Dielectric Properties of Water in the Microwave and Far-Infrared

2 Dielectric Properties of Water in the Microwave and Far-Infrared Regions Ε. H. GRANT, S. SZWARNOWSKI, and R. J. SHEPPARD

Downloaded by COLUMBIA UNIV on December 10, 2014 | http://pubs.acs.org Publication Date: August 4, 1981 | doi: 10.1021/bk-1981-0157.ch002

Physics Department, Queen Elizabeth College, London, W8 7AH, U.K.

The dielectric behaviour of pure water has been the subject of study in numerous laboratories over the past fifty years. As a result there is a good understanding of how the complex permittivity Ê = E' - JE" varies with frequency from DC up to a few tens of GHz and it is generally agreed that the dielectric dispersion in this range can be represented either by the Debye equation or by some function involving a small distribution of relaxation times. At these frequencies the variations of Ê with frequency are due to relaxation phenomena and may be described by equations appropriate to classical physics. In the far infrared and at higher frequencies resonance-type phenomena which can be accounted for on quantum principles, have been observed and well documented. Interest therefore lies in the frequency region between around 20-200GHz where measurements are sparse and, where they have been reported, relate to isolated frequencies and temperatures only. Prior to the commencement of the present work, the highest microwave frequency at which values of complex permittivity of water were available as a function of temperature was 35GHz (1). In the far infrared the principal work in existence was that of Asfar and Hasted (2) who measured the dielectric properties of water at 19-20°C in the frequency range 170GHz-13THz, using interferometric techniques. In the present investigation it was therefore decided to measure the complex permittivity of water at 70GHz over a range of temperatures in the hope that combining the 70GHz results with those previously obtained at higher and lower frequencies would enable a better understanding of the dielectric behaviour of water in this relatively unexplored frequency region, thus helping to fill the final gap in the knowledge of how the 0097-6156/81/0157-0047$05.00/0 © 1981 American Chemical Society

In Biological Effects of Nonionizing Radiation; Illinger, K.; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.


48

BIOLOGICAL EFFECTS OF NONIONIZING RADIATION

d i e l e c t r i c properties of water vary w i t h frequency from D C to the v i s i b l e r e g i o n . D e t e r m i n i n g the c o m p l e x p e r m i t t i v i t y of water at 7 0 G H z poses problems i n that the a t t e n u a t i o n i s very high w h i c h , c o u p l e d w i t h the s m a l l d i m e n s i o n s of the waveguides, n e c e s s i t a t e s the t a k i n g of s p e c i a l p r e c a u t i o n s i n order to o b t a i n h i g h a c c u r a c y . Experimental Procedure The w a v e g u i d e s y s t e m used to measure the d i e l e c t r i c parameters of water and other l o s s y l i q u i d s has been described previously (3). B a s i c a l l y it i n v o l v e s the measurement of the power p r o f i l e of a w a v e r e f l e c t e d from a movable short c i r c u i t as it t r a v e r s e s the l i q u i d under t e s t . The v a r i a t i o n of the a m p l i t u d e of the r e f l e c t e d w a v e w i t h the d i s t a n c e moved by the short c i r c u i t i s r e l a t e d d i r e c t l y to the c o m p l e x p e r m i t t i v i t y of the l i q u i d c o n t a i n e d i n the experimental c e l l . A computer program has been w r i t t e n to e v a l u a t e E' and 6 from the measured a m p l i t u d e v a r i a t i o n . In order to o b t a i n high p r e c i s i o n it i s n e c e s s a r y to measure the d i s t a n c e s moved by the plunger very a c c u r a t e l y and t h i s has been a c h i e v e d by a worm d r i v e and l e v e r m e c h a n i s m of high m e c h a n i c a l s t a b i l i t y w h i c h e n a b l e s the plunger to be l o c a t e d to a p r e c i s i o n of 0 . 0 0 1 m m . Other important requirements are good frequency s t a b i l i t y , a h i g h s i g n a l to n o i s e ratio and detector c r y s t a l c h a r a c t e r i s t i c s w h i c h are independent of power l e v e l . A l l t h e s e features have been incorporated into the present s y s t e m (3). R e s u l t s and D i s c u s s i o n The v a l u e s measured for water at 7 0 G H z at different temperatures are shown i n T a b l e I. Both E' and E" are seen to i n c r e a s e w i t h i n c r e a s i n g temperature, w h i c h i s t y p i c a l b e h a v i o u r for a pure p o l a r l i q u i d at frequencies w e l l in e x c e s s of the r e l a x a t i o n frequency. In the first i n s t a n c e the data were c o m b i n e d w i t h t h o s e o b t a i n e d by other workers at l o w e r frequencies (4_-l_5) and an a n a l y s i s was c a r r i e d out. The c o m b i n e d v a l u e s of E' and E" were fitted to a C o l e - C o l e d i s p e r s i o n r e l a t i o n s h i p i n order to find the best v a l u e for uC and 6 a a t e a c h temperature. Thç C o l e - C o l e dispersion equation is

where

is the s t a t i c p e r m i t t i v i t y and

i s the d i e l e c t r i c



2.

GRANT E T AL.

49

Dielectric Properties of Water

T a b l e I. The c o m p l e x p e r m i t t i v i t y of water at 7 0 G H z . (The errors correspond to the 95% c o n f i d e n c e i n t e r v a l s )

0.3 1.0 5.0 10.2 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0

Δ β"

e"

T/°C 7.49 7

5

9

· ο ' R 8.60^ 9.20* 8

9

1

'

2

9 2

f ib

10.82 12.05; 13.00° 14.54^ 15.86* 17.41 8

2

0.08 0.09 0. 11 0.08 0. 10 0. 10 0. 14 0. 15 0.13 0. 16 0.22 0. 17

11.46 11.82 13.03Γ 14.46 15.8θ' 17.55 19.14^ 20.847 21.90 23.32^ 24.80^ 25.50^ ο U

6

0.09 0.10 0.11 0.12 0. 13 0. 14 0. 16 0. 17 0.20 0. 19 0.22 0.21


50

BIOLOGICAL E F F E C T S O F NONIONIZING RADIATION

r e l a x a t i o n time at the appropriate temperature. The parameter £ i s the v a l u e of the p e r m i t t i v i t y to w h i c h 6 would be a s y m t o t i c at h i g h frequencies i n the a b s e n c e of further dispersion. A s s h o w n l a t e r i n t h i s paper t h i s a s s u m p t i o n d o e s not hold for water and so i n p r a c t i c e never appears as a p l a t e a u and i t s v a l u e i s m o d e l - s e n s i t i v e . The parameter d e s i g n a t e d by & i s a measure of any d i s t r i b u t i o n of r e l a x a t i o n times w h i c h may e x i s t , and for the c a s e of a s i n g l e r e l a x a t i o n timeCk=0 and e q u a t i o n Q J becomes the D e b y e equation. The r e s u l t s of the fit are shown i n Table I where it i s s e e n that at most temperatures the 95% c o n f i d e n c e l i m i t for U o v e r l a p s zero and hence there i s l i t t l e d e v i a t i o n from s i n g l e r e l a x a t i o n time behaviour a c c o r d i n g to t h i s a n a l y s i s . Another inference i s that the v a l u e of - d e d u c e d from t h i s procedure probably d e c r e a s e s w i t h i n c r e a s i n g temperature but the most important c o n c l u s i o n i s that the r e l a x a t i o n time Τ i s not p a r t i c u l a r l y s e n s i t i v e to the model c h o s e n , the v a l u e s obtained a g r e e i n g to w i t h i n e x p e r i m e n t a l error i r r e s p e c t i v e of whether the v a l u e of & was a l l o w e d to be fitted a s an unknown parameter, or whether it was c l a m p e d at zero (Debye c a s e ) . T a b l e II therefore summarises the d i e l e c t r i c b e h a v i o u r of water as e v a l u a t e d from c u r v e - f i t t i n g the measured p e r m i t t i v i t y data t a k e n at frequencies up to 7 0 G H z . Extra information w h i c h emerges by c o n s i d e r a t i o n of the new 7 0 G H z r e s u l t s a l o n e w i l l now be o b t a i n e d by e x a m i n i n g the_variation w i t h temperature of the f u n c t i o n F = ( € - β' ) (CJ e ) . If the d i e l e c t r i c b e h a v i o u r of water at t h i s frequency c a n be e n t i r e l y a c c o u n t e d for by the D e b y e equation then F i s e q u a l to the r e l a x a t i o n time ( T ) . If there i s any departure from s i n g l e r e l a x a t i o n time b e h a v i o u r then F £ 'Ύ ; for example i f the e x p e r i m e n t a l data at 7 0 G H z c o u l d be d e s c r i b e d by a C o l e C o l e r e l a t i o n s h i p then F w o u l d e q u a l Ύ -f cf'tLfa U(t/F)] as shown p r e v i o u s l y (16). The v a l u e s of F a s c a l c u l a t e d from the r e s u l t s i n T a b l e Π are s h o w n i n Figure 1 together with the v a l u e s of the r e l a x a t i o n time Τ o b t a i n e d from T a b l e II, w h i c h are i n agreement w i t h the literature v a l u e s of r e l a x a t i o n t i m e . A c l e a r departure from D e b y e b e h a v i o u r i s o b s e r v e d b e l o w room temperature i . e . the v a l u e s of β and 6* at 7 0 G H z do not conform to a s i n g l e D e b y e d i s p e r s i o n . Therefore some other p r o c e s s must be i n v o l v e d w h i c h i s o p e r a t i v e at the h i g h frequency end of the p r i n c i p a l d i s p e r s i o n . The d i s c r e p a n c y i n Figure 1 cannot be a c c o u n t e d for by the p r o p o s i t i o n that the p r i n c i p a l d i s p e r s i o n c a n be c h a r a c t e r i s e d by a d i s t r i b u t i o n of


Λ

u

1

s



0 5 10 20 25 30 35 40 50

T/°C

ΔΤ/ρ

0.28 0. 14 0. 17 0.08 0.06 0.08 0.06 0.07 0.05

17. 64 14. 75 12.84 9.43 8.27 7.24 6.56 5.90 4. 78

5

17. 69 14. 71 12. 76 9.32 8.21 7. 19 6.52 5.87 4.77

r/ps 0.32 0.15 0.19 0.10 0.07 0.11 0.11 0.11 0.08

At/ps

Cole-Cole

5.02 6.09 5.20 5.35 5. 57 4.89 5.41 4.94 4.58

e*

Debye 6*

0. 63 0.36 0.56 0.40 0.30 0.50 0.42 0.57 0.52

Δ 5.37 5.85 4. 69 4.53 5. 11 4.48 5.01 4. 65 4.44

Λ

0.95 0.49 0.83 0. 60 0.44 0.85 0.89 0.93 0.85

Δβ

C o l e •rCole

W a t e r p e r m i t t i v i t y data fitted to v a r i o u s m o d e l s (The errors c o r r e s p o n d to the 95% c o n f i d e n c e i n t e r v a l s )

Debye Τ /ps

Table II


0.001 0.004 0.008 0.010 0.006 0.005 0.006 0.003 0.001

0.012 0.006 0.009 0.006 0.004 0.008 0.009 0.008 0.007

A*

C o l e - •Cole


BIOLOGICAL E F F E C T S OF NONIONIZING RADIATION

007

0-1

0-2

0-3

0-5 07

10

20

30

5 0 70

100

Frequency/THz

Figure 2. Complex permittivity of water at frequencies above 70 GHz ((V) from the present 70-GHz study; (O) e calculated from the Debye model using τ = 9.3 ps, € = 80.1, and e = 5.5; data of Asfar and Hasted (2): ( ) e'; (A) ") 8

x



2.

GRANT ET AL.


53

relaxation times. The v a l u e o required to a c c o u n t for the v a l u e of F at 0 C w o u l d be 0 . 0 6 w h i c h i s far i n e x c e s s of the v a l u e i n d i c a t e d i n Table I I . The e x p l a n a t i o n must therefore lie elsewhere. For a d i s p e r s i o n region h a v i n g a r e l a x a t i o n m e c h a n i s m u n d e r l y i n g i t a l o w e r i n g i n the temperature c o r r e s p o n d s to a d e c r e a s e i n the r e l a x a t i o n frequency. For water a t 0 e the r e l a x a t i o n frequency i s 9 G H z , i . e . a factor o f eight l o w e r than the frequency of measurement i n the present w o r k . Therefore measurements on water c a r r i e d out at 7 0 G H z at the l o w e r temperatures correspond to p o i n t s a p p r o a c h i n g the h i g h frequency t a i l of the main d i s p e r s i o n r e g i o n . The p r e s e n c e of a s m a l l s u b s i d i a r y d i s p e r s i o n i n the far infrared c o u l d therefore affect the v a l u e s of and b e l o w room temperature without h a v i n g a n y effect at higher t e m p e r a t u r e s . This p r o p o s a l w i l l now be examined i n further d e t a i l . D i e l e c t r i c d i s p e r s i o n i n pure water i n the far infrared R e l i a b l e p e r m i t t i v i t y data at frequencies i n e x c e s s o f around 1 0 0 G H z are rather s p a r s e but one study that h a s been c a r r i e d out i s that due to Asfar and H a s t e d (2) where occur measurements at a p p r o x i m a t e l y 19 C . Their v a l u e s of together w i t h the present 7 0 G H z point and v a l u e s o f c a l c u l a t e d b y a s s u m i n g a D e b y e type r e l a x a t i o n for the main d i s p e r s i o n are shown i n Figure 2 . The data of Asfar and H a s t e d are a l s o i n d i c a t e d i n t h i s F i g u r e , and i t may be s e e n that r e s o n a n c e a b s o r p t i o n due to effects s u c h a s hindered t r a n s l a t i o n b e g i n to o c c u r at around 2 T H z . At frequencies b e l o w t h i s , v a r i a t i o n s of aand are c o n s i s t e n t w i t h c l a s s i c a l - t y p e phenomena. To determine the nature of the d i s p e r s i o n c u r v e at frequencies between the m a i n d i s p e r s i o n a n d r e s o n a n c e regions the data c o n t a i n e d i n references (4-15) at 20 C were c o m b i n e d w i t h the 7 0 G H z v a l u e s and the p e r m i t t i v i t y r e s u l t s of Asfar and H a s t e d up to frequencies l e s s than 2 T H z . The c o m b i n e d data were then fitted to the equation

w h i c h r e p r e s e n t s a sum of two D e b y e d i s p e r s i o n r e g i o n s . The lower frequency d i s p e r s i o n , w h i c h h a s a r e l a x a t i o n time and a s t a t i c p e r m i t t i v i t y c o r r e s p o n d s to the m a i n dispersion region. The higher frequency r e g i o n has a l o w 2 frequency p e r m i t t i v i t y of and a s m a l l a m p l i t u d e of - n.



54


T a b l e III. Parameter

A two component fit to the m i c r o w a v e data and the infrared data of Asfar and H a s t e d Value

95% C o n f i d e n c e

80.10 5.74 3.34 9 . 4 8 ps 0 . 2 5 ps J

Τ*

Interval

Parameter h e l d c o n s t a n t 0.31 0.38 0 . 0 7 ps 0 . 0 8 ps



2.

GRANT ET AL.


55

The h i g h frequency l i m i t of for t h i s s e c o n d p r o c e s s i s therefore η . The r e s u l t of the fit i s shown i n T a b l e III where the mean v a l u e s of the v a r i o u s parameters and t h e i r a s s o c i a t e d 95% c o n f i d e n c e i n t e r v a l s are g i v e n . Considering the s m a l l a m p l i t u d e of the s e c o n d d i s p e r s i o n both i n a b s o l u t e te^rms and i n r e l a t i o n to the main d i s p e r s i o n the parameters η and are q u i t e w e l l d e f i n e d , and therefore it may be c o n c l u d e d that the d o u b l e D e b y e r e p r e s e n t a t i o n i s an a c c e p t a b l e d e s c r i p t i o n of the d i e l e c t r i c b e h a v i o u r of water up to around 2THz. Other a l t e r n a t i v e interpretations are c l e a r l y p o s s i b l e but no attempt has been made here to f o l l o w t h e s e up at t h i s stage. What i s c l e a r i s that a s m a l l s u b s i d i a r y d i s p e r s i o n r e g i o n i n the far infrared i s n e c e s s a r y to a c c o u n t for a l l the p r e s e n t l y a v a i l a b l e p e r m i t t i v i t y d a t a , and that s u c h a d i s p e r s i o n i s centred around 6 5 0 G H z and has an a m p l i t u d e of about 2 . 4 i n c o m p a r i s o n w i t h that of the p r i n c i p a l d i s p e r s i o n which is approximately 75. W h e t h e r or not a r e l a x a t i o n phenomenon c o u l d e x i s t at frequencies as high as hundreds of G H z c o u l d be a matter for discussion. H o w e v e r the i n e r t i a of the water m o l e c u l e i s f a i r l y l o w and at 20 C a number of zero bonded and s i n g l y bonded m o l e c u l e s ( w h i c h c a n rotate without surmounting an a p p r e c i a b l e p o t e n t i a l barrier) w o u l d be e x p e c t e d to e x i s t on t h e r m o d y n a m i c a l grounds i n s u f f i c i e n t q u a n t i t i e s to a c c o u n t for the s m a l l o b s e r v e d d i s p e r s i o n .

Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Grant, E . H . ; Shack,R.J. Brit. J.Appl. Phys. 1976, 18, 1807 Asfar, Ν . Μ . ; Hasted, T.B. J.Opt. Soc. Am. 1977, 67, 902 Szwarnowski,S.; Sheppard,R.J. J.Phys.E 1977. 10. 1163 Collie, C . H . ; H a s t e d . J . Β . ; Ritson, D . M . Proc. Phys. Soc. 1948, 60/ 145 Lane,J.Α.; Saxton,J.A. Proc. Roy. Soc. A. 1952, 213,400 Cook,H.F. Brit. J.Appl. Phys. 1952, 249 Hasted, J.B.; El Sabeh, S . H . M . Trans. Faraday Soc. 1953, 49, 1003 Buchanan,T.J.; Grant,Ε.Η. Brit. J.Appl. Phys. 1955, 6, 64 Grant,E.H.; Buchanan,T. J . ; Cook,H.F. J.Chem. Phys. 1957, 26, 156 Sandus,O; Lubitz,B.B. J. Phys. Chem. 1961, 65, 881 Van Loon,R; Finsy,R. J.Phys.D. 1975, 8, 1232


56



12. Bottreau,A. M . ; Marzat,C.; Lacroix , Υ . , Dutuit,Y. J.Microwave Power 1975, 10, 297 13. Schwan,H.P.; Sheppard, R.J.; Grant,E.H.; J. Chem. Phys. 1976, 64/ 2257 14. Pottel, H. In Hasted,J.B., "Aqueous Dielectrics"; Chapman & Hall: London 1973 15. Malmberg,C.G.; Maryott, A A. J.Res. Natl. Bur. Stand. (U.S) 1956, 56, 1 16. Grant,E.H. J.Phys. Chem. 1969, 73, 4386 RECEIVED

December 1, 1980.


Dielectric Properties of Water in the Microwave and Far-Infrared

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