r e P s , E ~ , a n d e ( 0 ) d e n o t e s t a t i c v a l u e s (v ---*0) .I n F i g . 2 , a s a n e x a m p l e , t h e d i e le c t r ic d e ca y f u n c t i o n
o f w a t er , c o r r e s p o n d i n g w i t h th e s p e c t r u m s h o w n i n
F i g . 1 , i s d i s p l a y e d v s t i m e t . T h e a u t o c o r r e l a t i o n
f u n c t i o n e x h i b i t s a s t e e p d e c r e a s e a t s m a l l t . S u c h
d e c r e a s e i s c o m m o n t o a l l l i q u i d s . I t r e s u l t s f r o m
d i s p l a c e m e n t p o l a r i z a t i o n m e c h a n i s m s r e s p o n d i n g
t o o f a s t ( < 1 0 - 1 2 s ) t o b e re s o lv e d b y m i c r o w a v e
s p e c t r o s co p y [ ( 2 ~ - 1 0 -1 2 s ) - l = 1 60 G H z ] . H e n c e t h e
i n i t i a l f a s t d e c a y i n t h e a u t o c o r r e l a t i o n f u n c t i o n
( F ig . 2 ) a n d , c o r r e s p o n d i n g l y , t h e e x t r a p o l a t e d
h i g h f r e q u e n c y p e r m i t t i v i t y e ( o o ) ( F i g . 1 ) a r e n o t
c o n s i d e r e d h e r e .T h e s l o w e r d e c a y i n g p a r t ( I ) r( t) o f ~ ( t ) r e p r e s e n t s
r e l a x a t i o n b e h a v i o u r . C r ( t ) o f w a t e r a t r o o m t e m -
p e r a t u r e c a n b e a l m o s t r e p r e s en t e d b y a n e x p o n e n t i a l
( K a a t z e , 1 9 8 % . 1 9 9 3 ) ,
@~( t ) = Or (0 )e - ' JL (5 )
T h e d e c a y t i m e z o f th i s f u n c t i o n i s c a l le d d i e l e c tr i c
r e l a x a t i o n t i m e . I f a c c o r d i n g t o e q u a t i o n ( 4) t h e
e x p o n e n t i a l d e c a y f u n c t i o n i s t r a n s f o r m e d i n t o t h e
f r e q u e n c y d o m a i n a D e b y e - t y p e r e l a x a t i o n s p e c t r a l
f u n c t i o n r e s u l t s ( D e b y e , 1 92 9). H e n c e t h e d i e l e c tr i cs p e c t r u m s h o w n i n F i g . 1 c a n b e a n a l y t ic a l l y r e p -
r e s e n t e d a s
E 0 ) - e o o )e ( v ) = e ( ~ ) + ( 6 )
1 - r i o g z
w h e r e co = 2 n v .
I n v i e w o f m o l e c u l a r m o d e l s o f l i q u i d s i t i s d e s i r -
a b l e t o d i s c u s s m a c r o s c o p i c a l l y a c c e s s i b l e p o l a r i z -
a t i o n r e l a x a t i o n p r o c e s s e s i n te r m s o f d i p o l e
a u t o c o r r e l a t i o n t i m e s % . H o w e v e r , t h e r e d o e s n o t
e x i s t a g e n e r a l l y v a l id r e l a t i o n , w i t h a l l q u a n t i t i e s
a c c e s s ib l e , b e t w e e n d e c a y t i m e s o f q~ ( t ) a n d t h o s e o ft h e ( n o r m a l i z e d ) d i p o l e a u t o c o r r e l a t i o n f u n c t i o n
~k~( t ) de f ined by
O , ( t ) . l , 0 ) >
¢ /~ ( t ) < / t (0 ) . / t (0 )> (7 )
T h e c o m p l e x i t y o f t h e p r o b l e m c a n b e r e a li z ed b y
u s i n g i n e q u a t i o n ( 2) t h e d e f i n i ti o n o f t h e p o l a r i z a ti o n
a s t h e t o t a l e le c tr ic m o m e n t p e r v o l u m e V o f a g i v e n
v o l u m e e l e m e n t ,
P ( t ) = ] ~ L / t , ( t ) . ( 8 )r i = i
I n t h i s e q u a t i o n , N v d e n o te s t h e n u m b e r o f m o l ec u -
l a r d i p o le s w i t h i n t h e s a m p l e v o l u m e . I n s e r t i o n o f t h e
s u m [ e q u a t i o n ( 8 ) ] i n t o e q u a t i o n ( 2) i n d i c at e s t w o
e s s e n t i a l l y d i f f e r e n t c o n t r i b u t i o n s to P ( t ) , p r o d u c t s
b e t w e e n i d e n ti c a l d i p o l e m o m e n t s ( s e l f t e r m ) a n d
t h o s e b e t w e e n d i f f e r e n t m o l e c u l a r d i p o l e m o m e n t s
( d i s t i n c t t e rm ) . H e n c e b e s i d es th e d i p o l e a u t o -
c o r r e l a t i o n f u n c t i o n q G ( t ) a l s o t h e c r o s s c o r r e l a t i o n
f u n c t i o n
< t ~ , t ) . Y ~ ~ , j 0 ) >
J* i (9)¢~ . ( t ) = O , (0 ) - ~ , ( 0 ) >
m a y a c t a n i n fl u e n c e o n ¢ ( t ) . O n e h a s t h u s t o b e
c a r e fu l w h e n i n t e r p r e t i n g m a c r o s c o p i c a l ly m e a s u r e d
c o l l e c t i v e r e l a x a t i o n t i m e s a s m o l e c u l a r a u t o c o r r e l a -
t i o n t i m e s . I t h a s b e e n s h o w n , h o w e v e r , t h a t w i t h
l i q u i d w a t e r
z . ~ z (10 )
i f , i n c o r r e s p o n d e n c e t o z , % r e f e r s t o t h e s l o w l yd e c a y i n g p a r t o f t h e a u t o c o r r e l a t i o n f u n c t i o n ( K a a t z e
a n d P o t t e l , 1 9 9 2 ) .
1 [
0 5
I E ~ { O ] I
0 I I I0 ¢ 8 1 2 p s 1 6
f
Fig. 2. D ielectric decay fun ctio n @(t ) of w ater at 25°Cdisplayed as a funct ion of t ime t .
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Microwave dielectric properties of liquids 551
2 E X P E R IM E N T A L M E T H O D S
Basically, the complex dielectric permittivity of a
liquid is determined by measuring either the complex
reflection coefficient at a sample/solid interface or the
complex tr ansmission coefficient of a liquid column,
preferably at variable thickness (Hill e t a l . , 1969;
Bennett and Calderwood, 1971; Gra nt e t a L , 1978;
Kaatze and Giese, 1980a; Bailey, 1985; Bryant, 1988;
Barthel e t a l . , 1991). Nowadays microprocessor con-
trolled network analyzer combined with broadband
coaxial line components are utilized up to consider-
ably high frequencies (v ~> 20 GHz) . These methods
compete with time domain techniques in which the
sample is exposed to a step voltage pulse (van
Gemert, 1973; Cole, 1977; Cole e t a l . , 1989; Clarkson
e t a L , 1977; Kaatze and Giese, 1980a; Gestblom and
Noreland, 1988; Gestblome t a l . ,
1987; Nozaki andBose, 1990). Time dom ain spectroscopy, however,
suffers from an unfavourabl e power spectrum (~tv-2)
of the exciting signal. Also in use above about 5 GHz
and mandatory at higher frequencies (v/> 20 GHz)
are smallband waveguide devices.
At microwave frequencies measurements of the
reflection coefficient are accompanied by a standing
wave pattern. Analysis of this pattern constitutes a
suitable method for the study of liquids with low
dielectric loss. Matched to high-loss liquids are
methods in which a travelling wave is transmitted by
the sample. This method normally involves a bridgecircuit in order to probe the electromagnetic field
within the sample interferometrically. If only small
amounts of liquids are available (Kaatze, 1973) or if
liquids of very low loss are investigated (Stumper,
1973) resonator techniques are to be preferred in
which the effect that the sample exerts on the electro-
magnetic field is substantially increased by multiple
reflections (Sucher and Fox, 1963).
The way in which complex permittivity measure-
ments on liquids are performed at microwave fre-
quencies may be illust rated by two different methods.
Up to a critical frequency which is mainly given bythe TM0rmode cut-off frequency vc of the circular
waveguide (Marcuvitz, 1951) of the specimen cell
(Fig. 3) the det erminat ion of the reflection coefficient
R (v) of this cell as a function of frequency v has
proven a powerful method. This cell essentially con-
stats o f a coaxial line/ci rcular waveguide transition.
The waveguide is excited below its cut -off frequency
re. When filled with a lossless sample (E = 0) vc is
given by the relation
2c(11)
vc v e/~72.61d
where c denotes the speed of light in empty space and
d the diameter of the cell.
The cut-off mode of operation results in a strong
decrease in the ampli tude of the electromagnetic field
within the waveguide. Hence the shape of the surface
of the liquid column does not affect the measurements
q /c i v i c
¢ F E ¢1vlC2
Co
P
Fig. 3. Sketch of the apparatus for broadband reflectioncoefficient measurements and electrical equivalent circuit ofthe sample cell (1). l(a), l(b), waveguide below cut-off andcoaxial line section of the cell, respectively; l(c), matcheddielectric window; l(d), coaxial feeding line; 2, reflection estset basically consisting of a directional coupler (2a) and a
beam splitter (2b); 3, network analyzer; 4, process controlcomputer.
so that easy to handle open cells can be used. In order
to reach optimum sensitivity in the measurements he
length l of the coaxial line (lb, Fig. 3) can be adjus ted
to the dielectric properties of the sample under test
and also to the frequency range of interest.
Via a reflection test set (2) the cell (1) is connected
to a computer-controlled network analyzer (nwa,3).
With the aid of the beam splitter (2b) of the test set
part of the signal from the generator (G) of the nwais fed back to the nwa reference port (R). The other
part is transmitted to the cell (1). A high-precision
directional coupler (2a) allows the wave reflected by
the cell to be received by the nwa (L6nnecke-Gabel,
1990).
In many cases it is sufficient to represent the cell
characteristics by two parameters to be found by
calibration measurements with liquids of well-known
dielectric properties. Suitable parameters for this
purpose are the length L of the feeding coaxial line
(ld, Fig. 3) and a n effective length le~ of the liquid-
filled part. Due to the aperiodic field in the waveguide
which acts like a capacity termination the effective
length exceeds/. At small I electric flux lines passing
the dielectric window (lc) and the sample liquid as
well become impor tan t (Kaatze and Giese, 1987a). As
has been shown by modal analysis it is sufficient to
consider the effect of these flux lines by two further
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552 UDO KAATZE
I
1~ .~ 16
3
I
1 2 3 3 3 1 /, 1 5 3 2
Fig. 4 . Schematic representation of autom atic microwave double -beam interferometers for off-balancemeasurem ents of the complex perm ittivity of liquids. I , mon och rom atic oscillator; 2, uniline; 3, directionalcoupler; 4, power sensor of 5, level meter; 6, frequency counter; 7, circular waveguide sample cell with8, dielectric window and 9 , shiftable pro be; 10, stepping m oto r with 1 , c ont rol unit; 12, digita l distancemeter; 13, flexible waveguide; 14, vari able attenu ator ; 15, varia ble phase shifter; 16, process con trol
computer.
p a r a m e t e r s a n d t o r e p r e s e n t t h e s a m p l e c e l l b y t h e
e q u i v a l e n t c i r c u i t s h o w n i n F i g . 3 . W e a p p l y t h i sm e t h o d i n m e a s u r e m e n ts b e t w e en 1 M H z a n d 3 G H z
w i t h t h e u p p e r l i m i t b e i n g g i v e n b y t h e f r e q u e n c y
r a n g e o f t h e a v a i l a b l e n w a .
A t f r e q u e n c i e s a b o v e s o m e G H z e l e c t r o m a g n e t i c
f i e ld s o f a g r e e a b l e w a v e l e n g t h 2 a r e s e t u p i n t h e
s a m p l e l iq u i d s. A s a c o n s e q u e n c e w a v e p r o p a g a t i o n
p r o p e r t i e s c a n b e d i r e c t l y m e a s u r e d . I n F i g . 4 a n
o u t l i n e o f a n a p p a r a t u s i s p r e s e n t e d w h i c h i n a
c o m p u t e r - c o n t r o l l e d m o d e o f o p e r a t i o n e n a b l e s t h e
f i e l d w i t h i n t h e l i q u i d t o b e p r o b e d i n t e r f e r o m e t r i c a l l y
( W a l l u s c h e t a L , 1 9 95 ). T h e s am p l e i s co n t a i n e d i n a
w a v e g u i d e ( 7 ) w h i c h i s s e a le d b y a d i e l e c t r i c w i n d o w
( 8 ). A r e c e i v i n g p r o b e ( 9 ), b a s i c a l l y a ls o a c i r c u l a r
w a v e g u i d e ( 9 ) i s i m m e r s e d i n t h e l i q u i d . P r o v i d e d
w i t h h i g h p r e c i s i o n b a l l b u s h g u i d e s t h i s p r o b e c a n b e
p r e c i s e l y s h i f t e d f re e o f b a c k l a s h a l o n g t h e d i r e c t i o n
z o f w a v e p r o p a g a t i o n . A n e s p e c ia l ly d e s i g n ed h e a d
o f t h e p r o b i n g w a v e g u id e p r e v e n t s w a v e s f ro m c o m -
i n g t o t h e u p p e r s u r f a c e o f th e l i q u i d c o l u m n . S i g n a l s
r e f l e c t e d f r o m t h i s s u r f a c e , t h e p o s i t i o n o f w h i c h
c h a n g e s w h e n s h i f t i n g t h e p r o b i n g w a v e g u i d e , h a v e
t h u s n o t t o b e t a k e n i n t o a c c o u n t i n t h e c o n s i d e r a t io n
o f t h e e l e c t r o m a g n e t i c f ie l d w i t h i n t h e s a m p l e v o l u m e .
D e p e n d i n g o n t h e d i e l e c tr i c p r o p e r t i e s o f t h e l i q u i d
u n d e r t e s t t h e s ig n a l m a y , h o w e v e r , b e m o r e o r
l e s s f r e q u e n t l y r e f l e c t e d b e t w e e n t h e w i n d o w / s a m p l e
i n t e r f a c e a n d t h e s a m p l e / p r o b e in t e r f a c e .
W i t h t h e a i d o f d i r e c t i o n a l c o u p l e r s t h e r e s u l t i n g
m i c r o w a v e s i gn a l t ra n s m i t t e d t h r o u g h t h e s a m p l e i s
c o m b i n e d w i th a r e f e r e nc e w a v e t h e a m p l i t u d e a n d
p h a s e o f w h i c h c a n b e a p p r o p r i a t e l y a dj u s t ed . F o r
t h i s p u r p o s e t h e r e f e r e n c e b r a n c h o f th e i n t e r f e r o m e -t e r is p r o v i d e d w i t h a v a r i a b l e a t t e n u a t o r ( 1 4) a n d a
v a r i a b l e p h a s e s h i f t e r ( 1 5 ) , r e s p e c t i v e l y . M e a s u r e -
m e n t s a r e p e r f o r m e d b y f i r s t a d j u s t i n g t h e r e f e r e n c e
w a v e a n d b y s u c c e s s iv e l y s h i f t in g t h e p r o b i n g w a v e g -
u i d e ( 9 ) a f t e r w a r d s . D u r i n g t h i s s h i f t t h e v o l t a g e U 0
o f t h e o u t p u t p o w e r s e n s o r is m o n i t o r e d u s i n g a
s u i t a b l e l e v el m e t e r . I n p u t v o l t a g e U~ i s m e a s u r e d b y
a n o t h e r p o w e r s e n s o r / l e v e l m e t e r u n i t . T h e r e f o r e ,
p o s s i b l e c h a n g e s i n t h e p o w e r o f t h e o s c i l l a t o r (1 ) c a n
b e c o n s i d e r e d b y f o r m i n g t h e r a t i o U o / U i . T h i s
r a t i o i s s t o r e d b y t h e p r o c e s s c o n t r o l c o m p u t e r ( 1 6 )
a s a f u n c t i o n o f d is c r e t e p r o b e p o s i t i o n s z,,
( n = 1 . . . . N ) . T h e c o m p l e x p e r m i t t i v i ty o f t h e
s a m p l e i s f o u n d b y f i t t in g t h e a n a l y t i c a l e x p r e s s i o n
f o r t h e i n t e r f e r o m e t e r v o l t a g e r a t i o t o t h e m e a s u r e d
U o ( z n ) / U ~ r e l a t i o n , a n e x a m p l e o f w h i c h i s d i s p l a y e d
in F ig . 5 .
3. DIP OLE FLUCTU ATIONS. ROTATI ONAL AND
TRANSLATIONAL MOTIONS
W i t h m a n y l i q u i d s o f i n t e r e s t d i e l e c t ri c r e l a x a t i o n
i s p r e d o m i n a n t l y d u e t o r e o r i e n t a t i o n a l m o t i o n s o f
d i p o l a r m o l e c u l e s . N e v e r t h e l e s s c h a n g e s i n t h e
a m o u n t # = i /l [ o f t h e d i p o l e m o m e n t m a y a l s o
c o n t r i b u t e t o t h e d i e l e c t r ic r e l a x a t i o n s p e c t r u m .
L e t u s f i r s t c o n s i d e r a p r o t i c d i m e t h y l s u l f o x i d e
( D M S O , F i g . 6) a s a n e x a m p l e . I t s d i e l e c t ri c s p e c t r u m
e x t e n d s o v e r a s o m e w h a t b r o a d e r f r e q u e n c y r a n g e
t h a n a D e b y e - t y p e r e l a x a t i o n s p e c t r u m . T h i s fi n d i n g
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M icrowave die lec t ric pro per t ies o f l iquids 553
1 m i m
-- 0.5o
0
0.5 ~.5
x
I ~ I/ I m
1.5 2.5 3.5
z l k
Fig. 5 . Plot of the interferom eter vol tage ra t io U0/Ui (Fig. 4)a s a func t i on o f t he p robe pos i t i on t o wave l eng th ra t i o , z / 2 .Figure sym b ol s m a rk d a t a m easured a t d i scret e z , va lue s(n = 1 . . . . N ) . T he fu l l cu rve i s t he g raph o f t he t heore t ic a lU o / U versus z / 2 func t i on wi th pa ram e te r s found by a
regression analysis (Wal lusch et al., 1995).
i s a n i n d i c a t i o n o f a n u n d e r l y i n g d i s t r i b u t i o n o f
r e l a x a t i o n ti m e s . I t w a s f o u n d t h a t t h e D M S O s p ec -
t r u m c a n b e w e l l r e p r e s e n t e d b y t h e f u n c t i o n p r o -
p o s e d b y D a v i d s o n a n d C o l e ( 1 9 5 0 ) w h i c h c a n b e
e x p r e s s e d a s
E o ) - ~ o o )e ( v ) = e ( ~ ) q (1 + ia~zs) (1-/~) (12 )
T h i s s p e c tr a l f u n c t i o n c o r r e s p o n d s w i t h a n u n s y m -
m e t r i c c o n t i n u o u s r e l a x a t i o n t i m e d i s t r i b u t i o n i n
w h i c h z s d e n o t e s t h e l a r g e s t r e l a x a t i o n t i m e . P a r -
a m e t e r f l m e a s u r e s t h e w i d t h o f th e r e l a x a t i o n t i m e
d i s t r i b u t i o n .
B e s i d e s t h e c h a r a c t e r i s t i c r e l a x a t i o n f r e q u e n c y
vs = (2nz~) - l t he f r eq ue nc y Ym= ( 2n Z m ) - 1 o f t h e m a x i -
m u m d i e l e ct r i c l o s s i s a l s o s h o w n i n F i g . 6 . T h i sf r e q u e n c y , d e f i n e d b y
d E ( v ) / d v l , , m = 0 , d 2 e ( v ) / d v 2 l v < 0 ( 1 3 )
i s p a r t i c u l a r l y u s e f u l i f r e s u lt s f o r u n s y m m e t r i c r e l a x -
a t i o n t i m e d i s t r i b u t i o n s a r e c o m p a r e d w i t h s u c h f o r
s y m m e t r ic d i s t r ib u t i o n s .
S i n ce m o l e c u l e s c a n n o t f o r m h y d r o g e n b o n d s i n
p u r e D M S O t h e a m o u n t o f t h e m o l e c u l a r d i po l e
m o m e n t m a y b e a s s u m e d t o b e i n d e p e n d e n t o f t im e .
L e t ® ( t ) d e n o t e t h e a n g l e t h r o u g h w h i c h t h e d ip o l e
c h a n g e s i t s d i re c t i o n d u r i n g t. T h e n o r m a l i z e d d i p o l e
a u t o c o r r e l a t i o n f u n c t i o n ( e q u a t i o n 7 ) c a n t h e n b e
w r i t t e n a s
~b~( t ) = ( c os O ( t ) ) (14 )
T h e d i e l e c t r i c r e l a x a t i o n t i m e (Zm ~ Z s) m a y t h u s b e
c o n s i d e r e d t h e r e o r i e n t a t io n t i m e % r. F r o m
( c o s ® ( % r ) ) = l / e f ol l o w s ® ( % r ) = 6 8 °. T h o u g h
D M S O i s a n o n - a s s o c i a t i n g l i q u i d th e r e e x is t in d i -
c a t i o n s f o r e f f e c t s o f a n t i - p a r a l l e l d i p o l e o r d e r i n g
( K a a t z e e t a l . , 1 9 8 9 b ) . A s a c o n s e q u e n c e , t h e s t a t i c
p e r m i t t i v i t y E ( 0 ) is d e t e r m i n e d b y
/tefr = g /2#. (15 )
A
x
50
4 0
30
20
10
25
20
15
10
5
~ o - ~ - o - o - ~ 6 ~ o '- c'(O)
ki k
I I I
/ ? \iI \m
, , o i i
~ 6 I I \
v
0
0.1 0.3 1 3 10 30 GHz 100
V
Fig. 6 . Co mp lex die lec t ric s pect rum of dim ethyl sulfoxide a t25°C (Kaatze et al . , 1989b).
I n e q u a t i o n ( 1 5 ) g d e n o t e s a ( s t a ti c ) c o r r e l a t i o n
p a r a m e t e r w h i c h m e a s u r e s t h e l o c al m o l e c u l a r o r d e r -
i n g o f t h e p e r m a n e n t d i p o le m o m e n t s ( K i r k w o o d ,
1 9 3 9 ) . A s i m i l a r s i t u a t i o n i s f o u n d w i t h v a r i o u s
l iq u i d s . F l u c t u a t i o n s e i th e r o f t h e a m o u n t o r o f t h e
o r i e n t a t i o n o f m o l e c u l a r d i p o l e m o m e n t s c o n t r i b u t e
p r e d o m i n a n t l y b u t n o t s o l e l y to t h e d i e l e c tr i c s p e c -
t r u m . D i e l ec t ri c s p e c t ra o f l i q u i d s c o n t a i n i n g n o n -
d i p o l a r e l e c tr o n d o n a t o r a n d a c c e p t o r m o l e c u le s h a v e
b e e n d i s cu s s e d a s to b e d u e t o f o r m a t i o n o f d i p o l a r
c h a r g e t r a n s f e r c o m p l e x e s ( G 6 t t m a n n , 1 9 7 6) . I f t h e
k i n e ti c s o f c o m p l e x a t i o n i s p r e s e n t ed b y t h e r e a c t i o n
s c h e m e
kr
a c c e p t o r + d o n a t o r ~ - - - c o m p l e x ( 16 )kb
t h e r e l a x a t i o n t i m e Zm i s g i v e n b y
1 t 1
- - - - ~ 1 7 )
T m Cohere Tor
w h e r e
I(18 )
Cchem - (1 -- x ) k f + k b
H e r e i n x d e n o t e s t h e m o l e f r a c t i o n o f c o m p l e x e s i n
t h e m i x t u r e . I n d e r i v i n g e q u a t i o n ( 1 8 ) i t h a s b e e n
t a c it ly a s s u m e d t h a t t h e r e o r i e n t a t i o n a l m o t i o n s o f
d i p o l a r c o m p l e x e s a r e c h a r a c t e r i z e d b y a d i s c r e t e
RPC 45 z~42
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554 U d o K a a t z e
Table 1. Diele ctric e laxatio n time ~m,chem ical relax ation time %h~m[equation (18)], and dip ole reorientation tim e zo, for som e liquid
electron transfer systems at 20 r'C (G6ttman n, 1976)
Tm T c h e m t'orAcceptor Dona tor ps ps ps
p-Benzoquinone p-X yle ne 4.9 6.7 19
p-Be nzo quin one Mesitylene 5.2 6.8 22D u r oqu inon e Mes i ty tene 4 .7 5 .6 22
r e o r i e n t a t i o n t i m e Z o r. T a b l e 1 p r e s e n t s d a t a f o r s o m e
c h a r g e t ra n s f e r s y s t e m s t o s h o w t h a t cm i s i n d e e d
m a i n l y g i v e n b y ~chem b u t t h a t i t i s a l s o i n f l u e n c e d b y
t h e d i p o l e o r i e n t a t i o n t i m e % r -
A n e x a m p l e o f re l a x a ti o n m e c h a n i s m i n w h i c h
b o t h c h e m i c a l k i n e ti c s a n d m o l e c u l a r d y n a m i c s c o u l d
a c t a s a n i n f l u e n c e o n t h e r e s u l t in g r e l a x a t i o n t i m e i s
t h e f o r m a t i o n o f io n c o m p l e x e s i n s o l u t i o n ( P o t te l ,
1 9 6 5, 1 9 66 ; F a l k e n h a g e n , 1 9 71 ; K a a t z e a n d G i e s e ,1 9 8 7 a ; K a a t z e e t a l . , 1 9 8 7 b ) . I n F i g . 7 t h e c o m p l e x
d i e l e c t r i c s p e c t r u m o f a 0.1 m o l a r s o l u t i o n o f
S c 2 ( S O 4 ) 3 i s d i s p l a y e d a s a n e x a m p l e . T h e d i e l e c t r i c
c o n t r i b u t i o n e ( v ) t o t h e t o t a l ( n e g a t i v e ) i m a g i n a r y
p a r t E '(o t(V ) o f t h e p e r m i t t i v i t y i s s h o w n i n t h a t
d i a g r a m . I t h a s b e e n c a l c u l a t e d u s i n g t h e r e l a t i o n
E t ' (v ) = e ' t' ot (Y) - - O (E0( .O) - I ( 19)
t o s u b t r a c t t h e c o n t r i b u t i o n d u e t o d r i f t o f i o n s. I n
e q u a t i o n ( 1 9 ) cr d e n o t e s t h e s p e c i f i c e l e c t r ic c o n d u c -
, , , , , 4 , H - - i, , - , - , 4 - t i /
°° . .
s ~fo6o
4
2
o J J p ~(, )-- - I00
~ 3
1
0 3 i
1 0 0 = t I I
10 .....
~ 3 l - ........ ( 2 n ~ ) - ' .) '::i.(2 rrr3 )-' ........1 / , I I ,':':. [ , I , 1
0 . 0 0 0 1 0 . 0 0 1 0 .0 1 0 .1 1 1 0 G H z 1 0 0
Fig . 7 . Complex d ie lec t r ic spec t r um w i thout conduc t iv i tycontr ibutions ( ful l points ; Kaatze and Giese, 1987a) and
ultrasonic excess abso rption spectrum (circles ; Bonson et al . ,1978) of a 0 .1 mo la r aqueou s so lu t ion of s candium su l fa tea t 25° C. D ashed cur ves ind ica te the subdiv i s ion of thedielectr ic spectrum into solvent water ( sw ) and solute ioncomplex ( u ) cont r ibu t ions . D ot ted cur ves a r e gr aphs ofthe D ebye- type r e laxa t ion te r ms w hich accor d ing to thechemica l equi l ib r ium be tw een d i f f e r en t ion complexconf igur a t ions ( equa t ion 20) cont r ibu te to the u l t r a sonic
absor p t ion .
t i v i ty . A l s o s h o w n i n F i g . 7 is t h e u l t r a s o n i c e x c e s s
a b s o r p t i o n s p e c t r u m o f t h e 0 .1 m o l a r S c2 (S O4 )3
s o l u t i o n . I t e x h i b i t s t h r e e D e b y e - t y p e r e l a x a t i o n r e -
g i o n s a t f r e q u e n c i e s b e l o w s o m e G H z . T h e s e u l t r a -
s o n i c re l a x a t i o n s h a v e b e e n a t t r i b u t e d ( B o n s e n e t a l . ,
1 9 7 8 ) t o t h e e q u i l i b r i a i n t h e E i g e n - T a m m m e c h a n -i s m ( E i g e n a n d T a m m , 1 9 62 ) o f st e p w is e d i s s o c i at i o n
o f s a lt s w h i c h m a y b e g e n e r a ll y r e p r e s e n t e d b y t h e
r e l a t i o n
kfl[Mm + ]aq q- [L~- ] aq- ~ [ Mm +( H 2 0) 2 L=- ]aq
kbl
k 2 kox . [ M m + ( H 2 0 ) t ' - ] a q . - [ M m + L ~ - ].q . ( 20)
k b k b 3
M m + a n d L I - a r c s h o r t h a n d n o t a t i o ns f o r t h c m + f ol d
c h a r g e d r e c ta l i o n a n d t h c I - f o l d c h a r g c d l i g an d ,
r e s p e c t i v e l y .
T h e d i e le c tr i c s p e c t r u m o f t h e S c 2 ( S O 4 ) 3 s o l u t i o n
r c v c a l s t w o r c l a x a t i on re g i o n s o n l y o n c o f w h i c h i s
c l e a rl y d u c t o t hc s o l vc n t w a t c r ( s w ) . H c n c c t h c
f o r m a t i o n a n d d c c a y o f i o n c o m p l c x c s a s w e l l a s t h ei r
r e o r i c n t at i o n a l m o t i o n s o b v i o u s l y r c s u l t i n a si n g l c
r e l a x a t i on o n l y . T h c r c a r c t w o r c a s o n s f o r t h i s ( o n a
f ir st l a n c e , s u r p r i s i n g ) r e s ul t . T h c r c o r i c n t a t i o n t i m e s
o f t h e d i p o l a r s p e c i c s i n t hc e q u i l i b r iu m ( 2 0 ) a p p e a r
t o b e m u c h s m a l l c r t h a n t h c r cl c v an t c h e m i c a l r c l a x -
a t i o n t i m es s o t h a t Z ~ > > Z ~ m t h r o u g h o u t . I n a d -
d i t i o n , t h c r c o r i c n t a t i o n t i m c s o f t h e d i f f c r e n t d i p o l a r
i o n s p e c i e s n e a rl y a g r e e w i t h a n o t h e r ( K a a t z c a n d
G i e s e , 1 9 8 7 a ) s o t h a t t h c d i f f c r cn t r c l a x a t i o n r e g i o n s
a s d u c t o f l u c t u a t io n s i n i o n c o m p l e x o r i e n t a ti o n
s t r o n g l y o v c r l a p .
A n o t h e r p h c n o m c n o n i n w h i c h d i e l e c t r i c r c l a x -
a t i o n m a y r c f lc c t r c o r i c n t at i o n a l a n d t r a n s l at i o n a l
m o t i o n s a s w c l l i s t h e s o l v a t i on b y d i p o l a r s o l v c nt s .
A p r o m i n c n t e x a m p l e i s t h e h y d r a t i o n ( h ) o f s m a l l
i o n s i n w h i c h d i e l ec t r i c s a t u r a t i o n c f f c c ts ( S e c t i o n 6 )
r c s u l t i n a n o r i e n t a t i o n c o r r e l a t i o n f a c t o r g h s m a l l e r
t h a n t h a t o f t h e b u l k s o l v c n t w a t c r g . F l uc t u a t i o n s i n
t h e d i e le c tr i c p o l a r i z at i o n o f t h c s o l u t i o n s m a y t h u s
n o t o n l y b c c a u s c d b y c h a n g e s i n t h e o r i e n t a ti o n o f
t h c d i p o l a r s ol v c n t m o l c c u l e s. S i n c e a n c x c h a n g c o f
a w a t e r m o l e c u l e b e t w e e n t h e h y d r a t i o n r c g i o n a n d
t h e b u l k p h a s e i s a c c o m p a n i c d b y a c h a n g e
( g l / 2 _ g ~ / 2 ) ~ i n t h e e f f e c t iv e d i p o l e m o m e n t , t r a n s l a .
t i o n a l m o t i o n s w i ll a l so c o n t r i b u t e t o p o l a r i z a t i o n
f l u c t u a t i o n s ( G i e s e e t a l . , 1970; Giese, 1972) . In
g e n e r a l t h e r e s u l t i n g d i e l e c t r ic r e l a x a t i o n t i m e Lm
d e p e n d s o n t h e r e o r i e n t a t i o n t i m e s T ho r a n d Vsw a n d
m e a n r e s id e n c e t i m e s , g a n d * ** o f w a t e r m o l e c u l e s
i n t h e h y d r a t i o n r e g i o n a n d b u l k p h a s e , r e sp e c t iv e l y ,
a n d a l so o n t h e o r i e n t a t i o n c o r r e l a t i o n f a c t o r gh a n d
n u m b e r Z h p e r i o n o f w a t e r m o l e c u l e s w i t h gh 4= g
( F ig . 8 ) .
H y d r a t i o n w a t e r e x c h a n g e i s c e r t a in l y o n e r e a s o n
f o r t h e r e m a r k a b l e f i n d i n g t h a t t h e d i e l e c t r i c r e l a x -
a t i o n t i m e o f w a t e r a r o u n d s m a l l io n s m a y b e s h if t ed
t o w a r d s m a l l e r v a l u e s w i th r e s p e c t to p u r e w a t e r
( T a b l e 2 ). A n e n h a n c e m e n t o f t h e r e o r ie n t a t i o n t i m e
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Microwave dielectric properties of liquids 555
of hyd ration molecules in strong Coulombic fields is
to be expected and has indeed been found by
measurements of nuclear magnetic longitudinal relax-
ation rates (1/TI) of electrolyte solutions (Hertz,
1973). The reorientation time for the proton-proton
vector of Li + hydra tion water, for example, has avalue of about 20 ps (Endom e t a l . , 1967)while the
residence time in the hydration region amounts to
about 40 ps (Hertz a nd Zeidler, 1963).
In Table 2 the shift in the dielectric relaxation time
of water around ionic solutes is expressed by individ-
ual B~ coefficients which have been calculated
according to the additivity rule
m + m -- - B ~ - + - - B 2 = B e 2 1 )m m
assuming Ba = --0.01 (mol/kg) -1. In equat ion (21)
m, m +, and m - denote the molal concentration ofsalt, cations and anions, respectively, and
1 /' dZm xBd = n l i ra / -/ . (22)
Cw m ~ 0 ~ , dm J
Subscript w is used to identify paramete rs of
non-affected water. The negative B2 values of small
bi- and trivalent cations, however cannot be due to
hydrat ion water exchange since the residence time of
water molecules in the dielectrically saturated hy-
drat ion region (g~ <g) is too long (e.g. ~* >/1/~s
aro und AP+; Hertz, 1973) to cont ribute to the dielec-
tric spectrum in the microwave region. Effects ofnegative hydration a nd of kinetic depolarization to be
discussed below may also reduce the dielectric relax-
ation time of water around ionic solutes.
4 . A S S O C I A T I N G L I Q U I D S . W T E R
Dielectric relaxation of associating liquids could
reflect changes in p resulting from the continuous
rupture a nd reformation of hydrogen bonds. Due to
recent advances, particularly in computer simulation
studies, a deeper insight in the microdynamics of
water is currently achieved. Much interest is thusdirected toward this unique liquid in the following.
Rather surprisingly, the microwave dielectric spec-
trum of water (v ~< 100 GHz, Fig. 1) at temperatures
between 0 and 60°C can be adequately represented by
a discrete relaxation term (Kaatze, 1989a). Small but
systematic deviations from the single Debye term
spectrum are found if near millimetre wavelength
data (100 GHz ~< v -%<410 GHz; Blue, 1980; Hasted e t
a l . , 1987) are also taken into account. The David-
son-Cole function (equation 12) and the double
Debye term relaxation spectrum
+ , ~ j AEw, (23)e ( v ) = ew (o o ) = 1 + iOOZw,,
appear to be more suited to describe the measured
water relaxation spectra in the extended frequency
range. Nevertheless deviations from a single Debye-
type relaxation are small (flw ~ 0.04 if equation 12 is
applied, Aewl/Aew2~ 0.02, %1 ~ 1 ps; Barthel e t a l . ,
1991; Kaatze , 1993). Since we are ma inly interested in
the microwave behaviour here it is sufficient to as-sume a discrete pure water relaxation time %(=%2,
Aewl = 0 in equati on 23). This relaxation time de-
creases from 17.7 to 4.0 ps if temperature is raised
from 0 to 60°C (Kaatze, 1989a).
The following picture on the interrela tion of struc-
tural a n d dynamic properties and the kinetics of
H-bonds develops from computer simulations of
water (Geiger e t a l . , 1986; Tan aka and Ohmine, 1987;
Ohmine e t a l . , 1988; Bertolini e t a l . , 1989; Sciortino
and Fornili, 1989; Sciortino e t a l . , 1990; Sciortino e t
a l . , 1991; Sciortino e t a l . , 1992). In liquid water
molecules are almost totally connected forming arandom H-bonded network well above the percola-
tion threshold. The strength of each bond fluctuates
rapidly within intervals between 0.1 and 1 ps. Rup-
ture of bonds, however, results in significant reorien-
tational motions of a molecule only if two
preconditions are simultaneously fulfilled. The mol-
ecule has, of course, to be unbonded or weakly
T a b l e 2 . C a t i o n i c a n d a n i o n i c r e l a t i v e m o l a l s h i f t s B ~ o f d i e l e c t r i c
r e l a x a t i o n t i m e s a t 2 5 C . T h e d a t a h a v e b e e n c a l c u l a t e d u s i n g
e q u a t i o n ( 2 1 ) a n d d a t a f o r a q u e o u s s o l u t i o n s w i t h o u t i n d i c a t i o n s
f o r n o t i c e a b l e e f f e c t s o f i o n c o m p l e x f o r m a t i o n ( G i e s e , 1 9 7 0;W e n a n d K a a t z e , 1 97 7; K a a t z e , 1 9 8 3 ). R a d i i r e f o r s m a l l i o n s
a c c o r d i n g t o C o n w a y ( 1 9 81 ) . R a d i i f o r la r g e i o n s e s t i m a t e d f r o m
t h e i o n i c a p p a r e n t m o l a r v o l u m e s ( K a a t z e , 1 9 83 ). X : X N ÷ ,
a z o n i a s p i r o a l k a n e s
r + BJ- r + BJ-
C a t i o n / ~ ( m o l / k g ) - t C a t i o n , ~ ( m o l / k g ) - i
L i + 0 . 6 0 - 0 . 0 4 M e 4 N + 3 . 22 0 . 1 7
N a + 0 . 9 5 - - 0 . 0 7 E t 4 N + 3 . 8 5 0 . 3 9
K + 1 . 33 - 0 . 0 7 P r 4 N - - 4 . 3 6 0 . 7 3
R b + 1 . 48 - - 0 . 0 7 B u 4 N + 4 . 7 5 0 . 8 8
N H ~ - 1 . 48 - 0 . 0 4 4 : 4 N + 3 . 6 5 0 . 2 9
C s + 1 . 69 - 0 . 0 5 5 : 5 N + 3 . 9 0 0 . 3 7
6 : 6 N + 4 . 1 3 0 . 4 3
r + B + r Bd-
C a t i o n /~. ( m o l / k g ) i A n i o n ~ ( m o l / k g ) i
B e 2 + 0 . 3 1 - - 0 . 0 8 F - 1 . 36 0 . 0 5
M g 2 + 0 . 65 - - 0 . 1 4 C I - 1 .8 1 - 0 . 0 1
C a 2 + 0 . 9 9 - -0 . 1 6 B r 1 .9 5 - 0 . 0 3
S r 2 + 1 . 13 - - 0 . 1 3 J 2 . 1 6 - 0 . 0 5
B a 2 + 1 . 35 - - 0 . 1 7 B F 4 2 . 8 - 0 . 1 3
A I 3 + 0 . 5 0 - - 0 . 0 7 N O { 2 . 9 - 0 . 0 5
y 3 + 0 . 9 3 - - 0 . 0 7 B ( P h ) 4 4 . 8 0 . 4 4
L a 3 + 1 .1 5 - - 0 . 1 0 C O ~ 2 . 5 0 . 1 8
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556 Udo Kaatze
bonded and, in addition, an extra site for a new
H-bond has to be offered by an appropria tely located
and orientated water molecule ( fifth neighbour ). If
these preconditions exist the reorientation of a water
molecule occurs within about 0.1 ps ( H-bond
switching ). Hence the dielectric relaxation time %(= 10ps at room temperature) of liquid water is
mainly given by the time passing until favourable
conditions to switch H-bonds follow from thermal
fluctuations.
Computer simulation studies furthermore showed
that an almost perfect tetrahedral structure exists at
reduced water density only (84% four coordinated
molecules at p = 0.75 g/cm 3 and T = 0°C; Sciortino e t
a l . , 1992). With increasing p more and more network
defects appear so that one third o f all water molecules
is five coordinated at p = 1 g/cm 3 and there is an
additional 12% with even six neighbours. The exist-ence of defects leads to bi furcated hydrogen bonds
and to a smearing of the binding energy. Hence
within the framework of these computer simulation
results the comparatively high mobility of water
molecules reflects the catalytic activity of network
defects. The dielectric relaxat ion time of aqueous
systems, and probably more generally of associating
liquids, should therefore significantly depend on the
density of relevant defects. There are in fact various
findings which may be taken to verify, at least
qualitatively, this statement. Some indica tions will be
presented below.First of all it is worthy to notice that the reduced
relaxati on time of water under high hydrostatic press-
ure (Pottel e t a l . , 1989) corresponds with the result
from compu ter simulation studies at reduced density.
As a result of high pressure the density of water
increases and, as a consequence, the density of appro-
5
s
2
0.5
0.2
0.1
0.05
0.02
0.01
0 0 0 5
5
I I I I
1 9
~,87 ~ H CH2)mOH
I I I I7 10 20 molll 50
C O H
70
Fig. 9. Relaxation time rm (equation 13) plotted versus theconcentration Coil of hydrogen bonding groups for theprimary alcohols (closed symbols; Gestblom and Sjrblom,
1984a, b) and water (circle; Kaatze, 1989a) at 20°C.
500
200
100
50
20~Y
1
5
2
1
I I I I I I I
I I I I t I I
1.5 2 3 ¢ 5 7 10
i~ 1
15
Fig. 10. Bilogarithmicplot of the relaxation time ratio cm/'Cversus the inverse H-bonding group density ~ -t (equation24) for monohydric alcohol water mixtures (open symbols)
at 25°C and for the primary alcohols (full points, Fig. 9) at20°C. El, isopropanol (Kaatze et al. , 1989c); A, primarybutanol (Mashimo and Kuwabara, 1989); V, tert.-butanol
(Kaatze, 1991a).
priate defects in the H-bond network. The variation
of the dielectric relaxation time within the homolo-
gous series of monohydric alcohols (Fig. 9) offers
another obvious confirmation of the above ideas on
the mechanism of dielectric relaxation in H-bonded
liquids. The relaxation time decreases drastically as
the concentration of hydrogen bonding groups isincreased, thus indicating tha t 17m i> 17w is predomi-
nant ly due to the smaller probabi lity for the existence
of network defects with the alcohols. Based on the
relaxation behaviour of mixtures much evidence for
the essential role of defects has been obtained recently
(Pottel, 1992; Kaatze and Pottel, 1992; Kaatze e t a l . ,
1994). In Fig. 10, as an example, relaxation times of
alcohol/water mixtures are compared with those of
the primary alcohols. For this purpose ratios Z m / Z w
are displayed as a function of ~ -t defined by
- ' = C w ( Cs w + cZ ¢ ) I . (24)Herein, Cw = Csw(C = 0) and Csw are the molari ty of
(pure) water and of the water in the mixtures, respect-
ively. Z~ denotes the number of H-bonding groups
per organic molecule and c the concentration of the
molecules which are capable of forming hydrogen
bonds. Again, with increasing t3 the dielectric relax-
ation time zm decreases. Part icularly striking, the
behaviour of the alcohol/water mixtures closely cor-
responds to that of the series of pure primary alco-
hols. Since, however C m / C w V S / ~ - l relations with
substantially different slope in the double logarithmic
plot have been found with other mixtures (Pottel,1992; Kaatze and Pottel, 1992) this correspondence
should not be overestimated. Lacking a well-founded
theoretical approach relating the kinetics of fluctuat-
ing H-bonded networks to the dielectric relaxati on of
associating liquids, parameter/5 is just an empirically
introduced quantity. It is indeed suited to indicate the
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Micr ow ave d ie lec t ric p r oper t ie s o f l iqu ids 557
/2 -
J
5
I I I/ O OO
t
2 2.5 3
Fig. 11. Relaxation t ime rat io [m/'l~was a functio n of #7 - j
( equa t ion 24) f or aqueous so lu t ions of qu inoxa l ine ( 25° C,A ; 3 5 ° C , O ; K a a t z e et al. , 1988).
t r e n d s i n t h e r e l a x a t i o n t i m e s b u t t h e r e a r e , o f c o u r s e ,
f a c to r s o t h e r t h a n t h e d e n s it y o f H - b o n d i n g g r o u p s
w h i c h a c t a s a n o t i c e a b l e i n f l u e n c e o n t h e r e l a x a t i o n
t i m e . W i t h m i x t u r e s o f w a t e r a n d q u i n o x a l i n e , f o r
e x a m p l e , t h e Z m / % r a ti o w h e n p l o t t e d v s / 3 -~ e x h i b i ts
a r e la t iv e m a x i m u m a t n e a rl y t h e e q u i m o l a r c o m p o -
s i t i o n ( F i g . 1 1 ) . T h i s b e h a v i o u r m a y b e t a k e n t o
r e f le c t t h a t ( p a r t i c u l a r l y a t l o w w a t e r c o n t e n t ) t h e
w a t e r h y d r o g e n s i n t e r a c t a l s o w i t h t h e d e l o c a l i z e d
e l e c t r o n s o f t h e q u i n o x a l i n e r in g s .
F o r r e a s o n s o f s i m p l i c it y t h e g l o b a l l y d e f i n e d r e l a x -
a t i o n t i m e Z m h a s b e e n u s e d i n t h e a b o v e d i s c u s s i o n .
I t is o n l y b r i e f l y m e n t i o n e d h e r e t h a t d i e l e c t r i c s p e c t r a
o f o r g a n i c so l v e n t s ar e n o r m a l l y m o r e c o m p l i c a t e d
t h a n t h a t o f p u r e w a t e r ( C r o s s l e y , 1 9 7 0 ). R e c e n t
m e a s u r e m e n t s b e t w e e n 1 M H z a n d a b o u t 7 0 G H z o f
s o m e a l c o h o l s s h o w t h a t t h e i r s p e c t r a c a n b e w e l l
r e p r e s e n t e d b y a s u m o f a D e b y e f u n c t i o n a n d a
D a v i d s o n - C o l e t e r m
£ a AEb
e ( v ) = e ( ~ ) = 1 + i o g z ~ F (1 -~-iog'Cb) {l-/~b) ( 25)
w i t h p a r a m e t e r v a l u e s g i v e n i n T a b l e 3 ( K a a t z e e t a l . ,
1989c , 1991a) .
A n i n t e r e s t i n g q u e s t i o n y e t t o b e a n s w e r e d i s t h e
a p p l i c a b i l i t y o f r e c e n t t h e o r e t i c a l a p p r o a c h e s ( e . g.
D i s s a d o a n d A l i s o n , 1 99 3) i n s te a d o f s e m i - e m p i r ic a l
r e l a x a t i o n f u n c t i o n s t o t h e m e a s u r e d d i e l e c t r i c
s p e c t r a .
x
A-x
¼
q
14
12
1 0
8
6
/
2 -
I05 h
4
3
2
1
00 . 0 0 0 1 0 . 0 0 1
I I
[
0 . 0 1 0 . 1 1 1 0 S H z 1 0 0
Fig . 12 . Complex d ie lec t r ic spec t r um of a mor phol ine /n-butanol mix tur e a t 30° C ( mole f r ac t ion o f mor p hol inex = 0.25; Kaatze et al. , 1991c) . The drawn curves representthe D avidson- Cole r e laxa t ion spec t r a l f unc t ion ( equa t ion12) with the fol lowing values for the parameters :
with respect to the dielectric properties are normally
found with completely miscible liquids. Other binary
mixtures of dipolar constituents may exhibit two
relaxation regions (Fig. 13) of which one can be
attributed to the solvent [Rs(v)], the other one to thesolute [Ru(v)]. Hence
E ( v ) = E ( o e ) + R ~ ( v ) + R u ( v ) (26)
where Rs(v) and R u ( v ) may be given by a Debye term,
respectively, or by terms reflecting a distribution of
relaxation times. Besides the unsymmetric David-
son-Cole distribution the symmetric Co le ~o le dis-
tribution is frequently applied. It corresponds with
the relaxation term (Cole and Cole, 1941)
A£~iu
R ~ , , ( v ) 1 + (icons,u) l- ~.~) (27)
Also used is an empirical relation which includes the
Davids on-Cole and the Cole-Cole behaviour as lim-
iting forms (Havriliak and Negami, 1966)
AE~,u
R s ' u ( V ) = [ 1 - ~ - ( i ( D ' ~ s ,u ) < 1 ~ s , u ) ] ( l -[ I s , u ) (28)
A clear subdivision of the spectra into cont ri-
butions R ~ ( v ) and R , ( v ) allow the effect of the solute
on the solvent relaxation and, vice versa, of the
solvent on the relaxation of the solute to be studiedseparately. Solvation phenomena have been exten-
sively studied in the past and are still of considerable
interest. Some illustra ting examples will be presented
in the following sections. The interrelation between
50c O)
~0
3 0 e s ( O )
2 0
10
0
15
.-~-- 1 0 I• 0 5
0
0 . 0 0 1
I I
c ( - )
I I I
I I I ~ I
_ s
~ _ . . , , 1~ ..'<.R ~ (v)
0 . 0 1 0 . 1 1 f i H z 1 0
V
Fig. 13. Complex dielectric spectrum of a 0.1 molarsolution of 1,2-dihexadecylglycero-L-3-phosphatidyl-N,N,N-trimethyl-N-hexanolamine (C~6-ether-PN6-1ecithin) inmethanol at 25°C (Kaatze et al. , 1985a). Dashed curvesindicate the subdivision of the spectrum into a solvent (% )
and solute ( u ) relaxation.
t~0
- - T o o o o o o o o o o o o o o
A C s l ° ' o \
1 0 - , A e f a
0 I I 1 I
1 0 i i l G o °oOO %
5
~ o I 1 I 1
0 _ ~ . ~ , . , ., ln n ~ o o o J I I I I0 . 0 0 1 0 . 0 1 0 . 1 1 1 0 G H z 1 0 0
V
Fig. 14. Complex dielectric spectrum of a 6.4 molar aqueoussolution of butyric acid at 25°C (Kaatze et al. , 1991d). Suffix
sl marks a slow relaxation, suffix fa a faster one.
solvent properties and the solute dielectric relaxation
becomes obvious if, for example, in solutions of
lecithins (Fig. 13) methanol is exchanged for water.
The dielectric relaxation time % of the zwitterionic
phospholipid head groups increases substantiallythereby thus reflecting the formation of extended
bilayer structures with restricted head group mob ility
instead of small molecular clusters (Kaatze e t a l . ,
1985a; Kaatze and Pottel, 1985b).
A special behaviour has been found with aqueous
solutions of carboxylic acids (Fig. 14). Again the
dielectric spectrum appears to be composed of two
contributions with relaxation characteristics. These
contributions, however, cannot be attributed to the
motions of carboxylic acid and water molecules,
respectively. Instead, evidence has been obtained
from the dielectric spectra of mixtures of acetic acid,propioni c acid and butyric acid with water that
the slow relaxation process ( sl ) is due to a mi-
crophase of low water content and that the faster
process ( fa ) reflects a subphase of higher water
content.
The system butyric acid/water exhibits an upper
critical demixing point at an acid concentration of
corit = 4.33 mol/1 and a temperature of Tcrit -- --3.0°C.
Obviously, in this system a microheterogeneous
structure exists al ready far away from Tcrit. Even
more surprising, indica tions for precritical behaviour
are also found with the lower homologues propionicacid and acetic acid that are completely miscible with
water. These indications from the dielectric spectra
are confirmed by results from broad-band ultrasonic
absorption measurements which are currently per-
formed on carboxylic acid/water systems (Kfihnel
e t a l . , 1995).
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M icrowa ve dielectr ic proper t ies of l iquids 559
6 . S O L V E N T C O N T R I B U T I O N T O T H E S T A T IC
P E R M I TT I V IT Y . O R I E N T A T I O N C O R R E L A T I O N ,
K I N E T IC D E P O L A R I Z A T I O N , D I E L E C T R I C
S A T U R A T I O N
I n b i n a r y s o l u ti o n s t h e e x t r a p o l a t e d s t a t i c
p e r m i t t i v i t y
e~(0) = l im R ~ ( v ) + e ( ~ ) ( 2 9)v ~ 0
o f t h e s o l v e n t c o n t r i b u t i o n t o t h e d i e le c t ri c s p e c t r u m
( F i g . 1 3 ) o f f e rs v a l u a b l e i n f o r m a t i o n o n s t r u c t u r a l
p r o p e r t i e s . F i r s t o f a l l , h o w e v e r , e ~ ( 0 ) r e f le c t s t h e
t r i v i a l e f f e ct o f d i l u t i o n o f a d i p o l a r s o l v e n t , n a m e l y
t h e r e d u c t i o n o f t h e n u m b e r N~, ( e q u a t i o n 8 ) o f
s o l v e n t d i p o l e m o m e n t s p e r v o l u m e b y t h e p r e se n c e
o f s o l u t e . V i a i n t e r n a l e l e c t r i c f i e l d s e ~ ( 0 ) o f h o m o -
g e n e o u s s o l u t i o n s d e p e n d s o n t h e s h a p e o f t h e s o l u te
p a r t i c l e s [ a n d e~ ( 0) o f d i e l e c t r i c a l l y h e t e r o g e n o u s m i x -
t u r e s d e p e n d s o n t h e s h a p e o f t h e s u b p h a se ] . F o r t h i s
r e a s o n m e a s u r e m e n t s o f th e s t a ti c p e r m i t t i v i ty a r e
u s e d a s a t o o l t o i n v e s t i g a t e s t r u c t u r a l a s p e c t s o f
m i c r o e m u l s i o n s ( S j r b l o m e t a l . , 1 9 9 1 ; S a e t e n e t a l . ,
1 99 2) A l s o f o r t h i s r e a s o n E ~(0 ) v a l u e s o f s u s p e n s i o n s
w i t h r e l e v a n c e in b i o l o g y a n d m e d i c i n e f r e q u e n t l y a r e
u n e x p e c t e d l y s m a l l. W a t e r t r a p p e d i n ce l ls ( L r n -
n e c k e - G a b e l , 1 9 90 ) a s in t h e c o r e o f p h o s p h o l i p i d
8
b i l a y e r v e s i c l e s ( P o t t e l e t a l . , 1 9 8 4 ; K a a t z e e t a l . ,
1 9 8 4 a) i s e x p o s e d t o s t r o n g d e p o l a r i z i n g f i el d s. H e n c e
t h i s w a t e r a d d s a r e d u c e d c o n t r i b u t i o n t o t h e d i e l e c -
t r i c s p e c t r u m e v e n i f i t s m o l e c u l a r p r o p e r t i e s a r e t h e
s a m e a s in t h e p u r e s o l v e n t . S o m e t i m e s t h i s e ff e c t i s
m i s i n t e r p r e t e d b y a s s u m i n g t h e d i e l ec t ri c d e c r e m e n to f h e t e r o g e n e o u s m i x t u r e s t o b e s ol e ly d u e t o i r r o t a -
t i o n a l l y b o u n d w a t e r .
U n f o r t u n a t e l y , e v e n f o r a h o m o g e n e o u s s o l u ti o n o f
e x a c t l y s p h e r i c a l l y s h a p e d s o l u t e p a r t i c l e s t h e e f fe c t o f
i n t e r n a l f i el d s c a n n o t b e r i g o r o u s l y t a k e n i n t o a c -
c o u n t . D i f f e r e n t a p p r o a c h e s t o a p p r o x i m a t e l y t r e a t
t h e s e f i e l d s l e a d t o a v a r i e t y o f m i x t u r e f o r m u l a e
r e l a t i n g t h e r e s u l t i n g p e r m i t t i v i t y Es 0 ) o f a c o m p o s i t e
d i e l e c t r i c t o t h e p e r m i t t i v i t i e s E a n d e2 o f t h e c o n s t i t u -
e n ts . T h e g r a p h s o f t w o c o m m o n l y u s e d m i x t u r e
r e l a t i o n s a r e d i s p l a y e d i n F i g . 1 5 t o i l l u s t r a t e t h e
d i s c r e p a n c y i n t h e p r e d i c t i o n s o f d i f f e r e n t f o r m u l a e .R e p r e s e n t e d f o r a q u e o u s s o l u t i o n s a t 2 5 ° C
[E l = E , ( 0 ) = 7 8 . 3 6 , e2 = 2 ] a r e t h e B r u g g e m a n m i x t u r e
r e l a t i o n ( B r u g g e m a n , 1 9 35 ) g i v e n b y
V l 3 = 1 v= (30 )EI--E~ LEAO
a n d a f o r m u l a o r i g i n a l l y d e r i v e d b y M a x w e l l ( 1 8 9 2 )
a n d W a g n e r ( 1 9 14 ) ( P o l d e r a n d v a n S a n t e n , 1 9 46 ;
B r o w n , 1 9 5 6 ; v a n B e c k , 1 9 6 7 : B r t t c h e r a n d B o r -
d e w i j k , 1 9 7 8 )
75
70
- 6 st
60
55
500 0 3
O~AO
\ t
0 1 0 2
V
Fig. 15. Solvent con tr ibutio n E~(0) (=Esw ) to the s tat ic
per mi t t iv i ty a t 25° C d isp layed as a f unc t ion of vo lumef r ac t ion v2 of so lu te f or aqueo us so lu t ions of low w eightnon -dipo lar molecu les [Es(0 = E (0) , ful l points ; P ottel andK aa tze , 1969; K aa tze an d W en, 1978a ; K aa tze e t a l . , 1988],of synthet ic polymers (circles ; Kaatze, 1975; Kaatze e t a l . ,
1978b), and of large ions (tr iangles ; Wen and Kaatz e, 1977~K aa tze , 1980b) . The f u l l and d ashed cu r ve a r e gr aphs o f themix ture relat ions def ined by equ ation s (30) and (31) , re-
spectively.
3v2 el I, 2 - ¢1 )
es(0 ) = El - 2EI ~- ¢2 -- V2(E2 -- El) (31 )
I n t h e s e e q u a t i o n s , Vz d e n o t e s t h e v o l u m e f r a c t i o n o f
c o n s t i t u e n t 2 , t h e s p h e r i c a l l y s h a p e d s o l u t e .
A l s o d i s p l ay e d i n t h e d i a g r a m a r e d a t a f o r a q u e o u s
s o l u t io n s o f o r g a n i c s o l u t e s f o r c o m p a r i s o n . M o s t
i n t e r e s t i n g l y , t h e s c a t t e r i n t h e e x t r a p o l a t e d s t a t i c
p e r m i t t i v i t y d a t a i s s m a l l i n d i c a t in g t h a t t h e o r i e n t a -
t i o n a l p o l a r i z a b i li t y o f w a t e r d e p e n d s w e a k l y o n l y o n
s p e c i a l i n t e r a c t i o n s w i t h t h e s o l u t e . T h e t e n d e n c y i n
t h e e x p e r i m e n t a l d a t a t o s l i g h t ly e x c e e d t h e p r e d ic -
t i o n s b y t h e m i x t u r e r e l a t i o n s s e e m s t o b e c h a r a c t e r -
i s ti c t o h y d r o p h o b i c h y d r a t i o n e f fe c t s ( S e c t i o n 7 ;
K a a t z e a n d P o t t e l , 1 99 2). I n t h e c a s e o f d i l u t e
s o l u t io n s i t is a n o b v i o u s a t t e m p t t o a s s u m e t h e
d i s t u r b i n g a c t i o n o f s o l u t e p a r t i c le s t o b e r e s t r i c te d t o
a l i m i t e d n u m b e r Z so lv o f s o l v e n t m o l e c u l e s a r o u n d
t h e s o l u t e a n d t o t h u s t r e a t t h e s p e c t r a i n t h e f r a m e -
w o r k o f a s o l v a t i o n m o d e l ( K a a t z e a n d P o t t e l,
1 9 8 5 b ) . A p p l y i n g s u c h m o d e l c h a n g e s i n t h e e f f e c t i v e
d i p o l e m o m e n t a r e a tt r i b u t e d t o a l t e ra t i o n s i n th e
d i p o l e o r i e n t a t i o n c o r r e l a t i o n f a c t o r ( g s o ~ v =~ g ) o f t h e
Z so l~ s o l v a t i o n m o l e c u l e s . F o r a q u e o u s s o l u t i o n s
gsolv = gh , Zs o l v = Zh ( F i g . 8 ) . I f r u l e o f t h r e e i s s i m p l y
u s e d i n s te a d o f t h e e x a c t t r e a t m e n t o f t h e p r o b l e m
( w h i c h d u e t o t h e i n t e r n a l d e p o l a r i z i n g f i e ld s i s
s o m e w h a t m o r e c o m p l i c a t e d ) r e l a t i o n
gs°i---Z= 1 + c ~ e . . . . . (0) -- Es,calc 0) (32 )
g c2Z~o lv Es ,ca lc (O - - e ( o 0 )
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560 Udo Kaatze
1.t~
1.3
1 2
1 1
1 0
0 9
I I
t . J A
~ ~
I
I
f
I
I
-r
0 8
0 7
I0 /*0
Z
\ : z : :
\
\
\
\
\
\
I I
8 1 2
~ 1
I
\ S
P
160 2 0 0
Fig. 16. Ratio gsoi~/g of solvation shell to solvent dipoleorientation correlation factor (equation 32) versus staticpermittivity E~ of solvent for solutions of TED in various
liquids at 25°C (Kaatze e t a l . , 1984b).
follows with E . . . . . ( 0 ) denoting the values extrapo-
lated from the measured spectra and E s , c a l c ( 0 those
predicted by a mixture relation (e.g. equat ion 30). In
equation (32), cl and c2 are the solvent and soluteconcent rations, respectively, and g is the dipole orien-
tation correlation factor of the undisturbed solvent.
For simplicity esolv(~) = E ( ~ ) has been used in deriv-
ing equation (32).
In Fig. 16, g s o l v / g ratios are shown for solutions of
in some dipolar solvents. Estimat ing these data it has
been assumed that the number of solvation molecules
nearly agrees with the numb er of neighbouring mol-
ecules. Hence the absolute values of the orientat ion
correla tion factor ratios should not be overestimated.Some interesting trends, however, are found in the
data.
Dimethyl sulfoxide (DMSO) and N,N-dimethyl-
formamide (DMF) are aprotic liquids and are thus
unable to form hydrogen bonds in the pure liquid.
Since TED offers also hydrogen bond accepting
abilities only g s o l v / g > 1 for DMSO and DM F
mixtures seems to most unambiguou sly reflect the
induction of enhanced solvent dipole orientation
correlation around non-dipolar solutes. It has been
briefly mentioned above that there are indi cations for
some antiparallel dipole alignment in pure DMSO(Kaatze e t a L , 1989b). Possibly the molecular order
resulting thereby is reduced by the presence of the
solute. It seems to be likely that a (rapidly fluctuating)
clathrate-like solvent structure is formed around
TED. Such structures are suggested to exist in
aqueous solutions of partly hydrophobic solutes.
TED is well-known to be especially effective in pro-
moting the structure of the hydration water around
it (Kaatze and Wen, 1978a). A similar behaviour with
a somewhat smaller g s o l v / g ratio, however, is found
with solutions of TED in formamide. Formamide
resembles water by its variety of H-bon ding capabili-ties. In contrast, methanol and N-methylformamide
offer just one H-bond donating site. These solvents
are therefore assumed to form chainlike associates
with preferentially ~r al le l orientation of neighbour-
ing dipole moments within the chains. Obviously,
addition of TED and other organic molecules
(Kaatze e t a l . , 1984b) leads to disrupture of highly
ordered chains and to a formation of structure with
a lower degree of orientation correlation (gso~v<g ;
Fig. 16).
Aroun d small inorganic ions the dipole orientation
correl ation factor g~o~v may be reduced by the orien-tation o f molecules in strong Coulombic fields. This
effect is called dielectric saturat ion or structure
saturation . Normally the extent of saturation is
expressed by a numbe r Z +- of apparently irrotation -
ally bound solvent molecules per cation or anion,
respectively. Besides dielectric saturation, however,
the extrapolated static permittivi ty of electrolyte
solutions reflects also the kinetic depolarizati on
mechanism (Hubbard e t a l . , 1977a; Hubbard and
Onsager, 1977b; Hubbar d, 1978; Hubbard e t a l . ,
1979). An ion which moves through a dipolar liquid
in an external electric field sets up a non-uniformhydrodynamic flow. The solvent molecular dipoles
are turned thereby in the direction opposed to that in
which they are orienta ted due to the external field. In
the Onsager-Hubbard continuum theory the kinetic
dielectric decrement is given by the re lation (Hubbard
e t a l . , 1979).
2 E ( 0 ) - q ( ~ )(~EH O = O 171 (33)
3 e0e I (0)
if perfect slip bound ary conditions on solvent flow are
assumed at the ion surfaces. Again, suffix l is used
to indicate parameters of the solvent. Hence 6eHo isexpected to predominan tly depend on the specific
electric conductivity of the solution and on the
dielectric relaxation time z~ of the solvent. Molecular
theories predict the kinetic polarization deficiency to
also increase with decreasing ion radius (Hubbard e t
a l . , 1979; Hub bard and Kayser, 1981; Wolynes, 1980;
Kusalik and Patey, 1983). Since dielectric satura tion
is expected to depend on the ion size in a similar
manner it is impossible to rigorously with respect to
es.ca~c(0) [calculated from equation (30) or equation
(31) wi thou t considering ionic field effects] divide
measured total dielectric decrements
6 E = E . . . . . (0) -- e~.cal~ 0) (34)
into contri buti ons from the different molecular mech-
anisms. Nevertheless, as shown by Fig. 17, the 5e
values derived from measured spectra may substan-
tially exceed the predictions due to either theoretical
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M icro w av e d i e l ec tr i c p ro p er t i e s o f l i q u id s 5 61
-t,
- 8
-12 -
- 1 6
..% . ~ U~'HO• x ~' - . .~
.. . ~
0 • . ~
0 ~. .
' - . °
tlO
o- 2 0 I I I
0 t~ 8 Slm 12O
F ig . 1 7. Th e d i e l ec t ri c d ec remen t ( eq u a t i o n 3 4 ) a t 2 5 ° Cp l o t t ed as a fu n c t i o n o f t h e sp ec i f i c e l ec t r i c co n d u c t i v i t y afo r a q u eo u s so lu t i o n s o f L iNO 3 ( fu ll p o in t s ) a n d L iC1(circles ; K aat ze an d Pot te l . 1984c I . Th e fu l l curv e repres en tst h e c o n t i n u u m m o d e l o f k i n e t i c d e p o l a r i z a t i o n ( e q u a t i o n3 3 ) . Th e d ash ed an d d o t t ed cu rv es i n d i ca t e t h e p red i c t i o n sb y t h e m o l e c u l a r m o d e l o f H u b b a r d et al. (1 9 7 9 ) an d b yth e mic ro sco p i c mo d e l o f Ku sa l i k an d P a t ey (1 9 8 3 ) ,
respect ively .
m o d e l . S i n c e t h e r e m a i n i n g d i f f e r e n c e is l i ke l y to b e
d u e t o d i e l e c tr i c s a t u r a t i o n , i t m a y b e a r g u e d t h a t
w i t h r e s p e c t t o t h e k i n e t i c d e p o l a r i z a t i o n s m a l l i o n s
t o g e t h e r w i t h i t s s h e l l o f s a t u r a t e d s o l v e n t m o l e c u l e s
m a y b e c o n s i d e r e d o n e m o v i n g u n i t ( K a a t z e , 1 98 3;
K a a t z e a n d P o t t e l , 1 9 84 c ). I t is t h e n j u s t if i e d t o a p p l y
t h e c o n t i n u u m t h e o r y ( e q u a t i o n 3 3 ) t o c a l c ul a t e t he
k i n e ti c d e p o l a r i z a t i o n 6eHo p r o d u c e d b y t he s e u n it s
a n d t o e v a l u a t e a r e m a i n i n g d i f f e re n c e f i e - & n o in
t h e d i e l e c tr i c d e c r e m e n t i n t e r m s o f s a t u r a t i o n e f fe c ts .
Z ÷ v a l u e s fo r s o m e c a t i o n s i n a q u e o u s s o l u t i o n s a r e
t a b u l a t e d i n F i g. 1 8. W i t h t h e e x c e p t i o n o f F - f o r
w h i c h Z - ~ 1 h a s b e e n f o u n d a n i o n s s ee m n o t t o
i n d u c e d i e l e c t r i c s a t u r a t i o n e f f e c t s .
T h e Z ÷ d a t a p r e s e n t e d i n F i g. 18 h a v e b e e n
d e r i ve d f r o m s p e c t r a f o r a q u e o u s s o l u ti o n s o f m o d e r -
a t e s o l u t e c o n c e n t r a t i o n ( c < 1 m o l / 1) w h i c h d i d n o t
r e v e a l i n d i c a t i o n s o f i o n c o m p l e x f o r m a t i o n . D u e t o
t h e i r r e d u c e d e l e c tr i c fi e ld i o n c o m p l e x e s f o r m a
s m a l l e r s h el l o f s a t u r a t e d s o l v e n t th a n c o m p l e t e l y
d i s s o c i a t e d c a t i o n s . A s t o b e e x p e c t e d a t e n d e n c y i n
t h e Z + v a lu e s o f m a i n g r o u p c a t i o n s e m e r g e s t o
d e c r e a s e w i t h d e c r e a s i n g c h a r g e o f t h e i on a n d w i t h
i n c r e a s i n g i o n i c r a d iu s . I t is i m p o r t a n t t o n o t i c e t h a t
s a t u r a t e d s o l v e n t m o l e c u l e s a r e f ix e d w i t h r e s p e c t t o
t h e i r e le c t r ic d i p o le a x i s o nl y . R o t a t i o n s a r o u n d t h i s
a x i s a r e s t il l p o s s i b l e a n d , a s m e n t i o n e d a b o v e , d e -
p e n d o n t h e i o n . T h e r e m a y o c c u r m o r e o r le ss fa s t
e x c h a n g e p r o c e s s e s b e t w e e n t h e s a t u r a t e d s o l v e n t a n d
t h e b u l k p h a s e .
7 . R E L A X A T I O N S P E C T R U M O F A Q U E O U S S O L U T I O N S .
H Y D R O P H O B I C A N D N E G A T IV E H Y D R A T I O N
I n t h e p a s t d e c a d e s a g r e a t v a r i e t y o f r e l a x a t i o n
s t ud i e s h a s b e e n p e r f o r m e d o n a q u e o u s s o l u t io n s o f
h i g h w a t e r c o n t e n t . T h e d i e l e c tr i c s p e c t r a n o r m a l l y
r e v e a l e d a s o l v e n t p a r t R~ ( v ) ( e q u a t i o n 2 6 ) w i t h a
d i e l e c t r i c r e l a x a t i o n t i m e z s d i f f e r e n t f r o m t h e p u r e
w a t e r v a l u e zw a t t h e s a m e t e m p e r a t u r e . I n a d d i t i o n ,
t h e R s ( v ) t e r m e x te n d s o v e r a s o m e w h a t b r o a d e r
f r e q ue n c y r a n g e t h a n t h e D e b y e t e r m c h a r a c t e r i z in g
t h e p u r e s o l v e n t . T h i s l a t t e r f i n d i n g c a n b e a l t e r n a -
t i v e l y d i s c u s s e d .
I t is c o m m o n p r a c t i c e t o c o n s i d e r s o l u t e i n d u c e d
c h a n g e s i n t h e d i e l e c t r i c r e l a x a t i o n o f t h e s o l v e n t
w a t e r b y a s s u m i n g a c o n t i n u o u s r e l a x a t i o n t im e
d i s t r i b u t i o n , h e n c e t o r e p r e s e n t R~ ( v ) b y a D a v i d -
s o n - C o l e , C o l e ~ o l e o r H a v r i l i a k - N e g a m i t e r m
( e q u a t i o n 2 8 w i t h a p p r o p r i a t e l y f i x e d c ~ a n d f l s ,
r e s p e c t i v e l y ) . T h e e f f e c t i n t h e r e l a x a t i o n t i m e a s i n
S e c t i o n 3 is c o n s i d e r e d b y i ts r e la t i v e m o l a l i n c r e m e n t
B d . N o r m a l l y , B d v a l u e s a r e g i v e n f o r t h e p r i n c i p l e
r e l a x a t i o n t i m e z s i n s t e a d o f t h e m o r e g l o b a l ~'m u s e d
i n e q u a t i o n ( 22 ). I n t h e c a s e o f d i l u t e a q u e o u s
s o l u t i o n s , h o w e v e r , d e v i a t i o n s f r o m s y m m e t r i c r e l a x -
a t i o n t i m e d i s t r i b u t i o n s a r e s m a l l s o t h a t z S ~ Z r~ .
B d - v al u es f o r s o l u t i o n s o f s o m e s e r ie s o f o r g a n i c
s o l u t e s a r e l is t e d in T a b l e 4 t o s h o w t h a t a g e n e r a l
t e n d e n c y o f z ~ e x i st s t o i n c r e a s e w i t h t h e n u m b e r o f
a l i p h a t i c g r o u p s p e r s o l u t e m o l e c u l e . T h i s f i n d i n g is
a r e f le c t i o n o f s o l u t e - w a t e r i n t e r a c t i o n s n a m e d
h y d r o p h o b i c h y d r a t i o n . I n t h e f ra m e w o r k o f t h e
a b o v e w a t e r r e l a x a t i o n m o d e l ( S e c t i o n 4 ) t h e i n c r e a s e
i n th e d i e l e ct r ic r e l a x a t i o n t i m e o f w a t e r a r o u n d
h y d r o p h o b i c p a r ti c le s p r e d o m i n a n t l y r e s u lt s f ro m t h e
r e d u c e d d e n s i t y o f h y d r o g e n b o n d i n g s i te s a t t h e
s o l v e n t - s o l u t e i n t e r fa c e . O f c o u r s e , f a c t o r s o t h e r t h a n
Li ÷ ~ Be2÷3.9 ~ 6.5
2.6 7.0 11.5
K+ ~ Ca2+
• ~
Rb @ st2. ~ y 3 .• 0 7.0 12.7
Cs+ ~ Ba2* ~ La3+• 0 5 . / 1 3 . /
F ig . 1 8. Nu m b er s o f d i e lec t r i ca ll y sa tu ra t e d wa t e r mo lecu l esp er i o n fo r t h e f i r s t t h r ee m ain g ro u p ca t i o n s as we l l a s y 3 +an d La 3 ÷. F u l l c i r cu l a r a r eas sh o w th e s i ze o f t h e b a re i o n s ,sh ad ed a r eas i n d i ca t e t h e sh e l l o f ap p aren t l y i r ro t a t i o n a l l y
b o u n d w a t e r m o l e c u le s .
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T a b l e 4 . R e l a t i v e m o l a l s h i ft s Be i n t h e p r i n c i p a l r e l a x a t i o n t i m e z s o f t h e s o l v e n t c o n t r i b u t i o n R ~ t o t h e d i e l e c t r i c
s p e c t r u m f o r a q u e o u s s o l u t i o n s o f s o m e c y c l i c s o l u t e s a n d f o r u r e a a n d d e r i v a t i v e s a s w e ll a s t h e c a t i o n i c p a r t
B~- to Bd fo r a q u e o u s s o l u t io n s o f t h e a m m o n i u m i o n a n d t h r ee s e r ie s o f o r g a n i c c a ti o n s . M o s t d a t a r e f e r to2 5 ' C . t h o s e f o r p y r i d i n e a n d d e r i v a t i v e s t o 2 0 ' C ( K a a t z e a n d P o t t e l , 1 9 9 2)
S o l u t e B d ( m o l / k g ) - ' S o l u t e B d ( m o l / k g ) i
P y r a z i n e 0 . 1 3 U r e a 0 . 0 3
M e t h y l p y r a n z i n e 0 . 19 M e t h ) l u r e a 0 . 0 82 . 3 - D i m e t h y l p y r a z m e 0 . 2 4 N . N - D i m e t h y l u r e a O . 1 7
2 . 5 - D i m e t h y l p y r a z m e 0 . 2 7 N . N ' - D i m e t h y l u r e a 0 . 18
2 . 6 - D i m e t h y l p y r a z l n e 0 . 2 5 E t h y l u r e a 0 . 1 3
E t h y l p y r a z l n e 0 . 21 T r i m e t h y l u r e a 0 . 2 4
2 . 3 . 5 -T r i m e t h y l p y r a z in e 0 . 3 2 N - P r o p y l u r e a 0 . 1 9
Q u i n o x a l i n e 0 . 1 9 T e t r a m e t h y l u r e a 0 . 3 0
2 - M e t h y l q u i n o x a l i n e 0 . 2 4 N . N - D i e t h y l u r c a 0 . 30
P y r i d i n e 0 . 1 9 N - B u t y l u r e a 0 . 2 l
2 - M e t h y l p y r i d i n e 0 . 2 7
3 - M e t h y l p y r i d i n e 0 . 2 2
2 , 4 - D i m e t h y l p y r i d i n e 0 . 2 7
2 . 6 - D i m e t h y l p y r i d i n e 0 . 2 8
C a t i o n B J ( m o l / k g ) - i C a t i o n B+ ( m o l / k g ) - i
A m m o n i u m - 0 . 04 T e t r a m e t h y l a m m o n i u m 0 .1 7
N - B u t y l a m m o n i u m 0 .2 8 T e t r a e t h y la m m o n i u m 0 .3 9
N - H e x y l a m o n i u m 0 .3 7 T e t r a p r o p y l a m m o n i n m 0 .7 3N - H e p t y l a m m o n i u m 0 .3 8 T e t r a b u t y l a m m o n i u m 0 .8 8
N - O c t y l a m m o n i u m 0 . 4 4 5 - A z o n i a s p i ro [ 4 , 4 ] n o n a n e 0 . 2 9
6 - A z o n i a s p i r o [ 5 , 5 ] u n d e c a n e 0 . 3 7
7 - A z o n i a s p i r o [ 6 , 6 ] t r i d e c a n e 0 . 4 3
the local concent ration of hydrogen bonding groups
are also important in determining the hydration
water properties. These factors may include the over-
all size and shape of the solute molecule, its flexibility
and the chemical nature and steric arrangement of its
H-bonding groups. The hydration water relaxationtimes are thus not only given by the chemical compo-
sition of the solute, as illustrated, for example, by
the Bd-values for the stereoisomers ethylurea
(0.13 mol/kg) and n,n' -dime thylurea (0.18 mol/kg).
Bd-values reflect both the relaxation time Zh and the
extent of the hydration region around a solute mol-
ecule. To achieve more detailed insights into the
effects of hydration and to be able to derive %, the
orientat ion correlation factor gh, and the num ber Z h
of hydration molecules per solute particle from the
measured spectra, Rs(v ) is frequently discussed in
terms of the aforementioned hydration model(Kaatze and Pottel, 1985b). With aqueous solutions
of organic solutes a simplified version of the hy-
drati on model is normal ly sufficient in which Rs(v) is
represented by a sum of two Debye relaxation terms.
T a b l e 5 . R a t i o z h / % o f t he h y d r a t i o n w a t e r t o p u r e w a t e r r e l a x a ti o n
t i m e f o r s o m e l o w w e i g h t o r g a n i c so l u t es ( 2 Y C ; K a a t z e a n d
W o e r m a n n , 1 98 2; K a a t z e a n d P o t te l , 1 9 8 5 b ; K a a t z e et a l . , 1986)
S o l u t e % / %
1 , 4 - D i o x a n e
E t h y l e n e g l y c o l
P y r i d i n eP y r a z i n e2 - M e t h y l p y r a z i n e
2 , 6 - D i m e t h y l p y r a z i n e
Q u i n o x a l i n e
2 - M e t h y l q u i n o x a l i n e
1 , 4 - D i a z a b i c y c l o - [ 2 , 2 , 2 ] o c t a n e ( T E D )
A c e t a m i d e
U r e a
1.6
2.1
1.81.5
1.61.9
1.9
2.22.22.12.1
One term, with the relaxation time r h, describes the
relaxation of the hydration water. The other term,
with relaxation time %, considers the non-affected
water in the solution. A few hydration water relax-
ation times are collected in Table 5. These data span
a small range of values only, though there may besignificant differences in the hydration behaviour of
the solutes. Urea, for instance, though Zh/Zw= 2.1,
does hardly affect water (B d = 0.03 mol/kg) while
TED with nearly the same relaxation time ratio
(zh/Zw -- 2.2) is particularly effective in promoting the
water structure around it (Bd- 0.36mol/kg). The
explanation for this striking result is as follows.
The most hydrophillic urea molecule exhibits a
remarkable structure as the six directions for the
formation of hydrogen bonds are restricted to a
plane. On its periphery the planar urea molecule thus
offers a high density of H-bond sites. It seems not tochange the relaxation time of water in this region.
Only in the directions perpendicular to the plane
defined by the structure of the urea molecule this
outs tand ing solute appears to be hydrophobic. Hence
only a few water molecules change the relaxation time
with respect to pure water. Nearly spherically shaped
TED, due to the positions of its H-bonding sites and
to the a rrangement of its hydrophobic groups relative
to these sites, obviously promotes a more extended
clathrate-like hydration structure around it and thus
produces a high Bd-value.
Aromati c rings induce smaller Bo-values tha n com-parable aliphatic molecules. This behaviour may be
taken to result from some weak H-bondi ng inter-
actions between water molecules and the delocalized
electrons which, as bifurcated H-bond in pure water
(Section 4) reduce the activat ion enthalpy or reorien-
tational motions. A similar situation may exist
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Micr ow ave d ie lec t ric p r oper t ie s o f l iqu ids
a r o u n d i o n s th a t e x h i b i t h y d r a t i o n w a t e r r e o r i e n t a -
t i o n t i m e s % d i s t i n c t l y s m a l l e r t h a n % ( Bd < 0 , n e g a -
t i v e h y d r a t i o n , T a b l e 2 ). A p r o m i n e n t e x a m p l e is
i o d i d e f o r w h i c h ( % / Z w ~ 0 . 5 ; P o t t e l e t a l . , 1974) .
A r o u n d t h is la r g e m o n o v a l e n t a n i o n a n e n h a n c e d
w a t e r m o b i l i t y m a y b e c a u s e d b y t h e c o m p a r a t i v e l ys o f t e l e c t r o n s h el l p r o v i d i n g r o t a t i n g w a t e r m o l e c u l e s
w i t h t r a n s i e n t H - b o n d - l i k e i n t e r a c t io n s . A l s o ( a t le a s t
i n p a rt s ) a r e fl e c t io n o f a n e n h a n c e d d e n s i t y o f
H - b o n d s it e s ( w h i c h c a n a ct as a p p r o p r i a t e n e t w o r k
d e f e c t s ) m a y b e t h e n e g a t i v e B J- v a l u e f o r s m a l l
c a t i o n s m e n t i o n e d i n S e c t i o n 3 . E l e c t r o s t r i c t i v e e f f e c t s
w i l l i n c r ea s e t h e d e n s i t y o f h y d r a t i o n m o l e c u l e s i n t h e
s t r o n g C o u l o m b i e f i el d s o f th e i o n . T o w a t e r m o l -
e c u l e s a t t h e i n t e r f a c e b e t w e e n t h e d i e l e c t r i c a l l y s a t u -
r a t e d s h e l l a n d t h e b u l k p h a s e a h i g h e r d e n s it y o f
H - b o n d i n g s i t e s i s t h e r e f o r e o f f e r e d t h a n i n p u r e
w a t e r a t t h e sa m e t e m p e r a t u r e a n d h y d r o d y n a m i cp r e s s u r e . A g a i n t h i s d e n s i t y e f f e c t m a y r e s u l t i n f a s t e r
r e o r i e n t a t i o n a l m o t i o n s .
8. COLLOI DAL SYSTEMS
A s b r i e f ly m e n t i o n e d a b o v e d i e l ec t r ic s p e c t r o s c o p y
o f e m u l s io n s , m i c r o e m u l s i o n s , a n d c o l l o i d a l s o lu t i o n s
h a s a t t r a c t e d m u c h a t t e n t i o n , p a r t i c u la r l y f o r re a s o n s
o f t h e o c c u r r e n c e a n d u s a b i li t y o f s u c h s y s t e m s i n
b i o l o g y a n d i n i n d u s t r i a l a n d m e d i c a l p r o c e s s e s.
M e r e l y t h e s p e ct r a o f t w o c o m p a r a t i v e l y s i m p l y
s t r u c t u r e d c o l l o i d a l s y s t e m s m a y b e d i s c u s s e d h e r e( F i g . 1 9 ), n a m e l y o f m i c e l l a r a q u e o u s s o l u t i o n s o f a
z w i t t e r i o n i c a n d a c a t i o n i c s u r f a c t a n t , r e s p e c t i v e l y .
1 0 0. . . _
%
\2O
0 I I ~ I I I I I
5 0 ~ ~ i i t i ~ i ~ I
f t . 105 o d ~ V l ~ ~ 2 / r r s ]
f J/ t , , 2 7 r u , , ' , , I
I
0 0 0 0 5 0 2 0 5 1 2 5 1 0 G H z 5 0
Fig . 19. Semi logar i thmic p lo t o f the r ea l pa r t U ( v) anddouble logar i thmic p lo t o f the d ie lec t r ic cont r ibu t ion ~ ( v)to the nega t ive imaginar y pa r t o f the d ie lec t r ic spec t rum a t25° C d isp layed f or a 0 .2 m ola r aq ueous so lu t ion of zw i t te-r ionic n-hexadecyl-sulfopropylbetaine (open circles ; Pottelet al. , 1978) and a 0 .155 mo la r aqu eous so lu t ion of n - hex-
adecyl - t r ime thylam monium br omide ( fu ll po in t s) .
563
B o t h s p e c t r a d i s p l a y e d i n F ig . 1 9 a r e c o m p o s e d o f a
s o l v e n t a n d a s o l u t e c o n t r ib u t i o n . A s d i s c u s se d a b o v e
f o r t h e m o l e c u l a r l y d i s p e r s e d s o l u t i o n s i n f o r m a t i o n
o n t h e h y d r a t i o n b e h a v i o u r o f th e m i c e ll e s c a n b e
d e r i v e d f r o m a n a n a l y s i s o f R ~ ( v ) . R u ( v ) f o r t h e
z w i t t e r i o n i c m i c e l l e s y i e l d s t h e m o b i l i t y u u a n d t h ed i p o l e o r i e n t a t i o n c o r r e l a t i o n f a c t o r gu o f t h e d i p o l a r
h e a d g r o u p a t t h e m i c e l l a r s u r f a c e ( P o t t e l et a l . ,
1 9 78 ). T h e e x i s t e n c e o f a s o l u t e t e r m R u ( v ) i n t h e
s p e c t r u m f o r t h e s o l u t i o n o f i o n i c m i c e l l e s i s a l e ss
o b v i o u s r e s u l t . R o ( v ) i s d u e t o t h e r e s t r i c t e d m o t i o n s
o f i o n s a t t h e s u r f a c e o f m i c el l es . A t t e m p t s t o t h e o r -
e t i c a l l y d e s c r i b e t h i s e f f e c t h a v e b e e n f i r st m a d e b y
S c h w a r z ( 1 9 6 2 ) w h o a s s u m e d t h e c o u n t e r i o n s t o b e
r e s t r ic t e d t o a n i n f i n i t e s i m a l l y t h i n l a y e r a r o u n d t h e
m i c e ll e s a n d w h o c o n s i d e r e d o n l y t a n g e n t i a l m o t i o n s
o f c h a r g e s w i t h i n t h i s l a y e r. S c h u r r ( 1 9 64 ) e x t e n d e d
t h is m o d e l f o r ra d i a l m o t i o n s o f c o u n t e r i o n s . I na n a l o g y w i t h t h e e x c h a n g e o f s o l v a t i o n m o l e c u l e s
d i s c u s s e d i n S e c t i o n 3 h e a l l o w e d f o r a n e x c h a n g e o f
i o n s b e t w e e n t h e th i n c o u n t e r i o n l a y e r a n d t h e b u l k
p h a s e. S i n c e t h e s e a n d m o r e r e c e n t t h e o r ie s ( D u k h i n
a n d S h i l o v ; 1 9 74 ; F i x m a n , 1 98 0; D e l a c e y a n d W h i t e ,
1 9 8 1; C h e w a n d S e n , 1 9 8 2; O ' B r i e n , 1 9 86 : G r o s s e a n d
B a r c h i n i , 1 9 8 6 ; G r o s s e , 1 9 8 8 ) d o n o t r e g a r d a p o s s -
i b l e l a r g e r a d i a l e x t e n s i o n o f t h e c o u n t e r i o n s h e l l
t h e s e m o d e l s a p p l y f o r s y s t e m s w i t h a d d i t i o n a l l o w
w e i g h t e l e c t r o l y t e o n l y . R e c e n t l y , h o w e v e r , a t h e o r e t i -
c a l m o d e l h a s b e e n p r e s e n t e d ( B a r c h i n i , 1 9 9 2 ) t h a t
d e s c r i b e s m i c e l l a r s o l u t i o n s a t l o w i o n i c s t re n g t h i nw h i c h d i f f u s e c o u n t e r i o n l a y e r s e x i s t .
A c k n o w l e d g e m e n t - - I am indebted to P r of es sor R . Pot te l f o rman y l ively and s t imulat ing discussions.
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