AC electroosmotic flow of Maxwell fluid in a pH-regulated parallel-plate silica nanochannel

Nomenclature

C _i

concentration of the ith ionic species (mol/m³),

C _i0

bulk concentration of the ith ionic species (mol/m³),

E

electric field (V m⁻¹)

e

elementary charge (C)

F

Faraday constant (C mol⁻¹)

h

half channel height (m)

K ₋

equilibrium constant (kmol/m³)

K ₊

equilibrium constant (kmol/m³)^–1

K

ratio of half channel height to Debye length

N _A

Avogadro’s number (mol⁻¹)

R

gas constant (J mol⁻¹ K⁻¹)

Re _ω

oscillating Reynolds number

T

absolute temperature (K)

t _m

relaxation time (s)

U ^*

dimensionless velocity amplitude

u

velocity component (m)

ω

frequency of electric field (s⁻¹)

y ^*

dimensionless coordinates

y

coordinates (m)

z _i

valence of the ith ionic species

λ _D

Debye length (m)

φ*, φ _s *

dimensionless electric potential, dimensionless surface potential

τ ₂₁

shear stress component (N m⁻²)

η

zero shear rate viscosity (kg m⁻¹ s⁻¹)

Γ _i

number site density of the ith species (nm^–2)

Γ _t

total number site density of SiOH molecules (nm^–2)

φ

electric potential (V)

σ _s

surface charge density (C m^–2)

ε

fluid permittivity (F m⁻¹)

ρ

fluid density (kg m⁻³)

Abbreviations

AC

alternating current

DC

direct current

EDL

electric double layer

EOF

electroosmotic flow

1 Introduction

Electroosmotic flow (EOF) is the movement of an electrolyte solution in a channel with charged surface under the action of an applied electric field. EOF has many applications such as micropumps, sample separation, and mixers in microfluidics and nanofluidics [1,2,3,4]. The charging mechanisms of the wall–fluid interface are complex, possibly including the dissociation or association of functional groups on the wall, the asymmetric dipoles of water molecules residing at the wall–fluid interface and ion adsorption, etc. [3]. The presence of surface charges affects the distribution of nearby ions in the electrolyte solution. Counter-ions are attracted toward the charged surface, while co-ions are repelled from the charged surface. These interactions between ions lead to the formation of the so-called electric double layer (EDL) in which the net electric charge density is not zero. Direct current (DC)/alternating current (AC) EOF are generated by a Columbic force which equals the product of the charge density and DC/AC electric field, respectively. AC EOF is also known as time periodic EOF. The problems on the velocity distribution, entropy generation, heat and mass transfers of EOF of Newtonian fluid in nanochannels (with at least one characteristic dimension below or in the order of 100 nm) have been studied extensively [5,6,7,8,9,10,11,12]. Recently, Shit et al. [10] investigated the linear stability of a rotating EOF in unconfined and confined configurations by using the Debye–Hückel approximation. In a curved rectangular microchannel [11], the effects of the slip length, the curvature ratio, and the aspect ratio on EOF were studied. Pan et al. [12] studied experimentally and numerically EOF and Joule heating in cellular micro/nano electroporation.

In practice, micro- and nanofluidic devices are frequently used to analyze the biofluids (such as blood) and polymer solution (such as polyacrylamide [PAAm] and polyethylene oxide [PEO]) [13,14,15,16,17,18,19,20,21,22,23]. These fluids exhibit non-Newtonian behavior. The blood and PAAm solutions can be described by Maxwell model [13,14,15] and the simplified Phan−Thien−Tanner (sPTT) model [16], respectively. The solution of cetylpyridinium chloride and sodium salicylate shows linear Maxwell fluid rheology at certain concentrations [14,15]. The ion transport and current rectification of sPTT and Sisko fluid in a conical nanochannels have been studied by Trivedi and Nirmalkar [18] and Tang et al. [19], respectively. Koner et al. [20] studied AC EOF of Maxwell fluid and mass transport of an electro neutral solute through polyelectrolyte layer-coated canonical nanopore. They also investigated the rheological impact on thermofluidic transport characteristics of fractional Maxwell fluids through a soft nanopore [21]. Siva et al. [22] investigated the EOF of couple stress fluid in a rotating microchannel, and found that the increase in the couple stress parameter accelerates the axial EOF inside EDL.

In the previous literature [4–23], it is assumed that the surface charge density remains unaffected by solution properties. However, the surface charge density in nanochannels made of materials like SiO₂ and Al₂O₃ is strongly dependent on hydrogen ions, which means it also depends on the solution pH. Hsu and Huang [24] derived analytical expressions for the electric potential and velocity distributions of EOF in a pH-regulated parallel-plate microchannels. Yeh et al. [25] analyzed the effects of the solution pH, bulk ionic concentration, and the applied gate potential on the zeta potential, surface charge density, and velocity of EOF in a charge-regulated parallel-plate nanochannel. They found that the magnitude of surface charge density increases with an increase in pH and the applied gate potential and a decrease in the bulk ionic concentration; the velocity of EOF can be modulated by adjusting the values of the solution pH, bulk ionic concentration and the applied gate potential. Sadeghi et al. [26] investigated EOF and ionic conductance in a pH-regulated rectangular nanochannel, and found that the mean velocity first increases with the background salt concentration up to a threshold value of this parameter after which the trend is reversed. Yang et al. [27] studied the AC EOF in a pH-regulated parallel-plate nanochannel made of SiO₂. The influences of the number of pH-regulated nanochannels and the distances between adjacent nanochannels on the surface charge density, ionic concentration, and velocity field were investigated [28].

The working fluids in the previous studies [24–28] are all Newtonian fluids in pH-regulated nanochannel. Hsu et al. [29], in 2019, studied the non-Newtonian (power law) fluid in a pH-regulated conical nanopore, and analyzed the effects of the solution pH, bulk salt concentration, and the power-law index on the ion transport in the nanopore. Moschopoulos et al. [30] conducted a theoretical and experimental investigation on EOF of polyethylene oxide in microchannels, taking into account the deprotonation reaction of silanol groups and the formation of a polymeric depletion layer. Recently, Peng et al. [31] in 2024 investigated the effects of electrolyte concentration, pH, and polymer concentration on EOF and ion transport of viscoelastic (Oldroyd-B) fluid in a pH-regulated nanochannel. Mehta et al. [32], in 2024, investigated the EOF characteristics of viscoelastic (PTT) fluid in a pH-regulated parallel-plate microchannel considering the effect of surface charge-dependent slip length.

The experimental and theoretical studies show that there exists fluids which exhibit linear Maxwell fluid rheology at certain concentrations [14,15]. Many researchers have studied EOF of Maxwell fluid and related heat, mass transport problems [13,14,15,20,33]. However, AC EOF of Maxwell fluid through a pH-regulated nanochannel has not been studied. The aim of the present research is to explore the effects of the solution pH, the background salt concentration, the relaxation time, and the electric field frequency on AC EOF of Maxwell fluid in a parallel-plate silica nanochannel.

2 Mathematical modeling

2.1 Electric potential distribution

We study AC EOF of Maxwell fluid through a slit silica nanochannel separated by 2h. The channel length and width are assumed to be much larger than the channel height 2h. The background salt in the fluid is assumed to be KCl. Cartesian coordinate system is placed at the middle of the nanochannel (Figure 1). The functional group SiOH is formed on the channel surface by the reaction between silica and water at wall–fluid interface. The deprotonation and protonation reactions between SiOH and H⁺ ions in the fluid give rise to the charged surface, as shown in Figure 1:

SiOH ⟷ K − SiO − + H + , SiOH + H + ⟷ K + SiOH 2 + .

Figure 1

AC EOF in a pH-regulated parallel-plate silica nanochannel subject to an AC electric field along the x-axis.

The equilibrium constants K ₋ and K ₊ of the deprotonation and protonation reactions are equal to

K + = Γ SiOH 2 + Γ SiOH [ H + ] s , K − = Γ SiO − [ H + ] s Γ SiOH ,

where Γ SiO − , Γ SiOH 2 + , and Γ SiOH represent the number site density (nm^–2) of SiO − , SiOH 2 + , and SiOH, respectively; [ H + ] s = 10 − pH exp − F ϕ s R T denotes H⁺ ion concentration (kmol/m³) at the channel surface, in which 10 − pH , ϕ s , F, R, and T are the bulk molar concentrations of H⁺, the surface electric potential, Faraday constant, gas constant, and the absolute temperature, respectively.

The surface charge density σ s (cm^–2) equals

(1) σ s = 10 18 e ( Γ SiOH 2 + − Γ SiO − ) = − 10 18 F Γ t N A K − − K + [ H + ] s 2 K − + [ H + ] s + K + [ H + ] s 2 ,

where Γ t = Γ SiOH 2 + + Γ SiO − + Γ SiOH ; e and N _A are the elementary charge and Avogadro’s number, respectively [26].

The relation between σ s and ϕ s can be obtained from the overall electroneutrality condition of the ions in the surface and the fluid.

(2) σ s = ε d ϕ d y , | y = h

where ε is the fluid permittivity and ϕ is the electric potential in EDL [26,27]. The K⁺, H⁺, Cl^–, and OH^– in an unidirectional flow through a parallel-plate nanochannel obey Boltzmann distributions C i = C i 0 exp − z i F ϕ R T , i = 1, 2, 3, 4 with bulk concentration C i 0 (mol/m³), respectively. These bulk concentrations satisfy the electroneutrality condition, i.e., C ₁₀ + C ₂₀ = C ₃₀ + C ₄₀. Here the valences z ₁ = z ₂ = −z ₃ = −z ₄ = 1, and we assume that ϕ | y = 0 = 0 [26,27].

When pH ≤ 7, the solution pH is adjusted with HCl, and hence, C ₁₀ = M _KCl × 10³, C ₂₀ = 10^–pH+3, C ₃₀ = C ₁₀ + 10^–pH+3 −10^pH−14+3, and C ₄₀ = 10^pH−14+3. When pH >7, the solution pH is adjusted with KOH, and hence, C ₁₀ = C ₃₀ − 10^−pH+3 + 10^pH−14+3, C ₂₀ = 10^−pH+3, C ₃₀ = M _KCl × 10³, and C ₄₀ = 10^pH−14+3. Here M _KCl (kmol/m³) is the molar concentration of KCl [26,27].

The electric potential in EDL satisfies the following equations and boundary conditions [3]:

(3) ε d 2 φ d y 2 = − ρ e ,

(4) ρ e = − 2 F C 0 sinh F φ RT ,

(5) φ | y = h = φ , s d φ d y | y = 0 = 0 ,

where C ₀ = C ₁₀ + C ₂₀ = C ₃₀ + C ₄₀. φ _s in Eq. (5) can be determined by solving Eq. (1) after substituting Eq. (2) in Eq. (1).

Introducing the following variables:

(6) y ∗ = y h , K = h λ D , φ ∗ = ϕ ϕ 0 , φ 0 = RT F , λ D = ε R T / 2 F 2 C 0 ,

where λ _D is the Debye length, we get the dimensionless Poisson–Boltzmann equation and boundary conditions for electric potential

(7) d 2 φ ∗ d y ∗ 2 = K 2 sinh ( φ ∗ ) ,

(8) φ ∗ | y = 1 = φ s ∗ , d φ ∗ d y ∗ | y ⁎ = 0 = 0 .

The dimensionless surface potential ϕ s ∗ can be obtained by solving the following dimensionless equation using MATLAB function fzero:

(9) K − − K + [ H + ] s 2 K − + [ H + ] s + K + [ H + ] s 2 = − 2 ε ϕ 0 K σ 0 h sinh ϕ s ∗ 2 ,

where [ H + ] s = 10 − pH exp ( − ϕ s ∗ ) , σ 0 = 10 18 F Γ t N A . From Eq. (7) and boundary conditions (8), we obtain

(10) ϕ ∗ ( y ∗ ) = 4 tanh − 1 exp ( K ( y ∗ − 1 ) ) tanh ϕ s ∗ 4 .

3 Results and discussion

In this section, we analyze the results of the solutions for electric potential and velocity distributions of AC EOF of Maxwell fluid in pH-regulated parallel-plate nanochannel, where the effects of the relaxation time (also known as memory span) t _m , the background salt concentration M _KCl, and the solution pH are taken into account. In the following discussion, the values of typical parameters are chosen as per the previous studies [20,24,26,27]: ρ = 1,000 kg m⁻³, η = 0.001 kg m⁻¹ s⁻¹, T = 300 K, R = 8.314 J mol⁻¹ K⁻¹, ε = 80 × 8.854 × 10⁻¹² F m⁻¹, h = 50 × 10⁻⁹ m, pK ₋ = 8, pK ₊ = 1.9 (pK _± = –logK _±), Γ _t = 6 nm⁻², N _A = 6.22 × 10²³ mol⁻¹, F = 96485.33 C mol⁻¹. The frequency ω of the external electric field varies from 0 to 10³s⁻¹. Consequently, Reynolds number Re _ω varies from 0 to 1 × 10⁻⁵. To guarantee the validity of fundamental assumption of undisturbed EDL, the relaxation time t _m should be smaller than 2π/ω [33]. Thus, in the following computations, we consider t _m to range from 0 to 10⁻² s.

The expression of electric potential (10) is similar to the result obtained in the study by Yang et al. [27]. When pH = 3.05, the electric potential is zero, and hence the surface charge density is zero (Eq. (2)). Figure 2 shows that the magnitude of surface electric potential in EDL increases with the distance from the pH value to the isoelectric point (pH = 3.05), and decreases with the background salt concentration M _KCl. From Figure 2, it can be observed that both the pH and concentration M _KCl of the solution have a significant impact on the surface potential.

Figure 2

Dimensionless electric potential distributions for different pH and M _KCl. (a) M _KCl = 0.002 M and (b) pH = 4.

The effects of t _m and pH on the velocity amplitude of AC EOF at fixed M _KCl = 0.002 M and ω = 10²Hz are depicted in Figure 3. Comparing these figures, it is found that the velocity amplitude of Maxwell fluid is larger than that of Newtonian fluid (corresponding to t _m = 0); the velocity amplitude of AC EOF of Maxwell fluid increases with the relaxation time t _m. The reason is that an increase in the relaxation time can result in enhanced fluidity. Figure 3(a) and (b) indicate that there is not much difference in the flow fields between Newtonian fluid and Maxwell fluid when t _m is small. The velocity amplitude increases with the distance from the pH value to the isoelectric point (pH = 3.05) due to the increase in surface potential magnitude (Figure 2). The change in pH value of the solution alters the surface potential and net charge density within EDL, thereby modifying the flow field. When pH = 3.05, the surface charge density is zero, and hence EOF is not generated.

Figure 3

The dimensionless velocity amplitude at M _KCl = 0.002 M and ω = 10²Hz for different pH and t _m: (a) t _m = 0 s, (b) t _m = 0.006 s, and (c) and (d) t _m = 0.02 s.

Figure 4 depicts the velocity amplitude of AC EOF of Maxwell fluid for different M _KCl and t _m when pH = 4 and ω = 10² Hz. The velocity amplitude decreases with the increase in M _KCl. This is because the surface potential decreases with M _KCl (Figure 2). The variations in velocity amplitude of Maxwell fluid with M _KCl and pH are similar to that of Newtonian fluid [26,27] (Figures 3 and 4). It also can be seen from Figure 4 that the velocity amplitude increases with the relaxation time t _m. Figure 4(a) and (b) also show that when the relaxation time t _m is small, the difference in the flow fields between Newtonian fluid and Maxwell fluid is small. Figures 2 and 4 illustrate the significant influence of the concentration M _KCl on the electric potential and flow field.

Figure 4

The dimensionless velocity amplitude at pH = 4 and ω = 10²Hz for different M _KCl and t _m: (a) t _m = 10^–4 s, (b) t _m = 0.006 s, (c) t _m = 0.02 s, and (d) t _m = 0.06 s.

Figure 5 shows the effects of the frequency of AC electric field on EOF in the pH-regulated nanochannel. Interestingly, it has been observed that the velocity amplitude of Maxwell fluid increases with an increase in the electric field frequency, while the velocity amplitude of Newtonian fluid remains unaffected by the changes in the electric field frequency. The reason is that the term t _m ω appearing in the governing Eq. (17) of EOF of Maxwell fluid gives rise to the variation in velocity amplitude of Maxwell fluid with the frequency; the term t _m ω is zero when the working fluid is Newtonian fluid. In addition, the velocity amplitude distribution is a plug-like curve. This is because the oscillating Reynolds number in the nanochannel is too low, i.e., the momemtum diffusion time scale is much smaller than the oscillation period, and consequently, the fluid momentum can diffuse from the EDLs into the middle of the channel.

Figure 5

The dimensionless velocity amplitude at M _KCl = 0.002 for different pH, ω, and t _m: (a) t _m = 0 s, (b) t _m = 0.001 s, and (c) t _m = 0.006 s.

4 Conclusion

In this work, AC EOF of Maxwell fluid in a pH-regulated parallel-plate silica nanochannel has been studied. The analytical and semi-analytical solutions for the electric potential and velocity are obtained, respectively. The effects of the relaxation time, solution pH, salt concentration, and the frequency of AC electric field on EOF have been analyzed. The absolute value of surface electric potential in EDL increases with the distance from the pH value to the isoelectric point (pH = 3.05), and decreases with the salt concentration M _KCl. The velocity amplitude of AC EOF of Maxwell fluid is larger than that of Newtonian fluid; the velocity amplitude of AC EOF of Maxwell fluid decreases with the salt concentration M _KCl and increases with the relaxation time t _m and the distance from the pH value to the isoelectric point (pH = 3.05), respectively. Interestingly, it is observed that the velocity amplitude of Maxwell fluid increases with the increase in electric field frequency ω, while the velocity amplitude of Newtonian fluid remains unaffected by the electric field frequency ω. Interestingly, the velocity amplitude of Maxwell fluid increases with electric field frequency ω, while the velocity amplitude of Newtonian fluid remains unaffected. The electrokinetic flow of the other types of non-Newtonian fluids in pH-regulated channels can be studied in the future.