Transient electrophoresis of an oil drop

1 Introduction

Electrophoresis of colloidal particles and liquid drops in electrolyte solutions has long been one of the central topics in electrokinetics. Classical theories developed by Smoluchowski, Henry, and many subsequent investigators have established a fundamental understanding of the steady-state electrophoretic mobility of charged particles under a constant electric field (von Smoluchowski, 1921; Hückel, 1924; Henry, 1931; Overbeek, 1943; Booth, 1950; Wiersema et al., 1966; O’Brien and White, 1978; Ohshima et al., 1983; Delgado, 2001; Dukhin and Goetz, 2001; Spasic and Hsu, 2005; Masliyah and Bhattacharjee, 2006; Ohshima, 2006; Lee, 2018). In parallel with these developments, the stability and electrostatic interactions of colloidal dispersions have been described by the classical DLVO theory (Derjaguin and Landau, 1941; Verwey and Overbeek, 1948).

The electrophoresis of liquid drops has attracted considerable theoretical attention since the pioneering work of Booth. Subsequently, Ohshima et al. (1984) developed a theory for charged mercury drops, and Baygents and Saville (1991) established a comprehensive theoretical framework for the electrophoresis of drops and bubbles in electrolyte solutions. Their theory demonstrated the important roles played by interfacial charge dynamics and electrohydrodynamic flow in determining droplet mobility. More recently, various extensions of droplet electrophoresis theory have been reported for dielectric, polarizable, hydrophobic, and surfactant-covered droplets.

Most theoretical studies of electrophoresis have focused on steady-state behavior. In contrast, the transient electrophoretic response following the sudden application of an electric field has received considerably less attention. When an electric field is imposed on a charged droplet, the ionic atmosphere surrounding the droplet requires a finite time to relax toward a new equilibrium configuration. During this process, the electrophoretic velocity evolves continuously from zero to its steady-state value. The transient response therefore contains valuable information regarding the dynamics of charge redistribution, electric double-layer relaxation, and electrohydrodynamic coupling.

Although steady-state electrophoresis has been extensively studied, transient electrokinetic phenomena have also received increasing theoretical attention. Early analyses of transient electrophoresis were presented by Morrison (1969), Morrison (1971), Ivory (1983) and Ivory (1984). Subsequently, transient electrophoresis was investigated for dielectric spheres with arbitrary electric double-layer thickness (Huang and Keh, 2005; Keh and Huang, 2005), concentrated suspensions (Chiang and Keh, 2015a), porous particles (Lai and Keh, 2020), concentrated suspensions of charged particles (Lai and Keh, 2021), cylindrical particles (Li and Keh, 2020), and salt-free suspensions (Luo and Keh, 2021). Related theoretical studies have also been reported for transient electroosmotic flows (Keh and Tseng, 2001; Chang, 2009, Chang, 2010, Chang, 2012; Chiang and Keh, 2015b), transient phoretic migration (Khair, 2012; Yu and Keh, 2024), and electrokinetic phenomena in porous media (Saad and Faltas, 2018; Saad, 2018; Saad, 2019; Sherief et al., 2021; Ragab, 2023; Ayman et al., 2024). In addition, recent reviews have summarized the current understanding of electrokinetic phenomena in concentrated and salt-free suspensions (Carrique et al., 2010; Delgado et al., 2016).

Recently, the present author has developed analytical theories for transient electrophoresis of spherical particles in salt-free media (Ohshima, 2002a; Ohshima, 2002b; Ohshima, 2003a; Ohshima 2003b), spherical particles (Ohshima, 2022a), cylindrical particles (Ohshima, 2022b), soft particles (Ohshima, 2022c), transient dynamic electrophoresis (Ohshima, 2023a; Ohshima, 2024a), particles with slip surfaces (Ohshima, 2023b), gel electrophoresis (Ohshima, 2023c), and concentrated suspensions (Ohshima, 2024c). These studies have demonstrated that transient electrokinetic phenomena provide valuable information on charge relaxation and electrohydrodynamic coupling that cannot be obtained solely from steady-state analyses. Despite these advances, transient electrophoresis of liquid drops has not yet been investigated. Because liquid drops possess mobile interfaces, interfacial charge transport, and Marangoni stresses, their transient response is expected to differ fundamentally from that of rigid particles. Furthermore, the relationship between transient electrophoresis and dynamic electrophoresis of liquid drops has not been clarified theoretically.

The purpose of the present study is therefore to develop a theory of transient electrophoresis of an oil drop based on the Baygents–Saville model (Baygents and Saville, 1991) and to clarify its relationship to dynamic electrophoresis. An analytical expression for the transient electrophoretic mobility is derived in the Laplace domain, and an inverse Laplace transform representation is obtained. It is shown that the resulting expression possesses essentially the same mathematical structure as the recently derived expression for the dynamic electrophoretic mobility and can be obtained from it through a simple parameter transformation. Numerical inverse Laplace transformation is employed to calculate the actual time-dependent mobility.

The effects of the electrokinetic parameter κa and the droplet viscosity ratio η _d/η on the transient electrophoretic response are investigated in detail. Comparisons are also made with the corresponding transient electrophoretic responses of a rigid solid particle and a perfectly conducting drop. These comparisons reveal the distinctive features of charge relaxation and electrohydrodynamic coupling in liquid drops and provide new insight into the relationship between steady-state mobility and transient relaxation behavior.

While our previous study focused on the frequency-domain response of liquid drops subjected to oscillatory electric fields, the transient response following the sudden application of an electric field has not yet been investigated. Analysis in the time domain provides direct information on the relaxation dynamics of the electric double layer and interfacial charge distribution, which cannot be obtained solely from steady-state electrophoresis. The present study therefore extends the theoretical framework to transient electrophoresis and derives analytical expressions for the time-dependent electrophoretic mobility of liquid drops. In addition, the relationship between transient and dynamic electrophoresis is clarified, providing a unified description of electrokinetic relaxation phenomena in both the time and frequency domains.

2 Theory

2.1 Governing equations

We consider a weakly charged spherical oil drop of radius a suspended in an electrolyte solution of relative permittivity ε _r and viscosity η. The surrounding solution contains a general electrolyte composed of N ionic species, each with valence z _*i*, bulk number density n i ∞, and friction coefficient λ _*i* (i = 1, 2, … , N). The drop possesses a uniform equilibrium surface charge density arising from the adsorption of ionic species at the oil–water interface. The viscosity of the drop phase is denoted by η _d.

At time t = 0, a uniform electric field *E* is suddenly applied to the system. The subsequent electrophoretic motion of the droplet is governed by the coupled electrostatic, ionic transport, and hydrodynamic equations.

For weakly charged systems, the Debye–Hückel approximation is applicable. The electrostatic potential satisfies the linearized Poisson–Boltzmann Equation 1.where κ = ∑ i = 1 N z i 2 e 2 n i ∞ ε r ε 0 k T 1 / 2 is the Debye-Hückel parameter (1/κ is the Debye length), ε _o is the vacuum permittivity, k is the Boltzmann constant, and T is the absolute temperature, while the fluid motion inside and outside the droplet is governed by the Stokes equations Here p (r, θ, t) and p _I (r, θ, t), respectively, denotes the pressure outside and inside the drop, ρ _e (r, θ, t) represents the charge density arising from the mobile ionic species. The electric potential in the region outside the drop is expressed as ψ(r, θ, t). Subject to the interfacial boundary conditions of Baygents and Saville (Baygents and Saville, 1991).

Δ ψ 0 r = κ 2 ψ 0 r , r > a (1)

ρ 0 ∂ ∂ t u r , θ , t + U t + η ∇ × ∇ × u r , θ , t + ∇ p r , θ , t + ρ e r , θ , t ∇ ψ r , θ , t = 0 , r > a (2)

ρ d ∂ ∂ t u I r , θ , t + U t + η d ∇ × ∇ × u I r , θ , t + ∇ p I r , θ , t = 0 , r < a (3)

The electrophoretic mobility is defined bywhere *U(t*) denotes the transient electrophoretic velocity.

U t = μ t E t (4)

2.2 Laplace-transform solution

To analyze the transient response, the Laplace transform.is introduced.

μ ^ s = ∫ 0 ∞ μ t e − s t d t (5)

Applying the Laplace transform to the governing equations are given by Equations 1–4 the final expression for the Laplace-domain electrophoretic mobility is given by Equation 6, and its correspondence with the dynamic electrophoretic mobility is presented in Equation 7.

Note that there is a simple correspondence between the Laplace transform μ ^ s of the transient electrophoretic mobility μ(t) and the dynamic electrophoretic mobility μ _D under an oscillating electric field of frequency ω. The Laplace transform μ ^ s of the transient electrophoretic mobility μ(t) can be obtained from the dynamic electrophoretic mobility μ _D by replacing–iω in th expression for μ _D by s.

Following a procedure analogous to that employed previously for dynamic electrophoresis, the electrostatic potential, ionic concentration, and velocity fields are obtained in closed form. After satisfying the interfacial boundary conditions, the Laplace transform of the transient electrophoretic mobility is obtained as.where s denotes the Laplace variable, and

μ ^ s = − ε r ε 0 ζ η κ 2 a 4 ν S 2 H a , S − Γ S [ e κ a + S { E 5 κ a + S + S E 4 κ a + S + S 3 E 3 κ a + S } ‐ H a , S e κ a E 5 κ a ‐ 2 S 3 κ a 2 1 + κ a S κ a + S ‐ 2 η S 1 + S 3 3 η d + 2 η κ a 1 + 1 κ a ] (6)

H r , S = 1 + S r a exp − S r a − 1 + S r 3 3 a 3 1 + η S 1 + S 3 η d + 2 η

H a , S = 1 + S + S 3 1 + η 1 + S 3 η d + 2 η

Γ S = − 2 S ρ p − ρ o 9 ρ 0 1 + η 1 + S 3 η d + 2 η

S = s ν a 2

The detailed derivation is omitted here because it closely parallels that of the dynamic electrophoretic mobility of an oil drop reported previously (Ohshima, 2025).

2.3 Relation to dynamic electrophoretic mobility

A particularly important result of the present work is that Equation 5 possesses essentially the same mathematical structure as the dynamic electrophoretic mobility obtained previously for an oscillating electric field.

The dynamic mobility may be written aswhere ω denotes the angular frequency and.

μ ω = ε r ε 0 ζ κ 2 η γ 2 L a − Φ [ e κ − i γ a E 5 κ − i γ a − i γ a E 4 κ − i γ a − γ 2 a 2 3 E 3 κ − i γ a − L a e κ a E 5 κ a + 2 γ 2 3 κ 2 1 − i γ κ a 2 κ − i γ a − η γ 2 Q 1 − i γ a κ a + 1 3 κ 2 3 η d + 2 η Q ] (7)

γ = i ω ρ 0 η = i + 1 ω ρ 0 2 η

γ d = i ω ρ 0 η = i + 1 ω ρ d 2 η d

L r = H r − Q η 3 η d + 2 Q η 1 − i γ a γ 2 r 3 3 a

L a = H a − Q η 3 η d + 2 Q η 1 − i γ a γ a 2 3

Q = 3 3 γ d a + γ d a 2 − 3 tan γ d a γ d a 3 − 6 γ d a − 3 γ d a 2 − 2 tan γ d a

H r = 1 − i γ r e i γ r − a − γ 2 r 3 3 a

H a = 1 − i γ a − γ 2 a 2 3

Φ = Γ 1 + Q η 3 η d + 2 Q η 1 − i γ a

Γ = 2 γ a 2 ρ 0 − ρ d 9 ρ 0

E n κ a = κ a n − 1 ∫ κ a ∞ e − t / t n d t is the exponential integral of order n. Comparison of Equations 5, 6 reveals that the transient mobility in the Laplace domain can be obtained directly from the dynamic mobility expression through the parameter transformation

i ω ↔ − s

Thus, transient electrophoresis and dynamic electrophoresis are shown to be two manifestations of the same electrokinetic response viewed in the time and frequency domains, respectively.

2.4 Numerical inverse laplace transformation

The actual transient electrophoretic mobility is obtained from Equation 5 by numerical inverse Laplace transformation.

The numerical inversion was performed using the built-in function NInverseLaplaceTransform in Wolfram Mathematica.

The transient mobility μ(t) is presented in normalized form aswhere μ _s denotes the steady-state electrophoretic mobility.

μ t / μ s

3 Results and discussion

3.1 Transient electrophoresis of oil drops

Figure 1 schematically illustrates the transient electrophoresis of an oil drop. Immediately after the electric field is applied, the ionic atmosphere surrounding the drop is not in equilibrium with the imposed field. Consequently, the electrophoretic mobility starts from zero and gradually approaches its steady-state value through a relaxation process.

Figures 2–4 show the normalized electrophoretic mobility μ(t)/μ _s as a function of the dimensionless time νt/a ² for viscosity ratios η _d/η = 1, 2, and 5, respectively.

In all cases, the mobility increases monotonically with time and eventually approaches unity, corresponding to the steady-state mobility.

A strong dependence on the electrokinetic parameter (\kappa a) is observed. The relaxation becomes progressively faster as (\kappa a) increases. The relaxation time decreases in the order κa = 0, 0.1, 1, 10, and ∞.

This behavior reflects the decrease in the characteristic thickness of the electric double layer with increasing (\kappa a). When the double layer is thick, charge redistribution requires a longer time, whereas the relaxation becomes increasingly rapid in the thin-double-layer limit.

The dependence on the electrokinetic parameter κa can be understood from the characteristic length scale of the electric double layer. For small κa, the double layer extends over a region comparable to the drop radius a, and the redistribution of ions required to establish the equilibrium charge distribution occurs over a relatively large volume. Consequently, the relaxation process is slow. As κa increases, the double layer becomes increasingly confined to a thin region adjacent to the interface, reducing the distance over which ions must redistribute. The characteristic charge-relaxation time therefore decreases, leading to a faster approach to the steady state. In the limit κa → ∞, the double layer is highly localized near the interface and the relaxation becomes almost instantaneous on the hydrodynamic time scale.

3.2 Effect of viscosity ratio

Comparison of Figures 2–4 reveals that the viscosity ratio exerts a significant influence on the transient response.

As η _d/η increases from 1 to 5, the relaxation becomes noticeably faster. The increase in the internal viscosity suppresses internal circulation within the droplet and reduces the time required for the coupled electrohydrodynamic system to reach equilibrium.

Consequently, droplets with larger viscosity ratios exhibit shorter relaxation times.

The influence of viscosity ratio is particularly pronounced in the intermediate range of κa, where both interfacial and double-layer relaxation processes contribute significantly to the transient response.

The effect of viscosity ratio can be interpreted in terms of the coupling between interfacial motion and charge relaxation. When the internal viscosity is low, significant internal circulation develops inside the droplet. The evolving flow field continuously redistributes interfacial charges and modifies the surrounding ionic atmosphere, thereby prolonging the adjustment toward the steady state. As the viscosity ratio increases, internal circulation is increasingly suppressed and the interface behaves more like that of a rigid particle. The weaker fluid motion reduces the hydrodynamic disturbance associated with charge redistribution, allowing the electrokinetic system to relax more rapidly.

3.3 Steady-state mobility

Figure 5 compares the steady-state electrophoretic mobility of oil drops and perfectly conducting drops (mercury).

For all values of κa, the mobility of the mercury drop exceeds that of the oil drop.

Furthermore, in the large-κa region, the mobility of the mercury drop increases approximately in proportion to κa, producing values substantially larger than those of oil drops. For this reason, a broken vertical axis is employed in Figure 5.

The higher mobility of mercury drops originates from their ability to maintain an equipotential interface and to support stronger electrohydrodynamic flow.

The ordering in relaxation time τ: τ _solid < τ _oil < τ _Hg can be interpreted as a consequence of increasing interfacial freedom. A rigid particle possesses no internal flow or interfacial transport, so the transient response is governed primarily by double-layer relaxation. Oil drops introduce additional coupling between charge relaxation and interfacial fluid motion, resulting in a longer relaxation process. For perfectly conducting drops, the equipotential condition requires substantial redistribution of interfacial charge to maintain electrical equilibrium, introducing an additional relaxation mechanism that further increases the relaxation time.

3.4 Comparison with solid particles

Figure 6 presents the transient electrophoretic mobility of a rigid solid particle.

The transient behavior is qualitatively similar to that of oil drops, with the mobility increasing monotonically toward its steady-state value.

However, comparison with Figures 2–4 shows that the relaxation process is considerably faster for solid particles.

This difference arises because the interfacial dynamics characteristic of liquid droplets are absent.

3.5 Comparison with perfectly conducting drops

Figure 7 shows the transient electrophoretic mobility of a perfectly conducting drop.

Although the steady-state mobility of the conducting drop is significantly larger than that of either the oil drop or the solid particle, the approach to equilibrium is slower.

The relaxation time is found to be longest for the conducting drop among the three systems examined.

The overall ordering of relaxation times τ is

τ solid < τ oil < τ Hg

This result demonstrates that the magnitude of the steady-state mobility and the relaxation time are governed by different physical mechanisms.

Whereas the steady-state mobility reflects the magnitude of the electrohydrodynamic driving force, the relaxation time is controlled by the dynamics of charge redistribution and interfacial transport processes.

Consequently, a larger steady-state mobility does not necessarily imply a faster approach to equilibrium.

3.6 Physical implications

The present results establish a direct connection between transient and dynamic electrophoresis of liquid drops.

The transient response provides information on the relaxation of the electric double layer and interfacial charge distribution in the time domain, while dynamic electrophoresis probes the same processes in the frequency domain.

The mathematical correspondence demonstrated in Section 2 indicates that these two descriptions are fundamentally equivalent representations of the same electrokinetic dynamics.

This unified viewpoint may be useful for interpreting both transient and frequency-dependent electrokinetic measurements of liquid droplets and related soft particles.

4 Concluding remarks

In this work, a theory for the transient electrophoresis of a weakly charged oil drop in an electrolyte solution has been developed on the basis of the Baygents–Saville model (Baygents and Saville, 1991). By applying the Laplace transformation method to the governing electrokinetic equations, an analytical expression for the transient electrophoretic mobility in the Laplace domain was obtained, together with an inverse Laplace transform representation of the time-dependent mobility.

The present theory assumes that the droplet remains spherical. This approximation is expected to be valid when the electric stress generated by the applied field is much smaller than the restoring capillary stress due to interfacial tension, so that field-induced deformation remains negligible. Under sufficiently strong electric fields or sufficiently low interfacial tension, droplet deformation may become significant and could affect the electrophoretic behavior. Extension of the present theory to deformable droplets is an interesting subject for future study.

A noteworthy result of the present analysis is that the derived expression possesses essentially the same mathematical structure as the dynamic electrophoretic mobility formula recently obtained by the author for an oil drop subjected to an oscillating electric field. It was shown that the transient mobility expression can be obtained directly from the dynamic mobility expression through an appropriate transformation of parameters. This establishes a direct correspondence between transient electrophoresis in the time domain and dynamic electrophoresis in the frequency domain.

The transient electrophoretic mobility was evaluated numerically by means of numerical inverse Laplace transformation. The results demonstrate that the relaxation process depends strongly on both the electrokinetic parameter κa, where a is the drop radius and κ is the Debye-Hückel parameter, and the viscosity ratio η _d/η. For oil drops, the relaxation time decreases with increasing κa and increasing viscosity ratio. In particular, the relaxation time was found to decrease in the order κa = 0.1, 1, 10, and ∞.

Comparisons with a rigid solid particle and a perfectly conducting drop revealed significant differences in the transient response. The relaxation time was found to be shortest for the solid particle, intermediate for the oil drop, and longest for the perfectly conducting drop. In contrast, the corresponding steady-state electrophoretic mobilities exhibit the opposite tendency. These results demonstrate that the magnitude of the steady-state electrophoretic mobility and the relaxation time are governed by different physical mechanisms. Whereas the steady-state mobility is determined primarily by the electrohydrodynamic driving force under equilibrium conditions, the relaxation time is controlled by the dynamics of charge redistribution and interfacial transport processes.

The present study provides a unified theoretical framework for transient and dynamic electrophoresis of liquid drops and contributes to a deeper understanding of electrokinetic relaxation phenomena in colloidal and interfacial systems.

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Footnotes

Edited by: Xiangchun Xuan, Clemson University, United States

Reviewed by: Teng Zhou, Hainan University, China

Kareem E. Ragab, Alexandria University, Egypt

Footnotes

a, drop radius; E, applied electric field strength; κ, Debye- Hückel parameter; ε _r, relative permittivity of electrolyte solution; η, viscosity of electrolyte solution; η _d, viscosity of oil drop; ρ ₀, mass density of electrolyte solution; ρ _d, mass density of oil drop; μ(t), transient electrophoretic mobility of oil drop; μ _s, steady-state electrophoretic mobility of oil drop.