Consider a system in which homogeneous particles (radius a, dielectric constant ε2, bulk layer conductivity σb, surface conductivity Kσ) are in a certain solvent (dielectric constant ε1, conductivity σ1) and an external electric field is applied. First of all, we consider the voltage inside and outside this particle. Considering the boundary conditions, refering to Böttcher's method85 and consider that each particle is suspended in a homogeneous solvent having bulk properties of this system. This is important in considering the influence of surrounding particles at an appropriate concentration on average. However, in the distribution of adjacent particles, the influence of local variation is not considered. Let us consider a polar coordinate (r, θ, φ) with the center of the particle as the origin. Assume that the z axis is the polar axis and the electric field strength E is applied in the z direction. The voltage ψ at a position separated from the particle by r is
𝜓 = −𝐸𝑟 𝑐𝑜𝑠𝜃 (2.67)
Under low electric field and high frequency condition, Joule heat and electrochemical influence are not important and can be ignored. It is thought that the particles and their
surroundings have uniform bulk conductivity and dielectric constant. Here, by applying the Laplace equation,
∇/𝜓/ = 0; ∇/𝜓 = 0 (2.68) The surface of the particles becomes
𝜓/ B = 𝜓 B (2.69)
𝜀/𝜕𝜓/
𝜕𝑟 B− 𝜀𝜕𝜓
𝜕𝑟 B = 4𝜋𝜎 (2.70)
σ is the surface charge density of the particle surface charge density, ε2, ε is the real part of the complex permittivity of the particle and solvent.
𝐸 = 𝐸F𝑒HIJ (2.71)
Here, a differential equation concerning σ is necessary. E0 is the average value of the electric field intensity in the suspension, ω is the angular frequency. The surface charge density varies with time due to two factors. (a) Conductivity of bulk results as a result of movement of ions to or from the particle surface. (b) Ions move along the particle surface, resulting in surface conductivity. Considering the isotropy of the medium, (a) is
𝑑𝜎B
𝑑𝑡 = 𝜎)𝜕𝜓
𝜕𝑟 B− 𝜎M𝜕𝜓/
𝜕𝑟 B (2.72)
Considering the charge transfer at the interface with respect to the surface conductivity Kσ, σ1=σb=0 σ1=σb=0. Since the electric field distribution and particle surface are objects in the axial direction, the continuous equation according to (b)
𝑑𝜎M
𝑑𝑡 = 𝐾N 𝑎/𝑠𝑖𝑛𝜃
𝜕
𝜕𝜃 𝑠𝑖𝑛𝜃𝜕𝜓/
𝜕𝜃 B (2.73)
Here, when handling Kσ, σb, σ1 as constants, the time function of the surface charge density is expressed by the sum of the equations (2.72) and (2.73) when the events (a) and (b) are simultaneously occurring.
𝑑𝜎
𝑑𝑡 = 𝐾N 𝑎/𝑠𝑖𝑛𝜃
𝜕
𝜕𝜃 𝑠𝑖𝑛𝜃𝜕𝜓/
𝜕𝜃 B + 𝜎)𝜕𝜓
𝜕𝑟 B− 𝜎M𝜕𝜓/
𝜕𝑟 B (2.74)
The voltage inside the particle is,
𝜓/ = − 3𝜀)∗𝑟𝐸𝑐𝑜𝑠𝜃 𝜀/∗+ 2𝜀)∗−8𝜋𝑖𝐾N
𝜔𝑎
(2.75) where Ks is the surface conductance of the particle, 𝐾N = 𝐾V 𝑎
The voltage outside the particle, 𝜓 = 𝑎'
𝑟' 𝜀/∗− 2𝜀)∗−8𝜋𝑖𝐾N
𝜔𝑎 𝜀/∗+ 2𝜀)∗−8𝜋𝑖𝐾N
𝜔𝑎 − 1 𝑟𝐸𝑐𝑜𝑠𝜃 (2.76) The surface charge density σ of the particles due to the applied electric field is
𝜎 = 3𝑖𝐸𝑐𝑜𝑠𝜃 𝜀/𝜎)− 𝜀) 𝜎M+2𝐾N 𝑎 𝜔 𝜀/∗+ 2𝜀)∗−8𝜋𝑖𝐾N
𝜔𝑎
(2.77) 𝜀)∗ = 𝜀) −4𝜋𝑖𝜎)
𝜔 (2.78)
𝜀/∗ = 𝜀/ −4𝜋𝑖𝜎M
𝜔 (2.79)
From (2.75), (2.76) and (2.77), 𝜀/∗ is always subtracted from 8"#$% &' . Comparing with the equation (2.89), it can be considered that the influence of the surface conductivity Kσ in this model is equivalent to the increase of Kσ due to 2𝐾N 𝑎. Therefore, in the case of particles, it can be considered in the same way as in the case of handling assuming that only the bulk conductivity is involved, and the conductivity σp of the particle at that time is
𝜎X = 𝜎M+ 2𝐾V (2.80)
where 𝐾V = 𝐾N 𝑎.
Considering the surface conductivity of the particles. Consider a case where a current with a surface current density of jσ flows on a certain conductor surface. Assuming that the electric field by this current is E, the current density jσ on the surface is
𝒋V = 𝐾V𝑬 (2.81)
Kσ is surface conductivity. Also, jσ is also expressed by the following equation.
𝒋V = _ 𝒋 𝑥 − 𝒋(∞)
F
𝑑𝑥 (2.82)
where j (x) is the current density in the distance from the conductor surface.
Also, for j(x) and j(∞), depending on electrolyte concentration c and ion mobility u, 𝒋 𝑥 − 𝒋(∞) = 𝑐H 𝑥 − 𝑐H(∞) 𝑧H 𝐹𝑢H(𝑥)𝑬
H (2.83)
where, |zi| is the ion valence number, and F is the Faraday constant. From these, the surface conductivity Kσ is
𝐾V = 𝐹 𝑧H 𝑐H 𝑥 − 𝑐H(∞)
_ F
𝑢H(𝑥)𝑑𝑥
H (2.84)
It is understood from this that the surface conductivity is related to the local electrolyte concentration and ion mobility.
In the conductor forming the electric double layer, the surface conductivity is,
𝐾V = 𝐾V,dJe4f+ 𝐾V,gHhh (2.85)
Here, Kσ,Stern are due to the Stern layer, and Kσ,Diff are due to the diffusion electric double layer. In the case of particles, it can be considered as well, and its surface conductivity is also represented by the equation (2.85). Using the equation (2.80) for the
conductivity σ p of the particle at that time, 𝜎X = 𝜎M+ 2𝐾dJe4f
𝑎 + 2𝐾gHhh
𝑎 (2.86)
Here, σb is the conductivity of the bulk layer, a is the particle radius, KStern and KDiff
are the conductance of the Stern layer and the diffusion electric double layer, respectively.
It is found that the conductivity near the surface of the particle becomes dominant as the particle radius becomes smaller according to the equation (2.86).
Generally, when different ions exist in the Stern layer, the conductivity Kσ, Ster of the Stern layer is
𝐾V,dJe4f = 𝐾HV,dJe4f
H = 𝜎HdJe4f𝑢HdJe4f
H (2.87)
σStern is the surface charge density in the Stern layer. Here Nernst-Einstein's equation 𝑢H = 𝑧H 𝐹𝐷H
𝑅𝑇 (2.88)
Equation (2.87) becomes
𝐾V,dJe4f = 𝐹
𝑅𝑇 𝑧H 𝐷HV𝜎HdJe4f
H (2.89)
Here, R is a gas constant, T is the temperature, and D is the diffusion coefficient of free ions.
Especially when there is only one type of free ions in the Stern layer, 𝐾V,dJe4f = 𝜎dJe4f𝑢dJe4f= 𝑧H 𝐹𝐷HV𝜎dJe4f
𝑅𝑇 (2.90)
Regarding the conductivity of the diffusion layer, not only the electrical conductivity but also the effect of electroosmotic flow must be considered. According to Bikerman, when there is one type of electrolyte ion, j(x)45 is
𝒋 𝑥 = 𝐹 𝑧l𝑐l 𝑥 − 𝑧m𝑐m 𝑥 𝜀F𝜀
𝜂 𝜁 − 𝜓 𝑥 𝑬 + 𝐹/
𝑅𝑇 𝑧m/𝐷m𝑐m 𝑥 + 𝑧l/𝐷l𝑐l 𝑥 𝑬
(2.91) η is the viscosity, ζ is the zeta potential, and ψ is the potential. One item in the equation is current density due to electro osmosis, and the second is due to electron transfer of the diffusion layer to the liquid. It is presumed that the diffusion coefficient of the ion is the same as that in the bulk. Using equations (2.82) and (2.88), the conductivity Kσ, Diff of the diffusion layer is
𝐾V,gHhh = 8𝜀F𝜀𝑐𝑅𝑇 𝑢m
𝐴 − 1− 𝑢l
𝐴 + 1−4𝜀F𝜀𝑐𝑅𝑇 𝜂𝑧𝐹
1
𝐴/− 1 (2.92)
𝐴 = 𝑐𝑜𝑡ℎ −𝑧𝐹𝜁
4𝑅𝑇 (2.93)
Due to equation (2.92),
𝐾V,gHhh = 2𝐹/𝑧/𝑐
𝑅𝑇𝜅 𝐷m 𝑒lstu /vw− 1 1 +3𝑚m 𝑧/ + 𝐷l 𝑒stu /vw− 1 1 +3𝑚l
𝑧/
(2.94)
𝑚± = 𝑅𝑇 𝐹
/ 2𝜀F𝜀
3𝜂𝐷± (2.95)
This represents the correlation of electro osmosis to the surface conductance. Here, when the cation and the anion have the same diffusion coefficient, the value of m is almost the same. At this time, the equation (2.94)
𝐾V,gHhh =4𝐹/𝑧/𝑐𝐷 1 + 3𝑚𝑧/
𝑅𝑇𝜅 𝑐𝑜𝑠ℎ 𝑧𝐹𝜁
2𝑅𝑇 − 1 (2.96)
where κ-1 is the Debye length
𝜅/ = 2𝑐𝑧𝐹/
𝜀F𝜀𝜂𝐷 (2.97)
2.6.2 Theory of dielectrophoretic property change due to DNA labeling
Since DNA has a negative charge in aqueous solution. It has one charge per single base length from its structure and has a large electric charge as a whole. Therefore, it can be expected that the surface conductance of the DNA labeled particle changes due to DNA attachment. Figure 2.24 shows the particle surface conductance dependence of Re[K] of dielectric particles. The physic properties used in the theoretical calculation is as following: the relative permittivity of the particles: 2.5; the relative permittivity of the solvent: 80; the conductivity of the solvent: 1.0 × 10-4 S / m; the applied voltage frequency: 100 kHz. From the figure, Re [K] changes from negative to positive as surface conductance increases.
2.6.3 Binding of particles and DNA46
In this study, streptavidin-modified particles were used, and DNA was modified at one end with biotin. DNA was bound to the particles via a specific binding, biotin / avidin bond.
The affinity between biotin with a molecular weight of 244.31 and avidin with a molecular weight of less than 70,000 is very high, comparable to almost 106 times the
affinity between antigen and antibody, which is said to have high specificity.
Avidin is composed of four subunits, each subunit binds to one molecule of biotin.
That is, one avidin molecule can specifically bind to four molecules of biotin. The binding between biotin and avidin has been shown to be noncovalent rather than covalent despite having high affinity, and the tryptophan residue of avidin is deeply involved in binding to biotin. Streptavidin derived from Streptomyces avidin was used in this study.
This is because streptavidin has not only high affinity with biotin but also nonspecific binding to cells etc.