In incompressible fluid dynamics, the behavior and velocity of the flow can be described by Navier- Stokes Equation:
--- (1)
where is the density of medium
, the velocity of flow( ), the
pressure , the viscosity and a body force term The Navire-Stokes equation is derivative from Newton’s Second Law, . On a hand, the left side of equation due to the inertia of whole fluid describes acceleration of transient and convection. On the other hand, the right side is the sum of pressure, viscous forces and the body forces. For laminar flow of microfluidic system the Reynold number expressing the ratio of inertial forces(convection acceleration) to viscous forces, is low, so the inertia terms are neglected. Considering the steady flow in addition, the Navier-Stokes equation is written:[27]
--- (2)
The equation presents the motion of fluid and gives the velocity for a given body force in the microfluidic channel.
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2-2 Electrical double layer
In an electrochemical system, an ideal polarized electrode involves no charge can be transferred across the electrode-solution interface. As a potential is applied to the electrode, charges (counter ions) will aggregate in the electrode surface until the potential drop across the capacitor satisfies the applied potential. The region of liquid near to the interface has a higher density of counter ions. The result in the change of distribution of ions near the surface is influenced by the distribution of electrode signal. The charges exist on the metal, the excess or deficiency of electrons and resides accumulating in a thin layer (< 0.01 nm) on the metal surface. The charges in the solution are made up of an excess of cations or anions in the vicinity of the electrode surface. The whole array of charged species existing at the metal-solution interface is called the electrical double layer (EDL).[28, 29]
Electrical double layer consist in the presence of several layers. In the further consideration of EDL model in Figure 2-1, the inner layer closest to the electrode contains specifically adsorbed solvent molecules and other species is called stern layer.
The stern layer can be subdivided into two regions, which is the inner and outer layer.
The inner surface of which is named inner Helmholtz plane, composed of solvent molecules and other specifically absorbed species. The bound solvent ions as outer layer, outer Helmholtz plane, of stern layer samples which are not absorbed specifically but attracted in electrostatic forces from the charged metal. Due to the thermal agitation of solution, excess charges distributed in the extend region from outer Helmholtz plane to solution bulk form a diffuse layer. This model is presented in Figure 2-1.
The volume of double layer is dependent on the pH and ionic strength of the solution. To calculate the potential distribution of the electrical double layer,
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Poission-Boltzman equation is given to describe the varieties of electrostatics potential of charged atoms in the space,
---- (3)
where is the electric potential due to surface charge, is electric charge, is the ion concentration, is the ion valence. The Poission-Boltzman equation can be solved by assuming generally the relation with respect to smaller , which leads to the Debye-Huckel approximation
--- (4)
and the potential distribution is
--- (5)
where
--- (6)
κ is called Debye-Huckel parameter. The potential decays exponentially in the diffuse
layer with a characteristic distance by Debye length . This value correlates the thickness of electrical double layer.[29]17
Figure 2-1 Model of the solid-electrolyte interface with corresponding potential v.s. the distance from electrode to wall. The electrode is illustrated with a negative surface potential . The inner Helmholtz plane layer
consists of nonhydrated coions and counterions, whereas the outer Helmholtz plane layer is built up of only hydrated ounterions.[29]
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2-3 AC electrokinetics:
2-3-1 AC Electroosmosis
When the potential is applied on a electrode in the solution, explained in Figure 2-2, the electrical double layer (EDL) is established and accumulated uniformly and the microfluidic cell is also supplied by external electrical power as a electrostatic body force . There is a tangential component of electric field, . Here considering the steady isobaric flow in x-direction along the surfaces (such as electrodes), with velocity and potential gradient remaining in y-direction, the equation (2) is simplified: [27]
--- (7)
Figure 2-2 Velocity, Coulomb force, and electrical potential in an electroosmotic flow.
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The term, , is induced Coulomb force by electric field applied on the electrical double layer. In further detail of Coulomb force, the charge density around the EDL can be gotten by Poisson equation, is the potential of bulk solution, the charge density of medium, and electrical permittivity of solution:
--- (8) defined as potential that represents the potential drop cross the electrical double layer.
is a potential at the surface and is another potential as a function of distance in y-direction. Equation (12) is the solution of the electroosmotic flow, that involving fluidic flow in the solution after the external work of electric applied. The free
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charges within the diffuse layer of electrical double layer move due to the electric field direction, tangential component, on the surface turning out electroosmosis. The mechanism was generally observed and accounted for in DC field. However, AC field providing a non-uniform field can also generate the fluidic flow locally relating to the charges of the double layer. As voltage is applied, it causes an electrical double layer outside the electrode surface. If ignoring the formation of stern layer in electrical double layer and assuming a linear approximation of diffuse double layer, like the chart in Figure 2-4, we consider a specific capacitance dropped across entire double layer, which is
--- (13)
here is surface capacitance(F/m2), the Debye length(m). Owing to the surface charge density, the capacitance can be rewritten as a ratio of charge to potential:
--- (14)
Where is the electrode surface charge density(C/m2) and is the zeta potential(V) which describe equivalent potential of diffuse layer. Here the zeta potential drop from the electrode surface to potential defined point, . The interval is same as Debye lenght(Figure 2-4). In the linear approximation the zeta potential is proportional to surface charge density in the diffuse double layer:
--- (15)
and
= --- (16)
where is the reciprocal Debye length. In term of velocity resulted from interaction
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of ions in double layer--- (17)
also
--- (18)
In microfluidic system, generating a non-uniform electric field between electrodes can model as a circuit, including a series of solution resistance and surface capacitance at either double layer on the electrode surface. Figure 2-3 show a two dimensional system with a semircular circuits.[31]
Figure 2-3 Approximate circuit diagram for two long coplanar plateeletrodes specrated by narrow gap.
The solution resistance and double layer capacitance is with a value
and
Here,
is
cross-sectional position and start from the center of the gap between electrodes andexpresses the length of electrodes into the page.
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Since eletroosmosis is affected exactly by the behavior of ions motion in the double layer. The potential with a time-averaged value, i.e. root mean square value cross the double layer can be driven via voltage dividing:
--- (19)
Resistance and capacitance parameters substitute the function
--- (20)
The total surface charge distribution and tangential electric field in x-direction can be evaluated from this model as and
Moreover, diagram outlining of AC electroosmosis is patterned at Figure 2-5. Its mechanism report the charges above electrode surface shift from the edge of electrode to the center. The ion movement induces and exerts net drag force due to medium viscoity. The forces acted on the double layer generate bulk fluid motion, rotational motion, on the electrode surface toward the center of electrode. The direction of the flow would not be alternative with electrical potential change since co-ions and counter-ions switch transiently as well due to applied electric field and the tangential component.
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Figure 2-4 Schematic diagram for concentration of ions close to a charged surface in solution. Also shown is the distribution of potential with distance from electrode surface.
Figure 2-5 Behaviors of ACEO on two electrodes applied different signal.
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2-3-2 Dielectrophoresis
Dielectrophoresis(DEP) describes the motion of polarizable particles under a non-uniform electric field. It always employs in an alternative current (AC) to produce the non-uniform electric field. Depending on electrical and polarized properties of the medium around particles and particles inside it can be attractive or repulsive, which termed negative and positive dielectrophoresis. Figure 2-6 shows that particles move toward the field in Positive DEP. The negative DEP causes dielectric constant as model and accounted for the solution permittivity , the force acting on a spherical particles is given by[27, 31]
--- (23)
where the Re is real part of Clausius-Mossotti (CM)-factor, which is frequency dependant. The CM-factor reflecting the effective polarizability of the particles and medium depends on the complex permittivity:
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strength in the direction, this is so-called the positive DEP, while negative DEP occur when the direction toward lower field strength and the CM-factor has negative value.
For ellipsoidal particles, such as E. coli and vibrio parahaemolyticus, it has an extend value range of CM-factor. The dipole of ions gets along with axes of ellipsoid.
An ellipsoid with three orthoghonal axes a, b, and c, the volume( ) is
= --- (25)
and the time average dielectorphoresis on an ellipsoid particle[20, 32, 33]
Re --- (26)
For the solid prolate ellipsoid, the CM-factor along axis is given by
--- (27)
α = a,b,c
is a component of the depolarization factor along any one of the three axis of ellipsoid(a,b,and c). The for ellipsoid can take account of ellipsoidal shape of cell. is for a prolate spheroid with semi axes a b = c has been described as
--- (28)
where
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Figure 2-6 Behaviors of particle movement under force of DEP.
2-4 Electrochemical methods
The electrochemical techniques here perform using a 3-electrode cell. The cell is consist of a working electrode, usually, a counter electrode, usually in Pt material, a reference electrode, usually in Ag/AgCl in saturated KCL, an electrolyte solution; and redox reactive species (Figure 2-7). Working electrode is usually Au surface in commercial assay, counter electrode in Pt material and a reference electrode Ag/AgCl in saturated KCL. Potential is measured between working electrode and reference electrode. However, the current is measured between working electrode and counter electrode.
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Figure 2-7 Electrochemical cell system. Three electrodes put in the electrolyte is made up by working electrode, Au patterned on SiO2 chip; counter electrode, Pt wire; and Ag/AgCl reference electrode in 3M KCl
solution.
The Cyclic Voltammetry can measure the varieties of current of distinguish analytes during pre-determined range of potential is applied. The voltage signal is triangular wave. In forward scan of input wave, cathodic potential increases and current increases as concentration of active species in solution become lower. After reaching the highest potential, the reverse scan provides the potential with anodic direction and the oxidation occurred in working electrode (Figure 2-8).
Figure 2-8 Cyclic voltammetry curves for (a) applied potential sweep and (b) measured current response.
(a) (b)
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2-5 Impedance measurement
Impedance measurement are made using AC signal, where the current oscillates in a sinusoidal function from as low as a less hundred to nearly 100,000 Hz in different measurements. As applied AC current to the electrode system and measure the AC voltage across the electrodes, the impedance is simply given by AC equivalent Ohm’s Law:
--- (29)
The instrument is capable of monitoring both magnitude of voltage and the phase change of voltage relative to the phase of current. Combining these parameters, the impedance can be broken down into two parts: one due to pure resistance and the other to the reactance of the system. The reactive part (Xc) in this case is due to the capacitance (C) associated with the medium around metal surfaces. The signal of impedance received from the electrode as a simple circuit both in series and parallel of resistances and reactive part.
For the circuit, the impedance of each element are given by:
--- (30)
--- (31)
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depends on the AC frequency(f) is given by--- (32)
and total impedance (Z) is given by
--- (33)