THE DIELECTRIC PROPERTIES OF WATER-DIELECTRIC INTERPHASES HUGO FRICKE
AND
HOWARD J. CURTIS
W a l t e r B. J a m e s Laboratory f o r B i o p h y s i c s , T h e Biological Laboratory, Cold S p r i n g Harbor, L o n g I s l a n d , N e w York Received February 6 , 1957
When a dielectric dispersed in water is placed in an electrical field a part of the current passes through the system at the interphases and these become polarized, as shown under certain conditions by the occurrence of electrochemical reactions ( 1 , 3 ) . For this reason the dielectric properties of such systems are generally different from what would be expected from the contributions of the single phases, on the basis, for example, of the Clausius-Massotti-Maxwell theorem, for the particular case of a suspension of spherical particles. This influence of the interphases is especially pronounced when measurement is made with direct current or with alternating current of low frequency, in which case the dielectric properties may be radically different from those of either of the components, as measured in the bulk. Recent measurements by Fricke and Curtis (11, 12) of the dielectric constant and electric conductance of suspensions show this influence of the interphases clearly; there is a rapid increase of the dielectric constant and a decrease of the conductance as the frequency decreases, the change in both being approximately as powers of the frequency. The high dielectric constant often found for colloidal solutions is undoubtedly also of this origin (6, 11, 14, 15, 16, 20, 21). This property was first observed by Errera on colloidal vanadium pentoxide (6), and explained by him as due t o electric moments of the colloidal particles. I n spite of the importance of these interphasial properties for the dielectric characteristics of a number of important systems, such as living cells ( 5 , 9), soil (7, 23, 24), and many of the insulating (hygroscopic) materials used in the electrical industry (paper, fabrics, rubber) (19, 25), experimental data suitable for reaching an understanding of their nature are largely lacking. The present study is based upon alternating current measurements of the dielectric constant and electric conductance of suspensions, particularly those with particles of spherical form. The measurements were made with a Wheatstone bridge by a substitution method which is based upon the comparison of the suspension with a 729
730
HUGO FRICKE A S D HOWARD J. CURTIS
potassium chloride solution. The solutions were measured in cells constructed as shown in figure 1. The cells are of Pyrex glass, cylindrical, and with one platinum electrode sealed into one end, while the other electrode is movable and mounted on a micrometer screw reading t o an accuracy of 0.002 mm. The purpose of the design is to make it possible to correct for electrode polarization by measuring a t different electrode distances. I n making the measurements two similar cells are used, one containing the suspension and the other a potassium chloride solution of approximately the same specific resistance. The cell with the suspension (the measuring cell) is placed in the bridge, and equilibrium established by varying a resistance box and a condenser in one of the other arms of the
rl-
MICROMETER S R E W
LASS TUBES OP L U C T R O C E
BOTTOM
E.LECTRODE
FIG.1. Electrolytic cell
bridge. The measuring cell is now replaced by the comparison cell and equilibrium reestablished for resistance, by varying the electrode distance, and for capacitance by varying a condenser (the measuring condenser) in parallel with the cell. The resistance of the comparison cell is thereafter measured by substitution against a resistance box, this measurement being made a t low frequency. The following terms are used: The complex admittance 1/Z is given by
1/Z
=
1/R + j o C
where R (ohms) and C (f) are termed resistance and capacitance of the suspension. The specific values of R and C are termed the resistivity
WATER-DIELECTRIC
731
INTERPHASES
R (ohms-em.) and capacity C (f-cm.-l), respectively, and the conductivity u
and the dielectric constant
E
are determined by (1)
u = 1/R e =
36nC X 10“
(2)
Xeglecting for the present the influence of electrode polarization, the capacitance of the electrolytic cell is the sum of two parts, of which one, C1, is the capacitance of the solution in the cell, and t h e other, Cz, t h e capacitance derived from the space outside the solution. The value of C1 i$
CL =
E
. 10/(36~.K).ppf
where K is t h e cell constant determined as the ratio of the resistance, measured with a standard potassium chloride solution in the cell, to the specific resistance of this solution. For the experimental conditions used here the value of C2 can with sufficient accuracy be taken as independent of the solution, and therefore C2 cancels out in the comparison between the two cells when the comparison is made with equal electrode distances. The dielectric constant, E , of the unknown solution consequently is determined by
~.10/(36~.K) = (~0.10/36a.K) f C’
+ C”
where K is the cell constant for the measuring cell: C’ is the difference in readings on the measuring condenser (in ppf), and eo is the dielectric constant of the potassium chloride solution, which in the present measurements can be taken as equal to that of water. The value of C” is the change in the capacitance of t h e comparison cell as the electrode distance is changed from that used in the measurement t o t h a t used in the measuring cell. The values of C” are obtained by a calibration in which the measuring and comparison cells are compared u ith potassium chloride solutions in both. The concentration of the solution in the measuring cell is kept constant, while the concentration of the solution in the comparison cell is changed t o give electrode distances over the desired range. One difficulty encountered in this work is that t h e dielectric comtant and particularly the conductance are not perfectly reproducible for suspensions because of settling, and for colloids apparently because of changein their internal structure. For this reason, relative values of e and IT, referred to those obtained a t some standard frequency, can be obtained with much greater accuracy than can their absolute values. Under the best conditions such relative values can be measured with an accuracy of one part in 100,000 for u and of 0.1 for e, and so perfectly satisfactory frequency
732
HUGO FRICKE AXD HOWARD J. CURTIS
curves for u and e can be obtained, although the absolute values of u and c are not reproducible with this accuracy. The inductance of the measuring condenher was shown to be negligible by comparing it against a specially built fixed condenser, consisting of two plates clamped close together and attached directly across the binding posts to the electrolytic cell; this condenser has a very much smaller inductance than the measuring condenser. The comparison of the two condensers showed no frequency dependence over the range of frequencies used. The capacity of the fixed condenser was 50 ppf, which is as large a capacity as is ever used a t the highest frequency. The influence of electrode polarization is eliminated by making measurements a t two different electrode distances. After completing the mea5ui-ements a t the greater distance, the electrodes are brought together to a suitable closer distance and measurements taken a t such frequencies for which the electrode polarization is appreciable. In order to take into consideration any slight change in the electrode polarization during these two sets of measurements, the electrodes are thereafter brought back t o the original distance and the measurements a t the low frequencies repeated. When the electrodes are moved, a slight change may occur in the absolute values of E and u. This error is eliminated by measuring the rates of change of E and u with frequency, rather than their absolute values. The electrode polarization acts as a capacity (C,) and a resistance (R,) in series with each other and with the cell. I n the present measurements l/C,o is always small compared to the resistance of the cell, which is the condition under which the forinulae now to be given can be used. Designate by AC1 and AR1 the non-corrected values for the difference in capacitance and resistance of the suspension, at the two successive frequencies w and 2 ~ and , for the greater electrode distance. Similarly, for the smaller electrode distance, ACz and ARz. Furthermore, designate by ACP and ARp those parts of AC and AR which are due to the electrode polarization, and by AC8 and ARs the true values of these two quantities. The following equalities hold:
WATER-DIELECTRIC
i33
ISTERPHASES
The ratios Ci C: and Ri R f are obtained as the values of Cz C1 and Rz R1at such a high frequency that the elertrodc polarization is negligible. Owing t o settling, in coarser suspensions these ratios may deviate sonipwhat from the ratio of the cell constants. In order to decrease the polarization, the electrodes are electrolytically covered with platinum black, from a 3 per cent platinum chloride solution containing 0.025 per cent of lead acetate. With palladium black the polarization may be still further reduced, but the greater constancy of the platinum black makes it more satisfactory. Most of the work has been carried out with two types of cell, of which one has a cross-sectional area of 1.0 cm.2 and a length of 20 em., and the and a length of 5 em. I n a few measurements other an area of 2.9
-
I
c'
lo"
w
w
lo-'
IO'
z8 WL 0
?
i
7"
I
4
16
64
1/4
I
4
16
64
KILOCYCLES PER SICOM
FIG.2. Logarithmic graphs of ( E - E * ) and ACTfor various suspensions. Curve I , 33 per cent cream (natural); curve 2, 12 per cent homogenized cream (commerrial brand); curve 3, 3 X latex (63 per cent); curve 4, 25 per cent mineral oil (diameter 1 per cent triethanolamine; curve 5, 20 per cent rouge 0 . 5 ~in ) 2 per cent oleic acid (diameter 2,) in 1 5 per cent isoelectric gelatin; curve 6 , 4 5 per cent kaolin (diameter 0 . 5 ~ in ) 0.0005 i r KCl; curve 7, 45 per cent kaolin (diameter 0 . 5 ~ in ) 0.5 per cent sodium oleate; curve 8, 5 per cent rellulose in water.
+
we have used a third pair of cells, the cross-sectional area of which is 10 and the length 20 em. Before using any pair of cells they were compared, a t all frequencies, with potassium chloride solutioiis in both. The measurements were usually made with the cells in air in a room kept a t the constant temperature of 21.4"C. I n some cases, when a steadier temperature was desired, the cells were kept in an oil bath. Data obtained for a variety of suspensions are shown in figure 2. The figure contains logarithmic graphs against the frequency of the quantities (e - em) and Au. The value E , is the nearly constant value of E obtained a t high frequencies and can be taken to rcpresent the contribution of the suspending and suspended phases to the dielectric constant, the value of (E - E,) therefore representing the coiitribution of the interphases. The value Ag(w) = u(2w) - u ( w ) represents the change in u rewlting from a
734
HUGO FRICKE AND HOWARD J . CURTIS
twofold increase in the frequency. The change of r with the frequency is small and has riot been measured for all systems. The values of ( E - EJ and A r usually vary approximately as powers of the frequency, although this is not so apparent in the particular measurements shown in figure 2. The exponent for ( E - E-) is negative arid is usually near -1. The difference between the two exponents is approximately equal to 1. The theoretical reason for this relation will be given below. For comparison, we have shown in figure 3 a series of dielectric measurements on soil, which were made by Smith-Rose (23). The moisture con-
l\4
FIG.3. Values of Hose (23).]
(e
-
E=)
112
I
2 4 yI?OCYCLL5
E
I6
32
for soil, moisture content 25 per cent.
1.40,
[From Smith-
1
I
I 1/4
I
64
PER SECOND
4 K 3 C Y C L E S PER SECOYD
6
FIG.4. Test of formula ( 5 ) for 45 per cent kaolin i n 0.5 per cent sodium oleate (values of u , e, and nz from figure 2). The left ordinates represent the values of [(uL - u l ) / ( e l - € , ) b J ] x 1Ol*(0) and (10/36~)[(2~-~ - 1)tan ( m ~ / 2 ) / ( 1- 2-m)](0).
tent of this soil was 25 per cent. In passing we may remark that the rapid increase in the conductance of soils observed at frequencies above 1000 kc. per second can also be accounted for as due to the influence of the interphases (12). The values of A(e - E,),'Aw and of Ar,'Aw obtained for any particular system are theoretically interrelated (10, 22). The relation is generally of a complicated form but becomes simple when ( E - E,) varies as a power of the frequency (10). Let us first consider this special case. Represent the complex impedance of the system by
z = R' + 1 j C f W
WATER-DIELECTRIC
IXTERPHASES
735
and assume
C’
=
Co’(&-“
The following equality holds (equation 6, reference 10) C‘wR‘ = tan (mn/’2) Introducing C and R (as defined above) instead of 6’ and R‘, we obtain
C
=
C’/[l
+ (C’wR’)*] = C O ’ ~ - mcos2 (m7rl2)
and
+
1 / R = 1/R’. [ ( c ’ ~ R ’ ) ~ / l(C’wR’)*] = C0’w (l-m) sin (m7r/2) cos (mx/2) Designate by R1, C1 and Rz, Cz the values of R and C a t t h e consecutive frequencies w and 2w respectively, and we obtain
c1 - c2 = Co’w-m ( 1 - 2-79 cos2 (mn/2) l/R1 - 1/Rz = ( C ~ ’ w ( l - ~ ) (1 ) - 2(l - m)) sin (mn/2) cos (mx/2) (l/Rl - 1/R2)/(C1 - C2)w = (1 - 2(’-”)) tan (m7r/2)/(1 - 2-m) Introduce
(E
-
EJ
and 1/R C
u instead
= u = [(e
- U l ) / ( € l - a)wl
E-)
( 1 0 / 3 6 ~ )X] lo-’*
(5)
and we obtain [(u2
-
of C and R :
x
10’2 = (10/36a) [(2(1-m)- 1) tan ( m n / 2 ) ] / ( 1- 2-m)
This formula was derived with the assumption that (e - e,) varies as a power of the frequency, (E - e = ) ~ w .- ~ Since the right-hand side of equation 5 is independent of the frequency, it follows that Au = u2 - u1 also varies as a power of the frequency with a n exponent equal to (1 m). The difference between the exponents equals 1, as remarked upon above. The variation of the observed values of (e - E,) is only approximately as a power of the frequency, but we can test equation 5 by using for each particular value of w, as the corresponding value of m, t h a t given by the slope of the logarithmic graph for (e - E,) between the frequencies w and 2w. The agreement hereby obtained is usually quite good, the difference between the experimental values of [(UZ - u1)/(q - e2)wI X lo1*and those given b y the right-hand side of equation 5 being seldom in excess of 10 per cent. A typical case is shown in figure 4. The following experiments were carried out to determine the dependence of (e - e,) on the size of the suspended particles of a suspension. Powdered Pyrex glass was separated into three different size groups by sedimentation, the average particle diameters being 80, 20, and 2 . 5 , ~respec~
736
HUGO FRICKE AXD HOWARD J. CURTIS
tively, and suspensions were prepared from each in 0.005 M potassium chloride. The measurements mere made on the settled suspensions, the values of (E - E,) as functions of the frequency being given in figure 5 . The graphs are parallel and the distances between them are AI = 0.50 and A, = 0.95, respectively. These values are within the experimental accuracy equal to the values log (80/20) = 0.60 and log (20/2.5) = 0.90, respectively, showing that ( E - EJ varies inversely as the particle diameter.
'B v w
1/4
FIG.5 . Values of
(e
-
e,)
1/2 1 2 4 8 KILOCYCLES PER SECCND
for suspensions of Pyrex of different particles diameter. (a) 8 0 ~ (b) ; 2 0 ~ (;c ) 2 . 5 ~
FIG.6
Consider a unit element A of interphase (figure 6) and represent the complex admittance of this element in the direction parallel to the interphase as 1/Z, = u s jCsw, where u s is referred to as the surface conductivity and C, as the surface capacity. The value of g S is to represent the total conductance a t the interphase including the conductance due to the ionic atmosphere present in the aqueous phase close to the interphase. I n general the complete representation of the dielectric properties of A
+
WATER-DIELECTRIC
737
INTERPHASES
requires the addition of 1,Z, as the complex admittance in the direction perpendicular to the interphase, this admittance representing the occurrence of electrolytic polarization resulting from the flov of current between the aqueous phase and the interphase. The simplest explanation of the finding t h a t ( E - E , ) varies inversely as the particle diameter is that 1,Z, is so high t h a t its influence is negligible, under the conditions of these experiments. Assuming this to be the case, we shall now proceed to calculate us and C, in ternis of u and ( E - e - ) , limiting ourielveb for the present to the case of a suspension of spherical particle-. The suspended particle itself is assumed to have a negligible conductance and dielectric constant, and the presence of surface admittance is introduced by surrounding each particle with a concentric region of thickness Au, the specific complex admittance of which is given by (1 Au) (us
+ jC,w) +
The admittance of the suspension and suspending fluid are 1, Z = u jC,w, respectively. For a suspension of this type, but for which the different phases are characterized by conductances alone, Maxwell (18) has derived the formula
j C w and l/Z1 = u1
+
(u/u1) ( U / d
- 1- (WuJ - 1
+2
(52,’d
(6)
+2
where u and u1 are the conductivities of the suspension and the suspending fluid, respectively, and a2 is given by u2
=
+ +
(243 (2~3
+ Aa13 - (a+ AaI3 + (us -
~ 2 ()a ~ 2 )
2(~3
62)
u2)
3
a
a3
The values u2 and u3 are the conductivities of the sphere and the surface region, respectively. Taking u2 = 0 and assuming for the present t h a t Aa/a is small, we obtain
a2 = 2u3 Aa a
(7)
We introduce now the complex admittances, as defined above, instead of conductances in equations 6 and 7 and obtain:
738
H U G O FRICKE AND HOWARD J . CURTIS
- P) u =
u1-
[(P
+
+
+ +
( 2 u , / d (1 2P)l [ ( P 2) (2flJad ( 1 - ,411 (1 - P I (1 2P) (2Cswlaud2 (10) - P I (2Cs@/adl2 2) (2da~lFFP)lz
+
+ +
+
+
where C, is the capacity of the quspension for C, = 0. Usually C,w is PO small (compared with aul) that the terms containing (2C,w/a~1)~ can be neglected, in which case we obtain
C-c,=
(18C,lU) . p
[(P
+ 2) + (2X/am) (1 - P)I2 2(1 + ( 2 u , / a d (1 + 2P) + 2) + (2Us’aal) (1 P)
u = u1’
(P
P)
(11)
(12)
Introducing dielectric constants instead of capacities we obtain from equations 11 and 12:
I n figure 7 the values of
(from equation 13) are plotted against 2u,/aul, taking as unity the value of ~ , ) ~ == l 72~.C,,’lOa)l. R e may now return to the experiments in which the relation of (e - E,) to particle diameter was determined. The conductivity of the suspending fluid (0.005 M potassium chloride) was c1 = 0.00067 ohms-‘. The value of us for Pyrex glass in 0.005 M potassium chloride is approximately u s = 1.5 X lop9 (13). The diameter of the smallest particles used was 2a = 2.5 X cm., giving CY = (2/1.25 X (1.5 X 0.03, while for the larger particles a! is still smaller. For 0.7 x such small values of a, equation 13 is approximately p , for different values of (e - e,) for p = 1. [(e -
(e - E,) CY
=
-
+
- E, = [(36a),’10I[(18pCs/a)/(p 2)*I (16) - 6,) varies inversely as the particle diameter 2a, in ac-
E
showing that (e cordance with the experimental result. For highly dispersed systems, according to equation 13, (E - e,) decreases with decreasing particle size, if we can assume C, and u s to be independent of the size. When p is small, (e - e,) reaches its maximal * In equations 13 and 14 C , is given in
ppf.
WATER-DIELECTRIC
ISTERPHASES
730
value for a = (2,a) (us/ul) = 2, or 2a = 2u, u1. For example, for u1 = lop4ohms-’ and u s = 0.5 x 10-9 ohms-1, the maximal value of (e1 - E ~ ) is reached for 2a = 0 . 1 ~ . The dependence of (E - em) on p has been determined experimentally for glass and for kaolin in various suspending fluids. At the higher volume concentrations, the rise of (E - em) with p is grnerally found to be slower than that given by equation 13. To some extent this is due to the non-spherical particle form of the suspensions used, but the chief reason may be the occurrence of aggregation. In heavy suspensions (E - em)
V0.UME
FIG.7 . Representation of
(E
C3YCCVT9ATIOU, p
- e , ) as a function of p , for different values of
a =
/
24 0C
e000
1600
3-
I200
UI
BOO
400 4
8
I2 16 20 24 20 32 VOLUME CONCENTRATION, X
36
40
FIG.8. PIIeasurements of (E - E*) for different volume concentrations of suspensions of ( I ) kaolin (diameter 0 . 5 ~ in ) (a) 1 per cent sodium oleate, (b) 000005 h’ sodium carbonate; and ( 2 ) Pyrex (diameter 1 OF) in 0 oooO5 Jf sodium carbonate. Frequency: 2 kc. per second. T h e broken curves are ralculated from equation (13).
may even decrease with increasing p. Representative data are shown in figure 8, in which the theoretical course of the curves, calculated from equation 13, is shown by the broken lines. The surface conductance is, to an important extent, derived from the ionic atmosphere present in the suspending fluid close to the interphaqe; this part (u,’) of the surface conductance ran be calculated when the electrokinetic potential is known (4, 13), and thus it is possible to obtain that part ( u s ” )of the surface conductance which is derived from the interphase itself. We shall refer to us” as the intrrphasial conductance and to uaN jC,w as the interphasial admittance.
+
740
HUGO FRICKE A N D HOWARD J . CURTIS
The interphasial admittance for Pyrex in solutions of potassium chloride was determined by measurements on suspensions of Pyrex spherules of diameter 1.71; the volume concentration was p = 0.21 and the potassium chloride concentration was varied between 0.0001 and 0.004 M . The results obtained at 0.50 kc. per second are given in table 1, the values of u 8 and C, being calculated from equations 14 and 15. Further details as to the experimental procedure will be found in the reference given (13). The values of C, as well as of us” appear t’obe independent of the salt concentration, their averages being respectively C, = 0.005 ppf and us” = 0.2 X ohms-’ a t 0.50 kc. per second. The values of C, and us’’ as functions of the frequency are given in figure 9. The interphasial conductivity us” can, a t the most, represent only in part a true conductance current a t the interphase. From the fact’ that C, decreases with increasing frequency it follows (10) that the polarization of the interphase is TABLE 1 Dielectric data jor suspensions of Pyrex spherules in solutions of p o t a s s i u m chloride p = 0.21; 2a = 1.7g; frequency 0.50 kc. per second
I
I
I
SVSPENDINQ FLUID
I
C,
m11-
moles
per liter
Conductivity
I
I
ohms-‘ cm.-l
1
ohms-1
1
ohms-1
ohms-’
PPf
0.60 X lo-* 0.52 X lo-’ 0.50 x 10-8 0 . 5 3 X 10-3 0 . 5 7 X lo-’
associated with an energy loss and this loss would be measured as a conductance. Because of the rapid variation of C, with the frequency (approximately as u-l), this conductance would be expected to be many times greater than the admittance C,W and may completely account for the observed interphasial conductance. I n several other cases we have determined the effect of adding potassium chloride to the suspending fluid of suspensions, increasing the electric conductance up to one-hundredfold, and have found C, t o remain wholly or nearly unchanged. It may be assumed t h a t changes would occur under conditions where the electrolyte is adsorbed a t the interphase and this deserves further study. What is immediately important is the fact that these experiments give added evidence t h a t C, originates a t the dielectric side of the interphase. That C, is greatly sensitive to the presence of surface-active molecules, such as soap, is shown by the data in
2 7 h p i e i i w t i i p indiratinn (14, 1.5) that ( ' ii llq2endent upon thc electrj(- chargc mi the interphase, (16 ),ring generally imalleqt when the chargp i. zero 'l'he follon-ing expcrimwits may he cited in support of thiq Pondered Pyres & i s was suipended i p = 0.23) i n three different iol11tio1ii of 1 per rent gelatin at p1-T = 3 0, 4 9, a i i d 7 0, respectively, the ~olufmiiat pH = 4 9 IXGIIEL p r e p a r ~ r frnin i i~nPlectricg~lntin,and the other f n o iolutioii,c by addmg Irydrocliloric~acid and sodium hydroxide, respecfigiii~
E I ( ~ 10 \ ,due i o f d i f f ~ i ~ ipi lt l W I I I P ~
(I
(8)
- e,)
pH
=
for iuqpciiiinni of hnolr t i i n 1 5 [ i f ! cent gelatln for 1 9 ; (11) p H = 2 78; (c) pH = 7 7 2
-oliitioti l'hc t e ~ i i l ta~l e h l i o n i ~111 h g i ~ i e In I'hc diclwtrir roii-txnt vna11wf fot thc i ~ o e l r ~ t ~~oi lcu t i o n hut , is
