(468b) Testing Fundamental Theories for Polyelectrolyte Electrophoresis: Comparing Theory and Experiment As Polyelectrolyte Charge Spacing and Solvent Dielectric Constant ARE Independently Varied

Standard electrostatic and electrokinetic models for polyelectrolytes depict the polymers as chains of uniform charge immersed in a continuous fluid electrolyte that dissolves point-like dissociated counterions alongside any small ions of added electrolyte. Under this depiction, the charge properties of single dilute polyelectrolytes are expressed through two dimensionless groups, a linear charge density parameter ξ and an ionic strength (or solute size) parameter aκ. The latter is the product of the polyelectrolyte’s backbone radius a with the inverse of the solvent’s Debye length κ1, and the former is the ratio e2/4πεεokTb, where e is the electron charge, e is the solvent dielectric constant, ε0 is the vacuum permittivity, k is the Boltzmann constant, T is the temperature, and b is the average contour spacing of polyelectrolyte unit charges; ξ is termed the ‘Manning parameter”, and the combination e2/4πεεokT is termed the Bjerrum length lb, the separation between unit charges at which their mutual electrostatic interaction energy matches kT. The dimensionless mobility μoʹ′, given by 3ηeμo/2εεokT, where μo is the polyelectrolyte mobility and η is the solvent viscosity, is thereby predicted by such models to be a unique function of ξ if aκ is fixed. Classically, the counterion condensation theory of Manning provides a functional form for this ξ dependence, although other forms are available.

Taken together, expectations of the previous paragraph frame a series of fundamental tests for the assumptions that underscore standard polyelectrolyte models, with actual trends potentially explored – as done here - by electrophoresis experiments on model polyelectrolytes. To compare electrophoresis theory and experiment, it is conventional to define an effective polyelectrolyte linear charge density ξeff. This parameter captures the nonlinear electrostatic and electrokinetic impacts associated with the clustering, or condensation, of counterions along highly charged polyelectrolyte backbones. This clustering, driven by strong electrostatic attractions, reduces the polyelectrolyte’s effective charge and thus its electrophoretic mobility. Clustering is absent when the polyelectrolyte linear charge is sufficiently low, and thus, ξeff=ξ for small ξ and ξeff<ξ for large ξ. Since the Debye-Hückel theory applies at low ξ, and μoʹ′ in this limit is proportional to ξ, a measurement of μoʹ′ at large ξ is easily transformed to a value of ξeff by extrapolating a mobility expression derived from Debye-Hückel theory.

Our key experimental strategy is to measure by capillary electrophoresis the free solution electrophoretic mobility of ionenes, the class of polyelectrolytes that contain strongly charged units (quaternized amines) in their flexible backbones. These polymers dissolve with unchanged b not just in aqueous buffers but also in other solvents of moderate to high ε; for the chosen ionenes, b itself is controlled by monomer choice, which sets the backbone chemistry between charges. Solvent mixing allows ε to be varied broadly while maintaining good solvent conditions for the polymers. Ionenes of this study dissolve in solvents such as water, ethylene glycol, acetonitrile, and n-methylformamide with 0.1<ξ<5, the domain of greatest experimental and theoretical interest. With these beneficial properties, ionenes are uniquely suited to the proposed tests, but trends for other polyelectrolytes prove similar to those found for ionenes. Measurements for ionenes with hydrophobic spacers between charge units are indistinguishable from those for ionenes with hydrophilic spaces between these units. Further, positively charged polyelectrolytes behave no differently than negatively charged polyelectrolytes.

In high ε solvents such as water, this study’s measurements uncover a linear rise of μoʹ′ with ξ at low ξ before an asymptotic μoʹ′ saturation as ξ exceeds approximately unity; further, as anticipated by standard theories, the measured variations of μo with b and ε collapse when expressed through ξ. Indeed, observed behaviors in high ε solvents are reasonably consistent with theoretical predictions made long ago by Manning. In moderate ε solvents, trends are more complex. Now, at fixed aκ, μoʹ′ is not a unique function of ξ, and μoʹ′ reaches a plateau at ξ different from unity. The unexpected behaviors suggest a new controlling parameter, which we propose is the counterion size d, expressed as the dimensionless ratio lb/d (more properly,
d is regarded as the distance of closest approach between polyelectrolyte and counterion). To test our hypothesis, experiments have been repeated with counterions of different hydrated size. As the hypothesis predicts, at constant aκ, μoʹ′ reduces to a universal function of lb/d and ξ. The measured function surprisingly shows counterion condensation above critical values of both ξ, and lb/d, the latter effect not predicted, to our knowledge, by any theory. The unexpected condensation reflects an onset of strong ion clustering as ε deceases.