Nonlinear Electro-Osmosis of Uncharged Polymer Solutions with Low Ionic Strength
Nonlinear electro-osmotic behaviour of dilute non-adsorbing polymer solutions with low salinity is investigated using Brownian dynamics simulations and a kinetic theory. In the Brownian simulations, the hydrodynamic interaction between the polymers and a
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Nonlinear Electro-Osmosis of Uncharged Polymer Solutions with Low Ionic Strength
Abstract Nonlinear electro-osmotic behaviour of dilute non-adsorbing polymer solutions with low salinity is investigated using Brownian dynamics simulations and a kinetic theory. In the Brownian simulations, the hydrodynamic interaction between the polymers and a no-slip wall is considered using the Rotne-Prager approximation of the Blake tensor. In a plug flow under a sufficiently strong applied electric field, the polymer migrates toward the bulk, forming a depletion layer thicker than the equilibrium one. Consequently, the electro-osmotic mobility increases nonlinearly with increasing electric field and becomes saturated. This nonlinear mobility does not depend qualitatively on the details of the rheological properties of the polymer solution. Analytical calculations using the kinetic theory for the same system quantitatively reproduce the results of the Brownian dynamics simulation well. Keywords Electrokinetics · Electrolyte · Polymer · Hydrodynamic interaction
3.1 Toy Model First, we propose a toy model for electro-osmosis of polymer solutions [1]. A dilute solution of non-adsorbing polymers is considered. The viscosity of the solution is given by (3.1) η = η0 (1 + ηsp ), where η0 is the viscosity of the pure solvent, and ηsp is the specific viscosity of the solution. The gyration length of the polymers is defined as δ0 , which is of the same order of the equilibrium depletion length. It is assumed that the polymers have δ0 ≈ 100 nm. Ions are also dissolved in the solution with the Debye length λ. When a well deionized water is considered, the Debye length is of the order of λ ≈ 103 nm although such a salt-free water is hardly realized owing spontaneous dissolutions of carbon dioxides. The interfacial structure near a charged surface is characterized by λ and δ0 . When an external electric field is applied, a shear flow is locally imposed within the distance λ from the wall, and the resultant shear rate is
© Springer Nature Singapore Pte Ltd. 2017 Y. Uematsu, Electro-Osmosis of Polymer Solutions, Springer Theses, DOI 10.1007/978-981-10-3424-4_3
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3 Nonlinear Electro-Osmosis of Uncharged Polymer Solutions …
γ˙ ≈
μ0 E , λ
(3.2)
where μ0 is the electro-osmotic mobility for the pure solvent and is estimated typically as μ0 ≈ 10−8 m2 /(V · s). According to the studies of the cross-stream migration in the uniform shear flow, [2] the depletion layer thickness depends on the shear rate, ˙ 2, δ ≈ δ0 (τ γ)
(3.3)
where τ is the characteristic relaxation time of the polymers, τ≈
η0 δ0 3 , kB T
(3.4)
where kB T is the thermal energy and is typically 10−4 s at room temperature. Using Eqs. (3.2) and (3.3), the depletion length in the presence of the applied electric field E can be expressed by ⎧ δ0 for E < E 0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 2 E δ≈ δ0 for E 0 ≤ E ≤ E 1 , ⎪ E ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎩ λ for E 1 < E,
(3.5)
√ where E 0 = λ/τ μ0 , and E 1 = E 0 λ/δ0 . Here, for simplicity, we assume that the depletion length does not exceed the Debye
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