MSc Chemistry
Analytical Sciences
Literature Thesis
Alternative micromixer designs employing axial mixing for
reduced dwell volumes in liquid chromatography systems
by
J. D. Kruijswijk
UvA 11856864
VU 2628251
March 2020
12 EC
Supervisor/Examiner:
Examiner:
I
Samenvatting
Sinds de opkomst van UHPLC technologie zijn de totale volumes, geïntroduceerd door chromatografiesystemen, steeds kleiner geworden. Om deze reden is er de behoefte ontstaan om in de chromatografiesystemen de extra-kolom bandverbreding en dwell volumes te verminderen. Dit realiseert men door het verkleinen van het totale volume in het systeem. Quaternaire pompen introduceren, in vergelijking met binaire pompen, grotere dwell volumes. De mixer levert meestal de grootste bijdrage aan het totale volume en zorgt voor een homogene mobiele fase en continuïteit van een gradiënt. Door het werkingsprincipe van quaternaire pompen wordt er een axiale inhomogeniteit gecreëerd die inherent is voor dit type pomp. Hierdoor is een effectieve axiale of longitudinale mixer in het systeem nodig ten opzichte van de beter bestudeerde radiale mixers. Een aantal ontwerpen worden hier geïntroduceerd die allemaal verschillen in volume, dit zijn zowel passieve als actieve mixers. Voor quaternaire pompen is het niet bekend welk volume nodig is om effectief axiaal te mixen, alhoewel het waarschijnlijk is dat de besproken mixers een te klein volume bevatten. Het effect van het opschalen naar grotere volumes van de mixers op de prestatie is niet bestudeerd. De besproken onderzoeken zijn vooral gericht op de numerieke stromingsleer (computational fluid dynamics) om het proces van mixen te visualiseren en de radiale mixer prestatie te bepalen. Een vergelijking tussen de verschillende mixers is hierdoor moeilijk. Een nieuwe methode is beschreven voor een kwantitatieve beschrijving van een axiale mixer. De onderzoeken waren vooral gefocust op het mengen van waterige oplossingen bij één of twee lineaire stroomsnelheden, waardoor het vergelijken tussen de prestatie van de mixers in een chromatografiesysteem nog meer bemoeilijkt werd. Helaas is voor geen van de mixers de drukval gemeten. Het is mogelijk om, ondanks de kleine afmeting van de mixers (met uitzondering van één ontwerp), ge-3D-print te worden.
II
Summary
The total volume inherently introduced by chromatography systems have been decreasing due to the advent of UHPLC technology. As such, there is a need for reducing extra-column band-broadening and dwell volumes in chromatography systems by lowering the overall volume of the system. In comparison to binary pumps, quaternary pumps exhibit larger dwell volumes. The largest contributor to the volume is generally the mixer, which ensures a homogeneous mobile phase composition or a homogeneous continuity of a gradient. The working principle of the quaternary pump generates significant axial inhomogeneities in the mobile phase, requiring an effective axial, or longitudinal, mixer over the more thoroughly studied radial mixer. Here, a range of designs are introduced, both passive and active mixers, all considerably differing in volume. It is unknown what mixer volume is required for effective axial mixing in quaternary pumps. However, the volumes of most introduced mixers are likely too small and the effect of scaling up the dimensions on the mixer performance was not described. The discussed studies were mostly restricted to computational fluid dynamics for simulating the mixing process, or determining the radial mixing performance. Accordingly, a comparison of various designs is difficult. As a result, a new method for quantitatively describing the axial mixing performance is described. The researches were limited to aqueous solvents and one or two linear flow velocities, limiting the assessment of the performance of a mixer in a chromatographic system. Unfortunately, for none of the mixers the pressure drop was determined. Moreover, the feature sizes of all mixers, except one, were in the range suitable for 3D printing.
III
Abbreviations and symbols
Abbreviations Definition
AMI Absolute mixing index
AMN Alternating multinode
ASM Asymmetric serpentine mixer
CFD Computational fluid dynamics
GPV Gradient proportioning valve
HPG High-pressure-mixing gradient
HPLC High-performance liquid chromatography
LC Liquid chromatography
LPG Low-pressure-mixing gradient
MI Mixing index
P-SAR Planar split-and-recombine
RMI Relative mixing index
TDM Time-difference-type mixer
TFA Trifluoroacetic acid
UHPLC Ultra-high-performance liquid chromatography Symbols
µ Mean flow velocity (m∙s-1)
c̄ Mean pixel intensity
c0 Pixel intensity of non-mixed region
c̄0 Mean pixel intensity of non-mixed region
c∞ Pixel intensity when homogeneously mixed
ci Pixel intensity
D Mass diffusion coefficient (m2∙s-1)
De or κ Dean Number
DH Hydraulic Diameter (m)
H Section width of mixer
L0 Periodic length of axial inhomogeneity
N Number of sampling points
nx Mole number of a component
Pe Péclet number
QT Flow rate
Re Reynolds number
v Kinematic viscosity (m2∙s-1)
vol% Percentage by volume
Xs Segment or segregation index
Y Mole number ratio of consumed proton
IV
Table of content
Samenvatting ... I Summary ... II Abbreviations and symbols ... III Table of content ... IV
1. Introduction ... 1
2. Theoretical framework ... 2
2.1. Liquid chromatography system ... 2
2.2. HPLC mixing ... 4
2.3. Fluid characteristics ... 6
2.4. Types of micromixers ... 8
2.5. Mixing efficiency definitions ... 10
3. Passive micromixers ... 15
3.1. The axial rearrangement micromixer ... 15
3.2. Time-difference-type mixer ... 17
3.3. Picoliter-volume mixer ... 19
3.4. Spiral-shaped channel micromixer ... 21
3.5. Rapid-expansion channel micromixers ... 23
3.6. Modified Tesla structure-based micromixers ... 25
3.7. Multivortex micromixers ... 27
4. Active micromixers ... 30
4.1. Time-pulsing micromixing ... 30
4.2. Active magnetic spheres micromixer ... 31
4.3. Acoustically oscillating sharp-edges micromixer ... 33
4.4. Standing acoustic waves micromixer ... 34
5. Discussion ... 37
6. Conclusion ... 40
1
1. Introduction
Liquid chromatography (LC) separations are influenced by a variety of parameters, including the chromatographic column and the characteristics of the system employed to perform the separation. The column defines the selectivity of the separation and the theoretical efficiency. However, the system inherently causes extra-column band broadening by introducing additional volume, which reduces the achieved efficiency. Additionally, the dwell volume—the volume between the points at which mobile phase mixing occurs and column head—depends on the system, which affects the total analysis time for gradient separations. For conventional high-performance liquid chromatography (HPLC) systems, the contribution of the extra-column band broadening on the total band broadening is limited compared to the band broadening occurring within the column. The van Deemter equation indicates that one way to reduce the band broadening in the column is by utilizing smaller particle diameters. With smaller particles, higher backpressures are produced, thus demanding new pump and system technologies. As such, ultra-high-performance liquid chromatography (UHPLC) has been introduced as the successor of conventional HPLC, as it is faster, more efficient and has a reduced solvent consumption. As the intra-column band broadening was significantly lowered, the extra-column band broadening introduced by the system in relation to the total band broadening became substantial. Therefore, the overall volume (including the dwell volume) of the system was minimized, as it is proportional to the extra-column band broadening.
The dwell volume is mostly of influence for gradient separations. Although the efficiency is relatively unaffected by a decrease in dwell volume, the time at the start of a separation at which isocratic conditions occurs is lowered. The volume is minimized so a higher throughput is achieved, especially for shorter gradients. Short gradients are often employed in UHPLC methods and are essential for the second-dimension separations in a two-dimensional liquid chromatography setup. The largest contributor for the dwell volume is the mixer volume (and depending on the pump, the pump volume as well).
This thesis explores developments for novel designs of (micro-) mixers for lowering dwell volumes in quaternary LC pump systems. Quaternary pumps employ a gradient proportioning valve (GPV) to create gradients by introducing solvents as a sequence of plugs. Therefore, effective axial mixing must occur to create a homogeneous mobile phase or a continuous gradient. Mixer designs resulting in axial mixing are favored in comparison to radial mixing designs, as the latter have been more thoroughly studied. Ideally, the mixers are assessed based on mixer characteristics and physical properties; however, these parameters are poorly studied and discussed. Current quaternary pumps (e.g. Waters H-class instruments) have dwell volumes in the order of 300-400 µL, so new designs must aim to achieve lower volumes.
2
2. Theoretical framework
To lay down a proper foundation to discuss new mixer designs, several topics are introduced. First, two main types of chromatographic gradient-inducing pumps are examined, and at which position mixers are positioned within the system. Next, the requirement for radial and axial mixing in an HPLC system is described. Current mixer designs are introduced, which includes various fluid characteristics and dimensionless numbers for comparison of designs. The mixer performance can be expressed in both mixing efficiency within the channel and for its homogeneous continuity of a gradient in a chromatographic system. Therefore, various equations for calculating the mixing efficiency are introduced. Additionally, a method to assess the mobile phase formation and chromatographic performance of a mixer is discussed.
2.1. Liquid chromatography system
Before discussing mixers and new designs, it is important to describe the chromatographic system in more detail, including the function and position of mixers in the system. In general, two main types are produced by various manufacturers, which are commercially available: high-pressure-mixing and low-pressure-mixing LC systems [1–5]. A schematic of a high-pressure-mixing gradient (HPG) pump is displayed in Figure 1a. To establish a gradient using this design, two pumps are employed. This design typically allows for the formation of binary gradients, which leads to the design often being called a binary pump. In contrast to the low-pressure-mixing gradient (LPG) pump (Figure 1b), the HPG is generally more expensive as the former operates with only one pump. However, the LPG is capable of generating binary, ternary, and quaternary gradients. The principal difference between the two designs is the location of mixing the mobile phases to create the programmed composition.
Figure 1 The schematics of two designs of solvent delivery systems for chromatography purposes. (a) A
high-pressure mixing (HPG) pump, often called binary pump. (b) A low-high-pressure gradient (LPG) pump, often called quaternary pump. Figure adapted from [6].
For HPG pumps, gradients are created in the mixer located after the two pumps. The composition is determined by the flow of the two pumps. For example, a mobile phase mixture of 80/20 (vol%) mobile phase A/B at a flow of 1.0 mL/min is produced by combining the ratio of the appropriate flows of the respective pumps. Therefore, mobile phase A is delivered at 0.8 mL/min and mobile phase B at 0.2 mL/min, which are mixed in
high-3 pressure conditions and results in a flow of 1.0 mL/min, hence the name high-pressure-mixing gradient pump. In an HPG system, a gradient is formed by software that controls the ratio of the flow rates of the independent pump channels throughout the analysis. It should be noted that due to excess volumes, upon mixing of solvents commonly used for gradient elution (i.e. water and acetonitrile or methanol) the total volume is not equal to the sum of the partial volumes of each solvent employed [1,6]. Consequently, the flow in a LC system after mixing is not (necessarily) equal to the sum of both partial flows. When mixing water with acetonitrile or methanol, the sum of the partial volumes is lower than the total volume. This phenomenon depends on several conditions, such as the type of solvent and mixing ratio. The volume deviation is non-negligible in chromatography, as a contraction of up to 4% can be observed for a 40/60 water-methanol mixture. Even though the sum of the partial flow rates of both pumps equals 1 mL/min, the flow rate after mixing delivered to the column deviates by the volume excess. Therefore, the flow rate of the mixture under high-pressure conditions deviates continuously and proportionally to the volume excess throughout the programmed gradient.
Gradients are formed differently in LPG pumps, namely by a sequence of solvents produced by a software-controlled gradient proportioning valve in the low-pressure conditions prior to the single pump. The mobile phase composition is controlled by the GPV, which ensures the portioning and (partial) mixing of the individual solvents entering the pump. To create a mobile phase mixture of 80/20 (vol%) mobile phase A/B at 1.0 mL/min, the LPG system pumps at the programmed flow rate. The GPV portions the flow into the respective mixture conditions. Therefore, within one cycle time of the GPV, it allows channel A to be open for 80% of the time and channel B for 20%. Upon proper mixing, the mobile phase mixture is created. To create a gradient, the ratio of channel opening time is changed throughout the analysis. If proper mixing occurs in the low-pressure conditions, the excess-volume effects described for the HPG pump does not influence the LPG pump to the same extent. The main difference is that the flow rate of the portioned mobile phase is controlled by a single pump. If the mobile phase that is aspirated by the pump is homogenously mixed, a constant flow rate is attained throughout the programmed gradient. However, to maintain the constant flow, it is critical that the individual mobile phases are homogeneously mixed before being subjected to high-pressure conditions in the pump. Consequently, incomplete mixing in low-pressure conditions and further mixing in high-low-pressure conditions results in a decrease in flow rate due to the volume-excess effect.
For LPG systems, the key role of mixing is the continuous homogenization of the sequential aspirated individual mobile phases due to the functioning of the GPV. As such, the more demanding axial mixing (Chapter 2.2) is necessary, preferably in low-pressure conditions without affecting the aspiration cycle of the pump. Effective mixing devices are difficult to implement prior to the pump, as a significant flow restriction is inherently introduced. In high-pressure conditions, the flow restriction is easily overcome; however, prior to the pump, only the hydrostatic pressure is present if the solvent reservoirs are in an elevated position. If the solvents are stored 1 meter above the pump, a pressure of about 0.10 bar is available. In combination with atmospheric pressure, which results in a total pressure of 1.10 bar, it is insufficient for overcoming the fluid restriction intrinsically introduced by mixers. Therefore, the true mixer is positioned in the high-pressure conditions after the pump, to correct for inhomogeneities in the mobile phase composition. A mixer is often not
4 located prior to the pump, and the GPV and pump itself are considered to mix the mobile phase sufficiently. The degree of mixing needed to achieve a fully homogenous mixture (throughout a gradient) and to avoid the volume-excess effects described before is different. The mixing occurring in low-pressure conditions by the GPV and pump are sufficient to mitigate the solvent-excess effect and a deviating flow rate. However, the mixer placed in the high-pressure conditions further homogenizes the mobile phase for a proper operation of the system.
One system characteristic which has not been discussed is the dwell volumes introduced by the HPG and LPG pumps [7,8]. Inherent to all HPLC systems is a time delay for programmed gradients, viz. between the point of mobile phase formation and the arrival of the mixed mobile phase at the column inlet. The time delay is a measure of the dwell volume present between the point of gradient formation and the components up to the column head, defined by the quotient of the volume over the applied flow rate. Recalling Figure 1, the components inside the gray areas contribute to the overall dwell volumes of HPG and LPG pumps. It is noticeable that LPG pumps have a higher number of contributing components and consequently have larger dwell volumes than HPG pumps. Contributors to the dwell volumes are the mixer, connection tubing, and the injection system. For low-pressure gradient pumps, the dwell volume also includes the GPV and the gradient pump volume. Generally, the largest contributor to the dwell volume is the mixer. Therefore, the focus here is on the type of mixers, mixer performance, and lowering mixer volumes.
2.2. HPLC mixing
One critical component for producing proper mobile phases and gradients in HPLC systems is the mixer [1,5,9]. A mixer is essential to homogenize the individual solvents for the mobile phase or producing continuous gradients. To identify the specific function of a mixer, it is necessary to distinguish between radial and axial (or longitudinal) mixing. Figure 2 shows a simplification of the produced mobile phases of an HPG and LPG pump before arriving to a mixer. In an HPG pump, the mobile phase is formed by the ratio of two high-pressure flow rates, which creates a composition as observed in Figure 2A. The wavelike pattern of the mobile phase obtained before mixing is an effect of the software controlling the individual pump channels to minimize pressure ripples in the system. As the two parallel streams cause a radial inhomogeneity, radial mixing is required. Even though a mobile phase with a radial inhomogeneity arriving at the column inlet would rather effectively mix radially. However, if the column is not sufficiently long, the chromatographic performance is negatively affected. The requirement for axial mixing emerges from a different source of inhomogeneity. An example is visualized in the LPG pump schematic of Figure 2B for which this effect is inherently existent. For an LPG pump, the desired mobile phase composition is created by the GPV, which results in a sequence of alternating solvents in the direction of the flow rate, or longitudinal direction. Therefore, an axial inhomogeneity (or an inhomogeneity in the time domain) is created and requires effective axial mixing for homogenization of the mobile phase. At ambient conditions, the diffusivity of liquids is particularly limited, thus mixing of the different solvents by diffusion is insufficient and will not approximate a homogenous mobile phase. Due to changes in flow direction, mixing occurs in the GPV, tubing, and pump. However, complete mixing and eliminating the axial
5 inhomogeneity takes place in the mixer located after the pump. It is incorrect to assume that an HPG and LPG pump only requires radial and axial mixing, respectively. An LPG pump benefits from slight radial mixing, due to the Poiseuille flow profile. Additionally, the GPV produces a gradient by opening each solvent segment from a different direction, inducing a minor radial inhomogeneity. Ideally, the HPG pump would only require radial mixing if no pressure ripples occur in the system. In reality however, all pumps exhibit some degree of pressure pulses. The pressure ripples in the two pumps of the HPG system are not synchronized and as a result, a ripple in composition ensues. Figure 2 visualizes the effect in the HPG pump, i.e. the uneven intersection of the two individual flows before arriving to the mixer. Axial mixing is required to homogenize the mobile phase in the direction of the flow rate. The mixer is a key component of an HPLC system and influences the chromatographic results. Besides instrumental failures of the pump and GPV, insufficient mixing of the mobile phase or gradient can lead to poor chromatographic performance. Poor chromatography might be observed as excessively tailing peaks. More specifically, incomplete axial mixing is more often the culprit than issues in radial mixing.
Figure 2 Schematic of (A) HPG pump and (B) LPG pump. The inhomogeneities produced in each pump are
illustrated in the enlarged tubing before the mixer. Figure adapted from [10].
To make a comparison between radial and axial mixing, the latter is more demanding. To reiterate, in an HPG pump a mobile phase mixture is created by two parallel streams of solvent that requires radial prior to delivery to the column inlet (see Figure 3A for an
6 illustration of the mixture at the point of convergence). The distances the fluids have traveled to mix effectively are mostly limited in the radial direction, or the diameter of the tubing, which is in the range of 100-200 µm depending on the vendor and system. However, in an LPG pump a mobile phase mixture is created by a discontinuous sequence of alternating solvents requiring axial mixing, as displayed in Figure 3B. As such, in axial mixing the fluids travel relatively long distances in the longitudinal direction of the connecting tubing, which exceeds the tubing diameter (as indicated by the arrow in the figure). This distance depends on a number of factors, including flow rate, the programmed mixture composition, and the diameter of the tube connecting the GPV to the pump. To estimate the distances two plugs of solvent have to overcome to mix completely, the lengths could be calculated in tubing of average size. A 5 µL solvent plug would fit into 32 cm of tubing with an internal diameter of 200 µm, which strongly exceeds the distances of the diameter as needed for radial mixing. In an aqueous microfluidic device, the distance travelled by diffusion in one second is 1 µm, or 1,000 s for 1 mm [11]. This clearly indicates that diffusion alone is insufficient as the driving force for mixing. Therefore, it is important to decrease diffusion lengths, so rapid mixing can occur, while still being governed by diffusion processes.
Figure 3 Simplified schematic overview of the produced gradients in (A) HPG pump, and (B) LPG pump. The arrow
indicates the distances the fluids have to travel for mixing. Schematic adapted from [9].
2.3. Fluid characteristics
Before discussing mixing and types of mixers, it is important to define a few dimensionless numbers that describe the fluid characteristics within a device. The ‘dimensionless’ states that the number can be utilized for comparison of fluid mechanics regardless of scale [12]. The relevant numbers are selected to display related fluid mechanical effects. As such, the fluids in systems differing in length scales are behaving similarly when the numbers match. Therefore, they are often employed for evaluating the operating parameters and flow behavior. Two numbers in particular are often encountered for comparing microfluidics, namely the Reynolds and Péclet number [13]. The Reynolds number is characteristic for
7 predicting fluid flow patterns, it is the ratio of inertial to viscous forces in a flowing liquid [12,14]. The Reynolds number is defined in Equation 1, where u is the mean flow velocity (m∙s-1), DH is the hydraulic diameter (m), and ν is the kinematic viscosity (m2∙s-1):
𝑅𝑅𝑅𝑅 =𝑢𝑢𝐷𝐷𝜈𝜈𝐻𝐻 Eq. 1
A turbulent flow is expected at high Reynolds numbers, as the inertial forces, which produce flow instabilities, are dominant. For turbulent flow to occur, a critical value has to be reached which is dependent on channel geometry. However, for low Reynolds numbers a laminar flow is predicted as the viscous forces governs the flow. This is characterized in a smooth and stable flow, as few to no instabilities are generated. Mixing in turbulent flows is inherently greater than in laminar flows, due to the chaotic instabilities present in the flow. By interpreting the equation, low Reynolds numbers, and thus laminar flows, are observed in fluids with a high kinematic viscosity, slow flow velocity, and/or systems with a small hydraulic diameter. Particularly the small scale in micromixers causes flows to behave laminar. Increasing the flow velocity will increase the Reynolds number, until the flow becomes turbulent. No well-defined value is reported at which a laminar flow transitions into turbulent flow [14]. However, for a Reynolds number lower than 1500, a laminar flow is often assumed. Higher values do not necessarily indicate a turbulent flow. Instead, a (slow) transition regime occurs from laminar to turbulent flow, in the transition both laminar and turbulent regions are present. Although, for Reynolds numbers exceeding 2300, turbulent flows are expected.
The Péclet number correlates the advective and diffusive transport rates [12,15]. Advection describes the transport of molecules within a bulk flow of a fluid due to a pressure gradient. In contrast, diffusion is the dispersion of molecules within a bulk due to a gradient (e.g. concentration or temperature), resulting in no net movement of the bulk. Diffusion inherently results in mixing of different fluid components. The term advection should not be confused with convection, which is used for the combination of both advection and diffusion processes. The Péclet number is defined in Equation 2, where u is the mean flow velocity (m∙s-1), DH is the hydraulic diameter (m), and D is the mass diffusion coefficient (m2∙s-1):
𝑃𝑃𝑅𝑅 =𝑢𝑢𝐷𝐷𝐻𝐻
𝐷𝐷 Eq. 2
In words, the equation is the ratio of the advective transport rate over the diffusive transport rate. Therefore, the equation is employed to determine if the rate of diffusion is comparable to the rate of advection. A Péclet number of 1 indicates that both the advective and diffusive transport are equal. A much lower number denotes that the diffusive transport rate dominates the advective transport rate and vice versa. The diffusion coefficient is a constant property related to the fluid and the hydraulic diameter a parameter of the system. Therefore, the flow velocity is the adjustable parameter within a system that affects the Péclet number. The Péclet number is directly proportional to the Reynolds number by the ratio of the kinematic viscosity (ν) over the diffusion coefficient (D), also called the Schmidt number. Therefore, the Péclet number, like the Reynolds number, is not a material constant, as both numbers are determined by the flow velocity and hydraulic diameter. The mixing
8 efficiency of various mixer designs can be assessed at various Reynolds and/or Péclet numbers. As the numbers are proportional to the flow rate and dimensionless, thus independent of the system scale, both numbers can be employed as operating parameters and are useful for quantitatively comparing various mixer designs.
2.4. Types of micromixers
After describing mixing and the dimensionless numbers, it is important to discuss mixer design and classification. The key difference between designs is the distinction of two types, namely passive and active mixers. Do note that in the context of this report, the discussed mixers are often classified as micromixers. Active, or dynamic, mixers are differentiated by requiring an external input of energy to perturb the solvents to enable mixing, by generating chaotic advection or increasing the contact area. The external energy generally involves moving components or various gradients in the mixing region. Active mixers are differentiated by the method of introducing the external energy [16]. The categorization includes pressure-field driven, electrical-field driven, acoustic-field driven, magnetic-field driven, and thermal-field driven mixers. In contrast, the passive, or static, mixers do not employ an external energy input, except for the pumping energy for the fluid to flow through the system. A distinction within the passive mixers can be made, namely by two-dimensional or three-two-dimensional structures. The dimensions are subdivided as well, which includes a range of structures and geometries, which typically modify the flow to decrease diffusion distances and increase contact areas between the liquids by inducing chaotic advection, flow lamination processes, and/or splitting-and-recombination processes. The fabrication and miniaturization of dynamic mixers is costly, and its integration into other systems and devices is challenging. Additionally, the external forces can be complicated to control in the dimensions of the microfluidic devices. As static mixers are more robust and easier to fabricate, these are more common and the only type of mixers employed in liquid chromatography systems. Each vendor employs varying technical and geometrical designs. None of the described mixers is defined in their functioning as a radial or axial mixer, or as a combination.
One classical static mixer utilized in an HPLC system is the mixing column, namely a typical HPLC column packed with 100-200 µm inert non-porous, silica beads [17]. As described by the A-term (parameter for eddy diffusion) in the van Deemter theory, larger particles in a column cause more turbulence, which corresponds to stronger axial mixing. In a separation column, dispersion is undesirable, while for a mixer this effect is required. Although packing the column with 100-200 µm particles is an effective mixing device, a mixer column is a relatively inadequate axial mixer in relation to its volume. Packing columns with diameters smaller than 2 mm, with 100-200 µm particles is ill advised. Packing columns with particles larger than a tenth of the channel diameter is unfeasible with regular packing methods. Additionally, reducing the column length negatively alters the mixing efficiency, thus a mixing column requires a typical length of 10 cm. Therefore, a column with a diameter of 2 mm and a length of 10 cm produces a volume of 200 µL, which does depend on the packing density. Current (U)HPLC systems typically do not employ mixer columns.
In contrast, cylindrical frit-based mixers are devices that are utilized in certain modern (U)HPLC systems and are available in various materials, e.g. stainless steel or nonferrous
9 alloys [5]. These mixers are marketed in various sizes, ranging from sub-100 µL to several mL. However, the smallest kinds perform more as filter frits than as mixers. A frit-based mixer is able to perform as both a radial and axial mixer, though it is more effective with regards to axial mixing. A suitable morphology of the material and a well-designed geometry of the frit are required for proper mixing, and in- and outflow, respectively. Significant axial mixing effects, essential in liquid chromatography systems, are difficult to attain at frit volumes lower than 100 µL, irrespective of design. Frit mixers with volumes ranging between 200 and 800 µL are typically the most effective in terms of volume and performance.
One version of a frit-based mixer is the one visualized in Figure 4, produced by Thermo Scientific [5,18]. The mixer combines both the radial and axial mixing principles in the first and second stage, respectively. The first stage that induces radial mixing contains a helical structure within the capillary, while the cylindrical frit-based mixer that follows ensures axial mixing. To discuss the basics for proper axial mixing, the second stage of Figure 4 is quite illustrative. For effective axial mixing to occur, it is essential that the liquid is able to flow through a range of paths differing in length, though it must be ensured that all paths are taken with equal probability. Therefore, by overcoming the distance in the device, the solvents travelling through the different pathways have varying flow-through times. As a result of the recombining of the various solvent flows into one flow, an axial mixing effect occurs. The second stage was carefully optimized in terms of aspect ratio. The different path lengths are decreased in a smaller diameter of the frit-based mixer, which causes less efficient axial mixing for a given length of the device. However, if the diameter of the device becomes too large, it is probable that the solvent will not flow to and through the channels near the outer surface of the device. Upon further increasing the diameter, scaling issues arise in terms of mixing efficiency and volume.
Figure 4 Schematic of a Dionex SpinFlow mixing design employed in Thermo Scientific products. The mixer
consists of two stages, combining a capillary with a helical structure embedded for radial mixing and a frit mixer for axial mixing in the second stage. Figure taken from [19].
Fundamentally, a packed mixer column and a frit-based mixer operate similarly; both introduce a randomized range of flow paths that differ in length. Therefore, the surface area between fluids is increased, especially in the axial direction. In turn, the diffusion distances are decreased and mixing can occur more rapidly. This is in contrast to other devices that
10 contain defined discrete flow channels in which a relatively steady partial flow distribution occurs. Thus, a device that operates by splitting the solvent stream into different sub-flows, then directing those along fixed, distinct paths differing in length and recombining them is often named a microfluidic mixer. Do note that the different paths and their lengths in microfluidic devices are often defined instead of random as mentioned in frit-based mixer and mixer columns. A carefully designed microfluidic mixer demonstrates effective mixing properties even at small volumes, already producing axial homogeneous mobile phases at volumes below 0.100 mL [1]. One requirement in microfluidic devices is the need for an inline frit, to prevent particulates from entering the channels that can lead to blockages, increased backpressures, and diminished mixing efficiencies. In general, the devices are fairly difficult and costly to manufacture and are restricted in scalability of producing a range of volumes each with an altered mixing efficiency.
2.5. Mixing efficiency definitions
To discuss, evaluate, and compare mixers, one must define the method of characterizing the mixer performance. This includes the extent of mixing, and as it is used in LC, it is important to discuss its performance in mixing realistic mobile phase commonly employed in (U)HPLC analysis. This would include the homogeneity of a mobile phase and the continuity of a gradient, especially in the presence of additives (e.g. tetrafluoroacetic acid). A crucial parameter for the performance of micromixers is the mixing index or efficiency. Hasmi and Xu reviewed several equations for defining the mixing efficiency, which will be discussed here [20]. To follow the extent of mixing of the fluids, the pixel signal intensities of a grey-scale image, obtained through an optical transparent material, is acquired at certain points of, or throughout, the mixer device. A simple method is basing the mixing index on the signal intensity differences of the fluids, and thus their separation [21]. An early equation of the mixing index was based on the standard deviation of the pixel intensities, as described by Equation 3:
Here, MI is the mixing index, ci is the pixel intensity, c̄ is the mean intensity, and N is the
number of sampling points. A mixing index of zero would be obtained for a device capable of homogeneously mixing the solvents. A maximum value is obtained for an unmixed state, which depends on the data. The standard deviation is a measure of the spread of data, without being a direct quantity for the mixing, as it is not dimensionless. As a result, this MI cannot be used for comparison purposes between different mixer designs. An altered, and arguable improved, dimensionless mixing index [22] is based on the quotient of the standard deviation of the intensity over the mean intensity, and introduced in Equation 4:
𝑀𝑀𝑀𝑀 = �𝑁𝑁 �(𝑐𝑐1 𝑖𝑖− 𝑐𝑐̅)2 𝑁𝑁 𝑖𝑖=1 Eq. 3 𝐴𝐴𝑀𝑀𝑀𝑀 = �1𝑁𝑁∑ (𝑐𝑐𝑖𝑖− 𝑐𝑐̅) 2 𝑁𝑁 𝑖𝑖=1 𝑐𝑐̅ Eq. 4
11 Here, AMI is the absolute mixing index. This definition of the mixing index is based on a zero-to-one scale, for a 50/50 mixture, where zero describes fully homogeneously mixed conditions and one unmixed conditions. The previous only holds true if one of the two fluids completely absorbs the light, thus achieving a minimum pixel intensity of zero. The AMI is a more direct quantity of the mixing index, though the AMI is still suitable for comparing varying devices. The values are highly dependent on the experimental details, such as lighting conditions and the UV/Vis-active compounds employed. One could modify the data by scaling and normalizing the data, making data processing and comparison laborious. Instead, the relative mixing index is introduced, which no longer requires pretreatment of the data [23]. The relative mixing index (RMI) is defined in Equation 5:
Here, c0 is the pixel intensity in the non-mixing section and c̄0 is the mean pixel intensity in non-mixing section. The same scale as described before applies, though a comparison between devices is possible. However, the ratio is quite difficult to assess intuitively. Alternatively, the RMI could be quantified as the percentage of (1 minus the ratio), which results in a mixing efficiency. Therefore, a mixing efficiency of 0% describes unmixed conditions and 100% fully homogeneously mixed conditions. The last mixing index is one based on the ratio of the integral calculation of the pixel intensities in the mixed and unmixed conditions within a device [11,24]. The definition is described in Equation 6:
Here, H is the section width of the mixer, and c∞ the pixel intensity when homogeneously
mixed. A mixing index of one indicates fully homogenously mixed conditions, while a value of one denotes unmixed conditions.
2.5.1. Villermaux-Dushman reaction protocol
Another method which is applied for the characterization of the mixing efficiency of a device is the Villermaux-Dushman reaction protocol, also called the iodide-iodate method [25–27]. In contrast to the previously described methods, the Villermaux-Dushman reaction scheme does not require the in-situ monitoring of the mixing. The protocol involves two reactions working in parallel, namely a neutralization and redox reactions that are in competition for a proton from a strong acid. The neutralization reaction, shown in Reaction 1 below, is quasi-instantaneous, contrary to the redox reaction (Reaction 2) which is fast and thus in relative terms slow. 𝑅𝑅𝑀𝑀𝑀𝑀 = �1𝑁𝑁∑ (𝑐𝑐𝑖𝑖− 𝑐𝑐̅) 2 𝑁𝑁 𝑖𝑖=1 �1𝑁𝑁∑ (𝑐𝑐𝑁𝑁𝑖𝑖=1 0− 𝑐𝑐� )0 2 Eq. 5 𝑀𝑀𝑀𝑀 = 1 −∫ |𝑐𝑐𝑖𝑖− 𝑐𝑐∞| 𝐻𝐻 0 𝑑𝑑𝑑𝑑 ∫ |𝑐𝑐0𝐻𝐻 0− 𝑐𝑐∞|𝑑𝑑𝑑𝑑 Eq. 6
12
H2BO3- + H+ → H3BO3 (quasi-instantaneous) (1)
5 I- + IO3- + 6 H+ → 3 I2 + 3 H2O (fast) (2)
I2 + I- ↔ I3- (quasi-instantaneous) (3)
For the method, a mixture of iodate and iodine in a slightly buffered alkaline-boric acid solution is mixed with a strong acid for the production of protons, often (diluted) sulfuric acid. One requirement is that the concentration of the acid needed for the neutralization of the borate ions should be a stoichiometric amount or be the limiting reagent. The concentration of the acid should never exceed the critical stoichiometric amount. In perfectly mixed and stoichiometrically-fitted conditions, the protons are fully consumed by the instantaneous neutralization reaction with the borate ions (Reaction 1). Therefore, the relatively slower Reaction 2 cannot occur and iodine is not formed. In contrast, if mixing is incomplete or slow, segregated regions of acidic and alkaline conditions appear. Within the acidic segregates, the concentration of protons exceeds the borate ions, the latter becomes depleted, and Reaction 1 is completed. The unreacted protons are consumed by the redox reaction (Reaction 2), causing the irreversible formation of iodine. The iodine is further converted to triiodide as given by Reaction 3. The triiodide can be straightforwardly measured using an UV spectrophotometer with a wavelength of 353 nm. A low absorbance indicates a low concentration of triiodide and therefore a high mixing efficiency.
For a more quantitative approach a segment or segregation index, Xs, can be calculated by
Equation 7 [28]:
Here, Y is the ratio of the mole number of the protons consumed by the slow, redox reaction and forming the iodine and triiodide products, and the total protons present. In Equation 8 the calculation for Y is given, where n is the mole number of the respective constituent of the reaction:
The superscript 0 signifies the initial reaction conditions. In Equation 7, the component YST is
described as the value of Y in an event where mixing is infinitely slow and thus fully unmixed or segregated conditions are maintained. As such, the neutralization and redox reaction appear both quasi-instantaneous in comparison to the mixing time. Therefore, the protons are depleted corresponding to the local concentrations of the borate ions and iodide/iodate buffer solution. Equation 9 defines the value of YST, where the brackets indicate the molar
concentration of the reaction components at the initial reaction conditions: 𝑋𝑋𝑠𝑠=𝑌𝑌𝑌𝑌 𝑆𝑆𝑆𝑆 Eq. 7 𝑌𝑌 = 2𝑛𝑛𝐼𝐼2+ 𝑛𝑛𝐼𝐼3− 𝑛𝑛𝐻𝐻+0 Eq. 8 𝑌𝑌𝑆𝑆𝑆𝑆 = 6[𝑀𝑀𝐼𝐼3 −]0 6[𝑀𝑀𝐼𝐼3−]0+ [𝐻𝐻2𝐼𝐼3−]0 Eq. 9
13 The values of the segregation index ranges between zero and one. An efficient mixing process is describes by a low value of Xs, as it indicates fast or homogeneous mixing, or a
combination of the two. Consequently, a segregation index of zero is obtained for highly efficient micromixing systems and a value of one for fully unmixed conditions.
Substituting the latter two equations into Equation 7 and rewriting both to equal terms, the following is obtained:
Here, the brackets indicate the molar concentration of the respective reaction components, n is the molar flow rate, QT is the flow rate, and the superscript 0 indicates concentrations of
the components at the initial reaction conditions. 2.5.2. Chromatographic performance
Besides the formulations of the mixing efficiency, the effect of the mixer on the chromatographic performance must be discussed as well. The homogeneity of the mobile phase in the axial direction is of importance for the composition ripple, which is the longitudinal continuity of the mobile phase. The mixer is a strong contributor to the composition ripple, though it is also affected by the intrinsic properties of the pump and/or GPV. Methods employing tetrafluoroacetic acid (TFA) as an additive to the mobile phases are highly sensitive to composition ripples that are observed by an UV detector [9]. For example, proteomics studies often use TFA in both the aqueous and organic mobile phase. As a strong acid, TFA completely dissociates in aqueous solutions. Simultaneously, the trifluoromethyl group causes the compound to be very hydrophobic, even as a conjugate base. As such, TFA strongly interacts with hydrophobic stationary phases in reversed-phase columns. This characteristic of TFA is beneficial for the retention of amphiphilic or hydrophilic cationic mixtures. Regardless of the actual retention behavior of the TFA and the analyte, either as a dynamic cation exchange or ion-pairing agent, the retention of the TFA on a reversed-phase column is controlled by the stronger eluotropic mobile phase, namely acetonitrile. An inconsistent mobile-phase composition of water and acetonitrile that varies in time (and longitudinally in space) corresponds to a discontinuity in concentration of TFA. As a result of the composition ripple, the TFA is enriched on and desorbed from the reversed-phase column in a continuous sequence. A lower acetonitrile composition indicates enrichment and a high acetonitrile content denotes mobilization of the enriched TFA region. At the end of the column, the elution of an enriched zone results in a concentrated TFA peak, this is detected by the UV detector as a composition ripple. Already a signal of the composition ripple can be obtained from an isocratic run of 0.1% TFA in 95/5% water and acetonitrile, as visualized in Figure 5. A comparison of the composition ripple performed with and without a reversed-phase column is presented in the results. The rippled baseline is clearly originating from the presence of the column, likely due to the interactions of the TFA with the column stationary phase.
𝑋𝑋𝑠𝑠=([𝑀𝑀2] + [𝑀𝑀3 −])𝑄𝑄 𝑆𝑆 𝑛𝑛𝐻𝐻+0 ∙ 6[𝑀𝑀𝐼𝐼3−]0+ [𝐻𝐻2𝐼𝐼3−]0 3[𝑀𝑀𝐼𝐼3−]0 Eq. 10
14
Figure 5 Composition ripple comparison as observed in a TFA-based application on a reversed-phase column.
Isocratic conditions of 95/5% water and acetonitrile, with 0.1% TFA added to both solvents. Figure taken from [1].
The composition ripple is dependent on the mixer volume and its efficiency in mixing longitudinally. Therefore, to evaluate an alternative mixer design the composition ripple of a TFA-based method can be assessed. Compared to larger volume mixers, a more effective longitudinal mixer results in reduced composition ripples, i.e. lower amplitude of the baseline ripple of the UV detector, with lower volumes.
15
3. Passive micromixers
3.1. The axial rearrangement micromixer
Goovaerts et al. [29] discussed an axial rearrangement mixer. The same mixing principle is utilized in the Jet Weaver mixer (Figure 6), a product of Agilent Technologies [30,31]. To mix the mobile phase longitudinally, the design takes advantage of axial dispersion besides diffusion alone. The mixer is fabricated with two distinct segments, namely the distributor and the mixing section (Figure 7A). The distributor comprises of 12 narrow channels, which are equal in length and have a height of 200 µm and a width of 250 µm. This section is the pressure-drop-determining component of the device. The channels in the mixing section differ in length and to establish an equally distributed flow in each mixing channel, the distributing channels are required to be uniformly long and narrow. This geometry of the distribution section results in it being the main contributor to the pressure drop and ensures an equivalent flow rate in all channels. As shown in Figure 7B, the mixing channels contain pots that add additional mixer volume (width of 300 µm and a height of 500 µm). The extra volume causes an increased difference in the residence times of the individual channels. As a result, a particular plug of solvent entering the mixer is distributed over a range of times, thus causing axial dispersion. The diameter of the mixing channels is relatively large to keep the pressure drop low. Moreover, the pots reduce the pressure drop even more, ensuring that the distributor is the main contributor to the pressure drop. Accordingly, the difference in volumes of the mixing channels causes the mobile phase to be axially rearranged, resulting in axial mixing.
Figure 6 A schematic design of a Jet Weaver Mixer, a
product of Agilent Technologies [30]. Figure 7 (A) An upper view of the mixer design. (B) A schematic representation of the mixing channels (1) connected to the pots (2). Figure taken from [29].
The mixing behavior was modelled and experimentally tested by using a single mixing channel at various flow rates. The single channel consisted of twelve alternating channels and pots, and fluorescence signals were obtained after injecting a fluorescent tracer. The
16 results revealed that after eight repeating channels and pots, the mixing was incomplete for both slow and fast flow rates, as the signal is quite defined in contrast to a flat profile expected for effective axial mixing (Figure 8). The tested flow rates were quite low from 50 to 400 µL/min and the results were normalized for easier comparison. However, the flow in the device is distributed over twelve channels, thus the total flow reaches the typical upper limit used in chromatography. As the flow is laminar, mixing is dominated by diffusion and increasing the flow rate results in less time for diffusion processes to occur. As observed in Figure 8, at increased flow rates, the obtained signals are comparable and mixing hardly takes place. As diffusion times are increased for lower flow rates, the signals are flatter and broader. In terms of mixing within a single channel, it is more efficient at lower flow rates.
Figure 8 The normalized signals obtained for various flow rates. The total areas were equalized for comparison
purposes. Figure reproduced from [29].
The device was modelled by examining the individual mixing channels that vary in length. The concentration profile for all 12 mixing channels were modelled, which were then summed to obtain the total outgoing profile of the mixer (Figure 9). The signal in each subsequent channel is more broad as increased mixing times are achieved in the longer channels, which allows for longer diffusion times and more mixing to occur. Besides the initial peak, the total signal is more longitudinally and homogeneously dispersed compared to the original plug entering the device. The authors of the paper recommended several changes to the design that would mitigate the initial high peak. By adding one channel and pot to the mixing channel length, the initial peak would be diminished. Additionally, a decrease in the overlap of the signal between neighboring channels is achieved that results in a higher mixing efficiency at the cost of a slight volume increase.
The current Jet Weaver mixers are employed in the binary (U)HPLC systems of Agilent, with a volume of 35 or 100 µL. It is postulated that the scalability of the device is possible by designing larger pots, or by adding more and longer channels. As is the case for all microfluidic systems, it is vital that the delivered solvents do not precipitate and are free of particles to prevent any blockages within the device, which strongly negatively affects the mixing characteristics.
17
Figure 9 The concentration profile of each mixing channel and the summed total signal. Results replicated from
[29].
3.2. Time-difference-type mixer
A time-difference-type mixer (TDM) is discussed by Hanada et al. [32,33] that is specifically utilized for in-line axial mixing, typically in industrial processes. The principle of operation relies on splitting the initial concentration profile into several branch paths, which are recombined later in the device. The time spent within each branch varies; hence, the recombination occurs sequentially and at various time-points. A schematic of the device is visualized in Figure 10. The figure includes the theoretical final conditions of the concentration profile (Figure 10B). The authors compared the mixing process to the principle of a moving average—a data-smoothing method by averaging values over a time period—as mixing occurs in the direction of the flow and thus temporally. The actual design of the device is displayed in Figure 11. The device comprises of a center flow path, surrounded by a spiral-groove channel embedded in the mixing element. The flow paths are interconnected by several periodically located branch paths. The solvent follows the path of the spiral groove encased by the housing, thus creating the main flow path. At each branch path, the solvent starts to flow into the relatively shorter central flow path towards the outlet, while the remainder continues its path in the spiral channel. Therefore, the residence time in the main channel after each branch path increases in comparison to the central path, resulting in a time difference. The solvent entering the central channel earlier, or farther upstream, exits the device the fastest and vice versa. As a result, an axial inhomogeneity is averaged in the direction of the flow, thus mixing in the longitudinal direction. The principle and final result of this mixer resembles the axial rearrangement mixer described before. Both operate by splitting the flow and distribute the flow over varying path lengths, which is then recombined.
18
Figure 10 Schematic visualization of a TDM,
reproduced from [33].
Figure 11 Design and structure of a TDM. (a) The mixing
element, and (b) the cross-section displaying the flow paths. Figure adapted from [32].
A tracer was introduced into the system to evaluate the axial mixing of the device. The pulse width of the tracer was 0.1 s at a linear velocity of 1 m/s. A clear broadening of the signal in all models of the TDM is observed in Figure 12A. For TDM No. 3 in particular, a Gaussian shape of the concentration profile is approximated, with a significant broad shape in comparison to the tracer pulse width. For this model, the individual concentration profiles at each branch path are displayed in Figure 12B. The profile in each repeating branch path is broader than before, due to the longer residence times, contributing to axial mixing. Moreover, the concentration profiles matches the ones obtained for the axial rearrangement mixer in Figure 9.
Figure 12 (a) Concentration profile of several TDM designs after a single injection of a tracer. (b) The individual
concentration profile at the branch paths in the design of TDM No. 3. Figure reproduced from [33].
To test the TDM mixer further, a tracer was continuously injected with a one-second interval. Figure 13A visualizes the tracer signal for two improved TDM mixers, showing an increased and continuous broadening of the tracer, while reducing the signal height. The latest version of the TDM mixer was compared to a standard mixer often employed in industry, namely the Kenics-type mixer known for its efficient radial mixing performance, in Figure 13B. In comparison to the Kenics-type mixer, the TDM significantly reduces the signal height of the tracer, by producing broader peaks. As such, the signal is temporally averaged.
19
Figure 13 Output concentration profile of sequential injection of a tracer. (a) Comparison of two TDM designs, and
(b) comparison of TDM No. 6 with a Kenics-type mixer, a standard industrial mixer. Figure taken from [33].
As mentioned before, the TDM is a mixer designed for industrial applications, with an approximated volume of 75 mL. However, by downscaling the device, it might be useful as an axial mixer in an LC system. Moreover, the volume could be modified by the number and outer diameter of the spirals. The device is typically operated under turbulent conditions, though instances of laminar flow occurred which did not deteriorate the concentration profiles. An important parameter for mixers is the pressure drop. However, the mixer was designed and optimized for reducing pressures. It was assumed that radial inhomogeneities were nonexistent, which would strongly affect the chromatographic performance. Therefore, further studies are required.
3.3. Picoliter-volume mixer
A picoliter-volume mixer was proposed by He et al. [34], designed as an extremely low volume mixer for electroosmotic flows. Initially, the mixing behavior was modelled for miniaturized particle-type bed mixers, also called a packed-bed mixer, and resembles a mixing column described in Chapter 2.4. For these mixers, transchannel coupling is the main process that induces mixing. Transchannel coupling occurs when two bordering flow paths merge after being split by a particle in the channel; the process is visualized in Figure 14A describing the process in a packed-bed mixer that is a single particle deep. Packed-bed mixers are generally efficient axial mixers. Although, radial mixing occurs steadily as the fluid is continuously split along the main flow path of the particle bed. For fabricated micromixers, the structure of the bed is displayed in Figure 14B. The same extent of mixing would occur as before. To decrease the influence of transchannel coupling, heterogeneities could be introduced to the packed bed, namely the density of the packing material of particle size. The heterogeneity introduces a range of channel sizes and in pressure-driven systems, both the radial and axial mixing performance would increase. Alternatively, by carefully arranged voids placed within the patterned-packed bed, an amplification of the mixing performance occurs (Figure 14C and D). The authors only studied the radial mixing performance, though the patterning could be optimizing for longitudinal mixing as well. Even in the proposed mixture (Figure 14D), longitudinal mixing occurs. The flow is effectively split at the small channel intersections. The main flow is essentially meandering throughout the
20 device, while the fluid in the narrower channels follows the longitudinal direction. Varying channel lengths introduces different residence times, while the main flow could take the longest time to traverse the device. Upon merging within the main channel, an axial rearrangement of the flow has occurred, which enhances axial mixing.
Figure 14 A particle-type bed mixer. (A) A schematic of a packed-bed mixer and the process of mixing by
transchannel coupling. The two fluids flow through the column and start to mix in the darkened section. (B) A microfabricated mixer on a chip. (C) A microfabricated mixer with arranged voids, enhancing the mixing effect. (D) The final design of the authors used in the study. Figure replicated from [34].
The radial mixing performance is qualitatively assessed for an electroosmotic flow at a linear velocity of 0.300 mm/s. A fluorescent tracer is added to one of the buffers to be mixed and two concurrent streams flow into the mixer as in Figure 15A. At the outlet the fluorescence signal is obtained for the cross-section (Figure 15B). If no mixing occurs, a distinct difference in signal intensity would be acquired for the clearly defined and unmixed buffers. Instead, a relatively flat profile is obtained, which indicates that nearly complete radial mixing has occurred.
In contrast to the TDM device described before, the volume of this mixer is very small at 100 pL. Even if the mixer would perform well as an axial mixer, it is significantly too small, as axial inhomogeneities in an (U)HPLC system are in the microliter range. However, the device appears scalable in terms of channel design and size. The linear flow velocities typically employed in liquid chromatography are one order of magnitude larger. Although, the mixing principle is independent of flow rate, as long as the flows are laminar. The experimented flow was electroosmotically driven, though the authors postulate that the design can be utilized for the preparation of mobile phases in pressure-driven LC systems.
21
Figure 15 The mixing of a two buffer solutions, one with and one without a fluorescein tracer in mixer with a width
of 100 µm, a height of 200 µm, and a depth of 10 µm. (A) Mixing of the two buffers within the device. (B) A relatively flat fluorescence concentration profile in the radial direction, obtained at the outlet. The flat profile indicates complete radial mixing. Figure taken from [34].
3.4. Spiral-shaped channel micromixer
A micromixer for reducing the periodic pump pulsation present in HPG pumps, which causes longitudinal fluctuations in the mobile phase composition, is designed by Tsukada et al. [35]. The principle of operation relies on utilizing multiple branched channels that differ in residence times. Moreover, the architecture of the channel produced a counteracting effect for the axial inhomogeneity. The basic structure of the micromixer is displayed in Figure 16A, where L0 is the periodic length of the longitudinal inhomogeneity in a system. Here, the
flow is bifurcated into two channels of unequal length. The length of channel 1 is significantly shorter than L0, while channel 2 has a length of one-half of L0. As a result, the fluids
experiences different, although defined, residence times in the two channels. In this design, upon recombination the axial fluctuation in concentration is the average of both channel outlets. In an HPG pump, the axial inhomogeneity is composed of a basic frequency and its harmonic overtones, which is a result of the pump pulsation caused by the pump mechanism. Therefore, a mixer is created with a series of branched-channel units varying in residence times, matching the various frequencies (Figure 16B). After passing a unit, one defined frequency of the axial inhomogeneityshould be diminished.
Figure 16 (A) Basic structure and principle of the spiral-shaped channel micromixer. L0 is defined as the periodic length of the longitudinal fluctuation in a channel. (B) Micromixer structure with serially cascaded channel units. Figure adapted from [35].
22 The three-dimensional structure of the micromixer is displayed in Figure 17A. The design is produced in several layers; each layer is visualized in Figure 17B for an easier interpretation of the various flow paths. The assumption was that the flow rates are equal in both channels. However, especially for higher total flow rates, a deviation occurs and the flow rate in channel 2 becomes higher than in channel 1. This effect is mostly caused by the flow resistance in curved channels, which is a function of flow rate and channel curvature.
Figure 17 (A) 3D-design of the spiral-shaped channel micromixer. (B) Overview of the individual sections within
the micromixer, the arrows indicate the direction of the flow. The figure is reproduced from [35].
The method for evaluating the axial mixing performance is studied by a TFA-method on a reversed-phase column that is also described in Chapter 2.5.2. The amplitude of the absorbance is a measure of the effectiveness of the mixer for mixing in the longitudinal direction. Figure 18 displays the results obtained for two flow rates, with and without a mixer present. The amplitude is clearly decreased, indicating a more axially homogenous mobile phase composition, thus less concentration fluctuations. The axial fluctuations decreased to 22 and 18% at flow rates of 0.500 and 1.000 mL/min, respectively.
Figure 18 The absorbance signal obtained of the resulting mixture for both with and without a mixer present at
23 The volume of the described spiral-shaped channel micromixer is 53 µL and was able to notably reduce axial inhomogeneities that were spread over a volume of 100 µL. The mixer was specifically designed for reducing selected frequencies of HPG pump pulsations. Few flow paths were employed, perhaps more are needed, at the cost of volume, that can be employed for more randomized axial inhomogeneities, e.g. for homogenizing the continuity in gradients in LPG pumps. As the flows are concurrently recombined, it is unknown if further radial mixing is required.
3.5. Rapid-expansion channel micromixers
Previously discussed micromixers relied on a split-and-recombine principle for mixing. In contrast, Coleman and Sinton [36,37] described a mixer that utilizes longitudinal diffusive mixing of a series of solvent injections. An electroosmotic flow is used for the transport of the fluids. The sequential injection effectively creates a flow that resembles an LPG pump. The width of the microchannel strongly affects the time required for mixing to occur by diffusion for two concurrent flows under laminar conditions. The time needed for mixing two streams is quadratically scaled with the channel width and inversely proportional to the diffusion coefficient (t ~ w2/D). For a flow such as in a sequential injection (or in a LPG
pump), the axial inhomogeneity creates a longitudinal concentration gradient. Therefore, the time needed to mix properly is quadratically scaled to the length of the sample in the channel instead. Upon entering an expansion chamber, the length of the injected plug adjusts inversely to the microchannel width (see Figure 19 for a schematic representation as an inset in the graphs). As such, the time necessary for mixing of a sequentially injected plug can be described by: t ~ L2/D ~ 1/(D∙w2). By introducing the sample into an expansion
chamber, the sample width is increased and, in turn, the sample length is decreased. Consequently, the diffusion distances are significantly reduced and increases the axial concentration gradient. As a result, for complete mixing to occur by diffusion alone, significantly less time is required. A similar strategy has been employed for the mixing of gas mixtures [38]. However, mixing by diffusion processes in gasses occurs more quickly, due to the inherently higher diffusion coefficients.
The mixer performance of an expansion chamber was modelled by computational fluid dynamics (CFD). Figure 19A displays the longitudinal concentration profile of the mixer along the middle of the channel for introduced sample lengths of 1 mm and a channel width of 2 mm. A reduction in the amplitude of the signal is observed, though insufficient mixing has occurred to yield a flat concentration profile. By increasing the expansion region to 5.5 mm (Figure 19B), an even lower amplitude is observed, though incomplete mixing still occurred. By increasing the injection frequency, so that introduced sample lengths are 0.1 mm (Figure 19C), a flat concentration profile is obtained quickly. The flat profile suggests that complete axial mixing has occurred and within only 0.2 mm of the expansion channel.
24
Figure 19 Modelled longitudinal concentration profiles of the channel of various expansion designs and sample
lengths: (a) 2 mm expansion channel and a sample length of 1 mm, (b) 5.5 mm expansion channel and a sample length of 1 mm, and (c) 2 mm expansion channel with a sample length of 0.1 mm. Figure taken from [36].
An alternative expansion-channel design was proposed that employed dual injection of the fluids (displayed in Figure 20). The design was created for sequential injection of two liquids by an electroosmotic flow directly within a microfluidic device. The figure visualizes the modelled results and the fluids appear fully axially mixed at the outlet. However, a radial imhomogeneity can be observed by the colour difference, caused by an injection bias, which is a result of unequal delivery of injected fluids into the two channels before the expansion channel.
Figure 20 Simulated concentration profile of the dual-injection micromixer
The volume of the mixer was not discussed in the paper. An accurate estimation cannot be made, as only one dimension of the expansion channel is known, namely the length. A ballpark figure would be based on an estimated channel width of 1 mm that is compared to the channel length on the schematic of Figure 19A. The depth of the channel is estimated as 0.1 mm, equal to the channel width of the outlet. As a result, the volume of the micromixer is approximated at 0.2 µL. A typical stroke volume of an LPG pump is 50 µL, so the axial inhomogeneity would surpass the mixer volume. Scaling up the design as to fit the pump stroke volume could be performed in all directions. However, the mixer might not perform as
25 well, or the volume of the mixer would be exceedingly large for LC applications. The study used electroosmotic flow for driving the fluids through the system. Whether pressure-driven flow and the associated Taylor-Aris dispersion improves or deteriorates the mixer performance, is not discussed in the article.
3.6. Modified Tesla structure-based micromixers
A micromixer based on two-dimensional modified Tesla structures was demonstrated by Hong et al. [39]. The design was improved upon by Yang et al. [40], by fabricating a three-dimensional configuration of the Tesla structures. The principle of operation of the Tesla-structured micromixer is relying on the Coandă effect—the tendency of a fluid to adhere to a convex surface. This effect produces transverse dispersion across a range of flow rates. The structure of the two-dimensional Tesla mixer is shown in Figure 21A. Here, a concurrent flow is unevenly split into a main and a minor flow path. Due to the Coandă effect, the main flow follows the sloped surface after the split and that flow is directed into the minor flow creating a transverse dispersion. The transverse dispersion is displayed on the side of Figure 21A. Mixing occurs in the radial direction as the split-and-recombination of the flow reduces diffusion distances by increasing contact surfaces of the fluids. The three-dimensional design was created to stimulate turbulences in the flow that would further increase the mixer performance. The authors only described the mixer for its radial mixing performance. However, inherent to the design, flow paths varying in length are present. Longitudinal mixing could occur as the flow is axially rearranged. By modifying the channel design, varying lengths and/or widths of the main or minor flow channel the extent of longitudinal mixing could be studied.
Figure 21 (A) Schematic and working principle of a Tesla-structured micromixer. (B) Design of the
26 For the two-dimensional Tesla micromixer a semi-qualitative mixing performance is displayed in Figure 22A. A blue and yellow dyed fluid was introduced into the mixer, upon mixing a green color appears. The mixing was studied for two flow rates, namely 1 µL/min and 100 µL/min. Mixing occurs in both, though more effectively in the latter. A more quantitative approach is performed by mixing an acidic and neutral fluid, which is divided at the outlet into 4 channels, collected and tested for the pH. The pH was calibrated to a mixing performance and several mixer designs were tested at various flow rates, as observed in Figure 23. The mixing performance is relatively constant for a range of flow rates. Unsurprisingly, more Tesla structures incorporated in a design creates a more effective mixer.
Figure 22 (a) Video frame of mixing a blue and yellow
dye. Video frame of flow rates (b) 1 µL/min and (c) 100 µL/min. Figure adapted from [39].
Figure 23 Radial mixing performance of T-type mixer
and two Tesla-structured micromixers. Figure taken from [39].
The CFD studies of the three-dimensional micromixer is shown in Figure 24A. A significant blurring of the interface of the fluids occurs after 10 Tesla mixing structures. The simulated mixing efficiencies are 95% or higher for a Reynolds numbers of 0 to 100. At a Reynolds number of 1, the radial mixing efficiency is plotted against the longitudinal channel length (Figure 24B). In a short distance, significant radial mixing has occurred.
