LITERATURES SURVEY - 開放式介電泳裝置應用於三維觀測技術

CHAPTER 1 INTRODUCTION

1.2 LITERATURES SURVEY

For measurement of small particles, there are few methods to control and image the micro or nano-particles. The dielectrophoresis was chosen to be our control method, and the AFM which can scan the contour of a particle in the liquid is chosen as the image method in this thesis. A closed chamber is needed when the dielectrophoresis is applied to perform a three-dimensional manipulation. But this closed chamber makes observation by using the AFM

become a difficult problem. This paper will present a viable open-chip device for the three-dimensional motion which is controlled by the dielectrophoretic force. A successful dielectrophoretic device had to complete three dimensional manipulations on the particle, and it should be trapped the particle at the center area.

The dielectrophoresis was first introduced by Phol (Phol, 1978). The further calculation was completed by Wang (Wang et al., 1994). He proposed a time averaged force calculation to unify the dielectrophoretic forces. These results made the calculation of the dielectrophoretic forces become easier.

Jones and Washizu followed Wang’s concept to complete the multipolar questions about the dielectrophoretic forces. The electrorotation was also mentioned in Jones and Washizu’s studies (Jones and Washizu, 1996;

Washizu and Jones, 1996). In 2003, Jones proposed a method to calculate the dieletrophoretic force on a biological specimen which was in the liquid environmental and experienced a non-uniform electric field (Jones, 2003).

Accordingly, there are two kinds of the particle rotation: the Quincke rotation and the Born-Lertes rotation which is also called electrorotation (see Figure 1.3). In Figure 1.3, (a) is a sketch of Quinke rotation, where ₁ and ₂ are the conductivity of the medium and particle; ₁　 and ₂ are the permittivity of the medium, and the particle,   ₁  ₁ ₁ and ₂  ₂ ₂. The quince rotation happens when ₂ ₁. (b) is a sketch of Born-Lertes rotation. It

happens when the electrodes are applied a current with the phase difference 90⁰.

The method in this thesis is mainly used by the electrorotation for the rotation and translation of a particle. Hölzel introduced a method for the calculation of the two- and three-dimensional distribution of the electric field in the arbitrary configuration of the electrodes. For any voltages, phase, and frequency being applied in the electrodes, this method can determine the distribution of the electric field (Hölzel, 1993). Hughes et al. also calculated the amplitude and the phase of an AC electric field, and the concept of the different position within this chamber (Hughes et al., 1999).

A widely adopted method for investigating the hydrodynamics caused by a vibrating probe in liquid is the boundary integral formulation, which is a semi-analytical method with a strongly theoretical background. Tuck (1969) proposed an integral method which has subsequently been applied to practical applications such as AFM and microelectromechanical systems. The tapping

mode has become a widely used technique for scanning bio-specimens (Tuck, 1969). In tapping mode of the AFM, when a tip taps the specimen, the system suffers interference from liquid in the small gap between the beam and the specimen. Meanwhile, both the beam and the tip apply pressure to the liquid, and this pressure works on the specimen. Möller et al. (1999) found that averaged heights of a topography of outer hexagonally packed intermediate (HPI) layer surfaces and extracellular cytoplasmic purple membrane surfaces measured by the tapping mode were up to 25% smaller than those measured by the contact mode (Möller et al., 1999).

1.3 Objective of this Thesis

An AFM can execute a scanning in normal temperature and pressure (NTP) condition or in a liquid environment. In the biological techniques, building a device that can use single process to complete a 3D imaging is not an easy task. We will develop an open-chip dielectrophoretic device and the procedures of manufacturing in this thesis. The open-chip dielectrophoretic device will be introduced in following chapters. The procedure mainly contained three parts. First, the theoretical analysis was introduced and built a two dimensional model for dielectrophoresis. Second, the simulation of the particle’s motion was done by finite element method. Third, the experiments

were done by applying devices which were manufactured by different methods.

Figure 1.1 Ten kinds of electrodes suggested by Hughes. (Hughes et al., 1998).

Figure 1.2 The basic combination of the AFM contains an AFM base, microcantilever with a tip, a laser, and a detector.

Figure 1.3 (a) A sketch of Quinke rotation. (b) A sketch of Born-Lertes rotation.

Chapter 2 Theory of Dielectrophoresis

The theory of dielectrophoresis introduces the dielectrophoresis from the electrostatics, through the conventional dielectrophoresis and the traveling-waved dielectrophoresis, to the fluidic force. The dielectrophoresis is a force induced by the effective dipole of the particle, and the electric dipole can be expressed by the electric potential. Those mean the dielectrophoretic force can be written as function of the electric potentials.

The particle needs to suspend in the liquid to achieve the manipulation, and so the fluidic force is also an important issue to suspend the particle.

Through solving the theoretical solution, we can set up a numerical model.

Then we could set up a correct FEM model for analyzing the dielectrophoretic problem by comparing the numerical model and FEM model. By building a FEM model, the dieletrophoretic problem could be analyzed with much less time.

2.1 The dielectrophoresis theory and the general solutions 2.1.1 Electrostatics

Being based on the electrostatic theory, the electric field, ˆE, can be

presented by the gradient of the electric potential, V^ˆ, i.e.

ˆ ˆ

E V (2.1)

By substituting the governing equation of the electrostatics into Eq. (2.1), we can obtain the formula,

2Vˆ 



  (2.2)

where  denotes the charge density, and  denotes the permittivity of the medium. Eq. (2.2) is known as the Poisson’s equation for the electric potential.

When it is assumed the charge density is zero, this equation can be reduced to the Laplace’s equation as

2.1.2 General solutions to the dielectrophoresis

Generally speaking, the general theory of the dielectrophoresis consists the traditional dielectrophoresis (the conventional dielectrophoresis) and the traveling-waved dielectrophoresis. The general dielectrophoresis is the convenient and the noncontact force to manipulate particles. It may be a novel technique for manipulating the biological cells or bio-particles. Here the traditional dielectrophoresis is one of the general dielectrophoresis and is a phenomenon of the particle experiencing a non-uniform electric field.

Whether the particle is charged or not, the particle will be effected by the dielectrophoresis. Once the particle is effected, the particle will be polarized.

After the polarization, the particle will start to move according to the shape of the electric field, as shown in Figure 2.1. Here the force of the dielectrophoresis is strongly affected by the arrangement of the electrodes, and different configuration of the electrodes may induce a maximum 50%

difference of the dielectrophoresis. For a quasi-static condition, the electric

19 field can be expressed like,

3 had pointed out the time-averaged general solution of the dielectrophoresis by effective moment method,

1 2 3

ˆ( , , , ) ( ˆ )ˆ

F x x x t  M E (2.9) where the effective dipole moment, ˆm, of a spherical particle with radius, r, and the center of the particle locate at ( , , )x x x can be presented by ₁ ₂ ₃

where ^{Re K}^_ ^*

 

^ ^_ ^and ^{Im K}^_ ^*

 

^ ^_ are the real and imaginary part of complex Clausius-Mossotti factor. It is defined by

  ^* ^* sphere and oblate spheroid as two models in normal saline, and it obtains _p and _p by the model proposed by Gimsa (Gimsa et al., 1996): electrorotation is also initiated by one of the dielectrophoretic forces, torque.

Torque T^ˆ can be calculated by using the effective moment approach (Jones, 1995),

1 2 3

ˆ( , , , ) ˆ ˆ

T x x x t ME (2.15) where ( , , )x x x is the axis which the torque act on the center axis of the ₁ ₂ ₃ particle. By applying Eqs. (2.2), (2.10), and (2.15), we determine the torque Tˆ

Equations (2.17) and (2.18) are more suitable for numerical method, because the sine function is a difficult part for the numerical calculation. An expression of the electric potential like Eq. (2.17) and (2.18) can avoid the calculation of the sine function term in Eq. (2.16). This characteristic makes the electric potential expression a better function for the numerical calculations.

The above equations are based on the spherical particle, but the actual shape of the erythrocyte is much like an ellipsoid. This kind of shape can calculated by the Eqs. (2.17) and (2.18). The ellipsoid has three semiaxes, a, b and c, and so the equations should be written as (Yang and Lei, 2007),

 

2.2 Viscous force and torque applying on the particle

Following these assumptions: (1) low Reynolds number flow, (2) rigid spherical particle, (3) dilute volume concentration of particles in suspension, and (4) laminar flow, a spherical particle with radius r is modified to moves in an infinite fluid with viscosity  where derived by Stokes (1851) and Kirchhoff (Lamb, 1932), respectively, as,

F = 6_  rU (2.24) and

8 3

T_   r  (2.25)

where U and  are the linear and angular velocity of the particle relative to fluid at the center of the sphere. Ganatos et al. (1980) studied the problem between two parallel walls by applying collocation theory, and they found the forces

2.3 Vibration theory of an atomic force microscope

Dynamic AFM has been widely used in high resolution images on a nanometer scale for decades. The most commonly used operating mode of dynamic AFM involves a feedback system of amplitude modulation and exploits the fact that the tip of the microcantilever oscillates with amplitudes of a few tens of nanometers. An interaction between tip and sample induces a strong nonlinearity in the motion of the tip; such nonlinearity includes tip-jump, bistability (Lee et al., 2002; Samtos et al., 2010), snapping, hysteresis, intermittency (Jamitzky and Stark, 2010), period doubling, and

bifurcation from periodic to chaotic oscillations (Hu and Raman, 2006).

These nonlinearities reduce the accuracy of the AFM. Shih (Shih, 2012) provided an easy method to understand the mechanism of the tip-jump. He also proposed that the tip-jump is a predictable motion, and the tip-jump always happens before the tip touched the sample.

2.3.1. Equations of the AFM tip motion

The elastic Bernoulli-Euler equation of the microcantilever motion is that

 

   

       

Where w(x,t) is the deflection; E_m is Young’s modulus of the cantilever; I(x) is the moment of inertia and is assumed to be constant; m(x) is the mass per unit length with the microcantilever assumed to be homogeneous; c is the damping coefficient, and q(x)p(t) represents the applied force per unit length on the microcantilever. Figure 2.4 is schematics of the deflection of AFM microcantilever. Δ1(t) is the elevated height of AFM base, Δ2(t) is static deflection caused by the force at the tip end, and ξ(l, t) is the dynamic deflection caused by the surface homogeneous wave. The deflection of the cantilever can be written as,

   

By substituting equation (2.29) into (2.28), the equation yields,

The deflection ξ(x,t) can be obtained under a constant boundary condition,

     

27 boundary condition of the tip-end and base-end. The coefficients, A tn

 

0 , and

 

B tn , can be obtained by applying the initial conditions of every linear segment. The numerical accuracy is affected by the integration interval in equation (2.38). A better setting of the integration interval should be relatively smaller than the jumping interval.

The boundary condition of a microcantilever at the end which is connected with the AFM base can be described as,

 

^0,

 

^, ⁰

w l t  (2.41)

in which S

 

ft is the excitation amplitude at frequency _f . The boundary condition at the tip-end is,

 

⁰

where Z is the distance between the AFM base and the sample, R is the radius of the tip, and A₁ and A₂ are the Hamaker constants. Here ^{F w}

 

^is So the boundary condition of the microcantilever can be described as,

 

   

Furthermore the eigenvalues _n can be calculated by,

    condition of the dynamic motion of the microcantilever in an AFM. Assumed the piezoelectric oscillator in an AFM applied a sinusoidal amplitude to the

microcantolever. The initial condition can be subscribed. The sinusoidal amplitude is subscribed as,

32 (2.31), and (2.32), the exact solution can be got for every segment.

Figure 2.1 The particle is polarized by the electric field and starts to move to the place where the force is balanced.

Figure 2.2 An illustrated image of the electrorotation. (Hughes, 2003)

Figure 2.3 Drag and torque coefficient for a sphere translating between two parallel walls in a fluid at rest (Ganatos et al., 1980)

Figure 2.4 Schematics of the deflection of AFM microcantilever.

Chapter 3 Numerical Simulation and Device Design

The FEM can simulate a complex problem including, electromagnetic problem, fluid-structure coupling problem, and other complex problems; and it can carry out a successfully functional dielectrophoretic device. Here the finite element method was applied to analyze the distribution of the electric field, the path of the particle, and other properties. These simulations provided us a preliminary assess for the device.

In this chapter, we lunched a three-step of studies in numerical calculation and in model simulations to test the feasibility of our FEM model to the experiments. Firstly, we solved the distribution of the dielectrophoretic force by numerical method. Second the commercial software of the FEM, COMSOL (https://www.comsol.com/), was applied to analyze the complex problem. Third, by comparing these two results calculated by the numerical method and FEM, the feasibility of the FEM could be proved.

3.1 A two-dimensional model - numerical method

Considered the particle is a circular specimen, Eq. (2.17) and Eq. (2.18) will be applied in the following calculations. For numerical analysis, we have the formula of the electric potential,

2 _r 0

   (3.1)

and

2 _i 0

   (3.2)

The boundary conditions of the electric potential are (0, ) 0

The electric potential can be determined as

4 ( ) ( )

We setup a two-dimensional model for the numerical analysis. In the model, there are a square observation area and four rectangular electrodes. Firstly, we set the square with the length equals a unit length as the observation area, and

the width of the electrodes is 0.35 in length. Then an electric potential 1 Volt is applied to the electrode at the left hand side, and -1 Volt is applied on the electrode at the right hand side. The electrodes on the top and the bottom are applied electric potentials, ^j and ^^j Volt. So the boundary conditions are shown as the Figure 3.1. Second, we calculated the distribution of the electric field with the iteration. Here, we meshed the whole observation area into 1000 1000 small elements. In order to calculate the first and the last nodes by the interpolation, the extra elements were needed. So there were total 1002

 1002 elements for the calculations. At every node the differential value was calculated. Third, we calculated the dielectrophoretic force by substituting the iteration into the previous result. And following the same procedure, the distribution of the dielectrophoretic force is shown as the Figure 3.2.

3.2 The Finite element method

3.2.1 A two-dimensional model – finite element method

In the commercial software, we set the electrostatics as the calculating physics in the software. The boundary conditions are the same as the numerical calculations (see Figure 3.1). After the calculation, the incorrect results was shown and it might come from the impropriated setting of the boundary (Figure 3.3), in figure which the dielectrophoretic force at the diagonal direction is not symmetric in this model. This error was induced by

the definition setting-up. In the software, the calculation of the electrostatics follows the point discharge theory. Under this setup, the discharged corner was not completely discharged, because the geometry of the electrodes were not a complete surface. The electric field happened an error, and this error made the simulation fail by applying this model. So that the model was modified to extend its area into a bigger square which was called chamber with 2 unit length. Still there was an observation area which is a 1 unit length square. And the electrodes were a rectangular with the length is 0.5 unit length, and the width was 0.35 unit length. The boundary conditions are shown as in Figure 3.4. The model has an outer square area called chamber, inner square area which is circled by four electrodes and called observation area, and four rectangular electrodes. The outer area is a square with length 2, and the inner area is a square with length 1. Blue rectangles are electrodes.

The length of the electrodes is 0.5, and the width is 0.35.  is the electric potential. The electrostatics was still the physics for the calculations.

In order to setup a viable model in COMSOL, we compared the results calculated by the numerical method and FEM. The results calculated by software are shown in the Figure 3.5. The comparison of these two results showed this configuration in the software is viable to analyze the following problems. The differences between these two results calculated by different methods are mainly affected by different assumption of the discharge. In the

numerical method, the whole edges of the electrodes can be discharged, but the corona discharge is only applied in the FEM analysis. After we modified these electrodes, the other two physics, the hydromechanics and the particle tracing, were also considered and added to the analysis. Here the medium was set to be static in the simulation, and the creeping flow and the zero velocity were set in the hydromechanics system. In particle tracing, accordingly to the software, we considered the red blood cell is a particle with single shell and circular specimen. The parameters of the particle and the medium are shown on Table 3.1. Here the time interval was set to be from 0 to 180 seconds, the electrodes were exerted an electric potential with 5 Volt, and each nearby electrode has a 90-degrees different phase. The solution is shown as Figure 3.6, and it shows the resultant force at the diagonal direction pushes the particle out of the observation area. And the particles which were very close the electrodes would stich onto the electrodes. At beginning, the particles except the particles, which were at the diagonal direction and very close to the electrodes, gathered fast to the center of the observation area. After 180 seconds, the particles became slow and still moved to the center.

3.2.2 A three-dimensional model

According to the previous study, a three-dimensional model was built for the analysis of three-dimensional movement. In three-dimensional analysis,

the effect of the AFM tip and the arrangement of the electrodes are the topic we concerned. The three-dimensional rotation and translation behaviors were also studied. A successful three-dimensional device should exert enough force on the particle, and also the forces should be symmetric. By applying such device, the particle would be trapped at the center of the device. From the improvement of the two-dimensional model, the three-dimensional model was set as shown in Figure 3.7. The outer cube is the chamber, in which the length of this cube is 2, the inner cube is the observation area with a length 1, the cuboid electrodes with 0.5 in long, 0.35 in wide, and 0.05 in high. The bottom surface. A device with four electrodes at the four sides of the device could fulfill the condition and the dielectrophoretic force was strong enough shown on the x-y plane. The electrodes at the top and the bottom had to be the same geometry, because the distribution of the electric field would be symmetry under this configuration. As the results shown in Figure 3.10, both of the force direction and the magnitude of the force show that the device is

able to manipulate the particle in the observation area. To avoid the particle escaping along the diagonal direction in any cut plane, we extend the electrodes at the four sidea one unit length high. Moreover, we added a tip into the three-dimensional simulation of the particle tracing, and we extended the whole model into a real size. So that the length is 100 times longer than the previous simulation. The tracking results of longer electrodes model are shown in Figure 3.11 and in Figure 3.12. In the figures, the particles were

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