2358
Langmuir 1991, 7, 2358-2361
Toward Self-consistent Characterizations of Low Conductivity Dispersions Ian D. Morrison* and Christine J. Tarnawskyj Webster Research Center, Xerox Corp., Webster, New York 14580 Received December 17,1990.I n Final Form: April 19,1991 The charge per unit mass, the electrophoretic mobility, and the size of particles in low conductivity dispersions can be measured independently. These three characterizationsof the dispersed particles are related to each other by a function that depends on the fluid viscosity and particle density. For monodisperse dispersions of insulating spheres this function is ( Q / M ) d 2 / p = 18q/p~. Therefore, in low conductivity dispersions three experimental values can be measured independently and checked to see if they are consistent. Such a test is particularly useful for measurements in low conductivity dispersions since these measurements are usually accepted only cautiously. Deviations of measured quantities from this simple relation can also be interpreted as an indication that the dispersiondeviatea from the assumptions of the simple model, and this is useful information. Several other equivalent functions are derived and all are applied to data taken on various low conductivity dispersions.
Introduction The characterization of nonaqueous dispersions is not nearly so advanced as the characterization of aqueous dispersions. The predominant processes in aqueous dispersions are ionic and these processes have been studied in detail for most of this century.* The predominant processes in nonaqueous dispersions are associated with solubilization, particularly polymer solubilization, and these processes have only been studied in the latter half of this century.2 Ionic processes in nonaqueous dispersions have been studiedw but the conclusions are much less certain than in aqueous dispersions. One reason for this uncertainty is the difficulty in obtaining reliable measurements of the electrical properties of low conductivity dispersions. This paper describes several testa for the consistency of experimental data as aids to judge reliability. Self-Consistent Relations between Measured Quantities The charge, qm, on a spherical, insulating particle, is related to its surface potential, fm, by where dm is the particle diameter, D is the dielectric constant of the medium, and €0 is the permittivity of free space. The mass, m, of a sphere is
where p2 is the density of the particle. Combining eqs 1 and 2 gives (1) Verwey, E.J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: New York, 1948. (2) Napper, D.H. Polymeric Stabilization of Colloidal Dispersions; Academic: New York, 1983. (3) van der Minne, J. L.; Hermanie, P. H. J. J . Colloid Sei. 1952,7,600. (4) van der Minne, J. L.; Hermanie, P. H. J. J. Colloid Sci. 1953,8,38. (5) Koelmans, H.; Overbeek, J. Th. G. Discuss. Faraday SOC.1954,18, 52. (6) Klinkenberg, A.; van der Minne, J. L. Electrostatics in the Petroleum Industry, The Prevention of Explosion Hazards; Elsevier: New York, 1958. (7) Fowkes, F.M.; Jinnai, H.; Mostafa, M. A.; Anderson, F. W.; Moore, R.J. ACS Symp. Ser. 1982, No. 200, 307. (8)Pugh, R.J.; Mataunaga, T.; Fowkes, F. M. Colloids Surf. 1983,7, 183. (9) Fowkes, F. M.; Pugh, R.J. ACS Symp. Ser. 1984, No. 240, 331.
(3) Equation 3 is true for each particle. Making the approximation that the relation also applies to the average of each of the experimental quantities, then (4)
where Q/M is the average charge to mass ratio, d is the average particle diameter, and l is the average surface potential. This approximation turns out to be quite sufficient for our purposes. When the ionic strength of the medium is low, the electrical double layers are extended, and the surface potential is related to the electrophoretic mobility, pm, by the Hiickel equation*O
where q is the fluid viscosity. Substituting eq 5 into eq 4 and using the average quantities for the surface potential, l, and the electrophoretic mobility, p, give
(Q/Wd2= P
(6)
p2
Equation 6 is particularly useful because it relates three quantities measured directly by various experiments to two material properties of the dispersion. One of the common techniques to measure the average charge to mass ratio of particles in a dispersion is to measure the current passing through a cell as particles plate against an electrode." This method assumes that the particles are the predominant charge carrying species. If they are, then the dispersion conductivity, C, can be approximated by C = npq (7) whsre n is the number of charges per unit volume. The term nq can be replaced by the charge to mass ratio to give
C where
Q,
@P~c((Q/W
is the mass fraction of particles and
PI
(8) is the
(10) Ross,&;Morrison,I. D. ColloidalSystemsand Interfaces;Wiley: New York, 1988, p 346f. (11) Novotny, V.; Hair, M. L. J. Colloid Interface Sci. 1979, 71, 273.
0 1991 American Chemical Society
Langmuir, Vol. 7, No. 10, 1991 2359
Characterizations of Low Conductiuity Dispersions density of the liquid. Substituting eq 8 into 6 gives (Q/W2d2
~
189
C @PIP2 Equations 4, 6 , and 9 are written with experimentally determined quantities on t h e left-hand side and known material properties on the right-hand side. Therefore, once t h e experimental quantities are measured, one of these equations can be used to check for consistency. Dispersions with distributions of sizes or mobilities must be described by modified equations, b u t much can be learned from t h e application of these simple expressions. Before giving examples of the applications of these three self-consistency relations, a short description of the usual experimental methods to determine the quantities on t h e left-hand sides is necessary. The experimental methods for making electrical measurements in low conductivity media are different than those for aqueous media.
Experimental Methods Average Charge per Unit Mass, QIM. The average Q / M can be measured directly for low conductivity dispersions but can only be calculated for high conductivity dispersions (especially aqueous dispersions). The average Q / M can be measured in two ways. The first is to plate out the particles on an electrode with an applied electric field." The electric current that flows and the mass of the particles that plates are measured. The integral of the current gives the total charge transferred. The average charge to mass is the ratio. Various approximations can be made to estimate the amount of current carried by ionic species even when the conductivity is moderate.12 The most likely source of error is ascribing too much of the current to particle motion and, therefore, obtaining too high avalue for QIM. The secondmethod to determine Q / M is to hold the particles on a porous support, wash the counterions away with a flow of solvent, collect them, and measure the total charge contained in the electrical double layers.lg The most likely source of error is that the flow of solvent does not eliminate all the charges in the double layer. In that case the Q / M value would be too low. Particle Size, d. The average particle diameter can be determined by a number of techniques.1' The particle sizing technique cannot use electrical sensing for dispersions of low conductivity. Also, the sizing equipment has to be compatible with organic solvents and many centrifugal and hydrodynamic techniques are not. Quasi-elastic light scattering (QELS) is appropriate when the size is small and the dispersions can be diluted.l6J8 Light scattering techniques are most sensitive to large particles so that the average size determined by light scattering may be more biased by large particles than the averages of the other experimental quantities. When light scattering averages are used in eqs 4,6, and 9, the most likely error is that the particle diameter is too large. Electrophoretic Mobilities, p, and Surface Potentials, t. The electrophoretic mobilities are measured in several ways: the usual four of electrophoresis, electroosmosis, sedimentation potential, and streaming potential10 as well as by optical transients" and by ultrasonics.1*21 It is impossible to say in general what the most likely direction of error will be for any of (12) Novotny, V. Colloids Surf. 1986,21, 219. (13) Morrison, I. D.; Graves, A. M.; Tamawekyj, C. J. Presented at the SPSEs 6th Int. Congress on Non-Impact Printing, Orlando,FL, October 24, 1990; submitted Langmuir, in press. (14) Barth, H. G., Ed. Modern Methods of Particle Size Anulysis; Wiley-Intemience: New York, 1984. (15) Dahneke,B.E.,Ed.Measurementofsuspendedparticles byquasielastic light scattering; Wiley-Interscience: New York, 1983. (16) Morrison, I. D.; Grabowski, E. F.; Herb, C. A. Langmuir 1985,1, 496. (17) Novotny, V. J. Appl. Phys. 1979,50, 324. (18) Debye, P. J. Chem. Phys. 1933,1,13. (19) OBrien, R. W. J. Fluid Mech. 1988, 190, 71. (20) OBrien, R. W. J. Fluid Mech. 1990,212, 81. (21) Marlow, B. J.; Fairhurst, D.; Pendee, H. P. Langmuir 1988,4,611.
these techniques. Some comparisons have been made.22 The electrophoretic mobilities discussed in this paper were measured by optical transients," electrophoresis," and ultrasonics.u Optical transients can be used when the charge carried by the particles is small compared to the total charge on the electrode. If the charge carried by the dispersion is too large, then space charge regions build and this method cannot be used. For electrophoresis, the measured velocity of the particles divided by the applied electric field is the mobility. Electrophoretic measurements in low conductivity dispersions require careful experimental design in order to maintain a known electric field." For the electrophoretic mobilities measured by ultrasound reported in this paper, the surface potential of the particles is calculated from the amplitude of the emitted sound.20 The mathematical description of the ultrasonic method is based on relating the force on a particle (mass times acceleration) with the force due to an electric field (charge times field).'* This method could be used in principle to measure the charge to mass ratio for particles at any conductivity. Conductivity, C. The electrical conductivity of nonaqueous dispersions has to be measured so as to avoid space charge and spurious capacitive effects.% Large area, parallel plate cells with small (about 1 mm) gaps are used to amplify the signal. The cells are often concentric cylinders. A common method is to apply a 5-50 V, 5-50 Hz,pulse. If the capacitance is small, then the conductivity is measured directly. Spurious capacitances can be nulled.% Using the conductivities obtained with this method with the static measurements assumes that conductivity is independent of frequency at low frequencies.
Examples and Discussion The experimentally measured data have t h e following units: conductivity, C, in pS/cm; charge to mass, Q/M, in pC/g; particle diameter, d, in pm; electrophoretic mobilities, p, in cm2/(V 8 ) ; surface potentials, l,in mV. The known material properties have the following units: viscosity, 7, in cP; density of liquid, P I , in g/cm3; the density of particle p2, in g/cm3. Mass fraction, 0,and dielectric constant, D , are unitless. If the data are reported with these units, then the following forms of the self-consistency relations can be used: (Q/M)d2 =-0.10620
r
Pz
(Q/M)'dZ
C
~
1.81 @PlP2
(4')
(9')
To demonstrate applications of eqs 4', 6 , and 9', we have gathered data for several low conductivity dispersions: dyed polymer colloids in dodecane, carbon black also in dodecane,and an electrophoretic liquid toner in a synthetic hydrocarbon. Three independent quantities were measured for each dispersion and those quantities are checked for self-consistency.
Dyed Polymer Colloids in Dodecane. Croucher et al. have described a general technique for preparing sterically stabilized polymer colloids in hydrocarbon^.^' T h e y provided us with two samples: (1)a 1.5 w t % poly(vinylpyrrolidone) (PVP) polymer dispersion, stabilized with poly(2-ethylhexyl acrylate) in dodecane, dyed with Ora(22) Kitahara, A. In Electrical Phenomenu at Interfaces; Kitahara, A., Watanabe, A., Eds.; Dekker: New York, 1984; p 133, Figure 5.6. (23) Kornbrekke, R. E.; Morrison, I. D.; Oja, T. Submitted for publication in Langmuir. (24) Matec Instruments, 75 South St., Hopkinton, MA 01748. (25) Novotny, V. ACS Symp. Ser. 1982, No. 200,281. (26) Scientifica, 340 Wall St., Princeton, NJ 05840. (27) Croucher, M. D.; Lok, K. P.; Wong, R. W.; Drappel, S.; Duff, J. M.; Pundsack, A. L.; Hair, M. L. J. Appl. Polym. Sci. 1986,30,593.
Morrison and Tarnawskyj
2360 Langmuir, Vol. 7, No. 10,1991
Table IV. Carbon Black Charged with a Stearic Acid Metal Salt
Table I. PVP Polymer Colloid Stabilized with Polyacrylate QIM,rC/g
d, r m
10% cmZ/(V 8)
(QlM)d21c
24.1
0.54
0.3
23
18~1~2 25
880
d, am
C , PSIcm
(Q/M)2d21C
11.1
1.11
15.8
9.6
d, rm
5; mV
(Q/M)d21t
-23
0.4
-28
0.13
3.3
0.1062Dl~2 0.21
sol Red G (Ciba-Giegy), and charged with 0.015 wt 5% lecithin and (2) a 1.5 wt 5% PVP polymer dispersion, stabilized with adsorbed Elvax 4320,an ethylene/vinyl acetate/acid terpolymer from Du Pont, in dodecane, dyed with Orasol Blue SGLN, and charged with 0.015 wt 5% lecithin. The characteristics of the red PVP dispersion are given in Table I. The average charge to mass ratio was measured by the washing technique.13 The dispersion was diluted in a lecithin solution of the same concentration and the particle size measured by QELS and the electrophoretic mobility measured by microelectrophoresis.25 As can be seen in Table I, the three measured quantities are quite consistent with the fluid viscosity and particle density. The characteristics of the blue PVP dispersion are given inTable 11. This dispersion is quite unstable, even diluted. The average charge to mass ratio was determined by the washing technique.13 We obtained some estimate of particle size by QELS but the dispersion flocculated too quickly to measure mobility by electrophoresis. Therefore we measured its conductivity and used eq 9' to check for self-consistency. The agreement is not so good, but considering the uncertainty in characterizing an unstable dispersion, this check for consistency increases our confidence in the measurements somewhat. One possible explanation for the discrepancy is that the particle size measured by QELS is too large because of flocculation. Carbon Blacks Dispersed in Dodecane. Morrison et al. have reported the average charge to mass ratios, measured by the washing technique and the surface potentials, measured with a Matec ESA 8000,for a number of 2 vol 5% Sterling R carbon black (Cabot) in dodecane dispersions as a function of added OLOA 1200, a poly(isobutylenesuccinimide) lubricating oil additive from Chevron.13 The charge to mass ratio and the surface potentials are measured on the dispersions without dilution. The average particle diameter was measured by QELS after dilution in dodecane. Table I11 shows the test of the consistency of the three measured quantities for the 24 wt ?6 OLOA/carbon dispersion with the dielectric constant of the dodecane and the density of carbon. The agreement is reasonable especially considering that this dispersion is certainly polydisperse in both size and charge and only averages are used in this comparison. Novotny has also studied the electrical properties of low conductivity dispersions and proposed a number of useful techniques to characterize their electrical properties.11J2J7~26 He characterized a dispersion of carbon black in Isopar G (Exxon) charged with a metal salt of stearic acid.28 The average charge to mass ratio was measured by the electrical plate out method, the mobility by the optical transient, and the particle diameter by QELS. The values (28) Novotny, V. Colloids Surf. 1981,2, 373.
26
18
1.8~16~1~2
Table 111. Sterling NS Charged with OLOA 1200
QIM,aC/g
3.1
Table V. Electrostatic Liquid Toner
Table 11. PVP Polymer Colloid Stabilized with Elvax 4320 Q I M , aC/g
0.3
56
5.5
2.4
705
39
are shown in Table IV and compared to the solvent viscosity and particle density. The agreement is reasonable. Novotny used eq 6 to calculate the particle diameter from the two electrical measurements and concluded that the light scattering results were a little off, possibly due to the presence of a few large particles. Particle sizing techniques have improved substantially since that publication and are now more reliable than the electrical measurements. An Electrostatic Liquid Toner. A new type of electrostatic liquid toner is being ~ t u d i e d . ~These Q toners are pigmented polymer particles dispersed in a high molecular weight alkane. The average charge to mass ratio was determined by the washing technique. The average particle size and electrophoretic mobility were measured after dilution in clear supernatant fluid. As shown in Table V the measured values are inconsistent with the liquid viscosity and particle density. The experimental measurements were checked carefully. The experimental inconsistency must follow from some systematic difference between the simple model and the way this sample behaves. These particles are believed to deflocculate in an electric field. If they do deflocculate in an electric field, then the mobility measurements are with deflocculated particles and the light scattering measurements are with flocculated particles. This could easily explain the inconsistency. This example shows the application of these self-consistency relations to obtain, or confirm, other information. One area of active research on nonaqueous dispersions is the mechanism by which surface active solutes impart dispersion stability. Often the simple question of whether the solute stabilizes the dispersion by forming a steric barrier or whether the solute stabilizes the dispersion by forming an electrostatic barrier, or both, is not easy to decide. A number of studies have found that the surface potential of the particle goes through a maximum with increasing concentration of ~olute.~OJ~ While a number of reasonable explanations have been put forth, another suggestion can be made by considering the ramifications of eq 4. Consider the effect of adding an ionic surface active solute to a dispersion. As the concentration of the solute at the particle surface increases, two simultaneous effects are possible: first, the charge per unit area on the particle could increase; second, the particle size could decrease because of deflocculation. Equation 4 can be rewritten to give ( Q / A ) d = 2DcJ (10) where A is the surface area of the particle. Equation 10 shows that the surface potential is proportional to the product of the charge per unit area and the diameter, both of which depend on the amount adsorbed. That is, the (29)Laraen, J. R. Presented at the 63rd Colloid and Surface Science Symposium, Seattle, WA, June 21,1989. (30) Kitahara, A.; Karaaawa, S.;Yamada, H. J. Colloid Interjace Sci. 1967,25,490.
(31) Kitahara,A.; Amano, M.; Kawaaaki,S.;Kon-no, K. ColloidPolym. Sci. 1977, 255, 1118.
Characterizations of Low Conductiuity Dispersiona surface potential is a quadratic function of adsorption. The surface potential might change in some apparently odd ways (e.g., have a maximum) but the adsorption process be quite simple. Studying processes by measuring more than one property of the dispersion might aid understanding.
Conclusions For a simple model for charged particles in low conductivity media, three relations have been derived relating experimentally measurable quantities: charge per unit mass, electrophoretic mobility, dispersion conductivity (when the counterion motion is insignificant),and average
Langmuir, Vol. 7, No. 10, 1991 2361 particle diameter, to known quantities such as solvent viscosity, density, and dielectricconstant, and the particle density. These relations provide a means to check experimentally measured quantities. This is particularly useful for dispersions of low conductivity because electrical characterizations of them are still so uncertain. If the experimentally measured quantities are believed to be reliable yet are inconsistent with these simple relations, then the inconsistency can be used to give additional information about the nature of the dispersion. Data on several systems were presented and analyzed. In general the published data are self-consistent. Inconsistencies are explainable in light of other information.
