5093
J . Phys. Chem. 1987, 91, 5093-5098
Surface Conductance and Other Properties of Latex Particles Measured by Electrorotation W. M. Arnold,* H. P. SchwanJ and U. Zimmermann Lehrstuhl fur Biotechnologie, Universitat Wurzburg, 0 8 7 0 0 Wurzburg, FRG (Received: March 13, 1987)
The dielectric properties of monodisperse latex particles with mean diameters in the range 5-10 wm have been studied by single-particlerotation. The surface conductance is readily deduced from the medium conductivity dependence of the rotation peak seen in the 30-1 000-kHz range. This peak satisfies the equations describing the surface-conductance-modified Maxwell-Wagner dispersion for these particles, with respect to both the optimum field frequency and the speed of rotation. The observed surface conductivities of the particles are constant over the medium conductivity range of 2-16 wS/cm. The magnitude of this conductance depended upon particle type and pretreatment, the range of values being 0.2-2.1 nS. The rotation spectra indicate the presence of an additional rotation peak at field frequencies below 1 kHz. Possible causes of this effect, such as the modification of the field seen by the particle by its own ionic atmosphere or a “negative” dielectric dispersion due to the frequency dependence of electrophoretic motion, are suggested.
Introduction The dielectric properties of polystyrene latex particle suspensions were first extensively measured by Schwan et al.’ The permittivities were very large at low frequencies and exhibited a low-frequency (0.5 lcHz for 1.17-llm-diameter particles) dispersion with a time constant proportional to the square of the particle diameter. The relaxation was interpreted as a frequency-dependent surface conductance. More recent measurement^^-^ on latex particles have given qualitatively similar results in the low-frequency region. However, the magnitude of the reported dispersion is variable, and the effect of medium conductivity is disputed. A smaller dispersion, which was at least related to the MaxwellWagner effect, was observed at higher f r e q ~ e n c i e s . ~ , ~ The early dielectric observations were at that time explained by Schwarz,6 on the basis of the polarization and diffusion-controlled relaxation of the distribution of a surface layer of counterions that are not free to exchange with the bulk medium. Only the tangential movement of ions was treated, although a modified theory’ allowed for radial transport. In addition, Dukhin and Shilovs presented a theory in which transport of ions within the diffuse double layer was also considered, leading to qualitatively similar results. However, the permittivities and time constants predicted by Dukhin differ greatly from those of the Schwarz model and from the experimental results. In the recent past several new theoriesg-I3 have appeared on the subject of counterion polarization. In these theories, the Debye-Huckel length is not regarded as very small; that is, ion transport and electric potential within a diffuse Gouy-Chapman double layer are considered. The importance of the {potential of the particles has been emphasized.I0 Numerical and analytical”J2 solutions are available. The surface conductance is of interest in the interpretation of the above theories because it should be partly determined by the increased ionic concentrations within the double layer. However, measurements of the high-frequency surface conductance have been lacking. Part of this paper concerns itself with such measurements. According to conventional dielectric theory,14J5 the real part of the surface conductance should increase above the dc value as frequency is raised through the counterion dispersion. We show later that one interpretation of the rotation seen in very low frequency fields is that the surface conductance decreases over a limited frequency range. However, there are other explanations based on the fact that the field seen by the particle is modified at low frequencies by the response of its own ionic atmosphere, which may also be undergoing a dispersion at these frequencies. The counterion polarization investigated by dielectric techniques Permanent address: Department of Bioengineering D2, University of Pennsylvania, Philadelphia, PA 19104.
on small particles should be well below the low-frequency limit of the present rotation work, because of the large size of the particles used. The above discussion shows that the surface charge is expected to have a strong influence on the low- and medium-frequency dielectric properties of the particles. In order to monitor the surface charges of the various particle preparations, we measured the electrophoretic mobilities. The results were expressed as { potentials and compared with the rotational results.
Theory General. The forces exerted by a field within a medium on a body within it are functions of the complex permittivities (t*I6) of the body and of the medium (both taken to be isotropic). The real and imaginary parts of complex permittivity, e*, are denoted by t and E’’, and the imaginary part results from the conductivity. At a given frequency w I -j p t -j u / w (1) where j v‘(-1) and u is the conductivity. The transformation between conductivity and permittivity uses the relationship u* jut*. Similarly, a complex conductivity can be defined as CT*
CT
+ ja”
= CT + jut
(2)
The minus sign present in eq 1 but not in eq 2 reflects the usual convention. We now consider p. body (denoted by subscript 2) immersed in a medium (denoted by subscript 1). The calculations that follow (1) Schwan, H. P.; Schwarz, G.; Maczuk, J.; Pauly, H. J . Phys. Chem. 1962, 66, 2626.
(2) Ballario, C.; Bonincontro, A.; Cametti, C. J . Colloid Interface. Sci. 1976, 54, 415.
(3) Sasaki, S . ; Ishikawa, A,; Hanai, T. Biophys. Chem. 1981, 14, 45. (4) Springer, M. M.; Korteweg, A.; Lyklema, J. J . Electroanal. Chem. Interfacial Electrochem. 1983, 153, 55. (5) Lim, K.-H.; Franses, E. 1. J . Colloid. Interface. Sci. 1986, 110, 201. (6) Schwarz, G. J . Phys. Chem. 1962, 66, 2636. (7) Schurr, J. M. J . Phys. Chem. 1964, 68, 2407. (8) Dukhin, S. S.; Shilov, V. N. Dielecrric Phenomena and the Double Layer in Disperse Systems and Polyelectrolytes; Wiley: New York, 1974. (9) Fixman, M. J . Chem. Phys. 1980, 72, 5177. (10) DeLacey, E. H. B.; White, L. R. J . Chem. SOC.Faraday Trans. 2 1981, 77, 2007. (11) Chew, W. C.; Sen, P. N. J . Chem. Phys. 1982, 77, 4683. (12) Lyklema, J.; Dukhin, S. S.; Shilov, V. N. J . Electroanal. Chem. Interfacial Electrochem. 1983, 143, 1. (13) Grosse, C.; Foster, K. R. J . Phys. Chem. 1987, 91, 3073. (14) Schwan, H. P. In Advances in Biological and Medical Physics; Lawrence, J. H.; Tobias, C. E., Eds.; Academic: New York, 1957; p 147. (1 5 ) Pethig, R. Dielectric Properties of Biological Materials; Wiley: Chichester, 1979. (16) We use c to represent the absolute permittivity. That is, c = eoq, where co represents the permittivity of vacuum and cr the relative dielectric constant.
0022-3654/87/209 1-5093$01 .SO10 0 1987 American Chemical Society
5094
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987
are simplified by the definition of a complex factor V . This is effectively a macroscopic application of the Clausius-Mossotti factor:
v =€2* C2*
-
Cl*
+ 2e1*
Arnold et al. and differentiation yields the field frequency cfo)giving maximum torque:
a2* - a,*
-
IT2*
+ 2u1*
(3)
If a rotating field of strength E is set up in the medium, a rotating dipole will be induced in the suspended body. For the case of a spherical particle of radius a this complex dipole p* will be given by (in the MKSA system) p* = 4 ~ a ’ E q V (4) Expressions essentially identical with this have been derived e 1 ~ e w h e r e . l ~The ~ ’ ~ torque ( L ) acting on the body results from the vector cross-product between field and dipole, so that only the quadrature (imaginary) component of the dipole is expected to contribute:
L = -Ep”
(5)
The negative sign indicates that the induced-dipole vector has an angle with respect to the field vector that is in the opposite direction to the direction of rotation. Consequently, cofield rotation is given by negative values of p” (and therefore of U”)and antifield rotation by positive values. The assignment of complex or real character to the e l in eq 4 is open to dispute. Rigorous derivations (quoted by Jones”) based on the use of the Maxwell stress tensor show that the appropriate quantity is indeed the real component ( e l , and not e l * ) . This is also consistent with the results reached in ref 18 and 19. Given that t l is not frequency dependent, then the induced-dipole theory shows very simply that the frequency dependence of the torque is directly due to the frequency dependence of U“. Observed rotation speeds (Q/2s)are less than a few hertz, so the surface of a microscopic particle attains very small Reynolds numbers (no turbulent flow). Therefore, the viscous drag is proportional to CLZ0
N = 87ra37Q (6) where 7 is the absolute viscosity. The steady-state rotational speed (Q,) is reached when N becomes equal to L. Combination of eqs 5 and 6 shows that Q, is independent of radius:
Substituting eq 10 into eq 9 shows that the torque at the maximum frequency will be proportional to
Thef, value (eq 10) is identical with that predictedr4.15for the Maxwell-Wagner dispersion of a very dilute suspension of particles. The numerator of eq 1i may be either positive or negative, and therefore the rotation corresponding to this Maxwell-Wagner dispersion may be either antifield or cofield, respectively. A Dispersive Particle in Purely Conductive Medium. Suppose that the particle itself exhibits a dispersion at a frequency so low that the permittivity of the medium can be neglected in relation to its conductance. If we restrict ourselves to frequencies well below the value given by a1/(2m,), then a >> (r” and therefore a* = u (independent of which considerably simplifies the interpretation. Therefore, at these low frequencies we use the conductivity forms of the equations derived above. Similar advantages have been noticed for conductivity plots in dielectric work.21 If the particle dispersion time constant is T , then
A,
+
u2* = uZt Aa2-1
jwr jwr
+
The term a2tdescribes the low-frequency limit of the particle conductivity, and Aa2 is the increment (usually positive) in conductivity due to the dispersion. The material basis of u21could be surface conductance or conductivity through the material of the particle itself. (In the case of liposomes or biological cells, it would include a term proportional to the membrane conductivity.) Using the same derivation as described earlier by one of us,22 we obtain the imaginary part of u* as
(7)
while the field frequency giving fastest rotation is given by The conductivity expression for U* (eq 3) can be written fr.
U*=1-3
- 1
u2*
*
+ 2a1*
This relationship can be used to transform the particle properties. If the particle shows a dielectric dispersion that appears in the complex conductivity plane as a circle, then the inverse transformation in eq 8 will result in a circle in the U plane (if ul* is frequency independent). The peak in the e” vs: frequency plot that indicates the presence of a dielectric dispersion will also give a peak when the rotation factor U”is plotted against frequency. Therefore, the rotation spectrum conveys the same information as the dielectric spectrum. A Nondispersive Particle. Consider the case where the properties of the particle (a2, e 2 ) and those of the medium (ul, e l ) are independent of frequency. The imaginary part of V is easily derived from eq 3: U“ = 3w
€201
(a2
-
tl‘T2
+ 2u1)2 + w 2 ( 9 + 2 4 2
(9)
(17) Jones, T. B. IEEE Trans. Ind. Appl. 1984, IA-20, 845. (18) Lovelace, R. V. E.; Stout, D. G.; Steponkus, P. L. J. Membr. Eiol. 1984, 82, 157. (19) Sauer, F. A.; Schlogl, R. W. In Interactions between Electromagnetic Fields and Cells; Chiabrera, A,; Nicolini, C.; Schwan, H. P., Eds.; Plenum: New York, 1985; p 203. (20) Lamb,H. Hydrodynamics; Cambridge University Press: Cambridge, U.K., 1906; 3rd ed.. Article 322.
It may be noted that the permittivity (el) of the particle is neglected in eq 12-14. This is illustrated by the consideration that the high-frequency limit of particle conductivity must be capacitive and not the purely real quantity predicted by eq 12. However, eq 2 shows that even at conductivities as low as 6 pS/cm, the error involved in neglecting e l is negligible (
