Langmuir 1993,9, 1009-1015
1009
An Ion Transfer Model for Contact Charging A. F. D i u ' and D. Fenzel-Alexander IBM Almaden Research Center K931801, 650 Harry Road, San Jose, California 95120 Received October 2,1992. In Final Form: December 18, 1992
The contact charge between a polymer containing ions and another polymer or metal is known to be affected by the type and concentration of the ions in the sample. A model is presented to describe the observed contact charge. The model is based on the transfer of ions between the two surfaces and is for the case where one of the materials contains a known concentration of ions provided by either a molecular salt or an ionomer. The model considers that the sign and the magnitude of the charge will depend on the ion content in the surface region of the polymer, the relative mobilities of the two ions in the salt, and the relative stabilities of the two ions on each of the two surfaces in contact. The model considers both the case where the charge is activatedby many contacts, for example, roll-mixing an ion-containing polymeric powder with beads of a different chemical composition,and the case where the charge is activated by single contacts as with two surfaces, planar or curved. From this model the relationships between charge and the physical characteristics of the powderlbead mix, such as,the weight ratios and the size of the particles and beads, are derived. This relationship parallels those derived from the electron transfer model. Introduction Organic salts are known to affect the contact charge that is generated when two polymeric materials are contactedand separated.l-17 We previously demonstrated that in those cases where one of the materials contains a salt, the contact charge is accompanied by ion transfer from one surfaceto the other. When the polymer contains an ionomer, such as, poly(styrene-co-N-methyl-4-vinylpyridinium toluenesulfonate) ([PI-PyMe+OTs-) or parthe sign of tially sulfonated polystyrene ([P]-PhS03- H+), the charge on the polymer is dictated by the sign of the ion which is covalently bonded to the polymer and which has no mobility. Also,the charge responds monotonically to the ion concentration in the lower concentration range, and there is a close correspondence between the surface charge density and the amount of ions transferred per unit area. When the polymer contains molecular salts, such as, tetraalkylammonium or N-methylpyridinium toluenesulfonates, the situation is more complicated because both ions can be mobile and the resulting sign and magnitude of the charge depend on the relative (1) Birkett, K. L.; Gregory, P. Dyes Pigm. 1986, 7 , 341. (2) Macholdt, H.-T.; Sieber, A. Dyes Pigm. 1988,9, 119. (3) Anderson, J. H.; Bugner, D. E. 4th International Congress on NonImpact Printing Technologies, New Orleans, LA, 1988; p 79. (4) Bugner, D. E.; Anderson, J. H.Polym. Prepr. (Am. Chem. SOC., Div. Polym. Chem.) 1988, No. 29, 463. (5) Anderson, J. H.; Bugner, D. E. US Patent 4,837,391, June 6,1989. (6) Anderson, J.H.;Bugner,D.E.;DeMejo,L.P.;Sutton,R. C.; Wilson, J. C. US Patent 4,837,392, June 6, 1989. (7) Diaz, A. F.; Fenzel-Alexander, D.; Miller, D. C.; Wollmann, D.; Eisenberg, A. J. Polym. Sci., Polym. Lett. 1990,28, 75.
(8)Diaz, A. F.; Fenzel-Alexander, D.; Miller, D. C.; Wollmann, D.; Eisenberg, A.; 5th International Congress on Non-Impact Printing Technologies, San Diego, CA, 1989. (9) Diaz, A. F.; Fenzel-Alexander, D.; Miller, D. C.; Wollmann, D.; Eisenberg, A. 63rd Colloid and SurfaceScience Symposium, ACS, Seattle, WA, 1989. (10) Diaz, A,; Fenzel-Alexander, D.; Wollmann, D.; Eisenberg, A. J. Polym. Sci., Part B Polym. Phys. 1991, 29, 1559. (11)Dim, A. F.; Wollmann, D.; Dreblow, D. Chem. Mater. 1991,3,997. (12) Wollmann, D.; Dreblow, D.; Diaz, A.; Eisenberg, A. Chem. Mater.
1991, 3, 1065. (13) Guay, J.; Ayala, J. E.; Diaz, A. F.; Dao, Le H.Chem. Mater. 1991, 3, 1068. (14) Saurenbach,F.; Wollmann, D.; Terris, B. D.; Diaz, A. F. Langmuir 1992,8, 1199. (15) Diaz, A. F.; Guay, J. IBM J. Sci. Technol., in press.
(16)Diaz, A.; Fenzel-Alexander, D.; Wollmann, D.; Barker, J. A. Langmuir 1992, 8, 2698. (17) Mizes, H.A.; Conwell, E. M.; Salamida, D. P. Appl. Phys. Lett. 1990,56, 1597.
transfer 'aptitude" of the two ions in the salt. In these cases, the usual result is the transfer of both ions in excessive amountswhich contaminates the second surface and produces a relatively low net contact charge.
strudues d
[q-pyMe+OTS- and [q-phwH+ -
The role of ions in contact charging is actually very complicated. Even when the source of ions in a polymer blend is an ionomer which dictates the sign of the charge, thus removing one variable, the magnitude of the charge can be difficult to predict because it is sensitive to factors beyond the average ion content in the surface region of the sample.lOJ1 With blends of a styrene-butyl methacrylate copolymer containing 1-10% of a [PI-PyMe+ OTs- ionomer, it was shown that even when the blends contain the same ion concentration, the magnitude of the charge will vary depending on the amount of phase separation in the bulk of the blend and the composition of the ionomer used in the blend.lOJ1 Both of these compositionalcharacteristics can affect the degree of ion pairing and ion aggregationin the polymer which in turn affects ion transfer. The dependence of the charge on the ion content of the polymer blend has been modeled on the basis that only dissociated ions transfer from one surface to the other to produce charge.1°J6 Much of the interest in contact chargingis in industrial laboratories and is related to electrophotography1&" because of the importance of the polymeric toner particles. The proper placement of toner on the photoconductor or paper is dependent on the charge to mass ratio of the particles,Q/M. Thischargeis activated in the "developerw by the the contacts made during the rolling motion of the powder and 'carrierw bead mixture (dual component developer) or by the contacts with the doctor blade and the roller duringthe passage of the powder through a roller(18) Dessauer, J. H.;Clark, A. E.Xerography and Related Processes; Focal Press: New York, 1965. (19) Diamond, A. Handbook of Imaging Materials; Marcel Dekker, Inc.: New York, 1991. (20) Gruber, R. J. SID 87 Digest 1987, 272.
0743-746319312409-1009$04.00/0 0 1993 American Chemical Society
1010 Langmuir, Vol. 9,No. 4, 1993
blade nip (monocomponentdeveloper). Organic salts are often used as additives to help control the charge on the toner. Typical salts are tetraalkylammonium salts' of arylsulfonates, arylphosphonates, or metal azobenzene complexes.2 Aside from the practical aspects, the mechanism of contact charging with polymers is also of continued interest. Unlike the case for charging between metals, with polymers the mechanism is not fully understood even though a substantial amount of experimental and theoretical work has been published on contact charging between polymers and between polymers and m e t a l ~ . ~ y 6For ? ~ reviews ~ - ~ on this subject, see the reports by D a v i e ~ , ~Harper,"M '.~~ Duke:' Seanor,32 and RoseInnes.33 In this report, we present a model which relates the charge with the ion content in the polymer. This model addresses the thermodynamic distribution of the ions between the two surfaces and does not consider the path or the kinetics of the exchange. In this regard, this model is similar to the application of surface state theory to chargingbetween toner particles and carrier beads reported by Lee34 and by Anderson,35 although in this report ion transfer from one polymer surface to another and not electron transfer is considered the basis for charging. Recently, Lee36 published a dual mechanism for metalpolymer contact charging which is based on the surface state model and which includes both electron and ion transfer. However in contrast with the models by Lee34136 and Anderson,35 the model presented in this report also applies to polymer-polymer, and it is further developed to consider the properties of the materials, in particular, those of the ions.
Experimental Section The materials used in this study were available from previous studies.'-12 The polymer blends were used as powders having 8-10 pm averagediameter particles, and they consist of a styrene butyl methacrylate random copolymer blended with an ionomer or molecular salt. The powders were charged using two different methods. In one case, the powder was mixed with ca. 0.2 mm diameter, irregularly shaped, ferrite beads and the mixture was rolled for 30 min at which time a stable charge was attained.7J0-12 The relative humidity was maintained at 48-52%. The charge of the powder and the beads was measured by the total blow-off method and is reported as Q/M ( ~ c / g ) . ~Alternatively, ' the powder was charged in a modified developer unit from an IBM 4019 printer. In this case, the powder is charged ae it passes between an elastomer roller and an elastomer blade. The powder was rolled for 5 min, ca. 150 passes through the roller-blade contact, at which time a stable charge level was attained. It was then removed from the roller by vacuum lift-off and analyzed. The powder from a vacuum lift-off was analyzed for positive and (21) Kornfeld, M. I. J. Phys. D Appl. Phys. 1976, 9, 1445. (22) Henry, P . S. H. Er. J. Appl. Phys. 1967,4, Suppl. 2, S6. (23) Lowell, J. J. Phys. D Appl. Phys. 1979, 12, 1541. (24) Hersh, S. P.; Montgomery, D. J. Text. Res. J. 1966,26,95. (25) Cressman, P. J.; Hartmann, G. C.; Kuder, J. E.; Saeva, F. D.; Wychick, D. J. Chem. Phys. 1974,61,2740. (26) Gibson, H. W. J. Am. Chem. SOC.1975,97, 3832. (27) Davies, D. K. Er. J. Appl. Phys. D 1969,2 (2), 1533. (28) Davies, D. K. Static Electrification;Inst. Phys. Conf. Ser. 1967, -No. - - 4. -,29. (29) Harper, H. R. Proc. R. S O ~A. 1961,205,83. (30) Harper, W. R. Proc. Static Electrification Conf. 1967, 4, 3. (31) Duke, C. B.; Fabish, T. J. J. Appl. Phys. 1978,49, 315. (32) Seanor, D. A. Physicochemical Aspects of Polymer Surfaces; Mittal, K. L., Ed.; Plenum: New York, Vol. 1, p 477. (33) Lowell, J.; Rose-Innes, A. C. Adu. Phys. 1980,29, 947. (34) Lee, L.-H. Photogr. Sci. Eng. 1978,22, 228. (35) Anderson, J. H. J. Imaging Sci. 1989, 33, 200. (36) Lee, L. H. Proc. ACS Polym. Mat. Sci. Eng. 1992,67, 4. (37) Schein, L. B.; Cranch, J. J. Appl. Phys. 1976,46,5140.
Diaz and Fenzel-Alexander Table I. Summary of Charging and Charge Dirtributionr QIM, &/g for powderlbeads 5-8 24-30 8% +/ion pair additive pmollg MX pmollg MX Cb chargesa -23 -42 -21 0.0610.94 [PI-PhSOa-H+ +3 0.10/0.90 dodecylPhS03- H+ -10 -11 -2 -5 -1 0.10/0.90 OTs- H+ +22 +13 +23 0.68/0.32 N-Me-4-EtPy+OTs+23 0.7910.21 E4N+ OTs+29 +21 P b P + OTs+36 +28 +39 0.95/0.05 +35 0.9610.04 [PI-PyMe+OTs+37 +74 a This powder was charge activated using a roller-blade in an electrophotographicdeveloper. b Blends contain 20-30 rmollg MX. 8% carbon black in all the samples. negative charged particles using a charge spectrometer.38 The relative humidity was between 45 and 60%. The charge values were corrected for the value acquired by the pure polymer under the same conditions.
Charging Results We previously showed that with polymer blends containing ions, the equilibrium charge that is attained after rolling a powderlbead mixture for 20-30 minutes had a differentresponseto the ion content depending on whether the source of ions was an ionomer or a molecular ~ a l t . ~ - ' ~ J ~ With ionomers in the polymer blend, the charge increases monotonically with ion content, while with molecular salts, the charge increases to a masimum and then decreases with additional amounts of salt. We now have charging results with a series of salts which have a wider range of molecular structure. With this series, a qualitative correlation is seen between the sign and magnitude of the charge and the relative size of the two ions in the salt. This can be seen in Table I. It may help to restate here that the charges reflect equilibrium values and not rates of charging. The chargevalues listed here differ slightly from our previously published values because polymer-coated beads were previously used to activate the charge. The QIM values at two concentration ranges, 5-8 and 24-30 rmollg are provided because there is a nonmonotonic response of QIM with salt content with the blende containing the molecular salts. The most meaningful comparisonis made with the blends containing5-8pmoYg salt because with the higher salt contents (24-30pmol/g) there is excessivetransfer of both ions from the salt to the second surface which produces a low charge. The highest charges are produced by the powders containingthe ionomera ( [PI-&Me+ OTs- and [P]-PhS03H+) and with these, the sign of the charge matches the sign of the covalently bonded ion. With molecular acids and salts, the blends acquire lower charges. With H+OTs-, the charge is negative and the sign is consistent with the ion transfer theory which predicts that the bigger anion preferentially remains on the surface of the blend while the smaller, more mobile proton transfers to the bead. The factors governing the retention of the larger ion are not fully understood but may be related to the 'entanglement" or 'solubilization" of the ion with or into the polymer chains in the vicinity of the surface. This would render it less "free" to transfer; the smaller ions are lees "entrapped" and preferentially leave the surface of the blend. DodecylPhSOs- H+shows a change in the sign with roller-blade charging. This unusual behavior may be related to the surfactant properties of this acid and remains to be understood. N-Methyl-4-ethylpyridinium+ (38) Terris, B. D.; Fowler, K. J.; Reiley, T. C.; Truong, T. 6th International Congress on Advances in Non-Impact Printing, 1990.
An Zon Transfer Model for Contact Charging
Langmuir, Vol. 9, No. 4, 1993 1011
Table 11. Summary of Charging Results with Other Ionomers ionomer QIM, pC/g Ionomers Containing PhaPMe+ OTs- Ions (ca. 0.04 pmol/g Ions)4 [P]-PhPPhzMe+ OTs>+60 [PI-PhSOs PhsPMe+ +14 PhsPMe+ OTa+20 Ionomera used to enhance charging rate (2-10% in Blend)4Oq4I [P]-(CHz)sNMea+MeOSOa+QIM [Pl-CONH-C(CHs)zCHzSOa-H+ -QIM
OTs- and Et4N+ OTs- produce a positive charge. These are interesting results because in these salta both ions have near equal size; thus size alone does not correlate with the sign of the charge and other factors such as ion-polymer compatibility must be important for determining the extent of transfer for each ion. A similar trend in the charges is observed with the blends containing 24-30 pmoVg ions except that now the charges are smaller due to the contamination of the bead surface by the ions. Finally, with the PhP+OTs- salt where the cation is the larger ion in the salt, a positive charge is observed. The simple correlation between the contact charge sign and the sign of the "anchored" ion in the ionomer is not observed with some of the ionomers from other laboratories. In the case of the ionomers with arylphosphonium arylsulfonate ions, both [PIPhPPhzMe+ OTs- (mobile anion) and [PlPhS03- Ph3PMe+ (mobile cation) induce a positive charge to the polymer when it is present as an additives4 These results are summarized in Table 11.The anomaly is thought to be the [PlPhS03- Ph3PMe+where the positive charge acquired by the polymer has been proposed to result from the presence of minor amounts of ionic impuritiesfound in the material.4 The corresponding molecular salt, PhPMe+ OTs-, also produces a positive charge. Our results with Ph4P+ OTs- also indicate a positive charge and this result is consistentwith the charge sign-ion size trend.
I+
Ph2PCH3 O b -
Q Q
s0,-
PhSPCH,
strudures Ot Ipl-PhPph2Me*OTs' and IpJ-PhS0,- ph3PMe*
On the other hand, with the ionomer [Pl-(CH2)3NMe3+ MeOSO3- (mobile anion) the polymer does acquire the expected positive charge. This material is a modified octene-maleic anhydride copolymer with pendent (3aminopropylene)trimethylammonium groups and it was used as an additive which enhances the rate of charge development by 15 times (Table II).39This study includes several materials with variations in the polymer structure and in the ion, including CH3OS03-, OTs-, C1-, and B r . Differences in the 'charge rate enhancement" were measured; in particular, ionomers containing an aromatic structure, either in the polymer or the ion, produced little or no rate enhancement. These differences may reflect surface concentration differences for the various materials.l2 With an "inner salt" ionomer where both ions are presumably "anchored", e.g., a styrene-2-acrylamido-2mathylpropanesulfonicacid copolymer,m a negative charge is acquired by the polymer. However, this ionomer has (39) Gruber, R. J.; Bo1te;S. B.; Agostine, D. US Patent 4,415,646, November 15, 1983. (40)Watanabe, M.; H. Hagaee, H. US Patent 4,883,735,November 28, 1989.
a mobile proton which can transfer out of the polymer whether it rests initially on the amide nitrogen or the sulfonate oxygen. Finally, the molecular "inner salt" N-stearyl-N-dimethyl-N-(3-sulfopropyl)ammonium sulfonate41 (no mobile ions) was reported to enhance the charging rate but not alter the sign of the charge produced by the accompanying charge additive. +CH~-CH,CH-~H~
I
I
I +
CH3-N-CH3
I
CH3
6
0 4 H
I
CH~-C-CH,-SO;H+
I
To determine the amount of positively and negatively charged particles in the powder, polymer blends containing 8% carbon black were charged by flowing the powder between a roller and a doctor blade to facilitate the charge spectrometeranalysis. These results can only be compared qualitatively with the charging results with the ionomerpolymer blends charged against the metal beads because the presence of carbon reducesthe magnitude of the charge, where the charging range is reduced by ca. 4 times, and the charging process with the roller-blade is much less efficient then with the metal beads. Although the charge values from the roller-blade charging are not listed in the table, the trend remains the same as with the charging with the beads but the range is from -7 to +11 pC/g. As seen in Table I, with the ionomers 94-95% of the particles have the sign of the "anchored" ion and only 4-5% have the opposite sign. The small amount of particles with the opposite sign may result from incomplete charging or possibly some chemical inhomogenieties in the surface of the particles. Moisture plays a crucial role in contact charging when the contacts are carried out in ambient conditions,and it may also affectthe sign of the charge?The powders containing the molecular salts have a significant amount of positively and negatively charged particles, but the greater portion of the particles have the same sign as the sign of the larger ion. Thus as previously proposed, with the molecular salts both ions transfer but the larger ion has more impedimentto transfer and remains behind in greater amounts.
Ion-Surface Interactions Model The model described here relates the contact charge with ion content for the case where one of the materials is a polymer containing ions. This model assumes that ions exchange between the two surfaces. For this model we specify the following conditions: (1) The charge results from transfer of ions across the contact interface. (2) Only the ions in the surface region of the particles are active for transfer. (3) The ions equilibrate between the two surfaces and equilibrium is governed by (41) Barbetta,A.J.;Agoetine,D.;Hoffend,T.R.;Manaca,R.D.;Tokoli, E. G.US Patent 4,752,550,June 21, 1988. (42) Mataui, N.; Oka, K.; Inaba, Y., The Sixth International Congrem on Advances in Non-ImpactPrinting Technologies Black & White and Color, 1990, 45. (43) Folan, L. M.; Arnold, S.;O'Keeffee, T. R.; Spock, D. E.; Schein, L. B.: Diaz. A. F. J. Electrort. 1990, 25, 155. (44) Pence, S.; Diaz, A. Unpublished results.
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1012 Langmuir, Vol. 9, No. 4, 1993
the ion-polymer interactions at both surfaces. This does not preclude the continued exchange of ions after the equilibriumdistribution is reached; however, this exchange does not change the ion distribution. (4) This is an initial situation where the surfacecompositionsare known. Once the surfacecompositionsare altered by extensive material transfer during the contacts, chemid change (oxidation45), or other contamination problems, the modeling becomes more complicated. Because of the importanceof contact charging to "toner" particles and "carrier" beads in an electrophotographic developer, we can consider that one of the surfaces is provided by a powder, p, with a small particle size (10 pm diameter) and the other are beads, P, with a larger particle size (100-200 pm diameter). The particles in the powder have a surface area a, and the beads have a surface area A. The powder is a polymer blend containing the salt with a surface ion concentration, I', and for a salt which has two monovalent ions, M+X-, then I'M+ = rx- = r. When the powder and beads are mixed, rolled gently until the equilibriumcharge is established, and then separated, the ions which transferred between the surfaces give the powder and the beads opposite sign charge. This is symbolized in eq 1 for (n)particles interacting per bead
P + n(p) = P(P), (1) After the particles are removed from the bead(& the surface ion concentration of the particles is reduced to (1 -fir, where f is the fraction of the ion content transferred to the beads. For a mixture containing (n)particles per bead, the total number of ions remaining on the powder particlesis (n)(l-f)ra,and thetotalchargeon (n)particles is (n)ql. The equal and opposite charge is on the bead, -q2, and satisfies charge neutrality. The charge is related to I' as in eq 1, where z is the valency of the ion and F is the Faraday constant (n)ql = ( n ) z ~ -bra (i
(2)
and the surface charge density is (n)qi ( n ) z F ( l -firs -= (3) (n)a (n)a In most practical cases monovalent ions are used where z, the magnitude of the charge, is 1. Correspondingly,the surface charge density of the bead is
is rarely true since the two surfaces will at least have different local compositions due to impurities and morphology variations. Relating Charge to QIM. Substituting eq 6 into eq 3 gives (7)
using the total mass (m)l for all the particles, the QIM is defined as
The corresponding expressions for the beads are
and
Since (n)ql equals -q2, (QIM)1 can also be defined from eq 9 as shown in eq 11. In practice, it is the charge of the carrier which is measured and used with the weight of the powder to calculate qIm for the powder
From eqs 8 and 10, the equilibrium constant for the distribution of the ions between the two surfacesis related to the ratio of the surface areas normalized for the weight ratio, which reduces to the form shown in eq 5 q 2 h = KAIm, (12) (n)q,/(m), ( n ) a / ( m ) , Two Flat Surfaces. The correspondingequations can be written for the case where the contact is between two flat surfaces with known areas, a1 and up, and with an ion concentration on surface 1 equal to I'. After contact and separation, the number of chargeson surface 1 is q1= zF(1 - f)I'al and on surface 2 it is q2 = zF(f)ral. Correspondingly, the charge densities are (13)
and
(4)
Because of charge neutrality (n)ql equals -q2, and the equilibrium distribution of the ions between the two surfaces is given by the ratio of eqs 3 and 4
-
q2JA (n)(f)I'alA (5) (n)q,l(n)a ( n ) ( l -f)raI(n)a f is defined by eq 6 and indicates that the ion distribution responds to K and the relative areas K=
In analogy with eqs 5 and 6
KaZIa, (16) = 1 + K(a2/al) Finally, substituting for f in eqs 13 and 14 provides
AK
(6) (n)a + AK The difference in the ion-polymer interactions is reflected in K. For the unique case where the ions are equally stable on both surfaces, the enthalpy for the process is zero and the free energy simply reflects the entropy of the process. That is, ion exchange is simply a dilution of the surface ion density and is driven by entropy. Needless to say, this (45) Hays,
D.J . Chem. Phys. 1974,61, 1455.
(17)
and
Application of the Model To get a net charge when a salt is present on one surface, selective ion transfer between the two surfaces must be
Langmuir, Vol. 9, No. 4, 1993 1013
An Ion Transfer Model for Contact Charging
introduced. In many cases only a small net charge will result because each ion in the salt will have a different stability on the two contact surfaces, i.e., a different K for each ion, and the ions will become distributed to give an imbalance of ions and charge on each surface. The differential stability will be affected by the surface ion concentration and decrease at the high ion concentrations to produce a minimal charge, even though an extensive amount of both ions may have transferred. The net charge transferred to the second surface when there are i number of different ion types can be expressed from eq 9 for particles and eq 18 for flat surfaces
In practice, most of the salts used contain monovalent ions, M+X-, where r M + approximates r x - approximates r,and i equals 2. For this case, eq 19becomes (21). Other things being equal,the sign of 42 at equilibrium will depend on K2 and K1, which reflects the relative stabilities of the M+ and X- on the two surfaces.
With this expression we see that there are several unique situations, as follows: (i) K1 is finite and K2 is zero; the greatest charge will result. An example of this is the use of ionomers where one of the ions in the salt is covalently bonded to the polymer chain and has no mobility. (ii) K1 equals Kz; the net charge will be near zero (other things being equal) since both ions, of opposite charge, will transfer to the same extent. (iii)K1 and K2 are finite but not equal. This is probably the most common occurrence with molecular salts, and the charge can be anything depending on the relative K values. The magnitude of the charge can be enhanced by selecting materials which favor (stabilize)one of the ions in the salt by, e.g., hydrogen bonding, dipoleion interactions, or immobilization of one ion by size or bonding to the polymer. The charges produced by the additives listed in Table I show this behavior. In these materials the additive is melt blended with styrene-butyl methacrylate (5-30 pmol/g) and contacted with ferrite beads. The greatest charge is produced by the ionomers (case i) while the molecular salts with ions of near equal size produce small charges (case iii). The ion transfer process must have an additional selectivity whereby not all of the ions have the same "aptitude" for transferring or equilibration between the bead surface. This suggests that the OTs- ions reside in different chemical environments which control the propensity of the ion to transfer and equilibrate between the two surfaces. This variationmust be related to ion pairing. Ion Pairing. We previouslyreported that the curvature in the Q / M vs [MXI plot could be fit by a model where only the dissociated ions, M+ and X-, of the highly ion paired salt, M+X-, contributed to the charge.16 The amount of free ions can be related to the formal ion content by eq 23 where [MX] (in mmol/g) is the bulk concentration of ions and K, is the equilibrium constant for ion pair dissociation. The ion distribution is established during the melt mixing of the blend when the media is fluid.
K,
+
M'X- + M+ X[X-I = [M+l = (Ka[MXl)1/2
(22) (23)
In this medium the salts are mostly ion paired as in aprotic solvents and solvents with low dielectric constants,& and they have small dissociation equilibrium constants, K,. For example, the dissociation constants of n-BudN+salts are M in anisole (D= 4.3) and ca. 10-4 M in THF (D = 7.4).& Thus, the concentration of dissociated ions is only a small fraction of the formal concentration. The situation actually gets more complicated because the ion pairs are also aggregated as ion pair dimers, trimers, etc.16 The presence of these ion pair aggregates further reduces the dissociated ion content. Dissociated ions are very important for charging since only those ions which are dissociated will transfer to the contacting surface to impart charge, while ions which are associated in ion pairs will remain paired by the Coulombic attraction and will not impart charge even if they transfer. Ion pairs which transfer will mostly contaminatethe second surface. For this modelwe need to consider the ions in the surface region with a thickness 6r (for a particle with radius r ) which can effectivelyescape to the contactingsurface. The concentration of ions in the surfaceregion (r)can be related to the bulk content, [MXI (or C M P ) ,as follows
r=
volume,,, dMX1 = brp[MXI = -20 x lO-'[MX] Surfaceparticle (24)
where the density, p, is ca. unity, the particle radius is 5 pm, and the "skin" thickness 10-20 A. By analogy with the case for the bulk
rx-= rM+ = (KarMx)1/2= (Ka6rp[MX1)1/2 (25) Substituting 'I into eq 11 yields
Reuse of Equilibrated Beads. In a previous paper, we showed that when beads recovered from a charging experiment (mixing, rolling, and blow-off of the powder) were mixed with a fresh sample of the powdered blend and the charging procedure was repeated, the resulting charge after each "reload" was incrementally lower but not zero. It even settled at an "apparent" steady level with high charge. In this experiment, the powder was a blend of styrene-butyl methacrylate and poly(styreneco-N-methylvinylpyridiniumOTs-) containing 52 pmol/g ions and the acquired charge on the powder for the first rolling was +88 pC/g. By the second reloading (third charge)the charge was +75 pC/g and it gradually decreased to 70 pC/g by the ninth reload (tenth charge). The XPS analysis of the recovered beads after some of the charging and blow-off of the powder revealed the presence of S for the OTs- anion in increasing amounts. In terms of the model, when ion pairing is important, the equilibrium charge is limited by the number of dissociated OTs- ions on the surface and is not a simple response to the relative areas of the two contactingsurfaces. The charge does not represent the transfer of all the ions or even a statistical distribution of the ions between the available surface areas (powder vs beads). Instead, it is only a small fraction of the charge that would result if all (46) Pettit, L.D.;Bruckenstein, S.J. Am. Chem. SOC. 1966,88,4783.
1014 Langmuir, Vol. 9, No. 4, 1993 the ions on the surface were to be transferred. Equation 25 shows that the amount of dissociated OTs- ions is a small fraction of the total OTs- content in the blend, and from Figures 4-7 in ref 16, it can be estimated that the fraction of dissociated ions in the sample [OTs-I/ [PyMeOTs] is in the range 0.0002 to 0.02 when Ka is 1o-S to lo4 M. This estimate is applied to both the bulk and surface ion content, since with ionomers the surface ion content can be the same as bulk content. This estimate can also be reasonably applied to molecular salts even though, in this case,the surface can have a 3-5 times higher ion content.7-9 Thus, in the first rolling cycles a very small fraction of the ions actually accumulates on the surface of the bead and the amount is given by the product of ([dissociated ion]) ( A K / ( ( n ) a+ AK)). Since the latter term is unity or less, it onlyservesto reduce the number of ions transferred. In each of the subsequent rolling cycles, the amount of OTs- transferred becomes progressively less because the accumulationof ions on the surfaceof the beads will reduce the value of K. In principle, mixing the recovered beads with freeh powder for charging could be repeated to 50 times (for Ka = 1V M) before the surface of the beads saturates with ions. In practice, the limit would be reduced to a smaller number of cycles due the accumulation of ionic impurities and the effect of moi~ture.~Mz Comparison with Literature Models. The equations resulting from this model have the same general form as the equationsresultingfrom the models developedby Lees4 and by Anderson.35 Starting with the reciprocal of eq 11
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