INTRODUCTION TO NANOTECHNOLOGY

An Overview of Fluid Mechanics for MEMS

-Reni Raju

MEMS (Applications)

Accelerometers for airbagsMicro heat exchangersSensorsActuatorsMicropumps

NEMS (Application)

Nanostructured CatalystsDrug Delivery systemsMolecular Assembler/ReplicatorsSensorsMagnetic Storage ApplicationsReinforced PolymersNanofluids

Fluid Mechanics of MEMS

Devices having a characteristic length of less than 1 mm but more than 1 micron.

10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 102

Dia. Of Proton H-Atom Diameter

Human Hair Man

NEMS MEMS

FLUID MODELLING

Conventional Navier Stokes with no-slip boundary conditions cannot be used.

Pressure Gradient is non-constant along a microduct and flowrate greater than predicted.

Surface to volume ratio is high of the order of 106 m-1 for a characteristic length of 1 micron.

Other factors like thermal creep, rarefaction, viscous dissipation, compressibility etc.

For Gases

Fluid Modeling

Molecular Modeling Continuum Models

Deterministic Statistical Euler Burnett

Navier StokesMD Liouville

DSMC Boltzmann

Either as a collection of molecules or as a continuum.

Mean Free path,

Characteristic Length,

Knudsen Number,

22

1

n

y

L

Re2

Ma

LKn

Local value of Knudsen Number determines the degree of rarefaction and the degree of validity of the continuum model.

Kn=0.0001 0.001 0.01 0.1 1 10 100

Continuum Flow

(Ordinary Density Levels)Slip-Flow Regime

(Slightly Rarefied)

Transition Regime

(Moderately Rarefied)Free-Molecule Flow

(Highly rarefied)

CONTINUUM MODEL

Local Properties such as Density and Velocity are averages over elements large compared with the microscopic structure of the fluid but small enough to permit the use of differential calculus.

Conservation of Mass:

Conservation of Momentum:

0

kk

uxt

ik

ki

k

ik

i gxx

uu

t

u

Conversation of Energy:

Closure:

k

iki

k

k

kk x

u

x

q

x

eu

t

e

kij

j

i

k

k

ikiki x

u

x

u

x

up

)(radiationFluxx

Tkq

ii

RTp

dTcde v

Euler’s Equation: Fluid is invisicid and non-conducting,

0

kk

uxt

ikk

ik

i gx

p

x

uu

t

u

k

k

kkv x

up

x

Tu

t

Tc

Compressibility

DENSITY CHANGES DUE TO TEMPERATURE Strong wall Heating or cooling may cause density change.

DENSITY CHANGES DUE TO PRESSURE Pressure changes due to viscous effects even for Ma<0.3.

Continuity Equation:

0

k

k

x

u

Dt

D

Dt

DT

Dt

Dp

Dt

D

1

p

T

TTp

pTp

1),(

1),(

For Adiabatic Walls;

0

0

0

Pr

*

pcu

TTT

*

**Pr

*

**

*

*

*

1 20 Dt

DT

A

B

Dt

DpMa

Dt

D

00

0000 ;

TB

TcA p

For Isothermal Wall;

0

0ˆTT

TTT

w

*

ˆ*

*

**

*

*

*

1

0

020 Dt

TD

T

TTB

Dt

DpMa

Dt

D w

Boundary Conditions

At the Fluid- Solid Interface No-slip and no-temperature jump is based on no discontinuities of

velocity/temperature. Continuum applicable for Kn<0.001

Tangential Slip velocity at wall,

For Real gases,

w

wallgasw y

uuuu

wv

vwallgasw y

uuuu

2

Slip velocity & Temperature Jump,

where

wwv

v

x

T

Ec

Kn

y

uuu

wallgas

*

*2

*

*** Re)1(

4

32

wT

Tgas

y

TKnTT

wall

*

***

Pr1

22

wi

riT

wi

riv

dEdE

dEdE

,

MOLECULAR BASED MODELS

Goal is to determine the position , velocity and state of all particles at all times.

DETERMINISTIC MODEL:

Particle described in the form of two body potential energy and time evolution of the molecular positions by integrating Newton’s Law of motion.

Shortcomings:

Need to choose a proper and convenient potential for a fluid & solid combination.

Vast computer resources.

STATISTICAL MODEL: Based on probability of finding a molecule at a particular position and

state. Six-dimensional phase space. Assumption, for dilute gases with binary collision with no degrees of

freedom.

Liouville equation, conservation of N-particle distribution function in 6N-dimensional space,

Boltzmann equation for monatomic gases with binary collision,

0..11

k

N

kk

k

N

kk x

Fxt

3,2,1

*),()()()(

j

ffJx

nfF

x

nf

t

nf

jj

jj

Non-linear collision integral, describes the net effect of populating and depopulating collisions on the distribution.

1

4

0

1*

1*2 )()(*),(

ddffffnffJ r

LIQUID FLOWS

The Average distance between the molecules approaches the molecular diameter.

Molecules are always in collision state. Difficult to predict. Non-Newtonian behaviour commences,

Contradictory results in experimental data and modelling. MD seems to be the best option available. Based on MD, the degree of slip increases as the relative wall density

increases or the strength of the wall-fluid coupling decreases.

12

y

u

Slip length,

cos LL

1

SURFACE PHENOMENA

Surface to Volume ratio for 1 micron is 106 m-1.

High Radiative and Convective Heat transfer. Increased importance to surface forces and waning importance of body

forces. Significant cohesive intermolecular forces between surface, stiction

independent of device mass. Adsorbed layer. Surface tension and nonlinear volumetric intermolecular forces.

Fluid Mechanics for NEMS

Nanofluids - thermal conductivity fluids.

Possibility of applying Continuum Model for low Knudsen number.(?)

Model applicability to Dense and rare gas.

Possible treatment of Liquids as dense gas at Nano scale.(?)

Importance of Quantum Mechanics.

Importance of Surface Phenomenon's.

TASKS AHEAD

Modeling using the Continuum model for the Slip Flow Regime Knudsen Numbers.

Understanding the mechanics of Nano-scaled Domains.

Arriving at a suitable modeling technique comparable with the experimental data (if available.)