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

1 Introduction

Optical tweezers (OTs) use radiation pressure from a focused laser beam to manipulate microscopic objects as small as atoms [2]. This technique has been used for more than 40 years in the physical sciences to study the behaviors and physical properties of micron and submicron-sized particles [1]. Free-space as well as fiber-optic versions [22, 24] of OTs have been used in the biological sciences to cause nanometer-range displacements of, and to apply piconewton-range forces to objects ranging from 10 nm to over 100 \(\upmu \)m. Optical forces have been used to measure the mechanical properties of DNA, cell membranes, whole cells [25], and microtubules; much research has been conducted on motor proteins such as kinesin, dynein, myosin, and RNA polymerase using optical forces [9, 10, 14, 18, 23, 27, 32, 34, 35].

Most studies in this field have investigated the behavior of microbeads around the focal line of the optical trap to determine important properties including the radial and axial trap strengths, spring constants, and force profiles [6]. However, the physical interaction of particles with their environment and its effect on their dynamic behavior during this trapping process at different length scales, micro and nano, have not been adequately studied. This issue leads to inaccurate inferences about the dynamic behavior of particles, living cells, and proteins at the micro and nanoscales [4, 5, 12, 13]. For example, overdamped behavior was observed for micrometer-sized objects, and was extrapolated to predict that even smaller particles would behave similarly [29].

In this report, novel experimental studies have been carried out on a series of beads at different length scales using OTs. The results show an interesting transition from the overdamped to the underdamped motions of particles at the micro and submicron scales. These unconventional results call into question the widely accepted notion of the overdamped motion of small particles in fluids characterized by a low Reynolds number and imply the relevance of inertial effects at the submicron scale. In other words, the mass properties should be retained in the equations of motion.

The observed underdamped behavior is confirmed using a planar dynamic model of a microsphere, under the influence of Gaussian-beam OTs. The beads’ motions from an initial position to the trap’s focal line are simulated. A multiscale modeling approach [4, 5, 12], based on a concept from the method of multiple scales [26], is used to generate a model that closely reproduces experimental observations of the microsphere’s motion. The multiscale features of physical phenomena in the bead model are caused by [12]:
  1. 1.

    Different structural length scales, and

     
  2. 2.

    Forces from interaction with the environment.

     
The disproportionality between the terms in the equations of motion, due to fluid–structure interactions with the environment, requires a multiple time scale model.

There are three primary methods for modeling optical forces. In the Rayleigh regime, the particle is approximated as a point dipole within an electromagnetic field. This approach is useful when the particle radius is much less than the wavelength of the laser. In the Mie regime, where the bead’s size is comparable with the wavelength, Mie scattering theory is used. Recent developments in computational techniques based on Mie scattering theory provide a capability to extend this method to larger particles up to 100 \(\upmu \)m diameter [33]. In the ray-optics regime, the laser beam is discretized into a number of rays, and geometric optics is used to calculate the forces exerted on the particle by each ray. This approach is useful for particles of radius larger than the wavelength of laser light used. In this article, a ray-optics approach is used for the sake of simplicity.

OTs simulations can be found on the Internet, such as [17] (a Java applet used for teaching purposes), [8] (a simple animation of a trapped bead), and [28] (an illustration of a DNA-stretching experiment in the Rayleigh regime), and a few theoretical studies such as [34]. These simulations model the system using a first-order differential equation, the overdamped Langevin equations, which ignore the mass and acceleration terms thereby implying overdamped motion. However, some issues arise when the mass is omitted concerning the violation of Newton’s second law [5]. In this article, a multiscale modeling approach, following that derived in [5, 12], yields a second-order model that retains the mass properties and thus can predict the experimentally observed underdamped motions.

2 Experimental observations

The bead’s motion is investigated using an OTs experiment. These experiments observed the oscillations predicted by the multiscale model. The setup for these experiments is discussed in the next section.

2.1 Experimental setup

Green-fluorescent polystyrene microspheres (Polysciences Inc.) were suspended in distilled water, and a small drop of this solution was placed on a coverslip. The OTs and the imaging platform are shown in Fig. 1. A Ti: sapphire laser (MaiTai HP, Newport Spectra-Physics Inc.) beam operating in cw mode at 800 nm was expanded using a beam expander (BE) and guided toward the sample via folding mirrors and into the rear port of an inverted optical microscope (Nikon Ti-U Eclipse). The laser beam was coupled to a 100\(\times \) microscope objective (MO) (NA: 1.3 or 1.4 in some cases) through the back laser port as shown. For temporal modulation of the tweezers beam (i.e., force), the initiation and exposure duration was controlled by an external shutter (S, Uniblitz).
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Fig. 1

Experimental setup: L1: Ti sapphire laser, BE beam expander, S shutter, P polarizer, WL white light, BS beam splitter, FL fluorescence excitation mercury lamp, ExF excitation filter, EmF emission filter, MO microscope objective, CL condenser lens, DM 1 & 2 dichroic mirror, and M mirror

Samples were illuminated using a high-pressure mercury lamp (Nikon) through blue/green excitation–emission filter cube to achieve high contrast for particle tracking. A dichroic mirror (DM1) was mounted to reflect the laser beam to the objective and to allow for the transmission of fluorescence excitation light (blue: BL). The fluorescent microsphere was trapped in the diffraction-limited spot, and the trapping plane was matched with imaging plane by varying the divergence of the laser beam. The power of the optical trapping beam at the sample plane was estimated by multiplying the transmission factor of the objective with the laser beam power measured at the back aperture of the objective using a power meter (PM100D, Thorlabs Inc.). Fine laser power control was managed by orienting the polarizer (P). Images were collected with a reverse-cooled high-speed digital camera (Hamamatsu C1140). The images were processed using ImageJ software. An IR cut-off filter (EmF) was used to prevent the backscattered laser light from reaching the camera.

2.2 Experimental results

In order to examine the effect of size on the dynamic behavior of objects in a trapping process, three beads with different diameters, 500, 990, 1,950 nm were used. Three experiments were carried out for each bead size for the reported laser/optical parameters, yielding a total of nine experiments. This investigation begins by considering the largest diameter bead, 1,950 nm, in Fig. 2. The circles mark the experimental results which show the overdamped behavior of microbead in response to the laser and viscous friction forces. This behavior has been observed for micrometer-sized objects and was extrapolated to predict that even smaller particles would exhibit overdamped behavior [29].
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Fig. 2

Experimental data for the motion of 1,950 nm microparticle in the horizontal direction. The data were captured at 2,134 frames/s using OTs with 149 mW power

The underdamped behavior (oscillations) begins to show for the 990-nm diameter bead in Fig. 3. The bead overshoots the focal line and oscillates a small amount before settling. The smallest bead, with 500-nm bead diameter, discussed in Sect. 3.2, shows a much larger overshoot of the focal line with larger oscillations in Fig. 4. These observations are corroborated using a new approach to multiscale modeling discussed in Sect. 3.
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Fig. 3

Experimental data for the motion of 990 nm microparticle in the horizontal direction. The data were captured at 5,132 frames/s using OTs with 156 mW power

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Fig. 4

Experimental data for the motion of 500 nm microparticle in the horizontal direction. The data were captured at 3,214 frames/s using OTs with 156 mW power

3 Bead model

3.1 Dynamic model

The simulations are based on a simple dynamic model of a planar bead in a fluid environment. The model was developed using Newton’s second law and Euler’s equations for rigid body dynamics. Fig. 5a shows the general setup for the simulation model. The inertial reference point, \({N_{\text {o}}}\), is defined as the center of the objective lens. The inertial reference frame, defined by the unit vectors \(\widehat{\mathbf{N}}_1\) and \(\widehat{\mathbf{N}}_2\), is composed of a right-handed, orthogonal set of axes. The virtual point F\(^\prime \) indicates the true point of convergence of the laser beam. The focal line is located at point \((0,f)\), with respect to the inertial reference frame:
$$\begin{aligned} f = \frac{n_\text {g} R_{\text {obj}}}{\text {NA}}, \end{aligned}$$
(1)
where \(n_{\text {g}},\,R_{\text {obj}}\), and NA are the index of refraction of glass, the radius of objective lens, and the numerical aperture of objective lens, respectively. The microparticle is defined as a body, S, with a center of mass, \(S_{\text {o}}\), and a body-attached frame, frame S. The generalized coordinates and speeds are defined as \(\mathbf{q} = \{q_1,q_2,q_3\}\), and \({\dot{\mathbf{q}}} = \{\dot{q}_1,\dot{q}_2,\dot{q}_3\}\), respectively. The dashed red (gray) line in Fig. 5 shows a sample of the laser beam, originating at distance \(\rho \) from \(N_{\text {o}}\), which intersects the bead.
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Fig. 5

Overall model setup: a coordinates, points, and frames; b forces and moments on the bead

The translational motion is defined by
$$\begin{aligned} \sum \mathbf{F} \ = \ m \, {\ddot{\mathbf{x}}} \ = \ m \left[ \begin{array}{l} \ddot{q}_1 \\ \ddot{q}_2 \end{array} \right] , \end{aligned}$$
(2)
where \(m\) is the total mass of the system, the vector \({\ddot{\mathbf{x}}}\) is the translational acceleration of the mass center, and the coordinates \(q_1\) and \(q_2\) are defined in Fig. 5a. The translational active forces are
$$\begin{aligned} \sum \mathbf{F} \ = \ \mathbf{F}_{\text {g}} + \mathbf{F}_{\text {drag}} + \mathbf{F}_{\text {buoy}} + \mathbf{F}_{\text {laser}} + \mathbf{F}_{\text {Brownian}}, \end{aligned}$$
(3)
where \(\mathbf{F}_{\text {g}},\,\mathbf{F}_{\text {drag}},\,\mathbf{F}_{\text {buoy}},\,\mathbf{F}_{\text {laser}}\), and \(\mathbf{F}_{\text {Brownian}}\) are gravity, viscous drag, buoyancy, optical, and random thermal forces, respectively. These forces are depicted in Fig. 5b and discussed in the Appendices.
Euler’s equation describes rotational motion as
$$\begin{aligned} \sum {M} \ = \ I_{33} \ \ddot{q}_3, \end{aligned}$$
(4)
where \(I_{33}\) is the moment of inertia of the bead about the \(\widehat{\mathbf{N}}_3 = \widehat{\mathbf{N}}_1 \times \widehat{\mathbf{N}}_2\) direction, and \(\ddot{q}_3\) is the angular acceleration of the bead corresponding to the \(q_3\) coordinate shown in Fig. 5a. The sum of moments acting on the bead is defined as
$$\begin{aligned} \sum {M} = {T}_{\text {laser}} + {T}_{\text {drag}} + {T}_{\text {Brownian}}, \end{aligned}$$
(5)
where \({T}_{\text {laser}},\,{T}_{\text {drag}}\) (see [20]), and \(\mathbf{T}_{\text {Brownian}}\) are beam, viscous drag, and random thermal moments, respectively. These moments are depicted in Fig. 5b and explicitly defined in the Appendices.
The combined system model can be expressed as
$$\begin{aligned} \left[ \begin{array}{lll} m &{} 0 &{} 0 \\ 0 &{} m &{} 0 \\ 0 &{} 0 &{} I_{33} \end{array} \right] \left[ \begin{array}{l} \ddot{q}_1 \\ \ddot{q}_2 \\ \ddot{q}_3 \end{array} \right] \ = \ A \, {\ddot{\mathbf{q}}} \ =\left[ \begin{array}{ll} {\sum } &{} \mathbf{F} \\ {\sum }&{} {M} \end{array} \right] , \end{aligned}$$
(6)
where A is the mass matrix. The terms on the left-hand side of (6) depend on mass and thus are referred to as generalized inertia forces. The forces on the right-hand side of (6) are called generalized active forces.

3.2 Multiscale analysis

A key issue with this model is that a very small body moves through a fluid environment yielding a situation which is characterized by a small Reynolds number, \(10^{-9} \le {\text {Re}} \le 10^{-4}\) for a 500-nm diameter bead, depending on the velocity. This number describes the relative importance of inertia versus drag forces. A small Reynolds number should indicate that the inertia forces have minimal impact on the bead’s motion. This implies that the inertia forces can be omitted from the model, as done by several simulations [8]. However, recent study has suggested that this approach may violate Newton’s second law, and offers an alternative modeling approach that retains the inertia forces [5, 12]. Herein, it will be shown that this new model closely predicts the bead’s behavior.

In order to investigate this model, it is necessary to examine the relationship between the generalized inertia forces and viscous drag forces and moments:
$$\begin{aligned}&m {\ddot{\mathbf{x}}} = \mathbf{F} \, - \, \beta _{v} \, \dot{\mathbf{x}},\end{aligned}$$
(7)
$$\begin{aligned}&I_{33} \ddot{q}_3 = {M} \, - \, \beta _{\omega } \, \dot{q}_3, \end{aligned}$$
(8)
where F and M contain all the forces and moments other than those related to viscous drag. The inertia properties for the 500 nm bead used herein are \(m = 0.0687\) pg and \(I_{33} = 0.00173 \times 10^{-6} {{\text {pg}}\,{\text {mm}}^2}\). In order to determine the viscous drag properties it is necessary to consider the characteristics of the fluid.

This can be accomplished using the Knudsen number, \(Kn\), which indicates whether the fluid must be considered as a continuum, or as discrete, individual molecules. It is the ratio of the fluid mean free path and the characteristic length of system. For the 500 nm bead in water, the mean free path is, \(\lambda _{\text {mfp}} = 0.3\,\)nm, and therefore, \(Kn = 0.0006\). Since the Knudsen number is less than 0.001, the fluid is considered as a continuum. Thus, it should be reasonable to use Stoke’s Law to calculate drag coefficients. However, because the bead is small, it is unclear how its surface interacts with the surrounding fluid to create drag. This is referred to as the no-slip boundary condition, which indicates whether the fluid sticks to the bead’s surface creating larger drag forces, or slipping occurs between the fluid and the bead, creating less drag. This phenomenon has been verified experimentally and theoretically [19]. This condition can be checked by calculating a slip correction factor [19], here \(C_{c} = 1/(1+2.52Kn) = 0.99995\), which multiplies the drag coefficients; this correction is negligible and thus is not used.

Based on these arguments, the translational drag coefficient in (7) is obtained using Stokes’ Law as \(\beta _{v} = 4.722 \times 10^{3}\) pg/ms. The rotational drag coefficient in (8) is obtained from an analysis of rotational diffusion [20] yielding \(\beta _{\omega } = 8\pi \mu _{\text {m}} r_{S}^{3} = 3.935 \times 10^{-4}\,{{\text {pg}}\,{\text {mm}}^{2}}/{\text {ms}}\) where \(\mu _{\text {m}}\) is the viscosity of the fluid medium, and \(r_S\) is the radius of the bead. Dividing through (7) and (8) by the drag coefficients yields
$$\begin{aligned} (1.46 \times 10^{-5}\,{\text {ms}}) \, {\ddot{\mathbf{x}}}&= \frac{m}{\beta _v} \, {\ddot{\mathbf{x}}} \ = \ \frac{\mathbf{F}}{\beta _v} - {\dot{\mathbf{x}}}, \end{aligned}$$
(9)
$$\begin{aligned} (0.44 \times 10^{-5}\,{\text {ms}}) {\ddot{q}}_3&= \frac{I_{33}}{\beta _{\omega }} \, {\ddot{q}}_3 \ = \ \frac{M}{\beta _{\omega }} - {\dot{q}}_3. \end{aligned}$$
(10)
The disproportionality between the mass and the viscous drag coefficients, \(O(10^{-5})\), creates large accelerations that require a small time step, yielding a long numerical integration time.
In order to reduce the run time, it has been suggested that the small coefficient of the acceleration terms in (9) and (10) implies that these terms can be omitted, yielding a first-order model. A second approach uses the multiscale analysis, as discussed in [12], which begins by determining a characteristically small number from the model in (9) and (10), \(m/\beta _v \approx 1.46 \times 10^{-5}\) ms. Using this in (7) and (8) yields,
$$\begin{aligned} \mathbf{0}&= \varepsilon \ (1\,{\text {ms}}) {\ddot{\mathbf{x}}} - \frac{\mathbf{F}}{\beta _v} + {\dot{\mathbf{x}}} = \varepsilon {\ddot{ \bar{\mathbf{x}}}} - \frac{\mathbf{F}}{\beta _v} + {\dot{\mathbf{x}}} \nonumber \\ \mathbf{0}&= \varepsilon \ (0.3\,{\text {ms}}) \ddot{q}_3 - \frac{M}{\beta _{\omega }} + \dot{q}_3 \ = \ \varepsilon \ddot{\bar{q}}_3 - \frac{M}{\beta _{\omega }} + \dot{q}_3 \end{aligned}$$
(11)
such that \(\varepsilon = 1.46 \times 10^{-5}\) is unitless. The small parameter \(\varepsilon \) is used to decompose time into different scales: \(T_i = \varepsilon ^{i}t\). This yields
$$\begin{aligned} \left\{ \begin{array}{l} \dot{\mathbf{x}} = \frac{d\mathbf{x}}{dt} = \varepsilon ^{0}\frac{\partial \mathbf x}{\partial T_{0}} + \varepsilon ^{1}\frac{\partial \mathbf x}{\partial T_{1}} + \varepsilon ^{2}\frac{\partial \mathbf x}{\partial T_{2}} + \cdots \\ \\ \displaystyle \dot{q}_3 = \frac{dq_3}{dt} = \varepsilon ^{0}\frac{\partial q_3}{\partial T_{0}} + \varepsilon ^{1}\frac{\partial q_3}{\partial T_{1}} + \varepsilon ^{2}\frac{\partial q_3}{\partial T_{2}} + \cdots \\ \end{array} \right. \end{aligned}$$
(12)
$$\begin{aligned} \left\{ \begin{array}{l} {\ddot{\bar{\mathbf{x}}}} = \frac{ d^{2} \bar{\mathbf{x}} }{dt^{2}} = \sum _{i=0}^{\infty } \sum _{j=0}^{\infty } \varepsilon ^i \varepsilon ^j \frac{\partial ^2 \bar{\mathbf{x}}}{\partial T_i \partial T_j} \\ \\ \displaystyle \ddot{\bar{q}}_3 = \frac{d^{2} \bar{q}_3}{dt^{2}} = \sum _{i=0}^{\infty } \sum _{j=0}^{\infty } \varepsilon ^i \varepsilon ^j \frac{\partial ^2 \bar{q_3}}{\partial T_i \partial T_j} \end{array} \right. \end{aligned}$$
(13)
when (12) and (13) are substituted into (11), and the terms are arranged in the order of increasing power of \(\varepsilon \), (11) becomes
$$\begin{aligned} \mathbf{0}&= \varepsilon ^{0}\left( -\frac{\mathbf{F}}{\beta _v}+\frac{\partial \mathbf{x}}{\partial T_0}\right) + \varepsilon ^{1}\left( \frac{\partial ^{2} {\bar{\mathbf{x}}}}{\partial T_{0}^{2}} + \frac{\partial \mathbf{x}}{\partial T_1} \right) + \cdots \nonumber \\ \mathbf{0}&= \varepsilon ^{0}\left( -\frac{M}{\beta _{\omega }}+\frac{\partial q_3}{\partial T_0}\right) + \varepsilon ^{1}\left( \frac{\partial ^{2} \bar{q}_3}{\partial T_{0}^{2}} + \frac{\partial q_3}{\partial T_1}\right) + \cdots \nonumber \\ \end{aligned}$$
(14)
The difference between \(\varepsilon ^{0} = 1\) and \(\varepsilon ^{1} = 1.46 \times 10^{-5}\) is fairly large, and so, it is likely that the generalized active forces in (6) and (11) must cancel to some extent to render the sum in (14) to be equal to zero. From a multibody dynamics standpoint, if a system of forces cancel, then they produce no motion and can be omitted from the equations of motion. The effort here is to remove these canceled forces from the model.
In this article, this is accomplished by decomposing the generalized active forces into large and small parts:
$$\begin{aligned} -\frac{\mathbf{F}}{\beta _v} + \frac{\partial \mathbf{x}}{\partial T_{0}}&= \left( a_1 + a_2\right) \left( -\frac{\mathbf{F}}{\beta _v} + \frac{\partial \mathbf{x}}{\partial T_0}\right) \nonumber \\ -\frac{M}{\beta _{\omega }} + \frac{\partial q_3}{\partial T_{0}}&= \left( a_1 + a_2\right) \left( -\frac{M}{\beta _{\omega }} + \frac{\partial q_3}{\partial T_0}\right) , \end{aligned}$$
(15)
where \(a_1+a_2 = 1\) and \( a_{1} \gg a_2\). Substituting (15) back into (14) yields
$$\begin{aligned} \mathbf{0}&= a_1 \left( -\frac{\mathbf{F}}{\beta _v} + \frac{\partial \mathbf{x}}{\partial T_0}\right) \, + \, a_2 \left( -\frac{\mathbf{F}}{\beta _v}+\frac{\partial \mathbf{x}}{\partial T_0}\right) \nonumber \\&+\varepsilon ^{1}\left( \frac{\partial ^{2} {\bar{\mathbf{x}}}}{\partial T_{0}^{2}} + \frac{\partial \mathbf{x}}{\partial T_1} \right) + \cdots \nonumber \\ \mathbf{0}&= a_1 \left( -\frac{M}{\beta _{\omega }} + \frac{\partial q_3}{\partial T_0}\right) + a_2 \left( -\frac{M}{\beta _{\omega }}+\frac{\partial q_3}{\partial T_0}\right) \nonumber \\&+\varepsilon ^{1}\left( \frac{\partial ^{2} \bar{q}_3}{\partial T_{0}^{2}} + \frac{\partial q_{3}}{\partial T_{1}}\right) + \cdots \end{aligned}$$
(16)
Here, it is assumed that the large forces, defined as,
$$\begin{aligned} \mathbf{\Gamma } \ = \ \left[ \begin{array}{l} \mathbf{\Gamma }_F \\ {\Gamma }_M \end{array} \right] \ = \ \left[ \begin{array}{l} a_{1}\left( -\frac{\mathbf{F}}{\beta _v} + \frac{\partial \mathbf{x}}{\partial T_0}\right) \\ a_{1}\left( -\frac{ {M}}{\beta _{\omega }} + \frac{\partial q_3}{\partial T_0}\right) \end{array} \right] \end{aligned}$$
(17)
cancel to the extent that they can be removed to prevent them from (16), yielding a second-order model of the form:
$$\begin{aligned} \mathbf{0}&= a_2 \ \left( -\frac{\mathbf{F}}{\beta _v}+\frac{\partial \mathbf{x}}{\partial T_0}\right) + \varepsilon ^{1}\left( \frac{\partial ^{2} \bar{\mathbf{x}}}{\partial T_{0}^{2}} + \frac{\partial \mathbf{x}}{\partial T_1} \right) + \cdots \nonumber \\&= m {\ddot{\mathbf{x}}} + a_{2} \ \beta _v \ {\dot{\mathbf{x}}} - a_{2} \ \mathbf{F} \nonumber \\ \mathbf{0}&= a_2 \left( -\frac{M}{\beta _{\omega }}+\frac{\partial q_3}{\partial T_0}\right) + \varepsilon ^{1}\left( \frac{\partial ^{2} \bar{q}_3}{\partial T_{0}^{2}} + \frac{\partial q_3}{\partial T_1}\right) + \cdots \nonumber \\&= I_{33} \ddot{q}_3 + a_{2} \ \beta _{\omega } \ \dot{q}_3 - a_{2} \ {M} \end{aligned}$$
(18)
assuming \(\frac{d \mathbf{x}}{dt} = \frac{\partial \mathbf{x}}{\partial T_0}\) and \(\frac{d {q_3}}{dt} = \frac{\partial {q_3}}{\partial T_0}\).

The scaling factor \(a_2\) is determined by matching the characteristics of the simulation with the experimental observations in Sect. 2.2; typically \(a_2 \ge \epsilon \). Since all of the terms in (18) will be in proportion, they can be numerically integrated in drastically less time. The results are used to check whether the forces in F sufficiently cancel, which provides a measure of the quality of (18) as a model of the system’s dynamics. This check should produce \(O({ \Gamma }_i) \le O(a_2)\), so that one may conclude that the forces in \(\mathbf{\varvec{\Gamma }}\) do not create significant motions beyond those produced by the \(a_2\) scaled forces.

In order to investigate the proposed approach, the different models for the 500 nm bead were coded in the C programming language and numerically integrated using a high-speed computer, a DELL PowerEdge 2900 III Server with two quad-core, 2 GHz processors, running the UNIX operating system, using an adaptive Runge–Kutta 45 algorithm. Numerical integration of the Newton–Euler model in (7) and (8) yields the overdamped motion shown in Fig. 6, which took \(\sim \)73 h, to obtain 20 ms of simulation time; (at 40 ms the laser is turned on.) In order to obtain any results, it was necessary to reduce the tolerance of the simulation, AbsTol = \(10^{-6}\), and RelTol = \(10^{-5}\). The lengthy run time stems from a reduction in the time step by the adaptive integrator to obtain the requested accuracy. Investigation of the step size reductions showed that the numerical integrator was proceeding with a picosecond step size.
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Fig. 6

Simulation data for \(q_1\) coordinate of microparticle using the Newton–Euler model (CPUtime \(\approx 73\,h, \mathtt{AbsTol} = 10^{-6}, \mathtt{RelTol} = 10^{-5}\), \(\Delta t\) = 0.001  ms)

The first-order model can be represented by (17) which yields the results shown in Fig. 7. The simulation run time is reduced to 45 min, with an increased tolerance AbsTol = \(10^{-8}\), and RelTol = \(10^{-7}\). Finally, the multiscale model in (18) is numerically integrated yielding the results in Fig. 8. The simulation run time is reduced to 21 min using the same tolerance. A comparison of the results from the Newton–Euler approach and the multiscale approach is shown in Fig. 9.
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Fig. 7

Simulation data for \(q_1\) coordinate of microparticle using the first-order model (CPUtime \(\approx \) 45 min, \(\mathtt{AbsTol} = 10^{-8}, \mathtt{RelTol} = 10^{-7},\,\Delta t = 0.001\) ms)

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Fig. 8

Simulation data for \(q_1\) using the multiscale model (CPUtime = \(21\,min\), AbsTol = \(10^{-8}\), RelTol = \(10^{-7},\,\Delta t = 0.001\) ms)

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Fig. 9

Overlap of simulation data for \(q_1\) using the Newton–Euler model and the multiscale model

The key thing to notice is the oscillations that appear in Fig. 8, when the bead reaches the focal line, which are absent from the simulation of the Newton–Euler model in Fig. 6 and the first-order model in Fig. 7. The bulk of the simulation is spent resolving the situation when the bead reaches the focal line because of the alternating forces that occur as it overshoots the focal line. The experimental data in Sect.2.2imply that the multiscale model represents the actual physical behavior of the 500 nm bead observed under the microscope.

4 Experimental and simulation results

The results obtained from the experimental and theoretical studies on three different sizes of microbeads are compared in Figs. 10, 11, and 12. The largest bead, 1,950 nm, displays the overdamped behavior predicted by the first-order model. The simulation data show that the multiscale model can predict this overdamped behavior with an appropriate choice of the scaling factor; the data for each of the simulations in Figs. 10, 11, and 12 are given in Table 1.
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-1185-0/MediaObjects/11071_2013_1185_Fig10_HTML.gif
Fig. 10

Comparison of experimental data (open circles) and simulation data (line) for \(q_1\) coordinate of 1,950 nm microparticle using the multiscale model (\({\hbox {CPUtime}} = 13\) min, AbsTol = \(10^{-8}\), RelTol = \(10^{-7}\), \({\Delta }t = 0.001\,\)ms)

https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-1185-0/MediaObjects/11071_2013_1185_Fig11_HTML.gif
Fig. 11

Comparison of experimental data (open circles) and simulation data (line) for \(q_1\) coordinate of 990 nm microparticle using the multiscale model (\({\hbox {CPUtime}} = 2.5 min\), AbsTol = \(10^{-8}\), RelTol = \(10^{-7}\), \({\Delta }t = 0.001\) ms)

https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-1185-0/MediaObjects/11071_2013_1185_Fig12_HTML.gif
Fig. 12

Comparison of experimental data (open circles) and simulation data (line) for \(q_1\) coordinate of 500 nm microparticle using the multiscale model (\({\hbox {CPUtime}} = 21_{min}\), AbsTol = \(10^{-8}\), RelTol = \(10^{-7}\), \(\Delta t = 0.001\,\)ms)

Table 1

Simulation data

 

Definition

Units

Bead diameter (nm)

   

1,950

990

500

\(m\)

Mass

pg

4.08

0.533

0.0687

\(I_{33}\)

Moment of inertia

\({\text {pg}}\,{\text {mm}}^2\)

\(1.5378\times 10^{-6}\)

\(0.052223\times 10^{-6}\)

\(0.00173\times 10^{-6}\)

\(Kn\)

Knudsen number

\(1.54 \times 10^{-4}\)

\(3 \times 10^{-4}\)

\(6 \times 10^{-4}\)

\(C_c\)

Stokes drag slip correction factor

0.99997

0.9992

0.99995

\(\beta _v\)

Translational drag coefficient

pg/ms

\(1.84\times 10^{4} \)

\(9.349\times 10^{3}\)

\(4.722 \times 10^{3}\)

\(\beta _{\omega }\)

Rotational drag coefficient

\({\text {pg}}\,{\text {mm}}^2/{\text {ms}}\)

\(2.33\times 10^{-2} \)

\(3.054\times 10^{-3}\)

\(3.935 \times 10^{-4}\)

\(a_2\)

Scaling factor

\(8 \times 10^{-3}\)

\(2 \times 10^{-4}\)

\(1.5 \times 10^{-5}\)

This overdamped behavior becomes underdamped in the experimental results of the smaller beads. The 990 nm bead overshoots the focal line and oscillates to a small extent before settling. The simulation result, solid line, shows that tuning of the scaling parameter, \(a_2\), allows the multiscale model to predict the small oscillations that cannot be predicted by the first-order model. The smallest bead 500 nm bead, discussed in Sect. 3.2, shows a much larger overshoot of the focal line with larger oscillations; see Fig.12. The successively smaller beads show the increasingly more oscillatory behavior predicted by the proposed multiscale model. Note that recent ray-optics simulations of nonspherical, microsized objects predict underdamped behavior of a cubic object [11].

Thus, there is general agreement between the experimental and computational results, which proves the presence of a transition in the dynamic behavior of objects under a trapping process at different length scales. However, the simulation results, especially for the 990 and 1,950 nm, show a deviation from corresponding experimental results, mostly during the transition from the initial position to the focus line. Possible reasons for these differences are as follows:
  1. 1.

    Uncertainties in the physical parameters used for the computational modeling; for example, the actual diameter of the microbeads may vary.

     
  2. 2.

    Approximations of the initial velocity of the microbead, required for the simulation, from the observed position data, and may contain some errors.

     
  3. 3.

    A three-dimensional bead model may more accurately capture the physical phenomena than the planar model used here.

     
  4. 4.

    Discretization of the laser beam into only 15 laser rays may introduce some error; however, this approximation is more accurate when the bead is near the focal line.

     
  5. 5.

    The force produced by the laser on the 500 nm bead may be better modeled using Mie scattering theories, rather than the ray-optics approach.

     
  6. 6.

    The scaling factor, introduced in the Sect. 3.2 for the simulation, may need finer tuning.

     
  7. 7.

    General numerical integration errors.

     
Still, this simple model does capture the key aspect of oscillatory behavior which was sought. This model will be continually refined and improved in the future.

5 Discussion and future study

The quality of the estimate provided by the multiscale model can be assessed by checking the assumption of small first terms in the relations of (17). This is done by calculating (17) during simulation. The simulation results in Figs. 13a, b for the 1,950 nm bead show that the orders of the first terms, both translational and rotational terms, are small, as expected to be on the order of the scaling value of \(a_2 = 8 \times 10^{-3}\).
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-1185-0/MediaObjects/11071_2013_1185_Fig13_HTML.gif
Fig. 13

Checking the assumption of force and moment cancelation for 1,950 nm. (a) The first term of forces (corresponding to the translational coordinates) in the asymptotic expansion, and (b) the first term of moment in the asymptotic expansion (14)

A similar conclusion can be drawn for the simulation of the translational coordinates of 990 nm bead in Fig. 14a; however, the rotational term is not small, as illustrated in Fig. 14b. The scaling factor for this case is \(a_2 = 2 \times 10^{-4}\). Some further investigation is necessary to explain this discrepancy. Similar results are obtained for the 500 nm bead, Fig. 15, with a scaling factor of \(a_2 = 1.5 \times 10^{-5}\).
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-1185-0/MediaObjects/11071_2013_1185_Fig14_HTML.gif
Fig. 14

Checking the assumption of force and moment cancelation for 990 nm. (a) The first term of forces (corresponding to the translational coordinates) in the asymptotic expansion, and (b) the first term of moment in the asymptotic expansion (14)

https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-1185-0/MediaObjects/11071_2013_1185_Fig15_HTML.gif
Fig. 15

Checking the assumption of force and moment cancelation for 500 nm. (a) The first term of forces (corresponding to the translational coordinates) in the asymptotic expansion, and (b) the first term of moment in the asymptotic expansion (14)

These checks indicate that a certain amount of refinement of the method for choosing the scaling factor is necessary. Still, it is interesting to note that the multiscale model can come quite close to predicting the behavior of these beads in the horizontal, \(q_1\), direction, and it is also still more interesting to note that the behavior predicted by the multiscale model can be observed experimentally.

Experimental results in the vertical direction are not examined in detail here because of time-scale disagreements between simulation and experiments, though spatial agreement is acceptable. One possible reason for this is that with the decreasing particle size, the mathematical treatment of light-particle interactions using ray optics becomes less accurate. Interactions on the scale of the 500 nm particles may be better treated with Mie scattering theories, but are significantly more complex. Second, the method used to track these fluorescent particles in the horizontal direction cannot be used for the vertical measurements. Further research is underway on new methods which will more accurately track the particle’s vertical position. Simulation data for the vertical and rotational directions are discussed in Appendix 6.

This article focuses on the behavior of the beads as they approach the trap. It is equally interesting to study the dynamics of the particle after it is trapped as done by most of the prior study discussed in Sect. 1. The retention of mass properties in the model used herein suggests a power spectrum of the trapped beads that will deviate from the Lorentzian profile predicted by the overdamped Langevin equations [7]. In future studies, using the proposed multiscale analysis and experimental position measurements with higher temporal resolution, this important phenomenon will be examined in the optical trap. Although, it is beyond the scope of this, this study will further enhance understanding of the behavior of nanoparticles and molecules in viscous environments such as biological cells.

6 Conclusions

This article presents experimental observation of overdamped to underdamped transition of microscopic particles in OTs, which is supported by numerical simulations using multiscale analysis. This study shows that the underdamped behavior observed experimentally and predicted by the new multiscale model actually does represent the physics of micro and nanosized particles in a fluid environment characterized by a low Reynolds number. In addition, the proposed model can be numerically integrated in a reasonable amount of time, which is drastically less than the equivalent Newton–Euler model. Here, data in the horizontal direction are examined because it is still difficult to analyze the bead’s motion in the vertical direction. Future study will involve improving the multiscale modeling techniques in terms of selecting the scaling parameter and modeling of forces. Improvements will also be made to particle tracking in the vertical direction to allow for further model validation.

Acknowledgments

This study is supported by the National Science Foundation under Grant No. MCB-1148541.

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$$\begin{aligned} \phi _k = \arctan \frac{\rho _k}{f}. \end{aligned}$$$$\begin{aligned} _\mathbf{N}^{\mathbf{F}_k}\mathbf{R} = \left[ \begin{array}{lll} \cos \phi _k &{} - \sin \phi _k &{} 0 \\ \sin \phi _k &{} \cos \phi _k &{} 0 \\ 0 &{} 0 &{} 1 \end{array} \right] . \end{aligned}$$$$\begin{aligned} F_{{\text {tot}}k} = \frac{n_{m} P_k}{c} \left[ 1 + R_{k} e^{2i \alpha _k} - T_{k}^{2} d \ \right] , \end{aligned}$$$$\begin{aligned} \displaystyle d \ = \ \frac{e^{2i(\alpha _k - \beta _k)}}{1 - R_{k} e^{i(\pi - 2\beta _k)}}. \end{aligned}$$$$\begin{aligned} d&= \frac{e^{i(2\alpha _k - 2\beta _k)}}{1 - R_{k} e^{i(\pi - 2\beta _k)}} \cdot \frac{1 - R_{k} e^{-i(\pi - 2\beta _k)}}{1 - R_{k} e^{-i(\pi - 2\beta _k)}} \end{aligned}$$$$\begin{aligned}&= \frac{e^{i(2\alpha _k - 2\beta _k)} \left[ 1 - R_{k} e^{-i(\pi -2\beta )}\right] }{\left[ 1 - R_{k} e^{i(2\alpha _k -2\beta _k)} \right] \left[ 1 - R_{k} e^{-i(2\alpha _k -2\beta _k)} \right] } \nonumber \\&= \frac{e^{i(2\alpha _k -2\beta _k)} - R_{k} e^{\left[ i(2\alpha _k -2\beta _k) -i(\pi -2\beta _k)\right] }}{1 + R_{k}^{2} -R_{k} \left[ e^{i(\pi - 2\beta _k)} + e^{-i(\pi -2\beta _k)} \right] } \nonumber \\&= \frac{e^{i{2\alpha _k - 2\beta _k}} - R_{k} e^{i(2\alpha _k - \pi )}}{1 + R_{k}^{2} - 2R_{k} \cos \left( \pi - 2\beta _k \right) } \end{aligned}$$$$\begin{aligned} \mathbf{F}_{tot,k} = F_{{\text {grad}},k} \widehat{\mathbf{N}}_1 + F_{{\text {scat}},k} \widehat{\mathbf{N}}_2. \end{aligned}$$

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