ROBERTS. ALWITT
1056
and are a function of l potential. Internally consistent application of these relations accounts reasonably well for the experimental results of Schwan, Schwarz, Maczuk, and Pauly2 without the introduction of arbitrary parameters.
More complete experimental results must be obtained in order to test rigorously both this theory and the basic Schwarz theory. A limitation of this theory is that it does not predict the observed frequency dependence of A& at large values of w ~ .
The Dielectric Dispersion of Paper Impregnated with a Glycol Electrolyte by Robert S. Alwitt Research and Development Laboratories, Sprague Electric Company, A'orth Adams. Massachusetts (Received September 1 3 , 1 9 6 8 )
01 $47
Paper impregnated with an electrolyte of sodium salicylate in ethylene glycol displayed a dielectric dispersion very similar to those observed for colloidal suspensions, except that A d was several orders of magnitude greater than previously reported values. The results were interpreted using the Schwarz theory for the dielectric dispersion of colloidal particles, modified by a relaxation spectrum derived from properties of the diffuse double layer. The large A d was apparently due to a high surface charge density, for which the temperature and concentration dependences were estimated. From the agreement of calculated and observed fiber diameters it was apparent that the processes responsible for dielectric dispersion occurred on the surfaces of the macroscopic fibers rather than on interior surfaces of the swollen fiber.
I. Introduction During the course of an investigation of the electrical propert,ies of paper in nonaqueous electrolytes it was found that the impregnated paper exhibited a very broad low-frequency dielectric dispersion, similar to that observed by Fricke and Curtis'J for aqueous suspensions of cellulose and other colloidal systems. Our interest in this property was with regard to its effect on the electrical behavior of electrolytic capacitors,a in which electrolyte-soaked paper serves as an electrode spacer. Because of the diverse uses for paper it mas thought that the results would be of general interest, particularly since no information has apparently been published on the dielectric properties of paper sheet in electrolyte solutions. The results are interpreted using a theory developed by Schwarz4 for the low-frequency dielectric dispersion of colloidal particles in electrolyte solutions, modified by introduction of a relaxation spectrum derived from properties of the diffuse double layer.
11. Theory It has long been recognized that the dielectric properties of colloidal suspensions arise from the motion of ions, under the influence of an external electric field, within the electric double layer at the solid-solution have shown that a tangential interface. Schwan, et surface conductance is insufficient to account for either the magnitude of the dielectric dispersion or the lowThe Journal of Physical Chemistry
frequency range in which it is found. Indeed, Fricke and Curtis2 had previously shown that a surface capacitance must also be operative. Schwarz4 introduced the idea of a surface diffusion flux acting counter to that produced by the impressed field. This results in a phase shift between current and field on the surface such that a surface capacitance results. This capacitance can occur at low frequencies and results in an apparent dielectric constant for the suspension that may be several orders of magnitude greater than that calculated from Maxwell-Wagner relations. Schwarzhasshown that therelaxation time for counterion motion along a sphere of radius R is r = R2/2ukT, where u is the ionic surface mobility. In general, there will be a distribution of surface mobilities giving rise to a relaxation spectrum. A set of equations describing the dielectric dispersion has recently been derived using the mobility distribution predicted by diffuse double layer theory.6 For small values of the l potential (1 t I < 25 mV) these expressions are, for the dielectric increment e' (1) (2) (3) (4) (5) 66, (6)
- e,
= Ae' =
AQ[I - W
arctan ( 1 / w r o ) ]
T ~
(1)
H . Fricke and H . J. Curtis, Phys. Rev., 47, 974 (1935). H . Fricke and H . J. Curtis, J . Phys. Chem., 4 1 , 729 (1937). R . S. Alwitt, J. Electrochem. Soc., to be published. G . Schwarz, J . Phys. Chem., 6 6 , 2636 (1962). H . P. Schwan, G . Schwarz, J. Maczuk, and H . Pauly, ibid., 2626 (1962). R . 8 . Alwitt, ibid., 7 3 , 1052 (1969).
DIELECTRIC DISPERSION OF PAPER IMPREGNATED WITH
( AeO/2)wro In [(1
+ w2r02)/w2r02]
(2)
where TO
= R2~v/2aokT
(3)
For larger values of the { potential the dielectric increment and dielectric loss are functions of {. For the purposes of examining our experimental results and estimating the magnitude of some physical properties, use of eq 1 and 2 will be sufficient. The paper sheet can be considered to be an array of cylinders (rods, fibers) aligned with their long axis in the plane of the sheet. Hydrogen bonding between fibers maintains the properties of a sheet. Although interactions between particles in such an array would be greater than in a suspension, it might be expected that the Schwarz theory would still describe the essential features of the dielectric dispersion properties of such material. Of course, some of the equations would have to be modified for a suspension of cylinders instead of spheres. Equations 1 and 2 are independent of particle geometry. Equation 3 is valid for a cylinder aligned with its long axis perpendicular to the impressed field. This is because symmetry in the axial direction permits reduction to the same two-dimensional problem as for a sphere. Although solution of Laplace's equation for the potential in the electrolyte and inside the cylinder must be developed in terms of Bessel functions rather than Legendre polynomials, this has no effect on the final relation between surface charge density and surface potential. The static dielectric increment (AtO) for an array of cylinders aligned with their long axis perpendicular to the field (e.g., a paper sheet between two flat electrodes) differs in two regards from At0 for a suspension of spheres. (See eq 36 of ref 4.) First, the additional dielectric constant due to the presence of a counterion layer is for a cylinder only half that for a sphere. This follows from the analysis by O'Konski' of the effect of particle shape and orientation on surface conductivity. Second, the functional dependence of A E ~on volume fraction (p) of particles is different for spheres and cylinders. Fricke8 has presented a general expression for AEOof a suspension of ellipsoids (his eq 28). With the conditions that the conductivity of the electrolyte is much greater than that of the particles and the permittivity of the particles plus counterion layer is much greater than that of the electrolyte, it follows that for spheres 9 =
GLYCOLELECTROLYTE
1057
and for cylinders aligned perpendicular to the field
and for t'he increment of dielectric loss
=
A
where €2 is the permittivity of a particle with its counterion layer. From these relationships and AQ for a colloidal suspension of spheres (eq 46 in ref 4),it follows that for a colloidal suspension of cylinders aligned perpendicular to the impressed field At0 =
When wro
2p (1
+
eo2Rco P ) t,kT ~
(4)
> 1, eq 1 reduces to At' S Aeo/3 ( 0 7 0 )
(5)
For the special case of a 1:1 electrolyte, for which the Debye length is 1 / ~= (DkT/87rneo2)1/2
(6)
substitution of eq 3, 4, and 6 into eq 5 yields
It has been found experimentally for colloidal suspensions that at large values of WT the slope of log Ae' US. log w often has an absolute value less than 2.6 The ramifications of this observation need not be considered in this present work since eq 7 is only used to estimate the temperature and concentration dependence of uo, and the observed average slope of -% was not so different from -2 as to affect significantly these estimates. 111. Experimental Procedure Two types of paper commonly used as electrode spacers in electrolytic capacitors were examined. (The author is indebted for the information in this paragraph to W. A. Selke, Schweitzer Division, Kimberly-Clark Corp., Lee, Mass. 01238.) One, commonly known as Manila paper, is made from the fibero of the abaca plant ( mum teztilus) . These fibers range in diameter from about 16 to 32 p. The pulp receives only a very light beating so these fiber dimensions are retained in the finished sheet, which is quite porous. The particular material used here had a dry thickness'of about 60 p and a void fraction of 0.79. The other paper is commonly known as Benares paper and is made from the fibers of the sunn plant (crotaleria juncea) . The plant fibers are about 15-4Op in diameter, but this pulp is more heavily beaten so that in the paper sheet most of the fibers have been broken down to fibrils of about 3-5 p diameter. The Benares paper is more dense than the Manila, and in these experiments had a
P
4 [l + (p/2)]+
(7) 0.T . O'Konski, J. Phys. Chern., 64, 605 (1960) (8) H.Fricke, ibid., 57, 934 (1953). Volume 73, Number 4
April 2989
ROBERTS. ALWITT
1058
void fraction of 0.65 and a thickness of about 25 H . Other properties of these papers have been reported else~here.~ The parallel capacitance and resistance of electrolyteimpregnated paper was determined at temperatures from -24 to +80° and at frequencies from 60 to 1000 Hz, Solutions of sodium salicylate in ethylene glycol, 0.015-0.500 M , were used as electrolytes. (Sodium salicylate is soluble in a number of organic solvents and was chosen in anticipation of extending this work to electrolytes made with other solvents.) Both papers swelled during immersion in glycol electrolytes. After prolonged immersion the Manila paper was 76 p thick and the Renares paper was 32 p thick.g In order to bring the measured capacitance and resistance within the range of available bridges, the paper was measured in series with an external capacitor, as indicated in Figure la. The bridge measured the equivalent series combination, as in Figure 1b. Values of C, and Rp were calculated from the expressions
The paper thickness was taken as the equilibrium swollen thickness in glycol. Most of the results reported were obtained with a single sheet of paper between the electrodes. Some measurements made with a stack of three sheets of paper yielded the same values for e' as with a single sheet, indicating that electrode polarization did not significantly affect results. The high-frequency limit of the dielectric constant, E,, was estimated from Maxwell-Wagner dispersion relations to be about 25 for Benares-glycol and 30 for Manila-glycol.
IV. Results Some dielectric dispersion curves, typical of those obtained over the range of experimental conditions, are shown in Figure 2. The important features of these data follow. (1) There are exceedingly large dielectric increments for both papers. The maximum values close to lo7 obtained in this work compare with values for Ae' of 104-105 observed in previous studies of colloidal
by means of a trial and error method using a digital computer.
Figure 1. (A) Experimental circuit of paper (Co, R,) in series with capacitor (Cext,Re,t). (B) Circuit measured by bridge.
The impregnated paper was placed between platinized foil electrodes which were connected in series to the external capacitor (1600 pF) and the measuring bridge. The area of an electrode was about 20 cm2. The electrode-paper assembly was placed between glass slides and kept in intimate contact with spring clips. The pressure was not sufficient to prevent swelling of the paper to its equilibrium t h i c k n e s ~ . ~Care was taken to exclude air from the sandwich. The papers were immersed in the electrolyte overnight at 25" prior to use. The parallel capacitance and resistance was related to the reel and imaginary parts of the complex dielectric constant by the expressions dielectric constant dielectric loss
= e"
The Journal of Physical Chemistry
(d/Ae,)Cp
(10)
( d / A t , ) (wR,)-'
(11)
= e' = =
I
I02
I
103
FREQUENCY (HL)
Figure 2. Frequency dependence of dielectric increment. Concentrations and temperatures as indicated. All data for single show results with sheets, except those indicated by three sheets of paper.
+,
(9) R . 9. Alwitt, Electrochem. Technol., 6, 172 (1968).
DIELECTRIC DISPERSION OF PAPER IMPREGNATED WITH A GLYCOL ELECTROLYTE suspensions.2J It was not possible to measure higher values of A& accurately. (2) A linear logarithmic relationship exists between the dielectric increment and frequency. Fricke and Curtis1f2found this dependency for a number of colloidal suspensions over a similar frequency range. ( 3 ) At all concentrations and temperatures the characteristic frequency, fo, was less than the lowest frequency measured. (By definition, f = fo when Ad = Aeo/2.) This follows from some general characteristics of dielectric dispersion curves which are that the maximum curvature of log Ad us. log f occurs in the vicinity of fo and approximations to linear log relationships, extending over several decades of Ad, can only be obtained at higher frequencies. Stated differently, the data in Figure 2 were all obtained a t conditions such that WT > 1. (4) The slopes of these lines ranged from - 1.64 to - 1-90,varying in a random fashion. This is less than the value of -2 predicted by eq 5 . ( 5 ) Both papers exhibited the same dielectric behavior, with Ad for Manila paper always about 7 times that for Benares paper. The temperature dependence of A 4 was examined using eq 7. No reliable data could be found on the viscosity of glycol below 20". Since it is wellestablished that the resistivity and viscosity of dilute electrolytes exhibit the same temperature dependence,I0 the change of resistivity with temperature was used as a measure of viscosity. The agreement with available viscosity data at higher temperatures was excellent. The term DT is almost temperature independent and need not be TEMP
lo-' 162
IO"
IO 0
106,
1059 I
/ 103
I
IOd
I
I
Io-'
IO0
CONCENTRATION
(M)
Figure 4. Concentration dependence of Ad for Benares paper a t 26" and 100 Hz.
considered. To compare results at several concentrations and with the two papers, the ratio of Ae' at two temperatures was evaluated.
k)
'
10
Figure 3. Temperature dependence of A d at 100 HI: 0.5 M ; A, Manila, 0.092 M ; 0 , Benares, 0.5 M .
IO2
0, Manila,
No information was available on the temperature dependence of surface charge density. In Figure 3 , the ratio of dielectric increments at a frequency of 100 Hz is plotted against the temperature and resistivity ratios of eq 12, using 25" as the reference temperature. If surface charge was independent of temperature, the data would be described by a line of unity slope passing through the reference point. The deviations of the data from this line are not great, but suggest a slight temperature dependence for surface charge density. For example, the deviation could be accounted for if the surface charge decreased by about a factor of 2 as the temperature increased from -25 to $80". In Figure 4 de' at 100 H E for Benares paper is shown as a function of solute concentration. The dielectric increment is obviously not inversely proportional to concentration as predicted by eq 7. Again, the deviation from the predicted behavior could be accounted for (10) H. 0 . Jones, "Conductivity and Viscosity in Mixed Solvents,'' Publication No. 80, Carnegie Institution of Washington, Washington, D. C., 1907. Volume 73, Number .I April 1969
ROBERT8. ALwiw
1060 if surface charge increased with increasing concentration according to r0a no.*,assuming the magnitude of 1: remained small. The results with Manila paper showed a similar coilcentration dependence. The total resistance of a colloidal suspension is made up of two parts, a frequency-independent resistance of bulk electrolyte and a frequency-dependent surface r e s i ~ t a n c e . ~ *In~ tgeneral, ~ the first is much larger than the second. The resistance of the electrolyte-impregnated papers that we studied had this same characteristic. Thus, the total resistance was proportional to electrolyte resistivity, over a wide range of concentrations and temperatures, but superposed on this was a small frequency dependence. Some typical results are shown in Figure 5. h similar low-frequency dispersion of electrical resistance was observed by Mason, et al.," for paper pulp in dilute aqueous KCl solutions. The surface resistance decreased with increasing frequency to a constant value at high frequencies.
30t 1
20
I 60'
1
I
1 I
400
I20
f
1
. 1
1000
(HZ)
Figure 5. Frequency dependence of resistance of the paper sheet, 1 cm* in area. Symbols are the same as in Figure 2.
The dielectric loss in eq 2 is a property of the surface process only, and the Ad' for the surface must be calculated from the bulk resistance and the total measured resistance. Reliable values for bulk resistance were not available and the surface dielectric loss of these papers was not of primary interest to us, so this line of experimentation was not pursued.
V. Discussion An evaluation can be made to see if the very large values of At' and small values of fo reported here are The Journal o/ Physical Chemistry
consistent with the known physical properties of the systems. Considering first the Manila paper, the largest At' was 2 X lo7 (Ae'max) obtained at conditions of 80°, 0.50 M concentration, and 120Hz. As we have shown, a t these conditions fo < 120Ilz so Aeo > 2A~'max and AEO/WO > 5.3 X lo4. From eq 1 it is easily determined that wo
0.43/70
(13)
Combining eq 3, 4, and 13 leads to the expression
which is independent of surface charge density. Inserting the right-hand side of eq 14 into the inequality for Manila paper leads to the result R > 5 p. This is consistent with the observation that the fibers of this paper have a diameter of 16-32 /I in the dry state and somewhat greater when swollen. This calculation for Benares under similar conditions leads to R > 3 p . This is slightly greater than expected for a system of fibrils of 3-5-1 diameter (dry), but may be due to the presence of some larger diameter fibrils because of incomplete beating of the pulp. In any event, for both papers the magnitude of Ae' at these frequencies and experimental conditions is consistent with the known physical properties of the paper. By choosing suitable values for R it is possible to estimate a lower limit for u from eq 4 and an upper limit from eq 3 and 13. With R = l o p for Manila and R = 4 p for Benares paper, these limits indicate that the magnitude of u,, is 10l6 omd2 for Manila paper and 10l6cm-2 for Benares paper. Previously reported values for several aqueous dispersions have been in the range 1012-1013cm-2.4J2 Thus, it appears that the large values of Ae' with these papers were due to a very high surface charge density. It was thought that the large uomight be the result of strong specific adsorption of salicylate anions. However, measurements made with KC1 substituted for sodium salicylate gave the same high values of At' and showed the same concentration dependence of As'. Thus, the magnitude of uo was not due to unique adsorptive properties of the anion. It might be noted in passing that the magnitude of At', according to eq 7, depends strongly upon the ratio uO/R. This ratio is estimated to be larger for Manila than for Benares paper, which probably accounts for the higher values of At' obtained with Manila paper. Schwan, et ~ 1 . ~found 5 that the dielectric data for a suspension of polystyrene spheres fell on a circular arc locus in a Cole-Cole plot. Colei3 has shown that (11) D. A. Goring, G. J. Biefer, and 9. G . Mason, Can. J. Res., 28B, 339 (1950). (12) 8 . M. Neale and R . H. Peters, Trans. Faraday Soc., 41, 478 (1946). (13) K.9. Cole, J. Chem. Phys., 9, 341 (1941).
DIELECTRIC DISPERSION OF PAPER IMPREGNATED WITH I
IO
I
I
IO 4
4
GLYCOLELECTROLYTE
A
1061
seemed possible that the dielectric properties of impregnated paper were due to the subfiber structure surrounded by imbibed electrolyte in the pores and channels of the fiber. If this had been the case, the effective particle diameter would have appeared to be several orders of magnitude smaller than was found. It is possible that effects due to this substructure are superposed on those of the macroscopic fibers. For example, the higher than expected values of Ae' obtained at higher frequencies (Figure 6) could be due to a dispersion involving the fiber substructure.
VI. Summary
IO
IO2
I
I
I02
IO4
IO
f
fHr\
Figure 6. Dielectric dispersion in 0.015 M solution 26": 0 , Manila; 0 , Benares.
this behavior is typical of many real dielectrics. The formula derived by Cole13 for the complex dielectric constant of such systems e* = e,
+ (eo
- e,)/[l
+
(iWT)'-"]
has the characteristic that a t intermediate values of WT (1 < WT < 100) a plot of log A d US. log UT has a linear portion with a negative slope with an absolute value less than 2, the value of the slope depending upon a (0 < CY < 1). At still higher values of UT the slope approaches the value of a - 1. For example, if CY = 0.02 the slope changes from -1.68 to -0.98. To see if the paper might show this change in slope, some measurements were made at conditions designed to give very high values of UT. The results are shown in Figure 6. A change in slope was observed but it was greater than predicted from the Cole theory. With Manila paper the slope changed from -1.70 to -0.74 and with Benares paper a still smaller slope was obtained a t the highest frequencies. It is of interest that the results obtained in these experiments indicate that the effective particle diameters are those of the macroscopic fibers or fibrils. Cellulose swells in electrolyte solutions made with suitable s ~ l v e n t s ,the ~ imbibed electrolyte causing a separation between adjacent microscopic strands of cellulose by rupture of hydrogen b0nds.1~ A priori it
Paper impregnated with a glycol electrolyte displayed a dielectric dispersion very similar to those observed for colloidal suspensions, except that Ae' was several orders of magnitude greater than previously reported values. Estimates of v0 suggested that a high surface charge density accounted for the large Ad. Using a distribution of relaxation times based upon ionic mobility in the electric double layer, it was estimated that uodecreased slightly with increasing temperature and increased with solute concentration. The calculated particle diameters were consistent with the picture that the processes responsible for the dielectric dispersion occurred on the surfaces of the macroscopic fibers rather than on interior surfaces of the swollen fiber.
Acknowledgment. The author is grateful to Mrs. E. Vigna for obtaining the laboratory results reported here.
VII. Appendix. List of Symbols A d eo I)
f k n
P R
T I*,
e',
elf
€* t o t em
Afo ST
7 K
P go 7
0
Paper area, omz Paper thickness, cm Electronic charge Dielectric constant Frequency, Ha Boltamann constant Concentration = cations/cm3 for a 1:1 electrolyte Volume fraction of particles Particle radius, cm Temperature, OK Ionic surface mobility Real and imaginary part of complex dielectric constant Complex dielectric constant Low- and high-frequency limits of dielectric constant Static dielectric increment = eo - em Permittivity of free space = 8.85 X F/cm and 7 . 9 7 X 10-2 esu/cm Viscosity, P Reciprocal Debye length of ionic atmosphere, cm-1 Resistivity, ohm-cm Surface charge density, charges/cm2 Relaxation time, sec Angular frequency, sec-1
(14) P. H. Hermans, "Physics and Chemistry of Cellulose Fibers,'' Elsevier Publishing Co., New York, N. Y., 1949.
Volume %.Y
Number 4 April 1969
