5036
Langmuir 1997, 13, 5036-5040
Particle Charge, Mobility, and ζ Potential in Nonpolar Colloids I. Chen,* J. Mort, and M. A. Machonkin Wilson Center for Research and Technology, Xerox Corporation, Webster, New York 14580 Received January 6, 1997. In Final Form: June 4, 1997X Nonpolar colloidal dispersions are used as inks for electrophoretic development of electrostatic latent images. The need for a modification of the classical relation between the charge and the ζ potential of particles in these dispersions is confirmed by an independent experimental determination of the particle charge and mobility, using the series-capacitor technique introduced in a previous publication. In addition to the charged toner particles, the inks contain surfactant micelle ions of both polarities with their mobilities, in general, differing by orders of magnitude. It is shown that, in spite of this complex composition of charge species, it is possible to obtain an estimate of the density and mobility of each species from an accurate interpretation of experimental results based on the theoretical model of the technique.
I. Introduction Electrical charges in nonpolar colloidal dispersions have received considerable attention in electric power and petroleum processing industries, mainly because of their detrimental effects.1 In contrast, they are an important factor in the emerging technology of liquid developers for electrophoretic imaging.2-4 These liquid developers (“inks” for short) are colloidal dispersions of pigmented resin particles (i.e. “toners”) in nonpolar fluids containing surfactants, known as “charge directors” (CD’s). In the fluid, the surfactants form molecular aggregates such as (inverse) micelles, which facilitate the stabilization of ionic species1 and give the fluid a finite conductivity. The interaction between the toners and the CD species yields charged toners, and in general, the conductivity of the dispersion is further increased.2-4 The three charged species, i.e. toners, co-ions, and counterions of the CD species, contribute additively to the conductivity. Thus, measurements of conductivity can only provide a value for the sum of three species. On the other hand, in electrophoretic development of electrostatic images, each species has its own role which can be influenced by the other coexisting species.5 Furthermore, the development process is known to be determined by the two factors affecting conductivity, namely the charge (volume) density and mobility, separately rather than by their product.5 Therefore, a complete electrical characterization of inks requires independent information on the charge density and mobility of the three species. While there are many techniques for the measurements of electrophoretic mobility, the series-capacitor experiment,6 depicted in Figure 1, distinguishes itself in the ability to determine independently the mobility and the density of the charge species, in colloidal dispersions of any solid content (in principle from 0 to 100%), under a condition which closely simulates the actual image development process. An exciting result from the series-capacitor measurements of typical inks is that the toner mobility, µt ≈ 10-4 X Abstract published in Advance ACS Abstracts, September 1, 1997.
(1) Morrison, I. D. Colloids Surf., A 1993, 71, 1. (2) Gibson, G. A.; Luebbe, R. H. J. Imaging Technol. 1991, 17, 207. (3) Larson, J. R.; Lane, G. A.; Swanson, J. R.; Trout, T. J.; El-Sayed, L. J. Imaging Technol. 1991, 17, 210. (4) Larson, J. R.; Morrison, I. D.; Robinson, T. S. IS&T’s Int. Congr. Adv. Non-Impact Print Technol., Final Program Proc., 8th 1992, 193. (5) Chen, I. J. Imaging Sci. Technol. 1995, 39, 473. (6) Chen, I.; Mort, J.; Machonkin, M. A.; Larson, J. R.; Bonsignore, F. J. Appl. Phys. 1996, 80, 6796.
S0743-7463(97)00020-6 CCC: $14.00
Figure 1. Schematic of the series-capacitor experiment.
Figure 2. New relation between the ζ potential and the particle charge to mass ratio, q/m, for three values of Debye length D in units of particle radius R (solid curves), and the classical relation for particles in charge-free media (dashed). The units for potential are kT/ze (≈25 mV), and those for q/m are kT/ zeR2Fm (≈0.5 µC/g for R ) 1 µm and mass density Fm ) 1 g/cm3) (reproduced from ref 8).
cm2/(V s), is about two orders of magnitude larger than that observed in diluted samples at lower fields.6 However, using reasonable values of permittivity and viscosity in the Huckel relation between the mobility and the particle surface potential, or “ζ potential”, the above value of µt corresponds to a physically unrealistic value of ζ > 3 V.7,8 This apparent conflict has been resolved by a re-examination of the relation between the particle charge and the surface potential, taking into consideration that the particles are surrounded by counterions forming an electrical double-layer.8 The new relationship, as shown in Figure 2, finds that, as the particle charge increases, the initial linear increase of the potential slows down and (7) Morrison, I. D.; Tarnawskyj, C. J. Langmuir 1991, 7, 2358. (8) Chen, I. Langmuir 1996, 12, 3437.
© 1997 American Chemical Society
Nonpolar Colloids
Langmuir, Vol. 13, No. 19, 1997 5037
approaches an asymptotic value. Thus, with a high particle charge density (about 200 µC/g), expected from the above mobility value based on Stokes’ law, the ζ potential remains in the realistic range of a few hundred millivolts for dispersions with a typical ionicity. The electrical double-layer model also predicts that the charge densities of particles and the counterions increase by orders of magnitude with the efficiency of charging (as represented by a large ζ potential), while the density of co-ions decreases slightly from the pre-interaction value.8 A similar conclusion has been reached from the chemical reaction kinetics of toner charging.4 This means that, if the fraction of co-ion charge is comparable to or larger than that of particle charge, the total charge (of one polarity) in the ink must not have increased much from the pre-interaction level, indicating a poor charging efficiency. Therefore for practically useful inks, the density of co-ion charge must be negligible compared to that of toners. The need for a modification of the relation between particle charge and ζ potential can be confirmed by an independent determination of the charge on the highmobility particle to be as high as that expected from Stokes’ law. This is an objective of the present investigation. For this purpose, in the next section, the theoretical model of the series-capacitor experiments is further developed for the interpretation of experimental observations, in particular, regarding the charge densities of various species in the samples. Representative experimental results obtained with typical inks are presented to demonstrate the application of the technique. II. Theoretical Model In the series-capacitor experiments,6 as shown schematically in Figure 1, the ink sample is sandwiched between a grounded perfect dielectric and an electrode held at the bias voltage Vb. The total current density JT due to the electrophoresis of toners and CD species in the sample capacitor is given by6
∫∑
JT(t) ) -[1/Ls(1 + Cs/Cd)]
Ls
0
k
(µkFk)E dx
(2)
Using the local values of charge densities Fk(x,t) and the field E(x,t), obtained by solving the continuity and Poisson equations, the current density JT can be calculated numerically as a function of time t. Examples of the results for samples containing only two species of opposite polarities in equal amounts, e.g. toners and counterions, have been presented previously.6 The time integral of current is often used as a measure of the total charge in the sample. From eq 2, the time integral can be related to the sample layer voltage Vs by
Qint ≡
∫J ∞
0
T
dt ) -Cd
∫ (dV /dt) dt ) C (V ∞
0
s
d
s0
- Vs∞) (3)
where Vs0 and Vs∞ denote the initial and the asymptotic values, respectively, of Vs. From the capacitive division of the bias voltage, the former is given by
Vs0 ) Vb/(1 + Cs/Cd)
If all the charge species can drift to the boundaries as t f ∞, Gauss’ theorem at x ) 0 requires
CdVd∞ - CsVs∞ ) Cd(Vb - Vs∞) - CsVs∞ ) Q0 ≡ F0Ls (5) where Q0 is the total charge per unit area of one polarity, with F0 denoting the volume charge density in the sample. Solving for Vs∞, this yields
Vs∞ ) [CdVb - Q0]/(Cs + Cd)
(4)
(6)
provided the expression is greater than zero (with the convention of Vb > 0). In other words, if Vb g Q0/Cd, the sample voltage decays from its initial value (eq 4) to the final value Vs∞ given by eq 6. However, if Vb < Q0/Cd, not all the charge can drift to the boundaries before the sample is fully “relaxed”, i.e. Vs∞ ) 0. Substituting these results into eq 3, the time integrals of total current density for these two cases can be expected as
Qint ) CdVs0 ) CdVb/(1 + Cs/Cd)
for
Vb e Q0/Cd with
(1)
where Cs ) s/Ls and Cd ) d/Ld are the capacitances, with ’s and L’s denoting the permittivities and the layer thicknesses, respectively. The subscripts s and d refer to the sample and the dielectric, respectively. µk and Fk are the mobility and the density, respectively, of the k-th charged species. The summation ∑k is taken over all charged species in the ink and gives the ink conductivity. It has also been shown that the time rates of change of voltages across the sample and the dielectric layers, Vs and Vd, respectively, are related to the total current density JT by6
dVs/dt ) -dVd/dt ) -JT/Cd
Figure 3. Time integral Qint of calculated current densities, as a function of bias voltage, Vb, with the counterion to toner mobility ratio µi/µt ) 1 (solid) and 0.1 (dashed). Qint is in units of total charge per unit area of sample, Q0, and Vb is in units of V0 ≡ Q0/Cs, where Cs is the sample capacitance.
Vs∞ ) 0 (7)
and, with Vs∞ given by eq 6,
Qint ) Q0/(1 + Cs/Cd)
for
Vb g Q0/Cd
(9)
According to these results, the current integral would increase with Q0 and could provide a measure of the total charge in the sample only in the latter regime of higher bias, which is denoted as the “Q-limited” regime. The lower bias regime (eq 7) can be designated as “CV-limited”, as the total charge transported is limited by the bias voltage (and the capacitances). The observation of an increase in current integral with Vb in this regime should not be interpreted as an indication that the total charge in the sample is a function of the bias voltage. On the other hand, the results of directly integrating the current densities, calculated from eq 1, are shown in Figure 3, as a function of the bias voltage Vb. In these numerical examples, for the reason mentioned in section I, a sample with only one species per polarity, e.g., toners and counterions, of total charge per unit area Q0, is considered. The capacitance ratio is assumed to be Cs/Cd ) 1, and Vb is given in units of V0 ≡ Q0/Cs. For the solid curve, the mobilities of the toners, µt, and the counterions, µi, have the same magnitude. The integration is taken up to a time when a sufficient saturation of the integral is obtained. In good agreement with the above analytical results (eqs 7 and 8), a transition from the CV-limited to the Q-limited regime can be seen at a bias value near Vb ) V0. The data for the dashed curve in Figure 3 are obtained similarly, except that the counterion mobility is assumed to be only one-tenth of the toner mobility, µi ) 0.1µt. The transition from the CV-limited to the Q-limited regime is not as sharp as in the previous case (solid curve) because the collection of charge is less complete in a finite integration time (same as above). This
5038 Langmuir, Vol. 13, No. 19, 1997
Chen et al.
Figure 4. Time integral Qint of calculated current densities (in units of Q0) as a function of time, for different counterion to toner mobility ratios µi/µt, at a bias Vb ) Q0/Cs. The unit of time is defined by eq 9.
Figure 5. Time integral Qint of calculated current densities (in units of Q0) versus time for samples containing fast and slow species with the mobility ratio µs/µf ) 0.01. The fraction of the fast species is varied from f ) 0 to 1. The bias is Vb ) Q0/Cs, and the unit of time is defined by eq 9. is a consequence of the increased space charge effect due to the mobility asymmetry. The integrals of current densities for different mobility ratios µi/µt are shown as functions of time in Figure 4. With the capacitance ratio of Cs/Cd ) 1, and the bias voltage Vb ) Q0/Cs, the discharge is at the onset of the Q-limited regime. The unit of time t0 is defined by
t0 ≡ s/µtF0 ) Ls2/µtVb
(9)
which can be recognized as the dielectric relaxation time of a material with a permittivity s and a conductivity µtF0 or as the transit time of the toners at a field Vb/Ls. For the mobility ratio µi/µt ) 1, several tens of time units are required for the current integral Qint to approach the saturation value expected from the analytic formula (eq 8). For mobility ratios as small as µi/µt ) 0.01, a plateau in the curve separating the arrivals of the two groups of species can be discerned. The fraction of the total charge that belongs to the fast species can be estimated from the plateau value of Qint, e.g. ∼50% in this case. This is further indicated by the results shown in Figure 5, in which samples containing fast and slow species of both polarities in equal densities and various partitions are considered. With a slow to fast mobility ratio of µs/µf ) 0.01, plateaus appear at Qint values proportional to the fraction of the fast species f. Thus, the current integral curves can be used to estimate the relative concentrations of species with different mobilities. The parallel-plate cell, which is a traditional tool for ink charge measurements,4 is a special case of the series-capacitor experiment, with the perfect dielectric eliminated. In this case, the current density and the current integral are given by the above
Figure 6. Calculated ratio of the current integral to the actual charge density in the sample layer, Qint/Q0, for a parallel-plate cell, as a function of Q0. Three cases with different counterion to toner mobility ratios, µi/µt, are shown. The charge density Q0 is in units of CsVb.
Figure 7. Experimental data of current integrals (at t ) 0.05 s) for inks containing CD’s of various molecular weights, as functions of bias voltage. expressions with the capacitance of the dielectric as Cd f ∞. Then, the transition from the CV-limited to the Q-limited regime occurs at a bias Vb ) 0. This means that the current integral is never CV-limited and always represents a measure of the ink charge density. However, this advantage is offset by two drawbacks. First, in the parallel-plate cell, the voltage across the ink layer remains constant at the applied bias value, and hence, the temporal and spatial variations of field do not simulate those in the actual image development process. The constant voltage also allows the (noise) current that may arise from thermal generation or injection from the electrodes at later times to flow continuously even after the originally existing charges are swept away. Second, although without the CV-limitation, the measurement is not free from space-charge effects. Figure 6 shows the calculated ratio of the current integral to the actual charge density, Qint/Q0, for a parallel-plate cell, as a function of Q0. The deviation of the ratio from unity is seen to occur as the latter exceeds the capacitance-voltage product of the cell, CsVb, and to begin at lower values for the larger mobility difference (µi * µt). To avoid this limitation, the cell has to be made very thin (large Cs) and/or a very high bias Vb has to be applied.
III. Experimental Results The experimental procedures, including the sample preparation and the data acquisition, have been reported in detail in a previous paper.6 The neglect of dielectrophoresis is justified by the observation that no current can be detected with samples containing no CD surfactants. In Figure 7, the experimental data of the current integrals (at t ) 0.05 s) versus bias voltage are presented for samples prepared with various CD’s, namely, an AB diblock quaternary ammonium polymer of various molecular weights, from 4 000 up to 93 000 (hereafter referred to as 4K Quat and 93K Quat, respectively).9 The fluid is
Nonpolar Colloids
Langmuir, Vol. 13, No. 19, 1997 5039
Figure 9. Same experimental data of Figure 8 replotted with the time axis in the logarithmic scale.
IV. Discussions
Figure 8. Experimental data of current integrals for two inks (solid curves) and CD solutions (dashed curves) incorporating either 4K or 93K Quat CD’s, measured at a bias voltage of 400 V: (a) integration over 0.05 s; (b) integration over 5 s.
an aliphatic hydrocarbon with a viscosity at 25 °C of ∼6 cP. The transition from the CV-limited to the Q-limited regime can be seen to occur at lower bias for the samples prepared with the higher molecular weight CD’s. The deviation from the linear increase in the CV-limited regime is also larger for these samples. On the basis of the facts that the toners are less charged and the CD counterion mobility is smaller in these samples, these observations are in good agreement with the expectation shown in Figure 3. In Figure 8 the experimental data of the current integrals versus time are presented for samples prepared with the 4K Quat and 93K Quat CD’s. Typical concentrations of the CD’s are 0.6 mg/cm3 for 4K Quat and 2mg/ cm3 for 93K Quat.9 Both samples with and without the toner particles (of diameter 1-2 µm, and 2% concentration by weight) are examined. They are represented by solid and dashed curves, respectively, in Figure 8. Figure 8a covers a time range out to 0.05 s, while Figure 8b extends out to 5 s. It is clear from Figure 8a that there is more charge in the samples (with and without toners) prepared with 4K Quat than in those prepared with 93K Quat in spite of the higher concentration (by weight but not by mole percent) of the latter. As the integration time is increased, a comparison of parts a and b of Figure 8 shows these trends are maintained. Although the collected charge appears substantially saturated at 0.05 s, further charge is accumulated if the integration time is extended. For example, in the sample with 4K Quat, after 0.05 s the current integral is 0.48 µC, whilst after 5 s this has grown to 0.78 µC, indicating the rate of accumulation, though finite, has decreased by essentially two orders of magnitude. With the applied voltage assured to be in the Q-limited regime described above, the current integral gives a measure of the charge density. (9) Page, L. A.; El-Sayed, L. Hard Copy Print. Mater., Media, Processes, Proc. SPIEsInt. Soc. Opt. Eng. 1990, 1253, 37.
From the known geometry of the sample, the above value of the current integral, considering some uncertainty in the capacitance values, corresponds to a charge density of F0 ≈ 5 µC/cm3 of the ink. With a negligible co-ion concentration, most of this charge is on the toner particles. Then, the toner charge-to-mass ratio (q/m) can be estimated from the toner concentration (2%) to be about q/m ≈ 250 µC/g. Using this value and Stokes law,7,8 the toner mobility is calculated to be µt ≈ 2 × 10-4 cm2/(V s), in good agreement with the value obtained from current transit measurements reported in the previous publication.6 As mentioned in section I, this independent confirmation of the charge density and mobility values provides strong evidence that the classical relationship between the charge density and the ζ potential needs to be modified for application to particles with electrical double-layer structures. An example of such a new relationship has been described.8 The time evolution of current integral can be more informative when the experimental data of Figure 8 are replotted with the time axis in the logarithmic scale, as shown in Figure 9. While the (solid) curves for inks are seen to increase monotonically with time, the (dashed) curves for CD-only samples, in particular for the one with 4K Quat, show a plateau from about t ) 0.03 to 0.1 s, before continuing to rise again. On the basis of the above theoretical analysis (Figures 4 and 5) this suggests that the mobility distributions in these samples are bimodal; namely, there are fast and slow species of either or both polarities, with the mobilities differing more than an order of magnitude. One can further attempt to determine a consistent identification of the polarity and relative abundance of the fast and slow species as follows. In general, the CD-only samples could contain four types of charge species, i.e. fast and slow co-ions (anions in this case) and fast and slow counterions (cations in this case). Similarly, the inks could contain the same four species plus the negatively charged toners. The monotonic increase of current integral for inks (solid curves) indicates that the charge species in inks are dominantly of single mobility (fast species). In other words, the concentrations of the slow species that may exist in the CD-only samples have decreased significantly in the inks. The decrease in concentration of slow co-ions when toners are efficiently charged can be expected, as described in section I,3,8 but there is no physical reason for the decrease of slow counterions at the same time. Therefore, one can conclude that the concentration of slow counterions in the CD-only sample has always been small compared to that of other species. It must be understood that this is not generally true and depends on the chemical characteristics of the
5040 Langmuir, Vol. 13, No. 19, 1997
Chen et al.
particular CD used in this sample. Denoting the concentrations by square brackets, the charge neutrality in the CD-only sample can now be written as
[fast counterion] ≈ [slow co-ion] . [fast co-ion] and [slow counterion] (12)
[fast co-ion] + [slow co-ion] ≈ [fast counterion], and [slow counterion] ≈ 0 (10)
Since the density of slow co-ions is decreased by the charging of toners, the major constituents of inks are the fast counterions and the fast charged toners. With the wide distributions in the sizes and shapes of toner particles and CD aggregates, one should expect similarly wide and overlapping distributions for the “fast” and the “slow” mobilities. Considering the possible smearing of features caused by these distributions, the above analysis satisfactorily accounts for the observations in this class of inks.
On the other hand, the plateau at approximately half the maximum in the (dashed) curves for CD-only samples suggests a bimodal mobility distribution, with approximately equal concentrations of fast and slow species. This can be expressed as
[fast co-ion] + [fast counterion] ≈ [slow co-ion], and [slow counterion] ≈ 0 (11) Adding the two relations, eqs 10 and 11, yields [fast co-ion] ≈ 0. This leaves only the fast counterions and the slow co-ions as the major constituents in the CD-only samples:
Acknowledgment. The authors wish to thank their colleagues F. Bonsignore, S. Chamberlain, G. Gibson, J. Larson, and J. Spiewak for the ink samples used in this investigation. LA970020N
