Microfluidics — Lab-on-Chip, Low-Re Flow — Engineering Reference

1. At a glance

Microfluidics is the engineering of fluid flow inside channels with at least one cross-sectional dimension between roughly 1 µm and 1 mm. At these scales the Reynolds number is almost always far below unity, so the flow is laminar, time-reversible, and surface-tension-dominated, and most of the intuition from macro-scale pipe flow breaks down. Three families of physics dominate the field: low-Re viscous flow, capillarity and wetting, and electrokinetics (electroosmotic flow + electrophoresis).

Practical microfluidic systems integrate sample handling, reagent metering, mixing, reaction, separation, and detection onto a single chip — the lab-on-chip (LOC) concept. The discipline sits at the intersection of fluid mechanics, chemical engineering, surface chemistry, and microelectronics manufacturing.

End-uses by market segment:

• In-vitro diagnostics (IVD) — lateral-flow immunoassays (LFA, e.g. SARS-CoV-2 antigen, pregnancy hCG), point-of-care molecular tests (Cepheid GeneXpert, Abbott ID NOW, Roche cobas Liat).
• Genomics and single-cell biology — Illumina library prep, 10x Genomics Chromium, Bio-Rad QX600 droplet digital PCR, Drop-seq / inDrops.
• Drug discovery and screening — Berkeley Lights Beacon (now Bruker), Sphere Fluidics Cyto-Mine, organ-on-chip (Emulate Bio, AlveoliX).
• Industrial chemistry — continuous-flow microreactors for kg-scale synthesis at seconds-residence time.
• Oil and gas — downhole microfluidic sensors, droplet-based core analysis.
• Printing — inkjet (HP, Canon, Memjet, EFI) is fundamentally microfluidic droplet generation at kHz–MHz.

Position in the engineering stack: fluid-mechanics (low-Re regime) → microfluidics → application domain (IVD / NGS / chemistry / printing). Companion disciplines: mems (the manufacturing substrate), heat-transfer (microchannel cooling), materials-polymers (PDMS, COC, PMMA chips), electromagnetics-engineering (electrokinetics and dielectrophoresis).

2. Why it matters

Microfluidics earns its place in industry through three orders of magnitude reduction in sample and reagent volume — µL → nL → pL — without losing assay performance. For a clinical sequencing run that was once 100 µL of reagent at $5/µL, a droplet-based protocol uses 100 pL at the same chemistry concentration: a million-fold cost compression per reaction. That arithmetic underpins the entire NGS, ddPCR, and single-cell genomics industries.

Three further drivers:

1. Integration of laboratory protocols. Sample prep, amplification, and readout fit on one disposable cartridge — Cepheid’s GeneXpert (TB, COVID, HIV viral load) is the canonical example. The cartridge is the laboratory.
2. Massive parallelism. A single 10x Chromium chip partitions a sample into ~80 000–100 000 droplets in minutes, each acting as an independent reaction vessel. ddPCR runs 20 000+ partitions per well for absolute nucleic-acid quantification.
3. Speed. Diffusion times scale as L², so a 100 µm mixing length completes ion mixing in seconds rather than the minutes a stirred 100 mL beaker needs.

Three failure modes that any production microfluidics program must engineer around:

• Bubble entrainment — compressible volume in an otherwise incompressible system; flow stops. Mitigated by degassing, bubble traps, and hydrophilic surface treatment.
• Surface adsorption — proteins (notably albumin) and DNA stick to bare PDMS and PMMA, shifting assay calibration; addressed with passivation (BSA blocking, PEG coatings, Pluronic F-127) or fluorinated wall chemistry.
• Lot-to-lot variability of polymer chips — surface energy of PDMS recovers from plasma activation over hours to days; cure ratio (10:1 base:crosslinker, Sylgard 184) affects modulus and bond strength.

3. First principles

3.1 Dimensionless numbers in the microfluidic regime

Number	Formula	Role at micro-scale	Typical value (10–500 µm channel, aqueous, mm/s flow)
Reynolds Re	ρ·U·D_h/µ	Inertia vs viscous → Re ≪ 1 ⇒ Stokes flow	0.001 – 10 (almost always laminar)
Péclet Pe	U·L/D_diff	Convection vs diffusion (mixing)	10² – 10⁵ — high; mixing is diffusion-limited
Capillary Ca	µ·U/γ	Viscous vs surface tension (droplet generation)	10⁻⁵ – 10⁻¹ (sets drop-vs-jet regime)
Bond / Eötvös Bo	ρ·g·L²/γ	Gravity vs surface tension	10⁻⁶ – 10⁻³ — gravity negligible
Weber We	ρ·U²·L/γ	Inertia vs surface tension (inkjet)	1 – 100 (inkjet)
Knudsen Kn	λ/L	Continuum validity (gas, sub-µm channels)	10⁻³ – 10⁻¹ (slip onset)
Dean De	Re · √(D_h/2R_c)	Secondary flow in curved channels	0.01 – 5
Elasticity El	λ_relax·µ/(ρ·L²)	Polymer viscoelasticity	depends on solution

3.2 Hagen–Poiseuille and hydraulic resistance

For a circular channel:

ΔP = 32·µ·L·U / D²        Q = π·D⁴·ΔP / (128·µ·L)        R_H = 128·µ·L / (π·D⁴)

For a rectangular channel of width w and height h with h ≤ w (the standard microchannel cross-section produced by photolithography), an excellent closed-form approximation (Bruus 2008) is:

R_H = (12·µ·L) / [ w·h³ · (1 − 0.63·(h/w)) ]

The h³ dependence is the dominant scaling — halving channel height multiplies resistance ~eightfold. This is the single most consequential design equation in the field: it sets pressure budget, flow rate, and pump selection.

3.3 Diffusion mixing

At Re < 1 there is no inertial mixing. Two co-flowing streams blend only by molecular diffusion across the interface. Time to diffuse across a half-width L:

τ_diff ≈ L² / D_diff

With D_diff for small ions ≈ 1 × 10⁻⁹ m²/s, for proteins ≈ 1 × 10⁻¹⁰ to 1 × 10⁻¹¹ m²/s, for mammalian cells ≈ 10⁻¹³ m²/s. A 100 µm channel needs 5–25 s to mix protein-scale species by diffusion alone — usable for slow assays, but unworkable for kinetic measurements. The remedy is either active mixing (acoustic, magnetic) or geometric chaotic advection (Stroock & Whitesides 2002 herringbone mixer).

3.4 Electrokinetics

A charged solid–liquid interface develops an electric double layer with zeta potential ζ. Applying a tangential field E gives electroosmotic flow (EOF) with a near-plug velocity profile:

v_EOF = − ε·ζ·E / µ            (Helmholtz–Smoluchowski, 1879/1903)

Bare fused-silica at pH 7 has ζ ≈ −60 to −100 mV, so a 1 kV across a 1 cm capillary drives 1–2 mm/s of plug flow with no dispersion — the foundation of capillary electrophoresis (Jorgenson & Lukacs 1981; commercialised in the ABI 3730 Sanger sequencer that completed the Human Genome Project).

Electrophoresis moves charged species through stationary fluid:

v_ep = µ_e · E              µ_e = q / (6π·µ·R)  (small species)

Electroosmotic and electrophoretic mobilities add vectorially in a capillary; the resolution depends on the difference in µ_e, not the absolute value.

3.5 Surface tension, contact angle, wetting

The Young equation defines the equilibrium contact angle θ on a smooth surface from interfacial tensions:

γ_sv − γ_sl = γ_lv · cos θ

Water on bare PDMS: θ ≈ 105° (hydrophobic). After 30 s air-plasma activation: θ ≈ 5–15° (hydrophilic), recovering to native over 24–72 h. This drift is the most common cause of irreproducibility in PDMS chips between fabrication batches.

Capillary pressure in a channel of half-width h:

ΔP_cap = 2·γ·cos θ / h

For water (γ = 0.0728 N/m) in a 50 µm hydrophilic channel: ΔP_cap ≈ 2900 Pa ≈ 29 mbar — enough to fill a chip without external pumping (the entire paper-microfluidics / LFA industry rests on this).

3.6 Stokes drag and particle handling

A sphere of radius R moving at velocity v in low-Re flow experiences:

F_drag = 6π·µ·R·v          (Stokes 1851)

This sets sedimentation in microchannels, hydrodynamic-focusing widths, dielectrophoresis force balance, and the Coulter principle for cell counting (Beckman Multisizer, Sysmex haematology analysers).

3.7 Knudsen / slip in gas microchannels

For gas in sub-µm channels, the mean free path λ becomes comparable to the channel size. Slip onset at Kn > 0.01, transitional 0.1 < Kn < 10, free-molecular Kn > 10. For air at STP λ ≈ 68 nm, so channels below ~7 µm see measurable slip. Pressure drop is reduced from the no-slip prediction by a factor (1 + 6·Kn).

3.8 Hydraulic–electrical analogy

For low-Re incompressible flow the Hagen–Poiseuille equation is linear in Q and ΔP, so an entire microfluidic network maps to a DC circuit:

Fluid quantity	Electrical analogue	Symbol
Pressure drop ΔP	Voltage V	V
Volumetric flow Q	Current I	I
Hydraulic resistance R_H	Resistance R	R
Compliance C_H	Capacitance C	C
Inertance L_H	Inductance L	L

Kirchhoff’s laws apply: parallel channels share ΔP and add Q; series channels share Q and add ΔP. SPICE-style circuit simulators (LTspice, Qucs) are routinely repurposed for microfluidic network design. Compliance arises from elastic-wall PDMS bulging (~10⁻¹⁵ m³/Pa per cm of 100 × 50 µm channel) and trapped air; inertance is usually negligible at < 100 Hz operation but matters for piezo-driven inkjet at 10–100 kHz.

3.9 Cell shear-stress budget

Wall shear stress in a rectangular channel with width w ≫ h (thin-channel limit):

τ_wall = 6·µ·Q / (w·h²)

For 1 µL/min through a 200 µm × 50 µm channel: τ_wall ≈ 0.2 Pa — safe for most adherent cells. Above ~1 Pa most mammalian cell lines start to detach or show stress-response gene expression; above ~10 Pa lysis follows within seconds. Endothelial cells, by contrast, are evolved for ~1 Pa physiological wall shear and require it to maintain phenotype on-chip.

4. Channel and chip fabrication

4.1 Substrate materials

Material	Typical use	Optical	Gas-permeable	Auto-fluorescence	Production scale
PDMS (Sylgard 184)	R&D, organ-on-chip, prototyping	yes	high (O₂, CO₂)	low–moderate	poor (manual)
Glass / fused silica	CE, optofluidics, high-T chemistry	yes	none	very low	moderate
Silicon	Integrated MEMS+fluidics, sensors	no (IR)	none	n/a	high (CMOS line)
PMMA	Production diagnostics	yes	very low	moderate	high (injection)
COC / COP (Zeonex, Topas)	Optical-grade diagnostic cartridges	yes	very low	very low	high
PC (polycarbonate)	Centrifugal disks (Gyros, Abaxis)	yes	low	high	high
Paper (Whatman)	Lateral-flow assays	n/a	n/a	n/a	very high
Hydrogel	Tissue scaffolds, organ-on-chip	yes	very high	low	low

4.2 Fabrication methods

Method	Min feature	Substrates	Throughput	Reference
Soft lithography (PDMS)	1 µm	PDMS on SU-8 master	hours / chip	Xia & Whitesides 1998
Photolithography + DRIE	0.5 µm	Si, glass	wafer batch	MEMS foundry standard
Hot embossing	5 µm	PMMA, COC, PC	minutes / chip	Becker & Heim 2000
Injection moulding	10 µm	PMMA, COC, PC	seconds / chip	production
Laser ablation (CO₂ / UV)	25 / 5 µm	most polymers	minutes / chip	rapid proto
Two-photon polymerisation	0.1 µm	photopolymer	very slow	Nanoscribe Photonic Professional
Micro-stereolithography	25 µm	resins	minutes / chip	BMF microArch, CADworks 3D
Roll-to-roll (R2R)	50 µm	thin polymer film	continuous	LFA mass production
Xurography (cutter)	100 µm	tape, film	minutes / chip	lab-grade prototyping

Sealing: oxygen-plasma covalent bonding of PDMS to glass, silicon, or PDMS forms the workhorse closure for research devices (siloxane Si–O–Si bonds, peel strength > 5 N/cm). Thermoplastic chips are closed by solvent-assisted thermal bonding (e.g. cyclohexane vapour + 95 °C for COC) or pressure-sensitive adhesive (PSA) layers. LFA strips are simply press-laminated.

Typical microchannel geometry: 10–500 µm wide × 10–200 µm deep × 1–100 mm long, aspect ratios h/w of 0.1 to 1.

5. Microfluidic unit operations

5.1 Pumping

Method	Pressure / flow control	Pulsation	Notes
Syringe pump	flow rate	high (lead-screw stepping)	Harvard PHD ULTRA, Cetoni Nemesys, KD Scientific
Pressure controller	pressure	very low	Fluigent MFCS-EZ / Flow EZ, Elveflow OB1
Peristaltic	flow rate	very high	Ismatec, Watson-Marlow
Capillary / wicking	passive	n/a	LFA, paper microfluidics
Electroosmotic	voltage	very low (plug flow)	CE instruments
Centrifugal	rotation rate	low	Gyros Bioaffy, Abaxis Piccolo Xpress
Acoustic / piezo	pulse	engineered	inkjet print heads
Vacuum pull	pressure	low	simple, no upstream contamination
Surface tension / Marangoni	passive	n/a	research

For sensitive cell or kinetic work, pressure control beats flow control: a Fluigent MFCS-EZ delivers settling-time < 100 ms with < 0.3 mbar pulsation, whereas a syringe pump produces lead-screw harmonics that can shake droplet-generation regimes off their stable point.

5.2 Mixing

At Re < 1, mixing is the bottleneck. Options, in order of complexity:

• Long serpentine diffusion — cheap, slow; works for ions in seconds, useless for proteins in centimetres.
• Lamination (split + recombine repeatedly) — passive, doubles interface count each stage.
• Chaotic advection (Stroock & Whitesides 2002 staggered herringbone, SHM) — passive, exponential mixing-length reduction; the de facto standard for PDMS chips. Critical bath dimensions: ridge height 17 µm in 200 µm channel, asymmetric herringbone period 100 µm.
• Active acoustic / SAW — surface-acoustic-wave transducer (Yeo & Friend 2014) couples MHz energy into the channel; sub-second mixing.
• Active magnetic — magnetic beads driven by external coils stir the channel.

5.3 Valves and pumps on chip

• Quake valve (Unger, Chou, Thorsen, Scherer & Quake 2000) — multilayer PDMS soft-lithography with a thin elastic membrane separating a flow layer from a control layer. Pressurising the control channel pinches the flow channel closed. Three valves in series form a peristaltic pump. The basis of large-scale integration (LSI) with thousands of valves on one chip (Standard Biotools / Fluidigm).
• Thermal phase-change — paraffin or thermo-responsive gel plugs.
• EWOD (electrowetting on dielectric) — Mugele & Baret 2005; voltage modulates contact angle to translate discrete droplets on an addressable electrode array (digital microfluidics — Sci-Bots, Volta Labs, Nuclera).
• Piezo / piezoelectric microvalves — Bürkert, Festo.

5.4 Droplet generation

Three canonical junction geometries: T-junction (Thorsen 2001), flow-focusing (Anna, Bontoux & Stone 2003), and co-flow (Cramer 2004). Drop diameter d in flow-focusing scales as:

d / w_nozzle ≈ f(Ca, Q_d/Q_c)

with w_nozzle the orifice width and Q_d/Q_c the dispersed-to-continuous flow ratio. Typical droplets: 20–500 µm diameter, 4 pL – 65 nL, generated at 0.1 – 30 kHz. Polydispersity below 1.5 % CV is routine.

Drop sorting: dielectrophoretic gating (FADS — fluorescence-activated droplet sorting, Baret et al. 2009), acoustic deflection, or pressure-driven gating.

5.5 Sensing and readout

• Optical fluorescence — by far the most common; excitation 405/488/532/635 nm lasers + PMT or sCMOS detector. Sub-pM LODs achievable.
• Bright-field / phase-contrast / DIC — cell morphology, organ-on-chip.
• Electrochemical — amperometric (glucose sensors), potentiometric ISFET (Sherrington 2011 → DNA Electronics → Nanopore concept), impedance spectroscopy.
• Coulter principle — resistive pulse counting through an aperture for cell or particle sizing (Beckman Multisizer, Sysmex XN).
• SPR (surface plasmon resonance) — Biacore / Cytiva; label-free binding kinetics.
• QCM / SAW mass sensing — pg-scale.
• Raman / SERS — Snowy Range, Metrohm.
• Mass spectrometry interfacing — chip-based nano-ESI (Advion TriVersa NanoMate).

6. Worked examples (carry units)

Example A — Hagen–Poiseuille flow rate, rectangular PDMS channel

Problem. Aqueous buffer at 20 °C (µ = 1.00 × 10⁻³ Pa·s) flows in a PDMS channel w = 100 µm wide, h = 50 µm deep, L = 10 mm long driven by a Fluigent pressure controller at ΔP = 100 mbar = 1.0 × 10⁴ Pa. Find flow rate and mean velocity.

Compute hydraulic resistance:

R_H = (12·µ·L) / [ w·h³ · (1 − 0.63·h/w) ]
    = (12 · 1.00×10⁻³ · 1.0×10⁻²) / [ 1.0×10⁻⁴ · (5.0×10⁻⁵)³ · (1 − 0.63·0.5) ]
    = 1.20×10⁻⁴ / [ 1.0×10⁻⁴ · 1.25×10⁻¹³ · 0.685 ]
    = 1.20×10⁻⁴ / 8.56×10⁻¹⁸
    = 1.40×10¹³ Pa·s/m³

Flow rate Q = ΔP / R_H = 1.0×10⁴ / 1.40×10¹³ = 7.14 × 10⁻¹⁰ m³/s ≈ 0.71 µL/s ≈ 43 µL/min.

Mean velocity U = Q/(w·h) = 7.14×10⁻¹⁰ / 5.0×10⁻⁹ = 0.143 m/s.

Reynolds number Re = ρ·U·D_h/µ with D_h = 2·w·h/(w+h) = 66.7 µm: Re = 998 · 0.143 · 6.67×10⁻⁵ / 1.0×10⁻³ = 9.5 — still firmly laminar but no longer trivially Stokes.

Example B — Diffusion mixing time, side-by-side laminar streams

Problem. Two streams co-flow in a 100 µm wide channel — half buffer, half protein-loaded sample. Find time (and channel length at U = 1 cm/s) to mix to within 90 % uniformity.

Half-width L = 50 µm. For a globular protein (MW ~ 60 kDa, hydrodynamic radius ~ 3.5 nm), Stokes–Einstein gives:

D_diff = kT / (6π·µ·R) = (1.38×10⁻²³ · 293) / (6π · 1.00×10⁻³ · 3.5×10⁻⁹)
       = 4.04×10⁻²¹ / 6.60×10⁻¹¹
       = 6.1 × 10⁻¹¹ m²/s

Mixing time τ ≈ L² / D_diff = (5.0×10⁻⁵)² / 6.1×10⁻¹¹ = 41 s. At U = 1 cm/s the required channel length is L_mix = U·τ = 0.41 m — completely impractical.

A staggered-herringbone mixer (Stroock 2002) achieves the same mixing in L_mix ≈ 1.5 cm, a 27× reduction. For salt ions (D ~ 1×10⁻⁹ m²/s) the diffusion time drops to 2.5 s and a 2.5 cm straight channel suffices.

Example C — Flow-focusing droplet generator

Problem. Aqueous-in-oil flow focusing with Q_aq = 10 nL/s (dispersed) and Q_oil = 50 nL/s (continuous) through a 30 µm × 30 µm nozzle. Estimate drop volume and generation frequency.

In the dripping regime (Ca ~ 0.01 – 0.1) drop diameter d ≈ w_nozzle gives volume ≈ (π/6)·d³ ≈ (π/6)·(30 µm)³ ≈ 14 pL. Slightly larger drops are typical due to neck pinch-off; assume d_drop = 40 µm → V_drop = 33 pL.

Generation frequency: f = Q_aq / V_drop = 1.0×10⁻¹¹ / 3.3×10⁻¹⁴ = 303 Hz.

Check oil consumption: each drop occupies a spacing of V_drop·(Q_oil + Q_aq)/Q_aq = 33 pL · 6 = 200 pL of total fluid → ~200 µm pitch in the post-nozzle channel.

Capillary number in the oil phase (µ_oil = 5 mPa·s, γ_aq/oil = 0.005 N/m, U_oil ≈ Q_oil / A_nozzle = 5.0×10⁻¹¹ / 9.0×10⁻¹⁰ = 0.056 m/s):

Ca = µ·U/γ = 5×10⁻³ · 0.056 / 5×10⁻³ = 0.056   → dripping regime, monodisperse drops expected.

For comparison, at Ca > ~0.3 the system transitions to the jetting regime with broader size distribution and downstream Rayleigh–Plateau breakup; above Ca ~ 1, a stable parallel jet forms and droplets disappear entirely. Designers stay at Ca = 0.01–0.1 for monodisperse production.

Example D — Capillary fill time of a paper-microfluidic strip

Problem. A nitrocellulose LFA strip with pore radius r = 4 µm, porosity ε = 0.7, contact angle θ ≈ 0° (fully wetting) is touched at one end to aqueous sample. Estimate time to fully wet a 40 mm strip. Sample: water at 20 °C.

Lucas–Washburn equation for capillary penetration length L(t) into a porous medium:

L²(t) = (r · γ · cos θ / (2·µ)) · t

With γ = 0.0728 N/m, µ = 1.0 × 10⁻³ Pa·s, r = 4 × 10⁻⁶ m, cos θ = 1:

L² = (4×10⁻⁶ · 0.0728 · 1 / (2 · 1×10⁻³)) · t = 1.456×10⁻⁴ · t   [m²]

For L = 0.04 m: t = (0.04)² / 1.456×10⁻⁴ = 11.0 s to traverse — consistent with the 10–20 s wicking time observed in commercial LFA prototypes before reaching the conjugate pad. The full assay clock (10–15 min) is dominated by antibody-binding kinetics, not transport, once the wave-front reaches the test line.

7. Specialised applications

7.1 Lateral-flow assay (LFA)

The dominant commercial format by unit volume — billions of units per year. A nitrocellulose strip wicks sample by capillarity past a conjugate pad (gold-nanoparticle or latex-conjugated antibody) to a test line of immobilised capture antibody and a control line. Yager & Bell 2006 provides the canonical review. Strip dimensions: 4 × 60 mm typical; read time 10–15 min; cost < $1/strip in volume.

Engineering levers: nitrocellulose pore size (8 µm and 15 µm grades, Millipore Hi-Flow Plus), conjugate-pad release efficiency, dried-reagent stability, line-printing precision (BioDot, Imagene). 2020–2022 saw the SARS-CoV-2 antigen wave (Abbott BinaxNOW, ACON Flowflex, SD Biosensor STANDARD Q).

7.2 Droplet digital PCR

Sample partitioned into 10 000–25 000 nL-scale droplets, each undergoing endpoint PCR; Poisson statistics give absolute quantification without standard curves. Bio-Rad QX200/QX600 reads ~20 000 droplets/well by flow cytometry; Stilla Naica images all droplets in a 2-D crystal in situ. Used for viral-load monitoring, rare-mutation detection (KRAS, EGFR liquid biopsy), and copy-number variation.

7.3 Single-cell genomics

10x Genomics Chromium (2016) uses microfluidic flow-focusing to co-encapsulate single cells with barcoded gel-bead-in-emulsion (GEM) reagents, partitioning 100–10 000 cells in 6 minutes. Successor platforms — Mission Bio Tapestri, Parse Biosciences (split-pool), BD Rhapsody (well-based). The underlying engineering is monodisperse droplet generation at 1–10 kHz with sub-CV 2 %.

7.4 Organ-on-chip

PDMS or hydrogel microchannels seeded with tissue-specific cell types under controlled shear and chemical gradients. Emulate Bio (lung, gut, liver, kidney chips), AlveoliX AXLung, Tissuse HUMIMIC. Used in pharma toxicology screening as alternatives to animal models. FDA Modernization Act 2.0 (2022) permits MPS (microphysiological systems) data in IND submissions.

7.5 Continuous-flow microreactors

Industrial chemistry at seconds-scale residence time with kilogram-per-day throughput. Used for hazardous chemistries (diazomethane, nitrations, organolithium), photochemistry, and continuous API manufacture. Vendors: Corning Advanced-Flow Reactors, Syrris Asia, Chemtrix Labtrix, Vapourtec.

7.6 Inkjet printing

Industrial inkjet generates 10–80 pL drops at 10–100 kHz per nozzle, with arrays of 10 000+ nozzles. HP thermal, Epson / Xaar / Konica-Minolta piezo, Memjet single-pass. Drop-on-demand physics is set by Weber and Reynolds; the Ohnesorge number Oh = µ/√(ρ·γ·L) gates printability (0.1 < Oh < 1).

7.7 Capillary electrophoresis

The Applied Biosystems 3730xl sequencer — based on capillary array electrophoresis in fused-silica capillaries — completed the Human Genome Project (Sanger chemistry). Modern descendants: Agilent Bioanalyzer / TapeStation (sizing), SCIEX PA 800 Plus (biopharma protein characterisation).

7.8 Reference data — physical properties at 20 °C

Quick-reference values for the most common microfluidic working fluids and substrate combinations.

Fluid	ρ [kg/m³]	µ [mPa·s]	γ [mN/m]	D_self [m²/s]
Water	998	1.00	72.8	2.3 × 10⁻⁹
Phosphate-buffered saline (1×)	1005	1.02	72.5	similar to water
Ethanol (absolute)	789	1.20	22.1	1.1 × 10⁻⁹
Glycerol (anhydrous)	1261	1410	64.0	1.0 × 10⁻¹¹
Whole blood (Hct 0.4)	1050–1060	3 – 4 (apparent, shear-dependent)	55–60	n/a
FC-40 fluorinated oil	1855	4.1	16 (air)	n/a
Mineral oil (light)	850	30	30 (air)	n/a
Silicone oil (10 cSt)	935	9.4	20 (air)	n/a
HFE-7500 (Novec)	1614	1.24	16 (air)	n/a

Interfacial tension of common immiscible pairs used in droplet microfluidics:

Phase pair	γ [mN/m]	Notes
Water / mineral oil	30 – 50	Span 80 surfactant drops to 5 mN/m
Water / FC-40 + 1 % EA-surfactant	4 – 6	RAN Biotechnologies 008-FluoroSurfactant
Water / HFE-7500 + 2 % surfactant	3 – 5	typical ddPCR / Drop-seq oil
Water / hexadecane	50	minimal surfactant
Water / decane	51

Diffusion coefficients in dilute aqueous solution (Stokes–Einstein, 20 °C):

Species	MW [Da]	R_h [nm]	D_diff [m²/s]
Na⁺ / Cl⁻	23 / 35	0.1	1.3 / 2.0 × 10⁻⁹
Glucose	180	0.4	6.7 × 10⁻¹⁰
Fluorescein	332	0.6	4.0 × 10⁻¹⁰
Lysozyme	14 000	1.9	1.1 × 10⁻¹⁰
BSA	66 000	3.5	6.1 × 10⁻¹¹
IgG antibody	150 000	5.3	4.0 × 10⁻¹¹
Viral particle (~30 nm)	~10⁷	15	1.4 × 10⁻¹¹
E. coli (~1 µm)	n/a	500	~4 × 10⁻¹³
Mammalian cell (~15 µm)	n/a	7500	~3 × 10⁻¹⁴

7.9 Point-of-care molecular diagnostics

• Cepheid GeneXpert — closed cartridge integrates lysis, nucleic-acid purification, qPCR; 30–90 min turnaround.
• Abbott ID NOW — isothermal nicking-enzyme amplification, 13 min flu/RSV/COVID.
• Roche cobas Liat — qPCR cartridge, 20 min respiratory panel.
• BioFire FilmArray — nested-multiplex PCR cartridge, 22 pathogens, 1 h.

8. Edge cases and gotchas

1. Bubble in channel is catastrophic. Gas is compressible — a single 1 mm bubble in a 50 µm × 100 µm × 1 cm channel stores enough volumetric capacitance to absorb several seconds of programmed flow. Use degassed buffers, in-line degassers (Biotech DEGASi), bubble traps, and hydrophilic surfaces that prevent pinning.
2. PDMS swelling by organic solvents. PDMS swells significantly in toluene, hexane, chloroform, dichloromethane; only mildly in DMSO, methanol, water. For organic chemistry use glass or COC, not PDMS.
3. PDMS gas permeability. A blessing for cell-culture chips (O₂ supply) but a curse for evaporation — droplets in PDMS shrink 5–20 %/h. Use paraffin-oil cap, glass capping, or a humidified housing.
4. Surface adsorption of proteins. Bare PDMS adsorbs albumin and IgG within seconds. Pre-fill with 1 % BSA, 0.1 % Pluronic F-127, or PEG-silane covalent coating.
5. Plasma-bond surface drift. Hydrophilic PDMS recovers to native hydrophobic over 24–72 h; conduct capillary-priming experiments within hours of plasma treatment.
6. Channel clogging. Particle aggregation, salt crystallisation at evaporation menisci, and biofilm growth all shut down channels. Use sub-channel-size filters (Vitrocom, Cytiva).
7. Cell viability under shear. Mammalian cells lyse above ~1 Pa wall shear stress; CHO cells and primary T-cells are particularly fragile. Compute τ_wall = 6·µ·Q/(w·h²) for a rectangular channel.
8. Joule heating in electrokinetic flow. Power per unit volume = σ·E² heats the channel; 100 V/cm in 100 mM PBS dissipates ~10 W/mL — boiling the buffer in seconds without a heat sink.
9. Pulsation from syringe pumps. Lead-screw harmonics destabilise droplet generation at kHz frequencies. Switch to pressure-controlled flow (Fluigent, Elveflow) for monodisperse drops.
10. Capillary number drift. Surfactant adsorption at oil/water interfaces lowers γ over seconds–minutes; Ca rises and drop size shifts. Use stable fluorosurfactants (e.g. 008-FluoroSurfactant from RAN Biotechnologies; or Krytox-PEG block copolymer).
11. Multilayer registration. Quake-valve chips need ≤ 5 µm registration between flow and control layers; use alignment marks and a mask aligner (OAI, SUSS MicroTec).
12. Lot-to-lot Sylgard variability. Sylgard 184 base:crosslinker ratio nominally 10:1; ratio drift of ±1 swings modulus 30 % and bond strength 50 %. Weigh, don’t volume-measure.
13. Cross-contamination. Shared inlets and luer connections retain reagent; use disposable cartridges or aggressive interstitial flushes.
14. Sample matrix effects. Whole blood (haematocrit, platelets) clogs sub-50 µm channels; design for plasma separation upstream (Daktari, cross-flow filtration).
15. Photobleaching of fluorophores. Confined volumes mean high photon flux per molecule; rotate excitation, use brighter dyes (Alexa Fluor 647) or quantum dots, or use frequency-domain detection.
16. Standardisation lag. ISO 22916:2022 finally fixed minimum connector and footprint dimensions; pre-2022 chips routinely use bespoke ports incompatible with COTS holders.

9. Tools and software

Design / layout

• L-Edit, KLayout, CleWin — mask layout for photolithography (GDSII output).
• Solidworks, Onshape, Fusion 360 — 3-D chip-holder and fluidic-manifold CAD.
• AutoCAD / DraftSight — legacy 2-D mask drawing.

Simulation

• COMSOL Multiphysics Microfluidics Module — the dominant commercial tool; couples Navier–Stokes, electrokinetics, transport of dilute species, two-phase level-set, and particle tracing.
• ANSYS Fluent + DPM (discrete phase model) — droplet and particle tracking.
• OpenFOAM (interFoam, electroFoam) — open-source two-phase and electrohydrodynamic solvers; see cfd-deep.
• Coventor MEMS+ — combined MEMS/microfluidic system simulation.
• Flow-3D — free-surface and droplet specialisation.

Photoresists and process chemistry

• MicroChem / Kayaku SU-8 — negative epoxy resist; the standard PDMS master material. SU-8 2050 / 3050 / 50 cover 5–200 µm films.
• AZ / merck-Performance positive resists — for hard-mask patterning of glass / silicon DRIE.
• ma-P 1275, ma-N 2410 (Micro Resist Technology) — alternative thick negative resists.

Components and instruments

• Pressure controllers — Fluigent MFCS-EZ / Flow EZ, Elveflow OB1, Dolomite Mitos P-Pump, LineUp Flow EZ.
• Syringe pumps — Harvard PHD ULTRA, KD Scientific Legato, Cetoni Nemesys (gas-tight syringe).
• Off-the-shelf chips and connectors — Dolomite Microfluidics, Micronit, ChipShop (microfluidic ChipShop GmbH), uFluidix, BlackHoleLab, ALine.
• Inverted microscopes — Nikon Eclipse Ti2, Zeiss Axio Observer 7, Olympus IX83, Leica DMi8.
• High-speed cameras — Photron FASTCAM Mini AX, Vision Research Phantom v2640, Mikrotron EoSens 25CXP.
• PDMS supply — Dow Sylgard 184 (industry-standard), Bluestar Silbione for medical-grade.
• Mask aligners — SÜSS MicroTec MA8/MA6, OAI Hybralign, ABM contact aligner.

Foundries / CROs

• uFluidix (Canada), BlackHoleLab (FR), ChipFlows, microfluidic ChipShop (DE), Micronit (NL), Dolomite (UK), ALine (US), Translume (US, fused-silica), FlexChip (US).

References on-screen

• Bruus, Theoretical Microfluidics — analytical resistance and electrokinetic formulae.
• Tabeling, Introduction to Microfluidics — concise pedagogy.
• Berthier & Silberzan, Microfluidics for Biotechnology — application-leaning second edition.

10. Cross-references

• fluid-mechanics — low-Re Stokes-flow foundation; the parent note.
• heat-transfer — microchannel single-phase and two-phase cooling for power electronics.
• mems — companion note; the manufacturing substrate for silicon-glass microfluidics.
• chemical-process-fundamentals — continuous-flow microreactor link.
• electromagnetics-engineering — EOF, dielectrophoresis, electrowetting underlying physics.
• materials-polymers — PDMS, PMMA, COC chip substrates and their cure/processing windows.
• cfd-deep — multiphase, level-set, and electrohydrodynamic CFD methods used to design droplet junctions and mixers.
• circuit-analysis — hydraulic-electrical analogy (R, C, L equivalents) for chip network design.
• Planned: biomechanics, bioinstrumentation, microfluidics.
• scientific — file formats for reaction/protocol description on chip.

11. Citations

1. Whitesides, G. M. “The origins and the future of microfluidics.” Nature, vol. 442, 2006, pp. 368–373. — canonical field overview.
2. Squires, T. M.; Quake, S. R. “Microfluidics: Fluid physics at the nanoliter scale.” Reviews of Modern Physics, vol. 77, 2005, pp. 977–1026.
3. Stone, H. A.; Stroock, A. D.; Ajdari, A. “Engineering flows in small devices: Microfluidics toward a lab-on-a-chip.” Annual Review of Fluid Mechanics, vol. 36, 2004, pp. 381–411.
4. Bruus, H. Theoretical Microfluidics. Oxford University Press, 2008. ISBN 978-0199235094.
5. Tabeling, P. Introduction to Microfluidics. Oxford University Press, 2005. ISBN 978-0198568643.
6. Berthier, J.; Silberzan, P. Microfluidics for Biotechnology, 2nd ed. Artech House, 2010. ISBN 978-1596934436.
7. Beebe, D. J.; Mensing, G. A.; Walker, G. M. “Physics and applications of microfluidics in biology.” Annual Review of Biomedical Engineering, vol. 4, 2002, pp. 261–286.
8. Xia, Y.; Whitesides, G. M. “Soft lithography.” Annual Review of Materials Science, vol. 28, 1998, pp. 153–184.
9. Unger, M. A.; Chou, H.-P.; Thorsen, T.; Scherer, A.; Quake, S. R. “Monolithic microfabricated valves and pumps by multilayer soft lithography.” Science, vol. 288, 2000, pp. 113–116. — Quake valve.
10. Anna, S. L.; Bontoux, N.; Stone, H. A. “Formation of dispersions using flow focusing in microchannels.” Applied Physics Letters, vol. 82, 2003, pp. 364–366. — flow-focusing droplets.
11. Thorsen, T.; Roberts, R. W.; Arnold, F. H.; Quake, S. R. “Dynamic pattern formation in a vesicle-generating microfluidic device.” Physical Review Letters, vol. 86, 2001, pp. 4163–4166. — T-junction droplets.
12. Stroock, A. D.; Dertinger, S. K. W.; Ajdari, A.; Mezic, I.; Stone, H. A.; Whitesides, G. M. “Chaotic mixer for microchannels.” Science, vol. 295, 2002, pp. 647–651. — herringbone mixer.
13. Pollack, M. G.; Fair, R. B.; Shenderov, A. D. “Electrowetting-based actuation of liquid droplets for microfluidic applications.” Applied Physics Letters, vol. 77, 2000, pp. 1725–1726. — EWOD origin.
14. Mugele, F.; Baret, J.-C. “Electrowetting: from basics to applications.” Journal of Physics: Condensed Matter, vol. 17, 2005, pp. R705–R774.
15. Baret, J.-C.; et al. “Fluorescence-activated droplet sorting (FADS).” Lab on a Chip, vol. 9, 2009, pp. 1850–1858.
16. Yager, P.; Edwards, T.; Fu, E.; Helton, K.; Nelson, K.; Tam, M. R.; Weigl, B. H. “Microfluidic diagnostic technologies for global public health.” Nature, vol. 442, 2006, pp. 412–418. — LFA review.
17. Macosko, E. Z.; et al. “Highly parallel genome-wide expression profiling of individual cells using nanoliter droplets.” Cell, vol. 161, 2015, pp. 1202–1214. — Drop-seq.
18. Zheng, G. X. Y.; et al. “Massively parallel digital transcriptional profiling of single cells.” Nature Communications, vol. 8, 2017, 14049. — 10x Chromium method paper.
19. Hindson, B. J.; et al. “High-throughput droplet digital PCR system for absolute quantitation of DNA copy number.” Analytical Chemistry, vol. 83, 2011, pp. 8604–8610. — Bio-Rad QX200.
20. Jorgenson, J. W.; Lukacs, K. D. “Zone electrophoresis in open-tubular glass capillaries.” Analytical Chemistry, vol. 53, 1981, pp. 1298–1302.
21. ISO 22916:2022 — Microfluidic devices — Interoperability requirements for dimensions, connections and initial device classification.
22. ASTM F838-20 — Standard test method for determining bacterial retention of membrane filters utilized for liquid filtration.
23. CLSI EP05-A3, EP09-A3, EP17-A2 — Evaluation of Precision, Method Comparison, and Detection Capability in Clinical Laboratory Diagnostics.
24. Becker, H.; Heim, U. “Hot embossing as a method for the fabrication of polymer high aspect ratio structures.” Sensors and Actuators A, vol. 83, 2000, pp. 130–135.
25. Yeo, L. Y.; Friend, J. R. “Surface acoustic wave microfluidics.” Annual Review of Fluid Mechanics, vol. 46, 2014, pp. 379–406.
26. NIST microfluidic flow-measurement standards: NIST SRM 1968 series; ongoing work on traceable nL/min flow standards.