Nonlinear Dynamics

, Volume 76, Issue 2, pp 1013–1030

Dynamics of microscopic objects in optical tweezers: experimental determination of underdamped regime and numerical simulation using multiscale analysis

Authors

  • Mahdi Haghshenas-Jaryani
    • Department of Mechanical and Aerospace EngineeringThe University of Texas at Arlington
  • Bryan Black
    • Department of PhysicsThe University of Texas at Arlington
  • Sarvenaz Ghaffari
    • Department of Mechanical and Aerospace EngineeringThe University of Texas at Arlington
  • James Drake
    • Department of BioengineeringThe University of Texas at Arlington
    • Department of Mechanical and Aerospace EngineeringThe University of Texas at Arlington
  • Samarendra Mohanty
    • Department of PhysicsThe University of Texas at Arlington
Original Paper

DOI: 10.1007/s11071-013-1185-0

Cite this article as:
Haghshenas-Jaryani, M., Black, B., Ghaffari, S. et al. Nonlinear Dyn (2014) 76: 1013. doi:10.1007/s11071-013-1185-0

Abstract

This article presents new experimental observations and numerical simulations to investigate the dynamic behavior of micro–nano-sized objects under the influence of optical tweezers (OTs). OTs are scientific tools that can apply forces and moments to small particles using a focused laser beam. The motions of three polystyrene microspheres of different diameters, 1,950, 990, and 500 nm, are examined. The results show a transition from the overdamped motion of the largest bead to the underdamped motion of the smallest bead. The experiments are verified using a dynamic model of a microbead under the influence of Gaussian beam OTs that is modeled using ray-optics. The time required to numerically integrate the classic Newton–Euler model is quite long because a picosecond step size must be used. This run time can be reduced using a first-order model, and greatly reduced using a new multiscale model. The difference between these two models is the underdamped behavior predicted by the multiscale model. The experimentally observed underdamped behavior proves that the multiscale model predicts the actual physics of a nano-sized particle moving in a fluid environment characterized by a low Reynolds number.

Keywords

Multiscale modelingRay-opticsDynamicsOptical tweezersLow Reynolds numberFluid dynamicsBrownian motionMethod multiple scales

Supplementary material

$$\begin{aligned}&\mathbf{F}_{\text {Brownian}} = C_{o1}(t) \, \widehat{\mathbf{N}}_1 + C_{o2}(t) \ \widehat{\mathbf{N}}_2 \nonumber \\&\mathbf{T}_{\text {Brownian}} = \bar{L}_S \ C_{o3}(t), \end{aligned}$$$$\begin{aligned} E \left[ C_{oi}(t) \right] = \langle \, C_{oi}(t) \, \rangle = 0 = \mu \end{aligned}$$$$\begin{aligned} E \left[ \ C_{oi}(t_1) \ C_{oj}(t_2) \ \right] = 2 \ \beta \ k_B \ \mathrm{T} \ \delta (t_1-t_2) \delta _{i,j}, \end{aligned}$$https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-1185-0/MediaObjects/11071_2013_1185_Fig18_HTML.gif$$\begin{aligned} E[ C_{oi}^2(t) ] \ = \ 2 \ \beta \ k_B \ \mathrm{T} \ = \ \mathrm{Var}(C_{oi}(t)) \ = \ \sigma ^2 \end{aligned}$$$$\begin{aligned} \varvec{\Gamma }_{\text {Brown}} = \hbox {R}_{\text {nd}} \ \mathbf{r}_{\text {nd}} = \hbox {R}_{\text {nd}} \ \left[ \begin{array}{l} {C}_{o1} \\ {C}_{o2} \\ {C}_{o3} \\ \vdots \end{array} \right] , \end{aligned}$$https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-1185-0/MediaObjects/11071_2013_1185_Fig19_HTML.gif$$\begin{aligned} \mathbf{F}_{\text {g}}&= -m_S g_0 \ \widehat{\mathbf{N}}_2\end{aligned}$$$$\begin{aligned} \mathbf{F}_B&= \rho _{\text {m}} g_0 V_p \ \widehat{\mathbf{N}}_2\end{aligned}$$$$\begin{aligned} \mathbf{F}_D&= -\beta _v^{N} \mathbf{V}_{S_0}, \end{aligned}$$$$\begin{aligned} \beta _v = 6\pi \mu _{\text {m}} r_S \end{aligned}$$$$\begin{aligned} \phi _{FS_0} \ = \ \arctan \frac{q_1}{\acute{f}-q_2}, \end{aligned}$$$$\begin{aligned}&\sin \theta = \frac{r_S}{\sqrt{q_1^2+(\acute{f}-q_2)^2}}\end{aligned}$$$$\begin{aligned}&\cos \theta = \sqrt{1-{\sin }^2 \theta } \end{aligned}$$$$\begin{aligned}&\theta = \arctan \frac{\sin \theta }{\cos \theta } \end{aligned}$$$$\begin{aligned}&\rho _{t1} = \acute{f} \ \tan (\gamma )\qquad \rho _{t2} = \acute{f} \ \tan (\delta ), \end{aligned}$$https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-1185-0/MediaObjects/11071_2013_1185_Fig20_HTML.gif$$\begin{aligned} \rho _k = \rho _{t1} + (k-1) \Delta \rho \qquad k = 1, \cdots , 15, \end{aligned}$$$$\begin{aligned} \Delta \rho = \frac{\text {span}}{14} \qquad {\text {span}} = \left| \rho _{t2} - \rho _{t1} \right| \end{aligned}$$$$\begin{aligned} {\text {span}} = 2R_{\text {obj}} \end{aligned}$$$$\begin{aligned} \mathbf{M}_{k} = \mathbf{P}_{S_{0} C_{k}} \times \mathbf{F}_{{\text {tot}},k} \end{aligned}$$$$\begin{aligned} {T}_{\text {drag}} = - \beta _{\omega } \dot{q}_3, \end{aligned}$$$$\begin{aligned} \beta _{\omega } = 8\pi \mu _{\text {m}} r_{S}^{3}. \end{aligned}$$$$\begin{aligned} {T}_{\text {Brownian}} = \bar{L}_S \ C_{o3}(t), \end{aligned}$$$$\begin{aligned} \cos \alpha _k&= \frac{\mathbf{P}_{C_{k}P_k} \cdot \widehat{\mathbf{n}}_k}{|| \mathbf{r}_{C_{k}P_{k}} || || \widehat{\mathbf{n}}_{k} ||} \end{aligned}$$$$\begin{aligned} \sin \alpha _k&= \frac{\mathbf{P}_{C_{k}P_{k}} \times \widehat{\mathbf{n}}_k}{|| \mathbf{P}_{C_{k}P_{k}} || || \widehat{\mathbf{n}}_k ||} \end{aligned}$$$$\begin{aligned} \alpha _k&= \arctan \frac{\sin \alpha _k}{\cos \alpha _k} \end{aligned}$$$$\begin{aligned} \sin \beta _k&= \frac{n_{m} \sin \alpha _{k}}{n_S} \end{aligned}$$$$\begin{aligned} \cos \beta _k&= |\sqrt{1 - \sin ^2 \beta _k}| \end{aligned}$$$$\begin{aligned} \beta _k&= \arctan \frac{\sin \beta _k}{\cos \beta _k} \end{aligned}$$$$\begin{aligned} R_k&= \frac{1}{2} \left[ \frac{n_{\text {m}} \cos \alpha _k - n_S \cos \beta _k}{n_{\text {m}} \cos \alpha _k + n_S \cos \beta _k} ^2 \right. \nonumber \\&\left. +\frac{n_{\text {m}} \cos \beta _k - n_S \cos \alpha _k}{n_{\text {m}} \cos \beta _k + n_S \cos \alpha _k} ^2 \right] \end{aligned}$$$$\begin{aligned} T_k&= 1 - R_k \end{aligned}$$https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-1185-0/MediaObjects/11071_2013_1185_Fig21_HTML.gif$$\begin{aligned} F_{\text {{scat}}} = \frac{n_{m} P}{c} \left[ \ 1+R\cos 2\alpha - a \ \right] , \end{aligned}$$$$\begin{aligned} a = \frac{T^{2}\left[ \cos (2\alpha - 2\beta ) - R\cos (2\alpha - \pi ) \right] }{1 + R^{2} - 2R \cos (\pi - 2\beta )} \end{aligned}$$$$\begin{aligned} F_{{\text {grad}}} = \frac{n_{m} P}{c} \left[ \ R \sin 2 \alpha - \ b \ \right] , \end{aligned}$$$$\begin{aligned} b = \frac{T^{2}\left[ \sin (2\alpha - 2\beta ) - R\sin (2\alpha - \pi ) \right] }{1 + R^{2} - 2R\cos (\pi - 2\beta )}, \end{aligned}$$https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-1185-0/MediaObjects/11071_2013_1185_Fig22_HTML.gif$$\begin{aligned} \phi _k = \arctan \frac{\rho _k}{f}. \end{aligned}$$$$\begin{aligned} _\mathbf{N}^{\mathbf{F}_k}\mathbf{R} = \left[ \begin{array}{lll} \cos \phi _k &{} - \sin \phi _k &{} 0 \\ \sin \phi _k &{} \cos \phi _k &{} 0 \\ 0 &{} 0 &{} 1 \end{array} \right] . \end{aligned}$$$$\begin{aligned} F_{{\text {tot}}k} = \frac{n_{m} P_k}{c} \left[ 1 + R_{k} e^{2i \alpha _k} - T_{k}^{2} d \ \right] , \end{aligned}$$$$\begin{aligned} \displaystyle d \ = \ \frac{e^{2i(\alpha _k - \beta _k)}}{1 - R_{k} e^{i(\pi - 2\beta _k)}}. \end{aligned}$$$$\begin{aligned} d&= \frac{e^{i(2\alpha _k - 2\beta _k)}}{1 - R_{k} e^{i(\pi - 2\beta _k)}} \cdot \frac{1 - R_{k} e^{-i(\pi - 2\beta _k)}}{1 - R_{k} e^{-i(\pi - 2\beta _k)}} \end{aligned}$$$$\begin{aligned}&= \frac{e^{i(2\alpha _k - 2\beta _k)} \left[ 1 - R_{k} e^{-i(\pi -2\beta )}\right] }{\left[ 1 - R_{k} e^{i(2\alpha _k -2\beta _k)} \right] \left[ 1 - R_{k} e^{-i(2\alpha _k -2\beta _k)} \right] } \nonumber \\&= \frac{e^{i(2\alpha _k -2\beta _k)} - R_{k} e^{\left[ i(2\alpha _k -2\beta _k) -i(\pi -2\beta _k)\right] }}{1 + R_{k}^{2} -R_{k} \left[ e^{i(\pi - 2\beta _k)} + e^{-i(\pi -2\beta _k)} \right] } \nonumber \\&= \frac{e^{i{2\alpha _k - 2\beta _k}} - R_{k} e^{i(2\alpha _k - \pi )}}{1 + R_{k}^{2} - 2R_{k} \cos \left( \pi - 2\beta _k \right) } \end{aligned}$$$$\begin{aligned} \mathbf{F}_{tot,k} = F_{{\text {grad}},k} \widehat{\mathbf{N}}_1 + F_{{\text {scat}},k} \widehat{\mathbf{N}}_2. \end{aligned}$$

Copyright information

© Springer Science+Business Media Dordrecht 2013