Water transport to the core–mantle boundary     

Michael J Walter 《国家科学评论(英文版)》2021,8(4)

Water is transported to Earth''s interior in lithospheric slabs at subduction zones. Shallow dehydration fuels hydrous island arc magmatism but some water is transported deeper in cool slab mantle. Further dehydration at ∼700 km may limit deeper transport but hydrated phases in slab crust have considerable capacity for transporting water to the core-mantle boundary. Quantifying how much remains the challenge.

Water can have remarkable effects when exposed to rocks at high pressures and temperatures. It can form new minerals with unique properties and often profoundly affects the physical, transport and rheological properties of nominally anhydrous mantle minerals. It has the ability to drastically reduce the melting point of mantle rocks to produce inviscid and reactive melts, often with extreme chemical flavors, and these melts can alter surrounding mantle with potential long-term geochemical consequences. At the base of the mantle, water can react with core iron to produce a super-oxidized and hydrated phase, FeO₂H_x, with the potential to profoundly alter the mantle and even the surface and atmosphere redox state, but only if enough water can reach such depths [1].Current estimates for bulk mantle water content based on the average H₂O/Ce ratio of oceanic basalts from melt inclusions and the most un-degassed basalts, coupled with mass balance constraints for Ce, indicate a fraction under one ocean mass [2], a robust estimate as long as the basalts sampled at the surface tap all mantle reservoirs. The mantle likely contains some primordial water but given that the post-accretion Earth was very hot, water has low solubility and readily degasses from magma at low pressures, and its solubility in crystallizing liquidus minerals is also very low, the mantle just after accretion may have been relatively dry. Thus, it is plausible that most or even all of the water in the current mantle is ‘recycled’, added primarily by subduction of hydrated lithospheric plates. If transport of water to the core–mantle boundary is an important geological process with planet-scale implications, then surface water incorporated into subducting slabs and transported to the core–mantle boundary may be a requirement.Water is added to the basaltic oceanic crust and peridotitic mantle in lithospheric plates (hereafter, slab crust and slab mantle, respectively) at mid-ocean ridges, at transform faults, and in bending faults formed at the outer rise prior to subduction [3]. Estimates vary but about 1 × 10¹² kg of water is currently subducted each year into the mantle [4], and at this rate roughly 2–3 ocean masses could have been added to the mantle since subduction began. However, much of this water is returned to the surface through hydrous magmatism at convergent margins, which itself is a response to slab dehydration in an initial, and large, release of water. Meta-basalt and meta-sediments comprising the slab crust lose their water very efficiently beneath the volcanic front because most slab crust geotherms cross mineral dehydration or melting reactions at depths of less than 150 km, and even if some water remains stored in minerals like lawsonite in cooler slabs, nearly complete dehydration is expected by ∼300 km [5].Peridotitic slab mantle may have much greater potential to deliver water deeper into the interior. As shown in Fig. 1a, an initial pulse of dehydration of slab mantle occurs at depths less than ∼200 km in warmer slabs, controlled primarily by breakdown of chlorite and antigorite when slab-therms cross a deep ‘trough’, sometimes referred to as a ‘choke point’, along the dehydration curve (Fig. 1a) [6]. But the slab mantle in cooler subduction zones can skirt beneath the dehydration reactions, and antigorite can transform directly to the hydrated alphabet silicate phases (Phases A, E, superhydrous B, D), delivering perhaps as much as 5 wt% water in locally hydrated regions (e.g. deep faults and fractures in the lithosphere) to transition zone depths [6]. Estimates based on mineral phase relations in the slab crust and the slab mantle coupled with subduction zone thermal models suggest that as much as 30% of subducted water may have been transported past the sub-volcanic dehydration front and into the deeper mantle [4], although this depends on the depth and extent of deep hydration of the slab mantle, which is poorly constrained. Coincidentally, this also amounts to about one ocean mass if water subduction rates have been roughly constant since subduction began, a figure tantalizingly close to the estimated mantle water content based on geochemical arguments [2]. But what is the likely fate of water in the slab mantle in the transition zone and beyond?Open in a separate windowFigure 1.(a) Schematic phase relations in meta-peridotite modified after [6,10,12]. Slab geotherms are after those in [4]. Cold slabs may transport as much as 5 wt% water past ‘choke point 1’ in locally hydrated regions of the slab mantle, whereas slab mantle is dehydrated in warmer slabs. Colder slab mantle that can transport water into the transition zone will undergo dehydration at ‘choke point 2’. How much water can be transported deeper into the mantle and potentially to the core depends on the dynamics of fluid/melt segregation in this region. (b) Schematic showing dehydration in the slab mantle at choke point 2. Migration of fluids within slab mantle will result in water dissolving into bridgmanite and other nominally anhydrous phases with a bulk storage capacity of ∼0.1 wt%, potentially accommodating much or all of the released water. Migration of fluids out of the slab into ambient mantle would also hydrate bridgmanite and other phases and result in net fluid loss from the slab. Conversely, migration of hydrous fluids into the crust could result in extensive hydration of meta-basalt with water accommodated first in nominally anhydrous phases like bridgmanite, Ca-perovskite and NAL phase, but especially in dense SiO₂ phases (stishovite and CaCl₂-type) that can host at least 3 wt% water (∼0.6 wt% in bulk crust).Lithospheric slabs are expected to slow down and deform in the transition zone due to the interplay among the many factors affecting buoyancy and plate rheology, potentially trapping slabs before they descend into the lower mantle [7]. If colder, water-bearing slabs heat up by as little as a few hundred degrees in the transition zone, hydrous phases in the slab mantle will break down to wadsleyite or ringwoodite-bearing assemblages, and a hydrous fluid (Fig. 1a). Wadselyite and ringwoodite can themselves accommodate significant amounts of water and so hydrated portions of the slab mantle would retain ∼1 wt% water. A hydrous ringwoodite inclusion in a sublithospheric diamond with ∼1.5 wt% H₂O may provide direct evidence for this process [8].But no matter if slabs heat up or not in the transition zone, as they penetrate into the lower mantle phase D, superhydrous phase B or ringwoodite in the slab mantle will dehydrate at ∼700–800 km due to another deep trough, or second ‘choke point’, transforming into an assemblage of nominally anhydrous minerals dominated by bridgmanite (∼75 wt%) with, relatively, a much lower bulk water storage capacity (< ∼0.1 wt%) [9] (Fig. 1a). Water released from the slab mantle should lead to melting at the top of the lower mantle [10], and indeed, low shear-wave velocity anomalies at ∼700–800 km below North America may be capturing such dehydration melting in real time [11].The fate of the hydrous fluids/melts released from the slab in the deep transition zone and shallow lower mantle determines how much water slabs can carry deeper into the lower mantle. Presumably water is released from regions of the slab mantle where it was originally deposited, like the fractures and faults that formed in the slab near the surface [3]. If hydrous melts can migrate into surrounding water-undersaturated peridotite within the slab, then water should dissolve into bridgmanite and coexisting nominally anhydrous phases (Ca-perovskite and ferropericlase) until they are saturated (Fig. 1b). And because bridgmanite (water capacity ∼0.1 wt%) dominates the phase assemblage, the slab mantle can potentially accommodate much or all of the released water depending on details of how the hydrous fluids migrate, react and disperse. If released water is simply re-dissolved into the slab mantle in this way then it could be transported deeper into the mantle mainly in bridgmanite, possibly to the core–mantle boundary. Water solubility in bridgmanite throughout the mantle pressure-temperature range is not known, so whether water would partially exsolve as the slab moves deeper stabilizing a melt or another hydrous phase, or remains stable in bridgmanite as a dispersed, minor component, remains to be discovered.Another possibility is that the hydrous fluids/melts produced at the second choke point in the slab mantle at ∼700 km migrate out of the slab mantle, perhaps along the pre-existing fractures and faults where bridgmanite-rich mantle should already be saturated, and into either oceanic crust or ambient mantle (Fig. 1b). If the hydrous melts move into ambient mantle, water would be consumed by water-undersaturated bridgmanite, leading to net loss of water from the slab to the upper part of the lower mantle, perhaps severely diminishing the slab’s capacity to transport water to the deeper mantle and core. But what if the water released from slab mantle migrates into the subducting, previously dehydrated, slab crust?Although slab crust is expected to be largely dehydrated in the upper mantle, changes in its mineralogy at higher pressures gives it the potential to host and carry significant quantities of water to the core–mantle boundary. Studies have identified a number of hydrous phases with CaCl₂-type structures, including δ-AlOOH, ϵ-FeOOH and MgSiO₂(OH)₂ (phase H), that can potentially stabilize in the slab crust in the transition zone or lower mantle. Indeed, these phases likely form extensive solid solutions such that an iron-bearing, alumina-rich, δ-H solid solution should stabilize at ∼50 GPa in the slab crust [12], but only after the nominally anhydrous phases in the crust, (aluminous bridgmanite, stishovite, Ca-perovskite and NAL phase) saturate in water. Once formed, the δ-H solid solution in the slab crust may remain stable all the way to the core mantle boundary if the slab temperature remains well below the mantle geotherm otherwise a hydrous melt may form instead [12] (Fig. 1a). But phase δ-H solid solution and the other potential hydrated oxide phases, intriguing as they are as potential hosts for water, may not be the likely primary host for water in slab crust. Recent studies suggest a new potential host for water—stishovite and post-stishovite dense SiO₂ phases [13,14].SiO₂ minerals make up about a fifth of the slab crust by weight in the transition zone and lower mantle [15] and recent experiments indicate that the dense SiO₂ phases, stishovite (rutile structure—very similar to CaCl₂ structure) and CaCl₂-type SiO₂, structures that are akin to phase H and other hydrated oxides, can host at least 3 wt% water, which is much more than previously considered. More importantly, these dense SiO₂ phases apparently remain stable and hydrated even at temperatures as high as the lower mantle geotherm, unlike other hydrous phases [13,14]. And as a major mineral in the slab crust, SiO₂ phases would have to saturate with water first before other hydrous phases, like δ-H solid solution, would stabilize. If the hydrous melts released from the slab mantle in the transition zone or lower mantle migrate into slab crust the water would dissolve into the undersaturated dense SiO₂ phase (Fig. 1b). Thus, hydrated dense SiO₂ phases are possibly the best candidate hosts for water transport in slab crust all the way to the core mantle boundary due to their high water storage capacity, high modal abundance and high-pressure-temperature stability.Once a slab makes it to the core–mantle boundary region, water held in the slab crust or the slab mantle may be released due to the high geothermal gradient. Heating of slabs at the core–mantle boundary, where temperatures may exceed 3000°C, may ultimately dehydrate SiO₂ phases in the slab crust or bridgmanite (or δ-H) in the slab mantle, with released water initiating melting in the mantle and/or reaction with the core to form hydrated iron metal and super oxides, phases that may potentially explain ultra-low seismic velocities in this region [1,10]. How much water can be released in this region from subducted lithosphere remains a question that is hard to quantify and depends on dynamic processes of dehydration and rehydration in the shallower mantle, specifically at the two ‘choke points’ in the slab mantle, processes that are as yet poorly understood. What is clear is that subducting slabs have the capacity to carry surface water all the way to the core in a number of phases, and possibly in a phase that has previously seemed quite unlikely, dense SiO₂.  相似文献   

Polyphosphonium-based ion bipolar junction transistors     

Erik O. Gabrielsson  Klas Tybrandt  Magnus Berggren 《Biomicrofluidics》2014,8(6)

Advancements in the field of electronics during the past few decades have inspired the use of transistors in a diversity of research fields, including biology and medicine. However, signals in living organisms are not only carried by electrons but also through fluxes of ions and biomolecules. Thus, in order to implement the transistor functionality to control biological signals, devices that can modulate currents of ions and biomolecules, i.e., ionic transistors and diodes, are needed. One successful approach for modulation of ionic currents is to use oppositely charged ion-selective membranes to form so called ion bipolar junction transistors (IBJTs). Unfortunately, overall IBJT device performance has been hindered due to the typical low mobility of ions, large geometries of the ion bipolar junction materials, and the possibility of electric field enhanced (EFE) water dissociation in the junction. Here, we introduce a novel polyphosphonium-based anion-selective material into npn-type IBJTs. The new material does not show EFE water dissociation and therefore allows for a reduction of junction length down to 2 μm, which significantly improves the switching performance of the ion transistor to 2 s. The presented improvement in speed as well the simplified design will be useful for future development of advanced iontronic circuits employing IBJTs, for example, addressable drug-delivery devices.There has been a recent interest in developing diodes^1–4 and transistors^4–8 that conduct and modulate ion currents. Such non-linear iontronic components are, for example, interesting as they allow further control of ions in, for instance, electrophoretic drug delivery devices. A range of microfabricated diodes,^9–11 transistors,^12,13 and circuits^9,14 has been constructed using ion-selective membranes. These membranes contain fixed charges of either polarity, compensated by mobile ions of opposite charge (counter-ions). When immersed in an electrolyte, counter-ions can move through the membrane, while ions with the same charge as the fixed charges (co-ions) are repelled. This renders the membrane selective for the counter-ion and can therefore be considered as p- or n-type ion conductors. By combining two membranes of opposite polarity, a bipolar membrane (BM) configuration is obtained¹⁵ (Figure 1(a)). The BM junction can be biased by an ion current in the reverse and forward directions, respectively.^16,17 This modulates the ion concentration inside the BM, and thus the ionic conductivity, which then results in an current rectification.^2,18 In the three-terminal ion bipolar junction transistor¹² (IBJT), an ion-selective base (B) is connected to oppositely selective emitter (E) and collector (C), forming two BM configurations (EB and BC) (Figure 1(b)). pnp- and npn-IBJTs have been constructed¹⁴ from photolithography patterned poly(styrene sulfonate) (PSS, p-selective) and quaternized poly(vinylbenzyl chloride) (n-selective) as emitter, collector, and base. In these devices, a neutral poly(ethylene glycol) (PEG) electrolyte is typically inserted into the junction to separate the base from the emitter and collector,¹² in order to avoid¹⁹ electric field enhanced (EFE) water dissociation¹⁶ (Figure 1(a)). EFE water dissociation is typically observed in BMs²⁰ and produces water ions inside the BM under reverse bias, which prevents proper IBJT operation. In PEG-IBJTs, the current between the emitter and collector (I_C) is thus modulated by controlling the ion concentration inside the PEG-junction.²¹ Ions are injected or extracted into the junction depending on the bias of the base (V_EB). In a npn-IBJT, a positive bias is typically applied between emitter and collector (V_EC), thus allowing anions to migrate from the emitter to the collector. In the cut-off mode (Figure 1(c)), a negative bias V_EB is applied, resulting in reverse bias of both EB and BC. Cations in the junction will migrate into the base, while anions will primarily migrate into the collector, due to the higher collector bias. This base current (I_B) will extract ions from the junction, which decreases the ionic conductivity in the junction resulting in a low I_C. Eventually, the resistive characteristics for ion charge transport, between the emitter and collector, will be entirely dominated by the junction. This gives that most of the applied V_EC is consumed across the junction with only a minimal voltage potential drop across the emitter and base terminals.Open in a separate windowFIG. 1.(a) The modes of operation for a BM; forward bias (high conduction and ion accumulation), reverse bias (low conduction and ion depletion), and EFE water dissociation (high conduction, formation of ions). (b) Illustrations of an npn-IBJT, with anion-selective emitter (E) and collector (C) forming a junction with a cation-selective base (B). (c) In cut-off mode, the base and collector extract ions from the junction, prohibiting co-ion migration through the base. (d) In active mode, the forward biased EB injects ions into the base, thus allowing anions from the emitter to migrate as co-ions through the base into the collector.In the active-mode of the npn-IBJT (Figure 1(d)), the V_EB bias at the base is reversed (i.e., now positive). This causes injection of cations, from the base, and anions, from the emitter, into the junction. As the ion concentration increases, anions from the emitter can start to drift across the junction to the collector, thus a high I_C is obtained. The high concentration of ions inside the junction is reflected in a low resistive value for ion transport. This now causes the voltage to drop over the emitter and collector terminals, thus lowering the EB forward bias and the injection of ions from the base. At the collector-junction interface, the extraction of anions produces an ion depletion zone and a corresponding voltage drop. Thus, in the active-mode, the applied V_EC is primarily consumed across the emitter and collector terminals and also at the collector-junction interface.The switching speed of an IBJT should be strongly correlated to the distance separating the emitter and collector,¹⁴ as this length determines the volume that needs to be filled or emptied with ions causing modulation of ions in the junction. To achieve a fast-switching IBJT, the junction volume, i.e., the collector-emitter separation, should be as small as possible. However, EFE water dissociation must be avoided since this process ruin the IBJT operation. EFE water dissociation is, in part, driven by the appearance of a large potential drop across a small distance, as occurring at the interface of a BM under reverse bias, producing a high electric field that accelerates the forward reaction rate of water auto-dissociation.¹⁶ Miniaturization of the collector-emitter distance is therefore problematic, as the separation inside the EB and BC BMs evidently also mush shrink, resulting in higher reverse bias electric fields across the BMs and thus promoting EFE water dissociation. The problem of EFE water dissociation in an IBJT primarily manifests itself in the cut-off mode, as water ions are generated in the reversed biased EB and BC BMs. These ions produce an elevated cut-off I_C, and hence deteriorate the IBJTs on–off performance. Here, we report an IBJT, in which the EFE water dissociation is avoided by the use of a novel polyphosphonium-based anion-selective material, which previously has been shown to prevent EFE water dissociation in BM diodes.¹¹ This allows the collector and emitter to directly contact the base without an intermediate PEG-layer. Without the need for a PEG-separator inside the BMs, the collector-emitter distance is reduced to only 2 μm.Polyphosphonium-based npn-IBJTs were produced following the same manufacturing protocol as was reported for polyphosphonium-based ion diodes.¹¹ Conjugated polymer electrodes and cation-selective base was patterned from ∼200 nm thick poly(3,4-ethylenedioxythiophene):polystyrene sulfonate film on polyethylene terephthalate-sheets using photolithography and dry-etching. The base was rendered electronically insulating by chemical overoxidation via exposure to sodium hypochlorite through a mask. A 2 μm thick SU8-layer was patterned on-top of this configuration, with an opening defining the actual junction. 1 μm thick polyphosphonium-based anion-selective emitter and collector were deposited and patterned using photolithography and dry-etching, to overlap with the base at the opening of the SU8. Finally, a second 10 μm thick layer of SU8 was used to seal the junction. The membranes were hydrated by incubation in dH2O for 24 h before any measurements were carried out. Aqueous 0.1M NaCl electrolytes were used during the measurement. All electrical measurements were performed using a Keithley 2602 source meter.The switching characteristics of the npn-IBJT were obtained by applying V_EC of 10 V and alternating V_EB at ±3 V for various duration of time, see Figure ​Figure2.2. A periodic 5 s switching with 8 Hz measurement rate was used to record the dynamics of the turn-on/off characteristics of the device. When V_EB switches from −3 to +3 V, there is a quick increase in the I_B, as ions from the base and emitter migrate into the emitter/base junction. After a delay of ∼0.25 s, I_C starts to increase due to the increased ion concentration in the emitter/base junction and the subsequent diffusion of anions into the base. As the I_C increases, the I_B decreases as the voltage drop between the emitter and base decreases, and after ∼2 s I_C reaches 90% of the steady state on-current level. For longer on-switching times, the I_B and I_C stay stable over 30 s, after which a small increase is observed. This current-drift in both I_B and I_C is likely due to the contribution of co-ion migration. As cations from the base migrate into the emitter as co-ions, the conductivity in the emitter increases, leading to an increased I_C value. This increases the ion concentration at the base, which gives less selective ion injection and thus more cation injection from the base, i.e., a higher I_B.Open in a separate windowFIG. 2.Emitter-collector current response as the IBJT is switched between cut-off (V_EB = −3 V) and active mode (V_EB = 3 V) for V_EC = 10 V, at 5 s and 120 s periods.As V_EB is switched back to −3 V, there is a sharp negative peak in I_E as ions are extracted from the junction, which occur mainly through the base (cations) and collector (anions) terminals. As the ion concentration in the base drops, I_C decreases. The transistor turns off to 10% of the value of the steady state on-current within ∼2 s, regardless of the duration of the on-state. The constant turn-off time indicates that ions are not accumulating to a significant extent inside the junction during the on-steady state but are instead constantly transported out of the junction. When all co-ions have been extracted from the junction, the Donnan exclusion prevents subsequent injection of anions into the base, and I_C is therefore low. The on/off ratio of I_C reaches above 100.A transfer curve was obtained by scanning V_EB between −3 and +3 V while keeping V_EC at 10 V (Figure 3(a)). As expected, both I_C and I_B remain low for negative V_EB. In this range, both EB and BC are biased in reverse direction. As V_EB turns positive, the EB configuration is switched into forward bias and ions are injected into the junction. This leads to a linear increase in I_C vs. V_EB. For the reverse scan, a minor hysteresis is observed for both the I_C and I_B scans, again probably due to the contribution of co-ion migration due to long time operation of the device.Open in a separate windowFIG. 3.Transfer and output curves. (a) The transfer curve is low for negative V_EB and increases linearly for positive V_EB with approximately zero threshold. (b) The output curves show I_C saturating with respect of V_EC for positive V_EB.The transistor output characteristics were obtained by scanning V_EC at different V_EB values (Figure 3(b)). The saturation regime, i.e., the bias mode was both EB and BC are in forward bias, was avoided as this has negative impact on the stability of the device. As reported for previous IBJT devices, the output characteristics show a clear saturation behaviour of I_C across the entire range of V_EC. Further, the I_C increases linearly with V_EB. The increase of both I_C and I_B when operating for extended periods of time in the active mode is again attributed to the addition and inclusion of co-ions in the junction. The current gain (I_C/I_B) at V_EC = 10 V decreases with V_EB and reaches 43.9, 17.9, and 10.7 for V_EB = 1 V, 2 V, and 3 V, respectively. For higher base bias voltages, the ion concentration increases in the junction and thus the injection selectivity decreases.In comparison with previously reported IBJTs,^12,14,21 the lack of a neutral electrolyte layer in the junction has an overall positive effect on the device characteristics. Main performance improvements are found in a decrease in the turn-on time from 9 s (for npn-IBJT²¹) to 2 s, for devices with comparable junction widths and heights. The main contribution to the improved switching speed is likely the decreased length between the emitter and collector. Interestingly, simulations have shown that an extended space charge region (ESCR), for a PEG-IBJT in cut-off mode, can extend several micrometers away from the collector.²² Thus, a PEG-IBJT with an emitter-collector separation of single micrometers should show an increased cut-off current due to the ESCR overlapping in the junction. However, by omitting the PEG in the junction, the ESCR is reduced due to screening from the fixed charges in the BM layers. This enables the IBJT, reported here, to operate with retained low cut-off currents. On-off ratios and ion current gains are approximately equal to previous IBJTs,^12,14,21 at above 100 and 10, respectively. The on–off ratio and ion current gain are more dependent on the selectivity of the membranes and the charge of the junction.Further, the need to separate the layers in a PEG-IBJT puts high demands on the patterning resolution and alignment accuracy to reduce the separation between emitter/collector and base. As polyphosphonium allows the IBJT to be built without separation of layers, miniaturization of the junction is relatively easier to obtain. The switching speed can potentially be further improved by retaining the base material between the emitter and collector (see Figure 1(b)), thus allowing for a more direct pathway for I_C. This design would, however, require a much more accurate layer alignment or that the base patterned on top of the emitter and collector layers. In general, such modifications of device geometry are simpler to accomplish with the non-EFE water dissociating polyphosphonium as fewer active layers are used, suggesting a further use of polyphosphonium to improve switching speed and miniaturization of IBJTs. Such further advancement in IBJT performance would be welcomed, for example, in the continued work towards complex ionic circuits¹⁴ to regulate signalling in bioelectronics and in drug delivery applications, in which generation of dynamic and complex gradients, at high spatial resolution, is of generic interest.  相似文献   

Enhancement of biosensing performance in a droplet-based bioreactor by in situ microstreaming     

Olivier Ducloux  Elisabeth Galopin  Farzam Zoueshtiagh  Alain Merlen    Vincent Thomy 《Biomicrofluidics》2010,4(1)

A droplet-based micro-total-analysis system involving biosensor performance enhancement by integrated surface-acoustic-wave (SAW) microstreaming is shown. The bioreactor consists of an encapsulated droplet with a biosensor on its periphery, with in situ streaming induced by SAW. This paper highlights the characterization by particle image tracking of the speed distribution inside the droplet. The analyte-biosensor interaction is then evaluated by finite element simulation with different streaming conditions. Calculation of the biosensing enhancement shows an optimum in the biosensor response. These results confirm that the evaluation of the Damköhler and Peclet numbers is of primary importance when designing biosensors enhanced by streaming.It has been pointed out that biosensing performances can be limited by the diffusion of the analytes near the sensing surface.¹ In the case of low Peclet number hydrodynamic flows, typical of microfluidic systems, molecule displacements are mainly governed by diffusive effects that affect time scales and sensitivity. To overcome this problem, the enhancement of biosensor performance by electrothermal stirring within microchannels was first reported by Meinhart et al.² Other authors^{3, 4} numerically studied the analyte transport as a function of the position of a nanowire-based sensor inside a microchannel, stressing on the fact that the challenge for nanobiosensors is not the sensor itself but the fluidic system that delivers the sample. Addressing this problem, Squires et al.⁵ developed a simple model applicable to biosensors embedded in microchannels. However, the presented model is limited to the case of a steady flow. The use of surface-acoustic waves (SAWs) for stirring in biomicrofluidic and chemical systems is becoming a popular investigation field,^{6, 7, 8, 9} especially to overcome problems linked to steady flows by enhancing the liquid∕surface interaction.^{1, 10, 11} The main challenges that need to be addressed when using SAW-induced stirring are the complexity of the flow and its poor reproducibility. However, some technical solutions were proposed to yield a simplified microstreaming. Yeo et al. presented a centrifugation system based on SAW that produces the rotation of the liquid in a droplet in a reproducible way by playing on the configuration of the transducers and reflectors,¹² and presented a comprehensive experimental study of the three-dimensional (3D) flow that causes particle concentration in SAW-stirred droplets,¹³ revealing the presence of an azimuthal secondary flow in addition to the main vortexlike circular flow present in acoustically stirred droplets. The efficiency of SAW stirring in microdroplets to favorably cope with mass transport issues was finally shown by Galopin et al.,¹⁴ but the effect of the stirring on the analyte∕biosensor interaction was not studied. It is expected to overcome mass transport limitations by bringing fresh analytes from the bulk solution to the sensing surface.The studied system, described in Fig. ​Fig.1,1, consists of a microliter droplet microchamber squeezed between a hydrophobic piezoelectric substrate and a hydrophobic glass cover. Rayleigh SAWs are generated using interdigitated transducers (interdigital spacing of 50 μm) laid on an X-cut LiNbO₃ substrate.^{1, 15, 16} The hydrophobicity of the substrate and the cover are obtained by grafting octadecyltrichlorosilane (OTS) self-assembled monolayers (contact angle of 108° and hysteresis of 9°). To do so, the surface is first hydroxylized using oxygen plasma (150 W, 100 mT, and 30 sccm³ O₂) during 1 min and then immersed for 3 h into a 1 mM OTS solution with n-hexane as a solvent.Open in a separate windowFigure 1(a) General view of the considered system. (b) Mean value of the measured speeds within the droplet as a function of the inlet power before amplification.When Rayleigh waves are radiated toward one-half of the microchamber, a vortex is created in the liquid around an axis orthogonal to the substrate due to the momentum transfer between the solid and the liquid. This wave is generated under the Rayleigh angle into the liquid.Speed cartographies of the flow induced in the droplet are realized using the particle image tracking technique for different SAW generation powers. To do so, instantaneous images of the flow are taken with a high-speed video camera at 200 frames∕s and an aperture time of 500 μs on a 0.25 μl droplet containing 1 μm diameter fluorescent particles. Figure ​Figure11 shows the mean speed measured in the droplet as a function of the inlet power. The great dependence of the induced mean speed with the SAW power enables a large range of flow speeds in the stirred droplet. Moreover, the flow was visualized with a low depth of field objective. It was found to be circular and two dimensional (2D) in a large thickness range of the droplet.The binding of analytes to immobilized ligands on a biosensor is a two step process, including the mass transport of the analyte to the surface, followed by a complexation step,Abulk↔kmAsurface+B↔ka,kdAB(1)with k_m as the constant rate for mass transport from and to the sensor, and k_a and k_d as the constant rates of association and dissociation of the complex.At the biosensor surface, the reaction kinetics consumes analytes but their transport is limited by diffusive effects. In this case, the Damköhler number brings valuable information by comparing these two effects. Calling the characteristic time of reaction and diffusion, respectively, τ_C and τ_M, the mixing time in diffusion regime can be approximated by τ_M≈h²∕D with D as the diffusion coefficient and h a characteristic length of the microchannel. Calling R_T the ligand concentration on the surface in mole∕m², the Damköhler number (D_a) can be written asDa=τMτC=kaRThD.(2)Depending on the type of reaction, the calculation of D_a helps determine if a specific biointeraction will benefit from a mass SAW-based microstreaming. If the Damköhler number is low, the reaction is slow compared to mass transport and the reaction will not significantly benefit from microstirring. For example, the hybridization of 19 base single stranded DNA in a microfluidic system with a characteristic length of 500 μm is characterized by a Damköhler number of 0.07 and is therefore not significantly influenced by mass transport. On the contrary, the binding of biotin to immobilized streptavidin is characterized by a D_a number of approximately 10⁴. In this case, the stirring solution will significantly improve the reaction rate.COMSOL numerical simulations were carried out to study the efficiency of the SAW stirring in the case of a droplet-based microbioreactor with a diameter of 1 mm. Assuming a 2D flow, the simulated model takes into account the convective and diffusive effects in the analyte-carrying fluid and the binding kinetics on the biosensor surface. This approach was thoroughly developed by Meinhart et al.²On the biosensor surface, the following equations are solved:∂B∂t=kacs(RT−B)−kdB,(3)∂B∂t=D|∂c∂y|y=0(4)with c as the local concentration of analytes in the droplet and B as the surface concentration of bound analytes on the biosensor surface. Simulation results show that a depleted zone is formed near the biosensor in the case of an interaction without stirring. This zone is characterized by a low concentration of analytes and results from the trapping of analytes on the biosensor surface, thus creating a concentration gradient on the vicinity of the biosensor. When stirring is applied, the geometry of the depleted zone is modified, as it is pushed in the direction of the flow. The geometry of the depleted zone then depends on many parameters, among which the diffusion coefficient D, the speed distribution of the flow (not only near the biosensor but also in the whole microfluidic system), and the reaction kinetics on the biosensor. In our case, which is assimilated to a simple circular flow, the depleted zone reaches a permanent state consisting of an analyte-poor layer situated in the exterior perimeter of the stirred droplet. The diffusion of analytes is then limited again by diffusion from the inner part of the droplet toward its exterior perimeter (see Fig. ​Fig.22).Open in a separate windowFigure 2(a) Mean concentration of bound analytes vs time for different mean flow speeds. (b) The obtained concentration profiles with and without circular stirring, t=10 000 s.The initial analyte and receptor concentrations are, respectively, 0.1 nM in the solution and 3.3×10⁻³ nM m on the biosensor surface, the diffusion coefficient is D=10⁻¹¹ m² s⁻¹, and the reaction constants are k_a=10⁶ M⁻¹ s⁻¹ and k_d=10⁻³ s⁻¹. Simulations show that the mean concentration of bound analytes highly increases with the flow speed, improving the efficiency of the biosensing device. To evaluate the benefits of in situ microstreaming with SAW, the same simulations were conducted for D_a numbers ranging from 10⁴ to 10⁸ M⁻¹∕s, by ranging the diffusion coefficient from 4×10⁻¹² to 4×10⁻⁹ m²∕s, and the association coefficient k_a from 10⁴ to 10⁸ M⁻¹∕s. The enhancement factor of analyte capture, defined as the ratio of the binding rate with streaming B and the binding rate without streaming B0, is plotted in Fig. ​Fig.33 for different values of D_a. Calculations are done in the case of a mean flow speed of 0.5 mm∕s.Open in a separate windowFigure 3(a) Enhancement factor (defined as the ratio between binding rate with streaming B and binding rate without streaming B0) for different Damkhöler numbers and (b) normalized enhancement factor for different Peclet numbers.One can notice the saturation of the enhancement factor curve for large value of D_a to the value of 3.5 for high D_a. This can be explained by the fact that for large k_a∕D_a ratios, the analytes, which normally require penetration in the depleted zone by diffusion, do not have time to interact with the biosensor when they pass in the vicinity of its surface. The efficiency of the streaming is then reduced for large values of D_a. In the case of our specific flow configuration, the enhancement factor reaches 3.2 for the interaction of streptavidin on immobilized biotin (D_a=10³).The reported simulation results can be compared to an experimental value obtained using the droplet-based surface plasmon resonance sensor streamed in situ using SAW reported by Yeo et al.¹² By monitoring the streptavidin∕biotin binding interaction on an activated gold slide, they showed that SAW stirring brings an improvement factor of more than 2. This difference can be accounted to the high complexity of the induced 3D flow, which was modeled in a simple manner in our calculations.Other factors must be taken into account when optimizing the improvement factor, such as the flow velocity and the characteristic length of the mixing. To do so, the Peclet number allows the comparison of the convective and diffusive effects.¹⁷ For δC a typical variation in concentration on the distance h, the Peclet number is given byPe=UhD.(5)A significantly high Peclet number causes a decrease in biosensing efficiency as the analytes do not have enough time to interact with the biosensing surface by diffusion through the analyte-poor layer. On the contrary, the case of a low Peclet number corresponds to the diffusion-limited problem. Therefore, for each Damköhler number, there is a Peclet number optimizing this factor. To illustrate this fact, Fig. ​Fig.3b3b shows the calculation of the enhancement factor as a function of the Peclet number for a given D_a.In this paper, we showed that surface loading of typical analytes on a droplet-based biosensor can be highly increased by SAW microstirring. The system permits the enhancement of the biosensing performances by the continuous renewal of the analyte-carrying fluid near the sensing surface. Thanks to mean flow speeds measured up to 1800 μm∕s, the SAW microstreaming can be beneficial to the biosensing of a large range of analyte∕ligand interactions. In addition to the biosensing performance improvement, such a method can be easily integrated in micro-micro-total-analysis systems, which makes it a convenient tool for liquid handling in future biochips.  相似文献   

Mixed control of uncertain jumping time-delay systems     

Magdi S. Mahmoud  Fouad M. AL-Sunni  Yan Shi   《Journal of The Franklin Institute》2008,345(5):536-552

In this paper, we investigate a class of linear continuous-time systems with Markovian jump parameters. An integral part of the system dynamics is a delayed state with time-varying and bounded delays. The jumping parameters are modeled as a continuous-time, discrete-state Markov process. Employing norm-bounded parametric uncertainties and utilizing the second-method of Lyapunov, we examine the problem of designing a mixed controller which minimizes a quadratic performance measure while satisfying a prescribed -norm bound on the closed-loop system. It is established that sufficient conditions for the existence of the mixed controller and the associated performance upper bound could be cast in the form of linear matrix inequalities.  相似文献