Approaches to enable vortex chromatography in silicon devices

1 Introduction

1.1 Axial dispersion in microfluidics

The performance of a chromatographic column is limited by axial dispersion resulting from the effects of band broadening. A key contribution is related to limited transverse diffusion. Taylor first described this phenomenon, later refined by Aris for flow in cylindrical tubes, forming the basis of what is now known as Taylor–Aris dispersion (Taylor, 1953; Aris, 1956), also known as the C-term effect in a chromatographic context (mostly including retention as well). This effect is significant for solutes with low diffusivity (Baca et al., 2019). In conventional LC, axial dispersion is reduced through the use of packed columns containing smaller particles (Op De Beeck et al., 2013). Yet, further downscaling leads to excessive back pressures. Recent studies have focused on improving cross-sectional mass transfer by inducing secondary flows to suppress axial dispersion and overcome the operational limits in conventional LC, as discussed in the following sections.

1.2 Concept of vortex chromatography

Secondary flows can be conceived to enhance mass transfer across the channel, in first instance developed for mixing (Sumpter and Lee, 1991). Various approaches have been explored to generate lateral flow, these can be classified as passive and active methods. Passive methods use channel geometry or surface modifications to promote transverse flow without external stimuli. The secondary flow in passive methods is induced directly from the axial flow at high Re values, a condition that in general (without mixing), is unfavorable for its associated large axial dispersion. In active methods, on the other hand, externally applied forces drive the lateral flow independent from the primary flow.

Vortex chromatography can be integrated into systems fabricated from materials including silicon and polymeric substrates. Silicon provides long-term stability over thousands of runs with its high elastic modulus, thermal stability, and chemical resistance. The induced surface charge on silicon substrates enables robust and reproducible vortex formation and maintains reliable flow control over repeated operating cycles. As a semiconducting material, silicon allows straightforward integration of electrodes directly into the substrate. In contrast, polymer-based devices generally require an additional conductive layer, such as indium tin oxide (ITO), to enable electrical actuation for vortex generation (Tiflidis et al., 2021). The integration of elements on silicon, such as electrodes or piezoelectric elements capable of generating active mixing phenomena, is also possible (Westerbeek et al., 2022).

A theoretical framework is required to understand how secondary flows can be defined and optimized within LC systems to address the limitations of the methods under discussion.

2 Secondary flow induced dispersion: theory, mechanisms and applications

2.1 Dispersion theory under secondary flow conditions

The extended dispersion theory by van Deemter includes additional sources of dispersion by the plate height (H) model using A, B, and C terms. The A term is called eddy dispersion and reflects the distribution of (unequal) flow paths that solute molecules follow in packed columns. The B-term represents longitudinal dispersion and can be reduced by reducing the elution time. Additionally, dispersion due to the mass transfer resistance, so-called C-term dispersion, arises from the finite rate of solute exchange between the mobile and stationary phases. The plate-height relationship with mobile phase velocity (u) (Equation 1) is:

H = A + B u + C u (1)

When radial diffusion is slow, as in the case of macromolecules or proteins, the velocity difference results in noticeable peak broadening and a loss of separation efficiency. The A-term is absent in highly ordered (pillar array) columns or open-tubular channels, eliminating eddy diffusion from the plate height contribution. The B-term can be minimized by reducing longitudinal dispersion, for example, by using radially elongated pillar arrays in classical LC systems (Op De Beeck et al., 2013).

For an open-tubular column, the plate height can be expressed in dimensionless form as the reduced plate height, defined by h = H/l, where l is the characteristic column length. Following, the reduced plate height is given (Equation 2) by Westerbeek et al. (2023).

h = 2 pe + 2 * κ aris * Pe (2)

The first term accounts for axial dispersion arising from molecular diffusion, while the second term represents convective dispersion associated with cross-sectional velocity non-uniformity. The dimensionless parameter κaris characterizes how efficiently velocity heterogeneity contributes to band broadening and, together with the Peclet number, determines column performance. The contributions of molecular diffusion and convective dispersion lead to an optimal Peclet number at which the plate height reaches a minimum, with κ_{ari s} governing the magnitude of the convective contribution at this optimum.

In this context, the effect of lateral flow on dispersion as a separate parameter is not accounted for in the van Deemter theory. To address this limitation, the Generalized Dispersion Theory (GDT) can be employed by incorporating the influence of transverse transport on overall dispersion (Brenner, 1980). GDT extends the long-time solution of the convection-diffusion equation with the effect of cross-sectional velocity profiles. In GDT, the solute transport is characterized by the concentration distribution, c (x,y,z,t) follows the convection–diffusion equation (Equation 3) (Brenner, 1993):where the velocity field is u (x, y) = (u (x, y), v (x, y), U (x, y)) with axial velocity U and D_m is the molecular diffusion coefficient. In the long-time limit, cross-sectional averaging yields an effective transport equation for the mean concentration. Dimensionless plate height is expressed in Equation 4 as c ¯ (z,t) (Brenner, 1993):

∂ c ∂ t + u · ∇ c = D m ∇ 2 c (3)

∂ c ¯ ∂ t + U * ∂ c ¯ ∂ z = D * ∂ 2 c ¯ ∂ z 2 (4)

U* is the average axial velocity, and D* is the axial dispersion coefficient. In GDT, the effective dispersion coefficient is decomposed into a molecular and a convective contribution according to Equation 5,where D m ¯ is the molecular contribution to the axial dispersion coefficient and D c ¯ the Taylor-Aris dispersion coefficient. The cross-sectional coordinates (x,y) are grouped into a local coordinate q∈q₀, where q₀ denotes the bounded channel cross-section. For unretained conditions, the effective transport coefficients are given by Equations 6–9 (Brenner, 1993). The average velocity is defined as,

D * = D m ¯ + D c ¯ (5)

U * = ∫ q 0 c 0 q U q d q (6)

Here c₀(q) is the stationary distribution of solute across the local subspace and is defined as the long-time limit of the cross sectional solute distribution. It satisfies the normalization condition (Equation 7).

∫ q 0 c 0 q d q = 1 (7)

D m ¯ = ∫ q 0 c 0 q D m d q (8)

D c ¯ = ∫ q 0 c 0 q B q U q − U * d q (9)

B(q) is the auxiliary field describing the relative axial displacement of solute associated with a given transverse position. The auxiliary field B(q) is obtained from the transverse transport problem (Equation 10) (Brenner, 1993),where u_⊥ is the transverse velocity field. Secondary flows do not contribute directly to axial transport, but modify the redistribution of solute across streamlines and thereby alter the B-field. This reduces the correlation between axial velocity deviations and solute displacement while leading to a decrease in the D c ¯ .κ_aris appears in the reduced plate-height expression, quantifying how efficiently cross-sectional velocity heterogeneity is converted into axial band broadening. In classical Taylor–Aris dispersion, the κ_aris geometrical factor is determined by the channel cross-section and the corresponding axial velocity profile. In GDT, dispersion is expressed using the reduced plate height (Equation 11) in a phenomenological form as Westerbeek et al. (2023):

∇ ⊥ · D m ∇ ⊥ B − u ⊥ B = u x , y − U * (10)

h = 2 D * U * d h (11)

While κ_aris reduces to a geometry-dependent constant in classical Taylor–Aris dispersion, GDT generalizes this concept by allowing κ_aris to depend on transverse transport processes through their influence on D c ¯ . In the present framework, the plate height depends D*, which includes the contribution from convective dispersion. Since the convective dispersion coefficient is governed by the auxiliary B-field (Equation 10), the influence of transverse transport is implicitly embedded in the plate-height expression through κ_aris. Consequently, secondary flows do not alter the structure of the reduced plate-height expression but instead modify the effective value of κ_aris by reshaping transverse solute redistribution and reducing the correlation between axial velocity deviations and solute displacement. Within GDT, κ_aris is governed by the auxiliary B-field and, therefore, depends on secondary flows.

Studies have shown theoretical and experimental agreement with GDT in LC systems (Westerbeek et al., 2022; Bihi et al., 2024). Researchers have also demonstrated that secondary flows can reduce dispersion more than the improvements achievable through the geometrical modifications alone (Zhao and Bau, 2007). As the theoretical understanding of secondary flows developed, researchers began to examine whether κ_aris could be reduced by geometric optimization and actively altering the flow profile. The idea of inducing lateral flows for dispersion suppression was proposed earlier (Tijssen, 1978; Adrover, 2013). These developments motivated to explore practical mechanisms to generate secondary flows, as will be discussed in the following section.

2.2 Secondary flow mechanisms

Lateral mixing plays a crucial role in reducing axial dispersion by facilitating faster transfer of species across flow streamlines. In passive mixing, the flow pattern is determined by channel geometry and hydrodynamic effects, while active methods employ external forces to induce more controlled and efficient lateral mixing. The schematics of several mixing mechanisms are shown in Figure 1 and discussed below.

2.2.1 Passive mixing

Modification of channel geometry is used without external stimuli in passive mixing methods. The passive designs rely on changes in flow direction, including zigzag, serpentine, spiral, and T-shaped micromixers. These geometries guide the fluid to follow the paths. This way, the flow periodically reorients and enhances mixing through stretching and folding. While these methods are simple and easy to fabricate, the strength and uniformity of the induced secondary flows are generally limited. In addition, the performance depends on the flow rate and the aspect ratio of the channel. In a study, researchers used a 15 µm deep staggered herringbone mixer (Figure 1a) to create chaotic flow (Stroock et al., 2002) and showed a significant reduction in band broadening in 70 × 200 μm channels. This concept relies on an asymmetric groove patterned on the channel wall to create chaotic advection. The alternating ridge orientation generates transverse flow that folds and stretches fluid elements, resulting in an increase in interfacial contact and enhancing mixing. However, the performance of herringbone structures is highly sensitive to the groove dimensions and channel size. In wider channels, the vortices often fail to span the whole cross-section. The grooves must be made very small to remain effective. As a result, practical implementation can become difficult, particularly when long columns or fabrication constraints are involved.

Another hydrodynamically driven passive mechanism is called Dean mixing, which occurs in curved channels due to centrifugal forces acting on fluid elements. An example of this is the study of Tijssen and co-workers in which Dean vortices are investigated in curved capillaries (Tijssen, 1978). They proposed that the resulting secondary flow could mitigate Taylor–Aris dispersion by up to 5-fold (Johnson and Kamm, 1986; Jayaraman, 1998). Dean vortices only become significant at sufficiently high axial velocities because they arise from inertial effects. At low Reynolds number regimes, a characteristic of microfluidics, the induced secondary flow remains weak. In the initial stages of research, efforts were made to benefit from turbulent mixing in larger channels (Giddings et al., 1966; Bauer, 1989). These efforts showed stronger transverse transport. However, these mechanisms were not suitable for microscale systems. These systems need due to the necessity of higher flow rates to induce turbulence, resulting in higher axial dispersion. In summary, passive micromixers have proven instrumental in demonstrating that geometry can induce lateral flow. However, the resulting mixing is either weak, geometry-dependent, or contributes to axial velocity. These limitations have encouraged the development of generating controlled lateral flow.

2.2.2 Active mixing

Active methods use external stimuli to generate lateral flows, unlike passive micromixers. This allows secondary flows to be generated independently from axial flow at low Re values. Active strategies also offer greater control over the location of mixing, strength, and frequency of induced vortices. This makes them more versatile than passive approaches. Several principles have been used to introduce active methods, including magnetic, acoustic, and electroosmotic mixing.

Magnetic mixing depends on the interaction between magnetic fields and magnetic particles. When the oscillating or rotating magnetic field is applied, the particles induce local disturbances while generating transverse microvortices (Shanko et al., 2022). These can enhance mixing efficiency, especially in larger channels where inertial effects are minimal. A well-known example of this is shown in Figure 1b where the rotating magnetic fields are used to generate nanoscale vortices in a 1 mm diameter capillary (Hajiani and Larachi, 2012). Magnetic actuation enhanced the transverse mixing and lowered Taylor dispersion by a factor of five compared with the (uniquely applied) pressure-driven flow situation. The approach also allows the flow to be controlled remotely and reversed without direct contact with the channel walls. However, applying this method in LC systems brings several practical difficulties. Magnetic particles introduced into the stream can disturb separation, so they should be removed after the operation. More importantly, the column did not have any (stationary phase) particles and relevant (micron-scale) chromatographic dimensions, and it seems difficult to apply the method in such (chromatography compatible) conditions.

Acoustic streaming is another active mechanism in which lateral flows are induced through the interaction of acoustic waves and the fluid. Bulk acoustic waves have been used to generate streaming vortices. When the acoustic waves travel along the channel surface or through the fluid, they create localized oscillations that produce time-averaged flow patterns as vortices. This acoustic streaming can be achieved by connecting a piezo-ceramic element (PZT) to the silicon chip to generate a standing wave (Figure 1c). These vortices can be extended across the entire cross-section of the channel under frequency matched conditions. The efficiency of vortex formation depends on the relationship between the acoustic wavelength and channel dimensions to ensure stable streaming patterns. An initial study on the effect of acoustic streaming resulted in more than a two-fold reduction (Figure 2a) and demonstrated that acoustically induced lateral convection is a viable strategy for dispersion reduction (Gelin et al., 2021). A recent study introduced depth matched acoustic resonance as a new approach, compared with the traditional width matching, generating lateral vortices across a 10 μm × 75 µm silicon channel (Naghdi et al., 2025). These acoustic-induced lateral vortices resulted in a 10-fold reduction in Taylor-Aris dispersion and demonstrate the scalability and compatibility of the method in micron-scale dimensions (Figure 2b).

Electroosmotic flow (EOF) occurs when an electric field is applied along a channel whose walls carry a surface charge, generating fluid movement near the electrical double layer. Unlike pressure-driven flow, which produces a parabolic velocity profile with a shear near the walls, EOF exhibits a nearly plug-like profile across the channel. While in traditionally EOF is being used to generate axial flow, it can also be employed to produce lateral (vortex) flows.

The electrode configuration affects vortex formation in EOF-driven microfluidic systems. The most straightforward co-planar configuration typically generates symmetric flow patterns that primarily enhance mixing near the channel walls, therewith limiting the mixing efficiency. On the other hand, vertically integrated electrode arrangements can induce multiple vortices with much more pronounced mixing throughout the entire channel cross-section (Figure 1d). Researchers compared regular and chaotic electrode configurations were compared and found that chaotic patterns provided promising dispersion reduction, especially at higher flow rates and lower adsorption constants (Venditti et al., 2022). Also, the computational simulations showed that secondary flows driven by electroosmosis can reduce dispersion more effectively than geometrical modifications alone (Zhao and Bau, 2007). Recent experimental studies have demonstrated dispersion reduction up to 5-fold in vertically patterned electrode configurations (Westerbeek et al., 2020) in micron-scale dimensions. In these studies, the electrode design is critical.

In the presence of a non-uniform electric field, EOF can be used to enable controlled vortex formation within the entire channel cross-section. This ability to actively shape the flow field makes EOF attractive for dispersion control in LC. Early studies on electroosmotic mixing focused on direct current (DC) EOF where a constant electric field drives the flow. Theoretical studies showed that combining DC-EOF with hydrodynamic flow in open-channel hydrodynamic chromatography would generate lateral vortices strong enough to reduce the separation time by nearly 50-fold (Figure 2c) (Biagioni and Cerbelli, 2022). This shows the potential of electrokinetic forces to perform significant performance gains. However, in a practical application, DC-EOF limitations include electrolysis at the electrodes, bubble formation, and changes in ionic strength due to Faradaic reactions. These effects can impact the performance over time and complicate the practical implementation.

In AC-EOF, the electric field oscillates in time, which prevents permanent accumulation of ions at the electrodes (and a concomitant reduction of the effective electrical field) and reduces the occurrence of Faradaic processes. This allows for higher field strengths while still avoiding electrochemical degradation. AC-EOF enables control over the frequency of the electric field while affecting the structure of the induced lateral vortices.

At low frequencies, the flow behaves similarly to DC-EOF. However the electric field cannot fully penetrate the electrical double layer, leading to complex flow patterns at high frequencies. Higher frequencies produce stronger and stable vortices. The voltage amplitude also plays a role, as higher voltages increase the slip velocity in the electrical double layer and enhance the secondary flow. Ionic strength is another critical parameter salt concentrations compress the electrical double layer, reducing the EOF. On the other hand, a very low ionic strength may destabilize the flow. Therefore, AC-EOF offers a rich parameter space with frequency, voltage, and ionic strength as key parameters that can be optimized to generate appropriate lateral flows under chromatographic conditions.

As a first reference system, chip-based open-tubular chromatography have been successfully evaluated. AC-EOF was used to generate transverse vortices in 40 × 20 µm (w x h) microchannels and reported a three-to five-fold reduction in the κaris parameter (Figure 2d) (Westerbeek et al., 2020). In a subsequent study, vortex-induced flow was used to reduce the C-term in 5 × 20 μm (w x h) channels by up to five-fold in 3 µm channels, demonstrating strong scalability to dimensions relevant for high-resolution separations (Figure 2e.) (Westerbeek et al., 2023). This analysis was then extended numerically to include unretained (Figure 2f) and retained (k > 0) conditions and found that even when analytes interact with the stationary phase, lateral vortices significantly reduce dispersion, more than 5-fold (Bihi et al., 2024). Vortex-induced EOF performance is depended on channel aspect ratio (Bihi et al., 2026) and the largest gains have been, obtained with square channels (AR = 1). These results show the versatility of EOF-based methods and their ability to target both axial velocity gradients and mass transfer resistance. Numerical simulations demonstrated that electroosmotically induced secondary flows could reduce dispersion beyond the limits of geometric optimization (Zhao and Bau, 2007). Experiments showed that AC EOF-induced lateral vortices lowered κ_{a ri s} up to 80% and reduce the C-term up to five-fold in open-tubular columns (Westerbeek et al., 2020; Westerbeek et al., 2023). The electrode patterning has also been explored to alternate between regular and chaotic vortex structures, allowing mixing strategies to be adjusted to different separation conditions (Venditti et al., 2022). These studies shows that vortex-based lateral mixing is not only effective but also highly controllable. GDT also provides a quantitative link between mixing strength and column efficiency while enabling a predictive design of these lateral mixing strategies. When combined with retention models, GDT can be used to determine the optimal vortex amplitude and frequency for different solute-surface interactions. These experimental and theoretical support distinguishes vortex-based methods from earlier approaches and highlights their potential as a foundation for next-generation LC systems.

Passive and active mixing strategies represent two distinct approaches for generating secondary flows. In passive approaches, secondary flows arise from the axial flow. As a result, vortex generation scales with axial velocity, and effective mixing typically requires higher flow rates, resulting in (inherent) increased axial dispersion. By contrast, active approaches generate secondary flows independently of the axial flow. This allows lateral transport to be adjusted at the same axial flow. This, however, requires additional system integration. In both cases, dispersion reduction is associated with enhanced transverse transport, achieved through different physical mechanisms.

Vortex-based active methods minimize dispersion by decoupling the lateral flow from the primary axial flow. Instead of relying on extreme miniaturization or geometric complexity, they use flow-field engineering to enhance mass transfer. This concept forms the basis of Vortex LC, where lateral vortices are deliberately integrated into the separation process. The ability to generate strong andtunable secondary flows makes vortex-based methods one of the promising directions for achieving high performance in liquid chromatography.

3 Conclusion

In this review, secondary flow-induced mechanisms in LC are discussed with an emphasis on Vortex Chromatography as an emerging approach to enhance mass transfer. Electroosmotic and acoustic mixing strategies have demonstrated up to an order of magnitude reduction in Taylor-Aris dispersion. Compared with passive methods, these techniques offer better control over the formation of lateral vortices and enable much higher separation efficiencies.

Despite these advances, most implementations remain limited to single channel configurations with limited loadability. Scalability toward high throughput designs may be challenging, as active mixing must be enabled in each individual channel or array. In acoustic systems, commercially available PZT actuators are typically limited to 20 MHz, which sets constraints on the channel dimensions. While porous structures are widely used in conventional liquid chromatography to increase surface area, their role in vortex chromatography and their interaction with vortex-driven transport have not yet been studied. A next step would involve experimentally examining retained-dispersion behavior in vortex chromatography. Future applications may also benefit macromolecular separations, where low diffusivity amplifies the Taylor-Aris dispersion. Vortex chromatography could thus open a new perspective for these analyses in miniaturized LC systems.

Vortex chromatography holds strong potential to reduce Taylor–Aris dispersion by more than one order of magnitude and represents a promising framework for next-generation, high-efficiency liquid chromatography systems.

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Footnotes

Edited by: Pedro Estrela, University of Bath, United Kingdom

Reviewed by: Makoto Tsunoda, The University of Tokyo, Japan