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Evaluation of the electric force in electrophoresis
Jyh-Ping Hsu
∗, Li-Hsien Yeh, Ming-Hong Ku
Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617 Received 24 August 2006; accepted 29 September 2006
Available online 30 October 2006
Abstract
A new expression for the evaluation of the electric force acting on a colloidal particle in an applied electric field is derived under the condition of weak applied electric field. The expression derived, which is based on the Maxwell stress tensor, is applicable to both rigid and soft particles for various types of surface conditions and to both symmetric and asymmetric geometries. We show that, depending upon the electrophoresis condi-tions, the electric force evaluated by the methods commonly used in the literature can be overestimated, thereby leading to incorrect electrophoretic mobility.
©2006 Elsevier Inc. All rights reserved.
Keywords: Electrophoretic mobility; Maxwell stress tensor; Various types of charged conditions
1. Introduction
Electrophoresis is used widely by experimentalists in vari-ous fields to characterize the charged conditions on the surface of an entity of colloidal size[1,2]. It is also frequently used as a tool to separate entities of similar physical properties that can-not be differentiated by other typical analytical methods[3,4]. The most important output of electrophoresis measurement is the electrophoretic mobility, defined as the electrophoretic ve-locity per unit strength of an applied electric field. Theoreti-cally, this quantity is determined from the fact that the total force, which includes the electrical force and the hydrodynamic force, acting on an entity vanishes in the steady state. Several different expressions have been used to evaluate the latter in various types of problems. These include, for example, prob-lems of total symmetric nature such as an isolated sphere in an infinite medium[5]or a dispersion of spherical particles[6,7], and those not of total symmetric nature such as two spheres in an infinite medium[8]. Often, because detailed derivations of these expressions are not presented, their applicability needs justification.
The purpose of this work is to derive, in detail, the appro-priate expressions to evaluate the electrical force acting on an
* Corresponding author. Fax: +886 2 23623040.
E-mail address:jphsu@ntu.edu.tw(J.-P. Hsu).
entity under conditions of practical significance. The applicabil-ity of the relevant expressions in the literature is also discussed.
2. Theory
Let us consider the general case illustrated inFig. 1, where a colloidal particle of surface Ωpis placed in a medium, which
may have a boundary of surface Ωb. The space between the
particle and the boundary is filled with an aqueous solution
Fig. 1. Schematic representation of the problem considered where a colloidal particle of surface Ωpis placed in a medium, which may have a boundary of
surface Ωb. A uniform electric field E parallel to the z-direction is applied; θ is
the solid angle. 0021-9797/$ – see front matter © 2006 Elsevier Inc. All rights reserved.
containing z1:z2 electrolytes; z1 and z2 are, respectively, the
valences of cations and anions. A uniform electric field E of strength E is applied in the z-direction. We assume that the liq-uid phase is an incompressible Newtonian flliq-uid. The applied electric field is relatively weak compared with that established by a charged surface. This is realistic, since the surface poten-tial is on the order of 25.7 mV to 10 kV and the Debye length ranges from 10 nm to 1 µm, which implies that the strength of the electric field established ranges from 25.7 to 109kV/m. The strength of the applied electric field in practice is much lower than these values. Both the surface of the particle and that of the boundary are nonconductive and nonslip.
2.1. Electrokinetic equations
Since the Reynolds number in electrophoresis is very small, the governing equations for the present problem can be ex-pressed as (1) ∇ · nju− Dj ∇nj+ zjenj kBT ∇Ψ = 0, (2) ∇2Ψ = −ρe ε = − 2 j=1 zjenj ε , (3) ∇ · u = 0, (4) 0= η∇2u− ∇p + ρeE.
In these expressions∇ and ∇2are, respectively, the gradient operator and the Laplace operator, Dj, nj, and zj are,
respec-tively, the diffusion coefficient, the number concentration, and the valence of ionic species j , e is the elementary charge, kB
is the Boltzmann constant, T is the absolute temperature, Ψ is the electric potential, ε is the permittivity of the liquid phase,
ρeis the space charge density, and u, η, and p are, respectively,
the velocity, the viscosity, and the pressure of the liquid phase.
E= −∇Ψ and ρeE is the electric body force acting on the
liq-uid.
If the applied electric field is weak, u, p, Ψ , and nj can be
expressed, respectively, as[9,10] (5) u= u(0)+ δu, (6) p= p(0)+ δp, (7) Ψ = Ψ(0)+ δΨ, (8) nj= n(j0)+ δnj.
Here, a quantity with superscript (0) refers to the equilibrium value; that is, when E is not applied, a quantity with prefix δ denotes a perturbed value arising from E. Note that u(0)= 0. By substituting Eqs.(5)–(8)into Eqs. (1)–(4), the problem under consideration can be divided into an equilibrium problem and a perturbed problem.
2.1.1. Equilibrium problem
Let us consider first the quantities in the equilibrium state. For the electrical field, we have
(9) ∇n(0) j + zjen(j0) kBT ∇Ψ (0)= 0, (10) ∇2Ψ(0)= − 2 j=1 zjen(j0) ε , (11) 0= −∇p(0)+ ε∇2Ψ(0)∇Ψ(0).
Integrating Eq.(9)from Ψ(0)= 0 and n(j0)= n∞j to Ψ(0)= Ψ(0) and n(j0)= n(j0)leads to a Boltzmann distribution
(12) n(j0)= n∞j exp −zjeΨ(0) kBT ,
where n∞j is the bulk concentration of ionic species j . Substi-tuting Eq.(12)into Eq.(10)gives
(13) ∇2Ψ(0)= − 2 j=1 zjen∞j ε exp zjeΨ(0) kBT .
Combining Eqs.(10) and (11)yields
(14) −∇p(0)= ε 2 j=1 zjen(j0) ε ∇Ψ (0).
Integrating this expression from p(0)= p∞ and n(j0)= n∞j to
p(0)= p(0)and n(j0)= n(j0)yields (15) p(0)= p∞+ 2 j=1 kBT n(j0)− n∞j .
This is the equilibrium or static pressure distribution, where
u= 0.
2.1.2. Perturbed problem
Under the condition of weak applied electric field, O’Brien and White[9]proposed using the expression
(16) nj= n∞j exp −zje(Ψ(0)+ δΨ + gj) kBT , j= 1, 2,
to take the effect of double-layer polarization into account, where gj is a potential used to describe double-layer
polar-ization. The governing equation for δΨ can be obtained from Eqs.(13) and (16), and Ψ = Ψ(0)+ δΨ as
∇2δΨ = ∇2Ψ − ∇2Ψ(0) = − 2 j=1 zjen∞j ε exp −zje(Ψ(0)+ δΨ + gj) kBT (17) − exp zjeΨ(0) kBT .
Combining Eqs.(8), (12), and (16)yields
δnj= n∞j exp −zje(Ψ(0)+ δΨ + gj) kBT (18) − n∞ j exp −zjeΨ(0) kBT .
∇2g j− zje kBT∇Ψ (0)· ∇g j= 1 Dj δu· ∇Ψ + 1 Dj δu· ∇gj (19) + zje kBT∇δΨ · ∇gj+ zje kBT∇gj· ∇gj .
Because δu, δp, δΨ , and δnj are all on the order of E,
the products of any two of these quantities becomes negligi-ble. Therefore, the governing equations for the flow field, by collecting terms on the order of E only, are
(20) ∇ · δu = 0,
(21) 0= η∇2δu− ∇δp + ε∇2Ψ(0)∇δΨ + ε∇2δΨ∇Ψ(0).
For convenience, Ψ(0), δΨ , δu, and δp are replaced,
respec-tively, by Ψ1, Ψ2, u, and p, and Eqs.(13), (17), and (19)–(21)
are rewritten as (22) ∇2Ψ 1= − 2 j=1 zjen∞j ε exp zjeΨ1 kBT , ∇2Ψ 2= − 2 j=1 zjen∞j ε (23) × exp −zje(Ψ1+ Ψ2+ gj) kBT − exp zjeΨ1 kBT , ∇2g j− zje kBT ∇Ψ1· ∇gj= 1 Dj u· ∇Ψ + 1 Dj u· ∇gj (24) + zje kBT∇Ψ 2· ∇gj+ zje kBT∇gj· ∇gj , (25) ∇ · u = 0, (26) 0= η∇2u− ∇p + ε∇2Ψ1∇Ψ2+ ε∇2Ψ2∇Ψ1.
In the last expression, the sum of the last two terms on its right-hand side denotes the electric body force acting on the liquid.
2.1.3. Low surface potential
The special case of low surface potential is often discussed in the literature. In this case the effect of double-layer polarization can be neglected, δnj= 0[11], and Eqs.(22)–(26)become[12–
15] (27) ∇2Ψ 1= κ2Ψ1, (28) ∇2Ψ 2= 0, (29) ∇ · u = 0, (30) 0= η∇2u− ∇p + ε∇2Ψ1∇Ψ2.
The last term on the right-hand side of Eq.(30)denotes the elec-tric body force, where−ε∇2Ψ1= ρeis the space charge density
and−∇Ψ2= E is the applied electric field[12–15]. In the
lit-erature, the electric body force is represented by −ρe∇Ψ =
ε∇2Ψ∇Ψ in many studies[6,7,10,16–25], some of which are not of total symmetric nature[16,21,22,24,25]. That is, Eq.(26)
or Eq.(30)is replaced by
(31) 0= η∇2u− ∇p − ε∇2Ψ∇Ψ.
Note that for a system not of total symmetric nature, such as a sphere normal to a plane[22,25], an extraneous electrosta-tic force ε∇2Ψ
1∇Ψ1is included in Eq.(31). This implies that
the electric field established by the equilibrium electric poten-tial,−∇Ψ1, is also a driving force for the electrophoresis of a
particle and the mobility will be overestimated.
2.2. Scaled governing equations
For a more concise treatment, the scaled quantities are used in subsequent analyses. To this end, the following scaling fac-tors are chosen respectively for the length scale, the electrical potential, the concentration of ionic species, and the velocity: the radius of a particle a, the equilibrium surface potential ζa,
the bulk concentration of electrolyte n∞j , and the reference elec-trophoretic velocity UE= εζa2/ηa. In terms of scaled
quanti-ties, the governing equations for an arbitrary surface potential, Eqs.(22)–(26), can be rewritten as[25–28]
(32) ∇∗2Ψ∗ 1 = − 1 (1+ α) (κa)2 Ψr exp−ΨrΨ1∗ − expαΨrΨ1∗ , ∇∗2Ψ2∗− (κa) 2 (1+ α) exp−ΨrΨ1∗ + α expαΨrΨ1∗ Ψ2∗ (33) = (κa)2 (1+ α) exp−ΨrΨ1∗ g1∗+ expαΨrΨ1∗ αg2∗, (34) ∇∗2g1∗− Ψr∇∗Ψ1∗· ∇∗g1∗= Ψ 2 rPe1u∗· ∇∗Ψ1∗, (35) ∇∗2g2∗+ αΨr∇∗Ψ1∗· ∇∗g2∗= Ψr2Pe2u∗· ∇∗Ψ1∗, (36) ∇∗· u∗= 0, (37) 0= ∇∗2u∗− ∇∗p∗+ ∇∗2Ψ1∗∇∗Ψ2∗+ ∇∗2Ψ2∗∇∗Ψ1∗,
where a symbol with an asterisk denotes a scaled quan-tity; Ψr = ζa/(z1e/kBT ) is the scaled surface potential of
the particle; Pej = ε(z1e/kBT )2/ηDj, j = 1, 2, is the
elec-tric Peclet number of ionic species j ; α = −z2/z1; and
κ= [2j=1(n∞j (ezj)2/εkBT )]1/2 is the reciprocal Debye
length.
Similarly, the scaled governing equations for the case of low surface potential are
(38) ∇∗2Ψ1∗= (κa)2Ψ1∗, (39) ∇∗2Ψ2∗= 0, (40) ∇∗· u∗= 0, (41) 0= ∇∗2u∗− ∇∗p∗+ ∇∗2Ψ1∗∇∗Ψ2∗. 2.3. Boundary conditions 2.3.1. Electric field
Three types of boundary conditions are usually assumed: constant surface potential[8,12,13,16–20,25–29], constant sur-face charge density [15,16,21], and charge-regulated surface
[6,7,14,22–24]. If the surface potential of a particle and that of a boundary are maintained respectively at ζaand ζb, then the
boundary conditions for Ψ1and Ψ2are
(42) Ψ1= ζa on Ωp, (43) Ψ1= ζb on Ωb, (44) n· ∇Ψ2= 0 on Ωp,
(45)
n· ∇Ψ2= −E cos θ on Ωb,
where n is the unit normal vector directed into the liquid phase. If the surface of a particle and that of a boundary are main-tained at constant surface charge density, then the correspond-ing boundary conditions become
(46) n· ∇Ψ1= − σa ε on Ωp, (47) n· ∇Ψ1= − σb ε on Ωb, (48) n· ∇Ψ2= 0 on Ωp, (49) n· ∇Ψ2= −E cos θ on Ωb,
where σaand σbare respectively the surface charge density of
the particle and that of the boundary. For the case of a charge-regulated surface, let us consider, for example, the dissociation reaction below on the surface of a particle[7]:
(50) AH⇔ A−+ H+.
It can be shown that the charge density on the particle sur-face, σa, is (51) σa= −e[A−] = − eNS 1+ ([H+]b/Ka)exp(−(eΨ1/kBT )) ,
where Ka= [A−]S[H+]S/[AH]Sis the equilibrium dissociation
constant for the dissociation reaction represented by Eq.(50), and NS is the density of dissociable functional groups on the
particle surface. Here, a symbol with square brackets denotes the concentration of a species; the subscripts S and b represent respectively the surface property and the bulk liquid property. In this case, the boundary condition associated with Ψ1on the
particle surface, Eq.(46), needs to be replaced by
(52)
n· ∇Ψ1= −
eNS/ε
1+ ([H+]b/Ka)exp(−(eΨ1/kBT ))
.
If we let∇∗= a∇, Ψ1∗= eΨ1/kBT, A= e2NSa/εkBT, and
B= [H+]b/Ka, then this expression can be rewritten as
(53)
n· ∇∗Ψ1∗= − A
1+ B exp(−Ψ1∗).
If Ψ1 is low, it can be shown that the corresponding surface
charge density σais[6,14,22–24] (54) σa= − eNS {1 + [H+]b/Ka}2− (e2NS/kBT ){[H+]b/Ka} {1 + [H+]b/Ka}2 Ψ1.
In this case, Eq.(46)reduces to
n· ∇Ψ1= − eNS/ε {1 + [H+]b/Ka} (55) −(e2NS/εkBT ){[H+]b/Ka} {1 + [H+]b/Ka}2 Ψ1,
or in terms of scaled symbols as
(56)
n· ∇∗Ψ1∗= A
1+ B−
AB
[1 + B]2Ψ1∗.
Suppose that the ionic concentration reaches the equilibrium value on the boundary surface and the surface of a particle is
ion-impenetrable. Then the boundary conditions for gjare[9]
(57)
gj= −Ψ2 on Ωb,
(58)
n· ∇gj= 0 on Ωp.
2.3.2. Flow field
If we let U be the particle velocity in the z-direction and ez
be the unit vector in the z-direction, then the boundary condi-tions associated with the flow field are
(59)
u= Uez on Ωp,
(60)
u= 0 on Ωb.
2.4. Electrophoretic mobility
For the present case, only the z-components of the forces acting on a particle, including the electrostatic force and the hydrodynamic force, need be considered. The z-component of the former, FEz, can be calculated by integrating the Maxwell
stress tensor over the particle surface,
(61) FEz= x S σE· n· ezdS,
where S represents particle surface, σE= εEE − (1/2)εE2I is
the Maxwell stress tensor, E= −∇Ψ = n(∂Ψ/∂n) + t(∂Ψ/∂t) is the applied electric field, I is the unit tensor, t is the unit tan-gential vector on the particle surface, n and t are, respectively, the magnitude of n and that of t, and E2= E · E. Substituting
σE into Eq.(61)yields
(62) FEz= x S ε∂Ψ ∂n ∂Ψ ∂z − 1 2ε ∂Ψ ∂n 2 + ∂Ψ ∂t 2 nz dS,
where nz is the z-component of n. Since Ψ = Ψ1+ Ψ2, this
expression becomes FEz= x S ε∂Ψ1 ∂n ∂Ψ1 ∂z − 1 2ε ∂Ψ1 ∂n 2 + ∂Ψ1 ∂t 2 nz dS +x S ε∂Ψ2 ∂n ∂Ψ2 ∂z − 1 2ε ∂Ψ2 ∂n 2 + ∂Ψ2 ∂t 2 nz dS +x S ε ∂Ψ1 ∂n ∂Ψ2 ∂z + ∂Ψ2 ∂n ∂Ψ1 ∂z (63) −1 2ε 2∂Ψ1 ∂n ∂Ψ2 ∂n + 2 ∂Ψ1 ∂t ∂Ψ2 ∂t nz dS.
Because a particle cannot be driven by its equilibrium elec-trostatic field, the terms involving (∂Ψ1/∂n)2, (∂Ψ1/∂t )2, and
(∂Ψ1/∂n)(∂Ψ1/∂z) should vanish. Also, (∂Ψ2/∂n)2, (∂Ψ2/
∂t )2, and (∂Ψ2/∂n)(∂Ψ2/∂z) are all small. The magnitude of
each of the rest terms is on the order of E, and therefore, Eq.(63)can be approximated by
FEz= x S ε ∂Ψ1 ∂n ∂Ψ2 ∂z + ∂Ψ2 ∂n ∂Ψ1 ∂z (64) − ε ∂Ψ1 ∂n ∂Ψ2 ∂n + ∂Ψ1 ∂t ∂Ψ2 ∂t nz dS.
This expression is applicable to both symmetric and asymmet-ric systems at an arbitrary surface potential subject to all three types of surface conditions.
For a nonconductive particle, ∂Ψ2/∂n= 0, and Eq.(64)
be-comes (65) FEz= x S ε ∂Ψ1 ∂n ∂Ψ2 ∂z − ε ∂Ψ1 ∂t ∂Ψ2 ∂t nz dS.
If the surface of a particle is maintained at a constant potential,
∂Ψ1/∂t= 0, and Eq.(65)can further be simplified as
(66) FEz= x S ε∂Ψ1 ∂n ∂Ψ2 ∂z dS= x S σ EzdS,
where σ = −εn · ∇Ψ1= −ε(∂Ψ1/∂n) is the surface charge
density, and Ez= −∂Ψ2/∂z is the strength of the local
elec-tric field in the z-direction. Equation(66)was used by Shugai and Carnie[13]for the electrophoresis of a sphere with a thick double layer parallel to a plane, normal to a plane, and along the axis of a cylindrical pore, and by Hsu and Ku[17]for the electrophoresis of a finite cylinder along the axis of a cylin-drical pore. Both assumed a weak E and low, constant surface potential. It is interesting to note that for a symmetric system, since ∂Ψ1/∂t= 0, Eq.(66)is not limited to the case of constant
surface potential, and becomes applicable to all three types of surface conditions in this study. The electrophoresis of a finite cylinder along the axis of a cylindrical pore[15], for example, belongs to this category.
The expression (67) FEz= x S σ EzdS= x S σ −∂Ψ ∂z dS
is widely used to evaluate FEz[10,16,19–21,23,24,29], where
σ= −εn · ∇Ψ = −ε(∂Ψ/∂n) is the surface charge density. In
terms of Ψ1and Ψ2, we have
FEz= x S ε∂Ψ1 ∂n ∂Ψ1 ∂z + ε ∂Ψ1 ∂n ∂Ψ2 ∂z + ε ∂Ψ2 ∂n ∂Ψ1 ∂z (68) + ε∂Ψ2 ∂n ∂Ψ2 ∂z dS.
The presence of the first, the third, and the fourth terms on the right-hand side of this expression implies that it is only applica-ble to some special symmetric electrophoresis proapplica-blems such as a sphere moving at the center of a spherical cavity[19]or a sphere[20]or a spheroid[23]moving along the axis of a cylin-drical pore.
The hydrodynamic force acting on a particle in the z-direct-ion, FDz, can be calculated by integrating the hydrodynamic
stress tensor over its surface; that is,
(69) FDz= x S σH· n· ezdS,
where σH = −pI + 2ηΔ is the hydrodynamic stress tensor,
Δ= [∇u + (∇u)T]/2 is the rate of deformation tensor, and the
superscript T denotes matrix transpose. Note that Eq.(69) is suitable for both rigid[8,13]and soft particles[10,28,30]. If we
Table 1
Summary of the present method for the calculation of the electrophoretic mobility of a colloidal particle under the condition of weak applied electric field
1. Types of particle: rigid or soft 2. Governing equations:
(1) High surface potential[25–28]
(a) Electric field
Eqs.(22)–(24)or Eqs.(32)–(35)
(b) Flow field
Eqs.(25) and (26)or Eqs.(36) and (37)
(2) Low surface potential[12–15]
(a) Electric field
Eqs.(27) and (28)or Eqs.(38) and (39)
(b) Flow field
Eqs.(29) and (30)or Eqs.(40) and (41)
3. Types of boundary conditions for electric field:
constant surface potential, constant surface charge density, charge-regulated surface
4. Electrostatic force in z-direction: FEz= x S ε ∂Ψ1 ∂n ∂Ψ2 ∂z + ∂Ψ2 ∂n ∂Ψ1 ∂z − ε ∂Ψ 1 ∂n ∂Ψ2 ∂n + ∂Ψ1 ∂t ∂Ψ2 ∂t nz dS 5. Hydrodynamic force in z-direction:
FDz= x S tzη ∂u n ∂t + ∂ut ∂n + nz −p + 2η∂u∂nn dS 6. Force balance at steady state:
FEz+ FDz= 0
let u= nun+ tut = n(u · n) + t(u · t), then Eq.(69) can be
rewritten as (70) FDz= x S tzη ∂un ∂t + ∂ut ∂n + nz −p + 2η∂un ∂n dS.
For a rigid particle (∂un/∂n)S= 0 and (∂un/∂t )S= 0; this
ex-pression can be further simplified as
(71) FDz= x S η∂(u· t) ∂n tzdS+ x S −pnzdS.
Note that FDz comprises a viscous force and a pressure term
[14–17,20,21,23,24,31]. The electrophoretic mobility of a par-ticle can be determined from the fact that the net force acting on it in the z-direction vanishes at steady state; that is,
(72)
FEz+ FDz= 0. 3. Conclusions
The method proposed in this study is summarized in Ta-ble 1, and two methods commonly used in the literature for the case of low surface potential are summarized in Table 2. Since the present method is applicable to all types of elec-trophoresis problems, regardless of the surface conditions on a particle and the geometry of a system, it is the most general one among the three methods. As presented inTable 2, for an
Table 2
Summary of two widely used methods for the calculation of the electrophoretic mobility of a colloidal particle under conditions of weak applied electric field and low surface potential
Method I[14,15,17] 1. Types of particle: rigid 2. Governing equations: ∇2Ψ 1= κ2Ψ1, ∇2Ψ2= 0, ∇ · u = 0, 0 = η∇2u− ∇p + ε∇2Ψ 1∇Ψ2
3. Types of boundary conditions for electric field: (1) Arbitrary geometry
constant surface potential (2) Symmetry geometry
constant surface potential, constant surface charge density, charge-regulated model
4. Electrostatic force in z-direction: FEz= x S ε∂Ψ1 ∂n ∂Ψ2 ∂z dS= x S σ EzdS 5. Hydrodynamic force in z-direction:
FDz= x S η∂(u· t) ∂n tzdS+ x S −pnzdS 6. Force balance at steady state:
FEz+ FDz= 0 Method II[16,19,20,23,24] 1. Types of particle: rigid 2. Governing equations: ∇2Ψ 1= κ2Ψ1, ∇2Ψ2= 0 ∇ · u = 0, 0 = η∇2u− ∇p + ε∇2Ψ∇Ψ
3. Types of boundary conditions for electric field: (1) Arbitrary geometry
inapplicable, in general (2) Symmetry geometry
constant surface potential, constant surface charge density, charge-regulated model
4. Electrostatic force in z-direction: FEz= x S ε∂Ψ ∂n ∂Ψ ∂z dS= x S σ EzdS 5. Hydrodynamic force in z-direction:
FDz= x S η∂(u· t) ∂n tzdS+ x S −pnzdS 6. Force balance at steady state:
FEz+ FDz= 0
arbitrary geometry, Method I is suitable for the electrophore-sis of a rigid particle with a constant surface potential. For a symmetric geometry, it becomes applicable to all three types of boundary conditions. Method II is inapplicable to a problem of asymmetric nature, in general. It can be used for all three types of boundary conditions, however, if a problem is of symmetric nature.
Acknowledgment
This work is supported by the National Science Council of the Republic of China.
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