A. S. MICHAELS, G. L. FALKENSTEIN, AND N. S. SCHNEIDER
1456
Dielectric Properties of Polyanion-Polycation Complexes
by Alan S. Michaels, Gary L. Falkenstein, and Nathaniel S. Schneider Department of Chemical Engineering, hfaseachusetts Institute of Technology, Cambridge, Massachusetts (Received March 9, 1964)
Stoichiometric poly(vinylbenzyltrimethylammonium)-poly(styrenesulfonate) polysalts containing water and NaBr display very large dielectric constants and loss factors (lo2-lo5), and very marked dielectric dispersion, over the frequency range 102-105 C.P.S. Increasing NaBr concentration increases the dielectric increment, while increasing water content shifts the dispersion to higher frequencies. D.c. conductivities of polysalt containing high conohm-’ cm.-l). These properties are centrations of NaBr are surprisingly low (ca. postulated to arise from microion polarization in discrete, noninterconnecting, microionpolyion domains within the polysalt structure, analogous to the electrical double layers surrounding colloidal particles in aqueous suspensions. Domain concentration and size is determined by the NaBr concentration; microion mobility, by the level of hydration of the polyion network. The applicability of these concepts to interpretation of the anomalous dielectric behavior of biocolloids is suggested.
(I) Introduction
( A ) Background and Objectives. In an earlier study’ of the preparation and properties of polyion complexes, or “polysalts,” formed by the interaction of sodium (polystyrenesulfonate) (NaSS) with poly(vinylbenzyltximethylamnionium chloride) (VBTAC1), cursory examination of the electrical properties of these products revealed unusual dielectric phenomena. It was found that the stoichiometric polysalt, when microion-free and equilibrated a t 50% relative humidity and ambient temperature, exhibited a dielectric constant decreasing from 14 to 11 over the frequency range IO3 to lo6 C.P.S. ; in contrast, complexes containing an excess of one of the component polyions (with accompanying niicroions) , under corresponding environmental conditions, exhibited anomalously high dielectric constants (several thousand) at low frequencies, which decreased rapidly with increasing frequency. These results, coupled with the observation that the complexes have rather high d.c. resistivities, suggested that these polysalts display unique polarization phenoinena under electrical stress, associated in some fashion with the presence in the structure of microions. The objective of this investigation was to examine the dielectric properties of these polyion complexes in greater detail, in an effort to interpret The Journal of Physical Chemistry
these properties in terms of polysalt structure and composition. Anomalous dielectric behavior-in particular, extremely high dielectric constants at low a.c. frequencies (less than 100 c.P.s.) and very broad dielectric dispersion-is a common property of biocolloids, as is summarized in an excellent review by Schwan.2 Numerous explanations have been offered for these observations, many of which call upon the highly organized mosaic structure of cell membranes, or other aspects of structural periodicity in tissue. Another possible, and less elusive, explanation advanced by Schwan attributes the dielectric anomalies to ionic polarization processes in electrical double layers associated with cell membranes; considerable support of this hypothesis has recently been provided by Schwan, Schwartz, et ~ l . , ~ who report quite similar dielectric dispersion in ionic surfactant-stabilized polystyrene latices. This latter work, and its pertinence to the present investigation, will merit greater attention later in this report. (1) A. S. Michaels and R. G. Miekka, J . Phys. Chem., 65, 1765
(1961). (2) H. P. Schwan, “Advances in Biology and Medical Physics.” Vol. V , Academic Press, New York, N . Y., 1957, pp. 147-209. (3) H. P. Schwan, G. Schwarta, J. Maczuk, and H. Pauly, J . Phys. Chem., 66, 2626 (1962).
DIELECTRIC PROPERTIES OF POLYANION-POLYCATION COMPLEXES
Solid, essentially homogeneous, polysalts can be prepared by (1) dissolving the individual component polyelectrolytes in any desired stoichiometric ratio in a ternary solvent comprising water, a water-miscible organic solvent, and a strong electrolyte; (2) allowing the volatile components partially to evaporate; and ( 3 ) gradually leaching out the microsolutes by water washing. Thin films of the complex, suitable for dielectric measurement, can be formed in this fashion. In addition, by equilibrating the resulting complex with an aqueous solution of an electrolyte ( e . g . , NaBr), niicroions of known type and concentration can be sorbed into the polysalt structure. I t is thus possible to produce either stoichiometric or nonstoichionietric polysalts containing widely variable concentrations of counterion or indifferent electrolyte. The majority of the results reported herein relate to stoic’hionietric VBTA-SS polysalts containing variable proportions of sodium bromide and water. ( B ) Characteristics of Complex Dielectrics. By measuring the magnitude and phase angle of the oscillatory current which flows through a capacitor containing a given dielectric material in response to an impressed oscillatory potential, one can calculate for that material a complex dielectric constant, E * , which can arbitrarily be resolved into a real (e’) and an imaginary ( e ” ) component as follon-s E*
E*
=
=
I/EWCO -
ie’’
6 ‘ is defined as the dielectric constant; E”, as the factor. I , E , w , and Co represent, respectively, current, voltage, frequency, and capacitance of geometrically equivalent vacuuni capacitor. As a general rule, e * (and thus also e’ and e ” ) are frequency-dependent . In a hoinogeneous dielectric, this frequency dependence (or “dielectric dispersion”) arises from the fact that certain polarization processes (e.g., dipole rotation) occurring in the material in response to the impressed potential cannot take place instantaneously; hence, such processes will contribute significantly to the dielectric constant below a certain frequency range, but very little above that range. The frequency range over which the real dielectric constant ( E ’ ) changes rapidly defines the time constant of the polarization process. A heterogeneous dielectric, consisting of two electrically-dissimilar phases (one of which is discontinuous) will also exhibit dielectric dispersion even if neither component phase alone exhibits such dispersion. I n this case, charge storage is the result not of molecular or :ttoniic polarization, but rather of the
1457
transport and accumulation of electronic charge at phase boundaries within the material. The tinie constant for such “interfacial polarization” is dependent upon the dielectric constants and conductivities of the component phases and also upon the geometry of the discontinuous phase.5-’2 In their simplest form, the relations describing the frequency-response characteristics of complex dielectrics contain three parameters which are necessary and sufficient to define completely the frequency dependency of both the dielectric constant and loss factor. These parameters are: eo, the dielectric constant measured at low frequency, in a frequency range where e’ is independent of frequency; e,, the dielectric constant measured at high frequency, also in a range where E’ is independent of frequency; and T , the ‘‘mean relaxation time” for the polarization processes giving rise to the “dielectric increment” (to - e-). Of especial importance to this work are the relationships between these niacroscopic parameters and the microstructure of the dielectric : The dielectric increment is a measure of the volume density of polarizable elements within the material and of the polarizability of each element. The “mean relaxation time” is an inverse measure of the mobility of charge carriers within the dielectric. For polysalt compositions of known water, microion, and polyion constitution, dielectric measurements should make it possible to deduce how the various components are distributed in the structure and how they interact with one another. (11) Experimental Samples of sodium poly(styrenesu1fonate) (W, ca. 760,000) and poly(vinylbenzyltrimethy1animonium chloride) (ATw ca. 300,000) were supplied by the Dow Chemical Company, Midland, Mich. Aqueous stock solutions (containing 4Oj, solids by weight) of the two polymers were prepared and purified of electrolyte impurities by percolation through “Rexyn” mixedbed ion-exchange resin. The resulting free acid (HSS) and free base (VBTAOH) polymer solutions (4) C. 1’. S m s t h , “Dielectric Behavior and Structure,” McGraw-Hill Book Co., Inc., New York, N. I*., 1955, Chapter 11. (5) H. Fricke, Phys. Rei’., 2 4 , 575 (1924). (6) H. Fricke, ibid., 26, 678 (1926). (7) H. Fricke, J . A p p l . Phys., 24, 644 (1953). (8) H. Fricke. J . f h y s . Chcm., 5 7 , 934 (1953). (9) H. Fricke nnd H . ,J. Curtis. ibid., 41, 729 (1937). (10) J . C. Maxwell. “ A Treatise on Electricity and Magnetism,” Oxford Press, New l ’ o r k , N. J . . , 1873. (11) K . W. Wagner, A r r h . Elektrotech., 2 , 371 (1914). (12) K. W. Wagner, ihid., 3, 67 (1914).
Volume 69. .\lumber
6
M a y 1966
1458
A. S. MICHAELS, G. L. FALKENSTEIK,
were promptly neutralized to pH 7 with NaOH and HC1. The neutral polyelectrolyte solutions were diluted to 0.4 g./dl. , and the polycation-polyanion equivalence ratio determined by conductometric titration. Using this ratio, measured volumes of the two polymer solutions were mixed to yield a stoichiometric polysalt precipitate, which was recovered from the mixture by centrifugation. The precipitated, “neutral” polysalt was then dissolved in a ternary solvent comprising water, sodium bromide, and acetone in the approximate weight ratio 3.8: 1.5: 1.0. Polysalt concentration was varied between 4 and 18% by weight. Freshly flamed platinum slides (3 X 1.5 x 0.03 cni.) were dipped into these solutions, allowed to drain, and partially air-dried to yield a gelled coating. This coating was carefully stripped off all but a 1.5 cm.2 area, and the slide was washed thoroughly in distilled water to remove sodium bromide and residual acetone from the remaining film and then equilibrated for a t least 24 hr. in an aqueous NaBr solution of controlled concentration. The slide with adhering polysalt filni was removed from the equilibrating solution, carefully blotted free of adhering liquid, and transferred to a desiccator maintained at 51yo relative humidity and 27.8”. Partial dehydration of the polysalt gel layer was accompanied by its consolidation to a transparent, adherent, dense film of quite uniform thickness. By contacting a mercury column (confined in a glass capillary under a constant hydrostatic head) against the exposed surface of the polysalt film on the slide, it was possible to form a parallel-plate capacitor of constant plate area, with the polysalt as the dielectric. This operation was performed within a sealed chamber maintained a t controlled temperature and humidity. After the desired electrical measurements were made, the slide with adhering film was completely desiccated over Pz05, the dry weight of the film determined gravimetrically, and the thickness of the film estimated by microscopic examination (using a filar micrometer ocular) of film fragments which detached from the substrate on drying. The sodium bromide content of the polysalt was determined by flaniephotometric analysis of the desiccated film fragments. Dielectric nieasurements were made using a General Radio Corp. impedance comparator, Type 1605A. Impedances and phase angles of the polysalt films were determined by null balancing against an external resistance-capacitance parallel circuit, at frequencies of lo2, lo3, lo4,and lo5 c.p.s. At the outset of the investigation, several polysalt filnis of identical composition but differing thicknesses (between 11 and 55 p) were preThe Journal of Physical Chemistry
AND
S. S. SCHNEIDER
pared on platinum substrates and their electrical properties compared. It was found that, within the limits of experimental accuracy, film impedances were essentially directly proportional to film thickness over the entire frequency range, and loss tangents were essentially thickness-independent. In addition, it was found that neither variations in impressed potential gradient , nor electrode composition, nor a major change in sample geometry wherein unsupported polyion complex films were subjected to a voltage gradient in the plane of the film,I3 yielded values for e‘ and e ” and their frequency dependency substantially different from the values reported here. I t was thus concluded that neither electrode polarization nor electrolysis contributed significantly to the observed dielectric behavior, and thus that the measured electrical properties were essentially bulk properties of the polysalt materials. Water-sorption isotherms for polysalts containing variable proportions of sodium bromide were independently determined by casting films (as above) on 0.5-mil platinum foil and measuring the weight of the film-coated foil as a function of relative humidity using a standard quartz-helix balance. Flaming and weighing of the foil a t the completion of the isotherm measurements allowed determination of the dry weight of adhering polysalt. The accuracy of the dielectric measurements, based on the sensitivity of the impedance comparator and the accuracy of the precision resistors and capacitors used in the circuit, is estimated to be better than f1%. Since each series of measurements of e’, e”, and tan 6 us. frequency was performed upon a single polymer sample, the reported relative values for each sample are believed €0 be accurate to * l % . On the other hand, the cumulative errors in measurement of sample thickness, electrode contact area, and salt and water content are estimated to be *20oj,; hence, the absolute values of e’ and e ” reported have an accuracy no greater than this latter figure. The tan 6 values, however, which are directly read on the comparator, and which are independent of sample geometry, have a probable error of 10%.
*
(111) Results and Discussion ( A ) Water and NaBr Sorption by Neutral Polysalt. Figure 1 shows the sorption of sodium bromide by (13) Samples of film approximately 1 cm. wide X 3 cm. long X 40 mils thick were clamped a t their ends between brass-plate electrodes, leaving approximately 2 cm. of free film unsupported. Values of e’ and e” calculated from measured impedances and phase angles in the frequency range 100 c.p.s. to 1 Mc. for sodium-containing polysalt a t 50% relative humidity and 25’ were within a factor of two of the values obtained with thin films of comparable complex deposited on platinum.
DIELECTRIC PROPERTIES OF POLYANION-POLYCATION COMPLEXES
zn
- 0A 9 -9I 0.40 - _... -
Q- by flame-photometry
El- from
0 4
dielectric constant
P'
VI
E,
1459
1
NaBr Content 0.46 equiv NoBr per equiv PS 0.23 0.12 0.12 0.0 Temperature = 24.5'C
1
I 0.30-
-.-
I
nc 0.0
e I
I
0.1
0.2
NaBr concentration in solutlon
- equlv
u c c
0.3 per liter
Figure 1. Sorption of NaBr by polysalt.
stoichiometric VBTA-SS polysalt, from aqueous sodium bromide solutions at 23". XaBr contents of the two highest salt-content complexes were determined directly by flame photometry. The values for other three samples were too low for direct measurement, but were estimated from dielectric constant measurements (to be discussed below) which revealed that log E' was essentially linear in NaBr content. In confirniation of earlier results,' the polyion coniplex selectively sorbs S a B r from aqueous NaBr solutions, the selectivity factor decreasing as solute concentration increases. It is significant that, even when the contacting solution is as dilute as 0.1 M in XaBr, the corresponding KaBr content of the polysalt represents more than one niicroion for every four polyion sites. Figure 2 presents water-sorption isotherms (at 24.5') for polysah containing varying amounts of sodium bromide. In all cases, the isotherms are convex toward the sorbate-activity axis and are virtually linear in water activity at low relative humidities. This Henry's-law-like behavior suggests that sorption can be regarded essentially as a process of dissolution of water in the polymer. I;or SaBr-free polysalt, the equilibrium water content at 50% relative humidity corresponds roughly to 2.8 equiv./equiv. of polysalt, or 1.4 water molecules per ion; this is undoubtedly far less than the amount of water required to complete the priniary ion-hydration shell, whereupon a direct proportionality between water content and vapor pressure niight be expected. Indeed, even a t 1 0 0 ~ relao
" 0
t 0.20 c
z
0.10
0.0
-
I R I 0
20
I
I
I
per cent
I
60
40
relative
I
1
80
I
1
100
humidity
Figure 2. Water sorption by polysalt cont,aining different levels of NaBr.
tive humidity, the water content of the salt-free complex corresponds to only about 4 water molecules per ion-about what niight be anticipated for hydration of an associated ion pair. It thus appears that much of the water which is sorbed by these polyion complexes is associated with the ionic functions as hydration shells. The upward curvature of the isotherms a t high humidities is, however, suggestive of capillary condensation; the existence of inicrovoids within the polysalt structure, which will fill with liquid water at high humidities, is quite likely. As Figure 2 shows, water sorption (at a given rclative humidity) increases monotonically with S a B r content of the polysalt. This increased sorptive capacity may arise from two sources: first, from hydration of the microions, and second, from increased swelling of the polyion network due to reduced ionic cross linkage. By relating the increase in water sorption to the S a B r content of the complex, the relative importance of these two factors can be estimated. As Table I indicates, at low relative humidities (below 50y0),the incremental water uptake per equivalent Volume 69,Number 5 Mag 1QF5
A. S. MICHAELS, G. L. FALKENSTEIN, AND N. 5. SCHNEIDER
1460
Table I : Water Sorption by Neutral Polysalt at 24.5’
Relative humidity, %
11.1 33.0 51.0 64.4 75.3 92.5 100.0
NaBr = 0.0
-
-
-Water content, C, molecules/ion-----NaBr 0.23 equiv./ NaBr 0.46 equiv./ equiv. of PS equiv. of PS
AC/
AC/
C
C
microion
C
microion
0.33 0.85 1.5 1.8 2.9 3.3 4.5
0.32 0.86 1.4 1.7 2.7 3.8
0.25 0.90 1.1 1 :3 1.9 5.9
0.34 0.90 1.6 2.1 3.2 7.2
0.35 1.0 1.8 2.6 3.7 15.5
...
...
...
...
of NaBr is essentially equal to the water associated with each polyion pair, suggesting that the salt is functioning primarily as a source of additional ionhydration sites. At higher humidities, and particularly a t higher NaBr contents, the incremental water uptake per unit of NaBr is greater than that predicted for ion hydration alone, indicating that excess water is being accommodated into the polyion network. Thus, only at high water activities does the structure begin to behave as a classical solvent-swollen network, in which the degree of solvation is directly related to the network cross-link density. ( B ) Dielectric Properties of Neutral Polysalt. Effects of Water and N a B r Content. Figures 3-6 show the variations, with frequency, of the dielectric constant (e’), loss factor (e”), and loss tangent (€”/e’) for neutral polysalt films free of NaBr and those containing differing concentrations of NaBr, at 27.5” and 5ly0 relative humidity. (The water content of the samples under these humidity conditions corresponds roughly to 1.4 HzO molecules per ion in the structure, irrespective of NaBr content.) The KaBr-free polysalt exhibits a relatively low dielectric constant (ca. 30 a t 100 c.P.s.) which decreases slowly with increasing frequency. The loss factor is also small hut decreases more rapidly with increasing frequency ; hence the loss tangent (Figure 3) monotonically decreases with frequency in the range investigated. This behavior resembles that of many nonionic, dipolar polymers (e.g., polyvinyl chloride) when plasticized sufficiently to drop their glass-transition teniperature below the temperature of measurement.’* Hence, the dielectric increment of microion-free, neutral polysalt might be deduced to arise from minor displacements of the pendant ionic functions on the macromolecules, the mobility of which is facilitated by the presence of water. However, the dielectric dispersion in the present situation is far broader than that observed with nonglassy polar polymers. Since the The Journal of Physical Chemistry
Relative tiumldlty: 51 % Temperature: 27.5 * C Water Content: 0.14 gma. per gm. dry fllm 0-Dblectrlc Conrtant 0 - L o l a Factor A-Lorr Tangent L
!
3 log,,
Figure 3. Dielectric propertied
frequency
of
4 (cpr)
dL-free polysalt.
dielectric properties of the polysalt are profoundly dependent upon the presence of microion (see below), it must be allowed that “neutral” polysalt, no matter how carefully prepared, may contain trace quantities of residual microion and that the observed dielectric aispersion may be caused by these impurities. As NaBr is introduced in increasing concentration into the complex, the dielectric constant increases very rapidly and its frequency dependency in the lo2lo5 C.P.S.range becomes more and more marked. A value of ( E ’ ) in the neighborhood of lo5 (Figure 3) is observed a t 100 C.P.S. for polysalt containing 0.46 equiv. of NaBr/equiv., which decreases nearly 1000-fold as the frequency is increased to lo5 C.P.S. Furthermore, the dielectric constant-frequency curves are in every case convex to the frequency axis, indicating that, if inflection points are to be found in these curves, they must occur at frequencies one or more decades below 100 C.P.S. Zero-frequency dielectric constants of these materials may, therefore, be in the range of several millions. The loss-factor curves (Figure 5 ) behave similarly with increasing S a B r content; in(14) R . M . Fuoss, “The Chemistry of Large Molecules,” R. E. Burke and 0. Grummitt, Ed., Interscience Publishers, New Tork, N . T., 1943, Chapter IV.
DIELECTRIC PROPERTIES OF POLYANION-POLYCATION COMPLEXES
1461
IT---
L
2
Figure 4. Dielectric constants for polysalt with different contents of NaBr.
deed, changes in e” are nearly proportional to changes in e‘, so that the loss tangents are not particularly sensitive to NaBr content. As Figure 6 shows, the loss-tangent curves are quite flat, and detectable maxima are found in the frequency range studied only for materials containing the highest NaBr concentrations. The absolute values of the loss tangents (Figure 6) are, in nearly every case, extremely highof the order of 0.2-5.0. Certain aspects of these data merit special note: First, all the polysalt samples exhibit an extremely broad dielectric dispersion-extending over a t least five decades of frequency ; second, the dielectric increment appears to be directly related to the NaBr content of the complex and reaches astronomical magnitudes a t high NaBr concentrations; third, the loss conductivity (proportional to w e ” ) of even the highest NaBr-content complex is surprisingly low-of the order of (ohm cni.) -‘-and decreases continuously with decreasing frequency, indicating that d.c. conductivity is very low, and contributes in no significant way to the dielectric properties. This latter observation
I
3 loo
!
4
frequency
(cps)
Figure 5. Loss factors for polysalt with different contents of NaBr.
J IO’
0’ frequency
I0’
(cor)
Figure 6 . Loss tangents for polysalt with different contents of NaBr.
beconies even more striking when it is realized that neutral polysalt cont,aining ca. 0.5 mole of NaBr per equivalent of polysalt is effectively 1.5 m in SaBr. Volume 69,h’umber 6 M a y 1966
A. S. MICHAELS, G. L. FALKENSTEIN, A N D K. S. SCHNEIDER
1462
10.1
Q
0.1
0.2
sodium bromide
0.3
0.4
-
content cquiv per cqulv polyrall
\
\
loglo
frequency
(cps)
Figure 7 . Change in dielectric constant with NaBr content.
Figure 8. Dielectric constants for polysalt containing 0.23 equiv. of NaBr/equiv. of PS, at different water contents.
It is clear that (1) the anomalous dielectric properties of polysalt structures arise primarily from polarization processes involving microion movements; and (2) there is surprisingly little long-range mobility of microions within the matrix, even at high microion concentrations. Furthermore, short-range microion movements within the matrix would appear to be greatly hindered, in light of the extremely low frequencies over which dispersion is observed. The marked dependence and uniform variation of E ’ with S a B r content becomes more evident in Figure 7, where these quantities are plotted against one another with frequency as the parameter. Between lo4 and lo5 c.P.s., log e’ is essentially linear in NaBr content, while a t lower frequencies, deviations from linearity are appreciable, particularly at high salt contents. By making a few direct measurements of salt concentration in the complex along with the corresponding dielectric constant measurements, it is possible to estimate with quite high accuracy the salt content of a sample of material by measuring only its dielectric constant at frequencies of 104-105 C.P.S. This procedure was used to determine the NaBr contents of
polysalt film samples which contained microion concentrations too low for direct analysis. These observations suggest that neutral polysalt may serve as a very sensitive electrical measuring device for determining low salt concentrations in solution with which the complex is in sorption equilibrium. Figures 8-10 show the frequency dependence of e’, E”, and loss tangent for neutral polysalt containing 0.23 equiv. of IYaBr/equiv. of polysalt, as a function of moisture content a t 27.8’. As water concentration increases, e’ and its frequency dependency increase rapidly; the loss factor varies in a similar fashion. The loss tangents (Figure 10) change relatively little with water content at low frequencies, but increase rapidly with water content a t high frequencies. Loss tangent maxima (with one exception), and the frequency at which the maximum occurs, increase with increagng water content. Comparison of Figure 8 with Figure 3 reveals that an increase in KaBr content (at nearly constant HzO/ ion ratio) increases both the magnitude of e’ and the slope of the €’-frequency curve, whereas an increase in water content (at constant S a B r concentration)
The Journal of Physical Chemistry
DIELECTRIC PROPERTIES OF POLYANION-POLYCATION COMPLEXES
*:.
1463
5
I c 0
P c
dim Bromide content: 0.23 equiv per equk p ~ y s a l t lempwatvr: 27.8% 0 - 9 2 . 5 % RH 0 - 7 5 X RH A-64.4% RH 0-51 X RH 0-33 X RH
-: I as
s
1.
IO‘
IO’
frequency
I0’
(cpr)
Figure 10. Loss tangents for polysalt, containing 0.23 equiv. of NaBr/equiv. of PS, at different water contents.
0-51 ISI-33
X RH X RH
‘Ot 1
0
2
3 log,,
4 frequency
J
5
(cps.1
Figure 9. Loss factors for polysalt containing 0.23 equiv. of NaBr/equiv. of PS, at different water contents.
appears to cause primarily a horizontal displacement of the €’-frequency curve. In fact, the curves of Figure 8 can, by simple horizontal superposition, be made to fit a single master curve reasonably closely. It is thus inferred that the effect of water is to alter only the polarization relaxation-time spectrum, whereas that of NaBr is to alter eo and e, (and probably the relaxation-time spectrum as well). If, as suggested earlier, the microions in the polysalt structure constitute the polarizable elements, then surely an increase in their concentration would be expected to increase eo and the dielectric increment. Water evidently serves to “loosen” the structure and facilitate ion motion, thereby reducing the polarization time constant. The observed variations of the loss tangent-frequency curves with water content are also consistent with the plasticizing role of water in the structure. For polysalt containing 0.46 equiv. of NaBr/equiv. trends are comparable to those observed in Figures 8-10, except that both the frequency and water content dependence of the electrical parameters are less
marked, the loss tangents smaller, and the absolute values of e’ and e’’ somewhat larger. These differences are in the direction to be expected consequent to an increase in volume concentration of polarizable elements (hydrated microions), with the attendant increase in microion mobility, within the polysalt matrix. (C) A Model of the Polysalt Structure. The distinguishing electrical characteristics of the polysalts studied here are (1) the astronomical dielectric constants and dielectric increments when NaBr and water are present, (2) the probable existence of loss-factor maxima only at frequencies well below lo0 c.P.s., (3) the extremely broad (4-6 decades) dielectric dispersion, and (4) the presence of loss-tangent maxima in the range of 103-106 C.P.S. For homogeneous dielectrics, not only are static dielectric contents of the order of 105-106 improbable, but dispersion is invariable confined to about two decades of frequency; furthermore, the frequencies corresponding to the loss-factor maximum and loss-tangent maximum are related to one Since none of another through the ratio (eO/e,)’”. these criteria is met by the polysalts, it is not possible to consider them to be homogeneous dielectrics. Alternatively, these structures might be regarded as heterogeneous dielectrics of the Maxwell-Wagner type: that is, a dispersion of a highly conductive (microion-containing water) phase in a low-conductivity (polysalt matrix) phase. However, application of the Maxwell-Wagner-Fricke theory5-I2 to these systems, assuming reasonable values for the conductivity and dielectric constant of a dispersed aqueous electrolyte phase, and using the corresponding quantities for the microion-free polysalt phase measured in this work, leads to predicted values of e’, e ” , and dielectric dispersion which are orders of magnitude smaller than the observed values. Volum.6 69, Number 6 Mag 1968
1464
A. S.MICHAELS, G. L. FALKENSTEIS, A X D S. S.SCHNEIDER
A more fruitful approach to an explanation of these anomalous dielectric properties has followed from an extension of the model and theory developed by S c h w a r t ~ , ’which ~ was successful in explaining the similarly anomalous dielectric behavior of aqueous polystyrene latices. An earlier and somewhat related treatment was developed by O’Konski’6 to explain ionic polarization of polyelectrolytes in solution. Schwartz has shown that, if a nonconductive spherical particle surrounded by an electrical double layer is suspended in a conducting medium, a potential impressed across the medium will cause counterion migration in the plane of the double layer. The extent of counterion polarization within the double layer will depend upon the counterion mobility and surface concentration, and is opposed by the tendency of the ions to counterdiff use and eliminate the surface concentration gradient. This double-layer polarization leads to an increase in the dielectric constant of the system which can, under appropriate circumstances, be substantially greater than that contributed by the bulk phases. The static dielectric increment (per particle) is greater the higher the double layer charge density and the larger the sphere. Furthermore, the relaxation time for the polarization is inversely proportional to the counterion mobility and proportional to the square of the sphere radius; hence, variations in ion mobilities within the double layer, and a spread of particle sizes, can lead to a very broad relaxation-time distribution. Schwartz’s model, therefore, leads to prediction of essentially the same kind of dielectric behavior as is observed in this investigation. The primary requirements for an anomalous dielectric of the Schwartz-type are (1) the presence in the structure of mobile ions electrostatically associated with immobile counterions, ( 2 ) the isolation of regions containing such mobile ions from other similar regions, and (3) a continuous matrix whose conductivity is sufficiently high and/or dielectric constant sufficiently low to ensure that the potential gradient imposed upon the microion-containing regions is high enough to induce significant ion displacement. When the model is expressed in these terms, the similarity between an aqueous dispersion of colloidal particles with double layers and a hydrated, microion-containing polysalt becomes evident. When a simple electrolyte enters a hydrous polyion complex network, it is unlikely that the individual niicroions will distribute themselves homogeneously throughout the network, since the apolar (hydrocarbon) parts of the structure will be inaccessible to them; they will thus be forced to occupy the ionic portions of the matrix in which the ionic cross linkages are localized. The Journal of Physical Chemistry
The approach of a microion pair to an ionic cross link can (and clearly does) lead to a localized breakdown of the link and the formation of two polyion-iiiicroion pairs. Once this process is initiated, osinotic forces will tend to favor accumulation of increasing amounts of microion in that region, thereby causing polyion uncoupling over relatively long chain segments. Thereafter, it becomes possible for spatial rearrangements of these microion-polyion segments to occur (analogous to elastic relaxation), with the resultant formation of discrete and separate (microanion-polycatiori and microcation-polyanion) domains of very low mobility, isolated from one another by intervening, low-microion content, neutral polysalt. It mill be appreciated that this model represents a colloidal (or subcolloidal) dispersion of electrical double layers in a weakly-conducting matrix and differs from the Schwartz model only in the absence of nonconducting particles (whose contribution to the dielectric constant is negligible). According to Schwartz, the static dielectric increment and relaxation time (assuming no distribution) of a dispersion of spherical particles with double layers is approxiniated by
and 7=-
R2 2ukT
where P is the volume concentration of dispersed phase, go is the counterion density in the double layer (em.?), po is the microionic charge, R is the sphere radius, e~ is the absolute dielectric constant of vacuum, u is the microion mobility (cm. sec.-l dyne-’), k is the Boltzmann constant, and T is the absolute temperature. A polysalt containing 0.23 equiv. of NaBr/ equiv. is roughly 20% by volume microion, whereupon P 0.20. If it is assumed that the microion domains are of the order of 100 ‘k. in radius, that the ion density a t the domain boundaries is about loL4 cni?, and that the niicroion mobility is of the order of 104 (us. about 108 in dilute aqueous solution), calculated values of (eo - em) and T are roughly 4 X l o 5 and 1.2 x sec., respectively. The latter corresponds to a characteristic frequency of 130 c.p.s. While little quantitative significance can be attached to these figures, the fact remains that their order of magnitude is quite consistent with the experimental observations. In addition, the observed effect of
=
(15) G . Schwartz, J . Phys. Chem., 6 6 , 2636 (1962) (16) C. T. O’Konski, ibid., 64, 605 (1960).
DIELECTRIC PROPERTIES OF POLYANION-POLYCATION COMPLEXES
water content of NaBr-containing polysalt on the dielectric dispersion is also consistent with the above model; if the primary action of water is to increase the microion niobility (u),the result will be a shift of the dispersion spectrum to higher frequencies without altering the dielectric increment. Support for the premise that the polarizable domains in the polysalt contain niicroions of predominantly one sign of charge was provided by limited dielectric data on a nonstoichiometric polysalt containing 1.3 equiv. of sodium (polystyrenesulfonate) per equiv. of neutral polysalt. This structure contains only Na+ as microion, at a concentration of approximately 0.4 equiv. of Sa+/equiv. of polyion. At 50% relative humidity and 27.8") the values of e ' and e" were, respectively, 3 X lo5 and lo5 at 100 c.P.s., and 5 X lo3 and lo4at lo6C.P.S. These results parallel quite closely those obtained with SaBr-containing neutral polysalt. The absolute values of e ' and e'' are considerably larger, and their frequency dependence is smaller for the Sa-containing complex than for the NaBr-containing complex. The differences are probably a reflection of the higher concentration of the smaller, more mobile, and less polarizable microion (Na+) in the former product. Evidence that nonintercomniunicating, microioncontaining domains in the polysalt matrix are responsible for the anomalous dielectric characteristics is provided by the following observation: if neutral polysalt is equilibrated with a concentrated aqueous electrolyte solution such that the microion sorption corresponds to -1 equiv./equiv. of polyion or greater, the dielectric constant drops to a value well below 50, and dispersion (in the 102-105C.P.S. range) disappears. At the same time, the conductivity ceases to be frequency dependent and approaches in magnitude that of an aqueous solution of the same microion concentra-
1465
tion. Under these conditions, evidently continuous paths for microion diffusion are established throughout the polyion matrix, and normal ionic-conduction behavior results. In summary, the results of this investigation have revealed that microion-containing polysalt structures behave as anomalous dielectrics in the audiofrequency and near-radiofrequency range. Like biological hydrocolloids, they display astrononiically large dielectric constants and loss factors a t low frequencies, exhibit very broad dielectric dispersion, and show large loss tangents. Both dielectric constant and loss factor increase monotonically with water and microion content, the former altering primarily the relaxation time spectrum and the latter the dielectric increment. This behavior can be qualitatively, and in part also quantitatively, explained by postulating that niicroions localized within isolated domains in the polysalt matrix are displaced by an electric field, as are counterions within an electrical double layer. The dielectric increment is determined by the total microion concentration, the domain size, and the local microion density within the domain. The dispersion spectrum is determined by the domain size and the microion mobility. It is suggested that this microion domain concept may prove useful in interpreting the dielectric behavior of many biological systems where the existence of welldefined, double-layer-containing phase boundaries may be in doubt.
Acknowledgment. This investigation was supported by the National Institutes of Health, Biophysics and Biochemistry Subsection, Grant No. GM-08288, and by the Atomic Energy Commission Grant AT(30-1)2574. This paper is a condensation of the doctoral dissertation of G. L. Falkenstein, Department of Chemical Engineering, Massachusetts Institute of Technology, Aug. 1963.
Volume 60,Xumber 6
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