3D printed electro-femtofluidic flow sensor

Abstract

Disclosed is an electro-femtofluidic flow sensor that infers volumetric flow by translating pressure-induced membrane deflection into resistance changes measured with Wheatstone bridge circuitry (300). The device comprises first and second membranes (102, 104) enclosing an electrically conductive fluid (110); first and second Wheatstone bridges (300), each having a variable resistor Rv formed by a portion of the electrically conductive fluid (110) and fixed resistors R1-R3; and at least one processor (302) that, when the sensor is fluidically connected in series with a flow channel carrying a fluid, computes membrane deflection from bridge outputs and estimates flow rate. To address in-situ nanoprinting feedback and control on opaque substrates used in molecular self-assembly, the electro-femtofluidic flow sensor enables femtoliter-per- second measurements. By enabling real-time, ultralow-flow sensing under opaque conditions, the approach may support new nanofabrication modalities and the creation of molecular-level machines for biomedical and other applications.

Description

Docket No.26163PCT 3D PRINTED ELECTRO-FEMTOFLUIDIC FLOW SENSOR CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims the benefit of U.S. Provisional Application No.63 / 702,151, filed October 1, 2024, which is hereby incorporated herein by reference in its entirety. TECHNICAL FIELD

[0002] The present invention relates generally to the field of fluid flow sensors. More specifically, the present invention is related to a 3D printed electro-femtofluidic flow sensor. BACKGROUND OF THE INVENTION

[0003] Nanoprinting has attracted significant attention in recent history due to its potential to fabricate morphologically complex structures at the nanoscale, a functionality that is being increasingly investigated for the development of microrobots and other nanoscale machines [see US 2023 / 0347582 A1]. Such devices are dreamed to be able to perform a variety of tasks including targeted drug delivery, sensing, and information processing, akin to the famous nanorobots of science fiction [see paper to Hamoudi et al., “Building block 3D printing based on molecular self-assembly monolayer with self-healing properties”]. These innovative technologies, if possible, would be undoubtedly revolutionary for the medical sciences, but before they can be actualized, several breakthroughs are desperately needed to enhance the capabilities of present nanoprinting systems: multi- material deposition functionality, higher resolution, higher throughput, and faster printing speeds [see Meister et al., “FluidFM: combining atomic force microscopy and nanofluidics in a universal liquid delivery system for single cell applications and beyond”, and Grter et al., “Patterning gold nanoparticles in liquid environment with high ionic strength for local fabrication of up to 100m long metallic interconnections”]. As a first step towards these goals, we discuss the operation modes, capabilities, and initial results of a modular, first 1 1530512.1Docket No.26163PCT of its kind AFM-based nanoprinting system that we have functionalized for deposition of *technorganic* material.

[0004] However, unlike other nanoprinting systems, our deposition principle brings forth challenges not previously encountered in the field. Namely, most nanoprinting setups have some form of live monitoring of the printing process, typically in the form of an inverted microscope, but since our system necessitates the use of an opaque gold substrate, such monitoring means are not feasible. To address this, our group has proposed a new device, the Electro-Femtofluidic Flow Sensor, which has been designed to measure flow rates on the order of femtoliters per second, providing critical real-time feedback on the printing process. This conceptual framework presented here.

[0005] Embodiments of the present invention are an improvement over prior art systems and methods. SUMMARY OF THE INVENTION

[0006] In one embodiment, the present invention provides a fluid flow sensor comprising: (a) a first membrane (102); (b) a second membrane (104); (c) an electrically conductive fluid (110) enclosed within and defining an inter-membrane conductive-fluid channel between the first and second membranes; (d) a first Wheatstone bridge (300) operatively coupled to the first membrane (102) and configured to output a first bridge signal indicative of deflection of the first membrane (102); (e) a second Wheatstone bridge (300) operatively coupled to the second membrane (104) and configured to output a second bridge signal indicative of deflection of the second membrane (104); and (f) at least one processor (302) operatively coupled to the first and second Wheatstone bridges and, when the sensor is fluidically connected in series with a flow channel carrying a fluid, the at least one processor (302) configured to: (i) determine a first pressure at the first membrane (102) and a second pressure at the second membrane (104) from the first and second bridge signals, respectively; (ii) compute a pressure drop across the flow channel as a 2 1530512.1Docket No.26163PCT difference between the first and second pressures; and (iii) estimate a volumetric flow rate of the fluid in the flow channel based on the pressure drop and a predetermined hydraulic resistance of the flow channel according to an equivalent shunt hydraulic resistor (ESHR) principle. These values would be determined through extensive, methodical calibration experiments.

[0007] In one embodiment, at least one of the first and second Wheatstone bridges (300) comprises four resistive elements arranged in a diamond configuration, one arm including a variable resistor Rvformed by a portion of the electrically conductive fluid (110) within the inter-membrane conductive-fluid channel, and the remaining arms including fixed resistors R1, R2, R3, and wherein a change in bridge output voltage ∆v between the bridge sense nodes is given by: ^^1 1 ^^− )where ^^^^is a supply supply, etc.) applied to the bridge and ΔR is a change in the resistance of Rv caused by pressure-induced deflection of themembrane due to fluid pressure in the flow channel.

[0008] In one embodiment, a total electrical resistance of the variable resistor Rv formedby the electrically conductive fluid (110) within the inter-membrane conductive-fluid channel is obtained by modeling the inter-membrane conductive-fluid channel as M serial slices and summing their respective slice resistances,^^^^ = ∑^^ ^^=1 ^^^^^^^^^^^^(^^).

[0009] In one embodiment, ^^ ( )^^∆^^ ^^^^^^^^^^ ^^ =^^∆^^, where ρ is the electrical resistivity of the electrically conductive,and width of the inter-membrane conductive-fluid channel, respectively, ^^ =^^ is a∆^^number of discretization steps along a length L of the inter-membrane fluid 3 1530512.1Docket No.26163PCT channel, H is an initial inter-membrane conductive-fluid channel height, and δi, jaccounts for membrane deflection at position (i, j).

[0010] In one embodiment, for each of the first and second membranes, deflection δi, j varies along both a length and a width of the first and second membranes due to fixed boundary conditions and is given by: ^^ 2^^^^∆^^ 2^^^^∆^^ ^^^^,^^=^^^^^^4[cos (^^+ ^^) + 1] [cos (^^+ ^^) + 1]where L is thethe first and second membranes, and δmaxis a maximum deflection of the first and second membranes.

[0011] In one embodiment, ^^^^^^ ^^^^^^ = ^^432^^ℎ3, where p is pressure of the fluid in the flow channel acting on the first andE is the Young’s modulus of a material associated with the first and second membranes, h is the membrane thickness, and α is a correction factor to account for the membrane being fixed on all four sides (and can be determined through simulation or experiment).

[0012] In one embodiment, the Young’s modulus is about 3.2 GPa.

[0013] In one embodiment, h=H.

[0014] In one embodiment, h=H=5 µm.

[0015] In one embodiment, L=W.

[0016] In one embodiment, L=W=700 µm.

[0017] In one embodiment, the fluid flow sensor is configured to detect flow rates in the range of 1 to 1000 fL / s.

[0018] In another embodiment, the present invention provides a method for estimating a flow rate, the method comprising: (a) providing a fluid flow sensor, the fluid flow sensor comprising: (i) a first membrane (102); (ii) a second membrane (104); (iii) an electrically conductive fluid (110) enclosed within and defining an inter-membrane conductive-fluid channel between the first and second membranes (102, 104); (iv) a first Wheatstone 4 1530512.1Docket No.26163PCT bridge (300) operatively coupled to the first membrane (102) and a second Wheatstone bridge (300) operatively coupled to the second membrane (104); and (v) at least one processor (302); (b) fluidically connecting the fluid flow sensor in series with a flow channel carrying a fluid (e.g. for nanoprinting application, but this example is merely illustrative and non-limiting); (c) obtaining, from the first Wheatstone bridge (300) and the second Wheatstone bridge (300), a first bridge signal and a second bridge signal, respectively; and (d) using the at least one processor (302) to: (i) determine a first pressure at the first membrane (102) and a second pressure at the second membrane (104) from the first and second bridge signals, respectively; (ii) compute a pressure drop across the flow channel as a difference between the first and second pressures; and (iii) estimate a volumetric flow rate of the fluid in the flow channel based on the pressure drop and a predetermined hydraulic resistance of the flow channel according to an equivalent shunt hydraulic resistor (ESHR) principle.

[0019] In one embodiment, at least one of the first and second Wheatstone bridges (300) comprises four resistive elements arranged in a diamond configuration, one arm including a variable resistor Rv formed by a portion of the electrically conductive fluid (110) within the inter-membrane conductive-fluid channel, and the remaining arms including fixed resistors R1, R2, and R3, and wherein a change in bridge output voltage ∆v between the bridge sense nodes is given by: 1 1 ) where ^^^^is a supplyetc.) applied to the bridge and ΔR is a change in the resistance of Rv caused by pressure-induced deflection of the membrane due to fluid pressure in the flow channel. 5 1530512.1Docket No.26163PCT

[0020] In one embodiment, a total electrical resistance of the variable resistor Rvformed by the electrically conductive fluid (110) within the inter-membrane conductive-fluid channel is obtained by modeling the inter-membrane conductive-fluid channel as M serial ^^slices and summing their respective slice resistances, ^^^^ = ∑ ^^=1 ^^^^^^^^^^^^(^^).

[0021] In one embodiment, ^^ (^^) =^^∆^^ ^^^^^^^^^^ ∑^^ ^^=1[^^−2^^^^,^^]∆^^,the electrical resistivity of the electrically conductive fluiddiscretization steps along the lengthand width of the inter-membrane conductive-fluid channel, respectively, ^^ =^^ is a ∆^^number of discretization steps along a length L of the inter-membrane fluidchannel, H is an initial inter-membrane conductive-fluid channel height, and δ accountsi, j for membrane deflection at position (i, j).

[0022] In one embodiment, for each of the first and second membranes, deflection, δi, j, varies along both a length and a width of the first and second membranes due to fixed boundary conditions and is given by: ^^ 2^^^^∆^^ 2^^^^∆^^^^^^^^ cos ( + ^^) + 1 cos ( + ^^) + 1 [ ] [ ]^^ =^^,^^^^ ^^ where L is theof the first and second membranes, and δ is a maximum deflection of the first and secondmax membranes. 4 ^^^^

[0023] In one embodiment, ^^ = ^^, where p is pressure of the fluid in the flow^^^^^^channel acting on the first and second membranes, E is the Young’s modulus of a material associated with the first and second membranes, h is the membrane thickness, and L is the membrane length, and α is a correction factor to account for the membrane being fixed on all four sides.

[0024] In one embodiment, the Young’s modulus is about 3.2 GPa.

[0025] In one embodiment, h=H. 6 1530512.1Docket No.26163PCT

[0026] In one embodiment, h=H=5 µm.

[0027] In one embodiment, L=W.

[0028] In one embodiment, L=W=700 µm.

[0029] In one embodiment, the fluid flow sensor is configured to detect flow rates in the range of 1 to 1000 fL / s. BRIEF DESCRIPTION OF FIGURES

[0030] The present disclosure, in accordance with one or more various examples, is described in detail with reference to the following figures. The drawings are provided for purposes of illustration only and merely depict examples of the disclosure. These drawings are provided to facilitate the reader's understanding of the disclosure and should not be considered limiting of the breadth, scope, or applicability of the disclosure. It should be noted that for clarity and ease of illustration these drawings are not necessarily made to scale.

[0031] FIG.1 is a non-limiting example implementation of the Electro-Femtofluidic Flow Sensing device.

[0032] FIG.2(a) depicts an example CAD model of the present invention’s flow sensing sensor. FIG. 2(b) depicts several frames from our simulations relating pressure to membrane deflection.

[0033] FIGS. 3(a)-(b) depict Wheatstone Bridge circuit (300) for measuring resistance changes in the present invention’s flow sensor. FIG. 3(a) shows the circuit model, and FIG.3(b) shows an example PCB design corresponding to the circuit of FIG.3(a).

[0034] FIG.4 depicts a graph of total resistance (in ohms) versus pressure (in mbar).

[0035] FIGS.5(a)-5(d) depict optical micrographs of printed nanoscale features using conventional FluidFM cantilevers.

[0036] FIGS.6(a)-6(b) depict SEM image of the printed structures. 7 1530512.1Docket No.26163PCT DETAILED DESCRIPTION

[0037] While this invention is illustrated and described in a preferred embodiment, the invention may be produced in many different configurations. There is depicted in the drawings, and will herein be described in detail, a preferred embodiment of the invention, with the understanding that the present disclosure is to be considered as an exemplification of the principles of the invention and the associated functional specifications for its construction and is not intended to limit the invention to the embodiment illustrated. Those skilled in the art will envision many other possible variations within the scope of the present invention.

[0038] Note that in this description, references to “one embodiment” or “an embodiment” mean that the feature being referred to is included in at least one embodiment of the invention. Further, separate references to “one embodiment” in this description do not necessarily refer to the same embodiment; however, neither are such embodiments mutually exclusive, unless so stated and except as will be readily apparent to those of ordinary skill in the art. Thus, the present invention can include any variety of combinations and / or integrations of the embodiments described herein.

[0039] The nanoprinting system of the present invention leverages self-assembled monolayers (SAMs) and FluidFM technology to achieve high-precision deposition of materials at the nanoscale. The use of low-viscosity print fluids allows for smaller printing nozzles and finer feature sizes. Additionally, we introduce an Electro-Femtofluidic Flow Sensing device to monitor flow rates in real-time.

[0040] FluidFM Technology

[0041] FluidFM combines the principles of atomic force microscopy with microfluidics, allowing for the controlled deposition of liquids through hollow AFM cantilevers. This technology is central to our nanoprinting system, providing the means to deposit materials with nanometer precision. 8 1530512.1Docket No.26163PCT

[0042] Electro-Femtofluidic Flow Sensing

[0043] To tackle the challenge of real-time monitoring during nanoprinting, particularly when using opaque substrates where conventional optical methods are ineffective, we conceptualized an Electro-Femtofluidic Flow Sensing device. This is a 3D printed sensor designed to measure flow rates on the order of femtoliters per second, providing critical real-time feedback on the printing process. This device is a significant advancement in the field. The preliminary design principles and modeling of the present invention are disclosed herein.

[0044] 1) Working Principle: As illustrated in FIG. 1, the device operates on the fluidic analogue of Ohm’s Law, akin to how a shunt resistor functions in electrical circuits to measure current. We refer to this as the equivalent shunt hydraulic resistor principle (ESHR). In this embodiment, the present invention provides a fluid flow sensor comprising: (a) a first membrane (102); (b) a second membrane (104); (c) an electrically conductive fluid (110) enclosed within and defining an inter-membrane conductive-fluid channel between the first and second membranes; (d) a first Wheatstone bridge (300) operativelycoupled to the first membrane (102); (e) a second Wheatstone bridge (300) operativelycoupled to the second membrane (104); and (f) at least one processor (302) configured, when the sensor is fluidically connected in series with a flow channel carrying a fluid, to (i) determine pressures from the respective bridge signals, (ii) compute a pressure drop across the flow channel, and (iii) estimate volumetric flow based on the pressure drop and a predetermined hydraulic resistance of the flow channel according to the ESHR principle.

[0045] FIG.3(a) illustrates how the Wheatstone bridge and the electro-femtofluidic deviceare connected in principle. The electrically conductive fluid in the device (110) is essentially a resistive element, no different than a wire (labeled Rvin FIG.3(a)); the only difference is that, instead of copper, it is conductive fluid in a channel. This allows the cross-section of this “fluidic” wire to change as the membranes enclosing it (102, 104) are 9 1530512.1Docket No.26163PCT pressurized by the outer fluid whose flow rate is to be quantified. Each (110) would have its own Wheatstone bridge connected to it, and each bridge would output independent, real-time measurements of the pressure of the outer fluid at that point in the channel.

[0046] Each (110) would have its own Wheatstone bridge connected to it, and each bridgewould output independent, real-time measurements of the pressure of the outer fluid at that point in the channel. Although FIGS. 3(a)–3(b) depict a single Wheatstone bridge (300) for clarity, the device includes two substantially identical Wheatstone bridge circuits (300)—one operatively coupled to the first membrane (102) and one operatively coupled to the second membrane (104)—each providing an independent output.

[0047] The Wheatstone bridges are not pictured in FIG. 1, but it is implied that the electrically conductive fluid acts as the resistor Rvrepresented in FIG.3(a). Elements 106 and 108 simply represent solid walls / material to which the membranes are fixed. They serve no purpose other than to provide attachment for the membranes.

[0048] The reservoir membrane allows the user to tune the shunt hydraulic resistor value so it behaves like a fluidic potentiometer, and it can be pneumatically actuated independently by another fluid (such as air) so that the user can set the air pressure to some known value, which in turn determines the hydraulic resistance of that part of the device. This is what allows the user to estimate the flow rate of print fluid traveling through the device using the hydraulic analogy of Ohm’s law (Q=P / R), where P is the pressure between two points, R is the hydraulic resistance between the points, and Q is the flow rate. Recall P would be calculated from the measurements obtained from the two Wheatstone bridges.

[0049] Allowing R to be adjustable in this manner gives the user more control and helps account for potential manufacturing deficiencies / imperfections in that part of the device but also requires additional calibration experiments to determine the relationship between applied air pressure on the membrane (Pair) and the resulting hydraulic resistance of the 10 1530512.1Docket No.26163PCT device (R). Once the calibration experiments are performed, the user can set the air pressure on that reservoir membrane channel to a desired value, which in turn allows them to tune the hydraulic resistance of the device to whatever value they wish. This may change depending on how sensitive they want the device to be to changes in flow rate; for example, if they want to measure larger ranges of flow rate, they may use a smaller value for the fluidic potentiometer (also referred to as a tunable hydraulic shunt resistor), accepting less sensitivity by nature. In our case, we seek to measure fluid flow rates on the order of 1 – 10 fL / s (for printing applications) as well as on the order of 100 – 1000 fL / s (for general system performance monitoring and debugging), so this tunability is necessary since it would not be possible to manufacture the channel dimensions needed to get the hydraulic resistance required for this application. As is common in sensing applications, you can have large range or high sensitivity, but it is very difficult and often practically impossible to have both at the same time. This approach approximates that functionality through the use of a tunable hydraulic shunt resistor. In theory, one could measure even less than femtofluidic flow using this approach by using a higher value for Pair to increase the hydraulic resistance of the reservoir membrane part of the device. This is the basic principle.

[0050] By assessing the pressure differential before and after a fluid element of known hydraulic resistance, the flow rate can be accurately determined. The core component is a microfluidic channel equipped with two thin deformable membranes filled with an electrically conductive fluid. By way of non-limiting example, the electrically conductive fluid may comprise an aqueous saline solution (e.g., H2O with dissolved NaCl) or another ionic solution in which mobile ions permit charge transport under an applied potential difference. In other embodiments, the electrically conductive fluid may comprise a room- temperature liquid metal such as eutectic gallium–indium (EGaIn), which remains liquid at 11 1530512.1Docket No.26163PCT or near room temperature and is widely used in soft and stretchable electronics. These examples are illustrative and not limiting.

[0051] By placing these elements within a larger channel containing some fluid of interest under pressure, the membranes deflect inward, altering the channel’s cross-sectional area and thus its electrical resistance. This resistance change is detected using a custom Wheatstone Bridge circuit (300), converting minute resistance variations into measurable voltage differences. This setup enables real-time monitoring of the flow rate, allowing users to detect issues like probe clogging and to quantify the exact volume of material being deposited. As used herein, “deflection” refers to the deflection of both membranes (102, 104) in response to changes in the pressure of the outer fluid (indicated by arrows 112 and 114), consistent with the equivalent shunt hydraulic resistor (ESHR) principle.

[0052] 2) Design and Modeling: The design of the flow sensor involves a careful balance of several parameters to maximize sensitivity while ensuring structural integrity. Key design variables include membrane thickness (h), membrane length (L), membrane width (W), and the distance between the membranes (H), which defines the channel height. For clarity, the design parameters are depicted in FIGS.2(a)-(b). FIG.2(a) depicts an example CAD model of the present invention’s flow sensing sensor. First and second membranes (102, 104) lie within the left and right parts of a channel. FIG.2(b) depicts several frames from the present invention’s simulations relating pressure to membrane deflection.

[0053] FIG.2(a) shows the CAD model of a single pressure-sensing membrane element for the Electro-Femtofluidic Flow Sensor. Fig.2(b) shows several frames from simulations relating pressure to membrane deflection for pressures in the range of ~0–10 mbar. We ultimately settled on a design with membrane thickness h=5 μm, membrane length L=350 μm, membrane width W=800 μm, channel height H=10 μm, and Nmembranes=200.

[0054] In one implementation, four membrane design parameters were evaluated—membrane length, width, thickness, and the distance between adjacent membranes—to 12 1530512.1Docket No.26163PCT convert fluid pressure into a changing electrical signal. Parameter selection was guided by manual simulations (e.g., custom tailor-made ChatGPT Python environments) run alongside Fusion 360 simulations to balance sensitivity and structural integrity of the membrane design.

[0055] Increasing thickness improves structural integrity but reduces sensitivity; decreasing thickness improves sensitivity but risks durability. In unconstrained simulations, the maximum stress exceeded yield at ~200 mbar and reached ~40 MPa. With realistic constraints (adjacent membranes deflecting into one another), the stress is reduced and bounded to approximately 3 / 8 of the unconstrained maximum, and even in an extreme deflection case the strain is ~0.003, implying ~10 MPa (assuming linear stress–strain based on ~0.012 strain at ~40 MPa).

[0056] Manufacturing limits were set by the UpNano NanoOne system, which reliably produces ~3 μm features with ~5 μm spacing. Accordingly, a minimal feasible membrane thickness was selected with a ~1.8 safety factor, and membrane separation was chosen to respect spacing limits. Smaller separation increases sensitivity but also raises hydraulic resistance and makes the device harder to clear; calculations indicated clearing remains feasible (on the order of ~5000 s) with sufficiently high pressure.

[0057] Membrane length strongly influences sensitivity (four-side-fixed behavior), so small changes in length produce large changes in deflection and in the effective gauge area. The design targeted ~0.1 per mbar sensitivity in the 0–10 mbar range to enable femtoliter-level flow sensitivity, while supporting measurement over ~0–100 mbar (corresponding to ~1–100 fL / s). Excessive length was avoided to mitigate printing-induced warping; length was iterated to meet both sensitivity and range goals. Membrane width was set primarily by fabrication practicality—very wide membranes can warp or sag— recognizing that the double-fixed beam relation provides an upper bound for maximum deflection and the true maximum lies between upper and lower bounds. Finally, stacking 13 1530512.1Docket No.26163PCT more membranes increases total deflection area and signal strength but also increases channel hydraulic resistance and clearing time; a nominal selection of Nmembranes=200 balances these trade-offs, with scope for later tuning.

[0058] a) Membrane Thickness and Channel Height: Thinner membranes offer higher sensitivity due to increased deflection under pressure, leading to greater changes in electrical resistance. However, they are more susceptible to mechanical failure from stress concentrations at the fixed edges as well as manufacturing defects. We selected a membrane thickness of h = 5 µm to balance sensitivity and durability. The channel height (H) also plays a crucial role; a smaller H increases sensitivity but raises hydraulic resistance, potentially impeding fluid flow and making it more difficult to clear out uncured resin from the device, increasing manufacturing difficulty. We chose H = 5 µm to maintain a manageable hydraulic resistance to enable functionalization of the channel with electrically conductive fluid while still enabling high sensitivity.

[0059] b) Membrane Length and Width: The membrane length (L) influences both deflection sensitivity and electrical resistance. A longer membrane increases deflection under pressure but decreases electrical resistance due to a larger cross-sectional area. Conversely, the membrane width (W) affects electrical resistance without significantly impacting deflection sensitivity. Wider membranes increase the resistance, improving the sensitivity of the Wheatstone Bridge measurement. The optimal dimensions were determined to be L = W = 700, µm, considering fabrication constraints and the capabilities of our two-photon polymerization (2PP) nanoprinting system (Up- Nano).

[0060] c) Fabrication Constraints: The device is designed for fabrication using 2PP nanoprinting technology, which imposes certain limitations such as minimum feature sizes and maximum print field dimensions. Printing beyond a single field of view requires stitching, which can introduce mechanical weaknesses. Therefore, the membrane dimensions were optimized to avoid stitching while meeting the sensitivity requirements. 14 1530512.1Docket No.26163PCT

[0061] 3) Analytical Modeling: To predict membrane deflection in response to applied pressure, we employed an analytical model based on Euler-Bernoulli beam theory, adjusted for a membrane fixed on all four sides. The maximum deflection δmax is given by:

[0062] ^^ = ^^^^^^4^^^^^^32^^ℎ3(1)

[0064] • p is the applied pressure,

[0065] • E is the Young’s modulus of the membrane material (approximately 3.2, GPa for the UpPhoto material),

[0066] • h is the membrane thickness,

[0067] • L is the membrane length,

[0068] • α is a correction factor to account for the membrane being fixed on all four sides.

[0069] It was determined α = 1 / 2.75 through finite element simulations in Fusion 360, comparing the deflection of a beam fixed at two ends to a membrane fixed at four edges under identical loading conditions.

[0070] a) Deflection Profile: The deflection varies along both the length (x-direction) and width (y-direction) of the membrane due to the fixed boundary conditions. We approximate the deflection at any point (i, j) using a separable function based on cosine terms: ^^ 2^^^^∆^^ 2^^^^∆^^

[0071] ^^^^,^^=^^^^^^4 [cos( ^^+ ^^) + 1] [cos (^^+ ^^)+ 1] (2)

[0072] andy dimensions, respectively.

[0073] b) Electrical Resistance Calculation: The change in electrical resistance due to membrane deflection is calculated by discretizing the channel into small volumetric elements and summing their resistances. The resistance of each slice along the width is:

[0074] ^^^^∆^^ ^^^^^^^^^^(^^) =^^ (3)15 1530512.1Docket No.26163PCT

[0075] where:

[0076] • ρ is the resistivity of the conductive fluid,

[0077] • ^^ =^^ ∆^^ is the number of discretization steps along the length,

[0078] • H is the initial channel height,

[0079] • δi, j is the deflection at position (i, j).

[0080] The total resistance of the channel is then:

[0081] ^^^^^^^^ℎ^^^^^^^^^^ = ∑^^=1^^^^^^^^^^^^(^^)(4)

[0083] 4) FiniteModeling: To validate our analytical models and optimize the design parameters, we conducted finite element analysis (FEA) simulations using Python. The simulations accounted for material properties, geometric constraints, and the interaction between mechanical deformation and electrical resistance changes.

[0084] a) Resistance Change Simulation: The FEA also simulated the conductive fluid’s resistance change due to membrane deflection. By discretizing the channel and calculating the resistance of each element, we obtained a detailed resistance profile under various pressure conditions. This allowed us to predict the sensor’s electrical response under varying pressure conditions accurately.

[0085] 5) Measurement Circuit: A custom Wheatstone Bridge circuit (300) is employed to detect the minute changes in electrical resistance caused by membrane deflection. The Wheatstone Bridge (300) is an established method for converting small resistance changes into measurable voltage differences, commonly used in strain gauge applications.

[0086] a) Circuit Configuration: The bridge (300) consists of four resistive elements arranged in a diamond configuration. One arm contains the variable resistance of the flow sensing electrofluidic resistive element (110) (Rv), while the other three arms consist of 16 1530512.1Docket No.26163PCT fixed resistors (R1, R2, R3). The change in voltage across the bridge (∆v) is proportional to the resistance change in ∆R:

[0087] ∆^^ = ^^1 1 ^^ (^^^^+1 −^^^^+∆^^) (5)^^3^^3+1 bridge(300) is shown in FIG.3(a).

[0089] b) Sensitivity and Calibration: By carefully selecting the fixed resistors and balancing the bridge (300) under no-flow conditions, the circuit is highly sensitive to changes in Rv. Calibration procedure would involve applying known pressures and recording the corresponding voltage change ∆v, creating a calibration curve that relates voltage output to pressure applied to the membrane.

[0090] 6) Flow Sensing Capabilities: The flow sensing capabilities of the device will heavily depend on the sensitivity of the change in electrical resistance to the applied pressure. Our simulations have shown that we can measure a change of ∼ 200 µV for a pressure change of 1 mbar, and from previous studies, authors have found that a pressure of 1 mbar corresponds to a flow rate of 2 fL / s when using the Cytosurge probe. We have stress-tested the membranes in simulation and have found that they should be able to withstand around 500 mbar of pressure without yielding in the current configuration. Further, given that oscilloscopes are commonly designed to be able to detect voltage differences as small as 10 µV, our sensor can theoretically detect flow rates between 0 and 1000 fL / s with high fidelity (on the order of 1 fL / s).

[0091] Additional work focuses on the fabrication and experimental validation of the device first on the macroscale to develop the proof of concept, then refining the process for micro / nanoscale functionality and integrating the sensor into the nanoprinting system.

[0092] In one embodiment, the present invention provides a fluid flow sensor comprising: (a) a first membrane (102); (b) a second membrane (104); (c) an electrically conductive 17 1530512.1Docket No.26163PCT fluid (110) enclosed within and defining an inter-membrane conductive-fluid channel between the first and second membranes; (d) a first Wheatstone bridge (300) operatively coupled to the first membrane (102) and configured to output a first bridge signal indicative of deflection of the first membrane (102); (e) a second Wheatstone bridge (300) operatively coupled to the second membrane (104) and configured to output a second bridge signal indicative of deflection of the second membrane (104); and (f) at least one processor (302) operatively coupled to the first and second Wheatstone bridges and, when the sensor is fluidically connected in series with a flow channel carrying a fluid, the at least one processor (302) configured to: (i) determine a first pressure at the first membrane (102) and a second pressure at the second membrane (104) from the first and second bridge signals, respectively; (ii) compute a pressure drop across the flow channel as a difference between the first and second pressures; and (iii) estimate a volumetric flow rate of the fluid in the flow channel based on the pressure drop and a predetermined hydraulic resistance of the flow channel according to an equivalent shunt hydraulic resistor (ESHR) principle.

[0093] In one embodiment, at least one of the first and second Wheatstone bridges (300) comprises four resistive elements arranged in a diamond configuration, one arm including a variable resistor Rvformed by a portion of the electrically conductive fluid (110) within the inter-membrane conductive-fluid channel, and the remaining arms including fixed resistors R1, R2, R3, and wherein a change in bridge output voltage ∆v between the bridge sense nodes is given by: )18 1530512.1Docket No.26163PCT where ^^^^is a supply voltage (e.g. from a battery, power supply, etc.) applied to the bridge and ΔR is a change in the resistance of Rv caused by pressure-induced deflection of themembrane due to fluid pressure in the flow channel.

[0094] In one embodiment, a total electrical resistance of the variable resistor Rv formedby the electrically conductive fluid (110) within the inter-membrane conductive-fluid channel is obtained by modeling the inter-membrane conductive-fluid channel as M serial ^^slices and summing their respective slice resistances,^^^^ = ∑ ^^=1 ^^^^^^^^^^^^(^^).

[0095] In one embodiment, ^^ (^^) =^^∆^^ ^^^^^^^^^^ ∑^^ , resistivity of ^^=1[^^−2^^^^,^^]∆^^the electrically conductive fluid (110), ∆x and ∆y are discretization steps along the lengthand width of the inter-membrane conductive-fluid channel, respectively, ^^ =^^ is a ∆^^ number of discretization steps along a length L of the inter-membrane fluidchannel, H is an initial inter-membrane conductive-fluid channel height, and δi, jaccounts for membrane deflection at position (i, j).

[0096] In one embodiment, for each of the first and second membranes, deflection δi, j varies along both a length and a width of the first and second membranes due to fixed boundary conditions and is given by: ^^ 2^^^^∆^^ 2^^^^∆^^^^^^^^ cos ( + ^^) + 1 cos ( + ^^) + 1 [ ] [ ]^^ =^^,^^^^ ^^where L is the the first and second membranes, and δmaxis a maximum deflection of the first and second membranes. 4 ^^^^

[0097] In one embodiment, ^^ = ^^, where p is pressure of the fluid in the flow^^^^^^channel acting on the first and E is the Young’s modulus of a material associated with the first and second membranes, h is the membrane thickness, and α is a correction factor to account for the membrane being fixed on all four sides. 19 1530512.1Docket No.26163PCT

[0098] In one embodiment, the Young’s modulus is about 3.2 GPa.

[0099] In one embodiment, h=H.

[0100] In one embodiment, h=H=5 µm.

[0101] In one embodiment, L=W.

[0102] In one embodiment, L=W=700 µm.

[0103] In one embodiment, the fluid flow sensor is configured to detect flow rates in the range of 1 to 1000 fL / s.

[0104] In another embodiment, the present invention provides a method for estimating a flow rate, the method comprising: (a) providing a fluid flow sensor, the fluid flow sensor comprising: (i) a first membrane (102); (ii) a second membrane (104); (iii) an electrically conductive fluid (110) enclosed within and defining an inter-membrane conductive-fluid channel between the first and second membranes (102, 104); (iv) a first Wheatstone bridge (300) operatively coupled to the first membrane (102) and a second Wheatstone bridge (300) operatively coupled to the second membrane (104); and (v) at least one processor (302); (b) fluidically connecting the fluid flow sensor in series with a flow channel carrying a fluid; (c) obtaining, from the first Wheatstone bridge (300) and the second Wheatstone bridge (300), a first bridge signal and a second bridge signal, respectively; and (d) using the at least one processor (302) to: (i) determine a first pressure at the first membrane (102) and a second pressure at the second membrane (104) from the first and second bridge signals, respectively; (ii) compute a pressure drop across the flow channel as a difference between the first and second pressures; and (iii) estimate a volumetric flow rate of the fluid in the flow channel based on the pressure drop and a predetermined hydraulic resistance of the flow channel according to an equivalent shunt hydraulic resistor (ESHR) principle.

[0105] In one embodiment, at least one of the first and second Wheatstone bridges (300) comprises four resistive elements arranged in a diamond configuration, one arm including 20 1530512.1Docket No.26163PCT a variable resistor Rvformed by a portion of the electrically conductive fluid (110) within the inter-membrane conductive-fluid channel, and the remaining arms including fixed resistors R1, R2, and R3, and wherein a change in bridge output voltage ∆v between the bridge sense nodes is given by: ∆^^ = ^^1 1 ^^ (− ) where ^^^^is a supplyetc.) applied to the bridge and ΔR is a change in the resistance of Rv caused by pressure-induced deflection of the membrane due to fluid pressure in the flow channel.

[0106] In one embodiment, a total electrical resistance of the variable resistor Rvformed by the electrically conductive fluid (110) within the inter-membrane conductive-fluid channel is obtained by modeling the inter-membrane conductive-fluid channel as M serial slices and summing their respective slice resistances, ^^^^ = ∑^^ ^^=1 ^^^^^^^^^^^^(^^).

[0107] In one embodiment, ^^ (^^) =^^∆^^ ^^^^^^^^^^ ∑^^ ^^=1[^^−2^^^^,^^]∆^^, where ρ is the electrical resistivity of the electrically conductive fluidare discretization steps along the lengthand width of the inter-membrane conductive-fluid channel, respectively, ^^ =^^ ∆^^ is anumber of discretization steps along a length L of the inter-membranefluid channel, H is an initial inter-membrane conductive-fluid channel height, and δi, jaccounts for membrane deflection at position (i, j).

[0108] In one embodiment, for each of the first and second membranes, deflection, δi, j, varies along both a length and a width of the first and second membranes due to fixed boundary conditions and is given by: ^^ 2^^^^∆^^ 2^^^^∆^^ =^^^^^^ + + + + 1]21 1530512.1Docket No.26163PCT where L is the length of the first and second membranes, W is the width of the first and second membranes, and δ is a maximum deflection of the first and secondmax membranes.

[0109] In one embodiment, ^^ = ^^^^^^4^^^^^^32^^ℎ3, where p is pressure of the fluid in the flow channel acting on the first and E is the Young’s modulus of a materialassociated with the first and second membranes, h is the membrane thickness, and L is the membrane length, and α is a correction factor to account for the membrane being fixed on all four sides.

[0110] In one embodiment, the Young’s modulus is about 3.2 GPa.

[0111] In one embodiment, h=H.

[0112] In one embodiment, h=H=5 µm.

[0113] In one embodiment, L=W.

[0114] In one embodiment, L=W=700 µm.

[0115] In one embodiment, the fluid flow sensor is configured to detect flow rates in the range of 1 to 1000 fL / s.

[0116] METHODS

[0117] Nanoprinting Hardware Assembly

[0118] The nanoprinting hardware of the present invention consists of a custom AFM setup integrated with FluidFM capabilities. Key components include a probe gantry, an inverted microscope, and a laser alignment system using a LASOS LGK 7672 He-Ne laser and Thorlabs optical components. A PK523HPA-H50S 5-phase stepper motor controlled via a CVD507BR-K driver facilitates probe-substrate engagement.

[0119] Software Design

[0120] Custom LabVIEW Virtual Instruments (VIs) were developed, and a 7868R FPGA was utilized for data acquisition and control. The software manages the probe-substrate 22 1530512.1Docket No.26163PCT engagement protocols and nanoprinting sequences, including printing dot grids and NCode-based patterns using the Nanocube 3-axis piezo nanopositioner.

[0121] Substrate Preparation

[0122] 1) Substrate Pre-Marking Protocol: Gold-coated silicon wafers from Sigma Aldrich were cleaved into smaller strips using wafer-cleaving tools. The Keyence Laser Scriber at MIT Nano was employed to create precise markings on the substrate, aiding in alignment during printing.

[0123] 2) SAM Chemical Preparation Procedure: The marked substrates were further cut into 5x5 mm pieces. They underwent a SAM formation process involving immersion in a C9-nonanedithiol solution, followed by rinsing with hexane, ethanol, and DI water.

[0124] FluidFM Functionalization

[0125] We used hollow micro / nano-AFM cantilevers from Cyto-surge with 300 nm apertures [see FluidFM Nanopipette, Cytosurge AG]. The cantilevers were pre-filled with the print fluid using a pressure-controlled protocol, monitored via an inverted microscope setup. Custom printed cantilevers required additional cleaning and verification using the ZEISS Xradia 620 Versa X-Ray Microscope.

[0126] Substrate Printing Preparation

[0127] The SAM-formed substrates were mounted onto a 50 mm petri dish using UV glue. A custom 3D-printed petri dish holder secured the substrate within the nanoprinting hardware setup. An acrylic cover piece was used to form a meniscus column of silver nitrate between the substrate and the cover.

[0128] Laser Alignment

[0129] Laser alignment was performed using standard AFM procedures [see Thorlabs, EDU AFM1-Manual.LLS.pdf]. The setup included a PDQ80A photodiode and micropositioners for precise adjustments. The alignment ensured optimal laser focus on the cantilever and accurate deflection measurements. 23 1530512.1Docket No.26163PCT

[0130] Probe-Substrate Engagement

[0131] The engagement protocol was executed through high-speed FPGA VI in LabVIEW. The stepper motors controlled the approach and retraction of the probe with nanometer precision, critical for successful nanoprinting.

[0132] Printing Protocols

[0133] Two printing protocols were developed: one for printing dot grids and another for NCode-based patterns using the Nanocube 3-axis piezo nanopositioner. NCode serves as our version of GCode, adapted for nanoscale positioning commands.

[0134] PRINTING MODES

[0135] Conventional FluidFM Cantilevers

[0136] Using Cytosurge’s FluidFM cantilevers, preliminary printing results were achieved demonstrating the system’s basic functionality. These cantilevers are widely used in literature and offer reliable performance for single-material deposition.

[0137] Custom 3D Printed Multiscale Hollow Cantilevers

[0138] Custom 3D printed hollow cantilevers were developed and capable of depositing multiple materials through a single output channel. The fabrication process involved using the UpNano 2PP printing system with a specialized protocol:

[0139] a. Large Scale (mm) Print with 5X or 10X objective.

[0140] b. Align Corner and Switch to 20X objective.

[0141] c. Re-align and focus on the corner with 20X.

[0142] d. Arrange part in the design window.

[0143] e. Center the part laterally and vertically.

[0144] f. Start print (iterate back if polymerization issues occur).

[0145] EXPERIMENTAL RESULTS

[0146] Nanoprinting Outcomes 24 1530512.1Docket No.26163PCT

[0147] The system of the present invention successfully printed nanoscale features using both printing modes. Optical micrographs and SEM images corroborate the efficacy of the methods. The printed structures demonstrate the potential for fabricating complex technorganic machines. FIGS. 5(a)–5(d) and 6(a)–6(b) illustrate these findings. FIGS. 5(a)–5(d) are optical micrographs of printed nanoscale features produced using conventional FluidFM cantilevers: FIG. 5(a) shows the result after leaving the system printing in the same location for 8 hours (scale bar: 1 mm); FIG. 5(b) shows the same region as FIG.5(a) with the probe removed from the view (scale bar: 2 mm); FIG. 5(c) shows an overhead view of the printed structures (scale bar: 3 mm); and FIG.5(d) shows a close-up of the printed structures (scale bar: 1 mm). FIGS.6(a)–6(b) are SEM images of the printed structures, obtained when ethanol is used as the print solvent, confirming the technorganic morphologies described by Hamoudi et al., “Building block 3D printing based on molecular self-assembly monolayer with self-healing properties.” The sample imaged in FIGS.6(a)–6(b) is the same sample shown in FIG.5(d).

[0148] Functionalization Verification

[0149] The successful functionalization of custom 3D printed probes were verified using high-resolution imaging techniques. The ZEISS Xradia 620 Versa X-Ray Microscope provided detailed images of the hollow channels, confirming proper cleaning and fluid flow capabilities.

[0150] Modeling and Simulation Results

[0151] Simulation results indicate that the proposed sensor design can achieve the desired sensitivity for flow rates between 0 and 1000 fL / s. The stress analysis confirms the structural integrity under operational pressures. FIG.4 shows the simulated deflection profile of the membrane under pressure. From this plot, we can gather that the sensitivity is 100 milli-Ω per mbar of pressure in the range of 0 to 10 mbar. Beyond 100 mbar, the resistance increases rapidly and becomes highly nonlinear. Included in the figure are plots 25 1530512.1Docket No.26163PCT of how the membrane would deflect at various pressures (10, 30, ..., 90 mbar). In FIG.4, the inset boxes schematically illustrate cross-sections of the inter-membrane conductive- fluid channel (110) at successively higher pressures in the flow channel. As the flow- channel pressure increases, the membranes (102, 104) deflect toward one another, narrowing and reshaping the available cross-section through which the electrically conductive fluid conducts charge. This progressive geometric constriction provides an intuitive depiction of why the bridge’s variable resistor Rvincreases with pressure: the area available for electron / ion transport becomes more constrained, so the electrical resistance of the conductive-fluid path rises.

[0152] The above-described features and applications can be implemented as software processes that are specified as a set of instructions recorded on a computer readable storage medium (also referred to as computer readable medium). When these instructions are executed by one or more processing unit(s) (e.g., one or more processors, cores of processors, or other processing units), they cause the processing unit(s) to perform the actions indicated in the instructions. Embodiments within the scope of the present disclosure may also include tangible and / or non-transitory computer-readable storage media for carrying or having computer-executable instructions or data structures stored thereon. Such non-transitory computer-readable storage media can be any available media that can be accessed by a general purpose or special purpose computer, including the functional design of any special purpose processor. By way of example, and not limitation, such non-transitory computer-readable media can include flash memory, RAM, ROM, EEPROM, CD-ROM or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to carry or store desired program code means in the form of computer-executable instructions, data structures, or processor chip design. The computer readable media does not include carrier waves and electronic signals passing wirelessly or over wired connections. 26 1530512.1Docket No.26163PCT

[0153] Computer-executable instructions include, for example, instructions and data which cause a general purpose computer, special purpose computer, or special purpose processing device to perform a certain function or group of functions. Computer- executable instructions also include program modules that are executed by computers in stand-alone or network environments. Generally, program modules include routines, programs, components, data structures, objects, and the functions inherent in the design of special-purpose processors, etc. that perform particular tasks or implement particular abstract data types. Computer-executable instructions, associated data structures, and program modules represent examples of the program code means for executing steps of the methods disclosed herein. The particular sequence of such executable instructions or associated data structures represents examples of corresponding acts for implementing the functions described in such steps.

[0154] Processors suitable for the execution of a computer program include, by way of example, both general and special purpose microprocessors, and any one or more processors of any kind of digital computer. Generally, a processor will receive instructions and data from a read-only memory or a random access memory or both. The essential elements of a computer are a processor for performing or executing instructions and one or more memory devices for storing instructions and data. Generally, a computer will also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto-optical disks, or optical disks. However, a computer need not have such devices.

[0155] These functions described above can be implemented in digital electronic circuitry, in computer software, firmware or hardware. The techniques can be implemented using one or more computer program products. Programmable processors and computers can be included in or packaged as mobile devices. The processes and logic flows can be performed by one or more programmable processors and by one or more programmable 27 1530512.1Docket No.26163PCT logic circuitry. General and special purpose computing devices and storage devices can be interconnected through communication networks.

[0156] Some implementations include electronic components, for example microprocessors, storage and memory that store computer program instructions in a machine-readable or computer-readable medium (alternatively referred to as computer- readable storage media, machine-readable media, or machine-readable storage media). Some examples of such computer-readable media include RAM, ROM, read-only compact discs (CD-ROM), recordable compact discs (CD-R), rewritable compact discs (CD-RW), read-only digital versatile discs (e.g., DVD-ROM, dual-layer DVD-ROM), a variety of recordable / rewritable DVDs (e.g., DVD-RAM, DVD-RW, DVD+RW, etc.), flash memory (e.g., SD cards, mini-SD cards, micro-SD cards, etc.), magnetic or solid state hard drives, read-only and recordable Blu-Ray®discs, ultra density optical discs, any other optical or magnetic media, and floppy disks. The computer-readable media can store a computer program that is executable by at least one processing unit and includes sets of instructions for performing various operations. Examples of computer programs or computer code include machine code, for example is produced by a compiler, and files including higher- level code that are executed by a computer, an electronic component, or a microprocessor using an interpreter.

[0157] While the above discussion primarily refers to microprocessor or multi-core processors that execute software, some implementations are performed by one or more integrated circuits, for example application specific integrated circuits (ASICs) or field programmable gate arrays (FPGAs). In some implementations, such integrated circuits execute instructions that are stored on the circuit itself.

[0158] It is understood that any specific order or hierarchy of steps in the processes disclosed is an illustration of example approaches. Based upon design preferences, it is understood that the specific order or hierarchy of steps in the processes may be 28 1530512.1Docket No.26163PCT rearranged, or that all illustrated steps be performed. Some of the steps may be performed simultaneously. For example, in certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system components illustrated above should not be understood as requiring such separation, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products.

[0159] Various modifications to these aspects will be readily apparent, and the generic principles defined herein may be applied to other aspects. Thus, the claims are not intended to be limited to the aspects shown herein, but is to be accorded the full scope consistent with the language claims, where reference to an element in the singular is not intended to mean “one and only one” unless specifically so stated, but rather “one or more.” Unless specifically stated otherwise, the term “some” refers to one or more. Pronouns in the masculine (e.g., his) include the feminine and neuter gender (e.g., her and its) and vice versa. Headings and subheadings, if any, are used for convenience only and do not limit the subject technology.

[0160] A phrase, for example, an “aspect” does not imply that the aspect is essential to the subject technology or that the aspect applies to all configurations of the subject technology. A disclosure relating to an aspect may apply to all configurations, or one or more configurations. A phrase, for example, an aspect may refer to one or more aspects and vice versa. A phrase, for example, a “configuration” does not imply that such configuration is essential to the subject technology or that such configuration applies to all configurations of the subject technology. A disclosure relating to a configuration may apply to all configurations, or one or more configurations. A phrase, for example, a configuration may refer to one or more configurations and vice versa. 29 1530512.1Docket No.26163PCT

[0161] The various embodiments described above are provided by way of illustration only and should not be construed to limit the scope of the disclosure. Those skilled in the art will readily recognize various modifications and changes that may be made to the principles described herein without following the example embodiments and applications illustrated and described herein, and without departing from the spirit and scope of the disclosure.

[0162] While this specification contains many specific implementation details, these should not be construed as limitations on the scope of any invention or of what may be claimed, but rather as descriptions of features that may be specific to particular embodiments of particular inventions. Certain features that are described in this specification in the context of separate embodiments can also be implemented in combination in a single embodiment. Conversely, various features that are described in the context of a single embodiment can also be implemented in multiple embodiments separately or in any suitable subcombination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a subcombination or variation of a subcombination.

[0163] Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system components in the embodiments described above should not be understood as requiring such separation in all embodiments, and it should be understood that the described program components 30 1530512.1Docket No.26163PCT and systems can generally be integrated together in a single software product or packaged into multiple software products.

[0164] As noted above, particular embodiments of the subject matter have been described, but other embodiments are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As one example, the processes depicted in the accompanying figures do not necessarily require the particular order shown, or sequential order, to achieve desirable results. In certain implementations, multitasking and parallel processing may be advantageous. CONCLUSION

[0165] A system and method have been shown in the above embodiments for the effective implementation of a 3D printed electro-femtofluidic flow sensor. While various preferred embodiments have been shown and described, it will be understood that there is no intent to limit the invention by such disclosure, but rather, it is intended to cover all modifications falling within the spirit and scope of the invention, as defined in the appended claims. 31 1530512.1

Claims

Docket No.26163PCT WHAT IS CLAIMED IS:

1. A fluid flow sensor comprising: (a) a first membrane (102); (b) a second membrane (104); (c) an electrically conductive fluid (110) enclosed within and defining an inter- membrane conductive-fluid channel between the first and second membranes; (d) a first Wheatstone bridge (300) operatively coupled to the first membrane (102) and configured to output a first bridge signal indicative of deflection of the first membrane (102); (e) a second Wheatstone bridge (300) operatively coupled to the second membrane (104) and configured to output a second bridge signal indicative of deflection of the second membrane (104); and (f) at least one processor (302) operatively coupled to the first and second Wheatstone bridges and, when the sensor is fluidically connected in series with a flow channel carrying a fluid, the at least one processor (302) configured to: i. determine a first pressure at the first membrane (102) and a second pressure at the second membrane (104) from the first and second bridge signals, respectively; ii. compute a pressure drop across the flow channel as a difference between the first and second pressures; and iii. estimate a volumetric flow rate of the fluid in the flow channel based on the pressure drop and a predetermined hydraulic resistance of the flow channel according to an equivalent shunt hydraulic resistor (ESHR) principle. 32 1530512.1Docket No.26163PCT 2. The fluid flow sensor of claim 1, wherein at least one of the first and second Wheatstone bridges (300) comprises four resistive elements arranged in a diamond configuration, one arm including a variable resistor Rv formed by a portion of the electrically conductive fluid (110) within the inter-membrane conductive-fluid channel, and the remaining arms including fixed resistors R1, R2, R3, and wherein a change in bridge output voltage ∆v between the bridge sense nodes is given by: ∆^^ = ^^1 1 ^^− ) where ^^^^is a supplyis a change in the resistanceof Rv caused by pressure-induced deflection of the membrane due to fluid pressure in the flowchannel.

3. The fluid flow sensor of claim 2, wherein a total electrical resistance of the variable resistor Rv formed by the electrically conductive fluid (110) within the inter-membrane conductive-fluid channel is obtained by modeling the inter-membrane conductive-fluid channel as M serial slices and summing their respective slice resistances, ^^^^ = ∑^^ ^^=1 ^^^^^^^^^^^^(^^).

4. The fluid flow sensor of claim 3, wherein ^^ (^^) =^^∆^^ ^^^^^^^^^^ ∑^^ ^^=1[^^−2^^^^,^^]∆^^where: ρ is the electrical resistivity of the electrically conductive fluid (110), ∆x and ∆y are discretization steps along the length and width of the inter-membrane conductive-fluid channel, respectively, ^^ =^^ ∆^^ is a number of discretization steps along a length L of the inter- membranefluid channel, H is an initial inter-membrane conductive-fluid channel height, and 33 1530512.1Docket No.26163PCT δi, jaccounts for membrane deflection at position (i, j).

5. The fluid flow sensor of claim 4, wherein, for each of the first and second membranes, deflection δi, j varies along both a length and a width of the first and second membranes due to fixed boundary conditions and is given by: ^^ 2^^^^∆^^ 2^^^^∆^^ ^^^^,^^=^^^^^^4[cos (^^+ ^^) + 1] [cos (^^+ ^^) + 1]where L isof the first and second membranes, and δmaxis a maximum deflection of the first and second membranes.

6. The fluid flow sensor of claim 5, wherein ^^ = ^^^^^^4^^^^^^32^^ℎ3where: p is pressure of the fluid in the flow channel acting on the first and second membranes, E is the Young’s modulus of a material associated with the first and second membranes, h is the membrane thickness, and α is a correction factor to account for the membrane being fixed on all four sides.

7. The fluid flow sensor of claim 6, wherein the Young’s modulus is about 3.2 GPa.

8. The fluid flow sensor of claim 6, wherein h=H.

9. The fluid flow sensor of claim 8, wherein h=H=5 µm.

10. The fluid flow sensor of claim 6, wherein L=W.

11. The fluid flow sensor of claim 10, wherein L=W=700 µm.

12. The fluid flow sensor of claim 1, wherein the fluid flow sensor is configured to detect flow rates in the range of 1 to 1000 fL / s.

13. A method for estimating a flow rate, the method comprising: (a) providing a fluid flow sensor, the fluid flow sensor comprising: i. a first membrane (102); 34 1530512.1Docket No.26163PCT ii. a second membrane (104); iii. an electrically conductive fluid (110) enclosed within and defining an inter- membrane conductive-fluid channel between the first and second membranes (102, 104); iv. a first Wheatstone bridge (300) operatively coupled to the first membrane (102) and a second Wheatstone bridge (300) operatively coupled to the second membrane (104); and v. at least one processor (302); (b) fluidically connecting the fluid flow sensor in series with a flow channel carrying a fluid; (c) obtaining, from the first Wheatstone bridge (300) and the second Wheatstone bridge (300), a first bridge signal and a second bridge signal, respectively; and (d) using the at least one processor (302) to: i. determine a first pressure at the first membrane (102) and a second pressure at the second membrane (104) from the first and second bridge signals, respectively; ii. compute a pressure drop across the flow channel as a difference between the first and second pressures; and iii. estimate a volumetric flow rate of the fluid in the flow channel based on the pressure drop and a predetermined hydraulic resistance of the flow channel according to an equivalent shunt hydraulic resistor (ESHR) principle.

14. The method of claim 13, wherein at least one of the first and second Wheatstone bridges (300) comprises four resistive elements arranged in a diamond configuration, one arm including a variable resistor Rv formed by a portion of the electrically conductive fluid (110) within the inter-membrane conductive-fluid channel, and the remaining arms including fixed resistors R1, 35 1530512.1Docket No.26163PCT R2, and R3, and wherein a change in bridge output voltage ∆v between the bridge sense nodes is given by: ∆^^ = ^^1 ^^ (1 ^^^^−^^^^) where ^^^^is a supplyis a change in the resistance of Rv caused by pressure-induced deflection of the membrane due to fluid pressure in the flow channel.

15. The method of claim 14, wherein a total electrical resistance of the variable resistor Rv formed by the electrically conductive fluid (110) within the inter-membrane conductive-fluid channel is obtained by modeling the inter-membrane conductive-fluid channel as M serial slices and summing their respective slice resistances, ^^^^ = ∑^^ ^^=1 ^^^^^^^^^^^^(^^).

16. The method of claim 15, wherein^^=1∆^^where:ρ is the electrical resistivity of the electrically conductive fluid (110), ∆x and ∆y are discretization steps along the length and width of the inter-membrane conductive-fluid channel, respectively, ^^ =^^ ∆^^ is a number of discretization steps along a length L of the inter- membranefluid channel, H is an initial inter-membrane conductive-fluid channel height, and δi, jaccounts for membrane deflection at position (i, j).

17. The method of claim 16, wherein, for each of the first and second membranes, deflection, δi, j, varies along both a length and a width of the first and second membranes due to fixed boundary conditions and is given by: 36 1530512.1Docket No.26163PCT ^^ 2^^^^∆^^ 2^^^^∆^^ ^^^^,^^=^^^^^^4[cos (^^+ ^^) + 1] [cos (^^+ ^^) + 1]where L is width of the first andsecond membranes, and δmax is a maximum deflection of the first and second membranes.

18. The method of claim 17, wherein ^^^^^^4^^^^^^ = ^^32^^ℎ3where:p is pressure of the fluid in the flow channel acting on the first and second membranes, E is the Young’s modulus of a material associated with the first and second membranes, h is the membrane thickness, and L is the membrane length, α is a correction factor to account for the membrane being fixed on all four sides.

19. The method of claim 18, wherein the Young’s modulus is about 3.2 GPa.

20. The method of claim 18, wherein h=H.

21. The method of claim 20, wherein h=H=5 µm.

22. The method of claim 18, wherein L=W.

23. The method of claim 22, wherein L=W=700 µm.

24. The method of claim 13, wherein the fluid flow sensor is configured to detect flow rates in the range of 1 to 1000 fL / s. 37 1530512.1

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