J. Phys. Chem. 1995, 99, 780-789
780
Dielectric Relaxation of Colloidal Particle Suspensions at Radio Frequencies Caused by Surface Conductance G. BlumJ H. MaierJ F. SauerJ and H. P. Schwan*ftt* Mar Planck Institut j2r Biophysik, 6000 Frankfurt a. M. 70, Germany, and Bioengineering Department, School for Engineering and Applied Sciences, University of Pennsylvania, Philadelphia, Pennsylvania I9104 Received: August 17, 1994; In Final Form: October 28, 1994@ We measured the dielectric properties of suspensions of colloidal particles over a broad frequency range from 5 Hz to 13 MHz. The dielectric response is characterized by two distinct dispersions a and /? that occur at low kilohertz and meghhertz frequencies, respectively. The a-response is identical with the counterion relaxation previously observed. The b-response is a Maxwell-Wagner effect resulting from the interaction of the suspended particles surrounded by a surface admittance element with the medium. The particle and surface admittances display relaxation only in the a-range. The /?-effect is strong for particles suspended in an electrolyte of low salt concentration. It is weak for high concentrations and does not significantly contribute to the dielectric response of biological cell suspensions and tissues. The /?-response permits calculation of the surface admittance at high frequencies. The results show that surface conductance and capacitance are independent of frequency above the a-range.
Introduction We report on the dispersion of the dielectric properties of colloidal particle suspensions that occurs at low megahertz frequencies. The effect permits easy determination of the dielectric properties of the suspended particles at these frequencies. We also investigated for the same particles the a-dispersion previously reported and applied the Chew-Sen theory to determine relevant particle properties and surface conductances at low frequencies. We compare these properties with those observed at higher frequencies. The basic phenomenon is due to the existence of a region near the charged particle surface containing space charges. This region has a conductivity that is different from that of the medium surrounding the particle. Usually this region of disturbed electroneutrality extends over a distance that is small compared to the radius for typical colloidal particles. The related mathematical treatment shows that we can under such circumstances only detect the product of this region's conductivity with its thickness, i.e., the surface conductance. Surface conductance values for colloidal particles and biological cells have been reported at dc and low alternating (ac) frequencies. Frequency dependent values are characteristic of the surface of colloidal particle suspension^^-^ and biological cells.2*8-10These changes occur at low frequencies (LF) and coincide with changes in the surface capacitance as demanded They result by the Kramers-Kronig integral from the counterion displacement induced by an alternating electrical field. Considerable theoretical effort took place to analyze this e f f e ~ t . ' ~ - ~ ~ Our data cover the frequency range from 5 Hz to 13 MHz. We note a strong dispersion at higher frequencies. It is apparently a Maxwell-Wagner effect caused by the particles covered with a surface admittance and interacting with the suspending medium. The magnitude of this dispersion would be much smaller without the existence of the surface admittance element. The effect has been observed before by Ballario et ~~~~
t Max Planck Institut fiir Biophysik. t
University of Pennsylvania.
* To whom correspondence should be addressed at 99 Kynlyn Road, Radnor PA 19087. Abstract published in Advance ACS Abstracts, December 15, 1994. @
aL5 and by Sasaki et al.26We show that the surface conductance observed at high frequencies is responsible for this effect. Surface conductance values of colloidal particles are not easy to determine from conductivity measurements. Particle concentrations should be low to avoid particle interaction^.^^ The medium's conductivity must be precisely known since it contributes strongly to the total conductivity a if the volume concentration is low. This follows from eq 1, which is valid for low frequencies and derived by 0-0,
-a+2am
aP - a ,
(a suspension conductivity, am and op medium and particle conductivity, p volume fraction occupied by particles). We show that evaluation of the high-frequency dispersion permits surface conductance determinations at high frequencies without knowledge of the medium's conductivity. Experimental Procedures Particle Suspensions. Suspensions of polybead carboxylate microspheres were obtained from Polysciences Inc. Their surface charge is 0.12 mequiv of COO/g of polymer (manufacturer quotation). The particle diameter was 0.918 pm with a standard deviation of 0.02 pm. We received the suspension from the manufacturer with a highly conducting medium. This medium made the suspension unsuitable for dielectric studies since it caused the resolution to be too low to measure the dispersive effects of interest and increased electrode polarization contributions to intolerable levels. The particles were separated from the medium by centrifugation, resuspended in distilled water, and dialyzed against distilled water for several days. This was followed by centrifugation and drying in a dessicator with CaClZ. Finally, weighted particle samples were resuspended in low-conductivity media. Despite these precautions the sample conductivity increased with time, no doubt due to release of additional ions from the particles into the medium. This effect is particularly noticeable for low conductivity media and made it difficult to establish the correct value of the medium's conductivity. In Table 1 we present some data that compare several medium conductivities am. This includes data before preparation of the suspension and after measurement and
0022-3654/95/2099-0780$09.00/0 0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. 2, 1995 781
Dielectric Relaxation of Colloidal Particle Suspensions
TABLE 1: Comparison of Medium Conductivities a,,, in pS/cm for the L serie# L1 L2 L3
LA L5 L6
“before”
“after”
best fit
“Maxwell- Wagner”
5.1 8.9 13.2 17.8 22.1 26.7
10.4 16.8 18.9 23.3 28.5 36.3
7.7 14.0 18.1 22.1 27.1 35.3
7.3 10.0 18.0 18.2 22.9 30.5
a “Before” data are medium values measured before particles were suspended. “After” data are those of the supernatend fluids gained by centrifugation after measurement. “Best fit” data are obtained by a fitting procedure that adjusts experimental data to eq 12. “MaxwellWagner” values were calculated from the P-dispersion.
coolant inlet
/PC-copnqrs
various electrode distances with calculated values. Inductance effects were small and usually negligible due to the low conductance of the samples investigated. The impedance analyzer gave erroneous readings indicating a false frequency dependence above approximately 1 MHz. These errors were much smaller than the effects of interest to us. Differences between the dielectric constants of the samples and electrolytes were small above the a-dispersion range since the particle volume concentration was so low. Hence, the dielectric properties of the samples and electrolytes of equal conductivities were quite similar. This enabled us to use the following correction method for these small errors. We carried out measurements with several electrolytes with conductivities covering a broad range and characteristic of our particle suspensions. The correction was done by subtraction of the proper electrolyte conductivity variations with frequency from sample data. This correction is based on the valid assumption that the electrolyte conductivity is independent of frequency in our case. We wish to emphasize that the dispersion effect observed in the low megahertz range with electrolytes was very small, about 10-fold smaller in magnitude that noted with the samples. Thus, even complete neglect of the error introduced by the impedance analyzer would not significantly affect the high-frequency dispersion of the samples.
Theory I
connection to micrometer-screw Figure 1. Measuring cell. The cell was designed for shortest possible connections to the HP-impedance analyzer. This minimizes effects caused by lead inductances and stray capacitances. Temperature control was kept at 23 “Csince conductances and time constants change with temperature. Electrode distance variation was used to eliminate electrode polarization effects which are strong in the a-dispersion range.
collection of the supernatant following centrifugation. We also present medium-conductivity data obtained using a fitting technique described below and values derived from the analysis of the high-frequency data. The data demonstrate that agreement between “before” and “after” is poor. Agreement between the “best fit” data and “after” data was better. Still better agreement exists between the “best fit” and “M-W’ gained from the analysis of the high-frequency data. However, this agreement was still insufficient to determine static particle conductivity values apfrom eq 1. For example in the case of experiment L1 a variation of a m from 7.70 to 7.75 pS/cm changes ap from 76 to 49 pS/cm, for a low-frequency conductivity a = 8.05 pS/cm. Measurements. All measurements were carried out at 23 “Cand covered the frequency range from 5 Hz to 13 MHz. We used a Hewlett-Packard impedance analyzer HP 4192A. The thermostated sample cell is shown in Figure 1. The electrode distance could be varied with a micrometer drive. The two platinum electrodes were covered with Pt black. Preparation of electrodes and corrections for electrode polarization, stray fields, and inductances followed procedures outlined before.28 The electrodes were frequently platinized in order to reduce the effect of electrode polarization on the dielectric constant. This effect was noticeable below 100 Hz, and we could identify it by comparison with data obtained with electrolytes of similar conductivity. Correction for this electrode effect was possible using either such comparisons or complex plane plots as shown later in Figures 5 and 6 or by variation of the electrode distance. The effects of electrode polarization on the conductivity data were small. Stray field effects were determined by comparing high-frequency capacitance data obtained with electrolytes at
The dielectric response of the polybead suspensions is characterized by two dispersions as reported in the experimental section. The low frequency effect is the a-dispersion effect reported first by S ~ h w a n . ~The - ~ high-frequency effect p will be identified as a Maxwell-Wagner effect. We first summarize relevant relaxation theories and then methods used to evaluate the particle data and their reduction to surface properties. (A) Maxwell- Wagner Dispersion. This model expresses bulk dielectric data as function of medium and particle properties and explains the observed high frequency dispersion. It can be derived from the complex form of Maxwell’s mixture eq 1 for suspensions of spherical particles exposed to ac:
K-K, - Kp-K, K 2K, ’KP 2K,
+
+
The subscripts m and p indicate medium and particle. The admittivities are defined by K = a j c o ~with a conductivity, co angular frequency, and E dielectric constant. p is the volume fraction. This equation was obtained assuming low concentrations of the suspended phase. Experimental evidence suggests that it is also valid for high concentration^.^^^^ Equation 2 is derived using the Laplace potential equation. This approach neglects the existence of space charges near the interface between particle and medium. However the error resulting from this neglect has been found to be small for particle diameters that are large compared to the Debye length.30.31 Equation 2 can be separated into its real and imaginary parts for any value of 33
+
p:329
(3)
+ (a, - 00)1 +(wn2 (on2
0 = 0,
(4)
The subscripts 0 and 00 indicate limit values at frequencies below and above the characteristic frequencyfo = ‘/2nT. T is a time constant whose value is determined by the ratio of the two
Blum et al.
782 J. Phys. Chem., Vol. 99,No. 2, 1995
dispersion magnitudes AE = E
0
- E, and AU = U, - UO:
conductivity a0 and the additions due to the Chew-Sen and Maxwell-Wagner models as
a=ao+
9 p A 2 ~ , ~ t & ( c o ~ h C - 1)
(1 The properties of particle and medium determine the dispersion magnitudes AE = EO - E , and Au = u, - uo and the time constant T as follows:
+ wt)(l + &E) + Au
with I = dda and 5 = @/RT the dimensionless 5-potential. The first and third terms of this sum are those quoted in eq 4 (dc and Maxwell-Wagner contributions). The second ChewSen term is derived from eq 11 as follows: The complex dielectric constant E* of the suspension is related to the induced dipole moment P by
( * = E - j L L EP 1*+ 2 P w ,l-pP
(7)
Au =
The limit values
E,
and uoare given by
a,- a, - ap-am
a,
+ 2u,
+
(9)
pup 2u,
We shall use these equations to evaluate the @-dispersion. (B) Counterion Relaxation. The low-frequency dispersion is characteristic of suspensions of colloidal This dispersion is caused by counterion r e l a x a t i ~ n . ' ~ -We ~ ~ shall use the Chew-Sen model since it provides simple closed-form expressions of determined accuracy depending on the ratio of Debye length and radius. They are needed for the best-fit procedure outlined below. The theory derives the induced particle dipole moment
1 62 P=--+Ay+A2 2 l-t
3E -jwz2+6ln(l-?)-
(
with 2
T=.E 20'
t=tanh-
F4 4RT
a= ~
+ G i j G - j w z
l
+
G
where 4 is the electric potential at the particle surface, I the ratio of Debye length to radius a,z a time constant, and D the cm2/s here)). p ion diffusion coefficient (D+ = D and a,,,are as defined before. The Chew-Sen model predicts deviations from the circular arc behavior. However, deviations are usually not pronounced. We can now state the total dispersive response of the conductivity as a sum of three terms, representing the dc
and the suspension admittivity is determined by K = jwE*. Separation of real and imaginary parts yields the suspension's conductivity. We carried out this calculation assuming that p is small. The a0 term in eq 12 is determined solely by medium and surface conductivities if the particle core is nonconducting. It can be calculated from eq 17 for w = 0 to transform the surface conductivity into the equivalent particle conductivity up and inserting this particle conductivity in the mixture eq 1. However, this requires knowledge of 0,. Equation 12 is used to apply the best-fit technique described below approximating both a- and p-dispersions. It is used to demonstrate that the combination of the above stated a- and @-theoriescan describe the experimental data. However it is not needed to derive particle data obtained from the analysis of the @-dispersion. (C) Methods Used To Determine Particle Properties. Attempts to tract the dielectric properties of the suspended particles from measured suspension data using eqs 1 and 2 failed. The particle conductivity calculated from eq 1 is sensitive to that of the medium, and Om was difficult to establish accurately as discussed above. We chose therefore several other approaches. (a) We extract particle and medium properties from the @-dispersion. This dispersion is defined by the four limit values EO, E,, UO,u, and the characteristic frequency fo. These values are obtained from the experimental data and plots such as shown in Figures 4-6. They can be used to determine the four particle and medium properties cp,E,,, and up,0,. The dielectric constant of the medium is known to be close to 79 and does not change significantly with the electrolyte concentration. Its knowledge simplifies the procedure. The four limit values obtained from dielectric and admittance plane plots can be used to calculate the T value for the /?-dispersion, using eq 5 . The corresponding characteristic frequency can be compared with the peak frequencies for the circles shown in Figures 5 and 6. Agreement is obtained as demonstrated by the data shown in Table 2. This method to evaluate particle and medium properties assumes that medium and particle properties are independent of frequency through the /?-dispersion range. The validity of this assumption is supported by the one time constant behavior demonstrated by the location of the /?-circle centers on the real axis of the complex plane plots and the identical peak frequencies ~ o ( E )and fo(u) listed for the @-dispersionin Table 2. The numerical evaluation proceeds as follows: 1. We calculate the permittivity ep from e, using eq 10 and assuming the well-established value E , = 79 for water and weak
J. Phys. Chem., Vol. 99, No. 2, 1995 783
Dielectric Relaxation of Colloidal Particle Suspensions .
2 -
X I
X
-calculated from best fit measurement
5
2 0 W
,010~
5
L
- 2 - 5 2 2
102
102 10'
21.0 20.5
'
1
X
lo2
lo3
lo4 f/Hz
io5
ioe
- best fit
71
measurement
I t
n
6
I
20.0
$;-
,419.5 b 19.0
18.5
f/Hz
f/Hz
Figure 2. Comparison of experimental data with "best-fit'' calculation. The conductivity data were fitted since they are not affected by electrode polarization. Deviations of the dielectric constant from the calculation are caused by electrode polarization and agree with estimates of this effect. The figures show data for a suspension of microspheres for the L3 series. Volume concentration 2%. Good agreement was achieved in all series: (a, top) dielectric constant; (b, bottom) conductivity.
Figure 4. Dielectric constant E (top) and conductivity a (bottom) of a polybead carboxylate microsphere suspension as function of frequency. Volume concentration 7.5%. The data display two dispersions a and B centered at low kilohertz and low megahertz frequencies. The increase of E below 100 Hz is caused by electrode polarization not yet corrected for.
--_----__
dw
0.25E
1O-"Efdl8 = up(l - p)
Figure 3. Model used to calculate surface properties G, and C, from (equivalent homogeneous) particle properties E, and a,. The particle with core properties a, and cc is surrounded by a diffuse layer with average properties E, and a, and thickness 6~ . It is surrounded by the medium with properties and a ,.
electrolytes at 23 "C. We get ep = 25 k 5 . An alternate approach is the use of characteristic frequency plots as function of am(Figure 9). Values so obtained also range from about 20 to 30.
+
+
2. We introduce the value of E = ~ ~-(p )1 ~m(2 p ) and the experimental value for the characteristic frequency fo of the /?-dispersion into eq 6 and calculate the sum up(l - p) + om(2 + P). 3. The value for up(1 - p ) um(2 p ) and E are introduced in eq 8, resulting in the final equations
+
+
-(14)
P(1 - P )
+ ~ , ( 2+ p )
(15)
The first equation (eq 14) is solved for upand then the second used to determine Om. Typical a , values are low since we used low-concentration electrolytes to reduce electrode polarization. Thus the second term of eq 15 is often small and, hence, the accuracy of the Om values sometimes low (See Table 1, which compares values calculated from eq 15 with "best-fit" data.) (b) The following procedure is applied to evaluate the a-dispersion of u. We solve eq 1 for u:
with upthe only frequency dependent variable and apply it to the conductivity limits of the a-dispersion. This relates the dispersion magnitude Au for the a-dispersion to the dispersion magnitude of the particle conductivity Aup and eliminates the influence of the large first term of eq 16. Subtraction of this value from the particle conductivity Sph observed at high frequeycies yields the low-frequency limit particle conductivity 0 , .
(c) A best-fit routine was selected which adjusted both medium conductivity and particle properties to fit both dispersions. Adjustable parameters in the Chew-Sen theory include
Blum et al.
784 J. Phys. Chem., Vol. 99,No. 2, 1995
r-----l
6oo
60 X
50 . 0
V
n
\ 40 n x
b”30 I
I
b y 2 0
x
’
x
3
-600
200l 400
600 o 800 1000o 1200
I
n
i
de0 Figure 5. Cole-Cole (dielectric plane) plots of a polybead carboxylate microsphere suspension. Volume concentration 10%. (a) Plot for the a-dispersion (left), (b) #?-dispersiondata (right).The circles touch each other on the real axis as demanded by the model that assumes two relaxation effects. The a-data fit a circle whose center is located beneath the real axis, while the #?-datafit a circle with a center located on this axis. The circles peak at characteristic frequencies of 1.1 Wz and 0.6 MHz. The 00 values used to calculate the ordinate values are the low-frequency limit values of the a- and #?-dispersion,respectively. The huge inflection in Figure 3b is caused by the beginning of the a-dispersion shown in Figure 5a.
n
E
3.5 ’
n 1 4 .
3.0 .
x
\ 2.5 . v, p 2.0 . U
E u
0
”
\
12-
0
aI0.
n
U
I
4.w
-‘?’.O 7.5 8.0 8.5 9.0 9.510.010.511.011.512.0
MI Figure 6. Admittance plane plots. The sample is the same as used for the plots in Figure 5 . The plots are for a- and /?-dispersions(left and right respectively) and touch each other on the real axis. The circle peak frequencies for the #?-dispersionare identical for the plots in Figures 5 and 6. This is indicative for a one-time-constant process. The center frequency of the a-dispersion in Figure 4 is 5 kHz and significantly higher than that in Figure 5. This is typical for a dispersion caused by a distribution of relaxation times. For further details see text and the characteristic frequency data in Table 2. The 6- values that were used to calculate the ordinate values are those listed in Table 2 for a- and #?-dispersions,respectively.
I;, 2, z, and a,. Thus, the a-dispersion could be well fitted. The j3-dispersion could also be evaluated by the program.35We used the nonlinear technique proposed by Levenberg-Marquardt to fit the experimental conductivity data to eq 12.36We choose to fit the conductivity data since they could be measured with greatest accuracy. A comparison of the fitted curve with experimental conductivity data is given in Figure 2a. We next calculated the dielectric constant from the fitting function. Satisfactory agreement with experimental data was obtained if the contributions caused by electrode polarization were neglected (Figure 2b). We used all approaches (a)-(c). The best-fit routine covers the total frequency range, assuming validity of the Chew-Sen theory. The Maxwell- Wagner model covers the &dispersion. It permits particle property determinations from simple equations and is independent of Chew-Sen. (D) Transformation of Particle Properties. The particle properties can be transformed into values for surface fonductance Gs,surface capacitance C,, I;-potential, and surface charge y . We derive the required equations next. ( a ) Surface Conductance and Capacitance. We treat the individual particles as indicated in Figure 3. The core c
represents the particle, and the surrounding shell s that of the adjacent electrolyte where electroneutrality does not exist. It is the region with space charges caused by the charged particle surface and resulting from the application of the field. Its thickness compares with the Debye length &, i.e., is about 10100 nm, that is much smaller than the particle radius. The particle admittivity Kp = ap jmPis defined by the relationship
+
(17) with the subscripts p, s, and c indicating total particle, shell, and core. This relationship was derived for dcZ7and can be shown to be valid for ac. We set [al(a d)I3 = 1 - 3A/a with A being identical with the shell thickness d for d
