J . Phys. Chem. 1989, 93, 5865-5869 a . CH,CN reduced a t 563 K b. NH3 decomposed a t 573 K C . NH3 decomposed a t 4 7 3 K
IC\ ------1
1
-360
I
I
-180
0
I
I80
I
I
1
360
u (ppm)
Figure 11. 'H line shape spectra after reduction at 563 K showing the similarity to spectra for NH and NH2 species adsorbed on 7-Mo2N. All spectra were recorded at 298 K. Key: (a) after reduction of CH3"CN at 563 K; (b) after NH3 decomposition at 473 K; (c) after NH3 decomposition at 573 K.
surface species present, although it does affect their distribution. Reduction of the surface species at 563 K results in drastic changes in the species present on the y-Mo2N surface. The most important change is simply that the total numbers of adsorbed I3C and ' H are greatly reduced (see Figures 9 and 10 and Table I). After reduction at 563 K, a I3C spin count reveals that less than 4% of the original I3C adsorbed at 298 K remains. This corresponds to only 5 X 1OIs I3C spins, which makes it extremely difficult to obtain a line shape in a reasonable amount of time. Although the number of 'H present on the surface is reduced, a line shape is easily obtainable. Shown in Figure 11 is the 'H line shape after reduction at 563 K. This line shape appears to consist of a broad and narrow component. The narrow component of the 'H line shape is unaffected by cooling to 77 K. This type of line shape has been observed previously in a study of N H ,
5865
desorption and decomposition on y-Mo2N.l0 The ' H spectra arising from N H , decomposition under vacuum at 473 and 573 K on y-MozN are also shown in Figure 11 for comparison. For NH, decomposition on y-MozN at 473 and 573 K, we have shown that the narrow component of the line shape appears to be the result of isolated N-H surface species, while the broad component is actually a Pake doublet arising from rotating N H 2 groups. It appears that reduction of surface CH3I3CN at 563 K results in the same N H and NH, species as does N H 3 decomposition at similar temperatures. i t 'is important to note a t this-point that these sorts of species do not arise from simple adsorption of atomic H followed by evacuation at progressively higher temperatures.* This is strong evidence that reduction in H2 at 563 K results in cleavage of the C-N bond of adsorbed CH3I3CN and/or CH313CH,NH, species on y-Mo2N. Hydrogenation of the nitrogen atom in these species and subsequent cleavage of the C-N bond leaves the NH, species on the catalyst surface. It should be added that the IH spin-count measurements are consistent with this conclusion. After reduction at 563 K, 44% of the original amount of 'H appears on the surface as N H and NH2. This is what would be expected if, for each original methyl group in a CH3I3CNmolecule, we end up with an N H or N H 2 group after reduction at 563 K.
Conclusions I3C and 'H NMR measurements suggest that CH3I3CNadsorbs molecularly on y-Mo,N, with a fraction of the adsorbate being highly mobile. Adsorbed CH,l3CN can be reduced at 373 K to form adsorbed CH,13CH,NHy species. Reduction at temperatures up to 493 K appear to further hydrogenate the remaining CH3I3CN to CH3I3CH,NHt, species. Reduction at 563 K results in essentially no I3C-containing species remaining on the catalyst, but N H and NH, species remain after cleavage of the C-N bond. Further work utilizing nitrogen-containing molecules more representative of the compounds found in heavy oil feedstocks is being carried out. Acknowledgment. This work was supported by the Director, Office of Basic Energy Sciences, Materials Science Division, of the U S . Department of Energy under Contract DE-AC0376SF00098.
Permittivity of a Suspension of Charged Cylindrical Particles in Electrolyte Solution Constantino Grosse**t Department of Physics, The Catholic University of America, Washington, DC 20064 (Received: August I , 1988; In Final Form: February 24, 1989)
A simple model recently developed for the low-frequency permittivity of suspensions of charged spherical particles in electrolyte solution (Grosse and Foster) is extended to the case of cylindrical particles oriented perpendicular to the applied field. Expressions for the conductivity and the permittivity as functions of frequency are deduced. They show that the relaxation occurs at essentially the same frequency as in the spherical case but, at lower frequencies, the behavior for the two geometries is totally different. The low-frequency conductivity of a suspension of cylindrical particles has the same value as the conductivity of the electrolyte. The low-frequency permittivity diverges logarithmically, while its value at any given frequency scales with the conductivity of the electrolyte.
Introduction Suspensions of charged particles in electrolyte solution generally exhibit a low-frequency dielectric relaxation of high amplitude. It was first observed in suspensions of spherical micrometer-size
* Address for correspondence: Instituto de Fisica, Universidad Nacional de Tucumin, Avenida lndependencia 1800, 4000 San Miguel de Tucumin, Ar entina. 'Permanent affiliation: lnstituto de Flsica, Universidad Nacional de Tucumln, San Miguel de Tucumln, Argentina, and Consejo Nacional de Investigaciones CientTficas y TCcnicas de la Repfiblica Argentina. 0022-3654/89/2093-5865$01.50/0
polystyrene particles by Schwan et al.,' and given a theoretical explanation by Schwarz.' H e concluded that this relaxation is due to the diffusion of counterions moving along the surface of the particle. This interpretation was later objected by Dukhin and Shilov,, who introduced the concept of a diffused double layer ( 1 ) Schwan, H. P.; Schwarz, G.; Maczuk, J.; Pauly, H. J . Phys. Chem. 1962, 66,2626. ( 2 ) Schwarz, G . J . Phys. Chem. 1962, 66, 2636. (3) Dukhin, S. S.; Shilov, V. N. Dielectric Phenomena and the Double Layer in Dispersed Systems and Polyelectrolytes; Halsted: Jerusalem, 1974.
0 1989 American Chemical Society
5866
The Journal of Physical Chemistry, Vol. 93, No. 15, 1989
and concluded that the mechanism controlling this relaxation is the diffusion of ions in the bulk electrolyte surrounding the particle. This conclusion was supported by a series of theoretical studin which the field-induced changes of the concentration ie~,~-lO of ions were calculated taking into account conduction, diffusion, and sometimes convection. All these newer works have in common the approach of writing down the complete set of electrokinetic equations which determine the ion motion. Because of their nonlinearity, these equations can be only solved either numerically or with the help of different approximations. As a result, these theories are extremely complex and usually do not provide analytical expressions for the relaxation parameters. I n a recent study" we presented a simple model for the lowfrequency relaxation of suspensions of charged spherical particles in electrolyte solution, which could be solved analytically. It was based on the assumption that the charged particle is surrounded by a thin layer of counterions which can be represented by a surface conductivity. The value of this conductivity was first assumed to be infinitely high, which led to a dielectric behavior of the system characterized by a broad low-frequency relaxation of high amplitude. This relaxation exhibited a surprisingly good agreement with available experimental data of dilute aqueous suspensions of polystyrene particle^,'^-'^ without using any adjustable parameters. This model was later extended to take into account a finite value of the surface cond~ctivity.'~This was at the origin of an additional high-frequency relaxation of small amplitude. The case of asymmetric electrolytes was also considered. In what follows we shall extend this formalism to suspensions of charged cylindrical particles oriented perpendicular to the applied field. This geometry is important since it is commonly encountered in composite materials and is also applicable to the case of polyelectrolyte solutions. Extensive e ~ p e r i m e n t a l ~ studies * - ~ ~ have been carried out for these last systems. They show that their dielectric behavior is characterized by two relaxations: one in the kilohertz and the other in the megahertz range. Theoretical interpretation^^^-^^ mostly deal with the low-frequency term, which is determined by the motion of ions along the axis of the molecule, and depends on the polyion length. The high-frequency relaxation is usually interpreted as being due to short jumps of the counterions between sites located along the molecule or to the relaxation of "bound" water. I n our approach we concentrate on the high-frequency term and calculate the contribution to the dielectric properties due to the transverse motion of the counterions. This contribution was
(4) Fixman, M. J . Chem. Phys. 1980,.72, 5177. (5) Chew, W. C.; Sen, P. N. J . Chem. Phys. 1982, 77, 4684. (6) Fixman, M. J . Chem. Phys. 1983, 78, 1483. (7) O'Brien, R. W. Ada. Colloid Interface Sci. 1983, 16, 281. (8) Lyklema, J.; Dukhin, S. S.;Shilov, V. N. J . Electroanal. Chem. 1983, 143, I . (9) Chew, W. C. J . Chem. Phys. 1984,80, 4541. (IO) Mandel, M.; Odijk, T. Annu. Reu. Phys. Chem. 1984, 35, 75. ( I 1) Grosse, C.; Foster, K. R. J . Phys. Chem. 1987, 91, 3073. (12) Ballario, C.; Bonincontro, A,; Cametti, C. J . Colloid Interface Sci. 1976, 54, 41 5. (13) Ballario, C.; Bonincontro, A,; Cametti, C. J . Colloid Interface Sci. 1979, 72, 304. (14) Sasaki, S.; Ishikawa, A,; Hanai, T . Biophys. Chem. 1981, 14, 45. ( I 5) Springer, M. M.; Korteweg, A,; Lyklema, J. J . Electroanal. Chem. 1983, 153, 55. (16) Lim, K.-H.; Frames, E. I . J . Colloid Interface Sci. 1986, 110, 201. ( I 7) Grosse, C. J . Phys. Chem. 1988, 92, 3905. (18) Takashima, S. J . Mol. Biol. 1963, 7, 455. (19) Hams, M. Biopolymers 1966, 4, 1035.
(20) Sakamoto, M.; Kanda, H.; Hayakawa, R.; Wada, Y . Biopolymers 1976, 15, 879.
(21) Sakamoto, M.; Hayakawa, R.; Wada, Y . Biopolymers 1978,17, 1507. (22) Mandel, M . Ann. N . Y . Acad. Sci. 1977, 303, 74. (23) Takashima, S.; Gabriel, C.; Sheppard, R. J.; Grant, E. H . Biophys. J . 1984, 46, 29. (24) Cole, R . H . Ann. N . Y . Acad. Sci. 1975, 303, 59. ( 2 5 ) Oosawa, F. Biopolymers 1970, 9, 6 7 7 . (26) Van der Touw, F.; Mandel, M . Biophys. Chem. 1974, 2, 218.
Grosse generally neglected in the previous treatments.
Model We shall now extend the simple model presented in ref 11 to suspensions of charged cylindrical particles in electrolyte solution. We consider that the cylinders are infinitely long, that the concentration is so low that the interactions among particles are negligible, and that the electrolyte is symmetric. This last assumption greatly simplifies the calculations and corresponds well to the case of KCI electrolytes used in most experimental works. For any particle, the electric field vector can be decomposed into a component parallel to the axis of the cylinder and two components perpendicular to it. For linear response, these components can be then averaged over all the possible orientations of the particles in the suspension. The result for randomly oriented particles is that the relaxation spectrum can be obtained considering that one-third of the particles is oriented along each of three orthogonal axes.27 Particles oriented along the applied field will contribute to the very low frequency relaxation due to the motion of ions along the axis of the cylinder. We shall concern ourselves with the much higher frequency relaxation due to the motion of ions perpendicular to the axis of the remaining two-thirds of the particles. We so consider an infinitely long charged cylinder immersed in a symmetric electrolyte. Because of its fixed charge, the particle is surrounded by a layer of counterions. Their equilibrium density is some unknown function of the radial distance to the axis of the particle. We shall use the simplest model for this equilibrium distribution: an infinitely thin shell. We consider that the particle is negatively charged so that the counterions are positive. They can freely exchange with the positive ions from the bulk electrolyte, while the negative ions are excluded from the counterion layer. When a field directed perpendicular to the axis of the cylinder is applied to the system, the counterions redistribute, while charges from the electrolyte form diffuse ion clounds surrounding the particle. We shall assume that these latter field-induced ion distributions are located outside the counterion layer. The motion of the counterions along the surface of the particle in response to the applied field will be characterized by a surface conductivity. We shall assume that the value of this conductivity is infinitely high so that, in all the frequency range of interest, the particle is shielded from the external field. As it was shown in ref 11 and 17, this is a good approximation for most systems except in the limit of very high electrolyte conductivities. We shall use the following nomenclature. The applied field is designated Eo exp(iwt), where w is the angular frequency and t is the time. The particle, together with its counterion layer, is characterized by its radius R and by a surface charge density p exp(iwt) cos 8, where 8 and r are the cylindrical coordinates of a system with its origin located on the axis of the particle. The dielectric properties of the electrolyte are characterized by a generalized conductivity K , = u, + iwtotm, where CT, and t, are the frequency-independent conductivity and relative permittivity, while to is the absolute permittivity of free space. The ions have charges e' = -e- = e, and their mobilities are'U = u= u = eD/kT, where D is the diffusion coefficient, T the temperature, and k Boltzmann's constant. The equilibrium ion densities are Ni = N , while p* exp(iwt) cos 8 are the field-induced variations of these densities. The changes in ion density are related to the current densities J* exp(iot) cos 8, and to the electric potential in the electrolyte U exp(iwt) cos 8, by the continuity and the Poisson equations: &eiwp* = div J* J* = -Neu grad U & eD grad
For weak fields, eq 1-3 lead to2* (27) Fricke, H. Phys. Rec. 1924, 24, 57.
(1) pi
(2)
The Journal of Physical Chemistry, Vol. 93, No. 15. 1989 5867
Permittivity of Charged Cylindrical Particles
= y2p
(4)
027 = 627
(5)
v2p
REAL
/
U
where
I
ep = e(p+ - p-)
(6)
is the charge density, and 7 = p+
+ p-
(7)
is the total change of ion density. The other symbols are y2 = x2(1
+ iF)
(8)
a2 = X2iF
(9)
where
x
= [~m/(~m~)I"~
(10)
is the reciprocal of the Debye screening length, and
F=
(11)
WCOC,,,/CT~
is the angular frequency of the applied field divided by the characteristic frequency of the electrolyte. In all the above equations the coordinate system used has not been explicitly specified. Therefore, they are formally identical for the case of spherical or cylindrical particles. Nevertheless, their solution strongly depends on the particle geometry. The results for spheres are
U = (-Ear p
+ A/r2) cos 0 - ep/(cocmy2)
= B exp(-yr)[(yr)-'
7 = C exp(-dr)[(br)-'
LOG NORMRLIZED FREQUENCY
'
IV. Value of the radial component of the current density of co-ions. Since the negative ions are excluded from the counterion layer, the normal component of their current density must vanish at r = R:
These boundary conditions lead to the following set of equations:
(12)
+ ( ~ r ) - ~cos] O
(13)
+
(14)
cos O
'
Figure 1. Real and imaginary parts of the dipolar coefficient as functions of the normalized frequency, eq 11. Full lines: cylinders, eq 33. Dotted lines: spheres with the same radius as the cylinders, eq 34.
where A, B, and C are integration constants, and r,O are spherical coordinates. For cylinders we obtain
U = (-,For
+ A/r)
cos 0 - ep/(cotmy2)
(15)
= BK,(yr) cos 0
(16)
7 = CKl(6r) cos 0
(17)
p
where K1 is the modified Bessel function of the third kind and first order,29also known as Basset function of order l.30 For a complex argument, this function cannot be separated into real and imaginary parts, which makes the analysis of the cylindrical problem much more complicated than that of the spherical one.
Boundary Conditions The integration constants A , B, C, and the value of the surface charge density p are determined from conditions valid on the surface of the particle: r = R. We shall use the same boundary conditions as considered in the spherical case." 1. Continuity of the potential: U(R,O) = 0
(18)
11. Discontinuity of the displacement:
Z = XR
Dipolar Coefficient The solution of eq 22-25 for the dipolar field coefficient is x , = 1 -1 + H + 2iF(G + H ) (33) While this result is very similar to the one obtained for spherical particles 3 X1=l(34) 2 H iF(G H )
+ +
111. Value of the field induced change of the density of counterions. Since the positive ions can freely exchange with the counterion layer, the variation in their density must vanish at r = R:
(28) Trukhan, E. M. Sou. Phys.-Solid State 1963, 4, 2560. (29) Abramowitz, M.; Stegun, A. Handbook of Mathematical Tables; Dover: New York, 1972. (30) Spanier, J.; Oldham, B. An Atlas of Functions; Hemisphere: Washington, DC, 1987.
+
eq 33 and 34 have a significantly different behavior as functions of the frequency because of the difference in the expressions for the coefficients G and H which, in the spherical case, are]'
+ + 2 ] / ( y R + 1) H = [(SR)2 + 26R + 2]/(6R + 1)
G = [ ( Y R ) ~ 2yR
(35)
(36)
A plot of the real and imaginary parts of the dipolar coefficient as functions of the frequency is presented in Figure 1. For comparison, the corresponding results for spherical particles are also included. The values of the functions G and H , eq 30 and 3 1, have been calculated numerically by using the following expansions for the Basset functions of a complex argument z . ~ I
5868
The Journal of Physical Chemistry, Vol. 93, No. 15, 1989
a . For small arguments IzI K,(z) =
< 5:
1 n-1 (-l)"(n - m - I)!
-
2 m=O
In (z/2) -
c ,=om!(n + m ) ! m
+ (-l)n+'
m!(z / 2)n-z" @(m+ 1)
b. For large arguments IzI
Grosse
(2/2)n+2m
@ ( n+ m
2
+ 1)
2
]
X
(37)
> 5:
In these expressions r and @ are the gamma and the digamma function^,^^^^' respectively. Figure 1 shows that the most obvious qualitative difference between the cylindrical and the spherical cases is the low-frequency limit of the real part of the dipolar coefficient, which tends to zero in the cylindrical case, and to in the spherical one. A much more important difference, not readily seen in this figure, is the way in which the imaginary part of the dipolar coefficient approaches zero at low frequencies, since this behavior determines the static permittivity.
Dielectric Properties of the Suspension The generalized conductivity of the suspension of infinitely long cylindrical particles oriented perpendicular to the field can be easily deduced extending the Maxwell mixture formalism3*to this case:33 (39)
(40)
from which the following expressions for the specific conductivity and the permittivity increments are obtained:
The corresponding formulas for spherical particles only differ from these expressions by having a factor 3, instead of 2, in the right-hand term. I . Conductivity o f t h e Suspension. Equation 41 shows that, for low frequencies, the change of the conductivity of the suspension with respect to the conductivity of the electrolyte is simply proportional to the real part of the dipolar coefficient, Figure 1. This leads to the rather surprising conclusion that the static conductivity of the suspension should be the same as the conductivity of the electrolyte
and should not depend, therefore, on the amount of cylindrical particles present in the system (provided um is kept constant and u is small). On the contrary, for spherical particles, the static conductivity has a higher value:]' gs(0) - o m CUm
3 4
=-
with the same radius as the cylinders.
by a layer of counterions is equal to the conductivity of the electrolyte, while it is 2 times higher for a spherical particle. The high-frequency result is not immediate, since the limiting behavior of the imaginary part of the dipolar coefficient must be investigated. In this limit, the functions G and H reduce to the same expression, since their arguments, eq 8 and 9, tend to the common form Z(F/2)'/2( 1 + i ) . The real and imaginary parts of this argument have the same value, which makes it possible to separate the Basset functions into real and imaginary parts by using the Kelvin function^.^^^^^ The result, in the limit of high frequencies, is H
-
+ (1 + i ) Z F ' / 2
1
(45)
The high-frequency conductivity value can now be determined from eq 41. The result is
For low values of the volume fraction, eq 39 reduces to Ks = Km(l + 2uXl)
Figure 2. Specific permittivity increment of the suspension, eq 42, as a function of the normalized frequency, eq 1 1. The curves correspond to different values of the parameter Z, eq 32, and have been normalized to the square of this magnitude. Full lines: cylinders. Dotted lines: spheres
(44)
This means that, from the standpoint of a static field, the equivalent conductivity of a charged cylindrical particle surrounded (31) Watson, G. N . A Treatise on the Theory of Bessel Functions, 2nd ed.; University Press: Cambridge, 1966. (32) Maxwell, J. C. EIectricity and Magnetism; Clarendon Press: Oxford, U.K., 1892; Vol. I . (33) Rayleigh, J . W . Philos. Mag. 1892, 34, 481.
us(m) =
um( 1
+ 2u)
(46)
It shows that, in the limit of high frequencies, a charged cylindrical particle surrounded by a layer of counterions simply behaves as a homogeneous particle with infinite conductivity. The same interpretation holds true for spherical particles, for which the high-frequency conductivity is given by an equation similar to eq 46, with the coefficient 2 replaced by 3. The total specific conductivity increment can be finally obtained combining eq 43 and 46: us(m)
-40)
UO"
= 2
(47)
II. Permittiuity of the Suspension. The expressions for the permittivity of the suspension are more difficult to deduce since, at lower frequencies, the function G cannot be separated analytically into real and imaginary parts. The only simple result pertains to the high-frequency limit, for which the permittivity of the suspension reduces to t s ( m ) = c,( 1 + 20) (48) For spherical particles, the corresponding expression is similar, except for a coefficient 3 instead of 2 in the right-hand side. The interpretation of eq 48 is consistent with that of eq 46: a t very high frequencies the particle behaves as a perfect conductor. At lower frequencies, the numerical method must be used. The results obtained are represented in Figure 2, where they are compared with the corresponding results for spherical particles. The behavior is fundamentally different for the two geometries since, in the cylindrical case, the permittivity diverges in the limit of low frequencies, while it attains a finite value for spherical particles. This divergence occurs because the imaginary part of the dipolar coefficient, right-hand side of eq 42, tends to zero slower than F. The limiting behavior of the permittivity at low frequencies is
The Journal of Physical Chemistry, Vol. 93, No. 15, 1989 5869
Permittivity of Charged Cylindrical Particles
ion cloud surrounding the particle increases when the frequency decreases. Therefore, the above calculations should only hold while the thickness of this ion cloud is small compared to the cylinder length, and as long as the assumption that the interactions among neighboring particles are negligible is still reasonable. The permittivity changes predicted by the theory are illustrated in Figure 3 as functions of the frequency. The curves correspond to suspensions with a volume fraction u = 0.01, electrolyte conductivity u, = 0.1 S/m, and different values of the radius of the particles. The dependence of the curves on u, can be easily obtained by rewriting eq 49 in the following fashion:
e -
LOG
(FREQUENCY/Hzl
Figure 3. Change of the permittivity of the suspension with respect to the high-frequency limit, as a function of the frequency. The curves correspond to different values of the radius of the cylindrical particles and were computed for a volume fraction u = 0.01 and an electrolyte conductivity u, = 0.1 S / m .
where C, = 0.577 21 5 66 ... is Euler’s constant. Plots of this expression correspond to the stright lines appearing in Figure 2. It can be seen that, at least for 2 > 1, the intersections of these lines with the frequency axis provide a good estimate for a characteristic relaxation frequency of the system: 4 exp(-2Ce) F, = 2 2
which corresponds to a relaxation time r = 0.8R2/D
(51)
This result is practically the same as the one obtained for spherical particles =R ~ / D
(52)
for which the static permittivity is bounded, and is determined by %(O) - %(m) = 9 2 ’ 2- 2 (53) V h 16 Z + 1
+
Discussion The most striking feature of the above results is the divergence of the permittivity when the frequency tends to zero. This qualitative behavior is commonly encountered in a variety of porous materials such as or bone35and is often explained on the basis of fractal geometry models.36 A suspension of charged cylindrical particles in a conductive medium appears to be the only simple system which exhibits such a divergence. Actually, the frequency range for which the permittivity increases with decreasing frequency should be finite in any real situation. The reason for this is that the size of the field-induced (34) Kenyon, W. E. J . Appl. Phys. 1983, 55, 3153. (35) Kosterich, J. D.; Foster, K. R.; Pollack, S. R. ZEEE Trans. 1983, BME-30, 81. ( 3 6 ) Wong, P. Physics and Chemistry of Porous Media; AIP Conf. Proc. 154: Banavar, J., Koplik, J., Winkler, K. W., Eds.; New York, 1987.
At frequencies much below the relaxation, the change in permittivity is simply proportional to the conductivity of the electrolyte, except when 2 = 1 (very small diameters and low conductivities), The lowest curve in Figure 3, R = 10 A, can be compared to measurements on suspensions of DNA. The relaxation frequency of about 100 MHz is typical for these systems, and the dielectric increment of nearly one dielectric unit is lower than, but of the same order as, the observed value^.^^^^^ While it is certain that other relaxation mechanisms, such as ion diffusion along the axis of the molecules or the relaxation of “bound” water, could play an important role in the high-frequency dielectric behavior of these systems, it appears that the contribution of the counterion diffusion perpendicular to the molecule’s axis is not negligible. The highest curve in Figure 3, R = 10 pm, can be compared to measurements on muscle tissue with the fibers oriented perpendicular to the applied field, for which permittivity values as high as lo6 dielectric units at 10 Hz have been rep~rted.~’While the present model cannot be strictly applied to this system, it is interesting to note that it predicts even higher permittivity values (when the appropriate values for the volume fraction and the electrolyte conductivity are used). This suggests that counterion diffusion could have an important contribution to the dielectric properties of tissues, which are yet very poorly understood. In order to test our theoretical results, it would be necessary to perform measurements on well-characterized materials which reproduce, as closely as possible, the model used. A system which appears to be ideal for this purpose is a liquid crystalline suspension of poly(tetrafluoroethy1ene) “whiskers”,38which have a diameter of the order of 200 8, and a length to diameter ratio of the order of 1000. No dielectric measurements are yet available for this system, but they are currently being made at the laboratory of K. Foster and will be reported separately. Acknowledgment. We are grateful to K. Foster and T. Litovitz for valuable comments on the manuscript. This research was supported by contract DAMD 17-86-C-6260 from the Walter Reed Army Institute of Research (WRAIR). (37) Epstein, B. R.; Foster, K. R. Med. Biol. Eng. Comput. 1983, 21, 51. (38) Folda, T.; Hoffmann, H.; Chanzy, H.; Smith, P. Nature 1988, 333, 55.
