UNIVERSITE DE PARIS SUD

U.F.R SCIENTIFIQUE D’ORSAY

Licence et Magistere de Physique

2014-2015

Classical Electromagnetism

——-

Lecture Notes

26 CHAPTER 1. Introduction and overview - Electromagnetism

1.8 Additional reading

The following article 16, describes one application of radiation pressure.

16. Dachwald et al., Int. AAAF Symposium on Space Propulsion, 2002.

PERFORM

ANCE

REQUIR

EM

ENTSFOR

NEAR-T

ERM

INTERPLANETARY

SOLAR

SAIL

CRAFT

MIS

SIO

NS

BerndDachwald

1,W

olfgangSeboldt1

andBerndHausler2

1German

AerospaceCenter(D

LR),

Colog

ne

Institute

ofSpaceSen

sorTechnolog

yan

dPlanetaryExploration

Phon

e:+49

-220

3-60

1{300

1|302

8}Fax

:+49

-220

3-60

146

55E-M

ail:{b

ernd.dachwald|wolfgan

g.seboldt}@dlr.de

2Universitatder

Bundesweh

rMunchen

,Neu

biberg

InstitutfurRau

mfahrttechnik

Phon

e:+49

-89-60

0421

38Fax

:+49

-89-60

0421

38E-M

ail:

bernd.haeusler@unibw-m

uen

chen

.de

Solarsailcraft

provide

awide

range

ofopportu

nitiesforhigh-energy

low-cost

missions.

To

date,

most

mission

stu

dies

require

arath

er

demanding

per-

formance

that

willnot

be

realized

by

solar

sailcraft

ofth

efirst

generation.

However,

even

with

solar

sailcraft

of

moderate

performance,

scientifically

relevantmissionsarefeasible.This

isdemonstratedwithaNearEarth

Asteroid

sample

retu

rn

mission

and

variousplaneta

ryrendezvousmissions.

Introduction

Utilizing

solely

the

freely

available

solarradiation

pressure

forpropulsion,solarsailcraftprovideawide

range

ofop

portunitiesforlow-costinterplanetarymis-

sion

s,man

yof

whicharediffi

cultor

impossibleforan

yother

typeof

convention

alspacecraftdueto

theirlarge

∆v-requirem

ent.

Man

yof

thosehigh-energy

mission

sareof

greatscientificrelevance,such

asmission

sto

Mercury

and

toNearEarth

Objects(asteroidsan

dshortperiod

comets)

with

highly

inclined

orretro-

grad

eorbits1.W

ithin

theinner

solarsystem

(includ-

ingthemainasteroid

belt)solarsailcraftareespecially

suited

formultiplerendezvousan

dsamplereturn

mis-

sion

sdueto

their(atleastin

principle)unlimited

∆v-

capab

ility.

Evenmission

sto

theou

tersolarsystem

may

beenhan

cedbyusingsolarsailcraft,

albeitthe

solarradiation

pressure

decreases

withthesquareof

thesun–saildistance.

For

such

mission

ssolarsail-

craftmay

gain

alargeam

ountof

energy

when

first

approachingthesun,therebyperform

ingaso-called

’solar

photon

icassist’man

euverthat

turnsthetrajec-

tory

into

ahyperbolic

one[4][5][13].Such

trajectories

allow

reason

able

tran

sfer

times

totheou

terplanets

(andto

nearinterstellar

space)

withou

ttheneedto

perform

anygravityassist

man

euver.

How

ever,with-

outtheuse

ofad

ditional

propulsivedevices

and/oran

aerocapture

man

euverat

thetarget

body,

only

fast

fly-byscan

beachieved

dueto

theassociated

large

hyperbolic

excess

velocities.

1More

than

55%

oftheNEO

population

hasinclinations

larger

than

10◦,more

than

30%

hasinclinationslarger

than

20◦.Reach

ingsuch

inclinationswithspacecraft

requires

avery

large∆v.

Several

mission

studiesforhigh-energy

interplane-

tary

solarsailcraftmission

shavebeencarriedou

tat

DLR

[4][5][8][9]

andelsewhere[13][15].Mostof

them

requirearather

dem

andingsailcraftperform

ance

tokeep

mission

duration

sshort(see

Tab

le1).How

ever,

takingthecurrentstate-of-the-artin

engineeringof

ultra-lightw

eigh

tstructuresinto

account,

solarsail-

craftof

thefirstgenerationwillbeof

relativelymod-

erateperform

ance.For

such

near-term

solarsailcraft

few

mission

exam

plescanbefoundin

theliterature.

Theaim

ofthis

pap

eris

tonarrow

dow

nthis

gap

and

togetalower

bou

nd

onsolarsailcraftperfor-

man

ceforinterplanetarymission

sthat

areunder

con-

sideration.It

willbeshow

n,that

challengingscien-

tificmission

sarefeasible

atrelatively

low

cost,even

withmoderateperform

ance

sailcraftof

thefirstgen-

eration.This

willbedem

onstratedbelow

bythetra-

jectoryan

alysisof

aproposed

sample

return

mission

toNearEarth

Asteroid

1996FG

3(m

ission

duration

approx.9.4years).

SolarSailcraftOrbitalMechanics

Themagnitudean

ddirection

ofthesolarradiation

pressure

(SRP)forceactingon

aflat

solarsail

due

tothemom

entum

tran

sfer

from

solarphoton

siscom-

pletely

characterizedbythesun-saildistance

andthe

sailattitude.

Thelatter

isgenerally

expressed

bythe

sail

normal

vector

n,whosedirection

isusually

de-

scribed

by

thesail

clock

angleα

and

thesail

cone

angleβ

(Figure

1).

Figure

2givesapicture

ofthe

forces

exertedon

aflat

andperfectly

reflectingsolar

sail(idealsail)of

area

Abythesolarradiation

pres-

6th

Inte

rnational A

AA

F S

ym

posiu

m o

n S

pace P

ropuls

ion:

Pro

puls

ion f

or

Space T

ransport

ation o

f th

e X

XIs

t C

entu

ry,

Vers

aill

es,

Fra

nce,

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ay 2

002

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2002 b

y D

LR

Sailcraft

perform

ance

Transfer

time

Target

body

ac[m

m/s2]

σ[g/m

2]

[yr]

References

Mercury

0.5

16.0

1.4

[13]

Pluto

(fly-by)

0.7

11.4

10.4

[4][5]

(4)Vesta

0.75

10.7

3.3

[4][8]

2P/Encke

0.85

9.4

3.0

[4]

21P/Giacobini-Zinner

1.0

8.0

6.8

[15]

Venus

1.0

8.0

0.6

[13]

Mars

1.0

8.0

1.0

[13]

(433)Eros

1.0

8.0

1.2

[13]

(1566)Icarus

1.25

6.4

1.2

[15]

Tab

le1:

Fastsolarsailcraft

missionsusingadvancedsailcraft

(rendezvous,

ifnotstatedotherwise)

Figure

1:Definitionofthesailclock

angle

αandthe

sailconean

gle

β

Figure

2:Perfect

reflection

sure

Pactingon

thesail’scenterofsurface.From

the

geom

etry

ofFigure

2thetotalSRPforceFSRPcanbe

easily

calculated:

Fr=

PA(er·n

)er

Fr′=−PA(er·n

)er′

andmak

inguse

ofer−

er′=

2(er·n

)n:

FSRP=

Fr+Fr′=

=2P

A(er·n

)2n=

=2P

Acos2βn

Thus,

inthe

case

ofperfect

reflection,the

thrust

forceis

alwaysalongthedirectionofthesailnormal

vector

n.

At1AU,thesolarradiation

pressure

is(P

0) 1

AU

. =4.563·1

0−6N/m

2.

Therefore,theeff

ec-

tive

pressure

(forceper

unit

area)actingonanideal

sailnorm

alto

thesun-lineistw

icethesolarradiation

pressure,2(P

0) 1

AU

. =9.126·1

0−6N/m

2.How

ever,a

realsolarsail

isnotaperfect

reflectorand

athor-

oughtrajectory

analysismust

takeinto

accountthe

opticalproperties

oftherealsail.Since

inthiscase

asm

allbutsignificantfractionoftheincomingsunlight

isabsorbed

orreflected

non-specularly,

atangential

forcecomponentis

actingonarealsolarsail,so

that

FSRP

isnolonger

alongthedirection

ofn.

How

-ever,forpreliminary

missionanalysisthis

tangential

forcecomponentmay

beneglected,since

theresulting

smallangulardeviationofFSRPfrom

thesailnormal

canbecompensatedbythesailsteeringstrategyfor

interplanetary

transfer

trajectories

(whereβ>

55.5◦

isnotrequired

[10]).Nonetheless,

anoverallsaileffi

-ciency

parameter

ηshould

beused,whichtakesinto

accountthereduced

magnitudeofFSRP

dueto

the

non-perfect

reflectivityofthesailincludingitsdeflec-

tion/w

arpingunder

load.

Assumingaconservative

saileffi

ciency

ofη≈

0.85(aluminum

coatedplastic

film

),weget

(Peff) 1

AU=

2η(P

0) 1

AU

. =7.757·10−6N/m

2

fortheeff

ective

pressure

actingat1AU

onasolarsail

thatis

orientednorm

alto

thesun-lineand

FSRP=

(Peff) 1

AU

(

1AU

r

)

2

Acos2βn

fortherespective

SRP

forcein

adistance

rfrom

the

sun.

Thus,

toexperience

areasonable

acceleration,

solarsailcraft

must

belargeandvery

lightw

eight.

Theorbitaldynamicsofsolarsailcraftisin

manyre-

spects

similarto

theorbitaldynamicsofother

space-

craft,whereasm

allcontinuousthrust

isapplied

tomodifythespacecraft’sorbitover

anextended

period

oftime.

How

ever,other

continuousthrust

spacecraft

may

orientitsthrust

vectorin

anydesired

direction

6th

Inte

rnational A

AA

F S

ym

posiu

m o

n S

pace P

ropuls

ion:

Pro

puls

ion f

or

Space T

ransport

ation o

f th

e X

XIs

t C

entu

ry,

Vers

aill

es,

Fra

nce,

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6 M

ay 2

002

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(c)

2002 b

y D

LR

andvary

itsthrust

levelwithin

awiderange,whereas

thethrust

vector

ofsolarsailcraftisconstrained

tolie

onthesurfaceof

a’bubble’directedaw

ayfrom

the

sun(see

Figure

3).Nevertheless,

bycontrollingthe

sailorientation

relative

tothesun,solarsailcraftcan

gain

orbital

angu

larmom

entum

(ifFSRP·e

t>

0)an

dspiral

outw

ards–aw

ayfrom

thesun–or

lose

orbital

angu

larmom

entum

(ifFSRP·e

t<

0)an

dspiral

in-

wards–towardsthesun.

Figure

3:Spirallinginwardsan

dou

twards

SolarSailcraftPerformance

Parameters

Beforetalkingab

outperform

ance

ofnear-term

solar

sailcraft,

themostcommon

perform

ance

definitions

shou

ldbegiven.

Theperform

ance

ofsolarsailcraft

may

beexpressed

bythefollow

ingparam

eters:

•thesailassembly

load

ing

σ s=

ms

A

isdefined

asthemassof

thesailassembly

(the

sailfilm

andtherequired

structure

forstoring,

deployingan

dtension

ingthesail,index

’s’)

per

unit

area.Thus,

thesailassembly

load

ingis

the

keyparameter

fortheperform

ance

ofasolarsail

andtheeffi

ciency

ofitsstructuraldesign.

•thesailcraftload

ing

σ=

m A=

ms+m

p

A=

σ s+

mp

A

isdefined

accordingly

asthespecificmassof

sail-

craftincludingthepayload

(index

’p’).It

shou

ldbenoted,that

theterm

payload

stan

dsforthe

totalsailcraftexceptthesolarsailassembly

(i.e.

exceptthepropulsionsystem

).

•thecharacteristicacceleration

acisdefined

asthe

max

imum

acceleration

at1AU

solardistance.It

canbecalculatedvia

(Peff) 1

AUA

=mac=

σAac=

=(σ

s+

mp

A)A

ac

⇒ac=

(Peff) 1

AU

σ s+

mp

A

Using

thecharacteristic

acceleration

,theSRP

forceactingon

thesailcanbewritten

as

FSRP=

mac

(

1AU

r

)

2

cos2βn

•the

ligh

tnessnumber

λ,which

isindep

endent

from

solardistance,is

defined

astheratioof

the

SRPacceleration

experiencedbyasolarsailnor-

mal

tothesun

linean

dthesolargravitational

acceleration

(5.93mm/s2

at1AU)

λ=

ac

5.93

mm/s2

Usingtheligh

tnessnumber,theSRPforceacting

onthesailcanbewritten

as

FSRP=

λµm r2

cos2βn

whereµ=

GM

sun.

DLR

Ground-based

Demonstrationof

SolarSail

Technology

InDecem

ber

1999

agrou

nd-based

dem

onstration

ofsolarsailcrafttechnologywas

perform

edat

theGer-

man

AerospaceCenter(D

LR)at

Cologne,

wherea

20m×

20m

solarsailwas

successfullydeployedin

asimulated

zero-g

environ

mentan

dam

bientenviron

-mentalconditions(F

igure

4)[6][14].

Figure

4:Fullydeployed20

m×20

msolarsailat

DLR

Thesquaresolarsailconsisted

offourCFRP

(Car-

bon

Fiber

Reinforced

Plastics)

boom

swithaspecific

massof

101g/m

andof

fourtriangu

larsailsegm

ents

mad

eof

aluminum-coated(0.1µm)plastic

film

swith

6th

Inte

rnational A

AA

F S

ym

posiu

m o

n S

pace P

ropuls

ion:

Pro

puls

ion f

or

Space T

ransport

ation o

f th

e X

XIs

t C

entu

ry,

Vers

aill

es,

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nce,

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ay 2

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2002 b

y D

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athicknessbetween4and12µm.

Theboomscon-

sisted

oftw

oCFRP

shells

thatwerebonded

atthe

edgesto

form

atubularshape,

sothatthey

canbe

pressed

flat

androlled

up(F

igure

5).

Figure

5:DLR

deployable

CFRP

boom

Theboom

swererolled

up

ina60cm

×60cm

×65

cm-sized

deployment

module,

from

where

they

unfolded

automatically.

After

deploymentthey

re-

turned

totheirtubularshapewithhighbendingand

bucklingstrength.

Subsequently,

thefoursail

seg-

ments

weredeployedbyropes.Toassessthehand-

lingbehav

iorof

differentsailmaterials,thesailseg-

ments

weremad

eof

threedifferentaluminum-coated

plastic

film

s,12

µm

polyethyleneterephtalate

(PET,

Mylarr

),7.5µm

polyim

ide(P

I,Kaptonr)and4µm

polyethylenenap

hthalate

(PEN).

Allsegments

were

reinforced

alon

gthethreeedges

ofthetriangle

toprevent

rips.

The

specific

mass

ofthe

sail

film

was

18.9g/m

2fortheMylarr

-segment,12.4g/m

2for

theKap

tonr-segmentand

10.5g/m

2forthePEN-

segm

ent.

Thedeploymentmodule

andthecross

sec-

tion

oftheboom

sforthisground-baseddem

onstration

weredim

ension

edfora40m×

40m

solarsail,which

was

toolargeforan

in-doordem

onstration.Forthe

structuralsizingof

theboomstw

oloadcaseswerecon-

sidered,bending–dueto

theSRPforce–andbuckling

–dueto

sail

deploymentand

sail

tensioningforces.

Accordingto

FEM

(FiniteElementMethod)calcula-

tion

s,similar

boom

scould

beusedalsoforlarger

sails

[7].

DLR

MissionProposalforENEAS

NearEarth

Asteroids(N

EAs)

are

apromisingcate-

gory

oftarget

bodiesforafirstsolarsailcraftmission,

since

they

canbeaccessed

relatively

easily

andsince

they

areof

greatscientificinterest.Therefore,in

Au-

gust

2000,adedicated

missionfortheexplorationof

NEAswithsolarsailcraft

(ENEAS)wasproposedby

DLR

incoop

erationwiththeWestfalischeW

ilhelms-

Universitatat

Munster

(Germany)asa

candidate

within

theGerman

small

satelliteprogram

forex-

traterrestric

sciences[3][14].

Based

on

thesuccess-

fuldeploymentexperim

entdescribed

above,

ENEAS

(Figure

6)wasintended

tofeature

adeployable50m×

50m

solarsailthatwould

becapable

totransporta

micro-satellitewithamass

of65.5kgto

aNEA

within

less

thanfive

years.Table

2summarizestheENEAS

parameters.

Figure

6:DLR

ENEASsolarsailcraft

withdeployed

controlmast

(artist’sview)

Sailarea

A(50m)2

Sailassem

bly

loading

σ s29.2g/m

2

Sailassem

bly

mass

ms

73kg

Payloadmass

mp

65.5kg

Totalsailcraft

mass

m138.5kg

Sailcraft

loading

σ55.4g/m

2

Lightnessnumber

λ1/42.4

Characteristicacceleration

ac

0.14mm/s2

CharacteristicSRP

force

FSRP,c

19.5mN

Table

2:ParametersfortheENEASsolarsailcraft

Forpropulsionless

attitudecontrol,

thesolarsail

and

the

micro-satellite

would

be

separated

by

acommerciallyavailable

10m

collapsible

controlmast,

whichis

housedinsidethedeploymentmodule

inits

stow

edconfiguration.Thiscontrolmast

isattached

tothedeploymentmodule

via

atw

odegreeoffreedom

actuatorgim

bal,whichallow

sto

rotate

themast

in-

cludingtheattached

micro-satellitewith

respectto

thesail

(Figure

6).

Thus,

by

rotating

thecontrol

mast,thecenterofmass

(CM)can

beoffsetfrom

thecenterofpressure

(CP).

Theresultingexternal

torquemay

beusedto

rotate

thesailaboutanyCM-

intersectingaxis

parallel

tothesailplane(this

atti-

tudecontrolconceptwasoriginallyproposedby[1]).

1996FG

3waschosen

asthetarget

body

forthe

ENEASmission,since

1996FG

3hasorbitalelem

ents

nottoodifferentfrom

thatofEarthandsince

itisan

object

ofexceptionalscientificinterest.Observations

indicatethat1996FG

3is

abinary

asteroid,consist-

ingofacentralbodywitharotationperiodofab

out

3.60hours

and

asatellitewith

an

orbitalperiod

of

about16.15hours.Thedetermined

averagebulk

den-

sity

is1.4±

0.3g/cm

3whichis

highly

suggestive

of

a’rubble

pile’

structure

[12].

ENEASis

intended

to

6th

Inte

rnational A

AA

F S

ym

posiu

m o

n S

pace P

ropuls

ion:

Pro

puls

ion f

or

Space T

ransport

ation o

f th

e X

XIs

t C

entu

ry,

Vers

aill

es,

Fra

nce,

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ay 2

002

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(c)

2002 b

y D

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determinethephysicalproperties

andtheevolution

ofthe1996FG

3system

.Trajectoryop

timizationusingthecalculusof

vari-

ationsrevealed,that

theENEASsailcraftcanreach

1996FG

3in

4.5years(1640days),ifit

isinserted

di-

rectly

into

aninterplanetarytrajectorywithahyper-

bolic

excess

energy

ofC3=

4km

2/s2.How

ever,more

recentlyperform

edtrajectoryop

timization,based

onartificial

neuralnetworksan

devolution

aryalgorithms

producedabettertrajectoryforthesamelaunch

date,

whichiscloser

tothe(unknow

n)glob

alop

timum

(Fig-

ure

7)[2].

Figure

7:1996FG

3rendezvoustrajectoryfor

ac=

0.14

mm/s

2

Theflighttimecould

bereducedby45

days(3%),

reducingat

thesametimetheC3requirem

entfrom

4km

2/s2

to0km

2/s2,thuspermittingareductionof

launch

cost.Theaccuracy

ofthetrajectorygenerated

bytheartificial

neuralnetworkis∆r<

11000km

for

therelative

distance

tothetarget

bodyat

rendezvous

and∆v<

43m/sfortherelative

velocity

(evenwith-

outperform

ingalocalfinetuningof

thetrajectory)

[2].

Near-Term

SolarSailcraft

Performance

Look

ingat

theequationforthecharacteristic

acceler-

ationof

solarsailcraftwithasquaresail,

ac=

(Peff) 1

AU

σ s+

mp

s2

,

onecanseethat

theperform

ance

dep

endson

three

designparameters,

thesailassembly

load

ingσ s,the

payload

massm

pan

dthesidelengths(orarea

s2)of

thesolarsail,definingathree-dim

ension

alsolarsail

designspace.

Figures8an

d9show

param

etricsec-

tion

sof

this

designspaceforafixed

σ s=

29.2g/m

2

andafixed

s=

50m

respectively

(asfortheENEAS

sailcraft).Ascanbeseen

from

thediagram

inFigure

8,acharacteristic

acceleration

ofupto

0.265mm/s2

canbeachievedwithou

tan

ypayload

.For

asm

aller

acapositivepayload

masscanbeaccommodated,de-

pendingon

thesailsize.Toachieve

acharacteristic

acceleration

beyon

d0.265

mm/s2,thesail

assembly

load

inghas

tobefurther

reduced(F

igure

9).

Figure

8:Thecharacteristic

acceleration

acas

afunctionof

san

dm

pforσ s

=29.2g/m

2

Figure

9:Thecharacteristic

acceleration

acas

afunctionof

σ san

dm

pfors=

50m

Bydifferentcombinationsof

thethreedesignpa-

rametersan

ydesired

characteristic

acceleration

can

beachieved2.

Anincrease

inpayload

masscan,for

exam

ple,beoff

setwithaproportion

alincrease

ofs2

orwitha(not

inverselyproportion

al)decreaseof

σ s.

2It

should

benotedthatm

pandscanbechosenindep

en-

den

tly,

whereasσ s(s)is

afunctionofswith∂σ s/∂s<

0,since

themass

oftheboomsand

thedep

loymen

tmodule

scale

less

thanlinearlywiththesailarea.How

ever,weare

onthesafe

side,

when

weassume∂σ s/∂s=

0to

keepcalculationssimple.

6th

Inte

rnational A

AA

F S

ym

posiu

m o

n S

pace P

ropuls

ion:

Pro

puls

ion f

or

Space T

ransport

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e X

XIs

t C

entu

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Thosedesignsensitivitiescanbedetermined

quanti-

tativelyusingsensitivityfunctions,

whichprovidean

indication

oftherelative

importance

ofeach

design

param

eter

foragiven

pointin

thesolarsail

design

space[10].Thesensitivityfunctionforanydesignpa-

rameter

ν∈{σ

s,m

p,s}may

bewritten

as

∆ac

ac

=Λν∆ν ν

with

Λσ s

=−

1

1+

mp/σ ss2

Λm

p=−

1

1+

σ ss2/m

p

Λs=

2

1+

σ ss2/m

p

For

theENEASsailcraft,wehaveΛσ s

=−0.537,

Λm

p=−0.473

and

Λs=

+0.946.

Ascan

beseen,

thesidelengthof

thesailisthemost

criticalparame-

terwithrespectto

theENEASsailcraft

perform

ance.

Thus,

anincrease

inperform

ance

isbestdonebyin-

creasingthesize

ofthesolarsail.

Ifcostscanbedescribed

by(know

norestimated)

functionsof

thethreedesignparameters,then

theop-

timum

(costminim

al)sailcraftdesignforagiven

per-

form

ance

canbedetermined.

ENEASwithSampleReturn

The

ENEAS

sailcraft

wasintended

torendezvous

1996FG

3forremotesensingwithaminim

um

scientific

payload

massof

5kg(C

CD

camera+

IRspectrometer

+magnetom

eter).

Tostudythe1996FG

3system

inmoredetail,itwou

ldbenecessary

toplace

alander

on

thesurfaceof

theasteroid

(e.g.formass

spectrometry

and/oralpha-protonspectrometry).

Someinvestiga-

tion

s(e.g.micro-structure

andisotopeanalysis)

tode-

term

inetheagean

dtheevolutionof1996FG

3could

beachieved

only

by

takingsamplesoftheasteroid

backto

Earth.Dueto

theirunlimited

∆v-capability,

solarsailcraftareespeciallycapable

toperform

such

sample

return

missions.

How

ever,compared

tothe

ENEASrendezvousmission,thepayloadmass

hasto

beincreasedconsiderably.Thekey

questionsforthe

ENEAS-SR

(sam

ple

return)missiondesignare:

Q1:

What

isthemax

imum

acceptable

missiondura-

tion

Tmax?

Q2:

What

istheminim

um

characteristicacceleration

ac,m

into

perform

themissionin

Tmax?

Q3:

What

istheexpectedsailassem

bly

loadingσ s

and

saildim

ension

sfornear-term

solarsailcraft?

Q4:

What

isthemax

imum

payloadmass

toget

ac,m

in

forthespecified

σ sands?

Answ

erto

Q1:Atpresent,

themaxim

um

accept-

able

missiondurationseem

sto

bedetermined

bythe

trip

timerequired

withchem

icalpropulsion,includ-

ing

(eventually

multiple)gravity

assistmaneuvers.

Dueto

therelatively

large∆v-requirem

entofab

out

6−10km/sforamissioncomparable

toENEAS-SR,

butwithchem

icalpropulsion,such

amissionwould

requireeither

anexpensive

launch

vehicle

andheavy

spacecraft,resultingin

ashort

trip

timeofafewyears,

orseveralgravityassists,resultingin

alongtrip

time3.

Since

ourapproach

aim

satlow-cost

missions,only

the

gravityassistoptionseem

sto

beareasonable

conven-

tional

alternative.

Thus,

fortheENEAS-SR

mission

weassumeatotalmissiondurationofmore

thanten

years

asnotacceptable.

Answ

erto

Q2:Trajectory

calculationsshow

,that

anENEAS-SR

missionto

1996FG

3canbeachieved

even

withacharacteristicaccelerationof0.10mm/s

2

in9.40years,includingarendezvoustrajectory

of6.27

years

(2290days,

Figure

10),

340daysofoperations

attheasteroid

andanEarthreturn

trajectory

of2.20

years

(805days,

Figure

11).

Figure

10:1996FG

3rendezvoustrajectory

for

ac=

0.10mm/s2

Answ

erto

Q3:Thediagram

inFigure

12show

sthe

required

sailsize

fordifferentsailassem

bly

loadings

andpayloadmasses,to

obtain

acharacteristicaccel-

eration

of0.10mm/s

2.Basedontheexperienceswith

theground-basedsolarsailtechnologydem

onstration

described

above,

weconsider

amaxim

um

sailsize

of

70m×70m

withasailassem

bly

loadingof29.2g/m

2

(sail

film

+booms+

deploymentmodule)asare-

alistic

–how

ever

stillchallenging–baselineforthe

ENEAS-SR

mission.

Answ

erto

Q4:Thespecified

σ sandsyield

apay-

loadmass

of237kgto

get

acharacteristicacceleration

3similarto

the

Rosetta

mission

tocomet

46P/W

irtanen

,which

willhave

three

interm

ediate

gravity

assistmaneu

vers

(Mars-E

arth-E

arth)andatrip

timeofapproxim

ately

nineyears

6th

Inte

rnational A

AA

F S

ym

posiu

m o

n S

pace P

ropuls

ion:

Pro

puls

ion f

or

Space T

ransport

ation o

f th

e X

XIs

t C

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Figure

11:1996FG

3sample

return

trajectoryfor

ac=

0.10

mm/s2

Figure

12:Thesidelengthsof

thesolarsailthat

isrequired

toachieve

acharacteristic

accel-

erationof

0.10mm/s2

asafunctionof

σ san

dm

p

of0.10mm/s2.

Currentresearch

atou

rdepartm

ent

indicates

that

itshou

ldbepossible

torealizesuch

amission

within

thespecified

massbudget,includinga

lander

ofab

out140kgan

dasample

return

capsule

ofab

out40

kg.Tab

le3summarizes

theENEAS-SR

param

eters.

ENEAS-SR

MissionScenario

Since

forsolarsailcraftof

moderateperform

ance

gain-

ingorbital

energy

intheEarth’s

gravitational

fieldis

diffi

cult

andtimeconsuming,

thelauncher

willinsert

theENEAS-SR

solarsailcraftdirectlyinto

aninter-

planetarytrajectorywithahyperbolic

excess

energy

ofC3=

0km

2/s

2.

After

theinjection,thesailan

d

Sailarea

A(70m)2

Sailassembly

load

ing

σ s29.2g/m

2

Sailassembly

mass

ms

143kg

Payload

mass

mp

237kg

Total

sailcraftmass

m380kg

Sailcraft

load

ing

σ77.6g/m

2

Lightnessnumber

λ1/59.3

Characteristic

acceleration

ac

0.10mm/s2

Characteristic

SRP

force

FSRP,c

38.0mN

Tab

le3:

Param

etersfortheENEAS-SRsolarsailcraft

theattitudecontrol

mastaredeployedin

a3-ax

issta-

bilized

mode.

Then

thesail

isoriented

tofollow

apre-calculatedattitudeprofile,lead

ingto

anop

timal

interplanetarytran

sfer

trajectory.

Duringthetran

s-fer,theENEAS-SR

solarsailcraftwillrunalmostau

-tonom

ously,

sothat

grou

ndmon

itoringwillbecarried

outon

aweekly

basison

ly.Attheendof

thetran

sfer

trajectory

thesolarsailcraftwillbemak

ingaren-

dezvouswith1996FG

3within

itsgravitational

sphere

ofinfluence

(Hill-sphere)

ofbetween70

km

radius(at

perihelion)an

d150km

radius(atap

helion).

Evenin

thenear-field

oftheasteroid,theSRP

acceleration

ofbetween0.05mm/s2

(atap

helion)an

d0.21mm/s2

(atperihelion)is

larger

than

theasteroid’s

gravita-

tion

alacceleration

(0.01to

0.00005

mm/s2

inadis-

tance

rangingfrom

5to

50km),

sothat

thesailcraft

isab

leto

hover

onan

artificial

equilibrium

surfacein

thehem

ispherethat

isop

positeto

thesun(F

igure

13).

Figure

13:Hoveringat

theasteroid

Thosequasi-stationaryhoveringpositionsareun-

stab

lebutcanbestab

ilized

usingafeedbackcontrol

loop

tosailattitudealon

e[11].Hoveringneartheas-

teroid,the(likelycomplex)gravitational

fieldof

the

target

bodyis

studied,so

that

acoarse

gravitational

fieldmodel

canbedetermined.Thereafter,thelan-

der

withtheEarth

return

capsule

isseparated

from

6th

Inte

rnational A

AA

F S

ym

posiu

m o

n S

pace P

ropuls

ion:

Pro

puls

ion f

or

Space T

ransport

ation o

f th

e X

XIs

t C

entu

ry,

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es,

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thesolarsailto

gointo

closerorbit

abouttheaster-

oid.W

hilemeasuringtheasteroid’sgravitationalfield

withincreasingaccuracy,theorbitofthelander

iscon-

tinuou

slylowered

untilasafe

landingtrajectory

can

becomputed(som

eorallofthose

extensive

compu-

tation

smay

beperform

edonEarth).

Once

landed,

thesample

isfeddirectlyinto

theEarthreturn

cap-

sule

and

brough

tback

tothehoveringsailcraft.

Inthis

mission

phase,

thesailcraft

iswaiting

edge-on

(sothat

noSRP

forceis

actingon

thesail)atthe

L2Lagrange

pointforthelander

inorder

toassist

therendezvou

s.Thelander

design,thesample

ex-

traction

mechan

ismsandthesubsystem

srequired

torendezvousthewaitingsailcraft

requirefurther

stud-

iesan

darebeyon

dthescopeofthis

paper.

Since

1996FG

3is

abinarysystem

,it

would

beinteresting

tolandan

dextractsamplesfrom

both

bodiesto

in-

vestigatetheorigin

andthecollisionalevolutionofthe

1996FG

3system

.Since

thegravitationalacceleration

isvery

low

neartheasteroid

andtherequired

∆vfor

thelander

less

than

10m/s,

acold

gassystem

witha

propellantmassof

less

than4kgwillsuffice

toper-

form

allop

erations.

After

rendezvouswiththehover-

ingsailcraft,

there-docked

ENEAS-SR

solarsailcraft

returnsthesample

toEarth.Thereturn

trajectory

ismuch

faster

than

thetransfer

trajectory

to1996FG

3

since

norendezvousis

required

atEarth.Thus,

the

sailcraftmay

arrive

with

arelatively

large

hyper-

bolic

excess

velocity

ofabout8.4km/s.

Thegravita-

tion

alacceleration

ofEarthaddsanother

11.2km/s,

sothat

theEarth

reentryvelocity

may

reach

about

√8.4

2+11

.22km/s=

14.0km/s.

Finally,

just

before

thearrivalof

theENEAS-SR

solarsailcraft

atEarth,

thereturn

capsule

isseparatedfrom

thelander

and

injected

into

anEarth

reentry

trajectory,whereit

isdecelerated

by

atmospheric

friction

and

breaking

parachutes.

OtherPromising

Missionsfor

Near-Term

SolarSailcraft

Wehaveinvestigated

theperform

ance

ofnear-term

so-

larsailcraftalso

forrendezvousmissionswithcelestial

bodiesother

than

NearEarthObjects.

Table

4gives

theminim

um

rendezvoustimes

forsolarsailcraftwith

acharacteristic

accelerationof0.10to

0.20mm/s2

for

severaltarget

bodies.

Itshow

sthateven

withnear-

term

solarsailcraftplanetary

rendezvousmissionare

feasiblewithin

theinner

solarsystem

,ifrelatively

long

trip

times

canbetolerated.How

ever,acharacteristic

acceleration

of0.10

mm/s2

seem

sto

bealower

bound

forrendezvousmissionswithin

theinner

solarsystem

.

Summary

Wehaveinvestigated

theminim

um

solarsailcraftper-

form

ance

requirem

ents

forvariousinterplanetary

mis-

sion

s.Wewereab

leto

show

,thatthecharacteristic

Transfer

time[yr]

Target

∆v m

inforac[m

m/s2]

body

[km/s]

0.10

0.15

0.20

Mercury

20.1

8.3

5.9

4.2

Venus

6.8

4.6

2.9

2.0

Mars

6.2

9.2

7.5

5.1

Table

4:Minim

um

transfer

times

totheinner

planets

forsolarsailcraft

withacharacteristicaccel-

erationof0.10,0.15and0.20mm/s2

(∆v m

in

denotestheminim

um

impulsive∆v-values

forellipticalnon-coplanarHohmann-likeor-

bit

transferswithzero

hyperbolicexcess

ve-

locities

atboth

endsofthetrajectory)

accelerationmust

beatleast

0.10mm/s2

inorder

toavoid

unacceptable

longmission

durationseven

for

relatively

easily

accessible

inner

solarsystem

bodies.

A70m×

70m

solarsailwithasailassem

bly

load

ing

of29.2g/m

2(sailfilm

+booms+

deploymentmod-

ule)wasconsidered

tobearealistic

–how

ever

still

challenging–near-term

baseline.

Withthissolarsail,

acharacteristicthrust

of38mN

canbeachieved.The

characteristicacceleration–definingthemissiondu-

ration–dep

endsontheactualpayloadmass

mpand

ranges

from

0.265mm/s2

(mp=

0kg)to

0.10mm/s2

(mp=

237kg).

Wehavealsodem

onstrated,thata

sample

return

missionto

aNearEarthAsteroid

with

such

asolarsailisfeasiblewithin

amissionduration

of

approx.9.4

years.In

additionto

thescientificvalue

ofsuch

amission,thedem

onstration

ofthetechni-

calcapabilitiesofsolarsailpropulsionin

deepspace

would

beacentralobjective.

References

[1]F.AngrilliandS.Bortolami.Attitudeandorbital

modellingofsolar-sailspacecraft.ESA

Journal,

14:431–446,1990.

[2]B.Dachwald

andW

.Seb

oldt.

Optimization

of

interplanetary

rendezvoustrajectories

forsolar

sailcraft

usinganeurocontroller.Monterey,

Au-

gust

2002.AIA

A/AASAstrodynamicsSpecialist

Conference.AIA

A-2002-4989,in

preparation.

[3]E.K

.Jessberger

andW

.Seb

oldtet

al.ENEAS–

exploration

ofnear-earth

asteroidswith

asail-

craft.

Proposalfora

Small

Satellite

Mission

within

theExtraterrestric

SciencesProgram

of

DLR,August

2000.

[4]M.Leipold.

SolarSail

Mission

Design.

Doc-

toralthesis,Lehrstuhlfur

Flugmechanik

und

Flugregelung;TechnischeUniversitatMunchen,

1999.DLR-FB-2000-22.

6th

Inte

rnational A

AA

F S

ym

posiu

m o

n S

pace P

ropuls

ion:

Pro

puls

ion f

or

Space T

ransport

ation o

f th

e X

XIs

t C

entu

ry,

Vers

aill

es,

Fra

nce,

14-1

6 M

ay 2

002

Copyright

(c)

2002 b

y D

LR

[5]M.Leipold.Tothesunan

dpluto

withsolarsails

andmicro-sciencecraft.

ActaAstronautica,45(4-

9):549–555,1999.

[6]M.Leipold,M.Eiden,C.E

.Garner,L.Her-

beck,D.Kassing,

T.Niederstad

t,T.Kruger,

G.Pagel,M.Rezazad

,H.Rozem

eijer,

W.Se-

boldt,C.Schop

pinger,C.Sickinger,an

dW

.Un-

kenbold.Solar

sailtechnologydevelop

mentan

ddem

onstration

.Lau

rel,

2000.4thIA

AInterna-

tion

alCon

ference

onLow

-CostPlanetary

Mis-

sion

s.IA

A-L-0707.

[7]M.Leipold,C.E

.Garner,R.Freelan

d,A.Herr-

man

n,

M.

Noca,

G.

Pagel,

W.

Seb

oldt,

G.Spragu

e,an

dW

.Unckenbold.

ODISSEE

–aproposal

fordem

onstration

ofasolarsail

inearthorbit.ActaAstronautica,45(4-9):557–566,

1999.

[8]M.Leipold,E.Pfeiffer,P.Groepper,M.Eiden,

W.Seb

oldt,L.Herbeck,an

dW

.Unkenbold.So-

larsail

technology

forad

vanced

space

science

mission

s.Tou

louse,2001.52

ndInternational

As-

tron

auticalCon

gress.

IAF-01-S.6.10.

[9]M.Leipold,W

.Seb

oldt,

S.Lingn

er,E.Borg,

A.

Herrm

ann,

A.

Pab

sch,

O.

Wagner,

and

J.Bruckner.Mercury

sun-synchronou

spolar

or-

biter

withasolarsail.ActaAstronautica,39(1-

4):143–151,1996.

[10]

C.R

.McInnes.

SolarSailing.

Technology,Dy-

namics

and

Mission

Applications.

Springer–

Praxis

Seriesin

SpaceScience

and

Technology.

Springer–Praxis,Berlin,Heidelberg,

New

York,

Chicester,

1999.

[11]

E.Morrow,D.J.Scheeres,

and

D.Lubin.

So-

larsailorbit

operationsat

asteroids.

Journalof

SpacecraftandRockets,38:279–286,2001.

[12]

S.Mottola

andF.Lah

ulla.

Mutual

eclipse

events

inasteroidal

binarysystem

1996FG

3:Observa-

tion

san

danumerical

model.

Icarus,

146:556–

567,

2000.

[13]

C.G

.Sau

er.

Optimum

solar-sail

interplane-

tary

trajectories.

San

Diego,

Augu

st2000.

AIA

A/A

ASAstrodynam

icsCon

ference.76-792.

[14]

W.Seb

oldt,

M.Leipold,M.Rezazad

,L.Her-

beck,W

.Unkenbold,D.Kassing,

andM.Eiden.

Ground-based

dem

onstration

ofsolarsail

tech-

nology.

Rio

deJan

eiro,2000.51

stInternational

Astronau

ticalCon

gress.

IAF-00-S.6.11.

[15]

J.L

.Wrigh

t.Space

Sailing.

Gordon

andBreach

Science

Publishers,

Philad

elphia,1992.

6th

Inte

rnational A

AA

F S

ym

posiu

m o

n S

pace P

ropuls

ion:

Pro

puls

ion f

or

Space T

ransport

ation o

f th

e X

XIs

t C

entu

ry,

Vers

aill

es,

Fra

nce,

14-1

6 M

ay 2

002

Copyright

(c)

2002 b

y D

LR

Chapter 2

Electrostatics

Introduction

Chapter 1 contains in principle all laws and properties necessary to solve any problem in elec-tromagnetism. The rest of the course will therefore be applications of these laws and properties inspecific cases.

As stated in section 1.1.4.a, in the static regime (or steady state), where the sources {ρ,−→J } do

not vary in time, the coupling between the electric and magnetic field vanishes. The electric fieldis only due to static charges, whereas the magnetic field is only created by constant currents. Onecan therefore study the electric field - Electrostatics - and the magnetic field - Magnetostatics

- separately.In this chapter, we will concentrate on Electrostatics. In section 2.1, we will briefly go over the

relations valid in electrostatics. These are only the retranscription of the relations seen in chapter1 in the case of time-independent sources.

2.1 Maxwell’s equations applied to Electrostatics and con-

sequences

2.1.1 Coulomb force

The force exerted by a charge q on a charge q′, at a distance r apart is :

−→F =

qq′

4πε0r2−→u qq′ (2.1)

where −→u qq′ is the unit vector going from the source charge q to the charge q′ and ε0 the vacuum

permittivity or permittivity of free space :

ε0 ≃ 8, 85× 10−12 F.m−1 = 8, 85× 10−12 kg−1.m−3.A2.s4

In the case of multiple charges {qi}i acting on a charge q′, Coulomb’s law can be written as:

−→F =

q′

4πε0

∑

i

qir2i

−→ui (2.2)

where ri is the distance between the source charge qi and the charge q′, and −→ui is the unit vectorgoing from the source charge qi to the charge q′.

32

2.1. MAXWELL’S EQUATIONS APPLIED TO ELECTROSTATICS AND CONSEQUENCES33

In the case of a distribution of charges {ρ(P )} acting on a charge q′ placed at point M ,Coulomb’s law can be written as:

−→F (M) =

q′

4πε0

˚

(V )

ρ(P )

PM3

−−→PM dV (2.3)

Note that these three expressions of Coulomb’s law are strictly identical, albeit the form theyare written in has been chosen for convenience.

2.1.2 Maxwell’s equations in electrostatics

The decoupling between the electric and magnetic fields reduces the number of equations to betaken into account. The Maxwell’s equations of electrostatics are :

Maxwell-Gauss : div−→E =

−→∇ .−→E =

ρ

ε0(2.4)

Maxwell-Faraday :−→rot

−→E =

−→∇ ∧

−→E =

−→0 (2.5)

2.1.3 Electrostatic field

2.1.3.a Electrostatic field

The field created by a charge q at point M situated at a distance r is :

−→E (M) =

q

4πε0r2−→u r (2.6)

where −→u r is the unit vector going from the source charge q to point M.

In the case of multiple charges {qi}, the field created at point M can be written as:

−→E (M) =

1

4πε0

∑

i

qir2i

−→ui (2.7)

where ri is the distance between the source charge qi and point M , and −→ui is the unit vector goingfrom the source charge qi to point M .

In the case of a distribution of charges {ρ(P )}, the field created at point M can be writtenas:

−→E (M) =

1

4πε0

˚

(V )

ρ(P )

PM3

−−→PM dV (2.8)

2.1.3.b Typical values

The table below gives the orders of magnitude for a few electric fields :

FM antenna at 100 km distance 0.5 mV.m−1

100 W bulb at 1 meter distance 50 V.m−1

Between the atmosphere and the ground 1 - 100V.m−1

Laser pointer 102 V.m−1

Shorting in air 106 V.m−1

High intensity laser 108 - 1012 V.m−1

34 CHAPTER 2. Electrostatics

2.1.3.c Conservation of charge

The continuity equation (or law of conservation of charge), in statics, gives:

div−→J = 0 (2.9)

−→J is then a vector with conservative flux.

Figure 2.1: Flux tube for the current density.

♦ Proof : Figure 2.1 shows a flux tube for the current density. This means that for all points on

non-hatched surface of (Σ),−→J .−→dS = 0. In other words, no current density goes in or out of the

tube except at the two ends S1 and S2. Let us moreover suppose that the flux tube is sufficiently

small to have−→J = J1

−→n1 at all points of S1, with−→n1 the outgoing normal to S1, and

−→J = J2

−→n2 atall points of S2, with

−→n2 the outgoing normal to S2. Then :

0 =

˚

(V )

div−→J

=

‹

(S1)+(Σ)+(S2)

−→J .−→dS

=

¨

(S1)

−→J .−→dS +

¨

(Σ)

−→J .−→dS +

¨

(S2)

−→J .−→dS

= J1S1 − J2S2

Hence J1S1 = J2S2. The flux of current density going through S1 is conserved when going out ofS2.

This property is found for all vectors−→A having div

−→A = 0.

2.1.3.d Gauss’ law

Let us consider a closed surface (S) delimiting a volume (V ) which contains a charge volumedensity ρ and a total charge Qint. Maxwell-Gauss’s equation gives Gauss’ law:

‹

(S)

−→E .−→dS =

˚

(V )

ρ

ε0dV =

Qint

ε0(2.10)

This relation is one of the most useful relations in electrostatics. As we will see in section 2.2, alongwith the use of symmetries and invariances, it allows to determine the electric field in numeroussimple situations.

2.1. MAXWELL’S EQUATIONS APPLIED TO ELECTROSTATICS AND CONSEQUENCES35

2.1.3.e Circulation of electrostatic field

Faraday’s law in electrostatics gives the circulation of electrostatic field :

˛

(C)

−→E .−→dl = 0 (2.11)

There is no work of the electrostatic force around a circular contour.

2.1.3.f Discontinuity equations at interfaces

The discontinuity equation for the electrostatic field is :

−→E 2 −

−→E 1 =

σ

ε0

−→n 1→2 (2.12)

where−→E i is the electric field in the medium i, σ is the surface charge density at the interface

between the two media, and −→n 1→2 is the unitary vector, normal to the interface, directed frommedium (1) to medium (2).

2.1.4 Electrostatic potential

2.1.4.a Electrostatic potential

Definition : The electrostatic potential is defined by

−→E = −

−−→grad φ (2.13)

or equivalently :

φ(A)− φ(B) =

ˆ B

A

−→E .−→dl (2.14)

The potential created by a charge q at point M situated at a distance r is :

φ(M) =q

4πε0r(2.15)

In the case of multiple charges {qi}i, the potential created at point M can be written as:

φ(M) =1

4πε0

∑

i

qiri

(2.16)

where ri is the distance between the source charge qi and point M .

36 CHAPTER 2. Electrostatics

In the case of a distribution of charges {ρ(P )}, the potential created at point M can bewritten as:

φ(M) =1

4πε0

˚

(V )

ρ(P )

PMdV (2.17)

⋆ Note : The electrostatic potential φ(r) of a group of charges vanishes at infinity. This is theorigin of the convention we will often use :

φ(r)→ 0 when r → +∞ (2.18)

2.1.4.b Potential propagation

In electrostatics, the potential propagation (see section 1.3.4) is written under the form ofPoisson’s equation :

△φ = −ρ

ε0(2.19)

In the specific case where there is no charge (ρ = 0), the propagation equation reduces toLaplace’s equation :

△φ = 0 (2.20)

The Poisson (or Laplace) equation will have an unique solution φ if boundary conditions arespecified on a closed surface :

– Either the potential is defined on a closed surface. This is called the Dirichlet boundary

condition.– Or the electric field is defined on a closed surface. This is called the Neumann boundary

condition.

♦ Proof : Let us suppose that φ1 and φ2 obey to the same Poisson equation △φ = − ρε0

inside avolume (V ). Let :

U = φ2 − φ1

Let us impose Dirichlet boundary condition on the closed bounding surface (S) of the volume (V ).Then, inside (V ) :

△U = △φ2 −△φ1 = −ρ

ε0+

ρ

ε0= 0

and on (S), U = 0.We now need to prove Green’s first identity :

˚

(V )

(

ξ△ψ +−−→grad ξ.

−−→grad ψ

)

dV =

‹

(S)

ξ∂ψ

∂ndS

For this, let us apply the divergence theorem (˝

(V )div−→AdV =

‚ −→A.−→dS =

‚ −→A.−→n dS) to

−→A =

ξ−−→grad ψ :

˚

(V )

div(

ξ−−→grad ψ

)

dV =

‹

(S)

(

ξ−−→grad ψ

)

.−→n dS

˚

(V )

(

ξ△ψ +−−→grad ξ.

−−→grad ψ

)

dV =

‹

(S)

ξ∂ψ

∂ndS

2.1. MAXWELL’S EQUATIONS APPLIED TO ELECTROSTATICS AND CONSEQUENCES37

Now, let us apply Green’s first identity to ξ = ψ = U :

˚

(V )

(

U△U +(−−→grad U

)2)

dV =

‹

(S)

U∂U

∂ndS

Given the boundary conditions and the fact that △U = 0 in (V ) :

˚

(V )

(−−→grad U

)2

dV = 0

which means that−−→grad U = 0 inside (V ). Hence U is constant in (V ) and since U = 0 on (S),

U = 0 in all (V ). Therefore φ1 = φ2 and the solution is unique.

2.1.5 Field lines and equipotential surfaces

2.1.5.a Field lines

Definition : Electric field lines are the lines that are, at all points in space, tangent to

the electric field−→E . They can be determined through the relation :

−→E (M) ∧

−→dl =

−→0 (2.21)

where−→dl is an infinitesimal vector along the field line, centered around point M .

2.1.5.b Equipotential surfaces

Definition : Equipotential surfaces are surfaces where the scalar potential φ(M) isconstant.

2.1.5.c Properties

Property #1 : The electrostatic potential and the electric field reproduce all symmetries

and invariances of the charge distribution ρ(P ) that create this potential and this field.

Property #2 : Equipotential surfaces are normal to the electric field−→E at all points and

therefore are normal to the field lines.

♦ Proof : Equation 2.14 shows that if points A and B are points sufficiently close to one anotheron the same equipotential surface :

φ(A)− φ(B) = 0 =

ˆ B

A

−→E .−→dl

with−→dl defining the equipotential surface. Therefore

−→E⊥

−→dl .

This relation is only valid in electrostatics. In the general case, the term in ∂−→A∂t has to be

taken into account.

Property #3 : Field lines cannot be closed. They begin and end either on a charge or at

infinity.

♦ Proof : Let us suppose that (C) is a closed field line. In that case, by definition of a field line

(−→E parallel to

−→dl) :

˛

(C)

−→E .−→dl =

˛

(C)

E dl ≥ 0

38 CHAPTER 2. Electrostatics

However, equation 2.11 gives :˛

(C)

−→E .−→dl = 0

These two conditions impose :¸

(C)

−→E .−→dl = 0 and therefore

−→E =

−→0 .

Property #4 : In regions with no charge,−→E conserves its flux : div

−→E = 0

♦ Proof : The proof is the same as the one given in section 2.1.3.c.

In other words, when the field lines move apart, the norm of−→E diminishes.

Property #5 : The electrostatic potential cannot be maximum nor minimum in a point

where there is no charge.

♦ Proof : Let A be a point where the potential φ is maximum. Then there exists a volume Vdelimited by a surface Σ such that all points M in this volume have :

∀M φ(M) ≤ φ(A)

Let us chose the volume V such that Σ is an equipotential surface. Then :

‹

(Σ)

−→E .−→dS > 0

unless−→E =

−→0 . But Gauss’ law also gives :

‹

(Σ)

−→E .−→dS =

Qint

ε0

So that Qint > 0.

2.1.6 Energetics in electrostatic

2.1.6.a Electrostatic energy

Definition : The volume density of electrostatic energy is given by :

u =ε0E

2

2(2.22)

This is a particular case of equation 1.24.

As a consequence, the total electrostatic energy U in a volume V can be written as :

U =

˚

(V )

u dV

U =1

2

˚

(V )

ε0E2 dV ≥ 0 (2.23)

This energy is always positive. It implies that φ(+∞) = 0 and that the presence of a field adds

energy to the system.

2.2. EXAMPLES OF USUAL CHARGE DISTRIBUTIONS 39

2.1.6.b Relation with the electrostatic potential

In section 1.3.2, we have seen that qφ corresponds to the potential energy of a charge q placedin a potential φ. In other words, if the charge q is brought from infinity (where φ = 0) to a pointx where the electrostatic potential is φ, the work done on the charge is

W = qφ

If the potential is created by an ensemble of N charges qj (j = 1, 2, ..., N) at positions −→rj , thenthe potential energy created on the charge qi placed at position −→ri is :

Wi =qi

4πε0

N∑

j=1

qj|−→rj −

−→ri |

Then, the total potential energy for all charges is :

W =N∑

i=1

Wi =1

4πε0

N∑

i=1

∑

j<i

qiqj|−→rj −

−→ri |=

1

8πε0

∑

(i,j)

qiqj|−→rj −

−→ri |

where the terms i = j are not included in the sum. This can be rewritten as :

W =1

2

∑

i

qiφi

where φi =1

4πε0

∑

jqj

|−→rj−−→ri |

is the potential created by all other charges on qi at−→ri .

2.2 Examples of usual charge distributions

2.2.1 Example #1 : Uniformly charged infinite wire

Let us consider an uniformly charged infinite wire as schematized figure 2.2.1. The uniformlinear charge density is λ. Let us determine the electrostatic field and potential created by such asystem.

Figure 2.2: Uniformly charged infinite wire.

40 CHAPTER 2. Electrostatics

The problem is of cylindrical symmetry. We will therefore use the cylindrical coordinates. Sincethe charge distribution ρ(r, �θ, �z), and the electrostatic potential conserves the symmetries of thecharge distribution, φ(r, �θ, �z). Hence :

−→E = −

−−→grad φ = −

dφ

dr−→ur

Another way of obtaining this result is to say that all pointM of space belongs to two symmetry

planes : the one defined by {−→ur,−→uz} and the one defined by {−→ur,

−→uθ}. Since at these points M the

electric field must belong to all symmetry planes,−→E ‖ −→ur.

We will apply Gauss’ law on the following Gauss surface : a cylinder of height h and radius r :

‹

(SGauss)

−→E .−→dS =

Qint

ε0¨

(S1)

−→E .−→dS +

¨

(S2)

−→E .−→dS +

¨

(Slat)

−→E .−→dS =

λh

ε0

0 + 0 + 2πrhE(r) =λh

ε0

E(r) =λ

2πε0r−→E =

λ

2πε0r−→ur

From the relation between φ and−→E , one derives : φ = − λ

2πε0ln(r)+constant. If one imposes φ = 0

on the conductor, one obtains : φ = − λ2πε0

ln(

ra

)

.

2.2.2 Example #2 : Uniformly charged infinite plane

Let us consider an uniformly charged infinite plane as schematized figure 2.2.2. The uniformsurface charge density is σ. Let us determine the electrostatic field and potential created by sucha system.

Figure 2.3: Uniformly charged infinite plane.

2.2. EXAMPLES OF USUAL CHARGE DISTRIBUTIONS 41

The problem is of cartesian symmetry. We will therefore use the cartesian coordinates. Sinceρ(�x, �y, z), and the electrostatic potential conserves the symmetries of the charge distribution,φ(�x, �y, z). Hence :

−→E = −

−−→grad φ = −

dφ

dz−→uz

Moreover, the {Oxy} plane is a symmetry plane, so that - using−→E ‖ −→uz :

−→E (SP(x, y, z)) = SP

(−→E (x, y, z)

)

−→E (−x, y, z) = SP

(−→E (x, y, z)

)

−→E (−x, y, z) = −

−→E (x, y, z)

We will apply Gauss’ law on the following Gauss surface : a cylinder of height h and radius R,centered around the {Oxy} plane :

‹

(SGauss)

−→E .−→dS =

Qint

ε0¨

(S1)

−→E .−→dS +

¨

(S2)

−→E .−→dS +

¨

(Slat)

−→E .−→dS =

σπR2

ε0

E(z)πR2 − E(−z)πR2 + 0 =σπR2

ε0

E(z) =σ

2ε0−→E =

σ

2ε0

−→uz

From the relation between φ and−→E , one derives : φ = − σ

2ε0z + constant. If one imposes φ = 0 on

the conductor, one obtains : φ = − σ2ε0z.

2.2.3 Example #3 : Uniformly charged sphere

Let us consider an uniformly charged sphere as schematized figure 2.2.3. The uniform volumecharge density is ρ. Let us determine the electrostatic field and potential created by such a system.

42 CHAPTER 2. Electrostatics

Figure 2.4: Uniformly charged sphere.

The problem is of spherical symmetry. We will therefore use the spherical coordinates. Sincethe charge distribution ρ(r, �θ,�ϕ ), and the electrostatic potential conserves the symmetries of thecharge distribution, φ(r, �θ,�ϕ ). Hence :

−→E = −

−−→grad φ = −

dφ

dr−→ur

Another way of obtaining this result is to say that all planes containing −→ur are symmetry planes.

Since the electric field must belong to all symmetry planes,−→E ‖ −→ur.

We will apply Gauss’ law on the following Gauss surface : a sphere of radius r :

‹

(SGauss)

−→E .−→dS =

Qint

ε0‹

(SGauss)

E−→ur.dS−→ur =

4πR3ρ

3ε0ˆ π

θ=0

ˆ 2π

ϕ=0

E(r)r2sinθdθ dϕ =4πR3ρ

3ε0

E(r)4πr2 =4πR3ρ

3ε0

E(r) =R3ρ

3ε0r2

−→E =

R3ρ

3ε0r2−→ur

From the relation between φ and−→E , one derives : φ = − ρR3

3ε0r+ constant. If one imposes φ = 0 on

the conductor, one obtains : φ = − ρR3

3ε0r+ ρR2

3ε0.

2.3 Conductors

2.3.1 Electrostatic field and potential inside a conductor

An electrical conductor is a material that contains numerous “free” electrons. This means thatthese electrons - usually from the outer shells of the atoms constituting the conductor - are able tofreely move inside the material. In particular, if a conductor is submitted to an external electric

field−→E ext, the electrons will be set in motion by this field and will only stop once equilibrium is

achieved, that is to say that−→E total =

−→0 .

Property #1 : The electrostatic field inside a conductor is zero :

−→E conductor =

−→0 (2.24)

It follows that the scalar potential φ inside a an electrical conductor is constant.Property #2 : An electrical conductor is an equipotential region.

φ = constant (2.25)

2.3. CONDUCTORS 43

2.3.2 Charges inside a conductor

Let us consider an electrical conductor. As we have seen, the electrostatic field inside the

conductor−→E int is zero :

−→E int =

−→0 . Then Maxwell-Gauss equation gives that the volume charge

density inside the conductor is ρconductor = ǫ0div−→E = 0 :

Property #3 : In a conductor at equilibrium, there are no volume charges. All excess

charges are at the surface of the conductor :

ρconductor = 0 (2.26)

In practice, all charges accumulate on a few atomic layers at the surface of the conductor. Theseare therefore surface charges.

2.3.3 Electric field at the vicinity of a conductor

Let us consider the conductor schematized on the opposite figure anduse the discontinuity equations at its interface with another medium

where reigns an electrostatic field−→E ext :

−→E ext −

−→E int =

−→E ext =

σsǫ0

−→uz

Thus, under the action of an external magnetic field, charges accumulate at the surface of theconductor. These surface charge, in turn, create an electric field such that, in the conductor thetotal field is zero.

Property #4 : Outside, but at the immediate vicinity of an electrical conductor, the external

electric field is :−→E ext =

σsǫ0

−→n (2.27)

where −→n is the vector normal to the conductor surface.

2.3.4 Application #1 : radiation pressure for a conductor placed in a

zero electric field

We here consider the point Me external to the conductor, such asindicated in the opposite figure. One can decompose the electric fieldat point Me into two components :

i. The field created by the surface charges which immediately facethe pointMe on the conductor, which are on the surface dS. SinceMe can be taken infinitely close to the conductor, while still being

exterior to the conductor, it “sees” the surface dS under a solid angle of approximately2π, that is to say that the surface charges appear, for Me to be distributed on an infiniteplane. This component is therefore equal to the field created by an uniformly charged plane

:−→E1 = σs

2ǫ0

−→uz.

ii. The field−→E2 created by all other surface charges on the conductor.

We will call Σ the surface of the conductor from which the surfacedS is extracted.

Since the field in Me is :−→E =

−→E1 +

−→E2 =

σsǫ0

−→uz

44 CHAPTER 2. Electrostatics

one deduce that all charges on Σ create a field at point Me :

−→E2 =

σs2ǫ0

−→uz

Now, let us calculate the force exerted by the surface charges of Σ on those on dS :

−→dF = dq

−→E2(Me) = σsdS ×

σs2ǫ0

−→uz

This force is equivalent to a pressure since proportional to dS. The pressure therefore is P =σ

2

s

2ǫ0.

This pressure is called the radiation pressure.⋆ A few comments :

– To say that the field−→E1 is the same than the one created by an infinite charged plane might

look like an approximation. However, it can be shown that it is the exact result. Indeed, theelectric field outside the conductor is :

−→E ext =

−→E1(Me) +

−→E2(Me) =

σsǫ0

−→uz

The field inside the conductor (at Mi) can also be decomposed into a component due to the

field created by dS :−→E1(Mi) = −

−→E1(Me), and a component due to the field created by Σ

which is equal to−→E2(Mi) =

−→E2(Me)

−−→Eint =

−→0 = −

−→E1(Me) +

−→E2(Me)

Hence :−→E1(Me) =

σs2ǫ0

−→uz

−→E2(Me) =

σs2ǫ0

−→uz

– It is interesting to note that the field in the vicinity of a conductor, at pointMe only dependson the local surface charge density σs(Me), i.e. the one carried by the small conductor surfacedS. The charges of the rest of the conductor (on Σ), “organize” themselves so as to cancel,inside the conductor, the field created by those carried by dS.

2.3.5 Application #2 : radiation pressure for a conductor placed in an

external electric field

In the presence of an constant external electric field−→E 0, let us pro-

ceed with an analogous reasoning. The total external electric field canbe written :

−→E ext =

−→E0 +

−→E1(Me) +

−→E2(Me) =

σsǫ0

−→uz

The field inside the conductor, at pointMi can also be decomposed into

a component due to the field created by the charges on dS, which value is−→E1(Mi) = −

−→E1(Me),

and a component due to the field created by the charges on Σ, which value is−→E2(Mi) =

−→E2(Me) :

−−→Eint =

−→0 =

−→E0 −

−→E1(Me) +

−→E2(Me)

Hence :−→E1(Me) =

σs2ǫ0

−→uz

−→E0 +

−→E2(Me) =

σs2ǫ0

−→uz

2.3. CONDUCTORS 45

This time, the charges on Σ ”organize” themselves so as to exactly compensate the field createdby charges on dS, taking into account the external electric field. The radiation pressure exerted on

dS corresponds to the force exerted on the surface charges on dS due to the external field−→E0 and

due to the charges on Σ. Its expression still is P =σ

2

s

2ǫ0.

N.B. 1 : If−→E0 =

−→0 , the result is the same as in the previous paragraph.

N.B. 2 : A conducting object placed within an uniform electrostatic field−→E0 charges itself at

its surface, so that the electric field outside but at the immediate vicinity of this object is notnecessarily uniform !N.B. 2 : In this particular case, one cannot apply the principle of superposition in a simple manner

(i.e. a charged conductor in−→E0 = a charged conductor in zero field + a static field). This is due

to the fact that the boundary conditions, at infinity in particular, are not trivial.

2.3.6 High voltage breakdown

Let us now mention the dependence of the field with respect to shape. More specifically, wewill qualitatively discuss the field near a sharp conductor.

First, let us remark that if one takes two spheres of radius R1 and R2 ≪ R1, both charged withQ. According to the calculation made section 2.2.3, the electric fields respectively verify :

4πR21E1 =

Q

ε0

4πR22E2 =

Q

ε0

Hence :

E1 =Q

4πR21ε0

E2 =Q

4πR22ε0

As can be seen, the field near the smaller sphere is larger than the one in the vicinity of the

larger sphere. In other words, any sharp point in a material has a large electric field near its surface.This result is important for practical realizations. Indeed, above a certain electric field, the air

will break down. In this case, a loose charge - whether electron or ion - will be accelerated by thefield, will then collide with an atom, knocking out an electron of this atom. The additional ion andelectron will, in turn, be accelerated and a cascade process will begin, resulting in a great numberof charged particules in the air. Their motion creates a discharge or a spark.

♣ Application #1 : If one wants to put a conductor to a high potential, one must avoid theconductor surface to have any sharp surface features. Otherwise the air will discharge towards thistip.

♣ Application #2 : This field effect is used by field-emission microscopy to image the material ofa thin needle 1. This technique has enabled, since 1955, to image materials with a resolution of afew Angstroms, well before the advent of local probes such as the Scanning Tunneling Microscope(STM) or the Atomic Force Microscopie (AFM).

1. For explanations on the working principle, see for example : Field Ion Microscopy, B. Gault et al., “Atom

Probe Microscopy”, accessible on the Internet.

46 CHAPTER 2. Electrostatics

2.4 Electrostatic dipole

The notion of electrostatic dipole is important both in physics and chemistry. Indeed, inchemistry, it can model the behavior of polar molecules - such as HCl for instance - under amagnetic field. In physics, we will see in the next chapter that it is at the basis of the microscopicdescription of dielectric materials.

2.4.1 Definition

Definition : An electrostatic dipole is a distribution of static charges {qi}i placed atpoints Pi, of finite spatial extension δ around point A, and for which the electric dipole moment−→p (in C.m) is non zero :

−→p =∑

i

qi−−→APi (2.28)

♣ Example : The simplest electrostatic dipole is a charge doublet {+q,−q} (q > 0). If +q is placed

at point P , −q at point N , and A the center point of [NP ], and−−→NP = d −→u :

−→p = +q−→AP − q

−−→AN = q

−−→NP = qd −→u

Note : A dipole which moment is not affected by the application of an external electric field iscalled rigid.

Definition : The dipole approximation consists in considering the effect of an electrostaticdipole at point M such that r = AM ≫ δ.

In other words, one neglects the spatial extension δ of the real charge distribution {qi}i aroundpoint A and replaces it by a dipole −→p placed in A. Therefore, a dipole can always be modeled bytwo “equivalent” charges of opposite sign, centered at point A.

2.4.2 Electrostatic field and potential created by an electrostatic dipole

2.4.2.a Electrostatic potential created by an electrostatic dipole

In the dipole approximation, the electrostatic potential created at point M by an electrostatic

dipole −→p placed at point A such that−−→AM = r−→u , with −→u an unit vector, is :

φ(M) =−→p .−→u

4πε0r2(2.29)

2.4. ELECTROSTATIC DIPOLE 47

Figure 2.5: Schematic representation of a dipole.

♦ Proof :

φ(M) =∑

i

qi4πε0PiM

1

PiM=

(

PiM2)−1/2

=

(

(−−→OM −

−−→OPi

)2)−1/2

with the geometry described figure 2.5 :

1

PiM=

(

r2 +OP 2i − 2r−→ur.

−−→OPi

)−1/2

=1

r

(

1 +OP 2

i

r2−

2r−→ur.−−→OPi

r2

)−1/2

so that at first order in OPi

r :

φ(M) =∑

i

qi4πε0r

(

1 +OP 2

i

r2−

2r−→ur.−−→OPi

r2

)−1/2

≃∑

i

qi4πε0r

(

1 +r−→ur.

−−→OPi

r2+ ...

)

≃∑

i

qi4πε0r

+∑

i

qi−→ur.−−→OPi

4πε0r2

≃ 0 +−→ur

4πε0r2.∑

i

qi−−→OPi

due to the global neutrality of the dipole. In the present case :

φ(M) ≃−→ur.−→p

4πε0r2

≃p cos θ

4πε0r2

48 CHAPTER 2. Electrostatics

2.4.2.b Electrostatic field created by an electrostatic dipole

In the dipole approximation, the electrostatic field created at pointM by an electrostatic dipole−→p placed at point A such that

−−→AM = r−→u , with −→u an unit vector, is :

−→E (M) =

3 (−→p .−→u ) .−→u −−→p

4πε0r3(2.30)

♦ Proof : Taking the same definitions as above :

V (M) ≃p cos θ

4πε0r2−→E = −

−−→grad φ

−→E ≃

∣

∣

∣

∣

∣

∣

2p cos θ4πε0r3p sin θ4πε0r3

0

Moreover :

3 (−→p .−→u ) .−→u −−→p

4πε0r3=

3p cos θ−→ur − p cos θ−→ur + p sin θ−→uθ

4πε0r3

=

∣

∣

∣

∣

∣

∣

2p cos θ4πε0r3p sin θ4πε0r3

0

2.4.2.c Field lines and equipotential surfaces for an electrostatic dipole

From equation 2.29, the equipotential surfaces are such that r2 = k cos θ with k a contant.Field lines are obtained by taking, at all points, the normal to the equipotential surfaces. Fieldlines and equipotential surfaces are schematized figure 2.6.

2.4. ELECTROSTATIC DIPOLE 49

Figure 2.6: Schematic representation of the equipotential surfaces and field lines of adipole. Taken from http://www.pstcc.edu/departments/natural_behavioral_sciences/Web%

20Physics/Experim%2001.htm. The dipole is here modeled by two opposite charges.

Note : Bear in mind that the above expressions are obtained in the dipole approximation, whichmeans that they are not valid within the charge distribution.

2.4.3 Mechanical action of an external electric field on a electrostatic

dipole

2.4.3.a Electrostatic dipole in an external electric field

In the dipole approximation, an electrostatic dipole −→p , of center A, placed in an external electric

field−→E is submitted to forces such that :

– the resultant force is :−→F (A) =

(

−→p .−−→gradA

)−→E (A)

– the angular momentum is :−→M(A) = −→p ∧

−→E (A)

♦ Proof : Let us first calculate the resultant force (first order approximation) :

−→F (A) =

∑

i

qi−→E (Pi)

≃

∣

∣

∣

∣

∣

∣

∣

∣

∣

∑

i qi

(−→Ex(A) +

−−→APi.

−−→grad Ex(A)

)

∑

i qi

(−→Ey(A) +

−−→APi.

−−→grad Ey(A)

)

∑

i qi

(−→Ez(A) +

−−→APi.

−−→grad Ez(A)

)

≃

∣

∣

∣

∣

∣

∣

∣

∑

i qi−−→APi.

−−→grad Ex(A)

∑

i qi−−→APi.

−−→grad Ey(A)

∑

i qi−−→APi.

−−→grad Ez(A)

≃(

−→p .−−→gradA

)−→E (A)

Then the angular momentum is :

M(A) =∑

i

−−→APi ∧ qi

−→E (Pi)

≃∑

i

−−→APi ∧ qi

−→E (A)

≃ −→p ∧−→E (A)

2.4.3.b Application in the case of a rigid electrostatic dipole

If the considered dipole is rigid, the charges composing the dipole do not move with respect toone another. −→p is then constant in norm. Then the resulting force is :

−→F (A) = −

−−→gradA (Ue)

Ue = −−→p .−→E (A)

50 CHAPTER 2. Electrostatics

♦ Proof :

Ue =∑

i

qiφ(Pi)

=∑

i

qiφext(Pi) +∑

i

qiφint(Pi)

=∑

i

qi

(

φext(A) +−−→APi.

−−→grad φext(A)

)

+ constant taken to be 0

= 0 +−→p .(

−−→E (A)

)

The force exerted on the dipole therefore derives from the potential energy Ue. Then, the dipole

tends to align with the external field−→E (M).

♣ Example #1: Solvation of ions - Some solvent molecules, such as water for instance, are polar,which means they possess an electrostatic dipole moment (figure 2.4.3.b). When ions are dissolvedin a solvent, the solvent molecules organize themselves around the ion in order to minimize theelectrostatic energy (figure 2.4.3.b). In chemistry, the standard unit for molecular dipole momentsis the Debye (1 D = 3.33×10−30 C.m). A water molecule has a dipole moment of 1.83 D.

Figure 2.7: Schematization of electrostatic dipole moment of a molecule of water. Taken from http:

//guweb2.gonzaga.edu/faculty/cronk/CHEM101pub/L18-index.cfm?L18resource=water.

Figure 2.8: Schematization of the hydration of Na+ by water molecules. Taken from http://www.

science.uwaterloo.ca/~cchieh/cact/applychem/hydration.html.

♣ Example #2: Van der Waals interactions - Van der Waals forces 2 group different forces, allproportional to 1/r7. You will derive some of these in the Electrostatics exercise sheet. These

2. These intermolecular forces have first been predicted by the Dutch physicist J.D. van der Waals in his PhD

thesis, defended in 1873.

2.5. METHODS IN ELECTROSTATICS 51

forces originate from dipole-dipole interactions 3 and explain most of intermolecular interactions.They have numerous applications, from scotch tape to biology 4 and including chemistry or polymerscience.

2.5 Methods in electrostatics

There are numerous solving methods in electrostatics which we will not review here 5. For thepurpose of this course, you will mainly be using one of the following methods :

– Directly compute the electric field using symmetries, invariances and :

−→E =

1

4πε0

˚

(V )

ρ

r2−→u dV

– Directly compute the electrostatic potential using symmetries, invariances and :

φ =1

4πε0

˚

(V )

ρ

rdV

– Determine the electric field using symmetries, invariances and Gauss theorem :

‹

−→E .−→dS =

Qint

ε0

Also note that, for a problem with spherical symmetry, the method of separation of variables -as seen in Quantum Mechanics for example - can be used.

2.5.1 Method of images

We will however detail one important method : the Method of images. This method can beuseful in the case of one or a few point charges in the presence of boundary surfaces that imposediscontinuities of the electric field. For instance, point charges in the presence of conductors.

The method consists in replacing the real problem with discontinuities - for eg. the conductingplanes - by an equivalent problem in an enlarged region, with no boundaries, where one or a fewvirtual charges - the image charges -, are placed outside the region of interest so that :

1. these image charge simulate the boundary conditions ;

2. the electrostatics equation - the Poisson equation - remain unchanged in the region of interest.

The virtual problem with image charges is usually easier to solve. Then, thanks to the unicitytheorem (see section 1.6.2), the electric field that is found for the equivalent-charge problem is thesame, in the region of interest, than the one in the real problem.

♣ Example : single charge in the presence of an infinite conducting plane

3. See exercise sheet on electrostatics.

4. See section 2.6

5. An introduction to those can be found in chapters 2, 3 and 4 of J.D. Jackson, Classical Electrodynamicsand

in chapters 5 to 8 of E. Weber, Electromagnetic Theory.

52 CHAPTER 2. Electrostatics

Figure 2.9: Schematization of the equivalence between a real problem and an equivalent-imageproblem.

The method of images can be understood in the simple example schematized figure 2.9.

Let us consider a charge q sitting in vacuum in the presence of an infinite conducting plane,at a distance d, occupying a semi-infinite space. As stated earlier, the potential in a conductor isconstant and since it extends to infinity, the corresponding potential is φ = 0. In particular, φ = 0is imposed at the boundary between the vacuum and the conductor.

Moreover, in the region of interest - outside the conductor - the electric field is determined bythe Poisson equation : ∆φ = − ρ

ε0.

Let us now consider another problem where there is no conductor, but two point charges q and−q, separated by 2d. Then, in the middle of these two charges φ = 0 and in the region wherethere previously was no conductor, the electric field is determined by the same Poisson equationas before : ∆φ = − ρ

ε0.

The two problems therefore have the same Poisson equation in the volume of interest - thevacuum in the initial problem -, with the same boundary conditions - φ = 0 where the boundarybetween vacuum and conductor was before -, so that the solution, in the region of interest, is thesame for the two problems.

The corresponding field lines are schematized figure 2.10. The plain lines are the field lines forthe real problem, whereas the dashed lines correspond to the additional field lines correspondingto the virtual problem, outside the region of interest.

2.6. ADDITIONAL READING 53

Figure 2.10: Field of a charge near a plane conducting surface, found by the method of images.Taken from R.P. Feynman, The Feynman Lectures, Chapter 6.

2.5.2 Numerical methods

For more complex problem - with no obvious symmetries for example - different solving softwaresare available. They generally require the user to enter the geometry of the boundary conditionsand the position of charges. They compute the electric and/or magnetic fields. Some applicationsdirectly solve Maxwell’s equations, but these are practical for 2D systems, or systems with highsymmetries.

For problems where the exact solution would be too costly in computer time, softwares usingfinite element analysis (FEA) can be used. These sets of numerical methods are used to findapproximate solutions to partial differential equations - in electrostatics or any other field -.

2.6 Additional reading

The following article 6, describes the mechanism by which geckos adhere to the surface theywalk on. The ultimate mechanism at play implies van der Waals forces.

6. Autumn et al., Integr. Comp. Biol., vol. 42, p.1081, 2002.

1081

INTEGR.COMP.BIOL.,42:1081–1090(2002)

MechanismsofAdhesionin

Geckos1

KELLARAUTUMN2ANDANNEM.PEATTIE

3

DepartmentofBiology,Lewis&

Clark

College,Portland,Oregon97219

SYNOPSIS.

Theextraordinaryadhesivecapabilitiesofgeckoshavechallengedexplanationformillennia,

sinceAristotlefirstrecordedhisobservations.Wehavediscoveredmanyofthesecretsofgeckoadhesion,

yetthemillionsofdry,adhesivesetaeonthetoesofgeckoscontinueto

generatepuzzlingnewquestionsand

valuable

answers.Eachepidermally-derived,keratinousseta

endsin

hundredsof200nm

spatulartips,

permittingintimate

contactwithroughandsm

ooth

surfacesalike.Priorstudiessuggestedthatadhesive

forcein

geckosetaewasdirectlyproportionalto

thewaterdropletcontactangle( u),anindicatorofthe

freesurfaceenergyofasubstrate.In

contrast,newtheorysuggeststhatadhesionenergybetweenagecko

seta

andasurface(W

GS)isin

factproportionalto

,andonlyfor

u.60

8.Areanalysisofprior

Ï(1

1cosu)

data,in

combinationwithourrecentstudy,supportthevanderWaalshypothesisofgeckoadhesion,and

contradictsurfacehydrophobicityasapredictorofadhesionforce.Previously,weandourcollaborators

measuredtheforceproductionofasingleseta.Initialeffortsto

attachaseta

failedbecause

ofimproper3D

orientation.However,bysimulatingthedynamicsofgeckolimbsduringclimbing(basedonforceplatedata)

wediscoveredthat,in

singlesetae,asm

allnormalpreload,combinedwitha5

mm

displacementyieldeda

verylargeadhesiveforceof200microNewton(mN),10timesthatpredictedbywhole-animalmeasurements.

6.5

millionsetaeofasingle

tokaygeckoattachedmaximallycould

generate

130kgforce.Thisraisesthe

questionofhowgeckosmanageto

detachtheirfeetin

just15ms.Wediscoveredthatsimplyincreasingthe

anglethatthesetalshaftmakeswiththesubstrate

to30

8causesdetachment.Understandinghowsimulta-

neousattachmentandrelease

ofmillionsofsetaearecontrolledwillrequireanapproachthatintegrates

levelsrangingfrom

moleculesto

lizards.

INTRODUCTION

Geckosseem

todefy

gravityastheyrunalong

smooth

verticalsurfacesatupto

20bodylengthsper

second(Autumnetal.,1999a),andevenupsidedown

ontheceiling.Overtwomillenniaago,Aristotlecom-

mentedontheabilityofthegeckoto

‘‘runupand

downatreein

anyway,evenwiththeheaddown-

wards’’(Aristotle/Thompson,1918,BookIX,Part9).

Howgeckosadhere

hasbeenagrippingtopicofsci-

entificresearchforwelloveracentury

(Cartier,1872;

Haase,1900;Gadow,1901;Weitlaner,1902;Schmidt,

1904;Hora,1923;Dellit,1934;Mahendra,1941;

Maderson,1964;RuibalandErnst,1965;Hiller,1968,

1969,1975;Gennaro,1969;Russell,1975,1986;Wil-

liamsandPeterson,1982;Stork,1983;Schleichand

Kastle,1986;Irschick

etal.,1996;Autumn

etal.,

2000;Liangetal.,2000;Autumnetal.,2002).Rapid

locomotiononaverticalsurfacerequirestheabilityto

generateparallel(frictional)forcesequaltoorgreater

thanbodyweight(Fig.1A).Detachmentis

perhaps

evenmore

importantthanattachment.Afterall,even

themostcommonhousehold

adhesivesaremorethan

sufficienttoholdstaticallytheweightofalargegecko

(50g),butrepeatedandrapiddetachmentwithoutsig-

nificantdetachmentforcesisbeyondthecapabilityof

anycurrentsyntheticadhesive.Thesecretofgeckos’

adhesivecapabilitiesliesin

thestructureandfunction

1FromtheSymposium

BiomechanicsofAdhesionpresentedatthe

AnnualMeetingoftheSocietyforIntegrativeandComparativeBi-

ology,2–6January

2002,atAnaheim,California.

2E-mail:autumn@lclark.edu

3Presentaddress

ofAnnePeattie

isDepartmentofIntegrative

Biology,UniversityofCalifornia,Berkeley,California94720.

oftheirfeet(Russell,2002)andin

theadhesivetoe

padsborneontheundersideofeachdigit.These

pads

consistofaseriesofmodifiedlamellae(scansors;Fig.

1B),eachonecoveredwithuniform

arraysofsimilar-

ly-orientedhair-likebristles(setae;Fig.1C)formed

from

b-keratin

(Wainwrightetal.,1982;Russell,

1986).Asinglesetaofthetokaygecko(G

ekkogecko;

Fig.1D)isapproximately

100micronsin

length

and

5micronsin

diameter(RuibalandErnst,1965;Rus-

sell,1975;WilliamsandPeterson,1982).Thesetaeof

thetokaygecko(aswellasmostothers)branchatthe

tipsinto100–1,000structuresknownasspatulae(Fig.

1E).A

single

spatula

consistsofastalk

withaflat-

tened,roughly

triangularend,where

theapexofthe

triangleconnectsthespatulato

itsstalk.Spatulaeare

approximately200nmattheirwidestedge(Ruibaland

Ernst,1965;WilliamsandPeterson,1982).Whilethe

tokayis

currentlythebest

studiedofanyadhesive

geckospecies,there

existmanyhundredsofspecies

withadhesivetoepads,encompassinganimpressive

rangeofmorphologicalvariationattheseta,scansor,

andtoelevels,whichhasyettobefullycharacterized.

Theresultsdiscussedin

thispaperarebasedpredom-

inantlyonobservationsofisolatedtokaygeckosetae

andwhole-animaldynamicsofthehouse

geckoHem-

idactylusgarnoti,whichsharesthetokay’ssetalmor-

phology(butnotitstoemorphology).

Whilethestructuresofmanygeckosetaeare

well

documented,afullunderstandingoftheirfunctionhas

beenmore

elusive.Haase

(1900)notedthatadhesion

isload-dependentandonly

occurs

inonedirection:

proximallyalongtheaxisofthetoe.Haase

wasalso

thefirsttosuggestthatgeckosstickbyintermolecular

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K.AUTUMN

ANDA.M.PEATTIE

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1083

GECKOADHESIONM

ECHANISMS

← FIG.1.

Geckoadhesivestructuresandmethodsusedtomeasuresetaladhesivefunction.Images(A)and(B)byMarkMoffett.Figuremodified

fromAutumnetal.(2000),Nature.

A)Ventralviewofatokaygecko(G

ekkogecko)climbingaverticalglass

surface.Arrowsrepresenttheforcesactingonthegecko’sfeet

asitclimbs.

B)Ventralviewofthefootofatokaygecko,showingseta-bearingscansors.

C)Setaeare

arrangedin

anearlygrid-likepattern

ontheventralsurfaceofeachscansor.In

thisscanningelectronmicrograph,each

diamond-shapedstructureisthebranchedendofagroupoffoursetaeclusteredtogetherin

atetrad.Boxshowssetaenlargedin

(D).

D)Singleisolatedgeckosetausedin

measurementsin

Autumnetal.(2000).Boxshowsspatulaeenlargedin

(E).

E)Spatulartipsofasinglegeckoseta.

F)Isolatedsetaadheringto

amicro-electromechanicalsystem

(MEMS)cantilevercapableofmeasuringforcesparallelandperpendicular

tothesurface.Arrowshowsdirectionofmanipulationduringtheexperiment,simulatingparallelforcesgeneratedduringverticallocomotion

ofthegecko.

G)Singlesetaadheringto

a25

mm

aluminum

bondingwirecapableofmeasuringdetachmentforceperpendicularto

thesurface.Arrow

showsdirectionofmanipulationduringtheexperiment,simulatingperpendicularmovementduringdetachmentofthefoot.( a

istheangle

betweenthesetalstalkandthewire.

forces(Adhasion),notingthatunderthis

hypothesis

theattractiveforceshould

increase

asthespacebe-

tweenthefeetandthesubstrate

decreases.Setaeare

recurvedsuchthattheirtipspointproximally,leading

Dellit(1934)to

hypothesizethatsetaeactlikehooks,

catchingonsurfaceirregularities(microinterlocking).

RuibalandErnst(1965)laterpostulatedthat,whilethe

seta

isengaged,thespatulaelieflatagainstthesub-

strate.Itwasclearto

them

thatthese

flattenedtips

increasedtherealizedcontactarea,increasingfriction-

alforce.Thiswasanimportantstepin

understanding

setalmechanicsbutitdidnotsignificantlychangepre-

dictionsassociatedwithamicrointerlockinghypothe-

sis.Thisimpliedthatadhesionshould

bestrongeron

roughersurfaces,andthatinvertedlocomotionshould

bedifficult,ifnotimpossible,sincefriction(inthe

classicalsense)only

operatesparallelto

theplaneof

locomotion,leavingnoverticalforcecomponenttoop-

pose

gravity.

Theturningpointin

thestudyofgeckoadhesion

camewithaseriesofexperimentsbyHiller(1968),

whosuggestedthatthematerialpropertiesofthesub-

strate,ratherthanitstexture,determinedthestrength

ofgeckoadhesion.Indemonstratingthatadhesionwas

amolecularphenomenonratherthanamechanicalone,

hisdiscovery

effectivelyrefutedthemicrointerlocking

andfrictionhypothesesandpavedthewayforthere-

searchwedescribehere.

MECHANICSOFGECKOADHESION

Inaphylogeneticcomparisonoftheforcesproduced

bypad-bearinglizards,Irschicketal.(1996)showed

thattwofrontfeetofatokaygecko(G

ekkogecko)

produced20.1

Nofforceparallelto

thesurfacewith

227mm

2ofpadarea.Thefootofatokaybearsap-

proximately3,600tetradsofsetaepermm

2,or14,400

setaepermm

2(SchleichandKastle,1986;personal

observation).Consequently,asinglesetashould

pro-

duceanaverageforceof6.2

mN,andanaverageshear

stress

of0.090Nmm

22(0.9atm).(NotethatAutumn

etal.(2000)usedavalueof5,000setaepermm

2taken

from

RuibalandErnst(1965).In

thispaper,weuse

themore

accurate

valueof14,400setaepermm

2

(SchleichandKastle

[1986];wehaveconfirmedthis

figure

withourownobservations.)Thedifficultyof

firstisolatingandmanipulatingasingleseta,andthen

findinganinstrumentcapableofmeasuringmicronew-

tonforcesintwodimensions,complicatedourattempt

todemonstratehowthefunctionofasinglesetamight

contributetoadhesionattheorganismallevel.Wewill

nowdescribehowweconfrontedthese

difficultiesin

ourexperimentalprocedure.

Mechanism

ofsetalattachment

Using

new

microelectromechanical

systems

(MEMS)forcemeasurementtechniques(Chuietal.,

1998),Autumnetal.(2000)measuredtheforcepro-

ductionofasinglegeckoseta(Fig.1F).Initialefforts

toattachasinglesetafailedtogenerateadhesiveforc-

esabovethatpredictedbyfrictionbecause

wecould

notachievetheproperorientationofthesetain6de-

greesoffreedom

(i.e.,translationin,androtation

about,allthreeaxes).Theangleofthesetalshaftwas

particularlyimportantin

achievinganadhesivebond

(Fig.1G).Whenwesimulatedthedynamicsofgecko

limbsduringclimbing(basedonforceplatedata;Au-

tumnetal.,1999a,b),wediscoveredthatasm

allnor-

malpreloadforce(Fig.2A)yieldedashearforceof

;40

mN,sixtimestheforcepredictedbywhole-ani-

malmeasurements(Irschicketal.,1996).Properori-

entation,preload,anddragyielded10to20timesthe

frictionalforcemeasuredwiththeseta

orientedwith

spatulaefacingawayfrom

thesurface(Fig.2B).The

smallnormalpreloadforce(Fig.2A),combinedwith

a5

mm

displacementyieldedavery

largeshearforce

of200

mN,32timestheforcepredictedbywhole-

animalmeasurements(Irschicketal.,1996;Fig.2C).

Thediscovery

thatmaximaladhesioninisolatedsetae

requiresasm

allpush

perpendiculartothesurface,fol-

lowedbyasm

allparalleldrag,explainedtheloadde-

pendenceanddirectionality

ofadhesionobservedat

thewhole-animalscalebyDellit(1934),andwascon-

sistentwiththehypothesisthatthestructure

ofindi-

vidualsetaeandspatulaeissuchthatasm

allpreload

andrearward

displacementisnecessary

toengagead-

hesion(RuibalandErnst,1965;Hiller,1968).In

their

resting

state,setalstalksare

recurvedproximally.

Whenthetoesofthegeckoare

planted,webelieve

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K.AUTUMN

ANDA.M.PEATTIE

FIG.2.

Singlesetameasurements.FiguremodifiedfromAutumnetal.(2000),Nature.

A)Submaximalforceofsinglesetaparalleltothesurfacewithaknownperpendicularpreload,asafunctionoftime.Perpendicularpreload

isdesignatedbythedashedline.t srepresentsthetimewhenthesetabegantoslideoffthesensor.Theinitialperpendicularforceneednotbe

maintainedduringthesubsequentpull.Diagramsshow

thestagesofsetalmovementcorrespondingto

theforcerecord

from

theMEMS

cantilever(Fig.1F).Arrowsindicatethedirectionofappliedforcetotheseta.Verticalarrowindicatesaparallelforce,andahorizontalarrow

indicatesaperpendicularforce.

B)Setalforceparalleltothesurfaceduringattachmentasafunctionofperpendicularpreloadforce.Setalforcewastakentobetheadhesive

forceatthetimejustpriorto

sliding(ts;Fig.2A).Thesolidlinerepresentsasetawithspatulaeprojectingtoward

thesurface.Thedashed

linerepresentsthesetalforcewithspatulaeprojectingawayfrom

thesurface(parallelforce

50.25perpendicularpreload

20.09;r2

50.64;

F513;df

51,9;P

50.007).Theforceproducedbytheinactive,non-spatularregionincreasedwithnorm

alorperpendicularforce,typical

ofmaterialswithacoefficientoffrictionequalto

0.25.Theperpendicularpreloadingforcethatcould

beappliedattainedamaximum

(near

15

mN),because

greaterforcesresultedin

thesetalbuckling.

C)Maximalforceafteramaximum

preload(ø15

mN)ofasinglesetaparallelto

thesurfaceasafunctionoftime.Diagramsshowthe

stagesofsetalmovementcorrespondingtotheforcerecord

fromtheMEMScantilever(Fig.1F).Arrowsindicatethedirectionofforceapplied

totheseta.Verticalarrowindicatesaparallelforce;horizontalarrowindicatesaperpendicularforce.Themaximumforce( ;200

mN)following

thesm

allrearward

displacement( ø5

mm)was32timesthatpredictedfrommaximalwholeanimalestimates(seetext).Thelargeincrease

inforceduringtherearward

displacementmaybecausedbyanincrease

thenumberofspatulaecontactingthesurface.

D)Changein

theorientationofthesetaemayfacilitate

detachment.Setalangle

(a)withthesurfaceatdetachmentasafunctionof

perpendicularforce.Filledsymbolsrepresentsetapulledawayfrom

thesurfaceuntilrelease.Opensymbolsrepresentsetaheldataconstant

forceasangleisincreased.Eachsymbolshaperepresentsadifferentseta.Datacollectedwithwiregauge(Fig.1G).Setalangleatdetachment

changedbyonly15%

overtheentirerangeofperpendicularforces.Thisobservationisconsistentwithanadhesivemodelwhereslidingstops

whenpullingatgreaterthanthecriticalsetalangleandhencestresscanincrease

ataboundary,causingfractureofthecontact.

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GECKOADHESIONM

ECHANISMS

thatthesetaearebentoutofthisrestingstate,flatten-

ingthestalksbetweenthetoeandthesubstratesuch

thattheirtipspointdistally.Thissm

allpreloadanda

micron-scaledisplacementofthetoeorscansorprox-

imallymayserveto

bringthespatulae(previouslyin

avarietyoforientations)uniformlyflush

withthesub-

strate,maximizingtheirsurfaceareaofcontact.Ad-

hesionresultsandthesetaearereadytobeartheload

oftheanimal’sbodyweight.

All6.5million(SchleichandKastle,1986;Irschick

etal.,1996)setaeofa50gtokaygeckoattachedmax-

imallycould

theoreticallygenerate

1,300N

(133kg

force)ofadhesive

force—enough

tosupport

the

weightoftwohumans.Thissuggeststhatageckoneed

only

attach3%

ofitssetaeto

generate

thegreatest

forcesmeasuredinthewholeanimal(20N;Irschicket

al.,1996).Lessthan0.04%ofagecko’ssetaeattached

maximallyareneededtosupportitsweightof50gon

awall.Atfirstglance,geckofeetseem

tobeenor-

mouslyoverbuilt.Onfurtherconsideration,however,

therearesomeclearadvantagestopossessingasmany

setaeaspossible

(atwhichpointwemightask

what

factors

actuallylimitagecko’sadhesivecapacity).It

isunlikely

thatallsetaeare

ableto

achievethesame

orientationsimultaneously.Theproportionofspatulae

attachedmaybegreatlyreducedonroughsurfaces

(particularlythose

withroughness

onthesamescale

asspatulaeorsetae;AutumnandGorbinpreparation).

Ondustyorexfoliatingsurfaces,attachmenttoawell-

anchoredsubstratewillnotbepossibleforevery

seta.

Largeforcesgeneratedbyperturbationsduringloco-

motion(e.g.,recoveringfrom

afall,predatoravoid-

ance,orstation-keepingin

highwinds)mayalsouti-

lizeagreaterproportionofgeckos’adhesivecapacity.

Mechanism

ofsetaldetachment

Thesurprisinglylargeforcesgeneratedbysinglese-

taeraisedthequestionofhowgeckosmanageto

de-

tachtheirfeetso

rapidly(15ms;Autumnetal.,1999a)

withnomeasurabledetachmentforces(Autumnetal.,

1999b).Wediscoveredthatincreasingtheangle

be-

tweenthesetalshaft

andthesubstrate

beyond308

causeddetachment(Autumnetal.,2000;Fig.1G;Fig.

2D).Itis

likely

thatastheangle

ofthesetalshaft

increases,stress

increasesatthetrailingedgeofthe

seta,causingfracture

ofthespatula-substrate

bonds.

Similarly,atthescansorlevel,geckos’

unusualtoe

peelingbehavior(digitalhyperextension;Russell,

1975,1981)mayaid

inreducingoreliminatingde-

tachmentforcesbydetachingonly

asm

allnumberof

setaeatanymoment.Howthispeelingbehaviorresults

inreachingthecriticalangleofdetachmentisstillun-

clear,butthetwoare

almostcertainlylinked.

Integrationofsetalmechanics,functional

morphologyofthefoot,anddynamicsoflocomotion

Itisimportanttoemphasizethatwithoutintegrating

dynamicsatalargerscale(bodyandlegs),thefunction

oftheseta

would

likely

stillremain

unknown.This

underscorestheimportanceofanintegrativeapproach

toansweringbiologicalquestions(Lauder,1991;Sav-

ageau,1991;Ryanetal.,1998;Dickinsonetal.,2000;

Autumnetal.,2002).Howattachmentanddetachment

ofmillionsofsetaeduringlocomotionis

integrated

withthefunctionofthescansor,toe,foot,leg,and

bodyremainsatopic

ofgreatinterest

andongoing

research(Sponberg

etal.,2001;Russell,2002).Rus-

sellhassuggestedthatinthetokay(G

ekkogecko),the

perpendicularpreloadand5

mm

dragrequirements

(Autumnetal.,2000)are

controlledbyhydrostatic

pressure

inthehighly

derivedbloodsinuses,andlat-

eraldigitaltendonsystem,respectively

(Russell,

2002).

Sincegeckosetaerequireapreloadin

thenormal

axisforadhesion,largeforcescouldpotentiallybeas-

sociatedwithattachmentofthefoot.Thetremendous

adhesivecapacityofgeckosetaesuggeststhatlarge

forcescouldalsooccurduringdetachment.In

fact,no

measurable

groundreactionforceswere

associated

witheitherattachmentordetachmentduringvertical

climbingonaforceplate

ofthehouse

geckoHemi-

dactylusgarnoti(Autumnetal.,1999a,b),indicating

thatthese

actionsare

eithermechanicallydecoupled

fromthecenterofmassinthisspecies,orso

smallas

tobeundetectable.

Theabsenceofdetachmentforcesisconsistentwith

(1)themechanismofdetachmentand(2)theanatomy

ofthefoot.Geckospeeltheirtoesupandawayfrom

thesubstrate

(digitalhyperextension)ratherthanat-

temptingto

detachanentire

footatonce,muchlike

removingapieceoftape.Peelingminimizespeakforc-

esbyspreadingdetachmentoutovertime.Sincethe

musclesresponsible

fordigitalhyperextension(inter-

osseidorsales;Russell,1975)are

locatedin

thefoot,

detachmentdoesnothavetobecoupledmechanically

tothecenterofmass,aswouldbethecaseifthegecko

useditslegmusculature

tobreaktheadhesivebonds

inthefoot.

Theabsenceofattachmentforcesisamore

com-

plicatedissue,withatleastthreepossibleexplanations.

AsRussell(2002)hassuggested,inflationofthedig-

italbloodsinuses(suchasthose

presentinindividuals

ofHemidactylusandGekko)maysatisfythepreload

requirementofthesetaeduringattachmentwithout

generatingmeasurable

forcesactingonthecenterof

mass.However,controlofinflationanddeflationof

thesinusesremainsto

bedemonstrated.Thismecha-

nismwouldnotbeavailabletothose

speciesthatlack

bloodsinuses.

Asecondpotentialexplanationisthatsetalpreload

anddragareaconsequenceofforcedevelopmentdur-

ingthestride.Climbinggeckosuse

allfourfeetsim-

ilarlytoproducepositivefore-aftforcesparalleltothe

surfacethatpropelthegeckoupwards(Autumnetal.,

1999b).Leftlegsapplyaforcetotherightwhileright

legsapply

aforceto

theleft.Therefore,allfourfeet

pullmedially,probably

draggingthesetaeto

engage

themfully,increasingtheforceofattachment(Autumn

etal.,1999b).However,geckos’

frontlegspullthe

centerofmassinto,whilehindlegspush

thecenterof

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1086

K.AUTUMN

ANDA.M.PEATTIE

massawayfrom,theverticalsubstrate(Autumnetal.,

1999b),generatinganetmomentpitchingtheanterior

toward

thesurfaceandcounteractingthetendencyof

theheadto

fallawayfrom

thesurface(Alexander,

1992).Frontlegsdonotpush

intotheverticalsubstrate

duringorafterfootcontact.Thus,these

resultsdonot

supportthehypothesisthatthesetaebecomepreloaded

asaconsequenceofforcedevelopmentduringthe

stride.Whilethis

ispossible

forthehindfeet,itis

difficultto

reconcilewiththenegativenormalforces

producedbythefrontfeet(Autumnetal.,1999b),un-

less

theattachmentforceisso

smallasto

beunde-

tectable.Theforcenecessary

tobendeventhousands

ofsetaeinto

anadhesiveorientationisprobablyquite

small(byourestimate,atmost10mN).In

thiscase,

wemayhaveobservednomeasurableattachmentforc-

essimply

because

wecannotmeasurethem.

Athirdpossibilityisthatattachmentisareversalof

thepeelingprocess

oftoedetachment,whichwebe-

lieveto

bedecoupledfrom

thecenterofmass.The

gecko’sfootmayapproachthesubstratewithoutpress-

inginto

it,re-applyingitsadhesivebygraduallyex-

tending(unrolling)itstoesagainst

thesurface,at

whichpointtheyare

readyto

beartheloadofthe

animal’sweight.

Inthis

case,setalpreloadforces

would

bespreadoutovertime,andwould

likely

be

farbelowtheresolutionofourforceplate( 61mN).

MOLECULARM

ECHANISM

OFGECKOADHESION

Whilethemechanism

ofsetalattachmentandde-

tachmentin

geckosisnowunderstoodin

mechanical

terms(Autumnetal.,2000),themolecularmechanism

underlyingadhesionin

setaehasremainedunclear.

Adhesioncanbecausedbyatleast11differenttypes

ofintermolecularsurfaceforcesattheinterfacebe-

tweensolids(Israelachvili,1992;Gay,2002),andcan-

notalwaysbedistinguishedfrom

friction,acompli-

catedphenomenonin

andofitself(Persson,1999).

Thestudyofsurfaceforcesiscurrentlyanactivearea

ofresearch(Kinloch,1987;Israelachvili,1992,2001;

Christenson,1993;Noyetal.,1995;Thomasetal.,

1995;GayandLeibler,1999;Kunzig,1999).

The

complexityoftheproblemstemsfromthedifficultyof

knowingwhatmaterialsareactuallyinteractingatthe

molecularscale

(Israelachvili,1992;Persson,1999).

Thusitisnotsurprisingthatadhesionin

geckoshas

remainedachallengingproblem.

Unsupportedmechanisms:

glue,suction,

electrostatics,andmicrointerlocking

Sincegeckoslackglandulartissueontheir

toes,

stickysecretionswere

ruledoutearlyin

thestudyof

geckoadhesion(W

agler,1830;Cartier,1874;Simmer-

macher,1884).Theideathattheindividualsetaeacted

asminiature

suctioncupswasfirstdebatedin

thein-

sectadhesionliterature

(Blackwall,1845;Hepworth,

1854),butwaslaterproposedforgekkonidlizardsby

Simmermacher(1884).However,there

are

nodatato

supportsuctionasanadhesivemechanism,andthe

adhesionexperimentscarriedoutinavacuumbyDel-

lit(1934)suggest

thatsuctionis

notinvolved.Fur-

thermore,ourmeasurements

of10atm

ofadhesion

pressure(Autumnetal.,2000)stronglycontradictthe

suctionhypothesis.

Electrostaticattraction(Schmidt,1904)is

another

possiblemechanism

foradhesionin

geckosetae.Ex-

periments

usingX-raybombardment(Dellit,1934)

eliminatedelectrostaticattractionasanecessarymech-

anism

forsetaladhesionsincethegeckoswere

still

able

toadhere

inionizedair.However,electrostatic

effectscouldpossiblyenhanceadhesionevenifanoth-

ermechanismisoperating(M

aderson,1964).

Friction(Hora,1923)andmicrointerlocking(Dellit,

1934;the‘‘climber’sboot’’modelofMahendra,1941)

mayalsoplayasecondary

role,buttheabilityofgeck-

ostoadherewhileinvertedonpolishedglass,andthe

presenceoflargeadhesiveforcesonamolecularly

smooth

SiO

2MEMSsemiconductor(Autumnetal.,

2000)showthatsurfaceirregularitiesarenotnecessary

foradhesion,andmayin

factbeanimpediment(Au-

tumnandGorb,in

preparation)

Potentialinterm

olecularmechanisms:

vanderWaals

andcapillary

forces

Hiller(1968,1969,1975)showedthatadhesive

forcewascorrelatedwiththewaterdropletcontactan-

gleofthesurface,andthuswiththesurfaceenergyof

thesubstrate,providingthefirstdirectevidencethat

intermolecularforcesare

responsible

foradhesionin

geckos.Ongoingresearchisnowdirectedtoward

un-

derstandingtheprecisenature

ofthese

intermolecular

forces.

Intermolecularcapillary

forcesare

theprincipal

mechanism

ofadhesionin

manyinsects(Gillettand

Wigglesworth,1932;EdwardsandTarkanian,1970;

Leeetal.,1986;LeesandHardie,1988;Brainerd,

1994;Dixonetal.,1990),frogs(EmersonandDiehl,

1980;Green,1981;HannaandBarnes,1991)andeven

mammals(Rosenberg

andRose,1999).Unlikethese

animals,geckoslackglandsonthesurfaceoftheirfeet

(Wagler,1830;Cartier,1872;Dellit,1934;Mahendra,

1941).Thisinitselfdoesnotprecludetheroleofthin

film

capillary

adhesion(vonWittich,1854,quoteddi-

rectlyin

Simmermacher,1884;Stork,1980;Scherge

andSchaefer,1998)sinceamonolayerofwatermol-

ecules(presumably

presentin

theenvironment)can

cause

strongattractionbetweensurfaces(Baieretal.,

1968;Israelachvili,1992).Theapparentinversecor-

relationbetweenadhesiveforceandhydrophobicity,as

inferredfrom

thewaterdropletcontactangle(Hiller,

1968)suggeststhatthepolarity

ofthesurfacemight

beanimportantfactorinthestrengthofadhesion(Fig.

3).Twopoints

are

worthnotingwithregard

tothis

hypothesis:(1)Suchamonolayerwould

haveto

be

ubiquitousandrelativelypureincontinuouspatchesin

orderforthegeckoto

takeadvantageofit,and(2)

Hiller(1968)did

notfavoracapillary

mechanismfor

geckoadhesionsinceadhesiveforcedid

notdecrease

completelyto

zero

onallhydrophobicsurfaces.

Analternativemechanism

isthatgeckosadhereby

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GECKOADHESIONM

ECHANISMS

FIG.3.

Norm

alizedforceversussurfacepolarity(measuredbywa-

terdropletcontactangle,u)fromHiller(1968,1969).Hiller’s(1968)

measurementsofwholegeckosonsurfacesofvaryingpolarity(cir-

cles)showedacorrelationbetweenpolarityandforce(y

520.012x

11.26;R2

50.85).Inaseparatestudy,Hiller(1969)foundaweakly

significantcorrelationbetweenforceandsurfacepolarity

onpoly-

ethylene

filmsmodified

with

corona

discharge

(squares;

y5

20.012x

11.79;R2

50.69).

vanderWaalsinteractionsalone(Stork,1980;Autumn

etal.,2000).Thisisavery

intriguinghypothesissince

vanderWaalsinteractionsare

theweakestofallin-

termolecularforces,butalsothemostuniversal.An

adhesiveutilizingthese

very

weakinteractionswould

becapable

ofstickingto

nearlyanynaturalsurface.

However,avastnumberofthese

interactionswould

haveto

operatesimultaneouslyin

orderto

generatea

significantamountofforce.Thatis,there

mustbea

large,realcontactareabetweentheorganism

andthe

substrate.Thehighly

branchedsetaeongeckotoes

maybesuitedformaximizingcontactarea.Ifthisis

thecase,setalmorphologywouldhaveagreatereffect

onforceofadhesionthansurfacechemistryoftheseta

orsubstrate.In

otherwords,thegeometryofthead-

hesivewould

bemore

importantthanthechemistry.

Thestrength

ofvanderWaalsattractionsishighly

dependentonthedistancebetweensurfaces,increases

withthepolarizabilityofthetwosurfaces,andisnot

directlyproportionalto

surfacepolarity

(Israelachvili,

1992).Theobservationthatgeckoscannotadhere

toPolytetrafluoroethylene(PTFE;Hiller,1968)isconsis-

tentwiththevanderWaalshypothesis,sincePTFEis

only

weakly

polarizable.

Thefollowingequationestimatestheforce(per

area)dueto

vanderWaalsinteractionsbetweentwo

planarsurfaces(Israelachvili,1992):

A22

Forceperarea(N

m)

5(1)

36pD

Inthisequation,ArepresentstheHamakerconstant,a

functionofthevolumeandpolarizabilityofthemol-

eculesinvolved.Formostsolidsandliquids,theHa-

makerconstantliesbetween4

310

220and4

310

219

J,so

itcannotaffectanestimateofforcebymorethan

apoweroften.Farmore

importantto

theestimateis

theseparationdistance(D)betweenthetwosurfaces

sinceforceofadhesionscalesinverselywiththethird

powerofthisdistance.Thisimpliesthatatsm

allsep-

arations,strongforcesofadhesioncanoccur,butfor

eachpoweroftenincrease

inthedistance,theforce

ofadhesiondecreases1,000fold.AssumingtheHa-

makerconstantto

beofatypicalvalue(1

310

219J),

andthespatularsurfaceareatobeapproximately2

310

214m

2(RuibalandErnst,1965;WilliamsandPe-

terson,1982;personalobservation)theforceofad-

hesionofasingle

seta

with100spatulaeincreases

from

11

mNto

11mNastheseparationdistancede-

creasesfrom

1nm

to0.1nm.Theactualforceofad-

hesionofasinglesetacanreachalmost200

mN(Au-

tumnetal.,2000).Thiswould

correspondto

asepa-

rationdistanceof0.38–0.81nm(assuming100–1,000

spatulae/seta).Theactualmagnitudeofthisgapdis-

tanceremainsunknown.

VanderWaalsandcapillary

adhesionare

notmu-

tuallyexclusivemechanisms.Althoughwateratthe

seta-substrate

interfacewould

increase

thegapdis-

tanceandtherefore

reducethestrength

ofvander

Waalsadhesion,thediameterofasinglewatermole-

cule(0.3nm)remainswellwithintherangeofvander

Waals

attraction(Israelachvili,1992).

Thus,

inthe

presenceofathin

film

ofwater,itisconceivablethat

thetwomechanismsare

workingin

tandem.Asmore

layers

ofwatermoleculesintervene,however,resis-

tancetoshearforcesacrossthefluidisprovidedsolely

byitsviscosity

(Baieretal.,1968).Waterdoesnot

haveahighviscosity

andtherefore

cannotwithstand

high

shearforces.

Therefore,capillary

forcesare

stronginthenormaldirectionandweakintheparallel

direction,whereastheopposite

istrueofsetaladhe-

sion.Empirically,setaehavedemonstratedstrongerre-

sistancetoshearforcesthantonormalforces(Autumn

etal.,2000),suggestingthatifcapillary

adhesionoc-

curs,thefilmsinvolvedwould

haveto

berelatively

thin.

Theextentto

whichathin

film

ofwaterwillform

overtheavailableinterfacialareaisdependentonthe

relativevaporpressure.Atextremely

low

humidity,

capillary

adhesionwillbeweakdueto

lackofad-

sorbedwateronsubstrates.Athighhumidity,water

beginstosaturatetheintersticesofroughsurfaces,act-

ingasalubricant(Israelachvili,1992).This

canbe

describedasthesandcastleeffect:onecannotbuilda

sandcastleoutofsandthatiseithervery

dry

orvery

wet.Dry

sanddoesnotadherebecause

thewettedin-

terfacialareaistoolow.Very

wetsanddoesnotadhere

because

theradiusofthewatermeniscusbetweenpar-

ticlesapproachesthesizeoftheparticleitself,andcap-

illary

forcesdeclinetoward

zero

(Israelachvili,1992).

Ifgeckosweretorelysolelyoncapillary

adhesionfor

attachment,setalfunctioncouldbeconstrainedbyrel-

ativehumidityofthehabitat.However,sincepad-bear-

inggeckospeciesare

foundin

habitatsrangingfrom

tropicalrainforeststodry,rockydesert,humiditydoes

notseem

tohaveastronginfluenceoneffectivead-

hesionin

nature.

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K.AUTUMN

ANDA.M.PEATTIE

FIG.4.

ReanalysisofHiller’spreviousmeasurements

ofgeckos

adheringto

surfacesofvaryingsurfacepolarity

( uwaterdroplet

contactangleofsurface;Hiller,1968,1969).Norm

alizedforcever-

susadhesionenergyapproximatedby

.Theapparent

Ï(1

1cosu)

correlationbetweenforceand

u(Fig.3)suggestedthatadhesionin

geckoswasafunctionof

u.Itisnow

known(Israelachvili,1992)

thatifoneoftwoadheringsurfacesishydrophilic( u

,608;open

circles),theiradhesionenergybears

nosimple

relationship

tothe

liquid

(water)contactangleofeithersurface.However,forhydro-

phobic

surfaces( u

.608)adheringbyvanderWaalsforces(see

Eqn.2)theadhesion

energy

isapproximately

proportionalto

.Thus,thecorrelationsbetweenforceandadhesionen-

Ï(1

1cosu)

ergyforhydrophobicsurfaces(closedcircles:y

51.41x

21.27;R2

50.83;closedsquares:y

51.60x

20.88;R2

50.73)supportthe

hypothesisthatgeckosadherebyvanderWaalsforces.

Areanalysisoftherelationship

betweenwater

contactangle

andgeckoadhesion

SinceHiller’s(1968,1969)data

correlate

the

strength

ofsetaladhesionwiththepolarityofthesur-

faceasindicatedbythewaterdropletcontactangle( u;

Fig.3),itisreasonableto

considercapillary

actiona

likely

mechanism

ofadhesion.However,recentad-

vancesin

theunderstandingofintermolecularsurface

forces(Israelachvili,1992)allowusto

revisitHiller’s

data(Hiller,1968,1969)andask

iftheysupportonly

acapillary

adhesivemechanism.Theadhesionenergy

(W)betweentwoidenticalsolidsurfacesisrelatedto

thecontactangle(u)ofaliquiddropletonthesurface

via

theYoung-Dupre

equation,

gL(1

1cosu)

5W,

where

gListhesurfacetension(orenergy)oftheliquid

(L)inunitsofmN/m

(ormJ/m

2).However,ifthetwo

adheringsurfacesaredifferentmaterials,asforgecko

setae(G)onasubstratesurface(S),theinterfacialad-

hesionenergy(W

GS)bearsnosimplelinearrelationto

theliquid(water)contactangle.Therelationshipscan

nowbeexpressedas,

gL(1

1cosu

LG)

5W

GGand

gL(1

1cosu

LS)

5W

SS,andbythethermodynamicrelation

(Israelachvili,1992):W

GS

51/2W

GG

11/2W

SS

2gGS,

where

gGSistheinterfacialtensionofthesetae-sub-

strateinterface,andW

GGandW

SSarethecohesionen-

ergiesofsetalandsubstratematerials,respectively.An

approximaterelation,validwhenthetwoadheringsur-

faces

are

nonpolarand

interactwith

each

other

only

via

vanderWaals

dispersionforces,is

(Israe-

lachvili,1992):W

GS

5.ReplacingW

GGand

ÏW

WGG

SS

WSS

with

the

above

equations

gives:

WGS

5

gL

,and

acorrelationbe-

Ï(1

1cosu

)(1

1cosu

)LG

LS

tweenadhesionenergy(W

GS)andwaterdropletcontact

angleonthesubstrate(u

LS)maybeexpressedas

W5

ÏgW

(11cosu

)GS

LGG

LS

(2)

Thus,for

gL

572mJ/m

2(forwater)andW

GG

560

mJ/m

2(a

typicalvalueforanonpolarvanderWaals

solid)wetheoreticallyexpectW

GSto

vary

monotoni-

callyfrom

WGS

ø93mJ/m

2for

uLS

508to

WGS

ø66

mJ/m

2for

uLS

5908to

WGS

50for

uLS

51808.This

correlation,however,is

expectedto

hold

only

when

thesubstratesurfaceishydrophobic(u

.608).Are-

analysisofHiller’sdata

(Hiller,1968,1969;Fig.4)

usingEquation1in

factsupportsthevanderWaals

hypothesiswithastrongcorrelationbetweenforceand

adhesionenergiesfor

u.608.

TestingthevanderWaals

andcapillary

adhesion

hypotheses

Totestdirectlywhethercapillary

adhesionorvan

derWaalsforceistheprimary

mechanismofadhesion

ingeckos,wemeasuredthehydrophobicityofthesetal

surface,andmeasuredadhesionontwopolarizable

semiconductorsurfacesthatvariedgreatlyin

hydro-

phobicity(Autumnetal.,2002).Wemeasuredthepar-

allelforceofsinglegeckotoesonagallium

arsenide

(GaAs)

semiconductorsurfacethatis

highly

hydro-

phobic(u

51108).Asacontrol,wemeasuredparallel

forceonthestrongly

hydrophilic(u

508)silicondi-

oxide(SiO

2)semiconductorsurface.Wealsocom-

paredtheperpendicularforceofsingleisolatedgecko

setaeonhydrophilic

(SiO

2,

u5

08)andhydrophobic

(Si,

u5

81.9

8)micro-electro-m

echanical-systems

(MEMS)forcesensors.Ifwet,capillary

adhesiveforc-

esdominate,weexpectedalackofadhesiononthe

stronglyhydrophobicGaAsandSiMEMSsurfaces.In

contrast,ifvanderWaals

forcesdominate,wepre-

dictedlargeadhesiveforcesonthehydrophobic,but

polarizable

GaAsandSiMEMSsurfaces.

Ineither

case

weexpectedstrongadhesionto

thehydrophilic

SiO

2semiconductorandMEMScontrolsurfaces.

Weshowedthattokaygeckosetaeare

stronglyhy-

drophobic

(160.9

8;Autumnetal.,2002),probably

aconsequenceofthehydrophobicsidegroupsofß-ker-

atin(GreggandRogers,1984).Parallelstress

oflive

geckotoesonGaAsandSiO

2semiconductorswasnot

significantlydifferent,andadhesionofasinglegecko

setaonthehydrophobicandhydrophobicMEMScan-

tilevers

differedbyonly

2%.These

resultsrejectthe

hypothesisthatwatercontactangle

(u)ofasurface

predictsattachmentforcesingeckosetae,assuggested

byHiller(1968,1969),andare

consistentwithour

reanalysis(above).SincevanderWaalsforceisthe

only

mechanism

thatcancause

hydrophobicsurfaces

toadhere

inair(Israelachvili,1992),theGaAsand

hydrophobic

MEMSsemiconductorexperimentspro-

videdirectevidencethatvanderWaalsforceisthe

mechanismofadhesioningeckosetae,andthatwater-

basedcapillary

forcesare

notsignificant.

CONCLUSION

ANDFUTUREDIRECTIONS

Geckosetaeareanoveltypeofadhesiveinthatthe

strength

ofadhesiondependslargely

ongeometry

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GECKOADHESIONM

ECHANISMS

ratherthanonsurfacechemistry.Manysecretsofsetal

form

andfunctionremain:Thefeetofgeckoscaptured

innature

are

usuallyclean(e.g.,Stenodactyluskho-

barensis;Russell,1979),andthepossibilityexiststhat

geckosetaeare

actuallyself-cleaning.Wehaveyetto

uncoverthedesignprinciplesunderlyingthevariation

in,andmultipleconvergentevolutionsofsetal,scan-

sor,andfootstructuresingeckos(Russell,1972,1976,

1979;BauerandRussell,1990),andinconvergentse-

taladhesivesystemsinanoles(PetersonandWilliams,

1981),skinks(W

illiamsandPeterson,1982),andin-

sects

(GillettandWigglesworth,1932;Edwardsand

Tarkanian,1970;Nachtigall,1974;Bauchhenss

and

Renner,1977;Hill,1977;Rovner,1978;Stork,1980,

1983;Walkeretal.,1985;Wigglesworth,1987;Ros-

coeandWalker,1991;Betz,1996;Gorb,1998,2001;

Attygalleetal.,2000;EisnerandAneshansley,2000;

Koelsch,2000;Gorb

etal.,2001;Gorb

andBeutel,

2001;SchergeandGorb,2001).Itisclearthatgecko

setaearevastlyoverbuiltforadhesiontosm

oothideal

surfacesunderstaticconditions.Thismaynotbethe

case

fornaturalsurfaces,orfordynamic

conditions

suchasforrunning,jumping,andfalling.Character-

izationofthesurfacesandloadingregimesseta-bear-

inganimalsexperienceinnaturewillbenecessary

be-

fore

wecanaddress

thequestionofhow

muchofa

safety

factorexists

inthegeckoadhesivesystem.

Studyofthesetaladhesivesystemrequiresanextraor-

dinary

degreeofintegrationacross

scalesanddisci-

plines—

from

quantum

mechanicsto

biomechanics—

andhighlightsthebenefitsofanintegrativeapproach

tobiology.

ACKNOWLEDGMENTS

Theresearchdescribedin

thispaperwastheresult

ofcollaborationbetweentheAutumn,Full,Fearing,

andKennylabs,andwassupportedundertheauspices

ofAlanRudolph,directorofDARPA’sControlledBi-

ologicalandBiomimeticSystemsprogram,N66001-

00-C-8047andN66001-01-C-8072.Thanksto

Mi-

chaelBroide,WalterFederle,Valeurie

Friedman,

AmandaGassett,WendyHansen,TonyRussell,Simon

Sponberg,andtwoanonymousreviewers.Weare

es-

peciallygratefultoJacobIsraelachviliforEquation2.

REFERENCES

Alexander,R.M.N.1992.Exploringbiomechanics:Animalsin

mo-

tion.ScientificAmerican,NewYork.

Aristotle,HistoriaAnimalium,trans.D.A.W.Thompson.1918.Clar-

endon,Oxford,http://classics.mit.edu/Aristotle/history

panim.html.

Attygalle,A.B.,D.J.

Aneshansley,J.Meinwald,andT.Eisner.

2000.Defense

byfootadhesioninachrysomelidbeetle(H

em-

isphaerota

cyanea):Characterizationoftheadhesiveoil.Zool-

ogy(Jena).103:1–6.

Autumn,K.,S.T.Hsieh,D.M.Dudek,J.Chen,C.Chitaphan,and

R.J.Full.1999a.Dynamicsofgeckosrunningvertically.Amer.

Zool.38:84A.

Autumn,K.,S.T.Hsieh,D.M.Dudek,J.Chen,C.Chitaphan,and

R.J.Full.1999b.Functionoffeetinascendinganddescending

geckos.Amer.Zool.38:84A.

Autumn,K.,Y.A.Liang,S.T.Hsieh,W.Zesch,W.-P.Chan,W.T.

Kenny,R.Fearing,andR.J.Full.2000b.Adhesiveforceofa

singlegeckofoot-hair.Nature405:681–685.

Autumn,K.,M.J.Ryan,andD.B.Wake.2002.Integratinghistor-

icalandmechanisticbiologyenhancesthestudyofadaptation.

Q.Rev.Biol.(Inpress).

Baier,R.E.,E.G.Shafrin,andW.A.Zisman.1968.Adhesion:

Mechanismsthatassistorimpedeit.Science162:1360–1368.

Bauchhenss,E.andM.Renner.1977.PulvillusofCalliphora

ery-

throcephala

Meig

(Diptera:Calliphoridae).Int.J.InsectMor-

phol.6:225–227.

Bauer,A.M.andA.P.Russell.1990.Alternativedigitalscansor

designin

theNew

CaledonianGekkonid

genera

Bavayia

and

Eurydactylodes.Mem.QueenslandMus.29:299–310.

Betz,O.1996.Functionandevolutionoftheadhesion-capture

ap-

paratusofStenusspecies(Coleoptera,Staphylinidae).

Zoo-

morphology116:15–34.

Blackwall,J.1845.Onthemeansbywhichwalk

variousanimals

ontheverticalsurfaceofpolishedbodies.Ann.Nat.Hist.XV:

115.

Brainerd,E.L.1994.Adhesionforceofantsonsm

ooth

surfaces.

Amer.Zool.34:128A.

Cartier,O.1872.StudienuberdenfeinerenBauderHautbeiden

Reptilien.I.Abt.Epiderm

isderGeckotiden.Verhandl.Wurz.

Phys.-med.Gesell.1:83–96.

Cartier,O.1874.StudienuberdenfeinerenBauderEpiderm

isbei

denReptilien.II.Abt.UberdieWachsthumserscheinungender

OberhautvonSchlangenundEidechsenbeiderHautung.Arb.

zool.-zoot.Inst.Wurz.1:239–258.

Christenson,H.K.1993.Adhesionandsurfaceenergyofmicain

airandwater.J.Phys.Chem.97:12034–41.

Chui,B.W.,T.W.Kenny,H.J.Mamin,B.D.Terris,andD.Rugar.

1998.Independentdetectionofverticalandlateralforceswith

asidewall-implanteddual-axispiezoresistivecantilever.Appl.

Phys.Lett.72:1388–1390.

Dellit,W.-D.1934.Zuranatomie

undphysiologie

derGeckozehe.

Jena.Z.Naturw.68:613–656.

Dickinson,M.H.,C.T.Farley,R.J.Full,M.A.Koehl,R.Kram,

andS.Lehman.2000.Howanimalsmove:Anintegrativeview.

Science288:100–106.

Dixon,A.F.G.,P.C.Croghan,andR.P.Gowing.1990.Themech-

anism

bywhichaphidsadhere

tosm

oth

surfaces.J.Exp.Biol.

152:243–253.

Edwards,J.S.andM.Tarkanian.1970.TheadhesivepadsofHet-

eroptera:Are-examination.Proc.R.Entomol.Soc.London45:

1–5.

Eisner,T.andD.J.Aneshansley.2000.Defense

byfootadhesionin

abeetle(H

emisphaerota

cyanea).Proc.Nat.Acad.Sci.U.S.A.

97:6568–6573.

Emerson,S.B.andD.Diehl.1980.Toepadmorphologyandmech-

anismsofstickingin

frogs.Biol.J.Linn.Soc.13:199–216.

Gadow,H.1901.TheCambridgenaturalhistory.Vol.8.Amphibia

andreptiles.McMillanandCo.,London.

Gay,C.2002.Stickiness—Somefundamentalsofadhesion.Integr.

Comp.Biol.42:1123–1126.

Gay,C.andL.Leibler.1999.Theory

oftackiness.Phys.Rev.Lett.

82:936–939.

Gennaro,J.G.J.1969.Thegeckogrip.Nat.Hist.78:36–43.

Gillett,J.D.andV.B.Wigglesworth.1932.Theclimbingorganof

aninsect,Rhodniusprolixus(Hemiptera,Reduviidae).Proc.R.

Soc.LondonB111:364–376.

Gorb,S.N.1998.Thedesignoftheflyadhesivepad:Distaltenent

setaeareadaptedtothedelivery

ofanadhesivesecretion.Proc.

R.Soc.LondonB265:747–752.

Gorb,S.N.2001.Attachmentdevicesofinsectcuticle.KluwerAc-

ademicPublishers,Dordrecht.

Gorb,S.N.andR.G.Beutel.2001.Evolutionoflocomotory

at-

tachmentpadsofhexapods.Naturwissenschaften.88:530–534.

Gorb,S.N.,E.Gorb,andV.Kastner.2001.Scale

effectsonthe

attachmentpadsandfrictionforcesin

syrphid

flies(Diptera,

Syrphidae).J.Exp.Biol.204:1421–1431.

Green,D.M.1981.Adhesionandthetoe-padsoftreefrogs.Copeia

4:790–796.

Gregg,K.andG.E.Rogers.1984.Featherkeratin:Composition,

structure,andbiogenesis.

InJ.Bereiter-Hahn,A.G.Matoltsy,

by guest on January 16, 2015http://icb.oxfordjournals.org/Downloaded from

1090

K.AUTUMN

ANDA.M.PEATTIE

andK.S.Richards(eds.),Biologyoftheintegument2:Verte-

brates,pp.666–694.Springer-Verlag,NewYork,NewYork.

Haase,A.1900.UntersuchungenuberdenBauunddieEntwicklung

derHaftlappenbeidenGeckotiden.Archiv.f.Naturgesch.66:

321–345.

Hanna,G.andW.J.P.Barnes.1991.Adhesionanddetatchmentof

thetoepadsoftreefrogs.J.Exp.Biol.155:103–125.

Hepworth,J.1854.Onthestructure

ofthefootofthefly.Q.J.

Micros.Sci.2:158–163.

Hill,D.E.1977.Thepretarsusofsalticid

spiders.Zool.J.Linn.

Soc.60:319–338.

Hiller,U.1968.UntersuchungenzumFeinbauundzurFunktionder

HaftborstenvonReptilien.Z.Morph.Tiere.62:307–362.

Hiller,U.1969.Correlationbetweencorona-dischargeofpolyeth-

ylene-filmsandtheadheringpowerofTarentola

m.mauritanica

(Rept.).Form

aetfunctio.1:350–352.

Hiller,U.1975.Comparativestudiesonthefunctionalmorphology

oftwogekkonidlizards.J.BombayNat.Hist.Soc.73:278–282.

Hora,S.L.1923.Theadhesiveapparatusonthetoesofcertain

geckosandtreefrogs.J.Proc.Asiat.Soc.Beng.9:137–145.

Irschick,D.J.,C.C.Austin,K.Petren,R.Fisher,J.B.Losos,and

O.Ellers.1996.A

comparativeanalysisofclingingability

amongpad-bearinglizards.Biol.J.Linn.Soc.59:21–35.

Israelachvili,J.1992.Interm

olecularandsurfaceforces.Academic

Press,NewYork.

Israelachvili,J.2001.Tribologyofidealandnon-idealsurfacesand

fluids.In

B.Bhushan,Fundamentalsoftribology—bridgingthe

gapbetweenthemacro-,micro-,andnano-scales,pp.631–650.

KluwerAcademicPublishers,Dordrecht.

Kinloch,A.J.1987.Adhesionandadhesives:

Scienceandtechnol-

ogy.ChapmanandHall,NewYork.

Koelsch,G.2000.Theultrastructure

ofglandsandtheproduction

andfunctionofthesecretionin

theadhesivecaptureapparatus

ofStenusspecies(Coleoptera:Staphylinidae).Can.J.Zool.78:

465–475.

Kunzig,R.1999.Whydoesitstick?Bubbles,bubbleseverywhere,

eachoneasm

allsuctioncup.Discover.27–29.

Lauder,G.V.1991.Biomechanicsandevolution:Integratingphys-

icalandhistoricalbiologyin

thestudyofcomplexsystems.In

J.M.V.RaynerandR.J.W.(eds.),Biomechanicsin

evolution,

pp.1–19.CambridgeUniversityPress,Cambridge,Massachu-

setts.

Lee,Y.I.,M.Kogan,andJ.R.J.Larsen.1986.Attachmentofthe

potatoleafhoppertosoybeanplantsurfacesasaffectedbymor-

phologyofthepretarsus.Entomol.Exp.Appl.42:101–107.

Lees,A.D.andJ.Hardie.1988.Theorgansofadhesionintheaphid

Megoura

viciae.J.Exp.Biol.136:209–228.

Liang,Y.A.,K.Autumn,S.T.Hsieh,W.Zesch,W.-P.Chan,R.

Fearing,R.J.

Full,andT.W.Kenny.2000.Adhesionforce

measurementsonsingle

geckosetae.TechnicalDigestofthe

2000Solid-StateSensorandActuatorWorkshop.2000:33–38.

Maderson,P.F.A.1964.Keratinizedepiderm

alderivativesasanaid

toclimbingin

gekkonid

lizards.Nature203:780–781.

Mahendra,B.C.1941.Contributionsto

thebionomics,anatomy,

reproductionanddevelopmentoftheIndianhouse

geckoHem-

idactylusflaviviridisRuppell.PartII.Theproblem

oflocomo-

tion.Proc.IndianAcad.Sci.,Sec.B13:288–306.

Nachtigall,W.1974.Biologicalmechanismsofattachment:

The

comparativemorphologyandbioengineeringoforgansforlink-

age,suction,andadhesion.Springer-Verlag,NewYork.

Noy,A.,C.D.Frisbie,L.F.Rozsnyai,M.S.Wrighton,andC.M.

Lieber.1995.Chemicalforcemicroscopy:Exploitingchemical-

ly-modifiedtipsto

quantify

adhesion,friction,andfunctional

groupdistributionsinmolecularassemblies.J.Am.Chem.Soc.

117:7943–7951.

Persson,B.N.J.1999.Slidingfriction.Surf.Sci.Rep.33:83–119.

Peterson,J.A.andE.E.Williams.1981.Acase

studyinretrograde

evolution:Theoncalineagein

anolinelizards.II.Subdigital

finestructure.Bull.Mus.Comp.Zool.149:215–268.

Roscoe,D.T.andG.Walker.1991.Theadhesionofspiders

tosm

ooth

surfaces.Bull.Br.Arachnol.Soc.8:224–226.

Rosenberg,H.I.andR.Rose.1999.Volaradhesivepadsofthe

feathertailglider,Acrobatespygmaeus(M

arsupialia;Acrobati-

dae).Can.J.Zool.77:233–248.

Rovner,J.S.1978.Adhesivehairsin

spiders:Behavioralfunctions

andhydraulicallymediatedmovement.Symp.Zool.Soc.Lon-

don42:99–108.

Ruibal,R.andV.Ernst.1965.Thestructure

ofthedigitalsetaeof

lizards.J.Morph.117:271–294.

Russell,A.P.1972.Thefootofgekkonid

lizards:Astudyin

com-

parativeandfunctionalanatomy.Ph.D.Diss.,Universityof

London.

Russell,A.P.1975.Acontributiontothefunctionalmorphologyof

thefootofthetokay,Gekkogecko(Reptilia,Gekkonidae).J.

Zool.London176:437–476.

Russell,A.P.1976.Somecommentsconcerningtheinterrelation-

shipsamongstgekkoninegeckos.In

A.d.A.BellairsandC.B.

Cox(eds.),Morphologyandbiologyofreptiles,pp.217–244.

AcademicPress,London.

Russell,A.P.1979.Parallelism

andintegrateddesignin

thefoot

structureofgekkonineanddiplodactylinegeckos.Copeia1979:

1–21.

Russell,A.P.1981.Descriptiveandfunctionalanatomyofthedig-

italvascularsystem

ofthetokay,Gekkogecko.J.Morph.169:

293–323.

Russell,A.P.1986.Themorphologicalbasisofweight-bearingin

thescansorsofthetokaygecko(Reptilia:Sauria).Can.J.Zool.

64:948–955.

Russell,A.P.2002.Integrativefunctionalmorphologyofthegek-

kotanadhesivesystem

(Reptila:Gekkota).Integr.Comp.Biol.

42:1154–1163.

Ryan,M.J.,K.Autumn,andD.B.Wake.1998.Isintegrativebi-

ologyaluxury

ornecessity?A

considerationfrom

sexualse-

lection.Int.Biol.Iss.News&

Rev.2:68–72.

Savageau,M.A.1991.Reconstructionistmolecularbiology.New

Biol.3:190–198.

Scherge,M.andS.N.Gorb.2001.Biologicalmicro-andnanotri-

bology:Nature’s

solutions.Springer,Berlin.

Scherge,M.,X.Li,andJ.A.Schaefer.1998.Theeffectofwater

onfrictionofMEMS.Tribol.Lett.215–220.

Schleich,H.H.andW.Kastle.1986.UltrastrukturenanGecko-

Zehen(Reptilia:Sauria:Gekkonidae).

Amphibia-Reptilia.7:

141–166.

Schmidt,H.R.1904.ZurAnatomie

undPhysiologie

derGeckop-

fote.JenaZ.Naturw.39:551.

Simmerm

acher,G.1884.UntersuchungenuberHaftapparateanTar-

salgliedern

vonInsekten.Zeitschr.Wiss.Zool.40:481–556.

Sponberg,S.,W.Hansen,A.Peattie,andK.Autumn.2001.Dynam-

icsofisolatedgeckosetalarrays.Amer.Zool.41:1594A.

Stork,N.E.1980.ExperimentalanalysisofadhesionofChrysolina

polita

(Chrysomelidae:Coleoptera)onavariety

ofsurfaces.J.

Exp.Biol.88:91–107.

Stork,N.E.1983.Acomparisonoftheadhesivesetaeonthefeet

oflizardsandarthropods.J.Nat.Hist.17:829–835.

Thomas,R.C.,J.E.Houston,R.M.Crooks,K.Taisun,andT.A.

Michalske.1995.Probingadhesionforcesatthemolecular

scale.J.Am.Chem.Soc.117:3830–3834.

vonWittich.1854.DerMechanismusderHaftzehenvonHyla

ar-

borea.Arch.f.Anat.u.Phys.180.

Wagler,J.

1830.NaturlichesSystem

derAmphibien.J.

G.Cot-

ta’schenBuchhandlung,Munich.

Wainwright,S.A.,W.D.Biggs,J.D.Currey,andJ.M.Gosline.

1982.Mechanicaldesignin

organisms.

PrincetonUniversity

Press,Princeton,NewJersey.

Walker,G.,A.B.Yule,andJ.Ratcliffe.1985.Theadhesiveorgan

oftheblowfly,Calliphora

vomitoria:A

functionalapproach

(Diptera:Calliphoridae).J.Zool.London205:297–307.

Weitlaner,F.1902.EineUntersuchunguberdenHaftfußdesGecko.

Verhdl.Zool.Bot.Ges.Wien.52:328–332.

Wigglesworth,V.B.1987.Howdoesaflyclingtotheundersurface

ofaglasssheet?J.Exp.Biol.129:373–376.

Williams,E.E.andJ.A.Peterson.1982.Convergentandalternative

designsin

thedigitaladhesivepadsofscincid

lizards.Science

215:1509–1511.

by guest on January 16, 2015http://icb.oxfordjournals.org/Downloaded from