Contributions theoriques a l’etude des polymeres aux

interfaces

Manoel Manghi

To cite this version:

Manoel Manghi. Contributions theoriques a l’etude des polymeres aux interfaces. Analyse dedonnees, Statistiques et Probabilites [physics.data-an]. Universite Joseph-Fourier - Grenoble I,2002. Francais. <tel-00002111>

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Submitted on 10 Dec 2002

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−→R

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0

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−→R

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2a2

p∑

i

∫ N

0

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^ k ¦ kbh4l

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−→R −−→R′)

^ k ¦ kfkml¾®I VSCF

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b

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^ k ¦ k I l

n monomères

N-n monomères

(1)

(2)

x

z

y

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¾²R¢ ÍmX`RTI!Q I\]>RT dX Q XwSR Y g

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^ k ¦ km fl

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Φ(z) ∼= pN |ψ0(z)|2^ k ¦ k J l

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R Y XQ !>RT [X RT Gbg F ∼= kBTpN |E0|¦ ] X X Íf E0

X Q Qg¢gX Y XQ XI! ψ0

^ \ ¦ ^ k ¦ km fl lÒ!XQ! Φ(z) X!X RT Ó ^ k ¦ k J lU

FΦ = EΦ − TSΦ

= kBT

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a2

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gXgÓ ^VSCF

l ¦¾(-Q`Y>RTg¢dgQSRT VSCF (z)

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1 ≤ α ≤Ni

Rj£]Q ∑αmi,α = N

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P :SRF!b X ´!X& >RT -!X XX X!T¸ XQ-∫ N

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^ c ¦ kfiml

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PE = P (m, z)B ^ c ¦ cfeml

B =∑NC

i=1Ni -Xg- `RT!VX X!T¸ XQF X ZY RQ ¦ ¾®V

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2

vΣ

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2

kBT

a2ΣS0

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dn

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=

∫ N

0

3

2a2z2(n)S(n) +

v

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S2(n)

z(n)− S′(n) ln

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nP (m)dm

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W; ¢SRg ´RT G[gVRT XXFg¢g¢`X¢>RT¢`RT z ^ X>RT g XQ z lCU δFE

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(va2

6

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a3

Σ

B∑

α=1

∫ mα

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^ c ¦ cfiml

I c

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P (m) ∝ G(0, z(m);m) ^ c ¦ h eml

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¦V& Q ´Q SRT ¬ X X Íf ^ c ¦ c I l-X Q F!XRTX0QjRF!X Y X!b !b Q g

Y XQ ´! SFESkBTΣ

=

∫ N

0

3

2a2(a2S(n))3 − S′(n) ln

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′(n)S0

)dn

^ c ¦ h h4l

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0 z2(n)dn] = 0¦ ¾(IX X!T¸ >Qa

!X Xg_gIQ`Y>RDCX !XQX!X!XRT X `RTQXQ ¦ ! "#$

S(0) = a−2%'&()*,+,)* +-

z(n) ∼= a

1+(kn)1/2

.k ∼ 1 / +)0 &&1-%23 4

z ∼= an−1/2&(+

nÀ 1 5Ih

\ K ] /4^5_ $;* R0R$&0 %$&' 4*29A9;*

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〈Hex〉PE ∼=∫

coucheΠ(z)dz ∼= kBT

a2

∫ N

0

(a2S(n)

)9/4(z(n)

a

)−5/4

dn^ c ¦ h kml

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〈Hel〉PE ∼=∫ N

0−S′(m)E(m)dm

^ c ¦ h cml

E(m) ∼= 32kBTa2

∫m0 z2(n)dn X X Íf´SR \X´!X X´!g¢ ¸ XQdQX X´! n

gXgÓ V!XRT>F X RT GbgVRT !qQ`Y>RTg¢ g4Zm ¦ W¬X¢ £RT Q E(m) ∼= kBT

a2

∫m0

(dz/ξdm/g

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ξ2∼= kBT

a2Φ1/4 ¦Ç Íf`RT X \>RT ^ c ¦ h cml0*RT¢>RT V `RTSR

SRT ^ c ¦ kml X>: XX

〈Hel〉PE ∼=kBT

a2

∫ N

0

(a2S(n)

)5/4(z(n)

a

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^ hX¦ kml X Íf`RT `RTC : I!g¢ ¸6 >RQI X¬!SR ¦ ¾®\]>RT As

Am

QX `RT ! RTgVR $m Q £]! Zb Óg`£RT>RT *gXgÓ >RT ¦Ç q!´ X !b ! 10−21 ¤ ¦w `RQ X`RT[ ÓQ !b `RT>QÒ g! m`RTX; w!ÍmX`RT mQ Ò `RQ VQg¢g:X =Q >RT[\]Xg X RT;SRC g¢Ó wQXQ`Y>w!gXgÓ h#h ¦ ÐRTX;ÒQjR!X Y X!b!Z[gÓ ^

Φ = 1 l £m¢RT Fs = γm

γm = Am12πa2

¦ Ð&RT>C0QjR!X XF !0gXgÓ ¶Q X ÍfI IX Q X g¢g Fs

∼= γ0 − γ1a2S0 > Fs∼= γ0 − γ1 !XRTXIQjR!X

ZY RQI`RT ´gXgÓ ^S0 ∼= a−2 l ¦ ¾®\]>RT γ0 = As

12πa2 γ1 = As−Am

12πa2

e

e \ ] *5",%$&'P *`,%"DQ 0R!S*`"*5,7Q344,?"*`689;:4<L=-*-,

Q £mgSRF X ´ >Q `RT!: `RQ X `£RTa: X RTSR!b= XQ-! X d! ZY RQ-RT g¢SRQg[C!X XgQ-! £fRT:>RTgXgÓ !XRTXSRI g¢Ó &QXQ`YX ¦bÇ C ÍfX[XXÒ!£ X X!b &Í]RTg:Qg¢ 0 `RQ X -gXgÓ !-SR g¢Ó FQXQ`YX h#h ¦=Â SRV !b`RT!: g!X !b I ^ S2

0 ¦¦¦ lÒ:QX!b`RT 0X £RT>RT ! γ1 ¦

)7-<*8 #:7 .-)O<>+-,o%²)O.0)<#7-.0+-< .0)vD#1 ;,)<

W¬QjRTQVSR X ! ZY RQ!X Y X!b !Z[gÓ X V! !>RT I X X Íf !VSR´QXQ`YXqR! ^ c ¦ h h4l ¦¬Ç § ¶(SR´Q jRT !X Ä ZY RQ!XRTXY X!b !VZ[gÓ VSR´ Z[g !§ Zb ÓgVF `RDCXSR´Q jRT !VXQ X ZY RQ>Qg¢gg R¢¹¬Íf hX¦ c ¦

fondu

fondu

air

© Á hX¦ c SYuk;9/V0m$9/'(Da #<'h'2&Vk^_568 cO#t]@$'@s#<'hk;9/')#en,\9/ gO01&+569/')#<^nl`k;9/&0m$9/'Da #<'eg@/&013J9_#$5&M

¾8X X Íf ^ c ¦ h h4l( ;!XQQ ; \]XwR `QQ +Y gVRT F! X ZY RQ ¦Â X!XRT `a g Xag[`RT CCRT G !Q`Y>RDCXÒ!Ò_ CRx h i ¦ > Qg¢ !¢ X RT `RQ § Q>\X>RTSR ZY RQV!gXgÓ - >--!Q`Y>RDCX^ X >RTVR gXgÓ ^ l0 g¢>RT GXl ¦Ò dQ X T¸ RQ £mR! G ÍfXIU• ¾(Rd g¢Ó F Y>RT\X ¦Ç =¬0gXgÓ ^ lX Y gVRT¢\OX Xq

SRT` Q4£fRT dRj£mQ V£m X ;SRT RT ´ ¢ÍmÓ g!b¶ ´!QF!X&gXgÓ XRTg¢!X XFQ`Y>RDCX ^ gXgÓ ^ l l ¦ ;X X &X! g¢[ d!SR0 X ´! ZY RQ& `\]XwgXgÓ ^ lÒ R!X h i ¦

• ¥ RTXZRÍ]RTgXQ q!X ÍfX \X ¦]Ç =m \X:!T¸

[h

W/"0R6 $;*`e fD'6" $;^-^-,?*5",%$& 0R!#%$*-,. *-,.9A$O"$& *-,?68'9;:4<>=-*-,

Z[gÓ RTVQ[`RQ!X X ZY RQmaX Q XÍf`RT X g¢ QjRTSR&g !& X >RQaX: Ò !b ¦* >RT SR \]X Ò Qg¢ !XRTXw X SR?¸ Q q!0XQ \ ¦ ^ c ¦ h h4l ¦;Ç £fRTXQ`YX;Qg¢gXX¢ f X¢\]X FX X!T¸ XQÒ Qg¢ Ò!:SRg_g:gVRTÓ XX8Y RT XX ?¸6 gVRT U jRT =Q `RTX!X a¸6 RQQ Q`YXVSR ZY RQI>RT&gXgÓ ^ l ¦ÐRTXwQ&QjRRTXQX&>RT [ \]X!bX-SR- XQ!CSR0 ZY RQOX £[FgXgÓ ^ l- ¢!XQ Y Rj£] dSR ZY RQ ¦Â ¢ ¶Rd qg HG[ ¸g`RTg h[#[ ¦ Ð& g-SRT X ¥ a¸  RT hjefk !&QjRTQ P  ¹ h[fc g XIRTÍfg`RT !&SR¢!X !: !Q`Y>RDCX: Ó!SRF Y RQ ¦

ÐRTXXw! Q Vg`RT ]Q :RT `RQ Q>\]Xw!Íf g¢>RT G QjRT`RQ >RTw>RT`RTgÓ X\X ∆ !X®!&SRfY Rº £RT` ^ ¹¬Íf hX¦ h l&U[ \X gXgÓ ^ l SRTg >!XRTX:SRF ´ RTgXIRT´Q`RQ!SRF ZY RQ g¢SRQFgXgÓ ^ lR! F Q!Xag¢g¶SR!b¶ XQF!X Y>RTVR QF −∆kBT

¦ TX ∆ > 0 Íf>&\X& Zb ÓgÍ]RTÍfXCX&X Íf&w!Q w!XQC X RT `RQ !CHGb g¢ ¦ ¾(RV\>RT ∆ C!I X !b 0!I X 0d X RTX`XQ0!I Ig!b>QjRT Q`Yg¢\Xw! !wQ`Y>RDCX ¦ ¾®wQjR!Ò Z[gÓ Y XQ >RT C `RC `RT !XRTXCSRV>RT hX¦ ¦ = Q- f*g F\XI X ¶& \XF! !Q`Y>RDCX: Ó:!&SRF ZY RQC_ & : Qg¢ >RT: 0 Ó:Q !b `RTXQ!X RTg¢ X! kBT h[fc ¦

P f X\]Xw X !:Q`Y>RDCX VQ`RQRj£]QÒSR ZY RTQm!Ó\X ∆ > 0¦

 Y[Z YXÓ `R!b`QaX ¦ ÐRTX-QQjR ¶Xg- !¢gXgÓ ^ l ^ k>RTQ`Y>RDCX4lw 2Γ/N ¦ ¾8X X ÍfI!bX¢Q IRT `RQ d Q>\XX Q !XQ

Ft∼= −2∆kBT

Γ

N∼= −2∆kBT

∫ N

0

S(n)

Ndn

^ hX¦ cml:SRWY XQ X ^ c ¦ h h4lwR!XRTX FX ÓgI & Q ^ >RT: ! ZY RQ4l

FS ∼= kBT

a2

∫ N

1

[a2S(n)

]3+[−a2S′(n)

]ln[−a2S′(n)

]− 2∆

N

[a2S(n)

]dn

^ hX¦ h lÇ >RTQg¢ C!Q g] X \>RT q!X \] C! ÒQ`Yg¢\]X ^ c ¦ hjcml

g!b>IU(a2S(n))2 +

S′′(n)S′(n)

∼= 2∆

N

^ hX¦ I l g¢gI>RT`RTgÓ ∆ ! X !b -! X >gg- 0!Í]RTXQ`YX-! X \>RT ^ hX¦ I l Ò! X !b &! 1/N ¦ ¾(R- Seq(n) ∼ n−1/2 \ ¦ ^ c ¦ h4 fl ® Ò!XQ&RQQ`RT&`RTÒ\X

k

e \ ] *5",%$&'P *`,%"DQ 0R!S*`"*5,7Q344,?"*`689;:4<L=-*-,

air

liquide liquide

air

© Á hX¦ h ST9/&c#$s#<'019/'9/01&Da #<'+V:&0o^v&08,2`_56o#t'e019/'9/01& V^V#$3$#0o#Q`m5X'6W)Ag@/&01xU$U '+#<' &'&U 8&'2Xd,2cO#$

∆kBTM

a2Seq(n)2 À 2∆

N n¿ N

∆

¦b XÍ]RT C!X ÍfgC!X R! Y RT! !Q`Y>RDCX ^ ÍfgI! T ZY RQ`RT[FY RT q` SRF g¢XÍfI!Ð&fVX l ¦

¾® \X ∆ CXCÍf`RTX!\X- X ^ ÍfgF!T ZY RQ`RT Y l SRV` ^ c ¦ h4 flOX &X&£fRTSRT0 N

∆ < n < N¦ ÐRTXQ F Íf0 X Seq

RT &SRT i ¦=Ç ¶=F g¢!¢!b !F X \]>RT ^ hX¦ I lgY QFQ`Y>RDCXq Íf ¶ X SR0 XQ &!CSR0QXQ`YX :RT w g¢SRT -Q!X XC`f ¦X SR-:_a C&QjR! Y >!bX!FZgÓ I!& !¢Q`Y>RDCX[ Fg!b>IQ`Yg¢\]Xg-\]:X Ó:Íf`RTX!IR -Rj£]Q&SRF ZY RQ ¦ >: f X:>RT:SRF ∆ ≤ 1 ¦Ç `RT \X®ZRqX¢` Q Ä g`RT qX ¢g!Ó h i U=-!£R¸ - ÓfY RT-! ∆ ² VX7Y `RQ !-HGb g¢ 0 0QjRT >RT-SR´ ZY RQ ¦ ¾(Q VQk0 wRT XQ Q:X& X!&!`RT X R! ! w!CQ`Y>RDCX ÒXQ``RT ¦* X! Y RT !XRTX:SR0>RT hX¦ ¦ Ð&RTXQ\]® bXX: f >∆ ' 1 :XXwX ` X: `RT ^ c ¦ h4 fl ¦

¾® \OX QX !Ó I`Y >!bd!IZgÓ >SRVQX ! X ZY RQ0 SR¢g_g\XQ!b Y X!b ¦ ÐRTXQQjRÒÍf`RTX! I!X HGbQÓ Ωex

Ò` !b § X a>a Íf !X HGQÓ ¦ ÐC¢X¶!XRTX¢QjR!b !XFSR YX £RT SRT >a(YXgÍmX`- QjRT`RQ `:>RT:Q`Y>RTg¢ Seq

¦ ¾²RF X ´! ZY RQX Q !XQ

γ = FSeq − Ffondu^ hX¦ [ml

Ffondu d X X Íf§ !ÄX §`RT!Ä HY XQ&Q#X ?¸ ¸6!b §>Y X!bt!Z¸

gÓ ¦Â ¸6QOX R\]OX X´ ´Q X Ím \Xd\] X X ÍfdR \]X´!

c

W/"0R6 $;*`e fD'6" $;^-^-,?*5",%$& 0R!#%$*-,. *-,.9A$O"$& *-,?68'9;:4<>=-*-,

Q`Y>RDCXVRTX > ¦jÇ -RT >RT[( X X Íf YX g¢\]X kBT wQ`Y>R\XQ`Y>RDCXjX>® £mXFfondu ∼= kBT

ΓN∼= kBT

a2N1/2

¦*Ç > `RTSR ^ c ¦ h4 fl!XRTX X X Íf F *\>RT ^ hX¦ h l SRF X ´! ZY RQX Q RTX>RT

γ(N,T ) ∼= γ∞ + βkBT

a2ln (N/N∗)

N1/2

^ hX¦ fl

γ∞ = γm − C kBTa2

β C `I!IQ VQ I[g \]X-!¢ X !b !¢ X V!SR§£fRTaX HGRQ RÄ !`X!XRTXX ! Q !X Q`YX ¦ ¾(R \>RT N∗ ∼ e 83∆ \]bRT>RT`RDC;!XRTX² X \>RT ^ hX¦ fl® (-X!bQ!Z[g `RT FQjRT`RQ \]X!b ZgÓ IQX ! ¦>Ç >RT ∆ ∼= 1 XXw £mX N∗ ∼ 50 ¦

¥ RT YXgVRT \XgSR Y XQ ¢!X>RT; X \ ¦ ^ hX¦ fl¬R!bg;0gVRDGbg-gt Nmax∼=

e2N∗ ¦fÇ wQ C !X RT !¢!X!bQ Y NmaxTw!Q C g

x?X \]4X `Rd£fRTVR Zg¢` \]X γ∞ ¦ ¢Qg¢ X!b d>RT V!SRQg¢ Vk ¶ U®!X X¢>RT ® X RT `RQ Ä! I!Q`Y>RDCX£] SR ZY RQ¶!X RT >RT X !XQ ¦> Qg¢ ´ wRT &*RTÒ*RT`RTgÓ N∗ ¦! XÒ £mX\X X =Ò! Ò!Q`Y>RDCXÒ!g¢XC N < N∗ QjRT `\]X N RTÍfg [SR Y `RQ !FgXgÓ XRT G§HG[ `ag¢ !bg¢X F X X Íf!b Zb ÓgRTÍfg Qq\HGb\]X X RTÍfg`RT fd! γ N < N∗ ¦ ѬRTX!b\X& N > N∗ b X -R `QSRZb!b `RT!XQq!Xg¢ > ¦ WX N Íf`RTX!XdXg- !Q XÍf`RT XIRQQ` &Q>!b`RT SR!b gVRQ f Q\]X S g¢ `RT=Q\¶gVRDG[g¢` X a[ ¦ T > γ !bg¢>IRj£]Q N N > N∗ ¦

RT X!\]XF0>RT`RTgÓ N∗ ` Íf`RTX!SR>RT !ZgÓ X X&Xg!b>Q`Yg¢\Xg & HGb g¢ ¦ ¾(R!b= XQ γmax−γ∞ ∼ 1

eN∗1/2 :SR&£RT

gVRDGbgVRTw;SR£fRTR Zg¢ \Xw RT ;XÍfÍmjRT ¦ Ð X F!£[XYZb \]>XX£]X!XQ >RT Q Q I ! G Íf>IU• N < N∗ SR0 X d! ZY RQ wX Y XQ dQ `RT ! N • N > N∗ γ 0 ÓQX `RTÍ]RT F`R0£fRTR Z[g¢ \]X γ∞ ¦ >RT`RT £]gb X \]>RT ^ hX¦ fl !XQqRQQ !qRj£]QSR Y g- ^ hX¦ h4l bRj£mQ

HGbf`RT ν(N) X\® h k¢\(RTXÍfga[ -Rj£]Q N ¦XÇ >RT Q- `RT ^ hX¦ fl g!0Qg¢ X!b ¢\X- X X £]-HGbf`RT& ¶Q jY!¢k cVw N !¢h:wÍf`RTX! N ¦

¾²R- X !& ZY RQ ^ hX¦ fl!Q CwXjRT gRj£mQCSR- g¢`RT !b\! T ¦ WSR¸ºXXX!XRTXSRg¢ N →∞ ¦ ¾® g −C kBT

a2R! GV ÍfXU]!X X>RT SR>RT

h

e de ] *5",%$&'P *`,%"DQ 0R!S*` *-,7,9AO%$&'",?,*5<$ $A9A4^5*-,

SR \]X kBTa2

mR QRT GFgXgÓ X &SR ZY RQ:!Q XÍf`RT Xf `X !X RT >RT X &R QRT GQ XÍf`RT fX!XQ ¦Ç HY ¸Y w QX!F g:! ^ hX¦ h l;RT>RT `ÒRC X !w ZY RQÒ>RT`RT CXw >RT >Q £] SRF!b ´!XQ RG ¶: \X\(RTÍfg ´£!g¢gRj£]Q T ¦ ¾®QjRTQ®g I\]XQ QX! ¶ :!g¢>RT ¦

)7-<*8 #:7 .-)O<>+-,o%²)O.0)< <*#:1 +I'*8 #:70< <>) 8¡J].08 1 +0@)<

ÐRTXQ >RT XXÍmX`RT X QjRTQaX: Q!RT QjR!X X ` g¢ ¸!bX ¦ X0! g¢XXF 0!X RT !§SR´ XQ !VSRdQXQ`YXqR! Ä! *RT :>RT QÓ g X RQ d!&SR¢ ´\®Qg¢ gSR¢QXQ`YX ¦

Xq f Xq\X´SR g¢Ó QXQ`YX RT gXgÓ Ò S0 = a−2 Q#X ?¸ ¸6!b ¢\OX X RTÍf!X SR g¢XÍf!¢Ð&7V&X h#h ¦®Â YZ[ YXÓ ``R!b Q ¬!SRQg¢>RT`RT Rj£]Q¬ `RT ;HGb g`RT G ¦Ç XFXX f XÒ\]XÒQ`Y>RDCX ÒQ`Y>RDCX:R! XCX XÓ w>RÒ\OX `:!XQ´ >RT ¦

"*- '$&*a9A$ "* *S.68M*2%$&*59 %/"*- <L":440M<$"*¾8X X Ífd ^ c ¦ h [ml! g¢X!XRTXVdQjR!X XdQXQ`YXR! d £fRT

!F_ g!b>R X !V FQg¢ q!VSR´ f g \]>HG Q>RT0SR´ !b ` £m!&gXgÓ \OX QX X ¦ X Y RT`X:!XQ X X!!XRTX: X X g-Íf`RTX![¸6QjRTXX\]XI >RT ¦

Ð&0X =Qg¢g¢¢QjR! Y X!bX!FZgÓ =X Rj£]Ó ¢\]XFSR X T¸ RQ £ !V !´Q`Y>RDCXIx X g¢ `RTq!XRTXSR X !´ ZY RQ´! SR ¦Â X!XRT Ò!XRTX´`aÍfg! T ZY RQ`RT Y RT ^

n ¿ N∆

l !X& Q!g¢gwOX £[ >R!XRTXSR§ XQ !SR§QXQ`YX ¦ W;RT XQ! g¢Q w £fRT QX ! &Qg¢gRT YX g¢\XbC£mg&HGQ v ∼= a3

¦ ¾®&QjRÒÍmX`RT®!ÒXw £RT ^0 < v < a3 lQjR*RT Qw!b £RT Θ ^

v = 0 l QX ! !XRTX X RTXa>!bQ ¨&¦Ç `RT: X \]>RT ^ c ¦ h [ml X X Íf FS, z !&SRFQXQ`YXR!` X Q b>RT

I

W/"0R6 $;*`e fD'6" $;^-^-,?*5",%$& 0R!#%$*-,. *-,.9A$O"$& *-,?68'9;:4<>=-*-,

! ZY RQ

FS, z ∼= kBT

a2

∫ N

0

[(a2S(n)

)9/4(z(n)

a

)−5/4

+(a2S(n)

)5/4(z(n)

a

)7/4

+ (−a2S′(n)) ln(−a2S′(n))− 2∆Na2S(n)

]dn

^ hX¦ J l g¢gSRC ¢ g¢ ¸6!bXfÒHGb QX Ff g \X Πb

¬SRCQXQ`YXR! ¦ Ð&X ¬q Zb Óg¢Q`Y>RTÍmV!¢Q`Y>RDCX!Z[gÓ VRj£mQSR ¦TX XXÒ `Rj£RTX:!XRTX X X g-&Íf`RTX![¸6QjRTX\X&`>RT [SRI ` qÒ Q`Yg¢\X`RT RGbÒ>RTw w£fRT :!XRTXÒSRF Q £mgX Πb

µb¦ ¾®

Íf`RTX! ¶X Q w!XQ>RTÒ C!C ZY RQ Ω = F +ΠbH −µbΓ H X >RT`

!qSRQXQ`YX Γ qXg- !qgXgÓ R!` ¢>RTFX `!q ZY RQ ¦ ¾®q YX g!bZ[>RTg¢\]X0RT IX Q

ΩS, z = FS, z+Πb

∫ N

0z(n)dn− µb

∫ N

0S(n)dn

^ hX¦ iml¾+X X Íf !X X /QX XÄ!gXgÓ dX/QXQ dX Q F =

∫fvoldV =

∫vol(−Πos + µΦ)dV Πos

µ Φ Q £mgqSR f gT¸ \X*0 Q`Yg¢\]X¢CSR.Y `RQ £]g¢\X ¦ Ð X RT Ó&C&!X Q`YXRT\XRT G´ XC g¢ ¸6!bX h XSR¢ ` df g \X- Πb = Φ

2b∂(fvol/Φb)

∂Φb∼= kBT

a3Φ9/4b

Q\®QX!b fvol ∼= 4

5kBTa3Φ9/4b

µb∼= 9

5

kBT

a3Φ5/4b

^ hX¦ hjeml

"!#%4*`"*a9&0b!S4! /"* L9 cd^ O $&9A$ D*¾²Rwg¢g¢`RT F!bI Ω >RT²`RT ;RT GIQ`Y>RTg¢X S z QX! :! G\>RT X

!¾(RTÍf`RTÍm-R `Q ^ Q VQ g \]X!& X !b I! X g¢ l

(a2S(n)

)9/4(z(n)

a

)−9/4

−(a2S(n)

)5/4(z(n)

a

)3/4

= Φ9/4b

^ hX¦ hfh4l(a2S(n)

)5/4(z(n)

a

)−5/4

+(a2S(n)

)1/4(z(n)

a

)7/4

+S′′(n)S′(n)

= Φ5/4b +

2∆

N

^ hX¦ hjkml

¾(\>RT X ^ hX¦ hfh4l: ^ hX¦ hjkml:X `RT YZb \]XI g¢> Q`YXgVRT - SR¹¬Íf hX¦ Ib¦ ¾(R g¢Ó HGb gF X \] ¢gQjRT\]X!FSRQXQ`YX ¦ T$ X \ `¶X

[

e de ] *5",%$&'P *`,%"DQ 0R!S*` *-,7,9AO%$&'",?,*5<$ $A9A4^5*-,

`RTXQ`Y>!X >RT δz g¢ Q `RT qX gVRTUSRbY Q!´`RTSR \]X0!CX X!T¸ XQ!0QX `RT 0!- ` k ∼ Φ1/4 d>RT RT Ó FSRX f g \]XHG Q>RT(SR SRw 0f g \XHGb Q>RT²SR: `RTXQ`YXa¸ g_g\ X! CSR!bSRT ¦ ¾8X \>RT ^ hX¦ hjkml¬ `R!b X Í]RT :! Q`Yg¢\XÍmX`RT !SRF `RTXQ`YX!X >RT` δz !&SR¢ ¦

dz

kSz

Fb

9/4

F9/4

© Á hX¦ I :&&V9/' "5&m+a 6c#<;JL0165!'cO#$BDa #<'569_#$5:mG,\9/ gO01&G,\9/ g "F,\&&6Mq;.$a U ^Da #<'+!'@;,\9V^9/'zQa ,,29K5GEQV)'&F-oHA&''vl`m569_#$5A0m$FV`"66s&',\9/'2XQ,2`'?'k;&V#<s;'2v6cO#<;JL6W\9/'(65&V^a 6cO#<;JLoGk-9/56x,\9_#<+,2`' /#/,\&V#<MQ&^#<=F-5& /G69_#<0o"ml k;9/56 D`1,2&V9/' 9&019_cO#$$'@o`i569_#$5

Φ9/4(z)W)`G,2&&9/'?V2&566I,.n`69/^#9/'

Φ9/4b

\`+k-9/563&`L/c#$ G5X'kSz

M

XRTX X!b wQw Z[ ÓgQX !`RT ÍfgR Zg¢ \X`\]X& X S(n) z(n) &!&!- `RTXQ ¦ XC!b ÍfXXRTX ¬! Gd ÍfX!XRTX:SR¢QXQ`YXIR! VUbSR ,'", !S4! /4*a$A*- 4* :SR ,4, !S'"!/"* *-N*- 4**¦

W Ó!0SR ZY RQSRQXQ`YXFX£] ´>RCSR &C g VSR XQ!SR¢ ´ :XÍfÍmjRT ¦ ¾®\>RT X ^ hX¦ hfh4lÒ ^ hX¦ hjkmlÒ g¢>!XQ

(a2S(n))9/4(z(n)

a

)−9/4

= (a2S(n))5/4(z(n)

a

)3/4 ^ hX¦ hjcml

(a2S(n))5/4(z(n)

a

)−5/4

+ (a2S(n))1/4(z(n)

a

)7/4

+S′′(n)S′(n)

= 0^ hX¦ h h l

! X :w!& RTXQ

Seq(n) ∼=1

a2n6/5^ hX¦ h I l

f

W/"0R6 $;*`e fD'6" $;^-^-,?*5",%$& 0R!#%$*-,. *-,.9A$O"$& *-,?68'9;:4<>=-*-,

zeq(n) ∼= an3/5^ hX¦ h[ml

X¶ £]X®SRÒ >Qa !X XQXQ`YXR! Ò>RT (!X X -!bX4\ ¦ ^ c ¦ h[ml ¦¾® `\]OX X ÍfX-!SR ZY RQX X RT GbgVRT QX `RT ¢XÍfÍmSRQ !SR OX X£fRTSRT ¦¬Â QRT>RT`RDC-` \]X Seq(nc)

9/4zeq(nc)−9/4 ∼ Φ

9/4b¶Q\

!XVX!bQgXg \XQjRT`RQ \X nc∼= Φ−5/4

b

¦ ¾(RÍfXÒQ X!XRT C RT an3/5c

∼= aΦ−3/4b

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a3S(n)

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^ hX¦ h4 fl(a2S(n))2

Φ7/4b

+S′′(n)S′(n)

∼= 2∆

N

^ hX¦ h J l

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a2n1/2^ hX¦ hjiml

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Φ1/8b

^ hX¦ kfeml

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Seq(n) ∼

1a2n6/5

pour n < nc ∼ Φ−5/4b

Φ7/8b

a2n1/2pour nc < n < N

^ hX¦ kbh4l

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0

0

11

F

zz4/3

xbb

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chaînes adsorbées

chaînes libres

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nc b~ F-5/4

1

zn6/5

~

zn1/2

Fb

7/8

~

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0 n

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7/8b N1/2)

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f(n)

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a2dz/ξbdn/g

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¦ TX z < ξbSR´X X!T¸ XQ 0Xq `RT \X

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Rsd Φb > Φ2

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γ = ΩSeq, zeq − Ωsolution

^ hX¦ k h lJ c

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W;RTQX XQ Ωsolution = 0:gSRTÍm>RT £fRTgXgÓ ¦XÇ £RT>Q`YX

!XRTXIQjRI!X XV Ä!ZgÓ ® Y RT -Qg¢ V!SR XQ X!gVRQ gQS ¦ TX Ωsolution

!b w X ÒSR \X¬X `RTXQ`Y>! !X >RT H ¦ ÐC Y Rº g-SRTRT QjRV!b Y X!b !Z[gÓ XXV XX!XQΩsolution

∼= kBTΓN∼= kBT

Φ7/8b

a2N1/2

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7/8b N1/2 ¦ ´X C\]OX ´`RT \X

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SR \X: \X ¦ ¾(QRTQ¬RT

γ(Φb, N) ∼= γ0 − γ1 +kBT

a2

[−αΦ5/4

b + βΦ7/8b

N1/2ln

(N

Φ7/12b N∗

)+ δΦ

3/2b

]^ hX¦ k I l

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! gIQ X!XRT Φ3/2b

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!b ZgÓ IQX ! ¦Ñ;RT¢\XSR`Y `RQ £mg¢\X gXgÓ OX ¢>RF Íf`RTX! Φb ¿ 1 ¬XX

£]XXXÒQ w! gw!g¢>RT w! X \>RT ^ hX¦ k I lXXwRT GbgXÒSRY g- ^ hX¦ k I lÒ £fRT

γ(Φb, N) ∼= γext − αkBT

a2Φ5/4b + β

kBT

a2Φ7/8b

N1/2ln

(N

Φ7/12b N∗

)^ hX¦ k#[ml

γext ∼= γ0−γ1 SR£RT! γ HGb `RT!b X ^Φb → 0 l ¦ Ð& GVQjRg¢

: `RT ¦ÐRTX:SR0g¢ I!Q`Y>RDCX X N →∞ bXX: £]XIU

γ(Φb, N →∞) ∼= γext − αkBT

a2Φ5/4b

^ hX¦ km flJh

e B f'<L6 0R0M$;,%' 0 '*5!`9&*5,7D^5,% 9 0 ,.*SNO68^5$A<>*50M4N

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¦ µ ∼= kBTΦ

5/4 !XRTXCSR =!XQF Íf`RT!b Q g¶ &SRQ SR X Ä!V ZY RQ ∆γ ' −kBTΦ5/4

b /a2¦ X£]X-Qg¢ X!b !?Y RºÄ g¢

\X> ÍfXw XÍ]RT jYUf \]X ΦbRTÍfg`mw *Q`Yg¢\]X:!XRTXSR&

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6w X \]>F g Φ5/4b

Φ3/2b

!XRTX ^ hX¦ k I lF¢Í]RTga` £]>RT ¨ XQ`Y>RTX!ÐRX! k#[ `RTSR.Y g-¢!7VCX ¦ ÐRTXQ0 `Rj£fRT ¶ £]Í]RTgI&£RT RT >! γ(Φb, N)

!XRTX&F Ífg¢! X ¦®Ç £fRTXQ`YX¶!XRTX& Ífg`g¢ ¸6!bXIQXQ -!SR¢·HY ¦ k#[ `a gQSRT[ Φb

N ! X \]>RT ^ hX¦ k I lÒOX R>R ! g¢X ¦

Ç HG[ `RTSRT X \]>RT ^ hX¦ k I l!XRTXw ÍfgQXQ ^Φb → 1 l fXX £]XSR

Y g- ^ hX¦ fl\]XCXX:Rj£[XÒ £]CwSR- X ! ZY RQ&! Y X!bXw!CZ[gÓ ¦Ç XbX X:\Xb!XRTXÒ X \>RT ^ hX¦ k#[ml XX:Rj£]XÒXÍfÍm& g g`RT

!q X !b ! −kBTa2

2∆NQ\]: ÍfÍmÓ \XSR£RTHGb `RT γext !X!b !

N¦®Â 0 gF 4£!b h (´ gF!bgg- !F!b ¢!0 X \>RT ^ hX¦ kfcmlgVRT&

XÍfÍmjRT-!£RT[wRT : g! X \ ¦ ^ hX¦ k#[ml ¦

#Q DFow,o8 <>#:7 o9:)%v1 )< ,=@<>+-1 '*oÒ'*< )#"D@;,8 )7C'*o:+"

¨ a X!b;X RT Q`YX ¢! Z¢!X Q`YX ¢Xq` f q>R¢!! g¢XÒRT :Q`YXf :\Xw!!X!XRTXQ8Y gmQjRTXXXÒ£]X!X!w`a>RT GÄQ VQ Fg \]X¢\]XVXX0`a>X ¦ 6OX § V>R-gXF\OX - `RT!Q Y :Xfw `RT :!b Q gCRT Gq!XX:HGb g`RT ¦ ¾(R¢Q Y `RT d!

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YX !X Q`YX0Rj£mQHG[ XQf :X : Óg-U

• Â g¢gXX² X R4£mX² Q !XRTX(Q`Y>RT Chj \X !SR:Q`Y>RDCXZ[g \]XÒRX E>HGb X ¦ X!£]X! >ÒgXgÓ YX \]X ^ ÍfX a gVR` gQSRT Mn0

lRj£]QqgXgÓ ¶Q jY¬!C`RT C∞lw!gVR ` C∞m

[ l m Q £]gSR¢`RTISRFgVR` gQSRT 0!b´gXgÓ -Q`Yg¢\]X ¦ ¾®£fRT :! C∞

:w!b¶ wgXgÓ X!b C £]:!XRTXwSR0 `RT ¦Ç `w !XRTX:ѬRTjRT hX¦ h ¦

• ¾®&Q`Y>RT X&X0 x RTgVRT& Qa`ag[&gX!b ¦ `CQjRT`RQ `&>RT X!bQ!Zb!b I !X®>RT

I =〈Mw〉〈Mn〉

avec 〈Mw〉 =∑

i(niMi)Mi∑i(niMi)

et 〈Mn〉 =∑

i niMi∑i ni

^ hX¦ k J l

i ! Íf>RT X X!Qd!Q`Y>RDCX!gVR` gSRT Mi Xg- ni

¦Â VX!bQ I£fRT q hh ¦ ¢Q`Y>RT XV X!b ¦ ¾®Q`Y>RT XV QX ! Q£] XgQg¢g0gX!b 1 < I < 1.2

¦*¥ RTg_gWY RT*¬ f \XSRIZb!b EXX& `RT [QjRTXC Íf Í]RT !ÒQ`Y>RDCX :ÍfXRT>RT`RDC FSRF ZY RQ ¦

W Ç WÐ ¥ P WÑw¹ ÇC∞

I k ha

^ gl h ¦ e J e ¦ c c ¦ Mn0

^ Í g l i J cm e hjkfefeγ0

^ g gl I c ¦ [h cm ¦ J h c ¦ i#[−∂γ/∂T |Mn→∞

^10−2 g g )Il [ ¦ h J Ib¦ J [ [ ¦ cm f

β 0.85± 0.04 0.10± 0.03 7.9± 0.02N∗ 27± 3 48± 5 4± 1

°²Ï hX¦ h 2#<& V ,\&V01&'2; cO#_'2^aWMn0

Da ,2 k!hfhfh 6W γ0

−∂γ/∂T |Mn→∞ Da ,2 h[ h I,\9_#<nQ[*v>Z)WX[C \[D7DCAM

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Qg¢g

γ(Mn, T ) ∼= γ∞ + βM1/2n0

kBT

a2ln (Mn/M

∗n)

M1/2n

^ hX¦ kfiml

M∗n∼= N∗Mn0

¦ XRx¡X X;!XHG[ g RTHGRTg¢X XQQ £]gSR¢!X!XRTXQ-a gVR `&gQSRT I! γ wSR¢!X!XRTXQI g¢`RT ¦

^568*5""0M"!S*a*5@<>0R,,* <L9;^5!- 9;0M$&*P wSR¢¹²Íf` hX¦ J w w X X! ZY RQ γ(Mn)

: :ZgÓ !b¶ UjZm Y[ZÓX ^ W Ç l Zb!bg YZ GRTX ^ WÐ ¥ P l®²Z[ `RDEXX Y[Z]¸ÓX ^ WÑw¹ Ç l ¦]Â ;g HGb g`RT!SR X ¢!Ò ZY RQÒ Y RT `F `RTSRg YX!0!I X RTXjRT! YXg-Zh ! g¢`RT ¢SR g¢`RT ! `RTX ´£ X`!:ZgÓ X!b J ®hje#[!®hjc J ¦

¾(R Y g- ^ hX¦ kfimlÒQg¢ I >RT`RTgÓ -U a Mn0 γ∞ M∗

n β ¦*Â X!XRT

¬QjRT`RQ \X;!²gXgÓ a Mn0RTX [\XSR: X F! ZY RQ¬²gVR `

X γ∞ = γ0 + T∂γ/∂T |Mn→∞²`0HG[ `RT F!VSR `RT q-X!b\XF!XRTX-

ѬRTjRT hX¦ h ¦ ¬X- `0!XQF\XFk>RT`RTgÓ IRx¡X `RT β N∗ ¦ XRj£]X&Q`YX ;HGb XQ& `RT &QC &ZgÓ &&X &`RT X ¦ Ѳ!X RT !¶QF !¢ZgÓ X X¢¢g¢ `RT X!bX g ¦Ç X ;¢ >!bQV!Zb!b ?¸ - SRT £]g Y RT ^ !V X !b q!dh ¦ I l ¦¬Ç X¬HGb !Xg- X F!XHGb g`RT :WÐ ¥ P X Y X!bX´ ´ F!b¶ : g¢`RT ¦

¾(R¹¬Íf hX¦ J g \Xw X RQQ !\>RT `RT jY®!:SR YX Rj£]Qw!XHGb gb¸`RT gVRT \]>RT ¦ ¾8X \]>RT ^ hX¦ kfiml;X Rx?X Ò>RTZY RT gRj£]Qw! GF>RT`RTgÓ β N∗ >\]¬ !b C!IÍf`RTX!RT X!bX ^

β ∼ 1 N∗ ∼ 30 − 50 >QHY ¦ ѬRTjRT hX¦ h4l ¦¾(&*RT`RTgÓ N∗ :QjRT`RQ \XI!Q`Y>R\]X Z[!Z[gÓ ¦

X\XQjR!b¢WÑw¹ Ç ¬FF Íf ¦Ç ¢ ¶TѲHE> ;QX ! wQg¢g`RT g¢ ¸ E>HGb ^

C∞ = 24 l :£RT ! Mn0 β !XQ´!b¶ !Q

¢qW Ç 0qWÐ ¥ P ¦ (XXRT X!X\XQ VQ β FX!X!XRT

!bÄ Z!Z[gÓ QX ! ¦ W;--! G g¢ -Z[gÓ - -! X !b !´hRT I\]XF¢WÑ:¹ Ç Í]RT ¦ i ¦ X¢QjRTX ¢!Q ¢!b >RT ¢_ `F¢Q`YXjG

J

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0

10

20

30

100 1000 104

105

PE

PDMS

PTFE

Mn (g/mol)

g(m

N/m

)

© Á hX¦ J &'@V9/'Q/#<Vk^_563nk-9/')#tGP,\9/ g01&3&'%k-9/'59/'n`+0mLn019/65#<`,\9_#<?69/ Y&&'2po,\9/ g01& D,\9/ g)/p gO&' [CTl Da ,2 hjc#[ 6W,\9/ g 01F g"V9)' [*v>Z_ul Da ,2 hjc I I,\9/ g:.#$9/9$ g&' [D7TCTl Da ,2 hjcm MSY 569_#<J569/V;,\9/'ln`n9/69/Vc#$ OMPr# /6 L65I#LG,.01βWN∗ M

!bV£]g!X gXgÓ \ mVZ[gÓ g¢ ¸ E>HGb]OX >R a3 ^ YXÓ 4l ]gVRT QZ[X!b ¦ Q VQÍmg \X £[X!b`RTCRT !XRTXwQjRTQ ¦

Ç XfXXQX `RT X\]XÒ X Rx?X g!wSRC YX \XRT GFg HGb g`RT `RT ZY RT`RTg_g+Y RTgVR` gQSRT [` N ' 2− 10 ¦ ]X YX Rw SRT !XRTX®SRÒg¢ !¶ Ó®ÍfX²Q`Y>RDCX ¦T ²RQQ !I ¶ (>RT X!b0 ÍfÍmÓ \XV!I g0Q `Q jY 0£] X X `V YX VXV£] `RT0>R-SRQ γ(N) ¦

^568*5""0M"!S*`*2 *2<L68^-0 %"D*

¾²RÄ!X!XRTXQ! γ @Y XQ !dSR g¢`RT T Í]RTg Qg¢>RT RT Gg HG[ g RT ¦ P ;SR&¹¬Íf hX¦ iC`; ;£fRT !SR X ¢!XÒ ZY RQ

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!b WÐ ¥ P ! G g¢`RT !b¶ T = 20Â T = 30

ÂC¦ ¾(R¹¬Íf hX¦ hje ` &! G` Ò!X HG[ XQ:!b= w CWÐ ¥ P Ò \X£RT SRT X! γ(T ) Mn

RGF Fg ¦ Ð&RTX! GQjR=&ÍfXQ X [ X \>RT ^ hX¦ kfiml &Q Y ?¸6Q ¦Ç ¶]£fRT !*RT`RTgÓa N∗ β `w! \X FQ X Q!g¢g ¦

Q(XQ XXQX `RT X\X& X \>RT `a>! ÓwXdQg¢ FSR Y :!SR¢!b¸!XRTXQÒ0 g¢`RT w!SR X ¢! ZY RQ;!¬£fRT SRT X;!SR ∂γ

∂TRj£mQSRgVR

gQSRT Mn¦ W¬w:Q`Y>RT X: :!XRTXwSRFQg¢>RT`RT X´R Mn < M∗

n

¦ TX X RQ CS` g!& C \X Mn

RTÍfg ¢ R `Q V X =!C !0Q`Y>RDCX ¦¾( \]X T RTÍfg &SR-Q \X&!C X =w! w!&Q`Y>RDCXRTÍfg SRF X ´! ZY RQ!bg¢X ¦

Ѳ HY ]*%X RTÍf!QX ! Q ¶ÒRj£mQ QjRT bQjRTXXÒRj£mXQX ! C\]X∆ X!X!XRT->RI! T Q\ ¶SR Q § Y>RT\]XVOX I\XF`RT`ag[IQjR h ¦ TX >RT`RTgÓ ∆ £fRT VQg¢g ∆ ∼ A + B/T A ∼ 1

¦Â g¢gI®OX HGb -RTXQXI!X-HG[ g`RT- g `RTC!X :£fRT SRT X! ∆Rj£]Q T XXRj£]X: Xf ∆ QX `RT ¦

ÐRTXVQR-!I >I`g¢ ¸6!bX®Ig -HG[ g`RT0! X Ä!V ?¸Y RQ0 &`RT ¦ T X FQ>RT `RTXQ¶ UVRTX J hjk#[ X!bFQ Óg ¦ Ð&RT>Q! G X!XSR d I !b Zb!bg YZ[ GRTX ^ WÐ ¥ P l!XRTXCI XÓX*g RjZRTC UY RT Q £]ag[ 24 Â 19.5 ÂC¦ ¾²R X ! ZY RQ!b XÓXg :£[ γtoluene = 28 g g ¦

 qQg¢>RT`RT 0 F `RTF¢X YX qFX F RT`X ¦Ñ²I!X RT !®!XIHG[ g`RT-QXQ XIFg_g¢Z[gÓ ¶¢WÐ ¥ P =!b¶ &!CgQSRT !&QXQ `RT X!b¶ ¦ Ð&-X& C X! `Rj£fRT G!b¶ XXXX:£mXa> *QaC!SR¢Qg¢>RT`RT d!¡x Y RT Qa¸!g¢gÒÒ! Y X!bX!XWÐ ¥ P ¦Ç >RT QaX bXXÒRj£]X!¡x I! g¢XC£fRT !:>RT`RTgÓ N∗ β \X&XXRTX!XQ ¦

Ç V Q £fRT X \]>RT ^ hX¦ k I l7Y XQ !:SRgVR :gQSRT MnXX XXCU

γ(Φb,Mn) ∼= γext − αkBT

a2Φ5/4b + βM

1/2n0

kBT

a2Φ7/8b

M1/2n

ln

(Mn

Φ7/12b M∗

n

)^ hX¦ cfeml

J i

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15

17

19

21

100 1000 104

105

20°C30°C

Mn (g/mol)

g(m

N/m

)

© Á hX¦ i q-'.#$&'56+A`%&08,\&$V#t3/#<Q`%&'@V9/'iv/#<Vk^_568Ak-9/')#t ,\9/ g01& 6V`$9/'

γ&'~k-9/'59/' o`0mL&o019/65#<`8,\9/ g01&

Mn,\9_#<o[*G>?Zl

T = 20 Da ,2 hjc I 8Pl T = 30 Da ,2 ife MST 569_#<J 569/V;,\9/'@V'l`9/

69/Vc#$ OMv'@,.013r#t/;OJLM

¥ RTSR-Qg¢>RT`RT YX THGb >Q!XRTXÒ&QjRÒ!X X& `Ó£mC Óg g`RT >RT:`RT RT´QjR:!X Y X!b ¦

• ¾® XÓX- QX ! -Qg¢g-`RT´d £fRTIWÐ ¥ P *Rj£]Q´>RT`R¸gÓ !¹¬ Z χ = 0.45 km ¦ X² g¢g²QX!XRT¬ Ó(-!b0 £RT¬RT YX g¢\X\XXXCRj£m>QX ! :x?X \OX Q ¦ ¾(R¢ÍmX`RT`RT !XfQjRTQ v < a3 :!>RTX X RTX!bQ ¨&¦ Xw!X!XRTXQ v RT>RT`RDCfgVRTR&!HGbf`RT Y RT ^

v1/8 v1/12 l X\OX 4X [ `RDCX>R!Íf`RTX!Q`Y>RTÍmg C!XRTX: X \]>RT !X ¢!£Xg \X;£fRT`RT[; Q`YXw!&h ¦m X ; \]XXXXX>X 0SR Y g- YX \]X ^ hX¦ cfeml ¦

• ÐC-X>XX&X £]XC\X>&!IXg- GdZ[gÓ &!C-WÐ ¥ P *I>RT`R¸gÓ C!¹¬ Z χ !X!XRTXQ`Y>RT 0h]£fRT CRj£mQSR Y `RQ £]g¢\]X Φb

km Q\>OX RT>RT`RDC>R!XRTXSR YX !¹¬ Z¸ ÍfÍfX mSR\X v ∼= a3(1−2χ) ¦Â I ¶IR VRT !¢>RTI! V&X I h QX !`RTIg!ÓYXXgX¸Íf\X ^

n ¸6QX g! l ]X:!X!XRTXQ! χ RTXQ! Φb !b ¦

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T (°C)

8

12

16

20

24

0 50 100 150 200

75000 g/mol

3900 g/mol

770 g/molg

(mN

/m)

© Á hX¦ hje C jo`b&08,\&$V#<B/#<s`b&'@V9/' u/#<Vk^_561ok;9/')# B8,\9/ gO01& 6V`$9/'@v

γ(T )lMn

wO2,\9_#< 9/"G6#<&v Y&&'2 AU 019/ Da ,2 I i 6W U 019/ Da ,2 hji MPS\&h 9/^569/V;,\9/'&'2lz`z9/s69/Vc#$ OM%'),.01 r#t/;OJ<]M

XwOX Rj£]X:>Rw ´Qg¢ I!Q ¶:!XRTX:X YX ¦• Ç X=XXRj£]X f F\X-SR g¢Ó ¢QXQ`Y>F!0 X ZY RQ¢ `RT ¢agT¸

>gÓa` ¦Â dRT G[gVRT VQRY >RT!VHG[ XQ! HE>Q £ d!>aX X:! X!bX!WÐ ¥ P !XRTX:&` >Óa> J k ¦ ¾²R Y `RQ £]b¸g¢\X SR ZY RQ Φs

¶\Rq ¢g I!e ¦ i#[ ¦®Ç X!b\]X\X¢!XRTX¢QjR!OX XX0` !X0WÐ ¥ P !XRTXC- XÓX=CgXgÓ [gY gR!` X ZY RQ-RTT ¦

Ç ¢ XQm>RT`RTgÓ XQX!XRTX X \ ¦ ^ hX¦ cfeml; γext α β M∗n

¦]Â X!XRTXXCQg¢>RT XRj£mQ-!g C!I X >Q&!- XC!-WÐ ¥ P XX `Xq!XQd£fRT ! β M∗

n £]q!XRTXQjR!b@Y X!b !dWÐ ¥ P

M∗n = 17.6

$Í g² β = 0.12 ¦*Â :£RT QX _ X!X!XRT C! Mn Φb

¦TX `:! Gq>RT`RTgÓ Rx¡X `RTIU γext a α ¦

P -SR ¹¬Íf hX¦ hfh`I 0-£RT SRT X0! γ(Φb)0!- >0 g¢ ¸

!bX!WÐ ¥ P !b¶ Mn¦ XqX` £mXq@Y RTq ¢UqQ

ibh

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19

20

21

22

23

24

25

26

0 0.2 0.4 0.6 0.8 1

500 kg/mol

3.7 kg/mol

1.6 kg/mol

1 kg/mol

g (mN/m)

Fb

© Á hX¦ hfh &'@V9/' %#<Vkp_56h%9/^#9/'@%&0o=F- ^#$6h%[*G>?Z $'@B9/p#OV':?&'k-9/'59/'iv`sk]_59/'(<9/p#t0o]c#$6W

ΦbW,\9_#<8 j&&'2v0mL&Q019/65#<`M2ST Y&&'2p

g01J9/A569/V;,\9/'&'2$#Le9/''6AV ,\&V01&'2;v J T = 24 W Mn KU 019/f

hjk#[ T = 19.5 W Mn _U 019/W MKU 019/W M KU 019/fM@ST3569_#<J&A69/'2\"&/#$3

a 6c#_$9/' OM !M

!b¶ MnX`V!!b`0>R->RTF g¢ `RTX SRT ¦ 6ZRd!XQ§QSRTÍmq

£RT SRT X! ΦbQV! Mn

¦Â !X!XRTXQdHGbQ ´Rj£]Q Mn ¢ Ó

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\ ;XÒ£RTHGb `RT &!b ¢ X: !b= ! γtoluene ¦m w!b¶ XQ wRT X!bXbQjRTSRUY g- ^ hX¦ cfemlXCX RT\X>RÒRTqQjR!Ò XÒ!bX ^

Φb → 0 l X`RTg¢g>RT QC\]X!XRT>Q ÍfgSR f g \]X Πb

SR =Q`Yg¢\X µb

!£]&_ 0 £RTX >RTQg¢ F X F!- `RTX RT !&Q`Y>RDCX&!>RTXCSR ¦

ifk

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α e ¦ h [β e ¦ hjk

M∗n

^ $[Í g l h4 ¦ I#Jγext

^Mn = 1

$[Í g l ^ g gl k hX¦ J [γext

^Mn = 1.6

$[Í g l ^ g gl k hX¦ ifeγext

^Mn = 3.7

$[Í g l ^ g gl k Ib¦ efeγext

^Mn = 500

$[Í g l ^ g gl k Ib¦ cm °²Ï hX¦ k X#<&8V ,\&V01&'2;+ a W M∗

n

Wβγext

,\9_#<8v9/^#9/'@v&0o=F- ^#$6+[*v>Zo,\9_#<3 YV6V'n0mL&x019/65#<`M

T ΦbQX `RTwXX X £mXXRTÍfg`RT /! γ Rj£]Q Mn

¦ÒÇ ¶wQg¢gM∗

n∼= 17 $[Í g >0 QX!d g0!I X \ ¦ ^ hX¦ cfeml XUY XQ Q `RT 0! Mn

¦ W¬Mn = 500

$[Í g fw g:Q Q jY( XÍfÍmjRTXX £]XSR&R Zg¢ \X\ ¦ ^ hX¦ km fl ¦

XQX `RT XX´ÍmÓ ´£fRT SRT ! γext Rj£mQ Mn^ QHY ¦ ѬRTjRT hX¦ kml ¦ X´QjRTX

_ &gY RTw\XC X ´RTwXÍfÍm& g −2∆Mn0Mn

kBTa2

Íf>RT Q!Xag¢g ¦>Ç ¶ `\]X Φb

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E(∆) = −A∂ ln Ξ∂β

+ µΓt(∆) = −kBT∆Γt(∆)^ hX¦ c J l

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∆

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a2

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a2ln (N/τ ′)

N1/2− 2kBTe

∆

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γ ∼= γ∞ + βkBT

a2ln (N/τ ′)

N1/2

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Mnτ ′Mn0

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[1 + λ

(Mn0Mn

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3Φ(z)3 + fΨ(z)

¦ ¾8X X ÍfF `RTI!b Z[ Óg%X Q RT b>RT: !X ZY RQ

F = −kBTa2

1

2σΨ(0) +

kBT

a3

∫ ∞

0

Ψ(z)ρ+(z)−Ψ(z)ρ−(z)

dz

+kBT

a3

∫ ∞

0

ρ+(z)[ln ρ+(z)− 1] + ρ−(z)[ln ρ−(z)− 1]

dz

+kBT

a3

∫ ∞

0

1

3Φ3(z) +

a2

6

[d√Φ(z)

dz

]2+ fΦ(z)Ψ(z)

dz

^ Ib¦ h[ml

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¾® ²Q f `RT \XI w IRT GQ`Y>RT Ím:>RT: X \>RT d!W¬` U

d2Ψ

dz2(z) = −4π`B

a2[fΦ(z) + ρ+(z)− ρ−(z)

] ^ Ib¦ h4 fl¾²Rg¢g¢`RT q!: X X Íf ^ Ib¦ h[ml>RT`RT ρ± QX!bÒRT G! G\>RT X! ¨ ¸gVRTÄU

ln ρ±(z)±Ψ(z) = ln ρ±0^ Ib¦ h J l

ρ±0 ! G¢QX RT ! g¢X ¦Ç XmSRCg¢g¢ RT V>RT`RT Φ QX!b X \>RT d!X Ç !! ÒRT !>\® wX!XRT!& X \>RT d! ¨ gVRT´::gXgÓ

Φ(z)2 + fΨ(z)− a2

6√Φ(z)

d2√Φ(z)

dz2= µ

^ Ib¦ hjiml \]>RT SR&ÍmX`RT RT !: X \ ¦ ^ k ¦ cfkml!bQ`Y>RT CkRT G¢Z[gÓ Q`Y>RT Ím ¦ ¾( ²Q`Yg¢\]> µ ^ ! kBT l wSR0£fRTaXw !& X \>RT ´! P Q`Y!bÍmR` Q ¦

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¦¾+X Qa X RT !b Z[ Óg gVRT !q! g¢X¢QX `RT £RTX!XRTX

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SR §U µ = Φ2b ρ+0 = ns

ρ−0 = ns + fΦb¦ X f X\XSRQXQ `RT

Q a¸ XC!SR ZY RQ :XÍfÍmjRT ¦*Ç d ¶XI : X Σ/V Σ X RT C!SRI ZY RQ V £]g!SRI qaÒ!b >RT`RDCÒ!XQ!XRTXRI g¢ Σ/V → 0

¦¾( Z[ Óg!W¬` b¸ ¨ gVRTb¸ Ç !! wRT !CX Q !XQ

d2Ψ

dz2(z) = −4π`B

a2

[fΦ(z) + nse

−Ψ(z) − (ns + fΦb) eΨ(z)

] ^ Ib¦ kfemlρ+(z) = nse

−Ψ(z) ^ Ib¦ kbh4lρ−(z) = (ns + fΦb)e

Ψ(z) ^ Ib¦ kfkmla2

6√Φ(z)

d2√Φ(z)

dz2= Φ(z)2 − Φ2

b + fΨ(z)^ Ib¦ kfcml

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e`

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b!XRTXw X \>RT ^ Ib¦ kfcml ¦

Ç XQ! G QX!b XqRT G g¢ R Q X \>RT ^ Ib¦ kfeml Rj£]Q

Ψ(∞) = 0&" %2 + / + %2 1- ) ) ) ρ±0 5

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dΨ

dz(0) =

4π`Ba

σ^ Ib¦ k h l

dΨ

dz(∞) = 0

^ Ib¦ k I ld lnΦ

dz(0) =

1

d

^ Ib¦ k#[ml

d FYXgÍmÓX X!b `RTXQ ¦ ¾®V\>RT X ^ Ib¦ k h l0 ^ Ib¦ k I l0g¢f Q`Y>RTg¢ SR ZY RQ´V X Q XX `RT`!´SR `RTX!b\]X´ X \>RT ^ Ib¦ k#[ml ! \X X \ ¦ ^ k ¦ cbh4l ; `R!bF `RQ XQ !b `RTXQ q¢gXgÓ V¢SR ZY RQ`! ¦®Â FQX!b C `0>RT¤mmRTZ ifkgVRT&0_ 0`ag¢SRQ¢>RT&XQX!b d Φ(0) b IRT Qg¢g>RT`RTgÓ -Rx¡X `RT k h ¦

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Φ(0) = 0 k h k I hjk h ¦ : ´: X!X R! ´!!b!:!X Q`YX ¦ TX X!XRTX: Ífg!FY RTFY Q:\]X]:gVRDGbg-g2!:SR Y `RQ £]g¢\]XVgXgÓ £fRT Qg¢g |Ψs|2 :SR0ÍfXQjRT`RQ \XI!SRFQXQ`YXI w X (f |Ψs|)−1 ¦

¾® \Xq¢£RT SRT X!b Q f `RT \X Y RT¢!£fRTF X X Íf YX?¸g¢\X Ψ ¿ 1 |Φ − Φb| ¿ Φb

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d2Ψ

dz2(z) = −8π`B

a2f√Φbδϕ(z) +

(κ2 +

4π`Ba2

fΦb

)Ψ(z)

^ Ib¦ km flρ+(z) = ns [1−Ψ(z)]

^ Ib¦ k J lρ−(z) = (ns + fΦb) [1 + Ψ(z)]

^ Ib¦ kfimla2

6√Φb

d2δϕ

dz2(z) = 4Φ

3/2b δϕ(z) + fΨ(z)

^ Ib¦ cfeml

Ψ¿ 1 ϕ = √Φb+δϕRj£]Q δϕ¿ 1 κ2 = 8π`Bns/a

2 ²>RT`RTgÓ w!Ð&Z] ¸ Q$] RTX:!bgX ¦

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^ Xf Y Φel ¦

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F

kBT∼= −σΨ(0)

2a2+1

a3

∫ ∞

0

Ψ(z)ρ+(z)−Ψ(z)ρ−(z)

dz

^ Ib¦ cbh4l

+1

a3

∫ ∞

0

ρ+(z)[ln ρ+(z)− 1] + ρ−(z)[ln ρ−(z)− 1]

dz

+

∫ N

0

a6

3

S3(n)

z2(n)+ S(n)

(z(n)

a

)2

− S′(n) ln

(−S

′(n)S0

)+ fS(n)Ψ(z(n))

dn

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(a3S

z

)2

+

(z

a

)2

+S′′

S′ + fΨ = µ^ Ib¦ cfkml

2d

dn

[aSz − 1

3

(a3S

z

)3]− fa3SdΨ

dz= 0

^ Ib¦ cfcml

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-¢`a[ Q`Yg¢\X µ -! X !b ! g−1e

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Φ = Φb

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¦¾(C `RQ X!-£mgFHGbQX Q £m ∫ Πm(z)dz

Πm SRQ[ !

gXgÓ FSR0 ´f g \X

Πm(z) ∼=kBT

ξ(z)3∼= kBT

Φ3/2(z)

(aξe)3/2^ Ib¦ c h l

Ç `RT[: X \]>RT ^ c ¦ kml X g I g¢g\]X∫ H

0Πm(z)dz ∼= kBT

∫ N

0

(a2S(n))3/2

ξ3/2e z1/2(n)

dn^ Ib¦ c I l

H = z(N) w X >RT !SR¢QXQ`YXIR! ¦¾+X SR Q I!w> X!T¸ XQX Q F X Q`Y>aI ξ U

Fel = kBT

∫ N

0−S′(n)

[∫ n

0

(dz/ξ

dm/g

)2 dm

g

]dn ∼= kBT

a

∫ N

0ξ1/2e S3/2(n)z3/2(n)dn

^ Ib¦ c#[ml

RT Ó: Íf`RT ´>RT:>RT ¦

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F ∼= kBT

∫ N

0

(a2S(n))3/2

ξ3/2e z1/2(n)

+1

aξ1/2e S3/2(n)z3/2(n)− S′(n) ln

(−S

′(n)S0

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^ Ib¦ cm fl

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3

2

(a

ξe

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)1/2

+3

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S′′

S′ = µb

^ Ib¦ c J l

1

2

(a

ξe

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− 3

2aξ1/2e S3/2z1/2 = Πb

^ Ib¦ cfiml

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%X Q µb∼= 1

gb∼=a3/2Φ

1/2b

ξ3/2e

^ Ib¦ h eml

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)4$%,#?(-.!l2%, H%&%,4Z0%_ H;gD2$67!l)4$%,C9,42%, H%&%,4Z*+4*+%,#%,C)4?)48d580%_2?09C0)4F@ A0?9C0E%,46_<%,9C0E0%2?0&K?0%,H?gIY46_0%1%,(-9 %,9C0?)48da0%12?09C0)43@ A0?9C0r())*>)H0% 2?0N%'7%,44I \(48d>\0%&% H9&*>#4%,4$% H%,()(C $/%,4)4$_9&9&?4$ b 20V"!#%&Y#?#?:9C0%&1!)%,(6V%&%,4$()()!n)*>)(-19&49,()4

Q(/o

a 0$ b 0%6@g 4 6?%&9,)#?4Z9&4/%'v2?0%& b ()08d=0 2%&6k4kC9.())48d5d=Cg*+%,4$ b !%B(6? 9&%&9,1%&?() %,1%,*>#?0"!.% $G%$# K%.?%& '9C05>!%,* 4V%.())"! bK%&%9C0)4P%,46F9.4PC)454r%,9C0?)48d b #A0%,%.F0%>d=Cf%&6P%,46?%?46?%,F%,44 40%F*+$6?%,(]2$!%6V46 4 b 0%F"!$#?9.(]2?(2V"!.%+@K%&% c d=?% R:I\A(-d:%, d=%.%,67C49&%& K* 0%#;()( @f)#?#4(\N0%+67C49&% K* 0%+9&%,4$%,<A9,? ?%JIa 0$;($9.(a%,9C0?)48d>&#H%. $/%, GK* 0%'#A()( a 0?G%.?%F%,4?%&;($9.(a%&W$?)())2?)?*M0%<9C0)45C46? s0%Ud=C67%,4$)489&49&%,4$C)=4v46V0$1)48(#H;gD2$67!V)4$%,C9,4 %,46?V#?0k0%,* #;.! f* 0%#;()( b #A0%,%. 0%4d=C6?)%,4 )4 6?%d=%&%E%,9C0?)48dU%,46? #??()('0%,* 29$U)#;6?0%*#A()( +#H%$G$9,4v0%^(-2?(2 b 0%>%,(-)9 K9&% b d=/%,4v2$!0% m )49, (-#-, R ). b kBT/ξ(L) b #A0%,% ξ(L) 0%<"!.%3'0%<(-12?(2

h )49&%<)1(-d:% b0%%,(-)9K9&% F*_()('465 2(-49&%&6P2$!V0%*_()(d9&49&%,4$C4 d=C67%,4$30%%&6Cd:%F0%n2?0 a 0%V9&9&%&+F0?_9.())48d 4X+0$_67%nv0% G9,+0_9C0)4Cgi%,46?%NC%d=%d/%&6 )4 0%r(-^2?(2u46Z0%,!ZC)4[()*+=^4P%,9C0?)48d 0/%,49&% b 0%r29)48d=%&67%,4$%0$1%'#?()9,)%,()!N#?%&%,4$)4V0?C9.()t4Dd_6?%&9,)#?4 4/%')"!E]0% t4%,G9&%;$:%,! G%.?% )4r0?;9&9&%&H0%16@g 4n*+$6?%,('s/0%#?(-4Ud:%&*+%,!5<E*_d=)4(\9.%$K0% (%'46?%,1gi6?%32F%,4?4%&39.())48dV*+$6?%,( F!%,(6?%&%,4$()()!_0% C*+% %&?()@f2?0N%'%,44N46 K%&% %,4%,d=!I\\0% #?4?4%&%,)48d h c 6?%&1g9,)#?42$! h %,*+%,4 , R h. #A0?9C04#; K?0%,O6?% /%,(#%&6 )46?%,#%,46?%,4$()!<2$!(#HMd=?#&, R=o b T=q . /)#H% :%, b 0%n*+4*+%,r9&49&%,4$C4 #?(% K?46[)4[0% h c 0%&! +#C2()9U46=4?0%& 10%3%,468'0%32?0 a 0?167!a%,%,4$#K*M0%<%,#8#?(%<'0%3C9.())48dE6?%&1g9,)#?45#A0?9C0X)*>#=%&+50#X6?%,4) ! d=C67%,4$_^0%N%&6Cd:% l%r*>d=0$_d=%r0>0?d=C67%,4$E2(-49&%&_0%8%,(-9V%&)48d G9&%V%'?%,%&6u4[0%8%,9C0%&6u9C0)4 46[0_)%,4?%& 0% d=(2(d%,W$?)())2?)?* 9C0t46H? ) 0 2%&%,450)#A4U0 0? 4 0%<*+=C2?(%39,?%<)49&%0% h c 0%&!r!%,(6?1^()#H%,*>)4?)* ?* K%&%%,4%,d=!

a 0?)48d:Ud:%,7#HC%>t4l0%_9.%+N9&49./%4d:%,*+%,! a #HVC9.())48d86?%&9,)#?4 9.4k2%G?465)45())%,C%ns4% b 6#?%&6 f* 0%*+$6?%,('2$!6v46 48,)Q h b Q.o b R7Q b R . b0%<0%, b d=:%,482$! h % $9$9, R=q . b A2C%&6V4U4 (%'46?%,1gi6?%42F%,4?4%&12?0V%.%&68($9.(t()!+9&9&?4$ K1^9B!())4679.(_d:%&*+%,! a 0%%&%,4$-( G%.?%O0% bi)4:%,C%Bc:F6@g 4#?#?:9C0nH0H0%2?(2n"!.% ;)()(C $/%,4nH#?#4(a0%67C49,% K* 0%F9&%,4$%,9,? =?%Fse2%&9.%H0%H)4$%,G9&%Hd9&49./% b 0%\2?(2>"!.%H6?%&9,%.%&Y;#H% d: f?0%,Y#;.!K*M0%7#;()(H@ c d=?%$ R:I a 0%<%&%,4$-(%&?() '0?*+$6?%,(d% G()()# @ a 0?Dd=0?0? 9,(% b K%&%%,4%,d=%& F %)4n4?)d0%,*_(p%,4%,d=! kBT Is

Q =q

• 0%0?9$$4%&CO0%F2?0n LDC ' R[1− (1− aNσ1/3

R )3/5]

• 0% f%&%3%,4%,d=!r#%,19C0)4V FDC ' −Nσ5/6 ln[1−aNσ1/3/R]

aNσ1/3/R

• 0% :()?*+% KC9,48#?(%3 φ(r) ' σ2/3

(1−r/R)4/3#A0%,% r 0%C67-(O67C49&%3+0%

#?0%,9.(?D9&%+@KC%&% c d=?% R:I *+48d>0%&%#?C%,6?)9,4 b 0%F(-A4%F;9&%,C)4?()!r0%F*+=A?#?)48d H#?%&679,A0

0% /(t*+%KC9,4N)49,%.%& #A0%,4%#;% *+ /%! K* 0%?G9&% b 46E4C2?()! b H0%%&6Cd:%a0%2?0 b #H% 46 φ(L) ' σ2/3

(1−aNσ1/3/R)4/5 a 0$ G9&%&%'?%,%&6_4_0%(-2?(25@K=*+9

d=C67%,4$ 46N%,(-9 G9&%JIH% :%,!_)*>#C4$ /%,4$()()! b 0% /()?*+% KC9,4n67/%,d:%&0%^9&%,4$%, G R ' aNσ1/3 /%,49&% b % /%B4P\0%($9.('C2?)())"!U)46?%0%2?0Pf?(()(%&6 b)4n0%39&4/%'n9.% b #H%*>d=0$1%'7#%&9,00?A#(%36?$%&4fd=:%0%F2%&14 #H%, *+48dX0%P%&?()n!$%,(6?%&6 2$! 0% h % 9$ *+$6?%,( b 4%PN#9,?(-()! ?#?)48d sA

9&4C4$5d=Cf)48d36?%,4)"! b 0%H2?0^0%,d=0$'d#?%&679,%&6>6?%&9,%.%;0%H)4$%,D9&%;'9,? /%&6*+%48d=()! %,Y)*>#?(%d=?*+%,4$Y)4679.%0Y4>0%A9&4$C! b 0%;2?0>0%,d=0$Y0?(6)49,%.C% h ?#?#=%K>U*+*+%,4$^0^0%E0%,d=0$+6?$%&4>9C048d:%%#A0%,4 0%N9,? ?%r9C048d:%&6 ? )48d30%A)4$%,G9&%A*+%48d=()!0%,4+%&679&%&Y0% :%,C()( /(t*+%AS0%;2?0a #H;gD2$67! )4$%,C9,4Y%\0%,4*+%\)*>#C4$ b #A0?9C0^%,46?O #;%B()(0%\2?0 a 0%\0%,d=0$#A)()(]0%,%G%3C2?)()"!.%<1>*+%$#A0Md=%.%, =()%

0?)6k#?#?:9C0 9&46?%,30%>67#?(%, 9,? ?%_%!S%&9,<4v0%+%,* ?(4kC2?)()"!J42$!V67)2?($9$V9&#()!*+%,4, R=R . a 0%,!V67!n0%<*_()(O9&%&9,4F67%3E9,? ?% +0% 9.C%r)4 0%KC*+%$#H $Z 0% (%'46?%,1gi6?% 2F%,4?4%&>*+$6?%,( b #A0?9C0X ()6 G :%,!#H%. $9,? =?%&a V6?%&9,)2%+9C0k9&49./%>2?0%&3?46?%,<(-d:%+9,? =?%& b 4%>* 0%,%G% $:%&%,#l)4

*>)46 0>0%N9,? /%&6Z2?0 >*+% $)46 F59&*>#?%&%&6 2?0 ? )48dP0%Nt4%,G9&%)46?%&%&6 *+?4$rl)49,%.)48d 0%82?0 9&49&%,4$C=4 s0%U9&%&#467)48dX9&*>#?%&46?$%&F43d=t4%^)4l4P)4$%,C9,4kF0%^2?0v%&6Cd:% b 0)#H% /%, b 2?FC0%,F)4vC*+%$)46(-%,C(\)4$%,C9,4V@G46k0%+20k#?(%E /%,!v67!a%,%,4$ f* 0189&*>#?%&%&6#?(-4\2?0 b #H%)4679.%)4 #?#%,467' I 4>0?\9&4$%' b 4%%&?()\2?C)4%&6+2$! 2?)446 (%'46?%,A4n9&*>#?%&C%,6N#()!*+%,;2?0%&H;*+=;)*>#C4$ , T7Q . s$0%,!r0)#H%&6r0 b#A0%,4u0%V(-.!/%,+0?9$4%&+ G9&%&6Xl2%V*_()(%,_04u0E3 f%&%n2?0 @ #A0?9C0u0%,%9&%&#46?; %d=)*+%&1T46$I b 0%*_)4n9&4$)2??4n0%K%&%%,4%,d=!Ed 9&*>#?%&%&62?0[)4Gd:$$6 (:%,4$_+0%8=*+9U9&4$)2??4 a 0%8%,(-98%,9C0?)48dZ%,4%,d=![0%9C0)4FF9&*>#C/%,()!5*_()(%, b % /%,4508d=0v0%,!5%)()(d48d=()!U%,9C0%&6[@G(%.F)4

Q 7Q

%d=)*+%FT:I \3%&?() b )4_0%1())*>)H]48d 9&*>#?%&C4F@f%d=)*+%&;T 46& b R¿ aNσ1/3 I b0%V=*+9n#?%&?%n>0%N*+=()! 9&4$)2??)48d %,* 46 >0%,%G%n?4?G* :%,>0%#A0(%2?0 /()?*+% 4_0%, #H6? b 0%*+4*+%,H9&49&%,4$C4rY?4?G* /%,Y0%1%,4$)%2?0 46u0%82?0 9,?%8_($9.()()![0r k%,*> gi67)()?%5()?4 3%&W$( :%,C()(9&49&%,4$C4 a 0%r"!.%r 0%N%,*> gi67)()?%N2?(2^ ξ ' aφ−3/4 @ #A0%,% φ = 3aNσ/R 0% /()?*+%

KC9,4IA46n0% f%&%3%,4%,d=!Nd0% #A0(%2?0VA0%,%G%+s

F sph3−4 '

R3

ξ3' (N σ)9/4R3/4 @ h I

a 0?%'#?%&4 K0%^%,4%,d=!8 =()6#H%,()(d)46?%%d=)*+%>TEF)45%d=)*+%$ b b #A0%,4P0%2?0V48d=()!n9&*>#?%&%&6k@ R¿ Lflat

I ?U)* N d=/%lZ9&%&9,86?%&9,)#?4 0%v9C0)4 9,?%l46 f%&%v%,4%,d=! )4

%d=)*+%& R+468T a 0%#?%&%,4 #H $rd/4?"!.%&6U G()()#k%;#?%&%,4$Y)4 h %&9,4ER3 h c #?#?:9C0&GY9C0+9&49./%A2?0%&()6 G4%.1gD0%,C

(/%,4$46V48()%,4:%39&4/()?4V*+%,0$6NA?())4%&6n)4 #?#%,467' a 0%F#??#=C%h %&9,4uTU d=/%N59.())48dv#?9,?%n 9&49./%N2?0%&^2$!k%4?)48d h c %&?()^C $$)48d)4$ 9&9&?4$\9&%,(-4\2%,(#H%&%,4E*+4*+%, 679,4ES(-%,C( 9,4H d=Cf%&69C0)4d=/%,4P)4 #?#%,467' 6 b #A0%,%^)F0)#A4v0F0%,!56?r4<()%,d=4?9.4$()!U0%9C0)4 K%&%+%,4%,d=!l46l0$<6?n4 %,4648d:%, 0% h c #?#?:9C0 4 #?#%,467' b #H%:d=/%0%%&?()12?C)4%&6V)4n0%39,!())4679.(_d:%&*+%,!N0?8d=0U>9.())48dE#?#?:9C0

/O? S

4v%d=)*+%& Rn46kT b 0%^#()!*+%,<9C0)4 %+48d=()!P%,9C0%&6 46P)30$#=)2?(%V%+0%_C%,(gi9&4%,4 %,(6l*+%,0$6 , R h b R=o b T=q . l%+4% r 0%+g67-(\67g49&%+V0%#?0%,9.(p?D9&%+@K6?%4%&6V2$! r = 0 I

a 0% )* Y0?%&9,4U1E6?%,%,*>)4% 0% f%&% %,4%,d=! F 2$!n()48d&G10%<*+4*+%,/()?*+%#fC9,4 φ(r) 46&K0%167)2??)4 g(r) 9C0)4Cgi%,46E*+4*+%,HC%B(Jgi9&4%,4$()! 4n0%9,(-C9.(]())*>)1d48d+%,9C0?)48d , R=o . b 0% K%&%%,4%,d=!nO0%3!7%,*L

F =

∫ R

0g(r0)

∫ r0

0

[3

2a2e(r, r0) + v

φ(r)

e(r, r0)

]drdr0 −

1

2

∫ R

0

4π(R− r)2va3

φ2(r)dr @ o:I

Q =R

#A0%,% g(r0)dr0 ^0%r4?* 2%,+9C0)4[%,46?>)4 0%n#?0%,9.(0%,()( C67) R − r046

#A670 dr0 46 e(r, r0) =∣∣ drdn

∣∣ >0%V($9.(1%'7%,44u3v9C0)4 #A0=% K%&%V%,46XE r0@f0% -2?(% n 6?%,4%&_0%V9,?%,4$+*+4*+%,gI a 0% +%,* 9&%&#46?_kP?* )48d=(%(gi9C0)4 K%&%%,4%,d=%&\)4_0%#%,4$-( kBTvφ(r) b 0%1%&9&46+%,* 9&%&9,\0% G9,0*+4*+%,1gD*+=4*_%, t4%,C9,410:%2%&%,486??2?(%(gi9&?4$%.6 a 0%4*_()"!J=4U9&467)4

∫ r0

0

dr

e(r, r0)= N @)Q.q:I

∫ R

04π(R− r)2φ(r)dr = 4πR2aNσ @)Q=QJI

'+0%14$?* 2%,;]*+4*+%,H)4E4% 9C0)4n46_)4E0% #A0(% !7%,* a 0%1*+%.4%,(6E9&49&%,4CgC486?%,%,*>)4%&682$! δF/δφ = 0 b #A0?9C0V!%,(6?

φ(r) =a3

4π(R− r)2∫ R

r

g(r0)

e(r, r0)dr0

@)Q.R:I

l%F#%,K*M0?9.(9,?(-4V2!_(#;>%,# , R=o . s #H%?#?#=%F00%F2?0n0%,d=0$ bxR b '7%&6 b 46#H% *>)4?)*>"!.%^0% fC%,%%,4%,d=! d=/%,452$! YW d@ o:I #A)05%&#%,9,Fr9C048d:%&)4 0% %,(6 φ

a 0)N!$%,(6?n D*>)()!u<#(%& φmin a 0%,4 #H%P6?%,%,*>)4%50?r4 $$4)#A4

#C*+%,%, x @ #A)0 0 ≤ x ≤ 1 Id2$!^*>)4?)*>"!J4N F [φmin, x]#A0?9C0_!%,(6?Y0%2?0_0%,d=0$ b

x∗ b 1%&W$?)())2?)?*

a 0%Yb"%&W$( gD)*+%Bc##%,"!<9.42%H#?#?()%&6 Gp0%\C%B(Jgi9&4%,4$ %,(6^#;O6?4%\#?% $()!)480% 9.%<^*+%,()4, R h. '0% K()()#A)48dE(#HE?*>#?410(6 s F0%,%314_6?%.6 !.4%9&4$C)4?)48d84n9C0)4l%,46 b #A0?9C0k9&4l2%*#A)%,4 g(r0) 6= 0 G 0 < r0/R < x 46 0%E9C0)4 %,46?^%+4<?46?%, %,44 s e(r0, r0) = 0

c * 0%_%&W$( gD)*+%rd=?*+%,4$ b #H%$4)# 0 0%_*+4*+%,9&49&%,4$C)=4Z#?(%_<#C2()9N46#H% $4)# 0%&K* 0%($9.(p%'%,448d0%39C0)4ks

φmin(r) = Φ

[A−B

( rR

)2]Θ(r − xR) @)Q.T:I

B =π2R3

8σN3a3v=10

9

(R

R∗

)3 @)Q($I

5 ') - ' 1 *%2) ) + #2&& - )V7 %2 -+) ) 5 F ) &-)* 1 %2 * A 5

Q =T

#A0%,%

R∗ =

(80

9π2

)1/3

a(vσ)1/3N @)Q :I

e(r, r0) =π

2N

√r20 − r2 @)Q :I

#A0%,% Θ V%,# f?49,4 b Φ = 3aNσ/R b B <2?C)4%&6 f* 0%E%&W/( gD)*+%r&d=?*+%,4$@ YW @)Q.q:IIA46 A A%,(-%&6V x 2$!v@ YW @)Q=QJII

(3− 3x+ x2)xA = 1 +10

9

(R

R∗

)3

x3(1− 3

2x+

3

5x2)

@)Q I

a 0%E*>)4?)*>"!J=4u10%K%&%r%,4%,d=! #%,>9C0)4 F (x) = F [φmin, x]/f#A)0 %&#%&9,^

9C048d:%&;)4 x 46?%,H0% 0?%&%F9&467)4 0 ≤ x ≤ 1 b φ(x−) ≥ 0 46 φ(0) = ΦA ≤ 1\)()(tC%&6E)4 c d=?% a 0? d=/%&\%1367)49,4r2%,(#H%&%,4r(#H 9.%&H6?%,#%,467)48d>4#A0%,0%, R A*_()(%,;(-d:%,A04 R∗ b #A0%,% R∗ @K%&% YW ]@)Q :II\;0%F9,)9.(]C67) G#A0?9C0V0%3#?0%BC%F%,4$tC%B()! ()(%&6n?#V2$!r0%2?0ks

• K R > R∗ @f%d=)*+%NR:I b 0%E*>t4t*?* K%&%_%,4%,d=! F $9&9,? G x∗ 6?%4%&6 0<0%?4?W$%3()?48d0%3%&W/4

x3(4

9x2 − 5

3x+

20

9

)=

(R∗

R

)3 @)Q h I

9C0V;9&467)4 f?(()(%&6v@% 0;0?;9&467)4n;%&W$?(%,4$; 0%F*+$0=4?0?)48dn0% #()!*+%,9&49&%,4$C4 0%%&6Cd:%^0%2?05, T=R . I %,*_)439&%,4$C(\%d=4ZAC67) R(1 − x∗) )4k0%E9&%,4$%, 0%_#?0%,% #A0?9C0Z9&4$C)4 #??%(/%,4$

• K R ≤ R∗ @f%d=)*+%<T:I0%F*>)4?)* ?* F $9&9,? K x = 1 b 0%,%F;4^%,*>#?"!E%d=446n0%39&49&%,4$C4P0%9&%,4$%, O0%3#?0%BC%F 4tC%

6H! =!)48d R K* ∞ aN1/2, 0%(t)*>)3 ()67)"!50% h c #?#?:9C0 b 0%G()()#A)48d9&%,4_%B*+%,d:%&

Q

0.20

0

0.4 0.6 0.8 1

50

100

150

200

250

300

x

F~

R=2R*

R=0.8R*

R=R*

W@0 ,D\ 0 #% +\&,%&1.6Z!5&U'2#, F (x) \0 #8+\3 &&,$0 -)E@U R/R∗

N = 1000 [ σ = 0.03 X R∗/a = 300.07) 98002 02O02/! R = 2R∗ X , 2 `

R = R∗ R = 0.8R∗ X , 3 `< #% 3@;#%+3!_0550 #% '?.-; '+"!5,%302,e.,!$#68 '()&E0@0 , X^@0 #%&H@0 '())76[ '+,%3E0 , ?[ φ(x−) ≥ 0 B+V8.E-) &,$0M02 x ≤ x∗ `<

R > R∗

4 %d=)*+%;R b 0%2?0 0%,d=0$ x∗ d=:%,4 2$! W @)Q h Ip46 0%*+4*+%, :()?*+% fC9,4@ W @)Q.T:IIA)*>#?()%&)4$+@ c d=?% :I

φ(r) = ΦB

[x∗2 −

( rR

)2] @)Q.o:I

.8 &-& 4 "!')2# + ! +43&##%2 ) ;σ,R) 5 8

2

# "! ) %$ *L ≤ (2

√2/π√3)v1/2N

$ ) # !Rσ ≤ σ1 = (

√6/5π)v1/2.8

3

σ ≤ σ1 & "! ) +

R1 ≤ R ≤ R∗$

R1 =

(4√5/3π)v1/2N [cos( 1

3arctan

√(σ2

σ)2 − 1) −

√3 sin( 1

3arctan

√(σ2

σ)2 − 1)]

σ2 = (8

√5/27π)v1/2 5

σ > σ1 &# ( )* + %2%2 +%2 ,) # # !' ) ( )* / +# 1 )*+- 5' ($ ) %') %'# %)" 4 ) !)*%' $ + ) &##%2 )6;

σ,R)-)* %2 '*#%2 ) ,! %2 *$ ,+

v,# %2-- #

σ; %'# v2 ¿ σ ¿ 1) 5

Q

YW$4<@ tQ.R:IA46l@)Q.o:I\!%,(6N0%9C0)4V%,46n67)2??4 #A)%,4V)4n?4?)O9C0)4A#%,?4?)(%,48d=0 g(t) = g(r0)a

2/(4πRσ) s

g(t) = Bx∗2t[3(2− 3x∗ + 2x∗2)

√1− t2 − 8x∗2(1− t2)3/2

+ 6x∗ arg tanh√1− t2 − 9x∗t2 arg tanh

√1− t2

] @ R=q:I

#A0%,% t = r0/(x∗R)

a 0%_67t2?4Z9C0)4Z%,46? 9.4Z(V2%_%'7#?%&%&6l)4k%,*+14$?* 2%,9C0)4_%,46?d#%,'?4?) /()?*+%s ρ(t) = g(t)

3(1−tx∗)2@f)4>?4?)Y9C0)4'#%,'?4?) /()?*+% b

Φ/N I K 0 < t < 1 "!#?9.(?#?22?)())"!^6?%,4)"! K K%&%;9C0)4+%,46?d'0)#A4>)4 c d=?%#

0.2 0.4 0.6 0.8 1

0.25

0.5

0.75

1

1.25

1.5

1.75

r/R

f F/

R=2R*

R=0.8R*

R=R*

,%'+&,$0 /@0 , !$Q8) Z\0 #%R9;#=<MX?` 184"!$#%&]X R = 2 R∗ [ ` 0 #% !$Q8) MB !_/C98) ',%3 0 #%'+,%'+&,$0 /@0 ?,O-,8B&#%M@05+fQ8,8E023B?0",%'+ x∗ 0 #%.E'+ 0 #%f"!$#%&:XB+ VD<=X < ``<LX"9`fFU0 R = R∗ [ 0 #%'+,%'+&,$0 /@0 ?, -+,8B#%A!$+'&B&)76@0f0 #%'+&,$02&U*0 #% "!$#%&< X '`SPH))=!$#%& X R = 0.8 R∗ [T+` 0 .,%'(@02+3N!_/C98) '!$Q8) E0 # Q8,8E02U'+,%'+&,$0 /@0 ?,]@0f0 #%U'+&,$02& 0 #% "!$#%& XB+ V8< X < 1``<

c * 0%&% b #;%%.67)()!+6?%,/%0%#f%&%1%,4%,d=!+#%,\9C0)4 F2 = F (x∗) d=:%,4E2$!YW @ o:Is

F2 =9

10

(π2

4

)1/3

N(vσ)2/3 ψ

(R∗

R

)@ R7QJI

Q

0

0

0.2 0.4 0.6 0.8 1

5

10

15

20

r/R

r

0.75 R*

1.27 R*

$@0 ,Of0 #%,%H) +3]'2#, &,%3e38&,D&E06 3B0 "9C80 , ρ -;?< r0/R >0

3 &&,$05/3 [ R = 1.27R∗ X `,%3 R = 0.75R∗ X+` X N = 1000 [ f = 600 `<

#A0%,%

ψ(y) =10. 41/3

3

1

(3− 3x∗ + x∗2)

[1

4

(9

10

)1/3 y

x∗+1

3

(25

6

)1/3 x∗2

y2(1− 3

2x∗ +

3

5x∗2)

− 1

189

(25

6

)1/3 x∗5

y5(28− 63x∗ + 201

4x∗2 − 17x∗3 + 12

5x∗4)

]@ R=R:I

4>0%%,4$)%%d=t*+% R b %&W/45@)Q h Id0)#'0Y0%A4*_()"!.%&6_2?0+0%,d=0$ x∗ )49,%.%&#A0%,4k0%+9,? =?% R∗/R )49,%.%& b ?4$)(Y)3%.9C0%& x∗ = 1 K R = R∗ @ c d=?% I 40%,&#H6? b 0%12?0E0%,d=0$\)49,%.%&+#0%,4E0%1)4$%,G9&% \9,? :%&6_*+%48d=()! b (48d+0%,%n+*+% f%&%V(/%,4$>)4[0%n9&%,4$C%B 4X0%VC*+% #;.! b )_9.4[2%V0)#A4[0+0%4*_()"!.%&6 f%&%%,4%,d=! ψ @ YW S@ R=R:IIY)49,%.%&H*+44?9.(()!r;9&4C4$*d=CK)48d^6?%,4)"!#A0%,480%9,? =?%3)49,%.%& @ c d=?% h Ia 0%N=*+9r#?%&?%E2(-49&%&>0%N%,(-9r%&)48d K9&%N)49&%r0%r#(t!*+%,^2?0Z

%&W/?)())2?)?* ?46?%,14+%'7%,4(p9&*>#?%&4 a 0?%.67)()!V%'#?(-)4, R=o . 0% D9,010%*+4*+%, /()?*+% fC9,4 =4?0%&O*+$0?()!p0%H?%,d%&6Cd:%H0%2?0N@K%&W$4n@)Q h I%&W$?=(%,4$ φ(x∗) = 0 I

Q

0.50

0

1 1.5 2

0.2

0.4

0.6

0.8

1

R*/R

x*

0.250 0.5 0.75 1 1.25 1.5 1.75 2

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

R*/R

L/Lflat

@0 ,H 0 #% )E80 , x∗ $+V8@0 ,eX <+` - 0 #%L,%H) +3>'?.-+@0. R∗/R < #% ,T?0f#% 0 #%H-+@0 ,0I#%\,%H) +39;# #%& #$0 L/Lflat

- 0 #%\,%H) +3'?.-+@0.U 0 #% "!$#%& R∗/R <

! !.546PW/%&%, T=T . %&%,* 80:%r(8%&6Z4 h c #?#?:9C0 K #H()(%,4Z9&49./%2?0%&)4_0?%d=)*+% a 0%,!$46+0%1C*+%1%&W$4r W @)Q h I KY0%A2?0_0%,d=0$ b 2??0%,) K%&%^%,4%,d=!U67!a%,%,4$ f* 0%^4%Nd=/%,4P0%,%k%^6?r4?46?%,C46 #0$!50?C;d:% 4<0%Y())*>)O!.%, 9,? ?% b R→∞ b 0%Y2?0<0%,d=0$ L = x∗R d=:%,4 2$! YW $@)Q h I

9.4V2%3%'#?%&C%&6V3s

Lflat =

(9

20

)1/3

R∗ =

(4

π2

)1/3

a(vσ)1/3N @ R=T:I

#A0?9C0VA0%%&?() 8)()4%, b )%,4U46 %& , R=o . 5%& :%, b #H%F0/% ψ(0) = 1 46n0%%,4%,d=! YW @ R7QJIH2%&9&*+%&

F2flat =9

10

(π2

4

)1/3

N(vσ)2/3 @ R$I

#A0?9C0V (^0%39,(-9.(]%&?()

Q h

0.50

0

1 1.5 2

0.5

1

1.5

2

2.5

R*/R

Yos

Y

Yel

h $@0 ,KL0 #% '() +3 020")D+ &,%&1.6 !5 '2#, ψ XB++ V8<$X < `` ,: 2,%3 3 -; 0 #%5,%H) +34'?.-+@0. R∗/R <;F )B !$&,$02+3H0 #% &);0 'UX ψel

` ,%3N?0 'X ψos

` !_0I4X #%& ψ = ψos + ψel`<

2F)48d f?0%, b #H%N9.4[%'#46 0?^%&?() K /%,! *_()(A9,? =?%&YW$4 @)Q h I!%,(6?A0%<!*>#?9 =()%3O0%2?0V0%,d=0$

L ' Lflat

[1 +

1

4

Lflat

R+29

240

(Lflat

R

)2]

@ R :I

%r0_5)*>#?(%N9.())48dv*+%,0$6X6?$%&>4^!)%,(6 0%N9&%&9,=()%G0?+!$*>#?9

Q =o

)49,%.%)4V0%2?0V0%,d=0$l%39.48(C>%'7#46n0%K%&%%,4%,d=! G*_()(9,? ?%&

F2 = F2flat

[1 +

5

12

Lflat

R+73

336

(Lflat

R

)2]

@ R :I

k%>2%, :%>00?39&4%,4$7#A)0l0%^%&(tC3 U)()4%,<46 )%,48, T . #A0 K?46G<0%_9&4/%' d:%&*+%,!l0%_g*+%_6?% /%,(#?*+%,4$2$!P%,#?(-9,)48d R 2$! −R a 0? <2%&9.% b^%'7#?(-)4%&6Z)4 , T . b )4 0%N9,4 :%'Z9.% b 0%r6?%.6 !.4%r!%,(6?4%d=()d=)2?(%N%,*+$K^*_()(9,? =?%& 4_9&49,()4 b KY0?%d=4 2 b 0% h c 9.(9,?(-4;0)# :#;%&>)49,%.%1a0%2?0

0%,d=0$#A0%,4X0%Nt4%,G9&%N>9,? /%,6 *+%V46Z*+C% b #A0?9C0X)4/(/%&+4 )49,%.%n 20%,(-9<46V=*+93%,4%,d=!

aN1/2 ≤ R ≤ R∗

45%d=)*+%+T b 0%2?050%,d=0$ L '7%&6v R @ x = 1 I a 0%*+4*+%, /()?*+%KC9,42%&9&*+%&@ c d=?%:I

φ(r) = φ(R) + ΦB

[1−

( rR

)2] @ R I

[F !#"$%&'(&*)+&,(.-$%&/,0,1,23 "4&,5!6%&.%& 7 %&.89: 2;!<%&, $=,(8%&1>?

2$=@&,.$/(>?%&A BCED?2&F!G%:"$%F/(@ H F . &>?'(I FA /1 &(;&,J18,2FE%&A #!&,

-(%&/(,5,(8,2 $=@&,."$%& ?'(,%&F%: 8"$/=A 5-KJ>; "(F- >?>? #&,FI1EL M"$%N!O%&JE%& QP

F ∼= Φ1/4L2

a2N+NΦ5/4

$=,E%&ΦR%&1 &A"?&J&,=-(%&/,?,(8,2

L&,$%&8/(8,

Φ = 4πR2aNσ/V) $=,%&

V = 4/3π[R3− (R−L)3]+&,

1/(>?S2/'(1"5- ;&,S-(%&/(,*H4F=T(%&9I%:"(%A)(&,(S-(%&/,3,,2I%&>?2=A U-K $N%&@&. L/Lflat − 1 ∼=

13aNσ1/3/R

H%5,;%&/@3%%&A)N 3 V-KWV- 48>?' %&8 $=@&,XYC/ &8 ; Z H Z ? NEH%8["("\)G&,(>?8(8>?%82%: &8?+ />?"]&J-KN/(@!<%&>^_&,GA 1(#"(E%&1'$&8*H ' (8)&,N82%: &8?(A %&,FA"$8S!B&,F-(%&/,\) $=,:,52%&-/(&N1,2& 3>?%&F&#&,F'(%&CN%)`=!a/$%& &/(%&)$1N%&9&>; &"\H5,M =1&,( %&A 5!O%=&,(F%&9&>; &A";!b &%

13

9& 8"3!14

H45,F%%&E&15&#&,FA 1(c1 $Φ ∼= σ2/3)M =T%&9=%:"$%

Φ ∼= σ2/3 +2

3

(aNσ

R

)

H &S&, N&,I"(8'(>?2NI 11"313&,(F%&8>?#Zd&,M ==!O% RÀ aNσ1/3 ∼= R∗H

eEf c%&8%#&,c81"g%&/@]!=9: -(1@ != $>?>?E%&1 g'K8$>?%#"(%: &A">?>_-$%: (_/"(%JhM/(&/ BG&8(#!=1 %& $N A8&, ;i ] 3>; QTM1"g N j U H ' $R%A)\!<%]/(%!< : 2]11)B&,c%: !O&(W# &,E%&>?$" $M >??YC/(1-$%&1/(>UH 5,C/)klE%: b)a&,;%: !O&(g"(( 1]_ $>?>?E%&1 C Z HW5,]L$'1 $=, 3'K $>?%="(%: &"5>?>]-(%: R!O%&>m'K82: (8/(58H

Q =q

#A0%,% Φ(R) A0% /()?*+% fC9,45A0%39&%,4$%,1d0%3#?0%,%d=:%,4V2$!Ps

φ(R) = Φ

[1−

(R

R∗

)3]

@ R h I

#A0?9C0l)49,%.%&7#A0%,4 R 6?%&9,%.%& a 0$ b 0%^*+4*+%, :()?*+% fC9,4l%,46?3N2%&9&*+%*+%3?4?K* 0%FC67) R 6?%&9,%.C%&6 a 0%39C0)48%,46V67)2??48<@ t = r/R I

g(t) = vt

[−6(1− 9

10B

)√1− t2 − 8B(1− t2)3/2 @ R=o:I

+ 6

(1 +

1

10B

)arg tanh

√1− t2 − 9Bt2 arg tanh

√1− t2

]@ T=q:I

;A0)#A4n)4 c d=?% b 0%F*_ A4$?* 2%,1d9C0)4V%,46?A#%,A?4?)1?G9&%3%F($9.%&680%3%&6Cd:%3d0%2?0 a 0% f%&%3%,4%,d=!r#%,19C0)4V d=/%,4V2$!5s

F3 =

(9π2

10

)1/3

(vσ)2/3N

[3

4

R∗

R+1

6

(R

R∗

)2

− 13

756

(R

R∗

)5]

@ T7QJI

P0%,*_9.()()! b 0? K%&%+%,4%,d=! F3 f !.%,4Z F (x = 1) 2%&9.C%+0%_%&W/?)())2?)?* C%

x = x∗ > 1 F?\%.9C0 b x = 1 2%,)48d8_#?0$!79.(d2?46P\0%#?2?(%,* a 0?F)4F)*>)(-_0%9.% 2?0U?46?%,9,*>#%&4X@G_#?%&%<%'?%,%&6P)%&6Cd:%JI<s0%<9&49&%,4$C=4U#?(%<0 >?49.%&68#C2()90#% 46V) ()% 10%<%&6Cd:%<'0%2?0 b YW '@ R h I b )49,%.%& #A0%,4v0%>9&*>#?%&4v48d:%, , T . F)F0F2%&%,4l0)#A42$! Ut()4%, b 0?A*+%.400%,%39.4V2%F4bi)4$%,67d=)C4?c3'9C0)4 r = R , T=R .

9&4C4$fd=CK)48d+6?%,4)"! b F3 @ W @ T7QJIIH^4Dd=(t!r)49,%.)48d&f?49,48 R∗/R@ c d=?% h I 9.%K?(34()!N 0?n%,4%,d=! 0)#N0 K R ¿ R∗ b 0%5(-r%,*+4%d=()d=)2?(% b )_9&*+%& K* 0%n#C2()9V4?C%VF0%V*+4*+%,E9&49&%,4$C4 4[0%g*+%:#;.! b f* YW ]@ R h I #H%39.4V%.42?()!n9&46?%,100%*+4*+%, 9&49&%,4$C4U9&4C4$146N%&W$(S Φ = 3Naσ/R a 0%,%G% b 0% K%&%F%,4%,d=!r9.4n2%F%&%,4V;0%F(% :%,(a>)*>#?(% c (!r0%&!

F3 ' vaN2σ

R+

R2

Na2@ T=R:I

#A0%,%F0% C;%,* ;0%F=*+9F#?%&?%>@G1<*+%.4Cg%B(6n(% /%,(jIH46r0%%&9&46N4%F0%%,(-9,t !EO4N6?%.(a9C0)4 c R ∼= R∗ b 0%&%(#H%,4%,d=%&A2(-49&% b 2??&#A0%,4 R 6?%&9,%.%&K?0%, b 0%<%'?9,()6?%&6 :()?*+%3)4$%,C9,4F48d=()!V6?*>)4%46V0%<%,4%,d=!n#?C9,9.()()!%&W$(HU0%N=*+9r#?%&?%P@ c d=?% h I a 0%r#)4$^ 0^0%&%E(#HP%B4%B&d=)%&>6?542(-49&%32%&9.%0%F2?0n0%,d=0$1 d:%&*+%,9.()()!V9&4C)4%&6

Q 7Q

4l0%+#?% %&9,4 b #H%+%&6 nC%,(gi9&4%,4$ %,(6 #?#?:9C0 867!50%E9&4K1g*_4 n9&49./%+2?0 h 9C0Z4 #?#?:9C0Z6?%&9,)2%&<20l0%E:%,C;d:%_67)2??4ZK%&% 9C0)4P%,46?4680%/%,C;d:% #=)4PH4$!n*+4*+%, b 9&46?t)4(t(!80%<#=t)4v0%F%,46@gD*+4*+%,p0%9C0)4N4#A0?9C0N)H#%,C)4;_4)#A4 H_$4)#A4n9,(-9.()()!_)4r(#H/%,4 a 0%>*+%.4Cg%,(6 /%,4k9C0 30<%'#=C%&6v)4v0%>#?% $<C%&9,4l#? 6?%& *+%$#A0<%,4%,<%'#%&4 GF0%^2?0 f%&%>%,4%,d=! b #A0?9C0 =%&()b/% φ2 a 0? #)4$)49r0%*+%.4Cg%,(6l##? ')*_4 465#?%&%,43(r)450% c (!5#?#?:9C0 9&%&9,F%'#?C%,C4 G10% K%&%<%,4%,d=!89.4U2% 2?C)4%&6U)48dr9.())48dN9&49&%B#4, T . 46 =g%&()b/% φ9/4 :#; %&6P2$!56?%32F%,4?4%&)4kN9.())48d h c #?#?:9C0 K0%^#?2?(%,*Y#()!*+%,F6?#?4 , T . Ut()4%, b )%,4v46 %&F#?#?()%&6U) )456?%,#HAV? 0%9&4G*_4[ b #H()(%,4 #()!*+%,^2?0Z)4 d:$/6 C(:%,4$ , T=R . a 0%r%&?())48dP2?09&49&%,4$C4Z#?(%_2?C)4%&6 K%,3*>)4?)*>"!J4 <()d=0$()!v67!S%,%,4$ K* 0<2?C)4%&60?8d=0l0%^*+%.4Cg%,(6 h c 9.(9,?(-4 a 0?F)4l9&4$C #A)0v0%>d=)4( (%'46?%,1g6?%:2F%,4?4%&*+$6?%,( K0%12?03#A0%,%120E0%1*+%.4Cg%,(6N46_0%19.())48d G*+\]0% K%&%%,4%,d=!E!$%,(6r0% :%,!EC*+%2?0n9&4G*_4 b #A0=%F9&49&%,4$C4V#?(%FH?4?G* 40% h c #?#?:9C0 b 0%;44CgD4?K*>)"!+0%9&49&%,4$C4E'0%d=)4 G'0%()d=0$67!a%(g%,49&%;2%,(#H%&%,4>*+%.4Cg%,(6+46^C9.(t)48d 40?'%&9,4 b #;%\#?%&C%,4$d0%&%;9.())48dF$( GO0%=;%d=)*+%&p0%F#?0%,9.(]9&49./%F2?0 c A%,9C0%&6n9C0)4<@f%d=)*+%&1R^46nT:I b #;%)*>#?()!E9.(9,?(-%F0%FC9.()t4Dd$f%&%F%,4%,d=!&K* 0%2?0n9&4K*_486?%,%,*>)4%&6 K* 0%*+%.4Cg%,(6 h c #?#?:9C0r%'7#=%&6+)4+0%#?% \%&9,4 ()08d=0_)Y4HYd:H0%9.(9,?(-4rS%G%,%,49&% , T=R . b )0?(6+2%1_9,%,4$H#?#? '7)*_4NY(%.Y)4+%d=)*+% T#A0%,%0%^9&49&%,4$C4v#?(%F%&C%,4$-()()!V?4 G* b 1#H% *+%,44%&6 4v4$!89.C% b0% ()%^H0%f%&%^%,4%,d=!P2?C)4%&6P)4P0?:#A.!8)4P%d=)*+%&<RN46lT3#A)()(d2%)*>#? /%&6l9&*>#%&6V>0%*+%.4Cg%,(6 =(t%2?C)4%&6V)4V0%#?% $1%&9,4

k%1(36?%&9,)2%*+% f?()()!^9C0)4&9,4;?46+0%,)H/%,C;d:%9&4G*_4Hd=/%B4

O7C1(N&,=EL('K2&a!<%+&,N8/(>?!O%: E&1φc&,N1 9&F "cc&,N8>?&N&%&>?G!`&,N!6%&I(%&

%&N8,2&_,,%G_A S&,M _c>? 4GiT1"\)&,(-(%&/(,]&"(k&J A6"]'(%&2&1(J ],(1,c82%: &8%&13Pak%: &,E%G"$19%&-(/(&A"d>?%&+(2H%L( >?'(1)#&,R' %: -K1'(%&T'$%&2&A"d1d%&!<%& jZ )&,FA ?'$%&T1S=18,2&3>?%& hK &A"3&,S,(8,

φ%&8.(A %N&, $ \&,M 3&,(S>?A G T1";'(%&T8)

"3 = #8>?'K(& &8\)$@=N1,2& >?%& 8(2%: &"1&,(F $ 8/(>?I!O%: E&15%&885 %N&,(FA"(S!&,S-$%&/,\H

Q =R

2$!r0% h c %.*+%,4$ 44%&689C0)4<@f%d=)*+% $IH9.4?42%F%.%&6n2$!r0% h c 0%&!Ps#H%36?%&9,)2%30%9C0)4U9&4K*_446V9.(9,?(-%30%39.())48d K%&%%,4%,d=!

#

a 0%39.())48d K*Md0%K%&%%,4%,d=!d=:%,4V)4V%K%,%,49&%, T=R . 3s

F sc ' 1

2

∫ R

0

4π(R− r)2a3

φ9/4(r)dr +3

2a2

∫ R

0dr0 g(r0)

∫ r0

0e(r, r0)φ

1/4(r)dr @ T=T:I

#A0%,%F0% G9, φ9/4 )4r0%F=*+9F%,*L9&*+%& f* 0% kBT #%,;2?(2 4C ! b #A0?)(%F0%

φ1/4 D9,Y)4>0%%,(9;%,* d=)4%&)4>0%9&%&9,4_F0%;?4?#%,?2%&6E9&4K*_40%;9C0)4#A0?9C0+%; #H()(%,4^2%,()#X0%H(%B4Dd=0>9.(% ξ(r) = aφ−3/4 a 0%A?0')4$%,#?%(g%&6r0%%,(-9%,*LA<#%,?248p%.9C0r2?(2np"!.% ξ s$);9&%&#46?; kT (∆ξ/ξ)2#%,2?(2 b #A0%,% ∆ξ 0%^/%,C;d:% 2?(2P%,(48d/4 6H? 0?%,* 9.452%<)4$%,#?%,%&6P)4P*+%+)4$?)/% #;.! b )48d5N*+%>#?%&9,%_6?%&9,)#?4Z;0%>($9.(\9C0)4 9&4K*_4 *+%,4$4%&6 2?%!v2$!P0%r?0 b *_()((%,48d=0Z9.(%& b 9C0)4^%+()b/%>)4 nC%,*> gi67)()?%C()?4Z@K #H()(%,482%,()# ξ 46 2-4U2 /% ξ I b #0?)()%<0%,!n%<%,9C0%&65(-d:%3(%,48d=0C9.(%& 480%, #HC6? b 0%$1*+%.4UCW/%367C49&%32%,(#H%&%,4U(#H>*+4*>%, n 46 n′ (48d

0%39C0)4Z@K%&% c d=?% o:IH d=/%,4n2!5s√(r − r′)2 ∝ (n− n′)3/5 (

√(r − r′)2 < ξ = aφ−3/4)

∝ (n− n′)1/2 (ξ <

√(r − r′)2 < Λ)

∝ n− n′ (

√(r − r′)2 > Λ) @ T$I

#A0%,%E0%_)4$%,*+%&67-%E(%,48d=0 9.(% Λ b #A0%,%_0? C4)4 $9&9,? b 9.()(%&6k0%E%,(-)92?(2 @KC%&%F%G%,%,49&%& , T h b T=o . IFs

Λ(r, r0) ' a2∣∣∣∣dr

dn

∣∣∣∣−1

φ−1/4 @ T :I

;0?A(%,48d=089.(% b 0%F#()!$*+%,19C0)4n;414n6?%.(aC46?* #;(bn4$!E*+%>s)4%.6 b )0Y4<9C09&%2??Y*+ /% G #;6_\% :%,!>%,#U@K%'#4%,4$QJI/%B49,% b 0%9C0)4_Y*_)4?()!+())4%.\)48d S0%&%%,(-9A2?(2\%,9C0%&6E(48d30%AC67-(67)%&9,4 0%)*>#C4$#)4$2%,)48d 0\0%&%2?(2H% 6?%4%&6r(48d 4% 67)%&9,4N46_0)4_0%10%,(#H 67)%&9,4 b2?(21'67!S%,%,4$ 9C0)41 :%,(-#

Q =T

4N0%%,(-9F%,*Md%&W$4k@ o:I b 0%6?%d=%&%3d%,9C0?)48d e(r, r0) * A2%F%,#?(-9&%&62$!r4n%!a%&9,/%F6?%d=%&%O%,9C0?)48d eeff(r, r0) = a2/Λ(r, r0)

b #A0?9C0n!%,(6?;0% G9, φ1/4 )4%&W$4 @ T=T:I

%A0Y0%1=*+9A2?(2E"!.% ξ(r) 6?%,#%,46?\4?()!^4+0%#=)4 r b #0%,%.Y0%"!.%A4N%,(-9 2?(2 Λ(r, r0) 4n3#9,?(-A9C0)4n(C 6?%,#%,46?;4r0%1($9.(S6%(d=C%,%FC%BC9C0?)48dd019C0)4 b b 4n0%F#=t)4 r0 d) K%&%%,46 4#?)49,)#?(% b 0%;9C0)4>9&4G*_4'0?(62%H2?C)4%&6^0?8d=0>*>)4?)*>"!J4+%&W/g

4 @ T=T:I b ^)4 %G%,%,49&% , T=R . /%,% b 0)#H% :%, b #H%N)*>#?()! %E0% :()?*+% fC9,4 φ(r)@K%&W$4X@)Q.o:I1>@ R II b 0% f%&% %,46567)2??4 g(t) @K%&W$4X@ R=q:I1^@ R=o:II b 460%6?%d=%&%3d%,9C0?)48d_6%B /%&6n0?8d=0V0%F*+%.4Cg%,(6V*+%,0$6V6?% :%,(#%&6n)4n0%F#?% $%&9,4 @ W )Q :Ia 0%&K%&%E%,4%,d=!k0%E2?0Z)4 %d=)*+%&>RU46 T89.4Z2%E9&*>#??%&6Z4$?*+%,9.()()! K*

%&W$4 @ T=T:IA(48d_0%()t4%, #H%0/%F)4679.%&6 /%,% b G1)*>#?()9,)"! b #H% K$9,1480%3(t)*>)'#A0%,% R¿ R∗ @ #H%,()()4V%d=)*+% T:I a 0% f%&%3%,4667)2??4 d=/%,4V2$!r%&W$4 @ R=o:I !.%,+6?%,)4 R/R∗ 0%,4V2%&9,*+%& s

g(t) ' 6 targ tanh

√1− t2 −

√1− t2

@ T :I

4 0%HC*+%&#;.! b O0?d6?%, b φ(r) d9&4C4$#A0%,% φ(r) = Φ = 3aNσ/R )00%&%\(#H#?#? '7)*_4 b 0%;C(K%&%A%,4%,d=! '%&W$(?F0%Cd(#HF%,*+Y YW ?@ T7QJIO* ?())#?()%&62$! φ1/4 s

F ' 3

2

aN2vσ

Rφ1/4 +

3π2

80

R2

Na2φ1/4

'(3

2

)1/4(9π210

)5/12

σ5/6N

[3

4

(R∗

R

)5/4

+1

6

(R

R∗

)7/4]

@ T I

#A0%,%0%6?*>)44d9&4$)2??4O=*+9@ p%,*EI a 0%\C%&9&46<%,* 0%\%,(-9%,*@K9&*>#??%&6 2 /% G R/R∗ → 0 I a 0%>4%'744!.%,89&4$)2??4 f* 0%+=*+9_460%3%,(-9%,*+)4V0% R/R∗ @K B IH%'7#44U%3*_()(%,2$!nD9,'6?%, (R/R∗)3 c )4()()! b )456?%, 4d:%,)*>#?()!n0%<9.())48dr6?%,#%,46?%,49&%Y%&#%&9,:%,()!V0%<=*+946

%,(-9 %,4%,d=!>)4_%d=)*+%&;R<46ET_@ #A)0?)4r?4a$4)#A4E4$?*+%,9.(S9&$% _9,%,4$;]6?%,\?4?)"!?I b )A#=C)2?(% /%&%,#84?()!r0%39.())48d_6?%,#%,46?%,49&% d2?(2<s ξ ' aΦ−3/4 46 Λ ' NΦ−1/4/R46Xld=4X0%V%,4%,d=! kBT #%,+2?(2 a 0$ b )4[%d=)*+%8R b #H%n%&9& /%,>0% /%,!Z)*>#?(%%&)*_%>0%2(-49&%+9C0?% :%&652$!80%^%,9C0%&6P9C0)4#?%&%,4$%&6P2$! (%'?46%B, .

Q

46v6?%2F%,4?4%&, Q . a 0%>9&*>#?%&%&6P2?0P)4P%d=)*+%_TEF0%,4k4(.d:V 2?0)4n#?%&%,49&%'%,*> gi67)()?%3()?)48p*+4*+%, /()?*+%fC9,4 Φb = Φ

@K%&% c d=?% T:Ia 0%H#C*+%,%, R O%,(-%&6F )(7%'7%,4(?=*+9H#?%&?% Π ∼ Φ9/4

b#A0?9C0>9&*>#?%&

0%d=CK%&6n(-.!:%, , T h b Q .

G

a 0% 4?C(S"!.%p4 N gD*+4*+%,A9C0)4r)4n<%,*> gi67)()?%F()?4n/()?*+%fC9,4 φ;4)#A48>2% R(φ) ' aN1/2φ−1/8 = ξ(N/g)1/2 #A0%,% ξ = ag3/5 = aφ−3/4 0%3=*+92?(2 "!.%86?%4%&6 %&(t%,E)4u0?E%&9,4 4u%d=)*+%&rRk46 T b #A0%,4 0%8#?0%,%VC67)_6?%&9,%.C%&6 b 0%F9C0)4;%,*_)4n%,9C0%&6VH(48d>H0%,tA%,46@gDC;gi%B46n%'%,4)4NH(-d:%,;040%1%,(-92?(2E"!.% Λ a 0?Y2%.6? #A4#A0%,4_0%AC67)\S0%#?0%,%%.9C0%&0%4?C("!.%d0%9C0)4 b b R = R(φ) <s

R ∼= aσ−1/7N3/7 @ T h I

c *_()(%,\#?0%,%Ag67)p@ #A0)9C0r9&%&#46?3%d=)*+%# b C%&% c d=% $I b 0%fd=CK%&6+9C0t4%F9&44%&6 5% #?%&9,%,()!5@K%&% c d=?% o:I b 0%9C0)4% #H()(%,4V;*_()((%,48d=0n9.(%& b23-4p(%B4Dd=09.(%&2%,(#H%&%,4 ξ 46 R b 46 9&44%&6r@ !.%, %'7#4%,4$gI G(-d:%,p4$?* 2%,*+4*+%, a 0%\C4)4^2%,(#H%&%,4^0% 2-4>46 0%H9&44%&6^C9&d9&%&#46? G = φ1/4R2/a2 *+4*+%, AC9.(%&A(-d:%,04 G *+4*+%, b 2%&9.%3O0%3#?0%,9.(2?46! b 0%59C0t4 0r4 9C09&%52??N C.!ut4 0%PC*+%5%d=4 #9&%5)4%.6 #%,G*>)48dl8C46?* #A(bus=d=(2()()! b )^U)48dP N/G /%,(-#?#?)48dP2?(2^ "!.% R

)48dZ0%PC*+%vd=?*+%,4$8N%.()%,N)4 0?N%&9,4 b #;%59&49,()6?%P0N0%P9&*>#?%&49&4$)2?48>0%3 :%,C()(%,4%,d=!n9.4V2%%&)*_%&6U3s

F4el 'N

G' N3/4σ−1/4(R/a)−7/4 @ T=o:I

a 0? b 0)#H% /%, b )14?()!V>9&%&9,4U>0% f%&%3%,4%,d=! b #A0=%*_)4U9&4$)2??4U1=*+946n d=/%,4V2$!r%&W$4 @ h I

,##8>?$=,M ]>?1 %#&5&,(?81 '"9%&/(&/(%&c!= 5E, 5'K2%#82A) $=,:, 1S /1 -K&,.-K $^ M". -K

ξ)K ".8(TA"3!<%SC/(>_-KE%I!+>?>;E%&I1 %&%S&,

GH U&,dA ]!G&,(#8-/(

15'K2% 182A)(&,SL$'(%&1=!O%ξ ' aφ−1

M"G ' (R/a)2

"$ )`%I8>?%$=, =!O%&>m&,S@&/M &8U9&/M"$A",%&8H

Q

root mean squaredistance

number of monomersa

|dr/dn|

x= Fa-3/4

L

3/5

1/2

1

root mean squaredistance

number of monomers

a

x= Fa-3/4

R

3/5

1/2

0

G

(a) (b)

o (, +VC]3B0",%'+O98?0 +&, 0 ^,%&&R+ 0 #%R/^'2#, X/3)'+\!5,%&,$0J`+[%;MU1.,%'?0 ,] 0 #%f,$.98&_,%&& 98?0 +&, 0 #%& X) +@E0 #8: ' '() `<X?`",W [ 4KE-;&,G-.&/?8Z38+ 0 ?02'2#8,C |dr/dn| ,%3 -;)E.:J/'?0 , φXE0 #8,H),%`+[$0 #%0 #8 'N),% B 0 #%*'(),C!$ '?0.<FU0 ) ,C 3B0", '+[$0 #%M38!5&,%38&,%'+ B ),%([ #8 '2# %+'?0IN0 #%I'?00 #@0 0 #%>'2#, B0 ?02'2#%+3 X !5,%&,$0 (`<F0 H)) & 3B0",%'++[ 0 #%'2#, 0"C0 #%Z'+,8H@0 , E0 .) 3^#-.N, &:3)E802 )E80 , \-.)E.NJ/'?0 ,φ E0 B )) &, @0*&H))L) &,CT0 #^+'() &X !5,%&,$0 +`\,%3KE0 BMP&, @0 ) &,CT0 # '() )18&40 #, ξ = aφ−3/4 X !5,%&,$0 `<*X"9`, [A0 #%:'2#,D\:'+\!$&+3 0 #%S !8!%;'2#38PU,%0 !8!$)76O,%3O0 #% '(),C]!8!$;'2#^6C &) 340 # U)) ,CZ!$ '?0.< \E-.&, -.)E.fJ/'?0 , φ [$)) '2#,D 98&#-.)8< #%16e )) &, @05H)) ) &,CT0 #e'() X !5,%&,$0 +`f,%3f@&,@0=) &,CT0 #Z'() ) 80 #, ξ = aφ−3/4 X !5,%&,$0 U)8 ,&:3)E802 )E80 ,C`<F G J,%&*?0 /,%3>"!_,D .-.&M0 # f&,$0 *,$02& M 0I#% !$#%&M. R , 3K0 #$ )18& ?0 /,%3 38,%0 %01, 3 ,6H1.0 #%& 0 #%16R'+,8Q8,%+3 X !5,%&,$0(`<

! #" %$'&

4k0?9,(% b #H%+0:%4d=/%,4k20 4 h c 46ZV9.())48d567!v0%_9,?%r460%PC$9,-%&6 K%&%5%,4%,d=! <#()!$*+%,N2?0%&r)4 9&49.:% d:%&*+%,%& )*>#?(%59.())48d6?%&9,)#?4r9.4+2%6?%,/%&6 f* h c 9.(9,?(-4 a 0%1%&%,4$-( K%.?%&H%A0% G()()#A)48dEsG4#H%. 9,? ?%&8@ R > R∗ ' 1.3Lflat

b #A0%,% Lflat)0%E0?94%&^F8#?(-4^2?0

Q

#A)0V0%3C*+%d=Cf)48d_6?%,4)"!I b )4#;67()!N9,? /%&6n2?0%& %3*+%,%d:%,4%,C()"!J=4U 2?0%&s/0%1*+4*+%,A9&49&%,4$C4V6?%&9,%.%&A+#H% *+ /% ! f* 0%?G9&% #%&9,9C9.())48dU6?%&9,)#?4 #A0?9C0#H?(6Z9&9&?4$ G30%4d:%&*+%,!k46 G30?<()#A()!v6?%&9,%.)48d9&49&%,4$C=4 #H?(68!$%,(6V40?)48dN%,()C%32?? 9&4C4$ 2?(21)480% (%'?46?%,1gi6?%*2%,4?4%&6?%&C9,)#?4 320%& a 0?3%,(-:%,()!P#$<C9.())48d86?%&9,)#?4 9.4l2%>%,(-%&6ln0%D9,010%39C0)4U%,46867)2??45#?%.6? :%,0%3%,4$)%3(-.!/%, /)#H% /%, b #H%<d=%30_%,(Jgi)*>)(- #?(% Y)4/%,%&66@g 4P"!$#%<14 0% 9&%&9,F9.())48dr6?%&9,)#?4:%,4nO0% -48O0%0?9$4%& #A)0 R 6?$%&A4 (t()# ^2?)48d++*M9&49,()4 b0?<#?#?:9C0k6?$%&4(%.6PN0%^()#H%& f%&%>%,4%,d=!5)49&%434?()!5) /%,%&)*_%&30%9,( K%&% %,4%,d=!82??(C_)!%,(6?45t49BC%&)48dN*+4*+%,9&49&%,4$C49#A0?9C0v67/%,d:%&)4 0%E9&%,4$%,G R ' Lflat

(-d:%E9,? =?%&N@[/%&%,#?)48d50% d=Cf)48dP6?%,4)"!k9&4C4$gI b0%5*+4*+%,n9&49&%,4$C4 2%&9&*+%,n#?.d=%&/%,()![?4?G* b (%.67)48dX 0%59,?%PUW/%&%$!.%&6Z%,*> gi67)()?%N()?4 #A0%,%E*+4*+%,1gD*+4*+%&^)4$%,C9,4^#?%&6?*>)4%n :%,9C0)4P%,9C0?)48dN%,4%,d=! a 0?6?%&C9,)#?459,4%,4$'#A)050% D9, 0 *+=9C0)4P%,46?%N(/9.%&6u^0%N9&%,4$%,+10%N#?0%,% b ()b/%E)4[59&*>#?%&C%&6Z2?0 #A0%,%r0%,! %($9.%&6r0%%&6Cd:%1a0%2?0 c )4()()! b #H% (d=/%%,(%,*+%,4$\39&*>#??%14$?*+%,9.()()!+0%K%&%3%,4%,d=!nC)48d_)4$E9&9&?4$9&%,(-4 2%,(#H%&%,4U*+4*+%, b 2$!N)48d_0% h c %&?()G 0% /()?*+% KC9,4v46U0%9C0)4v%,46567)2??4 k%<0#%00%67-;d=C* 0%67!a%,%,4$FC%(d=)*+%&<9&9&67)48dnr0% =()%&F R 4650%>d=Cft4Ddn6?%,4)"! σ #A)()(d2%^9C0%&9:%&6%'7#%,)*+%,4$C()()!n)4n*>9,$%,* ?(44, /R . 'C$4_ R ' Lflat

b 0%67!*>*+%,!2%,(#H%&%,4_9&4/%'+46>9&49./%9.%& b #A0?9C0_d=)4%&)4 0%567!a%,%,49&%5=)(-2?(%5#9&% GE0%U2?0 b 2%&9&*+%&N)*>#C4$V46 #?(-.!7Nk444%d=()d=)2?(%(%<)480% C2?)())"!8'*>9&%,()(%&468#()!$*+%,F6?%&9&C%&6U*+%,* 2?C4%& a 0?F9,(%2?)48d:<0%+%&C%,4$-(%,(%,*+%,4$ G30%+6?%,%,*>)44 ;*+%,* 2?C4%E#4$C4%& 9,? =?% b*+%.4v46 23-48d=67)"!V*+$67?()O%&#%&9B-()()! G 6?%&9&C%&6 :%&9,(%,'0?d=059,? =?%&F0%,*+$67!4*>9r%&W/t())2?)?* @ #A0%,%+0%+C67) ) A6?%,Q.q=q ,)Q.R . I a 0?<0%&%,9.(#HA 0?(6 (Ck)()()?*>)4%P0%UC%U<0%P6?C#?4 <#()!$*+%,n4 8d=0 ?G9&%&9&4)48d_O2??*># 46n0()()# O0?d=089,? =?%& ,)Q . /)#H% /%, b 0%rW/%,4Z1*+%,* 2?C4%9,4*#A)0X804#;:%,(%,48d=0 λ < Lflat

)()(#;)>9&46?%,C4[)49&%E0%EC48d:%N %,9E)4%,C9,4^2%,(#H%&%,4X9C0t4 0%,4Z(48d:%,04 λ a 0%67)2??4r46+9&4K*_4r$d=CK%&6>#()!*+%,0?(6>0%,4+2%1()%,%&6+4?()!)4E0% $9,)4?)"!_a0%Md=CK)48d #)4$ 4679.4H]0?0/%2%&%,4Hd=/%,4r%'#%,)*+%,4$C()()!_2$!

Q

h )48d=0 s/0%1)4%,49&%p?G9&% 8d=0?4%&;4E#()!*+%,A6?#?4rH*_(t(%,&#A0%,4r0%8d=0?4%&01 /%,!r*_()( #;/%,(%,48d=0 ,)Q .

&

a # ! 6 "#!c ! $ ! # &% ' ( %$)*+ -,. + d / "" 0 ,21 b 1 .3( 4! -# d d 5/"+ 6#7 "!0 d8 9:# d #! "+360! d 6 ;7 <%

>=@? A ! B C = ! *D E $ E GF ! = $ E *D F F B IH $JC D

6 6">( "KL! "NMO5/"+!P Q 0"" dSR' TVU c0 R % T ' 5-, b # ;&; dS 5/QWX% # #5Y # $ U:"5/"[Z4! 3,."+ , \,#7. b ! ,,. #V]ZG 5/ # _^

φ(r) = Φ +1

10C − C r2

R2

K % G` M

# "Φ = 3aNσ/R -'&ZG"# d aZb 5/ # 0!c7 R 54 "! d!

# "C = 3π2R2/8a2vN2 ! 5e'fg0hG g 5/i d 5/"+@j#% kl dS 5/(W b

ZG 5/ # m'5/ (ZS n7.oa "+"φ(R) ≡ Φ − 9/10C % ap + d

!XZS+# d ' 0q0 b # q # +ZG r5/"+!c 0! s "" , W.3!""5 - 0V"" dSR K (5/& g !M9%_c5t # -> # g 7.+! R ' !0!ZG 5/ +"# 0]"" dSR ]7 _^

Fos =1

2

∫4π(R− r)2dr v

a3φ2(r) =

2πR3v

a3

1

3Φ2 +

13

2100C2

K % u M

d # φflat(r)7 R

4πR2φflat(r) ≡ 4π(R− r)2φ(r)K % G1 M

pO!" R , O7# 0!V7 R " d v &Zbw7 b ,,.0!V, &,, b # 5-, 6] "" dSR9, W.^

Fel = 4πR2 π2

8N2a5

∫ R

0dr

−r3dφflat(r)

dr

=

π3R5

2N2a5

1

10Φ− 13

700C

K % WbM

u h

2 " d 0Q0 b # !P 0Q"" dSRT nQ5/& ! ,,b rK 0h+ g K %2W u MM]^

F =2π

3

R3v

a3Φ2 +

π3

20

R5

a5N2Φ− 39π5

22400

R7

a7vN4

K % M

aO #" g "! !" R , 4" 7# 0! 7 R i 5-, +Zb # 'Zb 5/ # ', , W .$^

g(r0)

4π(R− r)2 =

∫ R

r0

−dφflat

dr(r)

r0

Na√r2 − r20

drK % M

# "6 06"!'! 7 >5/ 37 K ", g b ]Zb 5/ # ', [M9%TaO d + d # b # ; [ZG"0hG K % 1 M9%

>=@? A $ VC F C = $JC>= *D $ F D

aOPU c ,,b !0 7.04 R n, 54 l d 54 _n n! d 0 q,# _n S% cq7 d # 0! T +" b "# i7./!0 7.0! ZG" R 5-, "5/V 4 ,." 5/-a#!5 # 08! q^ 8#hG g &ZG"# d 0!,.S (

n g n5/5/" 5 # y ≡ an1/2 P > /5/ b !T5/d ""# R y ≡ an1/2φ−1/8 φ ]5/5/"ZG 5/ # <%

U+ "# $wv !"0! 4 R# R 7 / Ow,."+ #+ "# ! &%g8 &ZG7 b # "ZG" b # "a #" "# R b ' , 03 dS 0! S"3/ d # d b &ZG6 b !3,."+ $5/ # d 9$"S% k/ dS # "3 ,."+ &37. 5/0w !"

kBTb # Q ' ,. ]/0!X n "# b # P n",0#"+]

!! 0!X n "+, R % g', " , b 6,."+ !'" 7.Q "0!n (g ! d R % $ # 0 54#$ ! pw dS 5/W>K # " v !>7.$5/ 5-,.#+NM d - d !0 , P 54 &%

!">,. +-! # r

5 # # "6],."+ "# TZS

V (r)% " 4 #t !"8T,."+ q d P! #

z 5

,. + b d d ,.",."! R /6#! >^ 5 d 05/" R b 6 # ,. + S"4# 7 R ∆r ' z2/2(R − r) % > 0h+ " b 4,."+ &0! 7 R ∆r |dV/dr| %6wRP !" dT ]7." # 0"i5/5/" b >,."+ 0hG cK % u WbM ;5/+! 0!

u

0 ! d , W 1 .3^V (r′)/kBT = φ5/4(r′) = φ5/4(r′(R)) + 3π2/(8N2)(r′2(R) − r′2) # "

dr′ = φ1/8(r)dr!

φ(r′) ≡ φ(r) % / " b wZ / $,."+ >w,.",."! ! dS ZG"a7 R (dV/dr′)(dr′/dr)∆r %ga dS 5/W b ]Zb 5/ # 30#"+ R 5 ! # ]&ZG

r′ ' rφ1/8 +

V (r, z) ' V (r) + kBTφ1/4 z2 r

N2 (R− r)K % M

g6! > z b Q"ZG ZG" 6\,."+ dS ZG"67 R $7.[ZG\0h+ <% 54 " d 8 0 b 3,."+ & # & !4!+0\w ! 54 <% d " d n 0 b 0! T ,."+ % 3 S#[Zb"7." # 0"P0# # / dS 5/0 b ] R , ! #

z R ' , 0!'7 R * '3

5/5/">0hG 0! > 0/#!5 # [ZG">! # b & # 7. d V,."+ !"

V (r, z) b 0,.!-]-"" dSR !"kBT

d ""&% k-" # ! b dS ZG"(#! ,.S

r b "# i54 R 6 'X 00!z dS Zb"

7 R ^

kBT 'z2φ1/4

a2[V (r, z)− V (r)] i.e. z ' aN 1/2φ−1/8

[R− rr

]1/4 K % M

φ 60"+ R 5 b "# Q!",."!i ,.S _ R d i

[(R− r)/r]1/4 % 03 # b 6,.+#0 dS Zb"'7 R ^

z ' an1/2φ−1/8 (r ¿ R)K % hbM

; J # "z

% 6wR8 +# b S;> "+" b "# dS ZG"a7 R ^

z ' aN1/2φ−1/8[1− r/R]1/4 (R− r ¿ R)K % M

aO6"" dSRq +ZG Zb0! ' ! ZG "5/"+ kBT,."7 7T "! z bBb ,."

Gz ' (z/a)2φ1/4 ' N [(R − r)/R]1/2 5/5/"[% # b 5 4U c 54

r = R sin[πn/2N ]K 0-"! 6 "+"

r = RM b # -!0!X /

(R − r)/R ∼(N − n)2/N2 % / " b 04"" dSR !X 4'/0 "# dS ZG"a7 R ^

Fconf '∫ O(N)

0

d(N − n)Gz

' lnN K % S` M V 0h+ " b 099 dSR 0,.!-P80!X r "# (g d dS 7 S% aO 6 !P7. !0! b # "ZG" b R !0 7.Q 54 &3 "+"QK

R− r ¿ RM9%

u`

>=@? A = $J= F $

c # d q#5/q,,b c g++!X b P d ! d N5 7.q0# g7 0! &Zb "R !X 7 0 P"5/ "R !"#!

R!T d # d

!" R σ K c d % u&` M9% kp !Q7. 0!>p # # dS 5/0\$,0"+ b 5-,0!-6," d 05/" R ^

• dS 5/ u 7 # " "R !" $654 KR < aN3/5 Mv5Q +5/O 0!

7 R "R !"!q" 0!' d "R !"&%8aO " 07 7' "! ;0h+

R!a " d R‖ ' N/R2/3 , W .k%

• dS 5/ %# ;"5 g ! /! # " 6> "# Rn 5-,00! %%aO " d R‖ ' aN1/2φ−1/8 %

aO 5 7.9 # 0"'0#; # - dS 5/0 R ' 1/N3σ3%

N-6/5

N1/2

N3/5

N4/5

N

R

N-1/2

s1

s=

1

1

1bis

2

3

4

R=NsR=N

3/5

s=N-6/5

R=Ns1/3

R=N3/7 -1/7

s

R=N-3 -3

s

% u&` $ AA '16C),%3 '()_'+,%'(-.A/102+3!5)768& 9;# < # ,%-;&&f'?.-+@0. B R ,%3 0 #%M/10 ,C 38&,DE06 σ < #8BMQCT. B,8)76^V8)E0"@0 E-.<5X Y Z#8P4[9;B Y '+\!$&+3 Z#8P4[ Y (Z', '/-;9;&#%[ Y '+\!$+39;#%[ Y &:3)E8029;&#C`<

u:u

u1

;

, u . % ,,." b "!$# &% '()+*-, . '/ 10 &%32 4% "(5% K !"5 76 0 b ! bu hSWbM9%

,21 . 2 %98%Cc 0" b % % "PU+ b 8% % / % %U " " b a]% S d [ZG b 6 %;: "+ b<$=?> A@B( C* 1! B% K ,54n" / b ! b u WbM9%

, W . 6 % g 2 %! 2 "0 bDE/F$, . '/G@B( HC*I#J! LK b u h K u h M9%

, . 2 % % P b %9MV% b % 6 % 2 b 8% U.% / * d b % N-% b U.% %\U+5 bPO 1/ 0 C%H!HQ;%4%R<SIT u&1Xu K u M T !a "" "" <%

, . % / ,." T %VU T U]% 6 % +! d TDE/F W$ X#J! WY[ZZIT u K u u M9%

, . 6 % ' !X ! T >%<\ T 6 % !X54 TL] (;^_Q C`Y[aXT 1 S1 K u hbM9%

, .bM-% 6 !2 TXc Wedf?%g,hd <R<RXT u(u&` h'K u M9%

, h . 2 % " T %<M; T / % Vib! T 8%V:" ! 5/" T % 6w "+j U+ "54 T % kib"+j7." TDE(;^ l hm ?nokp $,hd WY Lq Y K T :u K u r M9%

, .b8%U 5/ T %[Mts" T / % Vib! T %UX 54 T,hd H Jedf?%] u2 "/?% K T 1u K u M9%

, u&` . 2 % 6 5/ T 2 % "Z Tv 'o! d Wv 'B2<df?%WDE! pY[Z RXT u K u ` M9%

, uSu .$N%<U # T %*U?i T U]% !0 T % j % 2 0!& wj 6 !9Z T :>%<x" T %<M6! T 8%.U:#yjZ Tedf?% Wz FI] <S T uSu W r K 1S`S`Xu M9%

, u&1 . %V8bb T % Z:# R T % Tedf?% Wz FI] K SIT W G`S1 K u M9%

, u W .-U.% %UX # T 6 % 6 T % ! 54 T xO% % k+S TVedf?%z HFJDa|IT u&` :u K u u M9%

, u( . 2 % 6 T % ib TXc $,hd Wedf?% IY[Z T G1 ` K u M9%

, u . %U ViS T %;M6 5 T xO% U.% 6 0 T %;U T M-%[x # # Tm 1!Hkp LK T 1 r K u M9%

, u . % < ib T % 7 " TWm 1!k~ k!Q B% |XT S` K u ` M9%

u W

, u . % % 6 ! T M-% / Vi T % % +54 TWm f!Hk~ !HQ % |IT u K u ` M9%, u r . % / "kib T % ,. # :R TWc Wedf?% W@k@`OL (J! T u( K u M9%, u .bM-% / [ZS T % 00!X5 TXc $, . "/_@B( *I#J! WY SIT W `S1 K u M9%,21S` . % %*U:"Z TLm 1!Hkp !HQ % RIT S1 K u M9%,21Xu . % T %VU TWm 1!k~ k!Q B% T W K u WbM9%,21S1 . % % 2 [ZG T N% 5/[Z TJ] (;^_Q C YIT r W 1 K u M9%,21 W . % 6 %*)*+ T , ZS 5-5Q S<%,21 . % 6 %*)*+ TI?%H?n?~ X#L / ( IT r W K u r WbM9%,21 . N-% :V% < R T U]% % 6w + T % 6 %*)*+ T K , 7 0!M9%,21 . % ! T 8% j 6 % TXc Iedf?% WOW (! $a|XT W u K u r 1 M9%,21 . 6 % 6\ TLm f!Hk~ k!Q %RXT W r K u M9%,21 r . % %UX95/" Z TW#JFWgdf?% AcW Y +aoT WSW'K u r M9%,21 .VU.%9U]% P " T U]% % l " T % % 0 TgQ;k 2<df?% I] IT u W'K u rr M9%, W ` . % %U ZG[Z T % % 2 7 Z T :V% % 6 &Z [Z T % 6 %)*+ T \>% :V% 6 [Z T :V% %

6 5 R TJ$= I#L! X#5#wz |ZIT u` K u rr M"%, W u . N-% 7 T U.% ' !" TgQ[kB2<df?%W] IY[| YoT K u ` M9%, W 1 .VU.%9U]% P " T U]% % l " T % % 0 TLm 1!Hk~ !HQ % YT 1 Xu&` K u rr M9%, WSW . % ! ! T % % h+ 0 TJm 1!Hk~ !HQ %E|<ZIT K u M9%, W .VU.%9U]% P " T U]% % l " TIc Iedf?% WOW (! $aRXT u Xu K u rr M9%, W .b8% N% T )p% j 2 % Vi TIc Iedf?% 7,hd R<RIT 1 r WSWTK u M9%

, W . 6 % j 2 %! 2 "0 TW#J! (;^ ,(! 32 % ( $= Htedf?% '! % T K ZJ% 6 #"# T k Tu M9%

, W . 6 % j 2 %! 2 "0 TLm 1!Hkp !HQ %-Y;a u W K u r u M m 1!Hk~ !HQ % TXYWT 1 K u r 1 M9%, W r . % 7. R T 6 <% %VUO0 T ZG9# 6 E: T u %, W . % 7. R T 2 % / %ox0!X # T 6 % 6 T % , T1m f!Hk~ !HQ % SIT 1 K u M9%, G` .VU.% ' !" TIc Wedf?%W &%$|<SIT r WaK u M9%, u . 6 % j 2 %! 2 "0 TLm 1!Hkp !HQ %-Y[|IT u&` K u r ` M9%, G1 . 6 % 2 " T , ZS 5-5Q S T 1S`S`Xu %, W . % 2 R TJm 1!H ~ !HQ B%E|<ZXT W K u M9%

u

"!

#%$'& ( $ >C F = !

)] , T +"0#$ +* -,0,. R 5/."0$"4#+$hG " , R h+ 0&% ; '0*"5-, 60O",0"+0-x iS % u T 21q &Zb!0 ; Z 0 / h+ /V ZG"+ iS03 006 "# i5Q74! -,054 0076 " $0*X"5 0&%98<0 -,0$+$# 00:6 ] ;4L 3! ;Z 0 54 ,,.Sh+ =< 0/+ 70/!>< 0*X, "/ ( ;4L a! ,. 7 S%@? / ,,.S/"r 'hG 0 -,0,." ZG"+54L5/" A4L4&ZG ' ;4 /!>< /Z 0 4!- 54 ."/h+ 0]!" B* 7 ";,." ZG"+5C"/"ihG h+ -ED 5 0GFsK " "ZG"+ "5/"+! 04L5 0NM, < +"5 0! 6!0w,. R 5/."0&%B? O#5-5/0 T " T , 0O,O!0O,7 ."5/0!>< i ;7 i h+ T "O#5-5/"+, 0hG 0 h+ ."ZG"+ <2!X0 IH2W TJ 1LK %M? &5/ T hG < "+Z #?ib0 T N<2?i # -!>< ",0"+N # !>< & 7 i h+ O.0 5-, 0*X T " 6,""!,h+ =< T ,7 ."5/> ,." "+! !XS%

P =< "(0Q,;5/ ]h+ (",0#"+0px iS % u ."ZbV!>5Q7" 0h+ 0 v i 0!X >,. +v!$Z: $! , R h+ $!0p,. R 5/."0&%RQw5-5/"+$", "+ 0 -,0w"+ 0w ;4L 0 00TS5U 0 < +"# a03J ZG"+ 0$7 "TSU 0 < ""ki !><2!X0 P!X 6 V,0" !0 " TS

8<0p",.0@6 0vh+ 0 v+p Zb"5/"+\ 5-, 0v!+.0 vh+ !" O!eiS03 ?ib0=V+*" 0 5/"5Q7#0v#+ BV 5/"+v i !0&%L8<0 " v# 5-,."+\ p0"wj "5/"+ 5-5/>!00 D 5/ &GFP"T,# . T " 0#&ZS B*T! 6 H u&1 -K T

u J

K %

(a) (b)

! " % u$#&%')(+*,)- /.021435)-76,218693;:<=,21>5@?BACD?B<BE;'F E8G,21H5JI'K9FL,218MN<=, 0=,218AO?:396,21PQ%;:(R,21>G,)- .7021435)-76,21S1=?BFLEUTB'<4<439AO0@,21WVXCY'<S6,21>5@?BFFL,@5)EJ,)-7<Z1[CD?B6]\AO^Z<=,21SAO?:396,21)P

_ 0*!">">!O`;"0 H T G1-K 0Q7S0Q!4,. R 5/."0Q,."5/""+ 5-5 0! "5/":!V,,.S"!0;",.#0&% 6 T 5-5/- - ,,.S T 0 a06+]5/7 0" 05/"5Q7#0V+Q# , 0 T '!" 4! iS03 ?ib4ZS'N<2!," "5/"+I6 T! # "+ 0O ;4L 0&%8 0w5/"5Q7#0ZG+ T "q" T !," " 4 5/ 6 >! 7 '!0 +# +0h+ 0 " !X "+&%

)]-#T, _iS#! 5-, 0* T ,7 ."5/q5/C P"5/+! R 5 hG '!0 0!,. R 5/."0 T 5 hG ! Z !X ]!0 a0O!,. R 5/."0 T 5 h+ ;!0w5/"5_j7#0 T " P, R ?j 5 q! <2 #?ibS% );8O.0>5Q7" 0 " !0-+ " # 00 B*a7S0!],. R 5/."0" 0* ] +"OC"EiS#! ##+,. 0 21T!03,.?j R 5/."0w+eiS03 0w $!0$5/"5Q7#0&% :6] N T =< R 6,$!#&Z B*0 h+ 0h+ "+"+Z #?i , ?ib! 0v \,. ! !O ;4 0 j00&%T)%.0 T >7 4$! , T 0; #+;!5/">"n, 0 0, gi ""# B*hG + 5-5Q 6 , ,!0$,7 ."5/0 T h+ !,,.S"!0$",.0

u

K Y

, " #0 6-!03 !00&%)] 0 0\ \5/"#+$" " /!0 " ,. R 5/."0\5/7 0 T ;!"

!)iS03 ?ib8ZS' <2!,"I6 BV5iS # r!0> ;4 "[%)]Q 8,"5 .", T " ! w! ! "# ], R hG 3! ! " w"/4 ! 6!" !iS03<?ibQ" 4! # Q"+ 0 ;4 0&% ! ! ?iS#5/>!0Hi 5/0 T ,. ZG"+Z #?ib" /, ,!0 ,# h+ 0&%T? " 0 !X 6/!" B* O."0&%

U $!><27.! T $!,. R 5/."0 #+O!" B*/ ;4L 0 +V 5/"+$ i !0$! 4L5/h+ h+ TK 6 %2WbM9% -!0 N 9 hG 0Q! "/ _06i0*XOC"5//"wj 7 6 4L5/4!0Q ;4L 0&%? &5/ T 5/+>h+ =< 0,.S 7 4!4! "!/54 ."8O.0 i ""# 8! ' 8! 8"A4L !0 ;4L 0Q"_HiG!"6 "# 5Q74! # " h+ 0 ZG" 0Q,,#a #"+> &% 6 T 6 "+; !!" B*'," !0 5 0&%T)] 21 0 #9 +"0h+ 7V&ZG P0"ZG ! a0;hG 5-,.SQ P,."+ 5 h+ T µpol T ! jZG 5-5/"+#>", #"+ " #9 " 4 _! q! # /"+ 067 "&%? Q! ZG HiG "5/"+ '! 7 !0 +# +#"V" < +"# "+ 0Q7 " !X 6, 0 " [% ? 3"T! 0!X -! ",."! Q!0 # " hG 0!X ',.?j"+ !>< +"# a03 4wK ,.0 T ,L4L!" NM$" 4L q!0$,#5/."0

µpolT R T "

N%

? ! "BVPh+ # ZG"5/"+!X P ! 7 Q" 5-,#+ !!" B*Z 0 0!;5/C"5/6# R T"3!]!" B*qZ 0 03!]# R O.0! 3 ""+[%

BV T 3"+Z #?ib 3!],. R 5/."035/7 0 #+!03 ;4L 0 BV 5/"+# , 0nK 6 % M9% ? 4 "+q 08,, "0a!X0 Zb0! "+3 3Z 0 , ,."5 &7 "\ 8 7## !S%9? 5/+$hG ,#5/." 5-,.#+]! ,7 ."5/>0 6N,,.;! /# >! 4 ;4L Q03 Zb"5/"+ 0 < ", " ! Q! " &% ) < T,. :!6Z: 65 hG T n0 # " 03, %4 h+ !3",# 8" < ""ki !><2!X0 T < 0+j 6yjk! #&ZS 0## ,. $"," ;Z 0 !34 7# T 0!" B*/hG + 0 "#+ ! ",."!+0 T 5-5/ < i iST! `;"0 H J 1-K %@? /5/+8hG T #" S0 "#+ HiG 04, " T /4 hG >!Q 9,*# (0, 4 7 Q! ! " ;5/7 0 Th+ ! V! " #7V+*0&%M8 #0 , ?iba"+ 0>ZS -! !" Q! iS03<?ibQ" <2 ViS5/"+# ! < ", " ;! %P"ZS T 5/C"5/ , ?ib !X 6> ""ki !><2!X0 T, "Z 0S%

u

K %

)]v Oh+ T iS 0! "# , R ?j 5 hG 0! <2 #?ib3!0v,. R j5/."0&% 8 Q h+ 0,.!q w! ViS 0 a0w R !X, 0w,.S0!+ ,." "5 q R !X,7.;9w!>< 5/"5Q7# 4L !>< 7 , ! h+ ! < & K "ZG"+ "5/"+a! ",.S0P 4 l 7#8 ![M9%7\ ,." 87 "l"+"!X "ZG !0 , 5-, 0*021 <2 #?ib> 4L"# , < +"5 0! V!5/ 0 5_j, "5/"+# 08K"x iS % 1 M9% _ T "# 0]5/ 0]0,.N7 "6! <2!X0 <2 54 + "08!q, 0 D iG![j "," 0Fb% 6 40*X"5-, T 0- "," 5-5/ < ( ^ ( "+Q!0] 0hG " 0],.", ! h+ 0, " Vh+ 0] 06h+ yjiR." V7 H :u0K %R)] ;, ,$!0 T $"!$!0 ""ki 0$!><2 #?ib

10 < |µancre|/kT < 50%TP 0,77 "5/":;# 7 V! n,"5 9;"5-,;!,

! "" 0\! "# \!X -5/+!3!><2# "5/"+ T 54 4 iG!" 6 < 0, \h+ !O Zb 0 0!;# 0 " 00 6 <2 #?ib],." ZG"+ +"ZG" 3! ;,7 ."5/6,. 3"a5/+! jV" "# ,. &%w !" +O.0 "5-5/"+3, Z ]h+ 5-,."5/0+!,. R 5/."0-iS03 06 6!0]5/"5Q7#/,." 5C"/ ViS .""5/"+Q ", 7 4 B* V006!X 5/+!6!><2# "5/0+ H 1 r K %

! " % 1 #*,)- E9\CD,21 G 'FL5Z<W')M,P M'-5JIL,6 'FL5Z<=,O,21BE$IN\LGN<=?WC IL?:,O,)E>1 39F 1=^Z<=, G'F 1 6'AO,ZAO:<W'FL,P GN<=?B3 EJ, 6 'FL5Z<=,U,21BES-7FL,+AO?B60@5B- 6 ,>1;CD0@5Z3 -,P

? 4 ,,.S4"BV h+ T ! (Hi ! 0V ', T 0 0 a0#]0* ", "+ 0]!" B*(7 ";, " 6!" B*n7. &%Qw ,,.S T 6q" Th+ 0 ""ki 0!><2 #?ibn "+8 4#5-5/"+)iS#!08,. h+ 4 # l! a04L54+ 0],.+6"+ 06!" B*(7 " T η T HiG /6 u %>)0 " !0], " 06! "+j5/+! R 5 hG Q!><2!, n!07. ! a0+3,."5 !6! ""5 "

η(µancre),.

kib iG5-5/Q!>< ""ki 0!><2 #?ib H T K % 05/+"+03J ZG"5/"+hG 6,. !0

u r

KL & B % N Q Q

""ki 0 5-,.#+0 T η # 76 u % ); 4 T Hi ib0 HiG "5/"+ 0 a0; 00 B*P!" B*'7. ; 5/C"5/>7 "&% P "" l "n+5/+>h+ 0EiS#!07. 0" 00V: 6 < i q!>< + 7 ", ZG8! < ""ki q!>< +"# s"+ 0!" B*' ;4L 0H u&1S`-K %

#%$ $ F F ! C?*C $JE F ! $

? "#7 T ! 9$, T w! ?iS#5-5/O!0pHi 5/0v,. O! j" ,. R 5/."0;! Zb"5/"+] 5-, >21 0!" B*P ;4 0"PHiG!n+!" B*, 3,# . 0[% _ T " 4 T! V! # ]!]",#

hK "a 03!

aMw"3!

!" Q! iS03<?ibQ!0,. R 5/."0σ

K "P 0!a−2 M T , " Hi 5/0#+! ViS 0&%

)]3 , T 03! # 0+"a 0!a T 0 ""ki 0"a 0!

kBTT 0

,0# "kBT/a

3 " 0 4L 0"kBT/a

% 6 # " -,7 ."5/ T -5/+!+. 8!,0 5/+!+. ]!>< _ 0*!"+jk!`;"0K , 1 M9% \a03 " T 0 a03+7iS03 00 6 9 !" B*P7. ; !" B*P, ,*# . 0]" N<2?i ;!>< bi ""# # i! Q5/G! . - 21

h03h+ h+ T < 0+j 6yjk! ! 3 ""+! V! # !>< 0hG 7

H ∼= aNσ1/3%

)] h+ i 5/ T ,."+ 5 h+ ;!>< a;" +# +0*" 0O, 0 a0\ ;4L S%U O.0i ""# "5/"+ T /! "5Q7 + 7 ',."+ 5 h+ 6!0 a0iS03 00^

µtot = µtrans + µancre + µpolK % u M

8 0-!" B* ,"5 "-"5/0-: 0 + 7 / 0V,. V h+ !q 5-, 6 1 ! j5/" T < 0+j 6yjk! < "+, 4!-#

µtrans ∼= lnσ" < ""ki 4!><2 #?ib

µancre%

Qw j " 0 #+h+ ! ",."!/!X - R ,.!><2 3" T "-,# h+ T 4 |µancre| ≤ 50%28<

!" "]"5/V! < 0h.%vK % u M0], " VhG - B*P,. R 5/."0&% P + "+6!" B* + 7 /^

µpol ∼= µosm + µel%T8 + 7 iS5/ hG T µosm T "+ 5-,>!0 +#9N 64!" B*

,[%>P < ?i ! < ""ki / 0# -,. h+ < , "#" >5/."- a 0/! 8 "+0S% 8 + 7 h+ 80 0 6 a0 !X _5Q7! BV5iS # Sq 0 0 7 04, 0 a0 h+ 0+j " + " 00 BV00&%:Qw0 + 7 4+ "Z 00/" -!>< '"r 7 +- ""ki kBT

,V7 7r! "+# SmK 7 7 DL)] ! Fi,- i q&ZG <2,, q! )] ! !"V wj ]"5 jk! 00/H2W K M T ,. + 7 iS5/ hG T "

kBT,;7 7(! BV"5/"+

u

K %

P7 7 h+ qK n" >7 7 D 6\ GF H u&1 -K M,. + 7 h+ qK 4@% yj, u M9% 6 08! ViS " T / # # , ."P Z+_^ 0/7 7! "+# m+8 TN ξ

9 + ""+g

5/5/."0 T #! h+ i# n!07 7 hG 00]0

Λ T " + ""+G

5/5/."0&% ?hG pi ""# "5/"+ T " ! 067S#0 6 < 0hG 78" T ! T "Q!" B* + 7 >#+HiG 0&%8 ! "+ 0>!" B*i R ,.0Q!47 7V0Q! ZG"+Q5 #S% )] T h+ ! " T "a"ZS T "! P030"+ S%

8 +# + T Π T 0*X" 0, 0 a0w 6 ;4L 3, HiG "5/"+w + 7 wj QS5/ hG T Πosm

T "< + 7 h+ T Πel%-Qw0 !" B* +# +0 #+<54 0

B*> ;4 0&%L8 ,0 -S5/ hG T Πosm0 # "+0wZG" < 0*:" " \!

KΠosm > 0

, +Zb"+ *M T $hG +# + hG T ΠelT 5ib!3 iS 4h+

0 a0w#+ BV00]KΠel > 0

M\ " 00;KΠel < 0

M9%XUX +Oh+ ZG 5/ 0 , , a80

σ/h T +# +Πosm

K 0,<%Πel

M]0 08 _,."+ 5 h+ T µosmK 0,<%µel

M T , V

Πosm∼= µosmσ

h

K % 1 M

|Πel| ∼=µelσ

h

K %2WbM

R

8 x iS %2W",0"+ 06! 3 ""+]Hi 5/06!0 " 6,. R 5/."06" 4L !

σ"h

%MQw8! ?iS#5-5/0Q! 5 8, ! Φ ≤ 1 T h ≤ Nσ

%);/, ! # V!>",# (54-* 54 >07"+ h+ Q 05/5/."0]#+ " 0 T ,.

h ∼= N%+? 3! ZG 03! 3 ""+3Hi 5/0"T! "# %

Y X R X 8 h+ P! # q"+ 0-!" B* 7 "V0 B4 " " 6 < 0*X" s a!><

an"l7.l# Z+ T RF∼= aN3/5 T 0 a0q+ BV00a" i# n!X l7 7d!

BV"5/"+ T Λ T 0:V+* 0;, -! # 6"+ "!" B*a ;4 06^

Λ ∼= hK % M

_ < B4 " " :6Λ T aw $# hG w ?j "Z #+O"

Λ ∼= G3/5 % 8 a!]7 7

Λ T "'"Z T ]54 & Q ?j "Z #+ 6 1 ! 5/" h+ =<

u r `

KL & B % N Q Q

hN=

s

s

1

N-1

N-1

N-1/2

N-3/2

N1/2

N-6/5

0NN

3/5h

3

21

4

5

6

1

h=()Ns -3

h=N3s

2

h=N

3 s2

h=N

s1/3

hN

=s

3/7

-1/7

! " %2W # *H3')MN<W'A AO,$<=, C <=021=,ZF E;'F E6,21 GN30Z<=,ZF E 1+<=04MN3 AO,21U,ZF ?BFL5)E3?BF G, h ,)E σ P

u r u

K %

0 # 0]" < 0*:" T! a! -! T,# . Q B*a ;4 030 H -K

R‖ ∼=(N

G

)3/4

Λ ∼= N3/4

h1/4K % J M

8<],."+ 5 h+ 6!>< '5/5/."QN< ! ]"

µpolN∼= 1

G+1

N∼= 1

h5/3+1

N

K % M

8< !/"5/! < 0h*%*K % M 0,.! 6; ""ki 7DS5/ hG FkBT

, a"$0 "$, " ! + + &ZG 0v #"vHi 5/0&%RQw$"5/w0 7 "-"+"!X VHi ib&7 ! 5 6!0 ViS 0 a0QK

N À 1M9%+);;5/C"5/ T +# +G*" 06 V ;4

N<

Π ∼= σµelh

+σµosmh∼= Nσ

h8/3+σ

h

K % M

8 h+ σ

ViS5/"+ T :Q Hi 5/8"5 jk! 6 1 ! 5/" Q,. 0h+ $ 0! 3 ""+0 a0 BV00 5-5/" "+56/C"Q" +# &%>QwHi 5/>,,#a,.

σ ∼=σ∗ ∼= 1/R2

‖∼= h1/2N−3/2 T 6hG ! 0V 74 + ." , " " ! VHi u ^

h ∼= σ2N3 K % r M

X R X B )] ]Hi 5/ T T !a R ,.6!]7 7n,,#a&% \T03 " T ,. !0! # 0!

! a,# . 56 Q ;4 T , " " 0 6 ViS " ! '!0 +"# 3!ZG 5/Q0* T ξ T 0 +"# 6/!" B* ,+ #+00" aQ 54 N! %\ T 6 1 ! 5/" T aqh+ > # hG `6 "8 Z:! 4 0r!"# ( 4L T

Φ ∼ N/R2G ∼ 1 T " hG a5/5/."Pn a,77

HiG 6 u !>< C"T" +# &ZG r5/5/."P! (5/C"5/ a H -K % Qw'0 #80 ,"+ T -O.04! 3 ""+4!X /! (! 5/" lWn,. hG n,77 Th+ !" B* 5/5/."0>!>< 45/C"5/ a8 "+Q" +# Q0 4L 7 PK ,,. 6

Φ ∼N/R3

G ∼ N−1/2 M9%8<0 a0!7 7O!# ξ "O! " h

K 7 7! "+# *M'N< " "ZRC""+8! , T i # / hG 5-,.#+&% 8 ViS "

ξ0 V+* 08, ! r!8 Z+"5/"+ 5-, -, 0>7 7V! P ;4L 0 , 0, a T σ−1 ∼= (N/g)ξ2

K"x iS % M9% 6 ; " T 05/5/."0; ZG"+ 54 & ?j "Z #+ 66W;! 5/" 6 < :" " $!X /7 7

ΛKΛ ∼= G3/5 Mv" 9 +*;j "

u r 1

KL & B % N Q Q

Zb"+ "a5/C"5/a# hG 6 1 ! 5/" 6 < +" " 8!X s7 7ξ T ahG 3 !X 6

ξ ∼= (g/G)3/4Λ % ? Zb

g ∼= h

N2σ2et ξ ∼= h1/2

N3/2σ3/2K % M

hL

x

! " % #3 1Z3?BF T :6?:NV G,21 5JI'K9FL,21 G'F 1S6'<=04MN3?BFP

8<],."+ 5 hG Q> + 7 n"+, hG 0 BV"5/"+ T N/G T " + 7 PS5/ h+ T N/g T

µpolN∼= 1

G+1

g∼= 1

h5/3+N2σ2

h

K % u&` M

8 :# +0*X" 0 V ;4L 60

Π ∼= σµelh

+σµosmh∼= Nσ

h8/3+N3σ3

h2K % uSu M

8 < 0*:" "# !>< a T ! ]Hi 5/ 1 0R‖ ∼= σ−1/2 %

8<h+ '!" 4!~iS03 ?ib ViS5/"+ T !><2,O.0qK % M T ViS " ξ

! a>" "+! ]Hi 5/W h+

ξ ∼= Λ T # h+

h ∼= 1

N3σ3K % u&1 M

u r W

K %

| X R X B | )] aHi 5/ T 0 a08+ BV00 T 54 # T!X c7 7

ξ0/V+*0',

4 # _ZG 5 h+ Φ ∼= Nσ/h T ,> i &ZG 0# Q"5 jk! 00qK"x iS %bM9%

8 V# 6! a7 7T! "+# n0

ξ ∼= Φ−3/4 ∼=(

h

Nσ

)3/4 K % u WbM

8<0,. R 5/."0 4L5/"+!0 a0 `] # "0v jk! 5/" 0!w7 7ξ T Λ ∼= (Gg )1/2ξ T" 6 < 8 , " " 6

h T !0 a0 `] "0]7 jk! 5/" 0 T R‖ ∼= (Ng )1/2Λ

% "T"+"!X T !], # 60 !"+ h+ 6! 03!" B* 3"

R‖ ∼=(N

g

)1/2

ξ ∼= N3/8h1/8

σ1/8K % u M

8<],."+ 5 h+ 6N<

µpolN∼= 1

G+1

g∼= 1

h7/4(Nσ)1/4+

(Nσ

h

)5/4 K % u bM

" +# +QG*" 06 V ;4 ]0

Π ∼= σµelh

+σµosmh∼= (Nσ)3/4

h11/4+

(Nσ

h

)9/4 K % u M

L

hx

[ ! " % # Q3 143?BF T :6?:NV G -7FL,U5JI'K9FL,UG'F 1S6' <=04MN3?BFP

8 h+ ! # O",#+ 0v!" B*Q, \ ViS5/"+"\ ! # w !>< aV"( ("5 jk! 0 T Rsd

∼= N1/2Φ−1/8 T 0 a0;>+, BV00&%=Qw ";,+!X h+

G ∼= N T

h ∼= N3/7

σ1/7K % u M

u r

KL & B % N Q Q

a BB & ) )] ;Hi 5/ T V! #

h0 , " " 6 V! # 6!>< 0h+ 76"' T"5 j

! 0 T RsdT " 0 a0-+ " 00 H u W `LK % 8 ViS "

ξ T ViS " 4! c!0 +"# !;Zb 5/]0* T 0O 3!0;, < 0hG * _K % u WbMw" < 0* j# Q!0 a0, < 0hG K % u M9%T8 / Q!>< a ! Z !X >0 iS 6 V!>< aX`] " " 0I6 ! ihG < ("54 4# V!>< (5/?j5/.";,

ξ%8 a 4 5/;! a & ]!7 7O!# Λ T ,,. 07 7

hG 0H r K K 4%XHi 5/WV!X , M T !+ ># 60:V+* 0;, ! 6 H u&1 -KK , # u M;^

h ∼= N

GΛ

K % u r M

Qw0a7 7'n Z:"+ "# "5/"+P"'+a!0 a0 `6 "0'!(7 7ξ T

Λ ∼= (G/g)1/2ξ K"x iS # % M9% _

Λ ∼= N3/4

h3/4σ1/4K % u M

8 ;,.#9: 5 hG N< !

µpolN∼= 1

G+1

g∼=(h

N

)7/4

σ1/4 +

(Nσ

h

)5/4 K % 1S` M

8 0 a0 "#+ " 00 T +# + h+ Viba!a iSa" +# +a# 0*" 06 - ;4 ] <

Π ∼= −σµelh

+σµosmh∼= −

(h

N

)3/4

σ5/4 +

(Nσ

h

)9/4 K % 1Xu M

8<h+ !" V!GiS03 ?ibV! 5 + T < ""ki -S5/ h+ V! 5 + h+ =< 6 >h+ ,0# S5/ hG T < "h+ 7P&ZG 4L h+ K "

Λ ∼= ξM9% 8 7S#'08 6

< 0h+ 7" < 0*X" T! >7#S]0!0], !>< _ 0*!"+jk! `;"0

h ∼= Nσ1/3K % 1S1 M

Qw"0hG P! 0V 74 + ." B4 " " ! >Hi %

BB 6 \!0",# \ , " " 0 6

Nσ1/3 T 7SO!w,. R 5/."0\" " 0S% 8 # !X 7 7 h+ T Λ T 0 \ V+*0O, ! 6 T 0h.%K % u r M9% 6 !0\! # 0

u r

K %

x

L

h

[ ! " % # Q3 143?BF T :6?:NV G -7FL,U5JI'K9FL,UG'F 1S6' <=04MN3?BF P

B4 " " 06Λ T 0 +"# !>ZG 5/V0* (-+;, #+00 T # hG -0

?j "Z #+4^Λ ∼= G3/5 % _ T

Λ ∼=(N

h

)3/2 K % 1 WbM

8<03,. R 5/."0 4 5/"+! !0 a0 0 0!]7 7Λ

%8 < "Z (! + 7 n!0 +"# !6Zb 5/0* T03, 7 S% "

h+ ! # O",#+ 0 a0# v , " " 6Λ T 0+j " $ "+ T < 0*X"

"# ! a" ]54 '#!'!]7 7Λ

R‖ ∼=(N

G

)1/2

Λ ∼= N3/4

h1/4K % 1 M

_ T ,. -!0-!" 0-! iS03 ?ib,V,4 7 0 T 0V7 7Λ

!0 a0/! "+0"+"+6" +# T >h+ ! " A68 + 7 iS5/ h+ S% 6 < "ZS " T , 6" 5-,T5/ R " 07 7+3! 0 H u LK ^X a60Z 0 5-5/ iG V!Q7 7;! 0]!QZG 5/>0*

vΛ ∼= Λ3 "n5Q7NB∼= h/Λ

, a>"" "+#

ΦB∼= NBσ/h

%B8<,."+ 5 hG 6 " ;N< µint∼= vΛNBΦB

" ,."+ 5 h+ ]# 0

µpolN∼= 1

G+µint

N∼=(h

N

)5/2

+

(Nσ2

h

)1/2 K % 1 bM

8 +# +QG*" 06 V ;4 ] <

Π ∼= −σµelh

+σµosmh∼= −

(h

N

)3/2

σ +

(N

h

)3/2

σ2K % 1 M

u r

KL & B % N Q Q

P 0w! " ;!><2 " "w 54?ib i 05 " h+ 6 <2 !;!7 7O! "+# '! ]Hi 5/S% 8 V# "# !X a7 7 T ξ‖ T 0 V+*0], -! # "+ a0 T

ξ‖ ∼= σ−1/2 K % 1 M

087 78 Z:#+ _ ;4 P!T54 ."n!"S%:Qw087 7#+ 0! a0`6 "0;!Q7 7

Λ,,"+64, " a06! "+0 T ξ‖ ∼= (g/G)1/2Λ T

QhG !X 7 " 61/g ∼= µint

N/g 7 7, aS%T8 -# Q! 07 7! ! T,.",."! Q0

ξ⊥ ∼=g

Nh ∼= h3/2

N3/2σ

K % 1 r M

9 "+"!X T 067 7Q7 VibQ#/0* h+ 0 T +6 h+ "5/"+ _ ,. ,."+j5/" "ZG"+ "5/"+!;" Zb" + 7 PS5/ h+ 56 <2 !6! < D _ # -F T kBT,37 7 T " =< +,3!]Z " #7 "&%

; BV T h+ < 0* "# (!0 a0a!"Z "+ B4 " 9 6 ! # n"+

a0 T a"+a! Hi T 21 0 a0 " 00a+ 00 0 0a!0 0[%TQw / Zb h+

R‖ ≤ σ−1/2 T # !><2,O.0QK % 1 MO,.

h ≥ N3σ2K % 1 M

8 0 a0P+n T h+ "5/"+ 00n!i7 7 hG 0 T !i# Λ !0,QK % 1 WbM9% 6 + + T ;,."+ 5 h+ 6N< T ! ]Hi 5/ T

µpolN∼= 1

G+1

N∼=(h

N

)5/2

+1

N

K %2W ` M

);]5/C"5/ T +# +0*9 0 - ;4L 6N<

Π ∼= −σµel

h+σµosm

h∼= −

(h

N

)3/2

σ +σ

h

K %2W u M

\Z " V T 6 !>< 0*"5-, T hG 0$,."+ 5 hG 0!0$Hi u " #+ HiG B*,.

h ∼= N3/5 T # T n!>< a 0 T Th+ V+*X 4 + ." u % 8<0Hi 5/06,." Zb"+C"-! Z 0]" !" B* iS ,.0 T ih+ +# + T Π T 0*" 0V, 0 " 03 ;4L 030", ZG/K Hi uW# M T P# Zb8K Hi -" M9%

u r

K %

# $ IC6? H $J= =#

8 047 "/ i !08+4 ,,.S0 5-5/"ki 08!/ '# c!a,. R 5/."0&% Qw5-5/ 0/ 0/,." Zb"+/0*X, " 7"5/"+ ( ;4 q! iS03<?ib T 0 a04+-"s5/0 !><2!," " B4L54 /" " v!" w!giS03 ?ib -VV!w5 5 #" " ""ki S%);3, T $ ,,.S#$hG 35Q7! " w,." ZS "$" #&ZS w! < ""5Q7 iS#! h+ S% _ T 6 < 0h+ 7#"5/+! R 5 h+ T 0>,. R 5/."0_iS03 0,O ;" ;0*:"5 ] 0O!" B*7 " "+ 0"ZG ! " # T h+ V+* ;Z " $!X /,."+ 5 h+ T µtot T !0 " &% ? \ ,,.S\, ] 3h+ < "+, a!# r0>Hi ib&7 '!"Z+

µosm"µel

%@Qw" R ,. .0q0>Z " V 0,. 03O.0 ViS 0 a0>K

N À 1M"%

_ ,.": 5 h+ T µtot H 0h.%XK % u M K T #+ T < ""5Q7 !0 BV5iS # 0 0 7 0! \ "5 4! O! ?iS#5-5/QK

σ T h M T ",0"+x iS % % -, " T -! 0V " h4 " # " h+ 0 T 0,.!+ B* #9 3",#+ 0! 3 ""+Hi 5/0,. "5 _^

h1 ∼=(

N

µ− 1

)3/5 K %2W 1 M

h2 ∼= N3/5

(2

µ

)3/5 K %2WSWbM

h3 ∼= N3/5

(µ− 1)1/10K %2W M

h4 ∼= N3/5(µ2

)2/5 K %2W bM

h5 ∼= N3/5(µ− 1)2/5 K %2W M

21T &ZG3 T µ = µtot − µancre T h+ 0 HiG "5/"+ #+ # " hG >!X ,. R 5/."S%

"54hG vh+ 0vHi 5/0 u " +p,* "[% -03 " T !><2,O.0 0 0h+ 3K % M"K % W ` M T h 0 ! ",."!+O!

σ! 0w!" B*/Hi &% _ T 0 " O 0 0 7 0O B*

,. R 5/."0 " +h1 ≤ h ≤ h5

% a& T ! 0OHi T ;"5/6"+, h+ 6 1 ! 5/" T lnσ T h+ Q ]&ZGHi i -!

µtotT 5-,.S> HiR."V! ",."! /!

u rr

K | = &

s

1

0 hh

5h

4h

3h

2h

1

3

2

1

4

5

6

1

10 20 30 40 50 600.00

0.02

0.04

0.06

0.08

0.10

σ

h

µ = 20

! " % # M'-5JIL, 1=5JIL0ZA ' <=, C <=021@,ZF E;'F E[6, 5JIL,ZA 39F CD?EJ,ZF E3,Z6[5JI 3 A 3 -, 5@?BF 1)E;'F E$G'F 16, GN3')MN<W'A AO, G, 6' 3 M -7<=,)P ,)EG0 F3 E3?BF G,21I'-EJ,)-7<Z1 5 '<W'5)EJ0Z<43 1BE3 -,21 hi P GN<=?B3 EJ,

<=021@?B6 -E3?BF F -7AO0Z<43 -,G'F 186,U5 '1 µ = 20 ,)E N = 100 P

h&Zb σ % ? Hi ib" >"5/ !X ! 68Hi ib" 0 &0]0*X"0 T 21 0,. R 5/."0

#+3! ;Hi 5/! S%Qw ! " i 05 " 6hG h+ T 5-5/;",0"+0] Gx iS % r %B8<0!" B*

;4L 0v">HiG!Qw+p,p,# . 0 T 7 "Vh+ ! # 0v",#+ T h(x) Z O&ZG O,.S

x% 8 0 " #v,.S "+p! -V!5 5 " " ""ki 7S% ?

4 <2,,N* 54 S ZS+6^ 0 " $,. R 5/."00"+ i 7 "5/"+O!X -! # ",#+ 0!" B*a7 "&% 6 !X T iS Vh+ ] 0* P!0,. R 5/."08K _!0 a0!V7 7 4 54: 0],. R 5/."0NM;0]Hi i 0S% 6 0hG "+ T - ,,.S-hG 0 a0Vq+V,74 "5/"+-54 04 B* ;4L 0- !0 T "h+ 0O7 70"+3!# #:]

h% Qw"; R ,. .0]!>< ]!0 , + :

0 V 0! 5 ;21 < # " h+ ;!Z a! " 0eiS#!!"ZS+ ># !>< aS%

u r

K %

x0

h(x)

![" % r # *,)- ?:B,)E 1 -,Z65@?BF-,21 5@?BFFL,@5)EJ021 CY'< G,21 CD?B6]\ AO^Z<=,21+AO?:73 6,21P

? ; #6! -! ?iS#5-5/ %2W T " ,,.S#+6h+ "5/"+> B*n,. +6!><2wj #?ib T 0 a0]+tiS03 00] ;!0]7 ";, ,N . 0[% 8 " " ]ZG+] > "]!; >Hi 0 T ! 0V >,

h1 ≤ h(x) ≤ h5% 03 " T ! 0Hi

! 0V 0O,h(x) < h1

"h(x) > h5

T O" ""ki 0# %6 " O,. ON< R, "30O, 5-,.#+" <2 6!X " #! 6!>< '!03!" B*q7 "&% 6

h(x) < h1T < 0 "+, h+ Q! n BV"5/"+;h+ 00,.#7 Q!X n! ?ib T

#! \h+ w,. h(x) > h5

T 0 a0\+v#,74L"5/"+ " 00[% _ T 6 < 0hG 7 T "&603Z !6! a0&%

8 ,.S , " " < ""!X ;! &03 ZG"5/"+ 0ZG+w! ",."!X;! 4L5/, .">!07 "[% n03 " T 0 "

h1"h5

#+ # " h+ 0;!0,.?j R 5/."0w"w+ ! ",."!+0! i 05 " ;! < 7 "&%U 0w ;4 0$+$,# h+ "5/"+,# . 0 T & 0;"# 5-,.#+ T 6 "

h(x)ZS +wO.0 4L 7 "5/"+

&ZG x % c"ZS T h(x) Z '#, !"5/"+ T <2 #T! ;4L 0P"#,." S% ? " ! ! >, Z+ , "!>< , ."6"3!>< ', <%

6 0; ;! i21 0;!" B*n7" 06+ 7";, "; 7"," h+ T !P# R

R T 54 :"+ 6i P! N P5 54 d

K"x iS % M9% ? " ! 6 6!><27.! B4L54 !0 #9 ,. R 5/."0 T , ] 0 5/ 4L ( 0# ,. T "," 0T!" B* 7@"&% ? "#7 #T 4 j! # h+ =< 4 VZ " P,.S r!0-7@"&%BV T -! 0!X AD ,."+

u `

K | = &

!>< +"# Fa03 4 ], 0 " [%

X ;

R

h(r)

r

d

q

dS

e^

! " % # H0@?BAO0)E<43,$1 C IL^Z<=, 9C 6'FP

8 -! # 6 ",#+ 03!" B*a7" T h T Z 6!d

K ! # ]5 54 [M 6d+R

h(r) = R+ d−√R2 − r2 K %2W M

21r

0 ! # 6 <2-*!O R 5 " O5/0 0 ;4L O, S% ? v 0v# R # " hG 0

ri!9#95 "Q,

h(ri) = hi21

i = uW# X% 4L ! q,.S !d

,3#,,. B*q " hi

T , " ;,0":"+QK"x iS % u&` M]^

• 8<h+ d < h1

T 0],. R 5/."0 4L5/"+6 "R !X hG 4!V# R +" Tr1

T "6!-# R _0*:" T r5 % iZ !-! a0Q06! ,0"+Q "+4!X R '.95/S%

• 8<h+ d > h5

T , a] # 03!" B*a7@"3h+ +# "5/"+ ",0&%

• + 06!" B*( 60*XO."5/0 T 0 a0]ZG+] 0 ,."6 0, >, T ": 03!" B*a7" T "! 5 6,3 "R !X6!]# R

r5%

u u

K %

d

r3 r

4r

5

d

r5

d

h5

h4

h3

h2

h1

r3

r2

r1

r4

r5

! " % u&`8#S.7?B6 - E3?BF G,Q6' 5@?-5JIL,[G, 5@?BFFL,@5)EJ,)-7<Z1R6?B<Z1-,Q6'>GN3 1)E;'FL5@,RA 39F39A '6 , d '-MNAO,ZF EJ,P

)] /!" " T , " &0Q+! ViS 00Q! 0hG 0 0 a0#+60+j,. ZG"5/"+ BV00 6 1 ! 5/" K

0 ≤ r ≤ r2 T Hi 5/ 1 M T 6-W>! 5/" Kr2 ≤ r ≤ r3 THi 5/3WbM T 4 5/"+ w7#S 5-,006K

r3 ≤ r ≤ r4 T Hi 5/ M " 0]Kr4 ≤ r ≤ r5 THi 5/ bM9%

N B N

U: ,,.SQhG =< , "#" >0*X" / 4L −→F

a, ."- Q" iG!+ < 7@", V+*X T -V'!]54 +" T 6 < 0hG 7 T 03!" B*a7@" 6> ]! # 6! " T d % U 0 < +" T F T ! 4 q!8& T 0*" 08,* 0>,. R 5/."0 " / 0!" B*a7" ! ], 0 S

QO5-5/O \ ,,.S\hG 0 a00"+,.",."! 0 6 ;4 w, T 08+Q,54 0I6 ', .9#% 6 0h+ "+ T ,. >!0# #Q!8 R 5 " T /6! ! "" 5-,.S#+-,.",."! 6 ;4L V, S% i03 " T q5-5/!0 5-,.S#+0$,# . 0,. /# R

r! 30\+ S%98 4L # T F T N< $

F =

∫

surface(Πosm +Πel) d

−→S · −→e ⊥

K %2W r M

)] " i 05 " K"x iS % M T cd−→S · −→e ⊥ = 2πR2 cos θ sin θ dθ

21 <2Vi θ

0! 0V ,

r = R sin θ% ) <2,O.0>K %2W M T d−→S · −→e ⊥ = 2π(R + d− h)dh % _ T 74L 6N< T

u 1

K | = &

h5

h4

h3

h2

h

rr

3r

4r

50

d

! " % uSu # 5JIL0ZA '/G,6'O<=0 CY'<BE3 E3?BF 6?5 '6, G,21 ?B<=5@,21U6?B<Z1 -, h2 < d < h3PR,21X5JI'K9FL,21

1Z3 E4-0@,21HG'F 1 6 ,21;CY'5@, G0Z6939A 3 EJ0[CY'< r4 <=, CD?- 121=,ZF ER6'1;C IL^Z<=, E;'FLGN3 1 -,>5@,Z696,218143 E4- 0=,21>G'F 1 6'5@?-7<=?BFFL, G0 F3,[CY'< r4 < r < r5

L'E4E39<=,ZF ED6' 1;C IL^Z<=,P

!%4 0aO.0i ""# T

F =

∫ h5

d2π(R+ d− h) [Πosm(h) + Πel(h)] dh

K %2W M

8 < +HiS# 8!8K %2W M60! , 04" ! 3 ""+0 +HiS# 0 0,.!+ 6 h+ / ji 5/ T 00*X,0 !

Πosm"Πel

"#+!00, 00h+ >K % uSu M T K % u M T K % 1Xu M"TK % 1 M T qHi 5/ ! "_K"x iS % uSu M"% );5/C"5/ T n

σ(h) 0#

,. " 0 +HiS# 00 ]!0 0hG QK % u&` M T K % u bM T K % 1S` M"K % 1 bM"%U+ Ex iS % u&1 : ",0"+00 0 ZS p!

F (d) h+ 0p! + 7 wj

qS5/ h+ T Fosm(d) " + 7 h+ T Fel(d) %+Qw" 7.,." :C"! Z 0"d!" B* , 0&% 6 0 0q! # 0 T d < d∗ T F (d) > 0 T nh+ iS V(hG 0 " $",. #"+ 0!" B*/7@"&% 6 0 iS#!0$! # 0 T d > d∗ T 4 !"Z "+HiG ZG T F (d) < 0 T " 0 " ]0*" "+ 54 >!#,,. "+ 0!" B*P7@"[%8 -! # 6!>< 0h+ 7 T d∗ T 03 0]"+

h3"h4

%_ R " 7., "P! "# %J\P7"ZG6h+ ],. 03O.0 0! # 0 T

d < h1T /4L - ViS5/"+S% 03J" T Q #"5-, 6! a0 4 ;" 685/" VhG

u W

K %

0 10 20 30 40 50-100

0

100

200

300

F

J

J

J

J

J

HQ

T

M$6

C

FC

! " % u&1 # '<43'E3?BF 1 G,21 5@?BF E<43;: -E3?BF 1 ?)14AO?E3 -, Fosm H% E <W'3 E C 6,Z39F (O,)E0Z6'1BE3 -, Fel % E<W'3 E ,ZF CD?B39F E39696021 (U,ZF ?BFL5)E3?BF G, d CD?-7< R = 55 N = 100 ,)E µ = 20 % h1 = 2.7 h2 = 3.9 h3 = 11.1

h4 = 39.8 ,)E h5 = 51.5 (P ' 5@?-7<):,,ZF&E<W'3 EDMN<W'1S<=, C <=021=,ZF EJ,$6' <=021B-76 E;'F EJ,G,U5@,21G,)- 5@?BF E<43;: -E3?BF 1P R,[CD?B39F E G 0 -739693;:<=, d∗ .@'-ER35Z3 d∗ = 31.96 P

d ViS5/"+" 4L 4S5/ hG a&% I4L / "5-, T 4

F! 5 + T

!>< O, T Fosm ! 5 + T 0 a0 "#+!w5/ "V5/ 5-,000\" T !><2 O, T 0 a0Q 0076 < 0*X" " >! 80*" "+ I4L hG 4!/#,,. !, w", 5-,.#+S% 6 0$ZS w!

FelN< +ZG""+ T 5Q7! a0 " 00

! 5 + +&% d = d∗ T 4 S5/ h+ " 4 h+ 5-,.""+;0* "5/"+&%

BV T ,. !0",# , iS#!0h+ d∗ T 4L /S5/ h+ /0]Hi ib&7 8"

4L V# /";HiG ZG h+ =< 6 <2+ "6"h5

T #hG V, a- 03!" B*'7@"&%

) 3 ""+0 7.0T! 4L T F (d) T +a",0":0"P x iS % u W ,. T! 3 ju

K | = &

0 10 20 30 40 50

0

200

400

600

800

1000

1200

HQT

M$6

C

FC

0 10 20 30 40 50

0

2

4

6

(4

FC

! " % u W # M'-5JIL, '<43'E3?BF 18G,6' ;?B<=5@, F ,ZF ;?BFL5)E3?BF G, d CD?-7< R = 55 % E<W'3 E CL6 ,Z39F ( R = 100 % E<W'3 E 1 ,)ERCD?B39F E 1 ( ,)E R = 200 % E<W'3 E+,ZF CD?B39F E396960Z14( % N = 100 µ = 20 (P GN<=?B3 EJ, '<43'E3?BF 1 G,+6' ?B<=5@,UGN3 .)3 1=0@,[CY'< R F/R CD?-7<S6,21 UAZAO,21S<W' \ ?BF 1)P

"+0VZ " V!R

%\ #h+ 4L q ViS5/"+a&ZG R T 6 d #+&% 03 " T!><2,O.0 < 0h.%;K %2W M T F (R, d) 0 4 m /(!

R",." a! N< (

4L5/Q^F (R, d) = RF1(d)+F2(d)

%9) <2,O.0;K %2W M T h+ R

ViS5/"+ T <2 ! 6 ;4 0 T Σ = πr25

T ViS5/"+ & "5/":6&ZG R % 8<Q5Q7V! " 6 ViS5/"+HiG "5/"+ T ah+ 3 !X 6( c 0 #"5/"+4!

F% 8 <2 R '!0 7.0/5/+ T ,

" T hG ],. d > h3

T nF2(d)¿ RF1(d)

" T ! " +"ZS T 0 0 7.0

F (d)/R#Q ,.",.S"+-K 4%x iS % u WbM9% _ T 4! # V!>< 0hG 7

d∗ T 0>"+h3"

h4T ]ZS ]h+ ;O.0 4L 7 "5/"+&ZG R %

B N 8<0Ex iS 0QK % u&1 M"QK % u WbMO5/+"+hG

h3 < d∗ < h4% 8 /! # !>< 0hG 7

d∗0

! ! 0V ],

0 = 2π

∫ h4

d∗(R+ d∗ − h)

[(Nσ4(h)

h

)9/4

−(h

N

)3/4

σ5/44 (h)

]dh

+ 2π

∫ h5

h4

(R+ d∗ − h)[(

N

h

)3/2

σ25(h)−(h

N

)3/2

σ5(h)

]dh

K % G` M

u

K %

21σ4(h)

"σ5(h)

#+>! 0V 0 5-, " "5/":V, 0 0h+ aK % 1S` M9aK % 1 bM9%M8@< 0h+ yj oK % G` MV0 4 "5/"+4wj & #T"-,. 4! 0!X a '0*:,0 c R h+ T,,?j 0!

d∗(N,µ,R) T V ,,.S>h+ 8! 4Hi 5/ H 0hG mK % 1S` M K T G/g >> 1 T < 0+j 6yjk! h+ < ""ki hG !>< a "PHi ib&7 !"ZS+ < ""ki S5/?j h+ S% Qw"] R ,. .0#]0$iSS ." T #5-5/"+3,'."! 4 + ." T < 0+j 6yjk! ],.

h ∼= h4 = N3/5(µ/2)2/5" >! " 2V?% ' %<\'7 "+3! "6,,N* j

54

σ4(h) 'µ4/5h

N9/5

[1− 2

6/5

5

(h

h4

)2]

K % u M

8@< 0h+ K % G` M !X T B* iS#!3# R 3! 7 R T 6

d∗ = A(µ)h4

[1 +B(µ)

h4R+ o

(h4R

)] K % G1 M

21A(µ)

"B(µ)

! ",."!"+ 4 7 "5/"+>!µ

"Q+>! 0V >!/54 ." 5-, " 8, 00h+ ZS+06^

0 =7.21/5

15A3 −A+ 1− 7.2

1/5

15

− 1

24/5

[(h5h4

)3

− 1]− 4

[(h5h4

)1/2

− 1]− 2

11

[(h5h4

)11/2

− 1]

K % WbM

B = − (1−A2)

2(A− 7.21/5

5 A3)

[1− 7

5.24/5(1 +A2)

]+

1

(A− 7.21/5

5 A3)

×

3

4.24/5

[(h5h4

)4

− 1]− 2

6/5

3

[(h5h4

)1/2

− 1]− 2

1/5

13

[(h5h4

)11/2

− 1]

K % M

?h+ h5/h4 = 2(1 − µ−1)

! T ! 5 µ À 1 T h5 ∼= 2h4"ql Zb

A∞ = 0.504"B∞ = 0.085

%8 x iS % u ",0"+ 0;ZS !

d∗&ZG µ "

R T 00#6 4,wj,N* 54 <%2? #vhG 0v! ",."! 0"

µ"R

! < 0hG /,, 0]K % G1 M+O7 "aZ " V 00&% Qw0 7.0O5/:"+, " 3hG 54 iS; 6,,N* 54 PiSS ."4! !TK % G1 M T 7" 0]7.Q iS06,. 0 + " "++ wj5 " h+ 0

A(µ)"B(µ) T 54 " wZS " w=< 0 T ""Z T , T O0*:OC"5/95/":

" 7 6 B*q,,N* 54 4 0"3 B* + " "+3+ 5 " hG 0!0 !>< 5-5/ " B*! < 0h.%JK % 1S` M9%B8<0OZ " !00 " jk!0 w#+OhG =< ! ZG06K 0wZS " 0! bx iS % u +

A(µ = 40) = 0.78"B(µ = 40) = 0.035

M9% _ T d∗ ZS hG O.0

u

K | = &

4 7 "5/"+>&Zb 7 4! a, ."/"i,." T " piG!+QhG 0 !, ## /^

d∗ ' A∞N3/5µ2/5

(1 +B∞

N3/5µ2/5

R

)K % bM

8 < +","# S !X ( iSQ! 8! ",."! V"1/R T ! ""5 >, 85-5/>!, "

+ " "+ T 0O.0! 9 S%

0 20 40 60 80 10010

20

30

40

50

60

F

C

µ M$6

0 4000 8000 12000 16000

41

42

43

44

F

C

4 C

! " % u # '<43'E3?BF 1 G, 6'/GN3 1BE;'FL5@, G 0 -739693;:<=, d∗ ,ZF ?BFL5)E3?BF G, µ CD?-7< R = 100 % M'-5JIL,4(,)E G, R CD?-7< µ = 40 % GN<@?B3 E , (P R,21$1J\AO:?B6,21+1=?BF ED6,21S<=021B- 6 E 'E 1SF -7AO0Z<43 -,21 '6?B<Z1-,S6,21 5@?-7<):,21+1=?BF ED6,2186?B3 1>G,[CR-73 1Z1W'F 5=,+<=, CD,@5)E3 .7,ZAO,ZF E µ2/5 ,)E 1/R % N = 100 (P

_ , $! %4 T F (d) T 8! ""5 3,."+ !>< +"# T W (d) T "+ 0$!" B*7" T 5-5//^

W (d) =

∫ h5

dF (l)dl

K % M

) <2,O.0 <2 !F (d)

",0#"+0 x iS % u&1 T j " 0 V5-5/6!6!" B*q"5/0^

• a, ]", Zb],. 0,." 03! # 0 T d T 06 >,0 'S5/ hG 6! 6h+ ",. 037"

• Q"5/w# 4,. 0 DyiS#!0GF! # 0 T d KL&ZG d ≤ h5M 6 4 h+

h+ 037"&%

u

K %

QO5-5/>,. 0O.0,." 0]! N " T 0 < d < h3T W (d)

0O.0", 4 T +"0#6 Hi

h3 ≤ d ≤ h5%>)] " +"Z T V,."+ !>< +"# N<

4L5/ "5/"+

W (d) = Θ(h4 − d)∫ h4

ddl

∫ h4

l2π(R+ l − h)

[(Nσ4(h)

h

)9/4

−(h

N

)3/4

σ5/44 (h)

]dh

+ Θ(h4 − d)∫ h4

ddl

∫ h5

h4

2π(R+ l − h)[(

N

h

)3/2

σ25(h)−(h

N

)3/2

σ5(h)

]dh

+

∫ h5

max(h4,d)dl

∫ h5

l2π(R+ l − h)

[(N

h

)3/2

σ25(h)−(h

N

)3/2

σ5(h)

]dh

K % M

21Θ(x)

0 ]54 !&&Z !IHΘ(x) = 1

,. x > 0

" ` ,. x < 0 K %

Qw" !>< ] Z ,W (d)

,. 07iS#!O# R O! 7 R

% 6

h3 < d < h5T 0! # 0

h"d

+! < !X>!N3/5µ2/5

% _ T "n"54 #+ 0

0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.0

0.1

0.2

H

FJ

! " % u # ?EJ,ZFLE3,46QG 39F EJ,Z<W'5)E3?BF <=,ZFL?B<4A '693 1=081@,Z6?BF 6 0 -'E3?BF % )P @(H,ZF ;?BFL5)E3?BF G, d/h4 P <W'3 EC 6,Z39F N = 100 µ = 40 R = 120 E<W'3 E ,ZF CD?B39F E39696021 N = 100 µ = 20 R = 55 P

u r

K | = &

! # 03,N3/5µ2/5

" 03!" 0!-iS03 ?ib T σ T ,(µ/N)6/5 T ' Zb

W (d) ' R µ13/5

N3/5f

(µ,

d

N3/5µ2/5

) K % r M

21f(µ, x)

0 %4 TN! 5/" T",0"+0 _x iS % u X%\'7#9Zb]h+

f! ",."!lO.0 4L 7 "5/"+a!

µ"q!

R T ! _5/C"5/P54 ." hG 0 + " "+A

"

B! ",."! "+-O.0 4 7 "5/"+-!

µ! < 0h+ mK % G1 M9% \ -! 8,."+

!>< +"# P, 5-, "5/"+ 5-5/

W (d) ' R µ13/5

N3/5f

(d

N3/5µ2/5

) K % M

\ 7"ZG4hG W

06,.S 4QK +"# ", ZG[M;,. d < 0.6h4

T , !"Z "+HiG 4K +"# # Zb[M9%8<-5 5Q 5 0 +6,.

d∗, W (d) a @ hG =< 6

<2+ "3,. d = h5

%

0 20 40 60 80 1000

5

10

15

20

25

) M$6

µ M$6

! " % u # DFL,Z< MN3, G 'GNIL021Z3?BF G ,ZF;?BFL5)E3?BF/G, µ P R,2181 \ AO:?B6,2181=?BF EQ6,21 <=021)-76 E;'E 1 F -7AO0 <43 -,21>,)ED6' 5=?- <):, 6' 6?B3 µ13/5 % N = 100 R = 100 (P

8@< ""ki !><2!X0 T G T !0$!" B*/7 @"$0! 0V 5-5/ 3#&ZS 0# 6 < , j#" /,. / "," 0/!" B* 7@"/"/!><2,O.0TK % M T 5-5/

d∗ ' N3/5µ2/5 T c B*u

K %

iS#!3# R 3! 7 0

G = −W (d∗) ∼ Rµ13/5N−3/5 K % ` M

U+ x iS % u +q",0"+00 0ZS '!G(µ)

% (5/+ihG "

µ13/5N<2 @ -7 " &Zb 00 #+ 5 " hG 0 T /hG , Zb4hG 4

f(µ, cte)V! ",."!ihG >O.0 4 7 "5/"+6!

µ% 6 6 < #V! 8Hi

h4 < d < h5T 5/

4L f(µ, d

N3/5µ2/5)

03 Zb"# T ;#,,. dN3/5µ2/5

!"ZG"+! ",."!+!µ

%8 < " !;!0$,." 0S " 3 O! ,.S a!>< 0hG 7 T d∗ T ,."5/"!!"!

!>< ! #+6!]# !" T k T ! 4L 6!;#,,. T F = 12k(d− d∗)2

k =d2W

dl2

∣∣∣∣d∗

∼= kBT

a3R( µN

)9/5 K % u M

QO5-5/9:#'5/+! V < ""5Q7 n!080 #8, 0!"+ h+ _ ;4L T, P 7.]&ZG 7

1/R′ S%P 0O#; !=< ""! 0w0 #3 O21 0O!" B*7"+ !"+ hG 0 T < 0+j 6yjk! h+

R = R′ % '03J" T ! T >! # T h(r) Th+ ", 03!" B*a7 @"3 <

h(r) = 2R+ d− 2√R2 − r2 K % 1 M

8 4 -"+/!" B*(, ."0 T Fs/sT -! 0!X Q 6! 4 -0*" 0-"+- -, ."/"

', T Fs/pT Vib"5/"+!;ZS 7 /^

Fs/s(R, d) =1

4Fs/p(2R, d)

K % SWbM

Qw5-5/P a&Zb8Z: , 0!"5-5/"+iK'x iS % u WbM4h+ Fs/p(R, d) ∼ RF1(d)

,. 0iS#!

R T 3 ZG!

Fs/s(R, d) '1

2Fs/p(R, d)

K % M

n03J" T ! !6!" B*T, ."0 T <2 , @"0Q! / ;4L 0Q0 B4 " " 4!>< T, ."T"/ c, K 6

N T µ "8! h5V+*09MV" a5Q7P! a08h+

"+ 04!" B*r7@"40-, 4,." &% Q < 0 (# c,. hG < +" T! 4L T h+ ! ",."! 4 "5/"+!X '5Q76! a0 00 T ! 5 + S%

1S`S`

K | = &

0 10 20 30 40 50

0

50

100

150

200

250

300

HQT

M$6

C

FC

! " % uU# '<43'E3?BF 1>G,S6' ?B<=5@, Fs/sL,ZF;?BFL5)E3?BF G, d G'F 1 6, 5 '1HG, 1;C IL^Z<=,21 % E<W'3 E[,ZF

CD?B39F E396960214(U,)E G -7FL,$1;C IL^Z<=,U,)E G -7F C 6'F Fs/p% E<W'3 EC 6,Z39F ( % R = 55 a N = 100 µ = 20 (P

U+ x iS % u #+8",0"+00 08ZS q!Fs/s(d)

"Fs/p(d)

,. R =

55 T N = 100"

µ = 20%-\ # h+ =< R c7 " #,,.A6c,." ,O.0 iG 6

1 "+ 0q!" B* 4L 0&% l"ZS T _! # n!>< 0h+ 7 T d∗ 0" 7 "5/"+ !"wj h+ T 5/C"5// d∗s/s

" HiR.""5/"+ B4 " " 6d∗s/p

%=) <2,O.0 < 0hG lK % G1 M T 7 "

d∗s/s(R) = d∗s/p(2R) / d∗s/p(R)%

BV T !><2,O.0 < 0h.%]K %SWbM T P,."+ !>< +"# P#;4L5/ T ,. 0 iS#!# R 3! 7 T "

Ws/s(R, d) '1

2Ws/p(d)

K % bM

? a,. Zb'! ! 0!X ih+ N Zb"5/"+ < 03J"T! 7 i BV5iS # !X R O."5/S%U+ ,,.SVh+ =< 7" _ T ,V0*"5-, q," hG T 6' 7" ,0"+#+a, " 4L "0a! 7 0'! 3 ""+0_K"x iS % u r M"% _ < 0h+ 7 T n21 < 7" m0 Q,# 4L " -, , Q" 4L&ZG 06,#,,.%6 21 d0 6V T!]0 D GF T ! 7 6, 5-,.#+S%

1S`Xu

K %

(a) (b)

A A

BB

! " % u r # *U'F 1 6, 5 '1 %')( 6 ?:@,)E C <=021@,ZF EJ, '- 5@?BFFL,@5)EJ,)-7<Z1 -7FL, 5@?-7<): -7<=,>C 6 - 1 '3;:6,-, G'F 1S6, 5 '1X%;:(P;6R,21BE[G?BFL5 'N.7?B<43 1=0P

)] "$, T p"," 0 0 #v0"+ T "+ 0 # " hG 0! /! " ,. R 5/."0"6! < +"# "+/!" B*(7"; i !0 !"wj h+ 0&%9? 5 66!0$0 #w @4L5/! w!>< "$Z 7 0 h+ , 3,." 3# R T! 7 !0!" B*a7" T R T 0iS#!'!"ZS+ V# # " hG !0 *a ",. R 5/."0&%

_ ,."+ 5 h+ !0 a0 T µ T #+ T 0 ,. Hi q 0"+ 0!" B*T7@"! 4L

Σ = πr25∼= πRh5 ∼= RN3/5µ2/5

% 8 /# 654-* 54 Q!0 a00

h5 ∼= N3/5µ2/5" 6! # !>< 0hG 7"+# 0!" B*/7 @"$N< d∗ ∼= N3/5µ2/5(1+

BN3/5µ2/5

R )21

B ' 10−2 H 0hG * K % G1 M K % Qw"! # !>< 0hG 7q! 5 + ! O.04 7 "5/"+&ZG R %

8 +# + 0*X" 0n, ( 0q!" B*l7"q0 4L"5/"+ 5/iR."b%U "54 h+ "5/"+ T ,." 6! Z " /! " 6 < 0h+ 7/"i!" B*nHi ! 0QK'x iS % u M;^

• 0 a0 4 54+ Q0*X" " K ,. h4 < h < h5

M0*X" "+ 6"

1S`S1

K | = &

h5

r5

d*

P

P

t

t

t

t

! " % u # Q' 5@?-7<=?BFFL, , EJ0Z<43,)-7<=,O, ,Z<=5@, -7FL, EJ,ZF 143?BF G, 693 MNFL, τ -73 6 0 -739693 :7<=,O,21BE04M'6,6' C <=,21Z143?BF <=0 CR-76 143 ., P L'=CC 693 -0@, CY'<S6'/T4CY'1)E39696, V 39F EJ,Z<4FL,P

! iS# ZG8K "a 0kBT/a

M

τ ∼= F (d = h4)

2πr5' µ2R1/2

N3/2

K % M

• 0 a0 00 "+:4 5/9: D , F-hG 0*" O ,0 T P T ", j ZbQhG 0! 0V T 07" "#+ 6 < "h+ 7 T ,

πr25P = 2πr5τT VK "P 0

kBT/a3 M

P '( µN

)9/5 K % M

U < 7"$0 !>< 35/"5Q7# 4 5 "3, 37 , ! hG T < 5/i j ! +# + !X # 54 hG "5/": 6O \ -* ZG"< "# ! 7 7.& ,4, \,." S%Qw00 #N<2,, h+ "+$! , 6;!07@" !0\ Zb"!6 ;4 #+3,."5/"#+ -iS03 ?ib6!" " &%

8<,."+ !>< +"# T W (d) T "+ "!" B*7"O0O!, < 0h+ _K % r M9%

P 0v "54 O x iS % 1S` %2P <2?i v!>< V," !?j ,."+ 6O.0 O! # S%bUX,.0V0

h5 ∼= N3/5µ2/5"W (d > h5) = 0

%TP 05 5Q 5t,. 4! # >!>< 0hG 7

d∗ T " D ,L4 !" 4!X s, GF 0 HiG 6 < ""ki '!><2!X0 c G ' Rµ13/5/N3/5 %

1S` W

K %

0

G

d*

W(d)

dh

5

![" % 1S`# +696 -7<=,UG -CD?EJ,ZF E3,Z6G 39F EJ,Z<W'5)E3?BF W (d) P

8 4L hG T FcT 0## 6,. "," "!" B*'7@"07"+ h+

F (d)0

5 5Q 5 ^

Fc = Fmin∼= R

µ11/5

N6/5

K % r M

Q03 " T < , "#" ,, h+ \ 4L , " " 6Fc

"Z " p7 T 3! # ",#+ 0>!" B* 7"QZSP ViS5/"+"&%( q!4,. R 5/."0>8,." - >, 0h+ 7"

Fc, hG T !><2,O.0 x iS % u&1 T F (d > d1) < Fc

%8<0 " >0*" "+ %4L ]!]# T, 4L 7 6" 037"ZG+3N< iS" "5 0! 7 "5/0+&%

# $ E $ O = ! kC H p?6=H

)] ";, T " ! 3 ;Z 0 T !+ >5/"5Q7#60 , T 0,6!0],. R 5/."06 i 7#], _"%V+*XS%>? " ! ]!>< /h+ /6Hi ib" < ""ki !! 04L54 8! ;5/"5Q7#S%98 ;Z 0 0\! ,,.S0 BV 5/"+0* 7 S% q& T 4L !X# ,"!X;" 5-, 0w,, "0 h+ 0! Q5/"5Q7# !I6 <2 !>!X n5/+!+. V!>< ""ki V! 7 H u W 1LK %( Q,7 ."5/V!"Z "+] O.0 5-, 0* T "3 3 :4L 3, V 6 0 " ,. R 5/."0[%

? 3 ,,.S HiG "5/"+h+ 03,. R 5/."03=< +"#?i ":, F >5/"5Q7#S%

1S`

K a & =

03 " T " >0*X" / 4 / -,. +Q!><2 #?ib4" q5/"5Q7#806! 04L5 0 "5/"+&%=Qw"! 04L54 ?i 6/T 0,. R 5/."0;&ZG +" !X +"# (N ZG"+ 03,. R 5/."0 T Z " 06, V5/"5Q7#S% Qw"03 " "9 wj! q"s! "# , " (hG 0 H 1Xu T 1S`-K " T ,- " !q 5-, " T V ,,.Sh+ =<2 +"# P ,, "5/"+# =< 0* ]"+ 0 a0&%

8 PZ 0 (KL [M>0- ,,.S0q! i 05 " hG T 5-5/", "#"+a x iS % 1Xu % w0 ,."5 &7 O" 0 63 , , $,p!0 ,. R 5/."0AiS03 0&% , "#" 60*X" T !4L0(h+ jk# h+ T 4L >!Q# T F T ,, hG 0- (5-5/"! T 6 < ! T ,q0*"5-, T !>< P, n, hG S% U PZS C" ! # (! ",# T h T " -!" 6!-iS03 ?ib T σ T 6 < 0h+ 7S

R

h

L

q

F

F

! " % 1Xu # *,)- X1Z3 EZ-'E3?BF 1SG 0 - 3 693;:<=,$G - FL,+5@,Z696 -76,[39F F39AO,ZF E1=?-BC 6, 1=?-7A 3 1=, -7FL, ?B<=5@,G,E<W'5)E3?BF F Y,BE 5@?BFFL,@5)EJ0@,O-7F C 6'F , CY'< G,21 CD?B6]\ AO^Z<=,21+AO?:396 ,21P

? " ! >, 0 685Q7-! a0 T f T #+&% _ T 0;Z 6! !" >!biS03 ?ib T σ = f/Σ T 21

Σ0 <2 V! ;4L 0 T #+ hG "5/"+

00 6 0!Σ

% 8 h+ =<2 4 =< 0$,, hG 0 T ;! # 3!>< 0hG 7 T h T #

1S`

K %

0 4 + ."0 u " V!X a! ?iS#5-5/6! Gx iS %2WX% '03J" T ! ";Hi T!X ! ?iS#5-5/ T +# +0*" 0, 0 " $ 30+ S%:U: ;Z " !f

K T!σ

M T Ph ∼= N3/5 ,.

σ ≤ N−6/5 "h ∼= Nσ1/3

,. σ ≥ N−6/5 %

8 h+ =< 4L T F T m+ 0T,, h+ 0 T _(! 04L5/S%E\ T ;4 TS T ! a0#+ #+ T <2 '! ;4L 0'ZS(! ! 5 + "4,. 7 " ! 04 54 Q03 ZG!0 a0&%N8 !" \!iS03<?ib ViS5/"+$ " 6 " +# !0 a0Z]N< ,,.S" ;,0 8S5/ hG !0\5/5/."0&% _ < 0hG 7 T R O."5/3!9#95 <2 0"85 5 N+ < ""ki 7! ]7S!3,. R 5/."0&%

? 6 ,,.S#6h+ =< 6 < #+ KF = 0

M T 0 " 54L5/"+6 -7S-!,. R 5/."0K

σ0 > N−6/5 M9%? 6 6, 0Q! 4 + ." a!X ! ?iS#5-5/8! x iS %2WX%\U: <2 c!

F T 0 " -,. R 5/."0/0*X" "+ T 6 < 0h+ 7 T +# +(h+ < ,,.S 6

F"8 q"+! iHi _! h+ 087S0

+" 00&%=?hG / σ0 < N−6/5 T 4L + 7 6 5-, 5/" iG 6 1

! 5/" ! a0 T h 0N+ 6,." -,O.0 #+ #hG =< 6 hG =< 4 iSσ0 ∼= N−6/5 %

? Q ,,.SQh+ =< 6 < #+ T 0>, "Qh+ <2 ! T ;4 00

Σ = S/2 = πR20

%+8<!X ', 0 T V ;4L 6! ]0 #+#/^

S = 2πR20 = πR2

(1 +

1

cos θ

) K % M

21θ

0 <2Vi 4L5 (, 5/"5Q7# 6 N 7rK"x iS % 1Xu M9% m#+x = R

R0

T < 0h+ K % MO,."5/"!] " <2Vi

θ6x

cos θ =1(

2x2− 1) K % S` M

_ T V " L

! 6 <

L = R tan θ = 2R0

√1− x2x

K % Xu M

Q$" >" T! iS T τ T h+ < 0*X" ] " 6! 5 #+ V ;4L 0S% a7 ""ki " h+ T! ", "5/"+Z

dL !X 6

FdL = −2πτdR K % S1 M

1S`

K a & =

Qw5-5/ - ;4 6! >Z 0 ]0 #: T -! " ZS P! < 0hG K % MO,."5/"! " 0ZS 7 0

R"θ T "$&ZG K % Xu M T 47 "+ < 0*:,0 4 Z+,. ;"

! iS/^

τ =(1 + cos θ)2

cos θ sin θ

F

2π=

2

x2√1− x2

F

2π

K % WbM

8@< ""ki 6! ]! " 3,. R 5/."03! Q i ># T 6 <2 !]! < 0h+ K % 1 bM 6>5Q76! a0 #+>K "a "

kBTM^

F ∼= Nf

[2

5

(h

N

)5/2

+ 2f

πR2

(N

h

)1/2]

∼= Nfσ5/60

(2

5h5/2 +

2

x21

h1/2

) K % M

"'#+

σ0 =f

πR20

et h =h

Nσ1/30

K % bM

8 0 0h+ !>< 0h+ 76! 6!],. R 5/."0+

∂F∂h

= FK % M

−∂F∂R

= 2πτ = F2

x2√1− x2

K % M

8 q,"5 ." 0h+ i#!X < 0h+ 74!0%4L 0! a! ,.",."! 6 ;4L ]^ 4L ,, h+ 0, < , "#" T F T 0 0hG 70, 4L O!O#,,. h+ 0*" 0q, 0 a0/ S%@8 (# ! 0h+ s#!X < HiG '!0V#9 ! iST ;4L T! T"cHiG!l&Zb a, V+*S% \ 03J" T " ! iS 5-,.S 0, < , "#" ;"! 6/! 5 + " <2 > 0 0 7 > B* " &%W\ T "P#0#"#+ T 0 a0 ViS5/"+"+ " +"# 3!ZG 5/;0* T " " a! iS0 0hG 706, "],0 TS5/ h+ S%TQw0 0hG !X #9: 6

h3/2 − 1

x2h3/2= F

K % r M

2√1− x2x

= `h1/2 FK % M

21 < '

F =F

fσ1/20

et ` =R0

Nσ1/30

K % ` M

1S`

K %

8<0! N " T ` "h

+! 0;"T 0!Nσ

1/30

h+ 0 / " ;! /7S6 < 0hG 7 T 9 74L T F T "' 0!

fσ1/20 ' f/ξ T 74L hG 60*X" 0

;4 56 < 0hG 7S%"54hG 3hG ;,.

F = 0 T a" Zb]7 " < "#!>< 0hG 7 K 4 + ." bM T < 0+j 6yjk! h = 1

"x = 1

%

8 R O."5/!>< 0hG -K % r M # K % M0 4L"5/"+wj & S% 8 0 x(F)

"

h(F) 00$+ 5 " h+ "5/"+O+",0"+00O x iS % 1S1 ,. $! 3 ""+0$Z " !`%

0.0 0.2 0.4 0.6 0.8 1.0 1.20

10

20

30

h

F

0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.2

0.4

0.6

0.8

1.0

Z

F

! " % 1S1&# '<43'E3?BF 1/G, 6' GN3 1)E;'FL5@, % 1W'F 1 GN39AO,ZF 143?BF ( 1=0 CY'<W'F E 6,21 G,)- ?:B,)E 1 h =

h/(Nσ1/30 ) ,)EYG -+<W' \ ?BF G,Q6'S1B-7< '5@, 5@?BFFL,@5)EJ0@, x = R/R0

,ZF ;?BFL5)E3?BF G,Q6' ?B<=5@,'=CC 693 -0@,1B-7<H6' 5=,Z696 -76, F = F/(fσ

1/20 ) CD?-7<>6,5 '1G -7FL, G,ZF 143 EJ0 G,HMN<=,Y')M, 5@?BF 1BE;'F EJ, ` = 0 % E<W'3 E

C 6,Z39F ( ` = 10 % E39<=,)E 14( ` = 50 % CD?BF E39696021 (,)E ` = 100 % CD?B39F E 1 E39<=,)E 1 (P

? Q&Zb6Z " V -+ 5 " h+ "5/"+>hG - 60 Q! -Hi 5/ a!X _! ?iS#5-5/! )x iS %2WhG >h+ > /4L

F% 8@< 0*:" (54-* 54 V!0 a0]07"+

,. h = N T h = σ

−1/30

% "54hG T -V ! 4L " " V! ! T h+ T!><2,O.0 0 0h+[%$K % r M6"qK % M T >!>< 8!" 8! iS03<?ib/V+* T x = 1 T 0,.!4L5/ "5/"+w \21

` = 0%2? \7"ZG\ tx iS % 1S1 hG T 6 F

V+* T x ! 5 + 3"

h ViS5/"+]&Zb ` % 03 " T ,. O !X 65/C"5/,0 S5/ hG ! !

" T Q#,,.R/R0

! 5 + >, #, !"5/"+6h+ !R0

0 iS#! % 8< , ?ib

1S` r

K a & =

"+ 4!" ! iS03 ?ibQ" - " h !X ; %6- Q ViS5/":# (!

h&ZG `

,. 3 4 FV+*0S% Q$" 3 ]0*X,0 T R hG 6!

h(F)"x(F)

! 0 0*XOC"5/0&%

6 0 4 0 4L 7 0 T F ¿ 1 T T

x ' 1− 12

(`F

2

)2 K % :u M

h ' 1

3F

K % 1 M

6 0 4 0 5-,.#+0>KF À 1

M T - P R 5-, hG !/K % r M # K % M30

h ' `F

2

K % WbM

x '(`F

2

)−2/3 K % M

?>hG ' "+h

"F

0V & S% Qw5-5/ /5/+ x iS % 1S1 T "Q,,N* 54 n03!><2 #+, 3Z " V 06h+

`0$iS#! %

8 -! # T d T h+ 60 " 0 N<

d = L+ h−Nσ1/30

K % bM

\'"54 #+ T d T ,d = d/(Nσ

1/30 ) T '7 "+

d = 2`

√1− x2x

+ h− 1

≈ 2`

√1− x2x

K % M

Qw">!" ."V,,N* 54 0@ V 0 T "P,# h+ T (-,. QZ ZS+ TR0∼= 5 − 6µ 5 T ` ∼= 100 − 1000 %=8 '! #

d0! "#"+ "5/"+V+* 0-,

0 # 6!0,. R 5/."0 6 -! 5 + n! <2 6! - ;4L 0S%

? " ! "-, < ""ki 4!><2!X0 _"+ Z 0 -" - 7#&%Qw+# "5/"+ ! Q, %2W T ;,."+ 5 hG ]!0 a0=< 0O,:V+*]" 4 ;! 0 5/" < ""ki V, a h+ T 0# >,. ]h+ =< a->! "# S%) <2,O.0 5/" H u S1LK T iS#!" 5-,.#+80 q" _! iS4hG N< 0*"

1S`

K %

1.0 1.2 1.4 1.6 1.80

2

4

6

8

W

hc

! " % 1 W # '<43'E3?BF G, 6 0ZFL,Z< MN3, G 'GNIL02143?BF W 'N.7,@5 6' GN3 1BE;'FL5@,>5Z<43 E3 -, hc NC <=?7.? -'F E6, G0)E;'5JIL,ZAO,ZF EG,21 5@?BFFL,@5)EJ,)-7<Z1 G'F 16, 5 '1XG -7F , :<=?)1Z1@,AO?:396,UCD?-7< GN3 0Z<=,ZF EJ,21O.@'6,)-7<Z1GN3 0Z<=,ZF EJ,21>G, ` 5 '1HG -7FL,:<=?)121=, XG,ZF 1Z3 EJ0 0@, ` = 0 % E<W'3 EC 6,Z3 F( ` = 10 % E39<=,)E 1 ( ` = 50% CD?B39F E39696021 (,)E ` = 100 % CD?B39F E 1 E39<=,)E 14(P

/7 7_"5 6 aZ 0 S%MQw5-5/4 Q#5-5/"> 0Q! /Hi 5/ '!X ! ?iS#5-5/ %2W T 9/4L h+ 80 h+ "5/"+># 0 6 '# !X 7 7 T Λ T hG j5/C"5/! ",."!qh+ !

hH 0h*%JK % 1 WbM K "O0O! ! ",."!+3!

σ% ? w ,,.SO!

h+ 0> 0Q!0 " >#/! "# "+>,. Q 8G*X" h+ T hc T h+ ! ",."!! (, R ?j 5 P! <2 #?ibS% 8 x iS % 1S1 5/+PhG ',. 4

hcV+* T 4L

0# T FcT ,. !X

hc! 5 + h+

` ViS5/"+ T "3054-* 54 6!

!>< 6]!" !iS03 ?ib #+>K` = 0

M9% 03J" T 5-,0 "# ! @ ;!>< 4L ;!# ' ,, "5/"+# ] 0w,. R 5/."0 H # !"5/!X (5/"5Q7V!GiG V! < 0h*%K % r M K T h+ N< +" VI685/0 VhG !" V!biS03<?ib ViS5/"+S%

) <2,O.0 < 0h+ rK % M T 6 hc #",.!n > " hG T dc % 8@< ""ki -!=< ! j T W T 0!0],

W(hc) =W

fNσ5/60

=

∫ dc

0F(l)dl

K % M

U: x iS % 1 WP+>",0#"+00 0 7.0V",0"+#+ < ""ki a!><2!X0 r"4L ! < G*X" hG T 0h*%*K % M T ! w!>< 7S5/7 ,. w! 3 ""+0ZS " !

`"! 6!>< /7S 6a!" /!piS03<?ib #+S%)] /!" "

1Xu&`

K a & =

T < ""ki N<

Wf (hc) =

∫ hc

1

(h3/2 − 1

h3/2

)dh

∼= 2

5(hc

5/2 − 1) + 2(hc−1/2 − 1) K % r M

? ##3hG < ""ki 6!><2!X0 P ViS5/"+&ZG ` T 21 V!" ]!tiS03 ?ib0 V+* " !X N+ 6Q ""ki ]!><2!X0 T Wf

T 7 "q, 4 7 S% Qw " 0 ,"+ ,"5 "]7.!T, h+ 6 ;&ZbZ: T, 0!"5-5/"+]hG I4 hG 0N 6,. ! "# " 0 " O! 5 + h+

` ViS5/"+S% n T !><2,O.0 < 0h+ (K % M T

#6h+ q! # dc

0]7 " , -iS#!-! ]! " 65/7 0" ViS5/"+ & "5/":$&ZG ` % _ T 5-5/ N< " T O#&ZS ! 4 N< 03 O, iS#! ViS " " < ""ki 3!><2!X0 ViS5/"+&ZG ` 6 hc V+* S% 6

`À 1 T ,. Zb " 4 5Q >,, 0qK % WbM3"!><2,O.0VK % M3">K % M T d ' 2 ` h3/2 %? 37"

W ' 12

5(hc

5/2 − 1) K % M

8 < ""ki ;!><2!X0 '!"Z "+3 ! ",."!+]!X ,#5/."` T <2 ViS5/":# '!

d&ZG

`"#+ 5-,."06, -! 5 : P! 4L

F"`−1 %

Qw00 #5/:"+>h+ T "_,# hG T /, 0# Q!5 +>! 4! "# "5/"+!>< -Z 0 -hG 0; 0 68 i 7#]# !V,6!0 " 5/7 0]=< 0;,6,,#95/": , " :4L !\# >0*X" 0$ 0 a0 54 , :4L !X , <2 ViS5/"+# ! ]!" 3!7iS03<?ib T < 0+j 6yjk! 5-,0 "# !0,. R 5/."0&%

6 Vh+ V R O."5/>0Hi ,]!0 0hG 4L"5/"+]wj & 0 T ";! " !>< 0*X# ! " !0\0 # 6 %4 w R h+ 0$"$h+ + # 4 [% 4"ZS T 0 +"0#+v!$! "vh+ N Zb"5/"+ 0 0 #ph+ \ v&Zbp7"+ &% _ #yj "5/"+ T $5-5/0"+0w!! ViS "w! 0,, "0!X0 Zb0$! @ T 6h+ ."Zb6!03,, "0! 6! " " ]h+ ."ZG]! VZ 0 S%

_ T $5-5/0$7 0:6 # " "w @ 85 S , h+ , ""ki , /!4 ;4L T W T !- /54 ."4h+ -#[Z ! < , "#" #hG a 4L ZS ]!

dA03 5-, "5/"+

WdA% n ; ]!0 , '=<2V!]"h+ ; W 0O

# " hG ! S%9);O5/C"5/ T &ZG -,. +!OZ: !O5 h+ 3!0\ !0 T

1XuSu

K %

" ;"+V! # " " V! " ],; (5/+!X h+ T ,, " + O.0h+ 8! 8! " &%M? ZG R QhG /=< 0 "5/"+>, , h+ 0 ,, "0p!X0 Zb0 !0 @ !v,. R 5/."0 5/7 0wK "6, " T #&Z ! < , "#" NM36:3 !0 # " h+ 0!0 " T !0 # " hG 0!0Z 0 0 T 54 p7 "V!X V R O."5/ " Z 0 S% _ T $,#5/."! ""5 +,. 4",.>!X ( R O."5/>" Q#,,.

` = R0

Nσ1/30

hG iG! 4# I6 /4L !0,, "0! >Z 0 4K

R0Mw"! 0!0 " >K

N"f

M9%)] 3,.",. Zb, Pi ""# T 0 +"0#+$!><2 R " 30 # 6 5 ."

!0#&ZS B*'!>< _ 0*!" -5 h+ Q!0 !05/ H K % 6 ,,.S B*a54 j B*- !0\!X T 054" B* D 5/ GF T 5-5/ 0ib \!O,. R 5/."0 05/ #0 T +!0 !0Q +# +8" " 5 hG 80! ""5 04, " +# + T , h+ ;, 0 # " h+ 0 + '.0hG 0!X a54" <% ); >5/C"5/]54 ."6h+ >",. h+ ;!=< #!]! iS #]0O!9#95 "6, >" 'hG =< J 5-,.S T ;5/+!X ! " # "5/"+$!>< ib 0\!3, ;,0 4S5/ h+ 3h+ 0 5-,.S0QK "\/,, V!9 Z 06 !6!X a,."+ !>< +"# n03J 4"+6!" B*+" !!X a0& *M9% 8 5 h+ Q! Q! " ,." 5-,";,,N* 54 ZG"5/"+ 6 Q!>< ib !>,. R 5/."0&%I\ 5-,"!(! h+ >#/",.#>"(!G4L54 Z8! ",."!X " yj "5/"+]! +# + Vh+ 0 5-,.S0qK ! h+ +"0#[M3, T 6#&Zb" q!" -!_iS03<?ib S% Qw" ! 0 T < 0 ",/! < 0*:#,. "6Q!0O O, 5-, 0*0 T 0w !>< "# iS95/":3!>< ,. +!Z: 7 ?j , R h+ T $ ViiR." 4, " ,.,.S 7 ! + !0,, "0!X0 Zb0w! @ 8, O j 5/C"5/S% iS w! ""+ 6 # "5/"+ D , R h+ F! ; R ,.]! <%

# $ F C= !

)]] "# /5/0 T V#&ZS ,0#"+-! , 4>, /"_5/+!0 O."0 0$, + 9#"##+0O"w w5-5/0O,.9 +j C",O" ",.S j 4!",.!X B*-h+ 0 0, ! 0$hG 0\ $ ."ZG"+&% _ T "5Q7 OhG 0p!" B*> hG $ &Zb ! "00"V! "# ,,. "+v "5/":!03!9ZG ,,."5/"+ " " &% ? 3"T! ViS hG /^

• ] !4Qo2 ^ F ! % / "*H '(V% / % Q;* ! %+P N<2?i # 3! 5-,"!X 5-5/"+

1Xu&1

K

+# + ++!X ;, 0 " 3 , 6 B*! 04L54 3! > ;4L #hG j " ,.S'."! 5/+!X ;! 7 V T h+ 0w7 "q"+"!X O",# hG S% ? ,. T ,>0*X"5-, T Zb" 4L5/4!>< 0h+ 78! aZ 0 T"O#5-5/"+ " <2Vi 6,,"+!# 0 !"5/"+! Z 0 ; q 7#3 B*,, "03! 6! " [% 6 " T 3#&ZGOh+ 0O5/"5Q7N0# , 0 "+># < 03 "Q! <2?i # "5 h+ S% P "# +"0N+Q! 5_j,"!X 5-5/"+ 0 " -3J "+ 0 &%

• DEQ[> / /QHQ %> % HQ %+? &Zb " ! Q a!>< 0h+ 7;"w w=<2&ZG,* ! " " " h+ 6! '"+ 03!" B*q7"&%<\ T "# 3 R ,.0!><2wj #?ib T 5-5/ , i7 jk",#&Z ! T +'!0 T# h+ 0'" 003 " " " h+ 0 "+3 O 5-,.#+&% );"0*:, " " 03 <2!X0 a"+ TZ 0 n"8 l 7#8 ZG"a!T "," H u K +5/+ < 5-,.# !","5/."0 " " h+ 0&% n, " T 8b",,.0" .+5 " "Z !" 6 4L Q+ 5 " h+ "5/"+/">0*X, " 5/"+# "5/0+/hG T! # 85 54 ,."5/"#+ G* P! ",."! 7.& ,P!X D "5-,! TFq"+ 03!" B*a7"&%

• ] B% ( ! "(5%b% Q 2f2 H H( B% %=)]] B4 +# _! 0 4 B*(0*X, j " 0>#-,.S#"# ThG 0 ! < 5-,.# q!><2 0 +"# Vh+ 0Q, j5/":O!X 0 B* " #&%B? $,."#O", " B* +"# 3# Zb0!V R ,.-Z_!" S 6!+ q,.0/0 5-,#7 6 < G*X" _ 4!>< *a T " 0 T ", Zb0 T !>< i #S# hG S%=P "# "# "5/": +"0+jN+4!>< " ! ",80*X"5-, 8!>< ZS+ +":!0 8"4!'Z 0 0 *ki 0"[% BV T ,. 0 0O.0 , 0 T #"# 0# >!>< < +"# j T"+ 0 " Z " 003, >5/"5Q7# T ]h+ < + R " (hG 0H 1Xu0K %

• ] B%p2V ! "k B% %:8<0q * 0! _7 i T ,aG*"5-, T,#7 .95/! < !0 T 5/""+8" @" c!0-, 0 :"+4 iS#!r5Q7! kib0&% ,"5 ." "#,. "# ! 6Q04L ! ?iS#5-5/6!0wHi 5/0! p21 0 ,. R 5/."0p: ki 0&%L8 3, $" 5-,w! ,0# >S5/ hG !" + j !X # 6> ;5Q , SP!"3Hi 5/0&%

1Xu W

K %

1Xu

!

_ v! "$ .0 T v&ZGv,,L4 ! """!X <2,, w, 95/9 i h+ T! "ZG ,, 0>,` T ,. < " !>0 h+ V!0 0]!Q,. R 5/."06 B* +";4L 0&%)] "',,# T a! *a "8!7 00/0VZ 0 5-5/q r""5Q7 !7. 0 4L5 00w, 0$,. R 5/."0O" +"# T 6 < 0hG 7"5/+! R 5 hG b% Qw"0 q8! ViS q! T0 5-, 5/ R " !>< \!O! T 6T#&Zb"> 4L54 5/ 5-, T 0v! 6 <2 !! 1 5-,^ ! 7 /"V# O!0v7. 0 T S T " < ! ;,#5 "#+ "5 '!0O7. 0 T z % _ T +# "5/"+ O! Q0 ]! 5-,(5/ R "(hG Q h+ "5/"+ I4 # nZG 5 hG T Φ T ,."5/"]!Q! ""5 " 0 # " hG 0$! 5-5/## 3 ;h+ + 3!5/5/."0w!7 0&% "ZS T < # P! 3!>< 6;,."5/"3,3!>< "Z "3, " "5/"+ 0 + " "++ 5 " h+ 0&%

? &Zb45/+Th+ "P,, P0/ '0 TZ S% 8@< "+, n!7. 0,,#aq 5-5/' ' ZG 4 0c!T! "5Q7" 04&#Vib"5/"+!0 a0 6 < 3! T !X , ,0" ! ;4L S%2)] 5 N →∞"! !>< P# Z+

Θ T " ZG -5/C"5/! N !Φ

! h+ 0 5-,q5/ R "<% Qw0O!" B*40 0O=< + ",."!+O, 0w5/C"5/0 5 # "a,,# #"+q! n 5-5/ 5-, "5/"+# &&%:8 _0 (" !>< (,."5/" T ,0*"5-, T !-! / 0 " ! !/ Z+" T ! 6! <2!, !>< # s"5 jk! 0a"V!>< +#9 4L q! 4 !X !,. R 5/."0 T !><2 0 0!"4 B* # " +j h+ 0Q! hG ! ",."!"+-!

N% ( Q T 0Q!" B*_0 0Q >"5Q7 "+

C"!" B*/Z w! 0w!0 0O!,. R 5/."0^: wh+ 60 " w!>< 0Q 4Z , D ,. R 5/.9#NF T #"5Q7 7 6 <2,, _ 0*!"+jk! `;"0Q,. 07S#0 T <2,, 5-, 5/ R " #8 4L54 5/ # ! T, R hG !0 h+ !0&%_ T 4 ,w!! q"+ " q" 4L $, < R ,. .0

1Xu

! < 0* " !;7. 0 6 < +";4 S%\d"ZS T hG 0 m! Z " a! < ""ki 7 6 < 0h+ 7 T < 0+j 6yjk!

" q! ;4 T !X 6!0$0 #w! 3 ""+&%P N<2?i 6!>< O.",."5/"#+!ZS !" <2,, /Z 4" 5-,#+> B*n5/0 00*:, " 5/"+# 0&%? ; 0Z ! " 4! ;4L 3&ZG 54#w5/ !0 a0 6 < kiG # !0 056 < +";4 V" 6 <2# , " VhG >!07. ! a0 T 5/C"5/>h+ %4 # 8Zb 5 h+ 0 #+#>K ,$0*"5-, ! $!>< /4L!X 5-,0# 7 [M9%? \,,.Sw 0*X, ! 0Z w7"Z "0$0*X, " 5/9:# "5/"+ T 8&Z 04!", 6, 9 Q00 9N+ a! ",."! 8!06,, "0 + O.0h+ 0Q!X h+ ! T 0 5-,0 7 T &Zb N %N8@< "+, !0<7. 0 !X M6w 0! # ! 6" ! ;4L ,. 0$O.0 ViS 0 *a " T hG 0 654h+ !>< ! 3 "" h+ # Zb;!><2&Zb 0$, 0!"+#"0 0&%? w0, "Oh+ ! ZG 00*:, " " 0 V" T!6 ;4L 6!0 h+ !03,. R 5/."0#"+35/9 "0" 6"&%

8 (0 '" 4!>< T Z+ HiG "5/"+8 /!><2 0/,.",. ZG0&%@8 0/ R O."5/0 ki 0"_+] _0*95-, S%? &Zb]Z: ih+ 5/+! # !- R O."5/0]5/"#+Q"" '!0,. R 5/."0 ki 0] 56V ]! iS 6! 5-, 0* 6" , " " &% 8 <2!! !>< 03 " SN hG 05Q , 0,#5/."0" 0Hi 5/0" 0 3 0N 6!7 "0V " "Q!-, " Q,, 0&%? &Zb]5/+/h+ q0 /" 6!>< 8,wj,.# w!0 "5/"+w!",. 9+ < !, !0,. R R 0 6], w!>< _"5 jk! 0S%? ]! ZG 5-, .""5/0+ 0]ZS 6!X i,."+ S#yj h+ 6! Q hG - ! VHi P21 ],LV ! "+# Sn0, &%

\BV T ;!54 ; ;"OZS ;!0 " 5/7 0 "]7.! T 6Q +j# T ,. O,"5 ." 4L &%N8<0<,, "0p!X0 ZG0 ! 0 @ ! ",."!"+ + 5/"5/"+! _i 05 " !07@" 0"3!0 # " h+ 0!0,. R 5/."0&% 8@< " !! 0 R O."5/0 Z:Q!Q ZG 0,.",. Zb056/, Vi"5/> <2!X0 V 7 ?j 5 5 " h+ S%

1Xu

Q Y

!

)] "0*X T 0 ZG0* "5/"+ < 0h+ (KLWX% u WbM T ! $!>< /4L!X

!,. R 5/."0 T h+ !X ] P,#LV !7. 0 T S %T? 5-,]" "> &ZG - P,, 0" !><

S(n) ∼ 1n1/2

%'#+

S = S/S0 < 0hG P! 3 ""+ Z " V 0],

SN< !

S′′

S′ + kS2 = 0K _ % u M

21k ∼ 1 %+8 0 ! !]7.!aZ " V 00,

S+^

S(0) = 1K _ % 1 M

S(N) = 0K _ %2WbM

;,"5 ." +HiS# n!/K _ % u M !X 6

−3kS′ = S3 +A(N)

K _ % M

8 - T! < 0h*% K _ % u MO03! ""5 0,

n+B(N) =1

kA2/3(N)ln

[√S2(n)−A1/3(N)S(n) +A2/3(N)

S(n) +A1/3(N)

]

−√3

kA2/3(N)arc tan

[2S(n)√3A1/3(N)

− 1√3

] K _ %bM

1Xu

Y

0 200 400 600 8000.0

0.2

0.4

0.6

0.8

1.0

S

n

1 10 1001E-3

0.01

0.1

1

S

n

! " _ % u8# ?BACY'<W'3 1@?BF G, 6'U1=?B6 -E3?BF , '5)EJ, S(n) 5@?-7<):,C 6,Z39FL, ,)EG,[6'U1@?B6 -E3?BF '=CCL<=?5JIL0@,+,ZFX6?B3YG 0@5JIL,Z696,5@?-7<:,+,ZFCD?B39F E396960@,21 '-O0@5JIL,Z6 6,21 6939FL0 '3 <@,+,)E 6?@M'<43 EIA 3 -, % N = 808

k = 1 (P

1Xu r

Q Y

21A(N)

"B(N)

+ 0 #:0/!>< +HiS# c! ""5 00 F 0 0h+ TK _ % 1 M"QK _ %2WbM9%

U+ _x iS _ % u +3",0"+00 03 0h.%<K _ %bMw"

S(n) =

√3

2

1(n+ 3

2

)1/2K _ % M

Z " V*+S(0) = 1 T ,.

k = 1"N = 808

%

QO < "+, Q " 06 V,. R ! ,." 6"'# 6!07. 0&% 6N<

S = −kBS0∫ N

0(−S′) ln(−S′)dn

= −kBS0∫ 1

0ln(−S′)dS

K _ % M

\' #+ < 0h+ K _ % u M T '7 9:

S = −kBS0[ln

(kA(N) + 1

3

)− 3 + kNA(N)

] K _ % r M

21A(N)

"]! 0V , 0!"5-5/"+&%T8 5-,# n! < "+, −S/(kBS0)! 0V 6,

< 0h+ K _ % r M&ZG ; " #,, lnN/N1/2 0:4 ] _x iS _ % 1 %

Qw0$0 #5/+#9:wh+ <2,,N* 54 D " !3, ## F-=<2 O."h+ 4L j7 "5/"+ 5-,."5/"+! < "+, T 0h*%K _ % r M T "74L -! < ! !w,. R 5 " # <%

1Xu

Y

0 200 400 600 800

-4

-3

-2

-1

0

entr

opie

N

10 100 10000.1

1

10

entr

opie

N

! " _ % 1X# ?BACY'<W'3 1=?BF G,U6 ,ZF E <=?WC 3,XG, :?-5Z6,21 , '5)EJ, D5@?-7<):,HC 6,Z39FL, D,)EHG,U6' 1=?B6 -E3?BF'=CC <=?5JIL0@, 5@?-7<):,$,ZF CD?B39F E396960@,21 ,ZF ;?BFL5)E3?BF G, N ,BER'-O0@5JIL,Z696,21 6939FL0 '39<=,$,)EY6?@M'<43 EIA 3 -,% k = 1 (P R,21.='6,)-7<Z1H'1J\ACREJ?E3 -,21X% N →∞ (+1=?BF E 5JIL?B3 143,21S3G,ZF E3 -,21)P

1S1S`

Y

Θ

) 0* T !"$#%&(')*!LV'+,')-.'/0#)!;412 ')34!3'/56!#)78 !#/ #)'9')3*,!:; '%78 <

Θ-*,!=>'978 </D?*!@A+,TF

Bv < a3 CD) E')F4G>< 3'978 <

Θ T ')#9<! #) 6B*H!*G#)*& ! #) < T ')#9<! #)6!#).!*."$#9< D _ #9# T &: 6&a)=#)' :.@A&#9"I<J#) 4 ') D 8E=!Hi8#9"I"$# jK#9'9L')4 ! !#)#)@A0#978 < H <M-K=N '/L<! #)I!7O!"I<0

Φ∗ ' N−1/2 T'/P')Vi8!$!!'/ #)Q

ξ ' aΦ−1 '/P*!#)R8"I#)@<IN< !#9Π ' kBT

a3Φ3 D 8G

<!#9+#)8"I#)@< B1S D AT C '/#)@< B1S D C <("I<# V (

Fvol∼= kBT

a2

∫Φ(z)3dz ∼= kBT

a2

∫ N

0(a2S(n))3

(z(n)

a

)−2

dnB D9UVC

Fel∼= kBT

a2

∫ N

0a2S(n)

(z(n)

a

)2

dnB D T C

QWX0**!#)X<G#9'9#)X G')Y 6& *#9! D ):F'/"/C"IF"5 # .!J@AY G')Y 6& *#9! S T(+('):*!LV'M63' < @A#9'9#9+!

Seq(n) ∼

1n pour n < nc

∼= Φ−2b

Φbn1/2

pour nc < n < N

B D S C

T8T U

Y

_ #9# T '/0#) !;41 B D T C

γΘ(Φb, N) ∼= γext +kBT

a2

(−α+ 2η lnΦb)Φ

2b + β

Φb

N1/2ln

(N

Φ2/3b N∗

) B D C

21η ∼ 1

α T β T N∗ <.')(< 5#)<- 0V#)( ') 6& *#9! D- '97 < *!@<L+, 0V#-*& !

v < a3B @<#-#9 T 0V#9! ')

*& ! "/.!=7 '9"I-0*'9 "$"Iv ≡ v/a3 CD QWF '9#)Y"$# jK#9'9WOF4 ! !#)

*& !(B*P')Vi8!-2!!'/ #) N ξ ∼ av−1/4Φ−3/4 *,!(').#9<! #)%60B*L!*(rB ∼ v−1 *,!:'):#%O! #)65!#)!* B

rB=#9B4 !#)!

ξ C H AM-K D _ B* 6'9')r < ξ TP 6&a)P "$*,! "$"I P 6&a) '9#9+! N *,!

r < rB'/ 6&a)P3@<&#9"I<

#) 4 ') T *,! rB < r < ξ'/I 6&a)Hi B ')#9<! #)=27A'%"I20*'9"$#9< CD

_ B* 6'9')r > ξ

'/0 '9#) . 4 L+')+("$# jK#9'9 D6 ! !#9!II 6 !+ 5 HH#9#)3I'978 < T B*"I<# V&4 #)

6#)@<#97 < C!0 **,! N *,!z < rB

B n < gB ∼ v−2 C T '):<!#9+#)

8"I#)@<- '/#9@<-<( "+'/ +')= B* @<& #) B D9UVC B D T C T *,!z > rB

')0**!#) B1S D AT C B1S D C <.78 '/ +'). <(@<:'/0 *, H0**'9#)#9H47A

vJ*!#)

"$* D _ #9# T ' < !ki8#)27A'9"$#)@<2N< !#9

Fvol∼= kBT

a2

∫ H

rB

v3/4Φ(z)9/4dz ∼= kBT

a2

∫ N

gB

v3/4(a2S(n))9/4(z(n)

a

)−5/4

dnB D C

8. <F ! !E 6&@A "I"/.!Y ')!kBT

a2gξ2∼ kBT a2

ξ2ξ5/3

a5/3v1/3∼ v−1/4Φ1/4 D

8$<!#9+#) '/#)@< B1S D C ."I<# V H

Fel∼= kBT

a2

∫ N

gB

v−1/4(a2S(n))5/4

(z(n)

a

)7/4dn

B D C

QW'/$#9= L*!LV'+,'9-#978 < N

Seq(n) ∼

1n

*,!n < gB ∼ v−2

1v2/5n6/5

*,!gB < n < nc ∼ v−3/4Φ

−5/4b

Φ7/8b v1/8

n1/2*,!

nc < n < N

B D M C

T8T8T

Y

8G0#) !;412N< !#9- ')!

γ(Φb, N) ∼= γext +kBT

a2

(−α+ 2η ln v) v3/4Φ5/4

b + βv1/8Φ

7/8b

N1/2ln

(N

v1/12Φ7/12b N∗

)

B D r C?=-!!7 T "$"I$ T ')4=; '97 <: 6!"5 ' @ D B D T C ;*!& <

v = 1')2( '97 <Θ@ D B D C L#)<# V <

v ∼= ΦbT < +j 6yjK#9! rB ∼= ξ D

T8T S

Y

T8T

Q Y

) 0 0* T 0#9'9#)3'/E4 #)'9') B D S UVC *,!0' < 3+!80*,' ')!'O i8!03 !25!;4 $! D ?-**,8H@<$'/L<! #) 'J3Hi8'9#.i 4 +') $@A ')$<! j #)$ < <$!3P#) &

λ À H T 21λ

! !"$#9I*& !'/P#9$

H' < *& #)!H0'/ +!8 D QJI **!N*#9"5 #) 7 '/ +')

*,!(')( 6&a) 4 #9+')"I<- 6& !ki T f ¿ 1 D #9"$*,8 <

S′(n) = σδ(n−N) J' < @<& #) B D S UVC T 21 σ J'/0#9ti8!03 ?i ( 6&a) T ' < !ki8#)H'/3+!8 N< !#9 T *& !#922!;4

F

kBT∼= 1

a3

∫ ∞

0ρ(z)[ln ρ(z)− 1]−Ψ(z)ρ(z) dz

B Q D9UVC

+

∫ N

0

a6

3

σ3

z2(n)+ σ

(z(n)

a

)2

+ fσΨ(z(n))

dn

?=G#)#9Vi8XB*2!Hi8#) N '/!Hi8#) B UVC 0V#)F*& !G' < G*#)3F'/.+!8 T 0 ≤ z ≤ H T'/0!Hi8#) B T C 21L ').')<! j #)-<*!< T z ≥ H D

8 "$#9#9"$#) #) B Q D9UVC *& !H! **,! 6ρ

H' < @<& #)QX#) !Ψ

#) < B* @<& #)(X#)wjJ'9 "5 ; '/3!Hi8#) B T C N

d2Ψ

dz2(z) = 4π`Bρ(z)

B Q D T Cln ρ(z)−Ψ(z) = µ2

B Q D S C21

µ2->"'9#9*'9#)4 !278G?i8! Vi $@A# ! H'/5!"5 '9#) #)

ρ D 8E#9#) B* '9#9"$#9 -'/5!Hi8#) B T C < N ρ(∞) = 0

dΨdz (H) = 4π`Bσeff

21σeff

-'/ 6& !ki

T8T

Y

03 #97 2'/3+!8 D 8$'9#) @<& #) B Q D T C B Q D S C

ρ(z) =a2

2π`B

1

(z −H + λ)2B Q D C

Ψ(z) = −2 ln(1 +

z −Hλ

)+Ψ(H)

B Q D C

21λ = (2π`Baσeff)

−1 D) '/3!Hi8#) B UVC '/3"$#9#9"$#) #); B Q D9UVC *& !.! **,! 6

z#9

6

d

dn

[2aσz − 2

(a3σ

z

)3]− a3fσdΨ

dz(z(n)) = 0

B Q D C8E*,<#)' ')!8 #)@<H#9' < @A& #)X#)

d2Ψ

dz2(z) =

4π`Ba2

[−a3f σ

z+ ρ(z)

] B Q D M C L#9<Hi8! <(') @A& #) B Q D C B Q D M C !7A

2aσz − 2(a3σ

z

)3+ 2π`Bf

2(a2σ)2n2 − 4π`Baσf∫ n

0

∫ z(n)

0ρ(z)dz = D

B Q D r C21

D '/@<& <#9(5"I"+!(iA 6 78 '%

n = N D #9"$*'9# V&Y' < @ D B Q D r C 5*Vj*,8 <(@A

λÀ H < +j 6yjK#9!H Hi8'9#.i 4 <

ρ(z) ')0!Hi8#) B UVCD _ "I

:*'9

@<0')#9<! #)07A'9"I$0*'9 6 S !*H#)<2Hi8'9#.i 4 +')7 <2')#9<! #)')!8 #)@< B

aσz À(a3 σ

z

)3 CD 8@< @<& #) B Q D r C #9"$*'9# V H ')!(

z(n) ∼= z(N) + π`Ba3σf2(N2 − n2)

B Q D C **'9#)@<& <('/0#9#)

z(N) = 0 L!7A2 ')!'/0! #9!H'/0 6&a)

z(n) ∼= π`Ba3σf2

(N2 − n2

3

)n

B Q D9U C' < *& #)!('/3+! 8560' < @<#9'9#9+!H( ')!

H ∼= 2

3π`Ba

3σf2N3 B Q D9U8UVC!7 #9#9#/ ')"I<*& ! #9 H U T r K D QW"$"I '/ 4 ! #)27A'9"$#)@A E *& !

Φ(z) = a3σ/z +#)<Φ(z) ∼= 1

π`Bf2 (N2 − n2(z))B Q D9U T C

T8T

?-H@<0' < 0*#)R$'/P+!8IN< +#)< A ')"I< "$#9#9"$#) <H*& !2! **,!76H

' < !8#) B Q D9UVC 7 '9=*,!z(n)

0V# *& ! B Q D CD PK'T41 J J:4. !JP!"I

−eπ`B(fNσ)2H *!& <0 "$* ' < #9<! #) ')!8 #)@A& <!5')<! #)0

')"I"/.! B "I< ')H !7H T8T-K CD_ 7A2'9#)

σeff ∼= fσN .'/0')8!-`: QW6& *"5 N< !#9

λ ∼= (2π`BafσN)−1 D 8G$&#9#)λÀ H

:! !#9(

σ ¿ 1

`Ba2f3/2N2

B Q D9U S C

W#) < .')!8#9"I+!8

'/0#928!03 2#9 C!'9')2@A

σ À L−2 21L

J'/3 #9'9'):& !'9')2>< *,' ')!'<#9

σ À(a2(`Bf

2)2/3N2)−1

@<# !!#9<')2"5 #9H>< **'9#)4 #);!8#9"I D

8E-!'9 @< D B Q D9U C B Q D9U8UVC B Q D9U T C <-#)<#)@< 6IB* J!#)V7L 6<

'9#9&H T8TLK 3' < **!N*#9"5 #) >< A ' #9!"I< D QW0!'9 0*,7A<IC! " '9#)!I!'/ 6& <F.<! #9<5H U M-K #) <F ')!@6-$*!LV' 4 ! #)$7A'9"$#)@A. < 2$'/P 65<'/P7 ')!H2$*& !2'/ "/C"I$')#F>< 6'9')5@A @ D B Q D9U T C Φ(z) ' (π`Bf2N2)−1 D 8E24

λ < H A ')"IO(! #92 '/ 04 D H U M-K D

T8T M

BB

T8T r

! ! !

H U K _ D D _ "IR _ D D ` D ! " #D #9') ?= %$Y!

# *6L#9#) U M D

H T-K ,D D _ 6& !# D'& ()*+ )( # -, N U MT8T)# U MT/. U . S DH S K _ D _ '9+,!10!;41!

D W!#9"5

D "5 D323 45#6!(87 9 ;: N T8T MB#OT S U

U . DH < K ,D _ ')0* ! D>=* 4? A@ B CD( FE!G N . S # . M U M8M DH K ,D _ ')0* ! D CDH8I J61*" ;K NML #OT S L U . DH L-K D _ ')0* !#)#)

D!N ')

D 8 #9"5 D'7 ?O* P,: N T/L8T MB#OT/L S . U . D

H M K D _ +, D 6 ) 6#)-#97 !#9)Q !#)>R%P

U DH .-K D _ +,

` D DDS !!#)

D #9 D - *6&*T' DF& U5*+ 5U # G N T M #T . U U D

H K D _ +, N3D `=#) '%#9

D - *6&*T' D'& U5*+ 5U # !K N MT/L U #MT/L/. U L D

H U K D _ +, D > 86#

D = *6&*T' DV CDH I W XY7 # ZG;: N < . .)# < ./L U

T DH U8U K D _ +,

D ; 86#

D = *6&*T' DA CDH I W X7 # 4G\[ N U M L T T U8D

H U T-K D _ +, D = *6&*T' D'& ()*+ )( # ]E\, N < S MB# < S L S U . DH U S K 8 D _ 7<!

D _ !

D QY! D>=4* ? ^@ _3CD ( a` N < S # D< U T D

H U < K 8 D _ 7<! b D D QW D'& (5*+ )( ;c N T T)#OT M U . M DH U K 8 D _ 7<!

D QY!

D _ ! D3& (5*+ 5U N < M*.)# < . U . D

H U L-K b D 8 D . !! D'& (5*+ 5U # Fd NM. S T)#4. S < U T DH U M K b D 8 D . !! - b D SJD b D 2' 5#6FCDH_7 # : N S8S8S # S8S . U S D

T8T

H U .-K b D 8 D . !! - b D SJD b D @ W ?] "? H ( K : N U #4L/L U L DH U K U D ,D W6& #/

b D D QY6

b D D -+,!#9

b D D 6

b D _ D "I! D8=* 4? 5 @ ? @ `#? " aP (?F \, c N S S # S U . D

H T K D W#) ' Q D b **, Q D > !@< D 23 45#6 ? CDH =4* 4#? 2 ^: N S8S # < S

T U8DH T U K D W#) ' Q D > !@A

EQ D b **, D 3CD I W X 2

N U8U T < # U8U T M T DH T8T-K N3D R D J!#)V7 D D &6O'9#9& D>=4* 4#? A@ B CD ( B`5` ][ N </< # < . U M DH T S K N3D R D J!#)V7

D D 6<'9#9& D D W#9!6<#9 D=4* 4#? ]@ > CD ( `5` /: N U S # T

U < DH T < K P D J! 6V7 ) D _ '9"5

D!N !'/ D & (5/ )( ]E\, N U L/L # U L M U U . D

H T K P D W! 6V7 ) D _ '9"5 D N !'/ D =/ ? CDH

, c E N < T)# M U DH T/L-K D W 6 D ) D>=4* ? ^@ _3CDU ;:FG N U U #OT U . M DH T M K b D W! !*I D D PK"$"I!8 D; * ( ? @ ) ?D #9')

?= $Y! 6#9!3#9#)

U . DH T/.-K D W!#)#) 6

D D ?=

D 8 #9*, DB23 45#6 ? 3CDH =* 4? 2

d N < S #< U < T U8D

H T K ) D W!8 ` D D4S !!#)

D =' 4 8 D 8E#9+')! D3& ()*+ )( # E N U S T)#

U S U DH S K b D D QJ 6 b D D -#9'9'9#/ ! D>=4* ? H CD G N T .)#OT/L M U . DH S U K D QJ '9') DZ #+ @ *? H ? @ ? `#? 5 @ *? Z +D HD #9')

?=

$F! L#9#) U . D

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