Electromechanical instability of nanobridge in ionic liquid electrolyte media: influence of electrical double layer, dispersion forces and size effect

Abstract

In this paper, the electromechanical response and instability of the nanobridge immersed in ionic electrolyte media is investigated. The electrochemical force field is determined using double-layer theory and linearized Poisson–Boltzmann equation. The presence of dispersion forces, i.e., Casimir and van der Waals attractions are incorporated considering the correction due to the presence of liquid media between the interacting surfaces (three-layer model). The strain gradient elasticity is employed to model the size-dependent structural behavior of the nanobridge. To solve the nonlinear constitutive equation of the system, three approaches, e.g., the Rayleigh–Ritz method, Lumped parameter model and the numerical solution method are employed. Impacts of the dispersion forces and size effect on the instability characteristics as well as the effects of ion concentration and potential ratio are discussed.

Appendix: Lumped parameter model

In order to develop a lumped parameter model, a trial solution for deflection of the nanobridge is selected as the following:

$$ W(X) = \frac{{W_{\max}}}{2}\left({1 - \cos \left({\frac{2\pi X}{L}} \right)} \right) $$

(40)

Taking the derivative from total energy of the system [Eq. (22)] with respect to W _max and setting the result to zero (e.g., $\frac{{{\text{d}}\varPi}}{{{\text{d}}W_{\max}}} = 0$), yields the load–deflection characteristic equation:

$$ \frac{{\pi^{4} W_{\max} E}}{{4L^{4}}}\left({16ID_{1} + 64I\pi^{2} D_{2} + AW_{\max}^{2}} \right) - f_{\text{ext}} = 0 $$

(41)

Substituting f _ext at W = W _max in the above relation one can obtain:

$$ \begin{aligned} & \frac{{\pi^{4} W_{\max} E(16ID_{1} + 64I\pi^{2} D_{2} + AW_{\max}^{2})}}{{4L^{4}}} + \frac{{b\varepsilon \varepsilon_{0} \kappa^{2} \psi_{1}^{2} \left[{2\frac{{\psi_{2}}}{{\psi_{1}}}\cosh \left({\kappa (g - W_{\max})} \right) - 1 - \left({\frac{{\psi_{2}}}{{\psi_{1}}}} \right)^{2}} \right]}}{{2\sinh^{2} \left({\kappa (g - W_{\max})} \right)}} \\ & \quad - \left\{{\begin{array}{*{20}c} {\frac{{\bar{A}b}}{{6\pi (g - W_{\max})^{3}}}} \\ {\frac{{\pi^{2} \hbar b}}{{240\sqrt {\tau \upsilon} (g - W_{\max})^{4}}}\,} \\ \end{array}} \right. = 0 \\ \end{aligned} $$

(42)

Using the definition of $w_{\max} = W_{\max}^{{}}/g$ the dimensionless relation can be written as:

$$ \pi^{4} w_{\max} \left({4D_{1} + 16\pi^{2} D_{2} + \frac{1}{2}w_{\max}^{2} \eta} \right) + \frac{{\beta^{2} [2\lambda \cosh (\xi_{0} (1 - w_{\max})) - 1 - \lambda^{2}]}}{{2\sinh^{2} (\xi_{0} (1 - w_{\max}))}} - \frac{{\alpha_{m}}}{{(1 - w_{\max})^{m}}} = 0 $$

(43)

By rearranging Eq. (43), the relation between applied voltage and the maximum deflection can be obtained in the form of Eq. (37).