The Electrophoretic Mobility of PPI Dendrimers: Do ... - ACS Publications

Modeling of solution properties thus begins with a simple question: does a charged dendrimer behave more as a linear polyelectrolyte or more as a char...
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The Electrophoretic Mobility of PPI Dendrimers: Do Charged Dendrimers Behave as Linear Polyelectrolytes or Charged Spheres? Cynthia F. Welch† and David A. Hoagland* Department of Polymer Science & Engineering, University of Massachusetts Amherst, Amherst, Massachusetts 01003 Received September 18, 2002. In Final Form: November 23, 2002 The electrophoretic mobilities µ0 of charged polypropyleneimine dendrimers were measured by capillary electrophoresis as a function of ionization R, ionic strength I, and hydrodynamic radius a. For the two generations examined, 3 and 5, µ0 trends reminiscent of a linear polyelectrolyte are found in R but not in I and a. Strikingly, all three trends accurately follow standard electrokinetic model predictions for a charged, ion-impenetrable dielectric sphere of radius a. Although the sphere depiction is physically inexact, the meshing of theory to experiment requires no fitting of parameters. For highly charged dendrimers across the accessible aκ range (0.2 < aκ < 2), the standard electrokinetic model quantitatively captures a large observed relaxation effect. No evidence for nonspecific ion binding was uncovered. Commercial availability and conformity to sphere theory suggest that charged dendrimers might serve as model electrophoresis systems. Limitations and alternatives to the sphere depiction are discussed.

Introduction Because of ionization of branch or terminal units, several common dendrimer species develop substantial charge when dissolved in aqueous media. If, as commonly speculated,1,2 the branching arms radiate symmetrically outward from the molecular center, charged dendrimers might be modeledsto first approximationsas charged spheres. Viewed differently, charged dendrimers are simply highly branched polyelectrolytes. If so, the molecules might be modeleds again to first approximations as branched line charges. Modeling of solution properties thus begins with a simple question: does a charged dendrimer behave more as a linear polyelectrolyte or more as a charged sphere? The electrophoretic behaviors in free solution of charged spheres and linear polyelectrolytes are distinct. This study’s objective is therefore to collect electrophoretic mobility data for well-defined charged dendrimers and ascertain how well these data correlate to theoretical predictions for sphere and chain models or to experimental data for linear polyelectrolytes. Studies of charged dendrimers by electrophoresis have ranged widely, from characterizations of their structure3-5 to examinations of their utility as transport agents in chromatography6 and drug delivery.7 Dubin et al.8 explored how dendrimer-counterion interactions affect the mobil* Address correspondence to this author. Tel: (413) 577-1513. Fax: (413) 577-1510. E-mail: [email protected]. † Current address: The Dow Chemical Company, Midland, MI 48674. (1) de Gennes, P.-G.; Hervet, H. J. Phys. Lett. 1983, 44, L351. (2) Lescanec, R. L.; Muthukumar, M. Macromolecules 1990, 23, 2280. (3) Pesak, D. J.; Moore, J. S.; Wheat, T. E. Macromolecules 1997, 30, 6467. (4) Brothers, H. M.; Piehler, L. T.; Tomalia, D. A. J. Chromatogr. A 1998, 814, 233. (5) Li, J.; Swanson, D. R.; Qin, D.; Brothers, H. M.; Piehler, L. T.; Tomalia, D.; Meier, D. J. Langmuir 1999, 15, 7347. (6) Khaledi, M. G. In Micellar Electrokinetic Chromatography; Khaledi, M. G., Ed.; John Wiley & Sons: New York, 1998; Vol. 146, p 77. (7) Ruponen, M.; Yla-Herttuala, S.; Urtti, A. Biochim. Biophys. Acta - Biomembranes 1999, 1415, 331. (8) Huang, Q. R.; Dubin, P. L.; Moorefield, C. N.; Newkome, G. R. J. Phys. Chem. B 2000, 104, 898.

ity of dendrimers possessing ionizable terminal units; to rationalize experimental results, appeal was made to nonspecific ion binding, a concept with lengthy development for linear polyelectrolytes but limited history or theoretical justification for dendrimers and other compact solutes. These authors also contemplated that the properties of a charged dendrimer might be explained through a charged sphere model, but the approach was not pursued. Colloid scientists routinely extract a particle’s surface charge density σ or surface potential ζ from measurements of the free-solution electrophoretic mobility µ0. The procedure relies on the “standard electrokinetic model”,9 which handles electrostatics through the Poisson-Boltzmann equation. Questions regarding the model’s validity persist.10-13 Rarely do properties inferred by electrophoresis match those from other techniques (i.e., conductivity, streaming potential, dielectric response), and many literature contributions allude to anomalies such as ion adsorption, particle surface “hairiness”, or conduction processes behind the shear plane.11,13-19 Even for welldefined latex particles, significant deviations from predicted behaviors have been noted.20-22 For linear polyelectrolytes, the situation is similar.23 (9) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607. (10) Lozada-Cassou, M.; Gonza´lez-Tovar, E.; Olivares, W. Phys. Rev. E 1999, 60, R17. (11) Elimelech, M.; O’Melia, C. R. Colloids Surf. 1990, 44, 165. (12) Midmore, B. R.; Hunter, R. J. J. Colloid Interface Sci. 1988, 122, 521. (13) Barchini, R.; Saville, D. A. Langmuir 1996, 12, 1442. (14) Ferna´ndez Barbero, A.; Martı´nez Garcı´a, R.; Cabrerizo Vı´lchez, M. A.; Higaldo-Alvarez, R. Colloids Surf., A 1994, 92, 121. (15) Seebergh, J. E.; Berg, J. C. Colloids Surf., A 1995, 100, 139. (16) Tuin, G.; Senders, J. H. J. E.; Stein, H. N. J. Colloid Interface Sci. 1996, 179, 522. (17) Folkersma, R.; van Diemen, A. J. G.; Stein, H. N. Langmuir 1998, 14, 5973. (18) Rasmusson, M.; Wall, S. J. Colloid Interface Sci. 1999, 209, 312. (19) Vorwerg, L.; Antonietti, M.; Tauer, K. Colloids Surf., A 1999, 150, 129. (20) Borkovec, M.; Behrens, S. H.; Semmler, M. Langmuir 2000, 16, 5209. (21) Antonietti, M.; Vorwerg, L. Colloid Polym. Sci. 1997, 275, 883. (22) Barchini, R.; Saville, D. A. J. Colloid Interface Sci. 1995, 173, 86.

10.1021/la026572+ CCC: $25.00 © 2003 American Chemical Society Published on Web 01/21/2003

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Charged dendrimers display properties advantageous to their possible use as model electrophoresis systems. Dynamic light scattering and viscosity studies find that the molecules behave hydrodynamically as spheres.24 The hydrodynamic radius, essentially independent of solvent conditions, corresponds to a shear plane located near the molecular periphery. Unlike typical colloids, this plane is not associated with a physical interface. In most instances, charge is not just precisely defined but also easily manipulated, with covalent structure and solvent pH dictating the magnitude and location of individual charges.25-28 Further, at least one commercial species, polypropyleneimine (PPI) or Astromol dendrimers, are essentially monodisperse in both size and structure.29,30 Last, dendrimers are in the right size range (order of nm) to maximize the magnitude of the electrophoretic relaxation effect. Theoretical Background. The force balance that determines µ0 for any dilute, charged solute dissolved in electrolyte may be expressed31

Fe - Fh - Fc - Fr ) 0

(1)

where Fe is the direct electric force due to the solute’s charge, Fh is the direct hydrodynamic force due to the solute’s motion relative to stationary solvent, Fc is the indirect hydrodynamic force transmitted to the solute because of hydrodynamic disturbances created by electric forces on the undistorted counterion cloud, and Fr combines electrical and hydrodynamic force corrections due to distortion of the counterion cloud. The term “retardation” is associated with phenomena underlying Fc, while the term “relaxation” is associated with phenomena underlying Fr. Motivated by earlier analyses by Hu¨ckel32 and Smoluchowski,33 Henry34 solved eq 1 for charged spheres, obtaining µ0 as a function of radius a and Debye screening length κ-1. In doing so, he applied the Debye-Hu¨ckel approximation and neglected Fr, thereby restricting his solution to small ζ at intermediate aκ or arbitrary ζ when aκ is small or large,

µ0′ ) ζ′β(aκ)

(2)

where µ0′ is the dimensionless free-solution mobility, defined as µ0 divided by 20kT/3ηe, and ζ′ is the dimensionless zeta potential, defined as ζ divided by kT/e. The function β(aκ) varies from 1.0 (aκf0; Hu¨ckel (23) Hoagland, D. A.; Arvanitidou, E.; Welch, C. Macromolecules 1999, 32, 6180. (24) Scherrenberg, R.; Coussens, B.; van Vliet, P.; Edouard, G.; Brackman, J.; de Brabender, E. Macromolecules 1998, 31, 456. (25) Koper, G. J. M.; van Genderen, M. H. P.; Elissen-Roma´n, C.; Baars, M. W. P. L.; Meijer, E. W.; Borkovec, M. J. Am. Chem. Soc. 1997, 119, 6512. (26) van Duijvenbode, R. C.; Rajanayagam, A.; Koper, G. J. M.; Baars, M. W. P. L.; de Waal, B. F. M.; Meijer, E. W.; Borkovec, M. Macromolecules 2000, 33, 46. (27) Zhang, H.; Dubin, P. L.; Kaplan, J.; Moorefield, C. N.; Newkome, G. R. J. Phys. Chem. B 1997, 101, 3494. (28) Chen, W.; Tomalia, D. A.; Thomas, J. L. Macromolecules 2000, 33, 9169. (29) Striegel, A. M.; Plattner, R. D.; Willett, J. L. Anal. Chem. 1999, 71, 978. (30) Bu, L.; Nonidez, W. K.; Mays, J. W.; Tan, N. B. Macromolecules 2000, 33, 4445. (31) Dukhin, S. S.; Derjaguin, B. V. In Electrokinetic Phenomena; Matijevic, E., Ed.; Surfaces and Colloid Science 7; John Wiley & Sons: New York, 1974. (32) Hu¨ckel, E. Phys. Z. 1924, 25, 204. (33) von Smoluchowski, M. Z. Phys. Chem. 1918, 92, 129. (34) Henry, D. C. Proc. R. Soc., Ser. A 1931, 133, 106.

Figure 1. Comparison of theoretical predictions for µ0′ as a function of aκ for charged dielectric spheres: (A) at constant dimensionless surface potential ζ′ and (B) at constant dimensionless surface charge σ′ (dashed portions of curves are approximations). In A, curves below Henry are from Wiersema et al.;38 in B, curves below Henry are calculated with the O’Brien and White MOBILITY program.9

result) to 1.5 (aκf∞; Smoluchowski result). Under the Debye-Hu¨ckel approximation, µ0′ is proportional to ζ′. Adding contributions from Fr, Booth,35,36 Overbeek,37 Wiersema et al.,38 and later O’Brien and White9 derived more complete solutions for µ0′ of charged spheres. The O’Brien and White numerical solution offers an exact description within context of the standard electrokinetic model. For spheres maintaining ζ′ as aκ varies, Figure 1A compares predictions for µ0′/ζ′ by O’Brien and White to those by Henry; constancy of ζ′ defines fixed potential. Fr lowers µ0′ at intermediate aκ in a manner dependent on both ζ′ and the mobility of small ions. More relevant to the current discussion are charged spheres maintaining σ as aκ varies; constancy of σ defines fixed charge.39 Using the dimensionless surface charge density σ′, defined as σ (35) Booth, F. Nature 1948, 161, 83. (36) Booth, F. Proc. R. Soc. London, Ser. A 1950, 203, 514. (37) Overbeek, J. T. G. Adv. Colloid Sci. 1950, 3, 97. (38) Wiersema, P. H.; Loeb, A. L.; Overbeek, J. T. G. J. Colloid Interface Sci. 1966, 22, 78.

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divided by 0kT/ea, Figure 1B compares predictions of µ0′/σ′ by O’Brien and White to those by Henry. At fixed charge, µ0′ falls as aκ increases, and relaxation corrections again are largest at intermediate aκ. Lozada-Cassou et al.10 recently analyzed electrophoresis within the context of the restricted primitive model, a framework accounting for the finite sizes of small ions; under some conditions, predictions of µ0′ deviate substantially from those of O’Brien and White. Lesser modeling efforts have focused on nonspherical solutes. Again under the Debye-Hu¨ckel approximation, Henry derived a µ0 formula for charged cylinders of infinite length.34 To understand linear polyelectrolytes such as DNA in the presence of relaxation effects, Stigter40,41 adapted to cylinders a computational scheme developed by Wiersema et al.38 For similar purposes, Manning42 proposed a line charge model incorporating counterion condensation; Cleland later elaborated upon Manning’s model.43 Muthukumar,44 and independently Barrat and Joanny,45 analyzed the electrophoretic motion of arbitrary charge assemblies characterized by structure factor S(k). Likewise, various investigators outlined porous sphere models that contain interior charge distributions.46-49 However, in pursuit of greater structural generality, the relaxation effect has nearly always been neglected. In a few instances, solutes of complex shape were studied after substituting a constraint of thin double layers for the Debye-Hu¨ckel approximation.50 Numerical methods were recently employed to analyze the electrophoretic motion of rigid, nonspherical bodies of simple shape.51 Notwithstanding all efforts, in absence of parameter fitting, quantitative agreement between theory and experiment has not been achieved even for simple linear polyelectrolytes.23 Counterion condensationsthe compensation of a portion of a polyelectrolyte’s fixed charge by electrostatic association of small ions from the surrounding electrolytes appears important, but the concept remains difficult to introduce rigorously.23,42,52-55 Experimental Section Materials. Near lack of structural defects, commercial availability up to generation 5, and extensive literature characterization data make PPI dendrimers a good system for charged dendrimer studies. A generation 3 dendrimer (G3) (DAB-Am16, Aldrich) was used as received, but an analogous generation 5 dendrimer (G5) (DAM-Am-64, Aldrich) required purification to remove a fraction of molecules with nitrile end groups. As recommended by the manufacturer (DSM), G5 purification was accomplished by vacuum filtration of 15 wt % aqueous solutions over Celite filter aid (Fisher). Filtration clarifies the cloudy (39) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989. (40) Stigter, D. J. Phys. Chem. 1978, 82, 1424. (41) Stigter, D. J. Phys. Chem. 1978, 82, 1417. (42) Manning, G. S. Q. Rev. Biophys. 1978, 11, 179. (43) Cleland, R. L. Macromolecules 1991, 24, 4391. (44) Muthukumar, M. Electrophoresis 1996, 17, 1167. (45) Barrat, J.-L.; Joanny, J.-F. In Advances in Chemical Physics; Prigogine, I., Rice, S. A., Eds.; John Wiley: New York, 1996; Vol. 94, p 1. (46) Hermans, J. J.; Fujita, H. Proc. Akad. Amsterdam 1955, B58, 182. (47) Hermans, J. J. J. Polym. Sci. 1955, 18, 527. (48) Overbeek, J. T. G.; Stigter, D. Recl. Trav. Chim. Pays-Bas 1956, 75, 543. (49) Imai, N.; Iwasa, K. Isr. J. Chem. 1973, 11, 223. (50) Fair, M. C.; Anderson, J. L. J. Colloid Interface Sci. 1989, 127, 388. (51) Allison, S. A.; Nambi, P. Macromolecules 1994, 27, 1413. (52) Manning, G. S. J. Phys. Chem. 1981, 85, 1506. (53) Stigter, D. J. Phys. Chem. 1978, 82, 1603. (54) Xia, J.; Dubin, P. L.; Havel, H. A. Macromolecules 1993, 26, 6335. (55) Stigter, D. Biophys. J. 2000, 78, 121.

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Figure 2. Dependence of R and µ0 on pH for (A) G3 and (B) G5 dendrimers (I ) 0.01 M). solutions, and the purified dendrimer is recovered by rotary evaporation in a SpeedVac Plus 110 (Savant). Titrations. With falling pH, both the primary amine termini and the tertiary amine branches of PPI dendrimers protonate, the former preferentially at higher pH because of higher pKa.25 A full protonation threshold, dependent slightly on ionic strength I, lies between pH 3 and 4.56 Figure 2 shows the degree of protonation, expressed as ionization R, during titration with HCl or NaOH. To accommodate the I dependence of R, aliquots of 0.01 M HCl or 0.01 M NaOH were titrated into 0.01 M NaCl solutions containing highly diluted dendrimer. Electroneutrality permits R to be calculated ([HCl] - [NaOH] - [H+] + [OH-])/ [amine], where [H+] and [OH-] are determined by pH measurement, and [amine] is the total concentration of primary and tertiary amine groups. Capillary Electrophoresis. Buffers were prepared from sodium chloride (NaCl, Sigma), hydrochloric acid (HCl, Fisher), sodium hydroxide (NaOH, Fisher), glycine (Gly, Sigma), formic acid (Fisher), acetic acid (HOAc, Fisher), sodium acetate (NaOAc, Fisher), tris(hydroxymethyl)aminomethane (Tris, Sigma), and potassium hydrogen phthalate (KHP, Fisher). Multiple buffers, all at I ) 0.01 M, were needed to cover the extended pH range (2.8-11.5) over which R varies: Gly-HCl-NaCl (pH 2.8-3.5), NaOAc-HOAc (pH 4.5-5.5), Tris-HCl (pH 7-9), and GlyNaOH-NaCl (pH 9.5-11.5). At R ) 1 (pH ) 2.8), two buffers of variable I were available, KHP-HCl and formic acid-NaOH. Applying fields of less than 250 V/cm (required for constancy of temperature in the face of joule heating), µ0 was determined at 20 ( 0.5 °C by capillary electrophoresis in 40-60-cm capillaries. The value µ0 did not depend on field strength. Dendrimer solutions (1-10 mg/mL) were injected electrokinetically (2-5 s), and elution was monitored at 205 nm where PPI weakly absorbs, by an ISCO CV4 UV/vis detector. Accuracy of µ0 was better than (3%, and at these high dilutions, µ0 did not depend on dendrimer concentration. PPI dendrimers adsorb to bare silica, so electrophoresis capillaries with coated walls were needed. Two coatings proved satisfactory, one of neutral polymer and the second of cationic surfactant. Poly(vinyl alcohol)-coated capillaries (50 µm i.d., (56) van Duijvenbode, R. C.; Borkovec, M.; Koper, G. J. M. Polymer 1998, 39, 2657.

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Table 1. Equivalent Ionic Conductivities of Buffer Ionsa cations K+

Na+ H+ Gly+ b Tris+

λ (cm2S/mol) 73.48 50.08 349.65 40c 40c

anions Cl-

acetateOHGlyformateHP-

λ (cm2S/mol) 76.31 40.9 198 35c 54.6 53.5

a Reference 58. b Gly ) glycine; Tris ) tris(hydroxymethyl)aminomethane; HP ) hydrogen phthalate. c Estimated value.

Agilent Technologies) functioned by “burying” the negative charge of silica beneath a permanently attached neutral polymer layer. Surfactant-coated capillaries (50 µm i.d., Polymicro Technologies) exploited the positive charge of cetyltrimethylammonium bromide (CTAB, Aldrich) to reverse the bare wall charge.57 The reversibility of the second coating mandated inclusion of CTAB in all running buffers and sample solutions. The CTAB concentration was 0.5 mM. For pH > 9, CTAB appears to associate with PPI dendrimers, raising their charge; consequently, only polymercoated capillaries were used at elevated pH. Regardless of capillary type, a dendrimer’s electrophoretic velocity Vel was deduced from its net velocity Vnet by first subtracting the electroosmotic velocity Vosm. The value of Vosm was measured by co-injecting a solute of known µ0; acetone (Fisher, µ0 ) 0) was chosen for surfactant-coated capillaries and phenyltrimethylammonium chloride (Aldrich, µ0 ) 2.91 × 10-4 cm2/Vs at 20 °C) was selected for polymer-coated capillaries. After obtaining Vel, µ0 was calculated according to its definition, µ0 ) Vel/E.

Results and Discussion Methods of Data Analysis. Adapting a charged sphere depiction for dendrimer electrophoresis necessitates a series of physical approximations. The O’Brien and White analysis, available as the program MOBILITY (University of Melbourne), provides µ0′ given input of aκ, ζ′, and the limiting equivalent conductance λi0 of each co- and counterion.9 The hydrodynamic radii measured by Scherrenberg et al.,24 1.18 nm for G3 and 1.98 nm for G5, can be assigned to a. Although Scherrenberg et al. examined PPI dendrimers in pure water, dynamic light scattering indicates that addition of buffer salts or variation of pH leaves the radii essentially unchanged. Table 1 lists literature values of λi0 for the buffer co- and counterions.58 Deducing ζ′ for PPI dendrimers is more difficult. Letting N be the total number of amine groups (N ) 30 for G3, N ) 126 for G5), the total charge Q equals the product RNe. Some charge resides in the molecular interior and some on the periphery, the split affected by pH and generation. To reconcile an interior charge with the sphere model, two limits are envisaged. In the first, the effective surface charge is equated to Q, and in the second, to the fraction of Q because of terminal amines (16Q/30 for G3, 64Q/126 for G5). For either limit, the dendrimer interior is modeled as a nonconducting, dielectric continuum. Then, σ′ is calculated by smearing the assumed charge over an area 4πa2, effectively mapping this charge onto the shear surface. Such simplified modeling prevents small ions from equilibrating across the shear surface. For practical reasons, adopting the charged sphere depiction adds one additional modeling step, transformation of σ′ to ζ′. A limited tabulation of ζ′, aκ, and σ′ values for spherical particles was published by Loeb et al.,59 and (57) Welch, C. F.; Hoagland, D. A. Polym. Prepr. 1998, 39 (2), 771. (58) Lide, D. R. CRC Handbook of Chemistry and Physics, 76th ed.; CRC Press: Boca Raton, FL, 1995. (59) Loeb, A. L.; Overbeek, J. T. G.; Wiersema, P. H. The Electrical Double Layer Around a Spherical Colloid Particle; MIT Press: Cambridge, 1961.

Figure 3. Dependence of µ0′ on R for (A) G3 and (B) G5 dendrimers (I ) 0.01M; aκ ) 0.387 and 0.649 for G3 and G5, respectively) compared to charged sphere predictions of Henry (dashed curve), O’Brien and White in KCl solutions (solid curve), and O’Brien and White in the various buffer systems (open circles). Open squares are the experimental data.

Stigter60 later fit Loeb’s values with a summation formula. More recently, Yoon61 showed how the solution of simple integral equation replicated the earlier σ′ versus ζ′ relationships while affording values of σ′ for ζ′ > 8, a parameter range important for fully charged dendrimers. Variation of Mobility with Charge. As shown in Figure 3, trends in µ0′(R) for G3 and G5 are similar, departing little from trends reported for linear polyelectrolytes.62-64 In all cases, µ0′(R) rises steeply at low R and then slowly, if at all, at large R. Measurements were obtained in three buffers with R ≈ 1.0, but Figure 3 plots data for only two, Gly-HCl-NaCl and formic acid-NaOH. The two are indistinguishable. The third buffer, KHPHCl, appears anomalous, with µ0′ lying approximately 10-20% below its expected level. This behavior may reflect specific ion binding of phthalate, the only divalent counterion in any of the buffers. For variably charged linear polyelectrolytes, some have ascribed the nonlinearity of µ0′(R) to Manning counterion condensation.63-65 The condensation hypothesis states that when the average distance between monovalent backbone charges is less than the Bjerrum length, counterions of a 1:1 electrolyte bind to the backbone in sufficient density (60) Stigter, D. J. Electroanal. Chem. 1972, 37, 61. (61) Yoon, B. J. J. Colloid Interface Sci. 1997, 192, 503. (62) Hoagland, D. A.; Smisek, D. L.; Chen, D. Y. Electrophoresis 1996, 17, 1151. (63) Whitlock, L. R. In New Directions in Electrophoretic Methods; Jorgenson, J. W., Phillips, M., Eds.; ACS Symposium Series 335; American Chemical Society: Washington, DC, 1987; p 222. (64) Gao, J. Y.; Dubin, P. L.; Sato, T.; Morishima, Y. J. Chromatogr. A 1997, 766, 233. (65) Klein, J. W.; Ware, B. R. J. Chem. Phys. 1984, 80, 1334.

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to yield an “effective” charge density of one electron charge per Bjerrum length.42,52 To connect experimental trends to condensation, one must argue that µ0′(R) manifests only the uncondensed charge density.52 This association has been criticized on theoretical grounds.53,55 The Bjerrum length in aqueous media is about 0.71 nm, while neighboring charges in fully ionized PPI dendrimers are separated by about 0.37 nm. Crudely then, counterion condensation in the dendrimers would commence at R ≈ 0.5, consistent with the onset of significant nonlinearity in Figure 3. However, inasmuch as µ0(R) varies smoothly with R, no clear sign of counterion condensation is discerned. In addition, whether µ0′(R) becomes flat above R ) 0.5 is arguable. Figure 3 also compares µ0′(R) for G3 and G5 to charged sphere predictions made with surface charge equated to Q. Predictions of Henry for a generic 1:1 electrolyte and O’Brien and White for KCl are shown as dashed and solid curves, respectively, at the experimentally relevant value of I, 0.01 M. The O’Brien and White prediction overlaps the Henry prediction at low R, and at this limit, both agree with dendrimer data. Not unexpectedly, as R increases beyond ∼0.2, the Henry curve begins to overestimate µ0′ grossly. The O’Brien and White curve, however, continues to track µ0′ into the high R range. Only a qualitative tracking can be expected, given the electrolyte mismatch between experiment (various buffers) and theory (KCl). Eliminating the mismatch, the O’Brien and White model was reevaluated for specific buffers of the experiments. Figure 3 displays the new predictions as open circles. Under most conditions, trends in buffer differ little from KCl. The predicted “dip” at intermediate R (0.6 < R < 0.9) can be ascribed to the small value of λi0 for acetate counterions in NaOAc-HOAc buffer.38 Except in NaOAcHOAc, the dominant counterion is chloride, the same as in KCl. The close agreement (deviations below 20%) between O’Brien and White predictions and data across the full R range is remarkable. Although the comparison entails crude physical approximations, no fitting parameters are introduced. The ability of the O’Brien and White theory to correlate successfully the nonlinear dendrimer behavior strongly suggests that nonlinearity emanates from the relaxation effect, which the theory incorporates, and not ion binding, which the theory ignores. Dubin et al.8 collected µ0(R) data for generation 2-5 carboxyl-terminated dendrimers, and a µ0 maximum was noted in the range R∼0.8-0.9. The O’Brien and White theory predicts a maximum in µ0(R) when aκ > 3, but Dubin et al. observed one for aκ as small as 1.25. No µ0(R) maximum is found with PPI dendrimers. Variation of Mobility with Ionic Strength and Dendrimer Size. Figure 4A and 4B shows how µ0 for G3 and G5 depends on I in buffers achieving full protonation, formic acid/NaOH and KHP/HCl. Although similar trends are observed, the former always yields higher µ0. As noted earlier, µ0 in KHP/HCl is believed to reflect specific dendrimer-counterion interactions, and data for this buffer are therefore discarded from subsequent theoryexperiment comparisons. Regardless of chemical structure or backbone flexibility, µ0 of large linear polyelectrolytes decreases nearly logarithmically with increasing I.23 When such chains are long enough, µ0 is independent of molecular size.66-73 Figure (66) Grossman, P. D. In Capillary Electrophoresis: Theory and Practice; Grossman, P. D., Colburn, J. C., Eds.; Academic Press: San Diego, CA, 1992; Chapter 4. (67) Dolnik, V.; Liu, J.; Banks, J. F., Jr.; Novotny, M. V.; Bocek, P. J. Chromatogr. 1989, 480, 321.

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Figure 4. Dependence of µ0 on I for fully charged (R ) 1) G3 and G5 dendrimers in (A) formic acid/NaOH buffers and (B) KHP/HCl buffers.

4 reveals different trends for PPI dendrimers: the function µ0(I) does not decay logarithmically but does depend on molecular size. Different physical dimensions might explain the differences between the two solute classes. PPI dendrimers are smaller than typical linear polyelectrolytes, suggesting that a comparison to charged linear oligomers might be more appropriate. Unfortunately, the electrophoretic behaviors of charged oligomers remain poorly understood.23 The product of molecular size and κ-1 is of order unity for both charged oligomers and charged dendrimers, a feature consistent with large relaxation effects. Applying the standard electrokinetic model to interpret trends in µ0 with respect to I and a, Figure 5 replots the formic acid/NaOH dendrimer data of Figure 4 in the dimensionless form µ0′/σ′ vs aκ, where as before, the assumed surface charge is equated to Q. Henry as well as O’Brien and White predictions are plotted alongside. Irrespective of aκ, Henry greatly overestimates µ0′/σ′, an unsurprising result given R ) 1.0, where the DebyeHu¨ckel approximation does not apply. On the other hand, O’Brien and White almost quantitatively matches experimental data over the entire measurement range, 0.2 < aκ < 2.0, again without any fitting of parameters. The relaxation effect dominates µ0′, as inferred by the large (68) Carney, S. L.; Osborne, D. J. Anal. Biochem. 1991, 195, 132. (69) Vo¨lkel, A. R.; Noolandi, J. Macromolecules 1995, 28, 8182. (70) Braud, C.; Vert, M. Polym. Bull. 1992, 29, 177. (71) Cohen, A. S.; Terabe, S.; Smith, J. A.; Karger, B. L. Anal. Chem. 1987, 59, 1021. (72) Cohen, A. S.; Najarian, D.; Smith, J. A.; Karger, B. L. J. Chromatogr. 1988, 458, 323. (73) Stellwagen, N. C.; Gelfi, C.; Righetti, P. G. Electrophoresis 1997, 42, 687.

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investigators have termed ion-permeable solutes as “porous spheres”, “spherical polyelectrolytes”, or “soft particles”.46-48,74 As shown by Hermans and Overbeek,75 when a fixed charge is spread uniformly across a spherical electrolyte domain of radius a, inclusion of counterions from the surrounding electrolyte effectively neutralizes some of the fixed charge. Under the Debye-Hu¨ckel approximation, the unneutralized charge fraction Qeff/Q depends on aκ,

[

Qeff sinh(aκ) 3 ) (1 + aκ)e-aκ cosh(aκ) 2 Q aκ (aκ)

Figure 5. Dependence of µ0′/σ′(aκ) on aκ for fully charged (R ) 1) G3 and G5 dendrimers (in formic acid/NaOH buffers) compared to predictions for dielectric spheres of fixed charge. For G3, σ′ ) 18.1 (a ) 1.18 nm, Q ) 30e), and for G5, σ′ ) 45.3 (a ) 1.98 nm, Q ) 126e).

gap between Henry prediction and experimental data. At all aκ, O’Brien and White predicts lower µ0′ for G5 than G3, in accord with experiment. Dubin et al.8 investigated µ0(I) for fully ionized, carboxyl-terminated dendrimers but concluded that the observed decrease of µ0 with generation is inconsistent with a charged sphere depiction. However, their analysis was inexact, relying on an inappropriate relationship between surface potential and surface charge while neglecting the relaxation effect entirely. Consequently, they attributed the variation of µ0 to nonspecific, electrostatically induced ion binding. We see no reason to assert such phenomena for PPI dendrimers, at least in ordinary buffers. The aκ range investigated by Dubin et al. (1.25 < aκ < 2.7) lies somewhat above that of the current study (0.2 < aκ < 2.0). Limitations/Alternatives to the Standard Electrokinetic Model. The O’Brien and White formalism models a rigid, nonconducting, ion-impenetrable sphere carrying a fixed surface charge or surface potential.9 A PPI dendrimer poorly matches this description. PPI dendrimers distribute fixed charges throughout their structure, and further, the dendrimers are not rigid, as thermal motion of the solvent creates incessant shape fluctuations. Even without the fluctuations, low generation dendrimers, the type examined here, appear spherical only after orientational averaging. These configurational complexities obscure the assignments of dendrimer radius and surface charge. Finally, small ions are able to diffuse or be convected into the dendrimer “interior”, a region better resembling a liquid electrolyte than a solid dielectric. Given such complexities, rigorous justification of the charged sphere depiction for dendrimer electrophoresis remains beyond reach. In his pioneering study, Henry34 envisaged that a charged sphere might not behave as a dielectric insulator, and thus, imposing the Debye-Hu¨ckel approximation, he calculated µ0 as a function of the conductivity mismatch between sphere and surrounding electrolyte. One of his results is simple: in absence of a mismatch, the external electric field is not disturbed and the Hu¨ckel formula for µ0 applies irrespective of aκ. Equation 2 transforms to

]

(4)

For the range of current attention, 0.2 < aκ < 2.0, eq 4 predicts 0.986 < Qeff/Q < 0.593. Combining eqs 3 and 4, µ0′ can be written,

µ′ ) ζ′ )

σ′eff (1 + aκ)

(5)

(3)

Presented by Hermans and Fujita,46 Hermans,47 and Overbeek and Stigter48 as the non-free-draining limit of µ0′ for a charged porous sphere, this formula is plotted alongside PPI dendrimer data in Figure 5. Agreement is worse than with O’Brien and White. Fault clearly lies in imposition of the Debye-Hu¨ckel approximation and attendant neglect of the relaxation effect. Except that segment distributions can be more complicated, the polyelectrolyte theories of Muthukumar44 and Barrat and Joanny45 bear much in common with the porous sphere theories, and they correlate the dendrimer data just as poorly. Oshima proposed an improved theoretical scheme for predicting the mobility of “soft” particles, that is, hard spheres coated with uniform polyelectrolyte layers.74 The approach incorporates the salient features of the O’Brien and White formalism while permitting penetration of small ions. In one limit, the spherical core is absent and the solute possesses a structure resembling a charged dendrimer. Unfortunately, the theory has not been fully analyzed in this limit. Thus, to address ion penetration at a level beyond Debye-Hu¨ckel, Figure 6 compares µ0′(aκ) data to O’Brien and White predictions made by counting in σ′ only terminal group charges, assuming that all interior charges are compensated by partitioned counterions. This calculation represents the worst-case scenario for the hard sphere depiction. The two predictions, for partially and fully charged dendrimers, nearly match, implying that µ0′ is insensitive to σ′ at these high σ′ values. The insensitivity is not unexpected, given that µ0′ (R) shows little increase for R above ∼0.4 (Q ) 12e) for G3 and ∼0.3 (Q ) 38e) for G5. Absent other complications, the impact of flexibility/ nonsphericity on dendrimer mobility remains difficult to assess. The difficulty has two aspects. First, the average configuration and configurational fluctuations of a charged dendrimer are poorly known (especially for low generation number dendrimers).76 Second, electrophoresis models accounting for solute flexibility are nonexistent, and those that account for nonsphericity are few. Calculations for charged spheroids reveal that random orientational averaging significantly suppresses the influence of slight nonsphericity on µ0′.77

Most evidence suggests that small electrolyte ions can freely explore the dendrimer interior, equilibrating in the local electrostatic field and carrying current. Various

(74) Oshima, H. J. Colloid Interface Sci. 1994, 163, 474. (75) Hermans, J. J.; Overbeek, J. T. G. Recl. Trav. Chim. 1948, 67, 761. (76) Welch, P.; Muthukumar, M. Macromolecules 1998, 31, 5892. (77) Yoon, B. J.; Kim, S. J. Colloid Interface Sci. 1989, 128, 275.

µ0′ ) ζ′ )

σ′ (1 + aκ)

1088

Langmuir, Vol. 19, No. 4, 2003

Welch and Hoagland

To explain the conformity of PPI dendrimers to the standard electrokinetic model in absence of any fitting, we believe attention should focus on the dendrimer shear surface. Located near the molecular periphery, this surface is not a physical interface, unlike traditional colloids. The usual problems of ion adsorption, finite ion size, anomalous surface conductivity, and so forth are consequently suppressed or absent. Detailed calculations will be required to understand fully the factors that control the shear surface’s position and shape relative to the dendrimer’s average configuration and hydrodynamic radius. The ability of electrolyte ions to penetrate and thereby compensate for a portion of the dendrimer’s charge seemingly has little impact on µ0′. This insensitivity may be a fortuitous consequence of the large charge densities created by PPI ionization. The magnitudes of ζ′ calculated for the dendrimer shear surface are considerable, in the extreme case up to ∼12. The standard electrokinetic model can be expected to fail at such potentials.13,80 Assuming validity of the model for such large ζ′, O’Brien and White predict little dependence of µ0′ on ζ′ and thus on σ′. Further variations in chemistry, ionization, and generation will enable dendrimer electrophoresis studies to be performed over an expanded parameter space. A systematic examination of the relaxation effect as a function of parameters (aκ, λi0, ζ′) appears possible. The largest obstacle to wider study may well be detection of motion; most dendrimer species lack the strong absorbance needed for capillary electrophoresis detection. Figure 6. Effect of assumed σ′ on O’Brien and White calculations for (A) G3 and (B) G5 dendrimers. In (A), open squares are for σ′ ) 9.65 (Q ) 16e), corresponding to terminal amines charged, and open circles are for σ′ ) 18.1 (Q ) 30e), corresponding to all amines charged. Likewise in (B), open squares are for σ′ ) 23.0 (Q ) 64e) corresponding to terminal amines charged, and open circles are σ′ ) 45.3 (Q ) 126e), corresponding to all amines charged.

Assessment of PPI Dendrimers as Model Charged Spherical Particles. Much effort has been devoted to the production of model colloids suitable for testing of electrophoresis theories, and for this purpose, the widespread acceptance of the O’Brien and White calculations makes sphere systems attractive. The most intriguing systems are those that span aκ of order unity, a parameter range where theory is most complicated and the relaxation effect plays its largest role. In absence of fitting parameters, few (if any) systems have been reported to display the near quantitative agreement between theory and experiment found with PPI dendrimers. Adjustments to the surface charge or displacements of the shear plane underscore most previous claims of “reasonable” agreement between theory and experiment.20,78 Results of Carbeck and Negin79 are the most similar to those presented here. By fitting the charge of an unmodified, nanometer-sized protein, they showed that the standard electrokinetic model could predict shifts in protein mobility as charge was increased by incremental acetylation; values of protein size obtained by fitting of theory to experiment depended slightly on the added electrolyte concentration. (78) Gittings, M. R.; Saville, D. A. Langmuir 1995, 11, 798. (79) Carbeck, J. D.; Negin, R. S. J. Am. Chem. Soc. 2001, 123, 1252.

Summary PPI dendrimers behave as model electrophoresis systems for aκ of order unity, a range where model systems have been scarce or nonexistent. Good accord of µ0′ data with the standard electrokinetic model predictions of O’Brien and White reveals that the model reliably captures the relaxation effect in the parameter range where the effect is largest. The observed conformity of PPI dendrimers to a charged sphere depiction is unexpected, given that electrolyte ions freely penetrate into the interior and solute shape is not spherical. In ordinary buffers, values of µ0′ for PPI dendrimers can be understood without account for counterion condensation or ion binding. Similarity of µ0′(R) for charged dendrimers and linear polyelectrolytes suggests to us that deviations from linearity in both cases may well be attributed to the relaxation effect, and not as commonly assumed, electrostatically induced ion binding. However, in absence of a suitable electrophoresis model for flexible polyelectrolytes, particularly one that includes the relaxation effect, our hypothesis remains untested. Acknowledgment. Financial support was received from the University of Massachusetts Materials Research Science and Engineering Center. The authors benefited from the many helpful comments of M. Muthukumar. LA026572+ (80) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces, 6th ed.; John Wiley & Sons: New York, 1997.