Basic Principles in Microfluidics

Basic Principles in Microfluidics

Newton’s Second Law for Fluidics

Newton’s 2^nd Law (F= ma) :

• Time rate of change of momentum of a system equal to net force acting on system

• Sum of forces acting on control volume =

Rate of momentum efflux from control volume +

Rate of accumulation of momentum in control volume

!F = dP

dt

Navier - Stokes Equation

• Navier-Stokes equation applies when:

(1) There are more than one million molecules in smallest volume that a macroscopic change takes place.

(2) The flow is not too far from thermodynamic equilibrium.

Navier - Stokes Equation

dU

dt = ! "P

# ^{+ g +}

$

# ^"

2U

! ^dU

dt = "#P + !^g + $#²U

!iU = 0

! ^dU

dt = "#P + !^g + $#²U + $

3 #(#iU)

For noncompressible Fluid

Navier - Stokes in Microfluidics

• Terms become dominant based on physics of scale

• In microfluidics inertial forces dominate due to small dimensions, even though velocity can be high

dU

dt = ! "P

# ^{+ g +}

$

# ^"

2U

dU

dt = ! 1

" ^#P

VISCOSITY

Viscosity

Viscosity is a measure of resistance (friction) of the fluid to the flow

This determines “flow rate”

Symbols: η and in some books µ Units: Poise (gram/sec ^* Cm)

Viscosity

Viscosity is a measure of resistance (friction) of the fluid to the flow.

This determines “flow rate.”

Units: Poise (gram/sec• Cm)

Basic Properties - Viscosity

Fluids and gases are very different

• Fluids become less viscous as temperature increases

• Gases become more viscous at temperature increases

Viscosity in Gases and Fluids

• Gases

• Fluids

η ∼ η₀ e ^{− (Τ − Τ}₀⁾

! = !₀ (T₀ - constant) (T₀ - constant)

T T₀

"

#$

%

&'

3 2

Interfaces and Surface Tension

Interfaces

• Interface: Geometric Surface that delimits 2 fluids

• Separation depends on molecular

interactions and Brownian diffusion

Interfaces

• Interface: Geometric Surface that delimits 2 fluids

• Simplified view:

Interaction between molecules

At interface:

different energies

Interfaces

• If U is the total cohesive energy per

molecule and d is a characteristic molecular dimension, d² is its surface, then the energy loss (surface tension) is given by:

! = U

2d

Laplace’s Law

• Minimization of surface energy, create

curvature of fluids on other surfaces (fluids)

• Curvature 1/R

• Laplace’s Law, the change in pressure is related to the curvature of the surface.

For a sphere: ∆P = 2 (γ/R) For a cylinder: ∆P = γ/R

Droplet on a Surface of Two Properties

Simulations

Coarsening

• Two Droplets linked by a precursor film

Coarsening

• Two Droplets linked by a precursor film

Contact Angle

• Surface tension (force per length)

• Angle is determined by the balance of forces at the point of interface

Hydrophilic Hydrophobic

Contact Angle

• Surface tension (force per length)

• Angle is determined by the balance of forces at the point of interface

Oil on Water

Hydrophilic - Hydrophobic

Surface Tension

• Droplet on a surface

– Forces on cross section of drop – Surface tension along periphery – Pressure on section area

– Pressure difference outside/inside drop

Force = !PA = "r2!P Surface Tension=2!^r"

! = r

2 "P

Forces - Capillary Effects

• A wetting fluid will rise in a capillary tube

• Equilibrium: pressure drop across meniscus

• Surface tension

• Viscosity

h = 2 ! ^Cos( " ⁾

# ^gr

Capillary Force

Capillary Forces

• Small Channel (capillary) - Surface tension draws fluid of density ρ into the channel of radius ( r)

• θ = contact angle

γ = surface tension (N/m)

• Height of Fluid in a tube in the presence of gravity

Capillary Forces

F = 2 ! ^r " ^Cos( # ⁾

h = 2! ^Cos("⁾

#^gr

Forces - Capillary Effects

Capillary Forces

Droplet on Surfaces

Droplet on Irregular Surfaces

r: roughness

f: ratio of contact angle to the total horizon surface θ) = (f-1) / (r-f)

Wettability and Roughness

Reynolds Number

Fluids - Types of Flow

• Laminar Flow (Steady)

• Energy losses are dominated by viscosity effects

• Fluid particles move along smooth paths in laminas or layers

• Turbulent

• Most flow in nature are turbulent!

• Fluid particles move in irregular paths, somewhat similar to the molecular

momentum transfer but on a much larger scale

• Reynolds Number

• Reis a measure of turbulence

Reynolds Number

Reynolds number (Re) = inertial forces / viscous forces

Re = Kinetic energy / energy dissipated by shear Implies inertia relatively important

V_D = Drag velocity, L = characteristic length, η= viscosity, ρ = density

Re < 2100 : laminar (Stokes) flow regime – slow fluid flow, no inertial effects – laminar flow in microfluidics

– slow time constants, heavy damping

Re > 4000 : unstable laminar flow - turbulent flow regime

Re = !^V_D ^L

"

Re = 1

2 mV_D² 1

2!^V_D^A Re = (!^AL)V_D

"A

High and Low Reynolds number fluidics

When the Reynolds number is low, viscous interaction between the wall and the fluid is strong, and there is no turbulences or vortices

Is this Flow Turbulent?

Channel Geometry - Use a characteristic length : D_h

D is a geometric constant Re = !

"^VDh

Is this Flow Turbulent?

Mixing

Re = 12 and Re = 70

Cycle 1 Cycle 2 Cycle 3

Microchannels Cross Sections

Re and Size

Re

Re - Some examples

Friction factor ~ 1/ Re

Human Circulatory System

Flow associated with Skin

Knudsen Number

• Knudsen number assumes that we can treat the material as a

“continuum”

• Continuum hypothesis holds better for liquids than gases also,

λ_mfp= mean free path of molecules, D_h = hydraulic diameter

• K_n measures deviation of the state of the material continuum K_n< 0.01 continuum

0.01 < K_n < 0.1 slip flow

0.1 < K_n < 10 transition region

10 < K_n molecular flow

K_n = !_mfp

D_h ^{Kn =}

!"

2 (M Re)

The Smallest Length Scale of a Continuum

Low Re High Re

K = M !"

Stokes - Einstein Diffusion

Stokes - Einstein Equation Diffusion of a particle

(gas, fluid)

Translational Diffusivity

Rotational Diffusivity

η

D_t = K_BT 6

!"

D_r = K_BT 8!"^a3

Diffusion in Fluids

• Very short diffusion times

D = diffusion constant X = diffusion length τ = diffusion rate

• Laminar flow limits benefits for fluid mixing.

• Highly predictable diffusion has enabled a new class of microfluidic diffusion mixers

x = 2D! ! = 1

2 x²

D

Fluid Squeeze

Squeezed film damping

• Squeeze a film by pushing on the plates (one is not moving) Viscous drag is opposing the motion of the fluid

• Beam displacement

• Flow of fluid (Reynolds equation) Knudsen number, K,

is the ratio of the mean free path to gap

• Squeeze number: relative importance of viscous to spring forces

! ^"2U

"t² + EI "⁴U

"u⁴ = P + F L

12! ^d(Ph)

dt = "{(1+ 6k)h³P"P}

P = b dU

dt ^{b =}

96!W ³

"⁴h³ L

Concluding Remarks

Summary

• Re = turbulent / viscous stresses

• Re < 2100 : laminar (Stokes) flow regime, slow fluid flow, no inertial effects

• laminar flow in microfluidics

• slow time constants, heavy damping

• Re > 4000 : turbulent flow regime

Fluid Behavioral

What happens when the fluid is on the micro - nano scale?

We discussed scaling - this is a review Quantities proportional L³

• Inertia, buoyancy, etc.

Quantities proportional L²

• Drag, surface charge, etc.

Quantities proportional L¹

• Surface tension

Who “Rules”

η