Low-Frequency Micromotion of DEP-Levitated Plant Protoplasts: III. Rationalization of Higher Harmonics

Abstract

Experimental observations, previously reported in this series of papers, in the low-frequency regime of the spectrum in dielectrophoretically levitated plant protoplasts, have revealed higher harmonics to the fundamental driving field frequency (when ≤50 Hz). Our previous modeling of the nonlinear dynamics of this micromotion have failed to reproduce or indicate the source of this phenomena. The existing dynamical micromotion model, a highly nonlinear second-order differential equation constructed on the basis of Newton's second law to the situation of a particle levitated in an inhomogeneous electric field, has been extended to include: (a) cubic nonlinearity in the dielectrophoretic forces, (b) convective flow contributions as described by Oseen's equation, and (c) a novel mathematical model to describe the nonlinear dependence of the particle surface charge on the electric field frequency of the driver based on the solution of a Langevin or stochastic equation. Numerical integration of the vector field by means of a Runge–Kutta sixth-order algorithm has shown that neither the inhomogeneity of the electric field, as realized in dielectrophoresis of the suspended particle, nor the nonlinear hydrodynamic interaction of the particle with the medium are responsible for the appearance of the higher harmonics. Incorporation of the frequency-dependent surface charge modeling, however, has given good agreement with the experimental observations. Fast Fourier transforms of the numerical solutions show the experimentally observed higher harmonics in the micromotion attesting to the significance of the electric field-dependent dissociation constant of the carboxylic groups in the surface proteins. Phase plane portrait solutions when examined over several frequencies provide a means of estimating the surface charge of the plant protoplasts.