Univerza v Ljubljani 
Fakulteta za matematiko in fiziko 
Daniel Svensek 
Vpliv hidrodinamicnih tokov na 
reorientacijsko dinamiko tekocih kristalov 
Disertacija 
Ljubljana, marec 2003

University of Ljubljana 
Faculty of Mathematics and Physics 
Daniel Svensek 
Back
ow-aected reorientation 
dynamics in liquid crystals 
Thesis 
Ljubljana, March 2003

Radovednostjelepacednost. 




Zahvaljujemseprof.SlobodanuZumru,kimejetistegadavnegapoznopomladneg. 
dne,kosemnenadomapotrkalnavratanjegovesobice,karhitrosprejelvsvoj. 
raziskovalnoskupino.Datasplohnimajhna,semspoznaljeseni.JureB.,Primo. 
z



Z.-Pizo,AnamarijaB.B.-Pika,GregorS.-Gregovec,AndrejaS.,FahimehK.P. 
H.,MatejB.-Buzec,BostjanM.,prof.SamoK.-KraljSamoin,seveda,MilanA. 
-McMillan,hvala!Zelosemhvalezentudiprof.AlojzuKodretuzanasvete,prof. 
HelmutuBranduzagostoljubnost,podporoinnasveteterkriticnimbralcemdi. 
sertacijevnastajanju.PriznanjeministrstvuMZ

S(prejMZT)RepublikeSlovenije,

S

kijenanciraloraziskovanje.HvalaprijateljeminBrunarci,preka(l)jevalniciidej. 
HvalaAPZ-ju|vsebibilodrugace,
.
cenebinaredilavdicije. 
NajvecjahvaleznostpagresevedaTanji,starseminvsemdomacim. 



Izvle

ce. 


Vnematskihtekocihkristalihintankihsmekticnihlmihsmoznumericnimisredstv. 
raziskovalidinamikoureditvenegaparametra,sklopljenoshidrodinamiko.Posluzil. 
smoseEricksen-Lesliejeveteorijenematskegadirektorjainraziskaliucinkehidrodi. 
namicnegatoka,kispremljapreklopneprocesenematikavcelicah.Opozorilismon. 
primere,kosotiucinkiodlocilnegapomena.Zupostevanjemcelotnegatenzorskeg. 
ureditvenegaparametrasmoresiliproblemanihilacijeparatopoloskihnematski. 
disklinacijskihlinijterpokazali,dahidrodinamicnitokpospe..
sidisklinacijospozi. 
tivnomocjogledenadisklinacijoznegativnomocjo,hkratipapospe..
situdiproce. 
anihilacijevprimerjaviznehidrodinamicnoobravnavanimprimerom.Izka..
zese,d. 
jetransportstokompomembeninjelahkonjegovprispevekhgibanjudefektovpre. 
vladujo
.
c.SposplositvijoEricksen-Lesliejeveteorijenacelotenvektorskiureditven. 
parametersmoresilitudiproblemanihilacijeparadisklinacijskihvrtincevvtanke. 
prostostojecemlmusmekticneC-faze,kijevnasemopisuustrezalXY-modelu. 




Rezultatisevkvalitativnempogleduujemajostistimiprinematikih.Studiral. 
smo..
serazpadnematskedisklinacijskelinijezmocjo1napardisklinacijzmocj. 
1=2.Vprimerjaviznehidrodinamicnoobravnavanimproblemomtokspetpospe.. 
siodbojnogibanjenastalihdefektov.Dabiraziskalistabilnost,smoresili
uktuacijsk. 
problemravnedisklinacijskelinijessplosnocelostevilcnomocjozacelotentenzorsk. 
ureditveniparameter.Naslismodvevrstinara.
s
.
cajocih
uktuacij,kivodijodoraz. 
padaoziromadopobegavtretjodimenzijo. 


Kljucnebesede:nematskiteko
.
cikristali,smekticniteko
.
cikristali,SmClmi. 
ureditveniparameter,nematodinamika,hidrodinamika,Ericksen-Lesliejevateorija. 
Freederickszovprehod,defekti,disklinacije,vrtinci,parskaanihilacija,
uktuacij. 


PACS:61.30.Dk,61.30.Gd,61.30.Jf,83.80.Xz,47.15.Gf,47.15.Rq,61.30.Pq,68.15.+. 



Abstrac. 


Dynamicsoftheorderparametercoupledtohydrodynamicsisstudiednumericall. 
innematicandsmecticthinlmliquidcrystals.TheEricksen-Leslietheoryfo. 
thenematicdirectorisemployedtodeterminetheback
oweectsaccompanyin. 
externaleldswitchingprocessesofnematicsconnedtocells.Itisdemonstrate. 
thattherearecaseswheretheseeectsarecrucial.Pair-annihilationoftopologica. 
nematicdisclinationlinesisstudiedusingthefulltensororderparameter.Iti. 
foundthatthehydrodynamic
owisresponsibleforthespeed-upofthepositiv. 
strengthdisclinationrelativetothenegativeone,andfortheoverallspeed-upofth. 
annihilationprocessascomparedtothenonhydrodynamictreatment.Moreover,i. 
isdemonstratedthatthe
owtransportissubstantialandcandominatethemotio. 
ofdefects.TheEricksen-Leslietheoryisgeneralizedtothecompletevectororde. 
parameterandusedtostudythepair-annihilationofvorticesinafree-standingthi. 
lmofthesmectic-CliquidcrystalasarepresentativeoftheXY-model.Theresult. 
agreequalitativelywiththoseofthenematic.Decayofthenematicstrength. 
disclinationlineintoapairof1=2disclinationsisstudied.The
owagainspeed. 
uptherepellingmotionofthedecayproductsifcomparedtothenonhydrodynami 
treatment.Asastabilityanalysis,the
uctuationproblemofastraightdisclinatio. 
linewithageneralintegerstrengthissolvedforthecompletetensororderparameter. 
Twotypesofgrowing
uctuationsarefound,leadingtothedecayandtotheescap. 
inthethirddimension,respectively. 


Keywords:nematicliquidcrystals,smecticliquidcrystals,SmClms,orderpa. 
rameter,nematodynamics,hydrodynamics,back
ow,Ericksen-Leslietheory,Freed. 
ericksztransition,defects,disclinations,vortices,pair-annihilation,
uctuation. 


PACS:61.30.Dk,61.30.Gd,61.30.Jf,83.80.Xz,47.15.Gf,47.15.Rq,61.30.Pq,68.15.+. 



Contents 
Razsirjeni povzetek (Abstract in Slovene) i 
1 Introduction 7 
2 Theory 11 
2.1 Order parameter and thermodynamic potential . . . . . . . . . . . . 11 
2.2 Free energy functional . . . . . . . . . . . . . . . . . . . . . . . . . . 12 
2.3 Dynamic equation for the order parameter . . . . . . . . . . . . . . . 15 
2.4 Coupling to the 
ow . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 
3 Nematic order parameter 19 
4 Director dynamics in nematics 23 
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 
4.2 Ericksen-Leslie theory . . . . . . . . . . . . . . . . . . . . . . . . . . 24 
4.3 Characteristic scales . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 
4.3.1 Comment on heat diusion . . . . . . . . . . . . . . . . . . . . 28 
4.4 Description of the problem and numerical implementation . . . . . . 28 
4.5 2D problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 
4.5.1 Mechanisms governing the problem . . . . . . . . . . . . . . . 31 
4.5.2 Results and interpretation . . . . . . . . . . . . . . . . . . . . 32 
4.5.3 Comparison with the simplied treatment . . . . . . . . . . . 43 
4.6 Amplifying the kickback eect . . . . . . . . . . . . . . . . . . . . . . 46 
4.6.1 2D problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 
4.6.2 Quasi-3D problem . . . . . . . . . . . . . . . . . . . . . . . . 46 
4.6.3 Axial magnetic eld . . . . . . . . . . . . . . . . . . . . . . . 49 
4.6.4 Oblique magnetic eld . . . . . . . . . . . . . . . . . . . . . . 51 
4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 
5 Dynamics of a vector order parameter 55 
5.1 Free-energy density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 
5.2 Coupling to the 
ow . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 
6 Defects 61 
6.1 Disclinations of a 2D director/vector . . . . . . . . . . . . . . . . . . 63 
6.2 Disclinations of a 3D director/vector . . . . . . . . . . . . . . . . . . 65 
5

. 
CONTENT. 


6.3Structureofdisclinationcores......................6. 


7Pair-annihilationofdisclinationlinesinnematic. 
6. 


7.17.27.37.47.57.6Introduction...........................
Dynamicequations.......................
Characteristicscales......................
Technicalitiesandmaterialparameters............
Resultsanddiscussion.....................
7.5.1The
owasymmetry..................
7.5.2Reorientation-drivendefectmotionvs
owadvection7.5.3In
uenceofthedirectororientationangleonthe
owSummary............................
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.6. 
7. 
7. 
7. 
7. 
7. 
8. 
8. 
8. 
8Pair-annihilationofvorticesinSmClm. 
8.1SmCorderparameter......................
8.2Dynamicequations.......................
8.3Technicalitiesandmaterialparameters............
8.4Resultsanddiscussion.....................
8.5Summary............................
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
8. 
8. 
8. 
8. 
8. 
9. 
9Decayofintegerdisclinationsinnematic. 
9.1Fluctuationproblem......................
9.1.1Fluctuationeigenmodes................
9.1.2Eigenmodesleadingtodecay..............
9.1.3Eigenmodesleadingtoescape.............
9.1.4Remarksonthe
uctuationproblem.........
9.2Hydrodynamicspeedup.....................
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
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.
.
.
.
.
.
9. 
9. 
9. 
9. 
10. 
10. 
10. 
10Conclusio. 
10.1Futureperspectives............................
10. 
11. 
Bibliograph. 
11. 



Razsirjeni povzetek 
(Abstract in Slovene) 
Uvod 
Tekoci kristali so bolj urejeni kot tekocine, a manj kot trdne snovi | so nekaksna 
vmesna faza. Potreben pogoj za obstoj tekocekristalne faze so molekule podolgovate 
ali ploscate oblike. Pri termotropskih tekocih kristalih je stopnja urejenosti odvisna 
predvsem od temperature, pri liotropnih pa od koncentracije tekocekristalne komponente 
v raztopini. Zaradi urejenosti imajo tekoci kristali anizotropne makroskopske 
lastnosti. Najblizji izotropni tekocini je nematski tekoci kristal ali kratko nematik, ki 
premore orientacijsko molekulsko urejenost dolgega dosega. Povprecno smer molekul 
oznacimo z direktorjem n, ki je del tenzorskega ureditvenega parametra nematika. 
Bolj urejena je smekticna tekocekristalna faza, za katero je poleg orientacijskega 
reda znacilen pozicijski red v eni dimenziji. Ploskvam maksimalne gostote pravimo 
smekticne plasti. V SmA fazi je direktor pravokoten na plasti, v SmC fazi pa je 
nagnjen. 
Modeliranje tekocih kristalov je zazivelo s prihodom dovolj zmogljivih in dostopnih 
racunalnikov. Najprej so obdelali staticne probleme, nato so se raziskave usmerile 
v studij dinamike, ki jo najveckrat obravnavajo v okviru direktorskega opisa 
in brez upostevanja hidrodinamike. Za opis defektov moramo nujno poseci po tenzorskem 
opisu ter upostevati tudi hidrodinamicni tok. Predvsem slednje je precej 
tezavno. 
V disertaciji se ukvarjamo s simulacijo dinamike v nematskem in SmC tekocem 
kristalu z upostevanjem hidrodinamike. V prvem delu obravnavamo relaksacijske 
procese direktorja pri preklapljanju celice z zunanjim poljem, kjer pokazemo, da 
vloga hidrodinamskega toka ni omejena zgolj na kvantitativne popravke. V drugem 
delu se posvetimo glavnemu izzivu | hidrodinamicni obravnavi dinamike defektov. 
Z nadgrajeno metodologijo iz prvega dela uspemo pokazati, da na anihilacijo para 
nasprotnih defektov kot tudi na odboj para enakih defektov mocno vpliva tok, ki 
pospesi procese in v prvem primeru povzroci asimetrijo pri gibanju defektov. 
Teorija 
Na kratko bomo predstavili dinamicno teorijo kompleksnih disipativnih tekocin, se 
prej pa bomo uvedli pojem ureditvenega parametra in vpeljali ustrezen termodii

i. 
Razsirjenipovzete. 


namicnipotencial.Teorijasezazdaj..
senebonanasalananekizbraniureditven. 
parameter,ampakbosplosna,karbokoristnovnadaljevanju,kobomoobravnaval. 
sistemezrazlicnimiureditvenimiparametri. 


Ureditveniparameterintermodinamipotencial

cni

Teko
.
cikristalisourejenisistemi.Uredijoseobfaznemprehodu,prikateremsezlom. 
simetrijasistema.Zaopisurejenostivpeljemomezoskopskokolicino|ureditven. 
parameter,kimorabitinicelnvvisokotemperaturnifazi,vurejenifazipaodni 
razliceninodvisenodurejenosti,ssimetrijskimilastnostmi,kotjihimaurejenafaza. 
Vsplosnemjelahkosistemkrajevnonehomogen,takodajeureditveniparamete. 
polje.Deniramogakotpovprecjevmezoskopskemvolumnu,kinajbibildovoljve. 
lik,dajepovprecjedobrodenirano,hkratipadovoljmajhenvprimerjavizznaciln. 
skalonehomogenosti,takodajesistemznotrajnjegadovoljhomogen.Pristudij. 
dinamikesezanimamozaneravnovesnelastnostisistema.Odmikeodravnovesj. 
opisemozneravnovesnimivrednostmiureditvenegaparametra.Topomeni,das. 
neravnovesnelastnostisistemadolocenezdinamikoureditvenegaparametra,kole. 
tegaizmaknemoizravnovesja. 


Tekocekristalnitermodinamicnisistemivsplosnemzdruzujejotermicne,elek. 
tricneinmagnetneprostostnestopnje,medtemkodeloprispremembiprostornin. 
zanemarimo.PrikonstantnitemperaturiTinkonstantnihjakostihzunanjegaelek. 
tricnegainmagnetnegapoljaEinHjeustrezentermodinamicnipotencia. 


F=U..TS..VE
P..
0
VH
M. 
(1. 
dF=..SdT..TdS
i
..pdV..VP
dE..
0
VM
dH+dA
0
;(2. 


kigaimenujemokarprostaenergija;Ujenotranjaenergija.Sprememboentropij. 
Ssmorazdelilinareverzibilni(dQ=T)inireverzibilnidel,dS
i
>0.Vravnovesj. 




jeprostaenergijaminimalna.CesistemprikonstantnihT,EinHspravimoi. 
ravnovesja,jespremembaprosteenergijeenakaminimalnemudeludA
0
,kijezat. 
potrebno,t.j.deluprinadomestnireverzibilnispremembi.Elektricnoinmagnetn. 
delosta..
zevkljuceniv(2),takodajeprostaenergijasistemavpoljuH,katereg. 
konguracijaustrezaravnovesjupripoljuH
0
,z. 


Z. 


. 



F=..
0
VdVM(H)
d. 
(3. 


. 


visjaodravnovesne(analognozaelektricnopolje). 


Zastudijdinamikejetorejtrebapoiskatiodvisnostprosteenergijeodure. 
ditvenegaparametraq,kiimavsplosnemveckomponentq
i
.Prinehomogeni. 
sistemihvpeljemogostotoprosteenergije,takodajeprostaenergijafunkcional. 


. 


F=dVf(q;rq). 
(4. 


Prifenomenoloskempristopugostotoprosteenergijerazvijemoposkalarnihinvari. 
antah,kijihsestavimoizq,rqinzunanjihpolj.Pritemnastopapetkategoricni. 



Razsirjenipovzetek(AbstractinSlovene. 
ii. 


tipovprispevkov:homogeni,elasticni,kiralni,prispevkizunanjihpoljterpovrsinsk. 
prispevki,kijihtukajnebomonatancnejeopisovali. 


Ravnovesnokonguracijopoi.
scemotako,daminimiziramoprostoenergijo. 


 . 


ZZ

@f@f@. 


AF=dV..r

Aq+dS

Aq=0;(5. 


@. 
@r. 
@r. 


odkoderdobimoEuler-Lagrangeveenacbezavolumskiinpovrsinskidel.Slednjeg. 
bomoodzdajnaprejopu.
scali.Kadarmorajokomponenteureditvenegaparametr. 
zado.
scativezem,vpeljemoLagrangevemultiplikatorjekotponavadi,karjeekviva. 
lentnoprojiciranjuEuler-Lagrangevihenacbvprostoruureditvenegaparametran. 
podprostor,pravokotennatistega,kigadolo
.
cajovezi. 


Hidrodinamikaureditvenegaparametr. 


Kosesistempriblizujeravnovesju,semuprostaenergijamanj..
sanaracunvecanj. 
entropije. 


 . 


Z

@f@. 


S_
i_

T=..F=..dV..r

.
_q;(6. 


@. 
@r. 


kjerpikaoznacujesubstancialniodvod.Priireverzibilnihpojavihvecanjeentropij. 
splosnoopisemostokovi
i
inpripadajocimisilamiF
i
. 


. 


S_
i

T=dVF
i

i
. 
(7. 


vrezimusibkihtokovpapredpostavimo..
selinearnozvezomedsilamiintokovi[17]. 


F
i
=K
ij

j
;K
ij
=K
ji
. 
(8. 


kjerjepoOnsagerju[18],[19,p.365]matrikatransportnihkoecientovsimetricna. 


Poenacbi(6)lahkoimamo.
_qzatok,izrazvoklepaju(zminusom),kigaoznacim. 
. 


@f@. 


h=r
... 
(9. 


@rq@. 


pazasilo.GibanjetekocineopisemosposplosenoNavier-Stokesovoenacb. 


.
_v=r
. 
(10. 


pricemersevsakompleksnostskrivavnapetostnemtenzorju.Polegtlacnegadel. 
tavsebuje..
seelasticnidel,kijeposledicadejstva,dadeformacijasistemaspremen. 
krajevneodvodeureditvenegaparametra,torejtudigostotoprosteenergije. 


@. 



. 


=..
@
j
q. 
(11)

ij
@(@
i
q. 


Hgostotiprosteenergijemoramozdajdodatitudikineticniprispevek
1
v
2
. 


. 


Entropijskiizvirjeste. 


. 


h. 


S_
i

T=dV..
. 
+h
. 
_q:(12)

(
ij
+pA
ij
ij
)@
i
v
j


i. 
Razsirjenipovzete. 


Knara.
scanjuentropijetorejprispevatadvatokova|casovniodvodureditveneg. 
parametra.
_qingradienthitrostirv.Pripadajo
.
cisilistageneraliziranasilan. 
ureditveniparameterhinviskozninapetostnitenzor=+pI..
e
.Ugodno


v

je,
.
cetenzorjerazcepimonasimetricniinantisimetricnidel.Gostotaentropijskeg. 
izvirajepote. 



s

T.
_s
i
=
ij
A
ij
+
a
W
ij
+h
i
.
_q
i
. 
(13)

ij


s

a

kjerstainsimetricniinantisimetricnidelviskozneganapetostnegatenzorja. 
AinWpasimetricniinantisimetricnidelgradientahitrosti.Splosnalinearnazvez. 



a

(8)medsilamiintokovi(enacbe(2.35)-(2.37))namdolo
.
cisile
s
,inh(slednj. 
imenujemotudiviskoznageneraliziranasilah
v
),odkodersledijogibalneenacbe. 


A. 


..+h
. 
=0. 
(14. 


A. 
.
_v=..rp+r
(
v
+
e
). 
(15. 
r
v=0. 
(16. 


Ureditveniparameternematik. 


Zaureditveniparameternematikaizberemoprvinetrivialninenicelnimomentpo. 
razdelitvesmerimolekuld=(sin
0
cos
0
;sin
0
sin
0
;cos
0
),tojekvadrupolnimo. 
ment.Prispevekle-tegakporazdelitvenifunkcijigponavadizapisemokartezicno. 


5. 


g
(2)

(e)=Q
ij
e
i
e
j
;Q
ij
=


(3hd
i
d
j
i..A
ij
);(17. 
4. 


pricemerjeeenotskivektor,kipodajasmer.Vpeljalismosimetricniinbrezsledn. 
tenzorskiureditveniparameternematikaQ.Vlastnemsistemugazapisemoko. 


2. 


..
1

(S..P. 
. 


67

..
1

Q=
. 
(S+P)
5
. 
(18. 


. 


. 


vsplosnempako. 


1. 


.
Q
ij
=


S(3n
i
n
j
..A
ij
)+
P(e
1
e
1
..e
2
e
2
);(19. 
22
ijij

kjerj. 
S. 
3hcos
2

0
i
. 
... 
(20. 
skalarniureditveniparameter. 
P. 
. 
. 
hsin
2

0
cos2
0
. 
(21. 


pastopnjadvoosnosti.Venacbi(19)smovpeljalitrojicoortonormalnihvektor. 
jev(n;e
1
;e
2
),kidolo
.
cajolastnisistemtenzorjaQ:nimenujemodirektor,pa

e
1

sekundarnidirektoroziromadirektordvoosnosti. 



Razsirjenipovzetek(AbstractinSlovene. 
. 


Dinamikanematskegadirektorj. 


PredstavilibomoEricksen-Lesliejevoteorijo|hidrodinamicnoteorijonematskeg. 
direktorja.Numericnobomoobdelaliprocespreklapljanja/relaksacijedirektorj. 
vmagnetnempolju.Ogledalisibomoosnovnemehanizmenastankatokazarad. 
vrtenjadirektorjainnjegovegapovratnegavplivanadirektor.Relaksacijostoko. 
bomoprimerjalispoenostavljenimprimerombreztoka. 


Ericksen-Lesliejevateorij. 


Grezaposebniprimersplosnihenacb(14)in(15),kojeureditveniparameterdi. 
rektorn|enotskivektorssimetrijon=..n. 


GostotoprosteenergijevzunanjemmagnetnempoljuzapisemopoFrankuko. 
[46{48],[49,pp.102,119. 


1. 
2
. 
2
. 
f.


.
K
11
(r
n)
2
+
.
K
22
[n
(rn)]+
.
K
33
[n(rn)]..

a

0
(n
H)
2
;(22. 
222. 
kjersoK
11
,K
22
inK
33
temperaturnoodvisneelasticnekonstantezapahljacno. 
zvojnoinupogibnodeformacijodirektorja,Hjejakostmagnetnegapolja,
a
p. 
razlikamedmagnetnimasusceptibilnostmavsmerehvzporednoinpravokotnon. 
direktor.Vpriblizkueneelasticnekonstante,tudienokonstantnempriblizku,s. 
izrazpoenostavi. 


. 


f
one

=


K(rn)
2
. 
(23. 
. 
pricemersenismomenilizapovrsinskeprispevke. 
Generaliziranoelasticnoinmagnetnosilonadirektordobimoz(9). 


 . 


@f@. 


h
e. 
. 
=..+@
. 
. 
(24. 


@n
. 
@(@
j
n
i
. 


Viskoznageneraliziranasilaslediizsplosnelinearnezveze(2.37)medsilamiintokov. 
[50,p.142]. 


..h
v
=

1
N+

2
A
n. 
(25. 


kjerje

1
=
3
..
2
rotacijskaviskoznost,

2
=
3
+
2
=
6
..
5
,
. 
pas. 
Lesliejeviviskoznostnikoecienti[49,p.206].Njerelativnisubstancialnicasovn. 
odvoddirektorjagledenavrtenjetekocine,en.(4.7).Zaradivezi=1jetrebaob. 


n
2

siliprojiciratipravokotnonadirektor,takodajenakratkozapisanaenacbagibanj. 


. 


h
em

+h
. 
=0. 
(26. 


?. 


PosplosenaNavier-Stokesovaenacba. 


". 


@. 


+(v
r)v=..rp+r
(
v
+
e
);(27. 


@. 



v. 
Razsirjenipovzete. 


kjerjegostotainptlak,vsebujedvaprispevkaknapetostnemutenzorju.Viskozn. 
sledipoenacbah(2.35)in(2.36),[50,p.142]. 



. 


=
1
n
n(n
A
n)+
2
n
N+
3
N
n. 

4
A+
5
n
(A
n)+
6
(A
n)
n. 
(28. 


elasticnipapoenacbi(11),[49,p.152]. 


@. 



. 


=. 
(29)

ij
@
j
n
k
. 


@(@
i
n
k
. 


Tlakvenacbi(27)dolocimotako,dazadostimopogojunestisljivosti(16). 
ZnacilnikrajevniskaliproblemastavelikostceliceLinmagnetnakoherencn. 
dolzina[49,p.123. 


. 


1K
11

m
=. 
(30. 


H
0
j
a
. 


znacilnacasapastarelaksacijskicasdirektorskegapolj. 


.
K


1

. 
=
. 
(31. 
1. 


inrelaksacijskicashitrostnegapolj. 


L
. 


.


0
=. 
(32. 
. 


zakateraveljaocena(parameternestacionarnostitoka. 


L
2

..
K
1. 

0
==L
2
=
2

10
..6
. 
(33. 


.

. 


. 
m

. 
. 


Adiabatnaaproksimacijazahitrostnopolje,kjerzanemarimocasovniodvodvenacb. 
(27),jetorejupravicena.OcenazaReynoldsovosteviloj. 


LK
1. 


Re=L=
m

10
..6
. 
(34. 




2

.

. 


. 


torejlahkozavrzemotudinelinearniadvekcijskiclenven.(27). 
Obstaja..
setretjiznacilnicas,tojerelaksacijskicastemperaturnegapolj. 


c
p
l
. 

Q
=. 
(35. 


. 


kjerjelznacilnadolzinatemperaturnihnehomogenosti,c
p
specicnatoplotnaka. 
paciteta,patoplotnaprevodnost.Primerjavacasovd. 


Kc
. 



Q
==. 
(36. 


1
. 


karjeredavelikosti5
10
..4
.Ocenajesplosnainveljatudi,koimamoopravka. 
defekti.Torejlahkoresvednoprivzamemo,dajetemperaturakonstantna. 



Razsirjenipovzetek(AbstractinSlovene. 
vi. 


Obdelaniprimer. 


Prviobdelaniprimerjedvodimenzionalen(slika4.1).Magnetnopoljele..
zivy-smeri. 
Privzamemomocnosidranje,ssmermirazvidnimisslike4.1. 


Oglejmosimehanizmenastankatoka.Direktorparametrizirajmoskotomn. 
(cos';sin';0).Priokvirniobravnavijepotrebnoupostevatisamoviskoznosilo. 
kijopodajaclenskoecientom
2
.Prispevektegaclenalahkorazdelimonadv. 
dela|silo,kijeodvisnaodgradientakotnehitrosti!vrtenjadirektorja,insilo. 
kizavisiodgradientadirektorskegapolja.Prvonajlep..
sevidimopri'=0. 


 . 


@. 


f
1
=
2
0;. 
(37. 


@. 


Tasilajetorejpravokotnanadirektor,njenavelikostpajesorazmernazodvodo. 
!vsmeridirektorja,n
r!.Drugosilopoglejmovprimerur'=('
x
;0). 


f
2
=..
2
!'
x
(cos2';sin2'). 
(38. 


Velikosttesilejeodvisnaleod!jr'j,njenasmerpajetaka,dazgradiento. 
r'oklepadvakratvecjikotkotdirektor.Pazitijepotrebnonanegativnipredzna. 
parametra
2
innato,dajesmersilodvisnaodpredznaka!. 


Sedajsioglejmose,kaksenjevplivtokanadirektor.Tokovnopolje,kiustrez. 
homogenemuvrtenju(W=60,A=0),povzroci,dasetudidirektorvrtienako,seved. 
.
cenanjnedelujejodrugesile.Tokovnopolje,kiustrezacistemudeformacijskem. 
toku(W=0,A=60),pasku..
saporavnatidirektorvtistilastnismeriA,kiustrez. 
raztegu.Vprimerustriznegatoka,kijevsotaobehpravkaromenjenihtokov,enacb. 
(26)d. 


 . 


1

2

'_=..


cos2'+1. 
(39. 
2

. 


kjerjevelikoststriznehitrosti,kot'pajemerjengledenasmerhitrosti,slika4.2. 
Stacionarnaresitevobstajale,
.
cejej

2
=

1
j>1,torej
.
ceje
3
<0,inseglas. 


j'
0
j1. 
(40. 


sajje

2
=

1
..1.Topomeni,dasevstriznemtokudirektorpribliznoporavn. 
ssmerjohitrosti.Resitevs'
0
>0jestabilna,tistas'
0
<0panestabilna.Z. 
nematikMBBAkot'
0
zna..
sapriblizno'
0
7A.Opozoritijetrebanato,das. 
direktorktemukotuvrtivnasprotnismeriurinegakazalcalezaj'j<j'
0
j,privse. 
drugihkotihpasevrtivsmeriurinegakazalca. 


Zmehanizmi,kismojihpravkarspoznali,jemogo
.
cepribliznopredvideti,kakse. 
bopotekrelaksacijedirektorskegapoljaobvklopualiizklopuzunanjegapolja,n. 
dabibilotrebanareditinumericniizracun(razdelek4.5.2).Glavnaugotovitevje. 
dalahkorelaksacijazaradihidrodinamicnegatokapotekavdvehdobrodenirani. 
korakih,kisemedsebojlocitaposmeriglavnegatokovnegavrtincainceloposmer. 
vrtenjadirektorja,slike4.3,4.5,4.9in4.10.Takosedirektornekajcasavrti. 
napacnismeri,karjeznanokotkickbackpojav[55,p.167],kigajemocopaziti. 
celicahtekocekristalnihzaslonovkotmigotanje. 



vii. 
Razsirjenipovzete. 




Cepopolnohidrodinamicnorelaksacijoprimerjamospoenostavljenimmodelo. 
breztoka,najveckratvelja,dajedinamikastokomnekolikohitrejsa,slika4.11.T. 
trditevpaneveljasplosno,enegaodprotiprimerovprikazujeslika4.12.Vecjerazlik. 
medobravnavamasepokazejo,
.
ceprimerjamovrtenjedirektorjavdolocenemdel. 
celice,karjeleporazvidnozgrafovkrajevnihFourierovihkomponent(slike4.12. 
4.14). 


Zzeljo,dabiseboljpriblizalisituacijam,kijihlahkouresnicijovposkusih,sm. 
simulirali..
sedvaprimera,kjersmodovolilivrtenjedirektorjaizxy-ravnine,pra. 
takopasmodopustilituditokvz-smeri,vkaterisicernibilonobenihgradientov. 
Geometrijatokratustrezakapilariskvadratnimpresekom,slika4.18.Sidranjej. 
spetmocno,atokratvzporednozosjokapilare.Obakratjebilvzorecnazacetk. 
poravnanzmagnetnimpoljemvsmeriosix.Potemsmopoljenenadomapreklopil. 
pravokotnogledenazacetnosmer,vprvemprimeruvz-smer,vdrugempaj. 
koncnopoljelezalovravniniyzinzosjozoklepalokot70A.Vobehprimeri. 
zaradihidrodnamicnegatokaprocespreklapljanjapotekazelonenavadno,sliki4.1. 
in4.20.Hkoncnemustanjuprispemovdvehizrazitihkorakih:najprejzaradivpliv. 
tokahitronastanedomenskastena,natopasenotranjadomenazaradiukrivljenost. 
domenskestenepocasimanj..
sa[56,p.213]inkoncnoizgine.Tuditukajsedapote. 
(nastanekdomene)pribliznozaslutitizzgornjimimehanizmi. 




Vescassmoracunalivlimitineskoncnomocnegasidranja.Cejesidranjekoncno. 
>0,a..
sevednomocno,=
m
1,veljaocena,dasehitrosthidrodinamicneg. 
tokazmanjsujekot1..3=
m
,stempatudinavortokanadirektor. 


Zdodatniminumericnimiracunismougotovili,daoblikacelicenimaodlocilneg. 
vplivanadinamiko,
.
cejelevelikostcelicevposameznihsmerehpribliznoenaka. 


Cetorejkvadratnadomestimoskrogom,nebovelikihsprememb,cistodruga
.
cep. 
je,
.
cecelico,znatnoraztegnjenovx-smeri,nadomestimostako,kijeraztegnjena. 
y-smeri. 


Dinamikavektorskegaureditvenegaparametr. 


Izpeljalibomodinamicneenacbezavektorskiureditveniparameterc.Grezapos. 
plositevEricksen-Lesliejeveteorijezdodatkomnehidrodinamicneprostostnestopnj. 
|dolzinevektorja.Opozoritijetreba,dajeEricksen-Lesliejevateorijavektorsk. 
teorija,zatojolahkokonsistentnoposplosimolenapopolnvektorskiureditvenipa. 
rameterinnenatenzorskega.Obravnavalibomosplosentridimenzionalniprimer. 
dvodimenzionalnarazlicicapaboprislapravpriopisutankegalmasmektika-C. 


Gostotaprosteenergij. 


Osredotocilisebomonahomogeniinelasticnidelgostoteprosteenergijef(c;rc). 


1
2
1
4
. 


f=

Ac+

Cc+

.
L
ijkl
(@
i
c
j
)(@
k
c
l
):(41. 


24. 



A
0

Homogenaclena,kjerveljaA=A
0
(T..T
0
),>0,C>0,opisujetafaznipreho. 


. 


indolocataravnovesnovelikostvektorjac,c
0
=..A=C.Velasticnidelvkljucim. 



Razsirjenipovzetek(AbstractinSlovene. 
i. 


samoclene,kvadratnevprvihodvodih,nepaclenovzdrugimiodvodiL
ijk
@
i
@
j
c
k
. 
kiprinesejosamododatnepovrsinskeprispevke. 


A
A
.
L
PoiskatijetorejtrebamatrikoL
ijkl
.Zahtevalibomo,dajeprostaenergijain. 
variantnanainverzijo(r!..r,c!..c),karpomeni,damorabitiL
ijkl
pose. 
bejinvariantnanatooperacijo.Polegtegamorazado.
scatipermutacijskisimetrij. 
ijkl
=L
klij
.MatrikoL
ijkl
smemosestavitileizkomponentvektorjactermatri. 
ik
in
ijk
.Posamezniprispevkisozbranivtabeli5.1.Vpeljalismofundamentaln. 
elasticnekonstanteL
i
,kisoneodvisneoddolzinevektorjac. 




ClenzL
1
jeizotropeninvsedeformacijepoljacobravnavaenako.ClenizL
2
. 
L
4
inL
9
ustrezajopahljacni,zvojni,oziromaupogibnideformaciji,clenzL
3
p. 
jepovrsinski.Ostaliprispevkisonenicelnile,
.
cesespreminjavelikostvektorjac. 
PosebejzanimivastaclenazL
6
inL
7
,kiustrezatasklopitvimedspreminjanjemve. 
likosticinpahljacnooziromaupogibnodeformacijo,slika5.1.Primerjavazizrazo. 
(22)pove..
zeFrankoveelasticnekonstantesfundamentalnimikonstantamiL
i
. 


.
K
11
=c
2
L
1
+c
2
L
2
. 
(42. 
c
2
+c
4

.
K
22
=L
1
L
9
. 
(43. 
c
2
+c
4

.
K
33
=L
1
L
4
. 
(44. 


pricemersmodolocilitudiodvisnostFrankovihkonstantodvelikostivektorjac. 
najnizjemredu. 


Sklopitevstoko. 


.

Najtimoramoizrazzaviskozninapetostnitenzorinviskoznogeneraliziranosilon. 
vektorc,karpomeni,damoramodolocitimatrikeS,M,R,C,DinBvenacba. 
(2.35)-(2.37)zlastnostmi(2.39).Tuditukajzahtevamo,dajegostotaentropijskeg. 
izvirainvariantnanainverzijo,torejmorajobitimatrikeS,M,RinBsodevc,. 
inDpalihi.MatrikesmemosestavitileizkomponentvektorjactermatrikA
ik
i. 
ijk
.Prispevki,kipodajajodisipacijovizotropnitekocini,sozbranivtabeli5.2. 
ostalipavtabeli5.3.Vpeljalismofundamentalneviskozneparametre
i
,kisospe. 
neodvisniodvelikostivektorjac. 


Zahtevatimoramo,dapritogirotacijisistemanidisipacije(2.38)insil(2.35). 
(2.37).Simetricniinantisimetricnidelviskozneganapetostnegatenzorjastatak. 



. 
. 


i. 
=
0
A
ij
+
1
(A
ik
c
k
c
j
+A
jk
c
k
c
i
)+
2
A
kl
c
k
c
l
c
i
c
j
. 


. 


1

4
(N
i
c
j
+N
j
c
i
)+
6
c
k
c_
k
c
i
c
j
. 


. 



. 
. 


. 


i. 
=
3
(N
i
c
j
..N
j
c
i
)+
4
(A
ik
c
k
c
j
..A
jk
c
k
c
i
);(45. 


2. 


viskoznageneraliziranasilanavektorcp. 


..h
v
=
3
N
i
+
6
Ac
j
c
k
c
i
+
9
c
j
c_
j
c
i
;(46)

i
+
4
A
ij
c
jjk

kjerj. 
N
i
=c_
i
+W
ij
c
. 
(47. 


relativnisubstancialnicasovniodvodvektorjacgledenavrtenjetekocine. 




. 
Razsirjenipovzete. 


PrimerjavazizraziEricksen-Lesliejeveteorije(4.13)in(4.6)pove..
zeLesliejev. 
viskoznostnekoeciente
i
sfundamentalnimikoecienti
i
. 



4
=
0
. 
c
2


5
+
6
=
1
. 
c
4


1
=
2
. 
(48. 
c
2



1
=
3
. 
c
2



2
=
4
. 


kjerje

1
=
3
..
2
in

2
=
3
+
2
kotobi
.
cajno.DolocilismoodvisnostLesliejevi. 
koecientovodvelikostivektorjacvnajnizjemredu. 


Vsplosnemvektorskemprimeruviskoznesiledolocatadvaparametrave
.
c,
6
i. 

9
,kistapovezanazdisipacijo,kadarjecvzporedensc.Pratakoprispevekclena

_vz
4
hgeneraliziranisilizdajnivecomejennasmer,pravokotnonac. 


Defekt. 


Nakratkobomospregovoriliotockovnihdefektihvektorja/direktorjavdvodimen. 
zionalnemsistemuinolinijskihdefektihdirektorjavnematiku.Poenostavljen. 
lahkorecemo,dajedefektnezveznostalinedeniranostureditvenegaparametran. 
nekimnozicitockvprostoru|tocki,krivuljialiploskvi.Sevedapavzicne. 
sistemunikolinesrecamonezveznosti.Vnematiku,naprimer,imamoopravka. 
nezveznostjole,doklervztrajamopridirektorskemopisu,kakorhitropauporabim. 
celotentenzorskiureditveniparameter,dobimozvezneresitve[61{63].Negleden. 
topajepojemdefekta..
sevednosmiseln,kerimale-tadaljnosezneposledicezapolj. 
ureditvenegaparametra|defektlahkodobrodeniramo,cetudisplohnepoznam. 
strukturenjegovegajedra.Vtanamensizamislimozanko,kijosklenemookro. 
sredi.
s
.
cadefekta,takodapotekapoobmocju,kjerveljadirektorskiopis(slika6.1). 
Koseponjejenkratsprehodimo,direktoropi..
sekot,kimorabitizaradizveznost. 
direktorskegapoljainsimetrijen=..nveckratnik,=2n.Pravkarsm. 
deniraliovojnosteviloalimocdefektan,kistopoloskegavidikadefektpopolnom. 
doloca. 



. 

3

n=0;

;1;

;::. 
(49. 


2. 


Pritemjeslozatockovnidefektvnamisljenemdvodimenzionalnemnematikual. 
pazapravilinijskidefektvtrehdimenzijah. 


Defekti,kijihlahkozzveznotransformacijopretvorimoedenvdrugega,so. 
topoloskemsmisluekvivalentni.Torejtaki,kijihlahkozveznotransformiramo. 
brezdefektnostrukturo,splohnisodefekti.Vdvodimenzionalnemprimeru,koj. 
direktordenirannaravnini,vsaovojnastevila(49)oznacujejorazlicnedefekte,sa. 




zveznetransformacijemednjimineobstajajo.Cebibilureditveniparametervekto. 
zn6..n,stevilskmoDruga
.
cejevtrehdimenzijah.

=bibiledovoljeneleceloeci.Ta. 
lahkovsecelostevilskedefektestakoimenovanimpobegomvtretjodimenzijo[68,69. 
pretvorimovbrezdefektnostrukturo.Velja..
seve
.
c:vsakemudefektulahkozzvezn. 
transformacijumocspremenimozacelostevilo.Topomeni,davnematikuobstaj. 
leentopoloskidefekt,tojedefektzmocjo1=2. 




Razsirjenipovzetek(AbstractinSlovene. 
x. 


Tockovnidefektiv2Dprimer. 


Ogledalisibomokonguracijoinprostoenergijotockovnihdefektovdvodimenzi. 
onalnegadirektorjavenokonstantnempriblizku,en.(23).Vsiizraziveljajotud. 
zaravnelinijskedefekte;tiste,kipodajajoprostoenergijo,vtemprimerupa 
razumemokotdolzinskogostotoprosteenergije.Obparametrizacijin=(cos;sin. 
seravnovesnipogojzadefektssredi.
scemvizhodi.
scuglas. 


r
2

=0. 
(50. 


Resitevsmebitiodvisnaleodpolarnegakota. 


. 


=n+
0
=narctg


+
0
;n=0;
1
;1;
3
;:::;(51. 
x
2. 


kjerje
0
prostiparameter.Polcelostevilonjemocdefekta.Deformacijskaprost. 
energijatestrukturej. 


2. 


 !
. 
 !
2

.
Z
. 
Z
2

K@@
2
. 


45

F
d
=rdrd+=Knln;(52. 


2r. 
. 
@x@yr
. 


kjerjeRvelikostvzorca,r
0
pamikroskopskadolzina,prikateriprenehaveljat. 
direktorskiopis.Vprvempriblizkuvpeljemoizotropnojedrozradijemr
0
,kis. 
takojnatonadaljujezdirektorskimpoljeminravnovesnovrednostjoskalarneg. 
ureditvenegaparametra.Prostaenergijajeizotropnegajedrajetak. 


F
c
=r
2

f. 
(53)

0

kjerje
frazlikagostotprosteenergijeizotropneinurejenefaze.Zminimizacij. 
celotneprosteenergijeF
d
+F
c
dolocimoradijjedra. 


. 


Kn
. 


r
0
=. 
(54. 


2
. 


kijetakosorazmerenzmocjodefektainjevelikostnegaredanematskekorelacijsk. 
dolzine(7.18).Celotnaprostaenergijadefektajekoncn. 


2
. 
1. 
. 


F=F
c
+F
d
=nK


+ln. 
(55. 


2r
. 




Ceimamoopravkazvecimidefekti,dobimoresitevzaradilinearnostiravnovesn. 
enacbe(50)karssestevanjemposameznihresitev(51). 


XX

y..y
. 


=(n
i

i
+
0i
)=n
i
arctg+
0
:(56)

0

i. 
x..x
. 


Prostaenergijadvehdefektovnarazdaljirje[65,p.529. 


. 


F=F
1
+F
2
+2Kn
1
n
2
ln. 
(57. 


. 



xi. 
Razsirjenipovzete. 


Prvaclenastaprostienergiji(55)posameznihdefektov,tretjipapredstavljain. 
terakcijskoprostoenergijo.Vidimo,dasedefektazmocmiistegapredznakaodbi. 
jata,defektazmocminasprotnihpredznakovpaprivlacita.Vposebnemprimer. 
n
2
=..n
1
jeprostaenergij. 


2
. 
1. 
. 


F=2nK


+ln;n=jn
1
j;r
1
=r
2
=r
0
;(58. 
2r
. 


takodaseznebimologaritemskedivergencevodvisnostiodvelikostivzorca.. 
splosnemsebododefektizmocminasprotnihpredznakov(nenujnoenakimipoab. 
solutnivrednosti)zdruzevali,sajbonatanacinprostaenergijamanjsa.Nasprotn. 
pajeugodno,
.
cedefektzvelikomocjorazpadenavecmanjsih,takodasetipote. 
oddaljijodrugoddrugegainzmanj..
sajoprostoenergijo. 


Anihilacijadisklinacijskihlinijvnematik. 




Cezelimostudiratistatikoalidinamikodefektovvnematiku,semoramoposluzit. 
popolnegatenzorskegaopisanematskefaze.Zavkljucitevhidrodinamike,kijo. 
direktorskemopisuobravnavamovokviruEricksen-Lesliejeveteorije,potrebujem. 
tenzorskorazlicicoteteorije[89,90,92]. 


Resilibomoproblemanihilacijeravnihdisklinacijskihlinijmo
.
ci1/2vne. 
matiku,pricemerbomoizhajaliiztenzorsketeorije[92].Vkljucilibomosamotist. 
disipacijskeclene,kivdirektorskemopisuskonstantnostopnjoureditvepreidejo. 
Lesliejeveclene.TakoboviskoznihparametrovtolikokotpriLeslieju,znjimip. 
bodopreprostolinearnopovezani. 


Dinamicneena

cb. 


GostotoprosteenergijevodvisnostiodQzapisemovpriblizkueneelasticnekon. 
stante[10,p.156]. 


111. 


)
2

f=AQ
ij
Q
ji
+BQ
ij
Q
jk
Q
k. 
+C(Q
ij
Q
ji
+L(@
i
Q
jk
)(@
i
Q
jk
):(59. 
234. 



Dabilocilimedpahljacnoinupogibnodeformacijo,bimoralivkljucititudielasticn. 
clene,kubicnevQ[99].Euler-Lagrangevaenacbazafunkcionalprosteenergij. 


. 


F=dV[f(Q;rQ)..Q
ii
..
i

ijk
Q
jk
];(60. 


pricemersmozahtevalisimetricnostinbrezslednostQ,namdahomogeniinelasticn. 
delgeneraliziranesilenaureditveniparameterQ. 


@. 


h
h. 
. 


i. 
=L@
k
Q
ij
..+A
ij
+
k

kij
. 
(61. 
@Q
i. 


Lagrangeovihmultiplikatorjevseznebimo,
.
ceenacbo(61)projiciramonasimetricn. 
inbrezslednipodprostor(odstejemonjensimetricniinizotropnidel).Elasticn. 



Razsirjenipovzetek(AbstractinSlovene. 
xii. 


napetostnitenzorjepoenacbi(11. 


@. 



. 
ij
=..@
j
Q. 
(62)

kl

@(@
i
Q
kl
. 


ViskozninapetostnitenzorinviskoznageneraliziranasilanatenzorQsta[92. 
. 



. 


=
1
Q
ij
Q
kl
A
kl
+
4
A
ij
+
5
Q
ik
A
kj
+
6
Q
jk
A
ki
+
2
N
ij
..
1
Q
ik
N
kj
+
1
Q
jk
N
ki
;(63)

ij
. 



..h
. 
=
. 

2
A
ij
+
1
N
ij
. 
(64)

ij
. 


kjerj. 


_

.
N
ij
=Q
ij
+W
ik
Q
k. 
..Q
ik
W
kj
. 
(65. 


_

.
Q
i. 
=@Q
ij
=@t+(v
r)Q
i. 
pajesubstancialniodvod.AinWstasimetricnii. 
antisimetricnidelgradientahitrosti.Medviskoznimikoecientiv(63)in(64)velj. 
zveza
2
=
6
..
5
. 


GibalnaenacbazaureditveniparameterQjeravnovesjebrezslednegasimetricneg. 
dela(?)generaliziranihsil. 


n. 
h
he
+h
. 
=0. 
(66. 
. 
zvezm. 
Q
ii
=0;
ijk
Q
j. 
=0. 
(67. 


HitrostnopoljejedolocenosposplosenoNavier-Stokesovoenacbo(15)inpogoje. 
nestisljivosti(16),pricemerjenapetostnitenzorpodanzenacbama(62)in(63). 


Znacilnakrajevnaskalaproblemajenematskakorelacijskadolzina(tipicnozna.. 
sanekajnanometrov. 


. 


3. 


=. 
(68. 


2f
00
j
S. 


f
00
j
S0

kjerjevrednostdrugegaodvodagostoteprosteenergijeposkalarnemured. 
itvenemparametrupriravnovesnivrednostile-tega.Znacilnicas,kiimapome. 
relaksacijskegacasaureditvenegaparametranakrajevniskaliinzna..
satipicn. 
nekajdesetnanosekund,paj. 


=

1

2
=K=
1

2
=L. 
(69. 


kjerje

1
direktorskarotacijskaviskoznost,Kpadirektorskaelasticnakonstanta. 


Rezultat. 


Slika7.2kaze,dazaradihidrodinamicnegatokaanihilacijapotekahitrejei. 
asimetricno,pricemerjedefektspozitivnomocjohitrej..
sioddefektaznegativn. 
mocjo.Nasliki7.3pavidimo,datokvplivapredvsemnadefektspozitivnomocjo. 
medtemkohitrostnegativneganitakomocnospremenjena.Izka..
zese,datoknade. 
fektevnajvecjimerivplivaprekadvekcije.Rezultatejemockvalitativnopojasnit. 
zupostevanjempoglavitnihprispevkovknapetostnemutenzorju,kizenejotok:t. 



xi. 
Razsirjenipovzete. 


soelasticninapetostnitenzorincleniz
1
in
2
vviskoznemnapetostnemtenzorju. 
Ssimetrijskimiargumentilahkopokazemo,dajetok,kigapoganjaelasticnitenzor. 
simetricenzaobadefektainjusevedazeneskupaj,medtemkojetok,kigapoganj. 

1
clen,tocnoantisimetricen,njegovasmerpajeodpozitivnegadefektaknega. 
tivnemu,slika7.6.Kpospesitvianihilacijetorejnajboljprispevajoelasticnesile,. 




asimetrijipaviskozniclenz
1
.Seve
.
c,prispevkasezapozitivnidefektkonstruk. 
tivnosestejeta,medtemkosezanegativnegaodstejeta,scimerpojasnimomocnej.. 
sitoknamestuprvega. 


Primerjavaslik7.4in7.5pokaze,dastaadvekcijskiprispevekkhitrostidefekt. 
inprispevekzaradireorientacijeureditvenegaparametraenakihvelikostnihredov. 
Primajhnihmeddefektnihrazdaljah(nekajkorelacijskihdolzin)prevladadrugi,pr. 
vecjihpajeadvekcijapomembnej..
sa(slika9.10). 




Clenvviskoznemnapetostnemtenzorjuz
2
nimaposebnesimetrijegledespre. 
membepredznakadefektov,razlikujepatudimedkonguracijami,kiselocij. 
pohomogenirotacijidirektorskegapolja,slika7.1(elasticniin
1
clenison. 
toneobcutljivi).Venokonstantnempriblizkubreztokasetaksnekonguracij. 
obna..
sajoenako,stokompane,slika7.2,zakarjenajprejodgovoren
2
clen. 


Zakljucimolahko,dajezaasimetrijoinpospesitevanihilacijepomembn. 
razmerje
1
=
4
,insicerzvecanjemrazmerjahidrodinamicniucinkinara.
s
.
cajo.Ra. 
zlikavdinamikikonguracij,kiserazlikujejopohomogenirotacijidirektorja,p. 
nara.
s
.
cazvecanjemrazmerja
2
=
1
.Pasivniviskoznicleniz
1
,
5
in
6
vkvalita. 




tivnempogledunisopomembni.Cespremenimoelasticnokonstanto,setopozn. 
samopriznacilnemcasuprocesa,takodazreskaliranjemcasovnedimenzijeanihi. 
lacijapotekaenako. 


AnihilacijadefektovvlmihSm. 


UreditveniparameterSmCfaz. 


VSmAfazijedirektor,kipodajapovprecnosmermolekul,pravokotennasmekticn. 
plasti,vSmCfazipajenagnjen.Zeksperimentalnegavidikajezelopripravensiste. 
zaopazovanjedinamikedefektovprostostoje
.
cilmSmC,debellenekajsmekticni. 
plasti.Projekcijadirektorjanasmekticnoravninojedvodimenzionalnivektor. 
takoimenovanic-direktor,kijeprimerenureditveniparameterSmCfaze.Tako. 
jetrebaopozoriti,dajec-direktorimenunavkljubvresnicivektor,c=6..c.Nje. 
govavelikost(nagib,amplituda)kondenzirainpostaneodnicrazlicnaobprehodui. 
SmAvSmCfazo,medtemkojesmer(faza)hidrodinamicnakolicinazGoldstoneov. 
ekscitacijo. 


Topoloskidefektic-direktorjasodisklinacijskelinijescelimiovojnimistevili. 
vrtinci.Daseizognemosingularnostivsredi.
scudefekta,moramodovoliti,das. 
spreminjavelikostc,takodasesistemlahkozate
.
cekSmAfazi. 


Poudaritijepotrebno,dastaureditvenaparametraSmCinnematskefaze. 
osnovirazlicnaindalahkoSmCsistem|sspodnjimiomejitvami|prevedemon. 
XY-model,nematikapane.Zatorejsezdi,daimadinamikavrtincevvSmCsistem. 
zelosplosnoveljavo.Opozoritimoramo..
senato,dasodisklinacijescelimimocm. 



Razsirjenipovzetek(AbstractinSlovene. 
x. 




vnematikunestabilneinrazpadejonavsaksebibe..
ze
.
ce1=2disklinacije.Cezelim. 
studiratidinamikovrtincev,semoramonujnozate
.
cikvektorskemuureditvenem. 
parametru. 


Dinamicneena

cb. 


IzhajamoizhidrodinamicneteorijeCarlssona,Leslieja,StewartainClarkazaSm. 
tekocekristalnofazo[108,109],kiprivzamesmekticneplastiskonstantnodebelin. 
terkonstantenpovprecninagibmolekul.Zaopisdefektovjetrebadovolitivsa. 
spreminjanjenagiba,torejmoramoteorijonekolikoposplositi.Podrugistranipaj. 
bomobistvenopoenostavili,sajbomoobravnavalisistemzvariacijamivsamodve. 
dimenzijahinzravnimismekticnimiplastmi.Takoizsistemadinamicnihkolici. 
izlocimonormalonasmekticneravnine,kinajbokar.
^e
z
.Kerimamoopravka. 
prostostojecimtankimlmom,vpeljemodveneodvisnikrajevnispremenljivkixi. 
y,zr=.
^e
x
@
x
+.
^e
y
@
y
,pravtakopatuditokomejimonaravnino,v=v
x
.
^e
x
+v
y
.
^e
y
. 
Stemsmona.
ssistemprevedlinaXY-model.Izka..
zese,dasevtemprimer. 
teorija[108,109]prevedenaEricksen-Lesliejevoteorijozanematskiteko
.
cikristal. 
Dodatnobomomoraliposkrbeti..
sezaspreminjanjenagibamolekul.Takosm. 
prispelinatancnododvodimenzionalnerazlicicedinamikevektorskegaureditveneg. 
parametra.Vdvehdimenzijahse,razenodsotnostizvojnedeformacije,nespremen. 
nicdrugega. 


Povrsinskeelasticneclenebomospustili,pravtakopatudivecinoprispevko. 
iztabele5.1,sajzanjenepoznamoelasticnihkonstant.Razlikovatizelimoleme. 
pahljacnoinupogibnodeformacijo,kerstalahkozaradispontanepolarizacijevSm. 
energijskozelorazlicni[110{112].Gostotoprosteenergijetakozapisemoko. 


1
2
1
4
1. 
f=


Ac+
Cc+
B
1
(rc)
2
+
B
2
(r
c)
2
:(70. 
242. 
Primerjavazelasticnimicleniiztabele5.1pokaze,da,nemenecsezapovrsinsk. 
prispevke,velj. 
B
1
=L
1
;B
2
=L
1
+L
2
. 
(71. 


karpomeni,dasmoclenezL
4
-L
8
spustilikonsistentno.Opozoritizelimo,dast. 
elasticnikonstantiB
1
inB
2
neodvisniodnagiba.Euler-Lagrangeovaenacbaz. 


. 


funkcionalprosteenergijeF=dVf(c;rc)dahomogeniinelasticnidelgeneral. 
iziranesilenavektorc. 


2
@
2

h
i
=..(A+Cc)c
i
+B
1
j
c
i
+(B
2
..B
1
)@
i
@
j
c
j
:(72. 


Elasticninapetostnitenzordobimopoenacbi(2.26)iz(70). 


@. 



. 


=. 
(73)

ij
@
j
c
k
. 


@(@
i
c
k
. 


Neokrnjenateorija[109]zajema20viskoznihclenov,odkaterihpavprimer. 
ravnihsmekticnihplasti,ravninskegatokainodsotnostigradientovvsmerinor. 


malenaplastiostanejozgoljLesliejevi.Ceupostevamo..
sespreminjanjedolzinec. 



xv. 
Razsirjenipovzete. 


nampravenacbe(45)in(46)podajajotocenopisdisipativnihsil.Kljubtemus. 
bomoomejilisamonastandardneLesliejeveprispevke,katerihviskozneparametr. 
poznamo,inspustiliclenesparametroma
6
in
9
.Viskozninapetostnitenzorj. 
tak. 



. 
=
0
+
2
c
k
c
l
A
k
c
i
c
j
+
1

3
(N
i
c
j
..c
i
N
j
)+
i. 
A
ijl
. 


111



4
(N
i
c
j
+c
i
N
j
)+(
1
..
4
)c
i
A
jk
c
k
+(
1
+
4
)A
ik
c
k
c
j
;(74. 
22. 


viskoznageneraliziranasilanavektorcp. 


..h
v
=+
4
=_(75)

i

3
N
i
A
ij
c
j
;N
i
c
i
+W
ij
c
j
;

kjerjec=@c=@t+(v
r)csubstancialniodvodvektorjacgledenavrtenje

_casovnitekocine,AinWpastasimetricnioziromaantisimetricnidelgradientahitrosti. 


Cetudistaenacbi(74)in(75)natancnotaki,kotvEricksen-Lesliejeviteoriji,j. 
pomembnarazlikata,datukajviskozniparametrinisoodvisniodkondenziran. 
kolicine|velikostivektorjac,medtemkoLesliejeviso,en.(48). 


Zavedatisemoramo,dasmosspreminjanjemvelikosticnaravnoposplosil. 
Ericksen-Lesliejevoteorijozaprimerneenotskegavektorja,scimersmotudiav. 
tomatskodolocilipravilneodvisnostisilodnagibamolekul.Nasprotnozanematik. 
seletenzorskateorijadapraveodvisnostisilodskalarnegaureditvenegaparame. 
trainstopnjebiaksialnosti.Naslismotorejsistem,zakateregaEricksen-Lesliejev. 
teorijatocnovelja. 


Gibalnaenacbazavektorcjenakratk. 


h+h
v
=. 
(76. 


inskupajsposplosenoNavier-Stokesovoenacbo(15)inpogojemnestisljivosti(16. 
predstavljasistemtrehparcialnihdiferencialnihenacb,kiopisujejodinamikomod. 
eliranegaSmCsistema.Spetvpeljemoznacilnodolzino,kijetokratkorelacijsk. 
dolzinanagibamolekul. 


. 


.
B
0

. 
. 
(77. 


(A+3Cc
2
)

0

tipicnonekajnanometrov,inkarakteristicnica. 



3

. 


=. 
(78. 


.
B
. 




kjerjeB
0
=(B
1
+B
2
)=2.Casjerelaksacijskicasdeformacijvektorjacn. 
dolzinskiskali,aliekvivalentno,casprilagajanjavelikostivektorjac,tipicnoneka. 
desetnanosekund. 


Rezultat. 


Najprejpoglejmoprimerzizotropnoelasticnostjo,B
1
=B
2
.Anihilacijapotek. 
kvalitativnoenakokotprinematskihdefektihspolovicnomocjo,slika8.3.Hidrodi. 
namicnitok(slika8.2)spetpospe..
siprocesinpovzro
.
ciasimetrijovgibanjudefek. 
tov.Tok,kigapoganjajoelasticnesile,dajeglavniprispevekkpospesitvi,tok,kig. 



Razsirjenipovzetek(AbstractinSlovene. 
xvi. 


poganjavrtenjeureditvenegaparametra(natancnejeclenz
3
vviskoznemnapetost. 
nemtenzorju),pakasimetriji(slika8.5).Tokjemocanobpozitivnemdefekt. 
insibakobnegativnem,kerseomenjenatokovnaprispevkaenkratkonstruktivno. 
drugicdestruktivnosestavita.Hitrostobehprispevkovgledenahitrostgibanj. 
defektasamozaradireorientacijec-direktorjajesorazmernaz
3
=
0
.Asimetrij. 
medkonguracijami,kiserazlikujejozahomogenorotacijovektorskegapoljac. 
spetprina..
savglavnemviskozniprispevekclenaz
4
,takodasetaasimetrijave
.
ca. 
vecanjemrazmerja
4
=
0
.Ostali(pasivni)viskozniclenikvalitativnonisopomembni. 
Reskaliranjeelasticnihkonstantgledenaviskoznostispetspremenisamoznacilnica. 
procesa.Vmanj..
simerikasimetrijiprispevatudielasticnaanizotropija,kotpoka.. 
zeslika8.6. 


Razpaddefektovscelimicmi

mo

Brezupostevanjahidrodinamikebomoresilisplosenproblemstabilnostineskoncnih. 
ravnihdisklinacijskihlinijscelimimocmivnematiku,takodabomopoiskalilastn. 
resitveperturbacijokrogtehstruktur.Vnelinearnemrezimubomopreucilivpli. 
hidrodinamikenarazpaddisklinacijezmocjo1terodbojnogibanjenastalegapar. 
enakihdefektovmo
.
ci1=2. 


Lineariziraniproble. 


Posluzimosecilindricnihkoordinat(r;;z)spripadajocimibaznimivektorji(.
^e
r
;.
^e

;.
^e
z
). 
pricemerdisklinacijskalinijale..
zinaosiz.Venokonstantnempriblizku(59)jeprost. 
energijainvariantnanahomogenorotacijotenzorjaQ.Topomeni,daselastnisiste. 
tenzorjavrtikot = 
0
+(s..1),kogremookrogdefektazmocjosvizhodi.
scu,pr. 
cemerje 
0
prostiparameterdefektnestruktureinpredstavljakotmeddirektorje. 
pri=0inosjox(zaradialnidefektje 
0
=0,zatangencialnegapa 
0
==2). 
Odkotajeodvisnasamoorientacijalastnegasistema,skalarneinvariantetenzorj. 
Qpane.Zaraditeposplosenecilindricnesimetrijejeproblemlastnihresitevmo 
dovoljenostavnoresiti. 


Denirajmo..
seenoortonormalnotrojicovektorjev(.
^e
1
;.
^e
2
;.
^e
z
). 


. 


.
^e
. 
cos sin .
^e
r

. 
. 
(79. 


.
^e
. 
..sin cos .
^e
. 


takodaprineperturbiraniresitvi(imenujmojoosnovnostanje)lastnisistemtenzorj. 
Qpovsodsovpadastotrojico.Vpeljemo..
sepetortonormalnihbaznihtenzorjevT
i
. 
en.(9.2),inzapisem. 


Q(r;t)=a
i
(r;t)T
i
(r);i=..2;..1;0;1;2:(80. 


Osnovnostanjezaradisimetrijevsebujelekomponentia
0
ina
1
,perturbacijepas. 
splosne. 


. 


q
i
(r)+x
i
(r;t);i=0;. 


a
i
(r;t). 
;(81. 


x
i
(r;t);i=..1;2;... 



xviii Razsirjeni povzetek 
q0;1 sta komponenti osnovnega stanja, xi pa so komponente perturbacij, xi  q0;1. 
Komponenti osnovnega stanja zadoscata enacbam (9.6) in (9.7) in ju dobimo iz 
numericne resitve. Iz enacb moramo izlusciti le njun potek v blizini r = 0. Linearizirane 
enacbe za perturbacije pa tvorijo dva sistema: 
x_ 0 = r2x0 .. f0(r) x0 + f01(r) x1; (82) 
x_ 1 = r2x1 .. 
4s2 
r2 
x1 .. 
4s 
r2 
@x..1 
@ .. f1(r) x1 + f01(r) x0; (83) 
x_..1 = r2x..1 .. 
4s2 
r2 
x..1 + 
4s 
r2 
@x1 
@ .. f..1(r) x..1 (84) 
in 
x_ 2 = r2x2 .. 
s2 
r2 
x2 .. 2s
@x..2 
@ .. f2(r) x2; (85) 
x_..2 = r2x..2 .. 
s2 
r2 
x..2 + 2s
@x2 
@ .. f..2(r) x..2; (86) 
kjer je r2 Laplaceov operator v cilindricnih koordinatah, fi(r) pa so polinomi druge 
stopnje v komponentah q0 in q1, en. (9.15). Sicer preprosta odvisnost od koordinate 
z nas zaenkrat ne bo zanimala. Poudarimo se to, da za defekta z mocema s in 
..s dobimo identicne enacbe, ce ustrezno popravimo bazne tenzorje: s ! ..s in 
T
..1;..2 !..T
..1;..2 ne spremeni enacb. 
Lastne resitve sistemov (82)-(84) in (85)-(86) iscemo z nastavkom 
8> 
<> 
: 
x0 
x1 
x..1 
9> 
=> 
; 
= 8> 
<> 
: 
R0(r) cos(m) 
R1(r) cos(m) 
R..1(r) sin(m)
9> 
=> 
;
exp(..t); (87) 
 x2 
x..2  =  R2(r) cos(m) 
R..2(r) sin(m) exp(..t); (88) 
kjer je m celo stevilo. Zaradi preglednosti smo izpustili prosto fazo v kotnem delu. 
Preostaneta lastna sistema za radialne funkcije Ri(r) z lastno vrednostjo : 
r2R0 +   .. f0(r) .. 
m2 
r2 !R0 + f01(r)R1 = 0; (89) 
r2R1 +   .. f1(r) .. 
m2+4s2 
r2 !R1 .. 
4sm 
r2 
R..1 + f01(r)R0 = 0; (90) 
r2R..1 +   .. f..1(r) .. 
m2+4s2 
r2 !R..1 .. 
4sm 
r2 
R1 = 0 (91) 
in 
r2R2 +   .. f2(r) .. 
m2+s2 
r2 !R2 .. 
2sm 
r2 
R..2 = 0; (92) 
r2R..2 +   .. f..2(r) .. 
m2+s2 
r2 !R..2 .. 
2sm 
r2 
R2 = 0: (93) 
Resimo ju z metodo streljanja [54, p. 582].

Razsirjeni povzetek (Abstract in Slovene. 
xi. 


Razpaddefekt. 


Lastneekscitacije,kivodijodorazpadadefekta,zaradisimetrijevsebujejolekom. 
ponentex
0
,x
1
inx
..1
.Zadefektzmocjo1najdemoenosamonara.
s
.
cajo
.
colastn. 
resitev(<0),insicerprim=2,karustrezarazpadunadvadefektazmocjo1=2. 
sliki9.2in9.3.Vplivhidrodinamicnegatokanacasovnokonstantonara.
s
.
cajo
.
ceeksc. 
itacijejepricakovanomajhen(manj..
siod5%)inpospe..
sinara.
scanje.Izjavanicist. 
trdna,sajnismoupostevalivsehviskoznihprispevkov,polegtegajesamkoncep. 
hidrodinamikepritehdolzinskihincasovnihskalah(1nm,10ns)precejvprasljiv. 


Pridefektihzvecjimicelostevilskimimocminajdemovecnara.
s
.
cajocihekscitacij. 
Ekscitacijez<0imajodiskretenspekterinsolokalizirane,spekteronihz>. 
pajezvezen.Takolahkonara.
s
.
cajo
.
ceresitveprestevamo.Izka..
zese,dazavsa. 
topoloskodovoljenrazpadobstajavsajenanara.
s
.
cajo
.
caresitev,
.
celenobenaodmo
. 
cinastalihdefektovniprevelika.Resitveimajoznacilnokotnosimetrijo,doloceno. 
m.Vsplosnemdefektzmocjosrazpadenamdefektovzmocmi1=2,simetricn. 
razporejenihokrogdefektazmocjosm=2,slika9.4.Vserazpadnemoznost. 
defektovzs=2ins=3sozbranevtabelah9.1in9.2.Razpadsamona1=. 
defektejevednonajhitrejsi. 


Pobegdefekt. 


Vneomejenemsredstvulahkocelostevilskidefektipobegnejovnedeformiranostruk. 
turo,kateredeformacijskaenergijajenicelna.Poglejmotorej,kakojesstabilnos. 
tjonamajhneperturbacije.Tokratnastopatalekomponentix
2
inx
..2
,priceme. 
pricakujemo,dabozapobegpomembnakomponentax
2
,kiustrezavrtenjudirek. 
torjaizravnine.Prim=0seenacbi(92)in(93)resrazklopitainugotovimo,das. 
vseekscitacijex
2
nara.
s
.
cajoce,hkratipatudilokaliziraneinzdiskretnimspektrom. 
Resitvex
..2
prim=0invseostaleresitveprim=60sopojemajoce.Izka..
zese,das. 
nara.
s
.
cajo
.
ceekscitacije,kivodijodopobega,velikorazseznej..
seinstempocasnej.. 
seodonih,kiprivedejodorazpada.Zadefektzmocjo1jerazmerjecasovnihkonstan. 
okrog53.Torejbodefektrazpadel,predenmubouspelopobegniti. 


Vplivhidrodinamitoka

cnega

Oglejmosi..
sevplivhidrodinamikenadefektazmocjo1=2,kinastanetazrazpado. 
1defekta.Tokratasimetrijeprigibanjusevedani,zatopaimatoktolikovecj. 
vplivnahitrostdefektov(slika9.7),sajobaprispevka,elasticniinviskozni(clen. 

1
),poganjatatokvistismeri.Vplivhidrodinamikejevecjiprivecjihrazmerji. 

1
=
4
,karjelepovidnotudinasliki9.8. 


Kervtemprimerunite..
zavzzacetnimpogojem(tukajjetokoncnostanje)ko. 
prianihilaciji,lahkozvemovecotem,kajsedogajashitrostjodefektovpriveliki. 
meddefektnihrazdaljah.Posebejzanimivjegraf9.10,kiprikazuje,kakoserazmerj. 
medadvekcijskohitrostjo(transportstokom)incelotnohitrostjodefektaspreminj. 
zrazdaljomeddefektoma.Razmerjezrazdaljonara.
s
.
cainznatnoprese..
zepolovico. 
Hidrodinamicnegatokatorejnegrezanemarjati,..
seposebej,
.
ceupostevamodejstvo. 



x. 
Razsirjenipovzete. 


dassimulacijamizaenkratdosezemolemajhnemeddefektnerazdalje|nagraf. 
9.10okrog80ali0.17m. 


Zaklju

ce. 


Vdisertacijisempredstavilnekajproblemovdinamiketekocihkristalovzupostevanje. 
hidrodinamike,kismojihizbralizvidikaeksperimentalneinteoreticnerelevantnosti. 
nenazadnjepasevedatudizvidikaresljivosti. 


Relaksacijskeproblemesmov4.poglavjuobravnavalizuveljavljenoEricksen. 
Lesliejevoteorijonematskegadirektorja.Sluzilisopredvsemkotpripravazakasnej.. 
setezjepodvige.Zizbirodovoljkompleksnegeometrijesmovseenouspeliopozoritin. 
primere,kohidrodinamicnitokpovsemspremenicasovnirazvojsistema. 


Dinamicneenacbezavektorskiureditveniparameter,izpeljanev5.poglavju. 
potrebujemozaopisdefektovvsistemustemureditvenimparametrom,vnase. 
primerujebiltolmSmCfaze.Zizpeljavosmohkratipokazali,dajeEricksen. 
Lesliejevateorijanatancnoteorijazaenotskivektorskiureditveniparameterinni. 
nikakrsnizveziznematskimureditvenimtenzorjem. 


Zastudijdinamikedefektovvnematikujetakotrebazacetiznovainizdelat. 
tenzorskoteorijo.Le-tojepotemmocpoenostavitidodirektorske.Obratnapot,kje. 
biEricksen-Lesliejevoteorijo,vkateridirektornastopalinearno,razsirilistem,dab. 
dovolilivariacijoskalarnegaureditvenegaparametra,seneobnese.V7.poglavjusm. 
zokrnjenotenzorskoteorijoresilihidrodinamicniproblemanihilacijepararavni. 
nematskihdisklinacijskihlinij.Pokazalismo,dajezaasimetrijoprigibanjudefekto. 
odgovorenpredvsemtoknematsketekocine. 


Kljubeksperimentalnipripravnostivliteraturinismozasledilinobenihnu. 
mericnihstudijdinamikedefektovvSmClmih.Edenodmoznihrazlogovj. 
zapletenostenacbzvelimistevilomsnovnihparametrov,povecinineizmerjenih.Pr. 
tembimoraliupostevati..
sespreminjanjedolzinec-direktorja.V8.poglavjusm. 
sevecinite..
zavizognili,stemdasmoobdodatnihpredpostavkahdinamicnoteorij. 
SmCfazepoenostavilidoEricksen-Lesliejeveteorije,sistemSmClmapaprevedl. 
naXY-model,kateregadinamikoopisujejovektorskeenacbeiz5.poglavja.Pokazal. 
smo,dajevplivtokanaanihilacijoparadisklinacijzmocjo1kvalitativnoena. 
kotprinematiku. 


Vnematikujedisklinacijazmocjo1nestabilnainspontanorazpadenapa. 
enakih1=2disklinacij.Dabiraziskalizacetnistadijrazpada,smov9.poglavj. 
studiralidinamikoperturbacijravnihnematskihdikslinacijskihlinijscelimimocmi. 
Vpriblizkueneelasticnekonstantesmouspeliresititenzorski
uktuacijskiprob. 
lemravnihdikslinacijskihlinijsplosnihcelostevilskihmoci.Naslismodvevrst. 
nara.
s
.
cajocih
uktuacij,odgovornihzarazpadnadisklinacijezmanjsimimocm. 
oziromapobegvtretjodimenzijo.Vobehprimerihjespekterdiskreten,
uktuacij. 




palokalizirane.Casovnakonstantaprvihjezaveckotredvelikostimanjsa,tore. 
disklinacijezvelikimimocmirazpadejo,..
sepredenjimuspepobegniti.Preverilism. 
tudivplivhidrodinamikenaodbojnogibanjedvehenakih1=2disklinacij,nastali. 
porazpadu,kijezaraditokavelikohitrejse. 



Razsirjenipovzetek(AbstractinSlovene. 
xx. 




Cebisiodpredstavljenegamoralizapomnitileenostvar,najbotopomembnos. 
hidrodinamicnegatokapridinamikidefektovvtekocihkristalih.Pokazalismo,d. 
jeprispevekadvekcije(transportastokom)hgibanjudefektapovsemprimerljiv. 
prispevkomzaradireorientacijeureditvenegaparametra.Vprimeruizbireviskozni. 
parametrov,kiustrezajonematikuMBBA,jeprirelevantnihmeddefektnihrazdalja. 
advekcijacelopomembnejsa. 



xxi. 
Razsirjenipovzete. 



1 
Introduction 
Liquid crystals are mesophases between the liquid and solid phases, in the sense that 
they are more ordered than liquids, yet less ordered than solids. There exist many 
liquid-crystalline phases as there are many steps in which the translational and rotational 
symmetry of the liquid can be reduced to that of the solid. Microscopically, 
the required condition for a material to exhibit a liquid-crystalline phase is that it 
consists of elongated or disc-like molecules/particles. In thermotropic liquid crystals 
the order is controlled by the temperature, whereas in lyotropic liquid crystals it is 
controlled by the concentration of the liquid-crystalline material in a solution. Due 
to the ordering, liquid crystals exhibit anisotropic properties on the macroscopic 
level. 
The least ordered among liquid crystals is the nematic [1], which possesses a 
long-range orientational order of the molecules, Fig. 1.1. The average orientation of 
the molecules is specied by the director n, which is a part of the nematic tensor 
order parameter. The name \nematic" was invented by Friedel [2] in the early 
twentieth century. It originates from the Greek word for a thread, many of which 
can be observed between crossed polarizers due to line defects in the nematic. 
A more ordered phase is the smectic phase [3], which, if present, occurs at a lower 
temperature than the nematic phase. The name is again due to Friedel and comes 
from the Greek word for soap (smectics can form thin lms). Besides orientational 
order, the smectic possesses also an one-dimensional long-range translational order 
in the cast of a density wave. The surfaces of maximum density are called smectic 
layers or planes. In the smectic-A (SmA) phase the director n is normal to the 
layers, while in the smectic-C (SmC) phase occurring at a yet lower temperature 
the director is tilted with respect to the layer normal. One must point out that 
the notion of the translational ordering is dierent than in solid crystals, where the 
molecules/atoms are actually bound to crystal sites and can merely oscillate around 
them. In smectics, the molecules are not bound to the layers. 
Owing to the optical anisotropy, thermotropic liquid crystals are used in applications 
related to optics | liquid crystal displays (LCD's), optical switches, shutters, 
polarization rotators, tunable color lters, etc. On the other hand, lyotropic liquid 
crystals are widely used in chemical and food industry. 
The modelling of liquid crystal systems has come to life with powerful enough 
7

. 
Introductio. 



(a)isotropi 
(b)nemati 
(c)smecti 


Figure1.1Schematicrepresentationof(b)thenematicand(c)smecti 


phases.Theaverageorientationofthemoleculesisspeciedbythedirec. 


torn,Eq.(3.10).Thesmecticphaseischaracterizedbyanone-dimensiona. 


periodicdensitymodulation. 


computers.Staticstructuresofconnedliquidcrystalshavebeenstudiedthor. 
oughlyinthelasttwodecades;thesamecanbesaidforthestructuresofthedefec. 
cores.Dynamicstudieshavefollowednext,eitherassimulationsofappliedorap. 
plicableset-ups,orarisingfromthesheertheoreticalinterestindynamicprocesses. 
Mostfrequently,thedynamicsofthenematicliquidcrystals,includingthatofth. 
defects,hasbeenstudiedinthedirectordescription,neglectingthehydrodynami 

owentirely.WhilethedirectordescriptioniseAcientandperfectlyadequatefo. 
theproblemsnotinvolvinganydefects,theneglectofthe
owisalwaysquestionable. 
Hence,inapropertreatmentofdefectdynamics,onemustbothusethecomplet. 
tensororderparameteraswellastakeintoaccountthehydrodynamicpartofth. 
problem.Theuseofthetensororderparameterdoesnotbringanysignicantcom. 
plications|itonlyyieldsricherstructures.Thedrawbackisthatitintroduces. 
microscopiclengthscale,whichsetsanupperlimittothe(possiblymacroscopic. 
lengthscalesthatcanbereachedinasimulation.Ontheotherhand,theinclusio. 
ofthehydrodynamicsmakesthetreatmentdiAcult,bothconceptuallyandcompu. 
tationally.Theproblemsareparticularlydemandingiftheyinvolvemorethanjus. 
onespatialcoordinate,becauseinthiscasetheincompressibilityofthe
uidmus. 
beensuredbyndingtheproperpressuredistribution. 


IntheThesis,westudythedynamicsofnematicandSmCliquidcrystals.Inpar. 
ticular,wefocusonthehydrodynamicphenomenaaccompanyingthetimeevolutio. 
oftheorderparameter.Moreprecisely,weareinterestedintheso-calledback
o. 
eects|thegenerationofthe
uid
owbymotionoftheorderparameter(e.g.. 
bythedirectorreorientation)andconversely,thein
uenceofthegenerated
owt. 
themotionoftheorderparameter.IntherstpartoftheThesiswestudydirecto. 
relaxationprocessesinliquidcrystalcellstriggeredbyexternaleldswitching[4,5]. 
One-dimensionalproblemsofthiskindwerestudiedinthe1970s(seetheIntro. 
ductiontoChapter4foramoredetailedreview).Ourgeometryismoreconne. 
leadingtothedependenceontwospatialcoordinates.Thismakestheproblem. 
muchhardertosolverequiringinvolvedcomputationapproaches.Weshowtha. 
theback
owcanbemorethanjustquantitativelyimportant|itcancauseth. 



Introductio. 
. 


switchingtooccurintwocharacteristicstepsratherthanjustinone. 


Thesecondpartisdevotedtothedynamicsofdefects,whichhasbeenth. 
primarychallengetous.Forthemotivationandashortreviewoftheprecedin. 
researchseealsotheIntroductiontoChapter7.Ourambitionhasbeentosolveaful. 
hydrodynamicannihilationproblemofadefect-antidefectpair.Forthispurposew. 
makeuseofthenumericalmethoddevelopedwiththeswitchingproblems,properl. 
modiedinordertobeabletodescribethedefects.Thedependenceontwospatia. 
coordinatesisaminimumfortheseproblems.Westudytwoliquidcrystalsystems. 
thenematicandtheSmCfree-standingthinlm,showingthatboththeannihilatio. 
[6,7]andrepulsionofdefectsarealwayssubjecttostrongback
ow,whichspeed. 
uptheprocessesremarkably.Inthecaseoftheannihilation,itintroducesalsoa. 
asymmetryindefectmotion,i.e.,itmakesonedefectmovefasterthantheother. 


ThestructureoftheThesisisasfollows.InChapter2thedynamictheor. 
couplingtheorderparameterdynamicsandthehydrodynamicsisreviewed.Th. 
theoryiskeptgeneralinthisChapter,notassuminganyspecicformoftheorde. 
parameter.Chapter3denesthenematicorderparameterinarigorousmanner. 
illuminatingitfromthephysicalandmathematicalviewpoints.InChapter4th. 
Ericksen-Leslietheory|thedynamictheoryofthenematicdirector|ispresente. 
andappliedtotheswitchingphenomena.InChapter5thedynamictheoryfor. 
vectororderparameterrepresentingageneralizationoftheEricksen-Leslietheor. 
isderived.ThestartingpointofthederivationisthegeneraltheoryofChapter2. 
Chapter6providesthebasicsondisclinationdefectsinliquidcrystals,neededt. 
understandtheforthcomingChapters.InChapter7thetensorialdynamictheoryi. 
presentedandappliedtothepair-annihilationofdisclinationlinesinnematics.I. 
Chapter8thevectorialtheoryofChapter5isusedtostudythepair-annihilatio. 
ofvorticesinamodelledSmCthinlmsystembelongingtotheclassoftheXY. 
model.Finally,Chapter9isdevotedtothe(in)stabilityanddecayofdisclination. 
withintegerwindingnumbersinnematics. 



1. 
Introductio. 



2 
Theory 
In this section, an outline of the dynamic theory of dissipative complex 
uids will 
be presented after an introductory discussion on the order parameter and the thermodynamic 
potential. At this stage the theory will be kept general, not depending 
on the specic choice of the order parameter. This will appear convenient later on 
as diverse systems with dierent order parameters will be studied. 
2.1 Order parameter and thermodynamic poten- 
tial 
Liquid crystals are ordered systems, the order of which emerges at a symmetry 
breaking phase transition. To describe the ordering, a mesoscopic quantity | the 
order parameter | is introduced, which must vanish in the high temperature phase 
and be nonzero in the ordered phase, depending on the ordering and re
ecting its 
symmetry properties. As the system may be spatially inhomogeneous in general, 
the order parameter is a eld quantity. It is dened as an average over the mesoscopic 
volume of the sample, which, ideally, is large enough to serve a well-dened 
average, and small enough compared with the inhomogeneities to contain essentially 
a homogeneous portion of the sample. In dynamic studies, one is interested in 
nonequilibrium properties of the system. Departures from equilibrium are described 
by the order parameter taking a dierent value than the equilibrium one. Thus, the 
nonequilibrium properties of the system are to be related to the dynamics of the 
order parameter when out of equilibrium. 
Thermodynamic systems to be considered will generally involve thermal, electric, 
and magnetic degrees of freedom. The total dierential of the internal energy, dU = 
dQ + dA, where 
TdS = dQ + TdSi; (2.1) 
is thus 
dU = T(dS .. dSi) .. p dV + V E 
 dP + 0V H
 dM+ dA0: (2.2) 
In Eq. (2.1) the entropy change has been split into reversible (dQ=T) and irreversible 
parts, dSi > 0. The last term in Eq. (2.2) represents the work of any contingent 
11

1. 
Theor. 


forcesnotaccountedforexplicitly.AtconstanttemperatureTandconstantelectri 
andmagneticeldstrengthsEandH,theproperthermodynamicpotential|th. 
freeenergyF|isobtainedbyaLegendretransformation. 


F=U..TS..VE
P..
0
VH
M. 
(2.3. 
dF=..SdT..TdS
i
..pdV..VP
dE..
0
VM
dH+dA
0
:(2.4. 


Changesinthevolumearenormallyneglectedinliquidcrystalsystems.Atconstan. 
T,E,andH,thefreeenergyisdecreasingwhenthesystemisapproachingequilib. 
rium,whereitreachesitsminimumvalue.Itistheorderparameterdependenceo. 
thefreeenergythatdescribesthisbehavior.Beforendingit,letusalerttoho. 
thefreeenergyofnonequilibriumstatescanbecalculated.Beingathermodynami 
variable,itdependsonthestateofthesystemandnotontheparticularpathleadin. 
toit.Hence,weareallowedtochooseadierent|reversible|pathrunningonl. 
throughequilibriumstates,forwhichdS
i
vanishesinEq.(2.4)enablingustoper. 
formtheintegration.ThefreeenergychangeisthusequaltotheminimalworkdA
. 
requiredtoputthesystemintothenalstateatconstanttemperatureandexterna. 
elds.Letusconsideranisothermalsystemofdipolesbelowthephasetransitiona. 
anexample.Clearly,theequilibriumstateishomogeneouswithalldipolespointin. 
inthesamedirectiononaverage.Nowthinkofaspatiallymodulatedconguratio. 
|anonequilibriumstatewithahigherfreeenergy.Todeterminethefreeenerg. 
increase,onecanimagineapplyingsomeforcesthatconveythesystemfromtheho. 
mogeneoustothemodulatedstateviaanequilibriumpath.Theincreaseofthefre. 
energyisequaltotheworkdA
. 
oftheseforces.Similarly,ifthesampleissubjectt. 
anexternaleldH,withacongurationcorrespondingtoequilibriumatadieren. 
eldH
0
,thefreeenergycostofthisnonequilibriumstateisgivenb. 


Z. 


. 



F=..
0
VdVM(H)
dH. 
(2.5. 


. 


wheretheintegrationisperformedovertheequilibriumpathM(H).Fortheelectri 
eldthesituationisanalogous. 


2.2Freeenergyfunctiona. 


InthisSection,thestandardLandauphenomenologicalapproach[8]willbetake. 
toderivetheequationofmotionforageneralorderparameter(theapplicationo. 
theLandautheorytothenematicphaseisduetodeGennes[9]).Lettheorde. 
parameterqbemulticomponent,withthecomponentsdenotedq
i
.Aslearnedi. 
thepreviousSection,thefreeenergydependenceontheorderparametermustb. 
obtainedinordertostudynonequilibriumdynamics.Itwillbeestablishedinth. 
formofanexpansionaroundtheequilibriumstate.Thefreeenergydensityf(q;rq. 
isintroducedasafunctionoftheorderparametereldanditsspatialderivatives. 
Aconsistentderivationofthisconceptonthethermodynamicbasisisgivenin[10. 
pp.143-153].Hence,thefreeenergyisafunctional. 


. 


F=dVf(q;rq). 
(2.6. 



Theor. 
1. 


Inthephenomenologicalspirit,thefreeenergydensityfunctionalisconstructe. 
fromscalarinvariantsformedwithq,rq,andanyexternalelds.Therearev. 
classiabletypesofcontributions. 


1.Homogeneousterms.ThesearethestandardLandautermsdescribingth. 
phasetransitionandconsistofscalarinvariantsofq,whicharethecondense. 
quantitiesexhibitingasoftmode,i.e.,theyspontaneouslybecomenonzeroa. 
thephasetransition.Schematically. 


11. 


f
hom

=

Aq
2
+

Bq
3
+

Cq
4
:(2.7. 


23. 


ThelineartermisabsentduetotherequirementthatFandsodoesfbe. 
minimumatequilibrium.Thequadratictermmodelsthetransition|atth. 
supercoolingtemperatureitchangessign,A=A
0
(T..T

).Thethirdorde. 


T
. 


termispresentonlyinthecaseq=6..q,otherwiseBiszerobysymmetry.I. 
isthistermthatgivesadiscontinuousphasetransition.Thefourthorderter. 
isnecessarytoprovidetheexistenceofaglobalminimum.TheconstantsA
0
. 
B,andCaretemperature-independent. 


2.Elasticterms.Theygivethefreeenergydensitycostofspatialinhomo. 
geneities.IntheoriginalspirittheyrelatetotheGoldstonedegreesoffree. 
dom[11],buthavebeengeneralizedtoapplytothecompleteorderparameter. 
Thistermsareinvarianttoinversionandalwaysgivepositivecontributions. 
Schematically. 


. 


f
elast

=

L(rq)
2
. 
(2.8. 


. 
whereLisatemperature-independentelasticconstant.Dependingonth. 
complexityoftheorderparameter,therecanexistmanyscalarinvariant. 
formedwithrandqandthusmanyelasticterms,eachwithitsownelasti 
constantL
i
.Themostgeneralexpression,second-orderinthederivative,i. 


. 


f
elast

=

.
L
ijkl
(@
i
q
j
)(@
k
q
l
). 
(2.9. 


. 


wherethefourth-ranktensorofelasticconstantsmustre
ectthesymmetryo. 
thesystemandthuscanbecomposedonlyoftheidentitymatrixA
ij
,theLevi. 
Civitaantisymmetricmatrix
ijk
,andtheorderparameterq
i
.Furthermore. 
thepermutationsymmetr. 


.
L
ijkl
=L
kli. 
(2.10. 


mustbeobeyedonthebasisofthedenition(2.9).Hence,thematrixL
ijk. 
canbediagonalizedinthesensethatL
ijkl
vanishesunlessi=kandj=l. 
i.e.,thequadraticform(2.9)canbewrittenasasumofsquaretermsonly. 
TheelasticcoeAcientsofthediagonalizedformmustbeallpositivetoyield. 
positivedenitefreeenergydensity.Inprinciple,onecanincludealsohighe. 
orderderivativesortermsofhigherorderinq. 




1. 
Theor. 


3.Chiral(Lifshitz)terms.Thesearepseudoscalarsofthefor. 


f
chir

=L
c

ijk
q
i
@
j
q
k
. 
(2.11. 


Astheychangesignuponinversiontheyareallowedonlyinchiralsystems. 
systemslackingtheinversionsymmetry. 


4.Externaleldterms.Theycoupletheorderparametertoexternalelds(elec. 
tric,magnetic,elasticdeformation,etc.).Inliquidcrystals,theaveragedensit. 
ofpermanentelectricormagneticmomentsisusuallyzero,whichimpliesth. 
couplingtobequadraticintheelds,schematically. 


1. 


f
em

=..

"
0
X
e
E
q
E..


0
X
m
H
q
H:(2.12. 


2. 



Theactualwayofcontractiondependsontheorderparameter.Theconstant. 
X
e
andX
m
aremicroscopicparametersrelatedtomolecularsusceptibilities. 
FormingscalarswithEandrqresultsintheso-called
exoelectrictermso. 
theformrq
E,[12,13].In
exoelectricphenomena,theinhomogeneityofth. 
orderparameterproducesapolarization,whichcouplestotheexternaleld. 
orviceversa. 


5.Surfaceterms.Someoftheelastictermscanbewrittenasadivergence,an. 
thereforeconvertedtoasurfaceintegralwhenperforming(2.6).Physically. 
theyappearonaccountofthereducedsymmetryduetothepresenceofth. 
surface,wherethereisanothervector|thesurfacenormal|thatcantak. 
partinthescalarcontraction. 


Anothertypeofsurfacetermsisdeliveredbytheanchoring,whichrepresent. 
acontributiontothefreeenergyofthesampleduetotheinteractionwit. 
thewallsaswellasduetotheconnement-reducedphasespace.Commonly. 
theanchoringismodelledbythesurfacefavoringacertainvalueoftheorde. 
parameterq
0
[14{16],schematically. 


1(q..q
0
)
2

f
anch

=W. 
(2.13. 
. 


Iftheorderparameterisxedatthesurfaceorifthereisnosurface,th. 
surfacetermscanbeignoredwhendeterminingtheconguration. 


Tondtheequilibriumconguration,thefreeenergyasafunctionaloftheorde. 


parameterhastobeminimized. 
 . 
Z
@f@f
Z
@. 
AF=dV..r

Aq+dS

Aq=0;(2.14. 
@. 
@r. 
@r. 


whichyieldstheEuler-Lagrangeequationsforthebulkandthesurface.Inth. 
following,thesurfacecontributionswillbeomittedforbrevity.Incasetheorde. 
parameterissubjecttoanyconstraints,Lagrangemultipliersareintroducedin. 
standardmanner,whichisequivalenttoprojectingEq.(2.14)intheorderparamete. 
spacetothesubspacenormaltotheonedenedbytheconstraints.Thelatternotio. 
isquiteconvenientandwillbeusedinthenumerics. 



Theor. 
1. 


2.3Dynamicequationfortheorderparamete. 


Outofequilibrium,thefreeenergydensityfailstosatisfy(2.14).Whenthesyste. 
isapproachingequilibrium,thefreeenergyisdecreasingonaccountoftheincreasin. 
entropy. 


 . 


Z

@f@. 


S_
i_

T=..F=..dV..r

.
_q;(2.15. 


@. 
@r. 


wherethedotstandsforthematerialtimederivativeandthesurfacecontribution. 
havebeenomitted. 


Ingeneralirreversibleprocesses,theentropyproductionisexpressedintermso. 

uxes
i
andforcesF
i
. 


. 


S_
i

T=dVF
i

i
. 
(2.16. 


Inthelimitofweak
uxestheforcesdependlinearlyonthe
uxes[17]. 


F
i
=K
ij

j
;K
ij
=K
ji
. 
(2.17. 


wherethematrixoftransportcoeAcientsissymmetricaccordingtoOnsager'sreci. 
procityprinciple[18],[19,p.365].CombiningEqs.(2.16)and(2.17),aquadrati 
formresultsfortheentropyproduction. 


Z. 


S_
i

T=dV2D=dVK
ij

i

j
. 
(2.18. 


wherewehavedenedthedissipationfunctionD[19,p.368].Theforcescanb. 
obtained,butneednot(Eq.(2.17)isjustasgood),directlyfromthedissipatio. 
functiona. 


@. 


F
i
=. 
(2.19. 
@
. 


ThesymmetricmatrixK
i. 
canbediagonalized,i.e.,suchlinearcombinationso. 
the
uxescanbefoundthatthedissipationisexpressedasasumofsquaresofth. 

uxes. 


AccordingtoEq.(2.15),wecanidentify.
_qasthe
ux.Thesimplest(thelowes. 
orderinq)choiceforthecoeAcientmatrixisK
i. 
=
A
ij
,where
isamateria. 
parameterwiththedimensionoftheviscosity.Now,comparingEqs.(2.15)an. 
(2.18)theequationofmotionfortheorderparameterisobtained[20]. 


@f@. 


r
..=
.
_q. 
(2.20. 


@rq@. 


Frequently,Eq.(2.20)isinterpretedasabalanceoftwogeneralizedforces,h+h
v
=0. 
wherehisthedrivingforce,tobedenotedbrie
. 


@f@fA. 


h=r
..... 
(2.21. 


@rq@. 
A. 


h
v

and=..
.
_qistheopposingor\viscous"force. 



1. 
Theor. 


2.4Couplingtothe
o. 


Ingeneral,theorderparameterdynamicsiscoupledtothe
uid
ow,whichmean. 
thateithercangeneratetheother.The
uid
owisgovernedbyageneralize. 
Navier-Stokesequatio. 


.
_v=r
. 
(2.22. 


whereisthestresstensortobedetermined.Besidesthepressure,ifthefre. 
energydensitycontainsgradientterms,thereisanelasticcontributiontothestres. 
tensor.ItarisesbecauseachangeinthedeformationofthesystemAu(whilekeepin. 
Aq=0)changesthegradientofq. 


@. 
@q@u
j
A=..A. 
(2.23. 
@x
. 
@x
. 
@x
. 


andthusthefreeenergydensity. 


@f@f@u
j

Af=
A@
i
q=..
@
j
qA:(2.24. 


@(@
i
q)@(@
i
q)@x
. 


ComparingEq.(2.24)withtheconstitutiverelatio. 


@u
. 
Af=
ij
A. 
(2.25. 
@x
. 


theelasticcontributiontothestresstensorfollows. 


@. 



. 


ij
=..
@
j
q. 
(2.26. 
@(@
i
q. 


LetusinspecttheequilibriumconditionAF=0,allowingalsoadeformationA. 
ofthesystem(omittingthesurfacetermseverywhere). 


". 


. 


A. 


0=AF=dV
ij
A(@
i
u
j
)+
Aq:(2.27. 


A. 



e

Putting
ij
=
ij
..pA
ij
,wherepisthepressure,andinsertingEq.(2.26),wege. 


(. 
#. 


Z

@f@. 
A. 


0=AF=dV@
. 

@
j
q+
@
j
@
i
q+@
j
pAu
j
+
Aq:(2.28. 


@(@
i
q)@(@
i
q. 
A. 


Expressingthersttermbytheequilibriumconditionfortheorderparameter. 
Af=Aq=0,theexpressioninthebracket|thebodyforce|simpliest. 


". 


ZZ

@f@. 


0=dV
@
j
q+
@
j
@
i
q+@
j
pAu
j
=dV@
j
(f+p)Au
j
;(2.29. 


@. 
@(@
i
q. 


i.e.,inequilibriumthepressureissuchthatf+pisconstant. 



Theor. 
1. 


Outofequilibrium,inpresenceofthe
uid
owthedensityofthekineticenerg. 
1
v
2
mustbeaddedtothefreeenergydensity;nowtheentropyproductioni. 


. 


Z. 


d. 


S_
i

T=..dV


v
2
+f(q;rq):(2.30. 
dt. 


InsertingEq.(2.22)anddroppingsurfaceterms,wege. 


". 


Z

AfA. 


S_
i

T=dV
ij
@
i
v
j
..
.
_q..@
i
v
. 
;(2.31. 


A. 
A(@
i
u
j
. 



. 
ij
pA
ij
.

productioni. 


wheretherstfactorofthelasttermisrecognizedas..Theentrop. 


". 


. 


A. 


S_
i
..
e

T=dV(
ij
+pA
ij
ij
)@
i
v
j
..
. 
_q;(2.32. 


A. 
recallthat.
_qisthematerialtimederivative:.
_q=@q=@t+(v
r)q.Thus,therear. 
two
uxesthatraisetheentropyinthe
ow-coupledsystem|thetimederivativ. 
oftheorderparameter.
_qandthevelocitygradientrv.Theforceconjugatedt. 
thelatteristhedierencebetweenthetotalstresstensorwithoutthepressur. 
contributionandtheelasticstresstensorandwillbecalledtheviscousstresstenso. 

v
=+pI..
e
.Conveniently,theentropysourcedensity.
_s
i
isexpressedbysplittin. 
thetensorsintosymmetricandantisymmetricparts. 


T.
_s
i
=
s
A
ij
+
a
W
ij
+h
i
.
_q
i
. 
(2.33)

ijij


s

a

whereandarethesymmetricandantisymmetricviscousstresstensorparts. 


whileAandWarethesymmetricandantisymmetricpartsofthevelocitygradient. 


.
A
1. 
ij
=


(@
i
v
j
+@
j
v
i
);W
ij
=
(@
i
v
j
..@
j
v
i
):(2.34. 
2. 
Theforcesrea. 



. 
i. 
=S
ijkl
A
kl
+M
ijkl
W
kl
+C
ijk
.
_q
k
;(2.35. 



. 
i. 
=M
klij
A
kl
+R
ijkl
W
kl
+D
ijk
.
_q
k
;(2.36. 


..h
v
h
. 
=C
kli
A
kl
+D
kli
W
kl
+B
ij
.
_q
j
;(2.37)

i

andtheentropysourcedensityi. 


T.
_s
. 
=S
ijkl
A
kl
A
ij
+M
ijkl
W
kl
A
ij
+C
ijk
.
_q
k
A
ij
. 


.
M
klij
A
kl
W
ij
+R
ijkl
W
kl
W
ij
+D
ijk
.
_q
k
W
ij
+(2.38. 
.
C
kli
A
kl
.
_q
i
+D
kli
W
kl
.
_q
i
+B
ij
.
_q
j
.
_q
i
. 


ThecoeAcientmatricesobeythefollowingpropertiesregardingthepermutationo. 
indices. 


.
S
ijkl
=S
klij
;S
ijkl
=S
jikl
. 
.
R
ijkl
=R
klij
;R
ijkl
=..R
jikl
. 
.
M
ijkl
=M
jikl
;M
ijkl
=..M
ijlk
. 
(2.39. 
.
C
ijk
=C
jik
;D
ijk
=..D
jik
. 
.
B
ij
=B
ji
. 



1. 
Theor. 


Recallthatthequadratic(2.38)canbediagonalized.Furthermore,theentrop. 
productionmustbeinvariantunderthesymmetryoperationsofthesystem.As. 
consequence,thecoeAcientmatricescanconsistonlyofquantitiescharacterizin. 
thesystem|theorderparameterq,theidentitymatrixA
ij
,andtheLevi-Civit. 
antisymmetricmatrix
ijk
.Eachtermcomeswithitsownmaterialparameter.On. 
canverifythattheentropyproductionisalsoinvarianttotimereversal.Finally. 
theremustbenoentropyproduction(2.38)andnoforces(2.35)-(2.37)forahomo. 
geneousrotationofthesample,whichimposesessentialrelationsonthemateria. 
parameters,reducingthenumberofindependentconstants. 


Tosummarize,inabriefformthedynamicequationsfortheorderparamete. 
coupledtotheincompressible
uid
owread. 


A. 


..

+h
. 
=0. 
(2.40. 


A. 
.
_v=..rp+r
(
v
+
e
);(2.41. 
r
v=0. 
(2.42. 



v
h
v

e

wheretheforcesandaregivenbyEqs.(2.35)-(2.37),isgivenbyEq.(2.26). 


andthefunctionalderivativein(2.40)hasbeendenedinEq.(2.21).Adetaile. 
derivationoftheequationsforavectorialorderparametercoupledtothehydrody. 
namic
owiscarriedoutinChapter5. 




3 
Nematic order parameter 
In this Chapter we will dene the order parameter of the nematic liquid crystal. 
Nematic substances consist of molecules with an elongated or a disc-like shape, which 
are eectively cylindrical objects due to their rotation. Let us assign a unit vector 
d = (sin 0 cos 0; sin 0 sin 0; cos 0) to the long axis of the liquid crystal molecule. In 
case of disc-like molecules d is the normal of the disc. The distribution of molecular 
orientations, i.e., the distribution of vectors d, dened in the mesoscopic volume, is 
naturally specied by an angular probability distribution function 
g(e) = g(; ); (3.1) 
where e is a unit vector e = (x; y; z) = (sin  cos ; sin  sin ; cos ). In nematics, we 
nd empirically that g(e) = g(..e), or 
g(; ) = g( .. ;  + ); (3.2) 
i.e., there is no polar ordering. The distribution function g carries to much information 
on the ordering | it cannot be determined experimentally, neither is it 
convenient for analytical work. Therefore, the lowest nontrivial moment of g is chosen 
to serve as the order parameter. The probability density g can be expanded in 
spherical harmonics [21, p. 338]: 
g(; ) =Xl 
g(l)(; ) =Xl;m 
gl;mY R l;m(; ); gl;m = Z d
Y R l;m(; )g(; ); (3.3) 
where real combinations of the spheric functions Yl;m have been used: 
Y R l;m =8> 
><> 
>: 
Yl;m ; m = 0 
1 
p2 
(Yl;m + (..1)mYl;..m) ; m >0 
1 
p2i 
(Yl;m .. (..1)mYl;..m) ; m <0 
: (3.4) 
Due to (3.2) gl;m is zero for l odd. Thus, the lowest nontrivial moment is the 
quadrupole, l = 2, and the set of 5 quantities q2;m represents the nematic order 
parameter. In the isotropic phase, where g = g(0) = 1=4, the order parameter 
is zero as required. In the ordered phase, the quadrupole moment is nonzero, and 
19

2. 
Nematicorderparamete. 


g
(2)

correspondstothedeviation(uptothequadrupolemoment)oftheprobabilit. 
distributiongfromtheisotropicdistribution.ItiscustomarytousetheCartesia. 
notatio. 


. 


. 


g
(2)

R

=g
2;m
Y
2;m
. 


m=... 


qq. 
i

3z
2
..1

. 
1. 
(x
2
..y
2

g
2;. 
+g
2;1
2zx+g
2;..1
2zy+g
2;2
)+g
2;..2
2xy. 


416

. 


.
Q
. 
ij
e
i
e
j
. 
(3.5. 
4. 


e
2
x
2
y
2
z
2

Recallingthat=++=1,wehavedenedthenematictensororde. 


parameterQ.Beingsymmetricbydenition,thequadraticformcanbediagonalize. 
forbriefness. 




. 


g
(2)

z
2
x
2
y
2

=Q
z. 
+Q
xx
+Q
y. 
. 
(3.6. 
4. 


wher. 


. 

0

.
Q
3hcos
2
i... 
z. 
=
4
g
2;0
. 
. 
(3.7. 


. 
. 


q. 

0

. 
13hcos
2
i..1

3. 
14. 




0

g
2;0
=


hsin
2
cos2
0
i. 
;(3.8)

.
Q
x. 
=g
2;2
..

. 
2. 


. 
2. 


q. 

0

. 
13hcos
2
i..1

3. 
14. 




0

.
Q
y. 
=..g
2;2
..g
2;0
=..

hsin
2
cos2
0
i. 
:(3.9. 


. 
2. 


. 
2. 


NotethatQistraceless.Brie
y. 


2. 


..
1

(S..P. 
. 


67

..
1

Q=
. 
(S+P)
5
;(3.10. 


. 


. 


whichcanbeexpressedalsoinageneralcoordinatesystema. 


.
Q
1. 
ij
=S(3n
i
n
j
..A
ij
)+
ii

P(e
1
e
1
..e
2
e
2
);(3.11. 
22
jj
wher. 
3hcos
2
i..1


0
S. 
(3.12. 
. 


isthescalarorderparameter,alsocalledthedegreeoforder,an. 


. 




0

P=

hsin
2
cos2
0
. 
(3.13. 
. 
isthebiaxiality.InEq.(3.11)wehaveintroducedanorthonormaltriad(n;e
1
;e
2
. 
specifyingtheQ-tensoreigensystem:nisthedirector,ande
1
isthesecondarydirec. 
tor.Usually,thedirectornrepresentstheeigenvectorwiththelargestinabsolut. 
eigenvalue,butnotnecessarily(asinChapter9).Inageneralcoordinatesystem,i. 
followsfromEq.(3.5)tha. 


5. 


g
(2)

(e)=Q
ij
e
i
e
j
;Q
ij
=


(3hd
i
d
j
i..A
ij
);(3.14. 
4. 



Nematic order parameter 21 
(a) uniaxial, P = 0 (b) biaxial, P 6= 0 
Figure 3.1 The quadrupole contribution to the probability distribution func- 
tion, g
(2), in the case of (a) uniaxial and (b) biaxial ordering. In Cartesian 
coordinates, g
(2) is given by the quadratic form g
(2)(e) = 5 
4
Qij eiej, where e 
is a unit vector. 
and 
g(2)(e) = 
5 
8 3h(d 
 e)2i .. 1: (3.15)

2. 
Nematicorderparamete. 



4 
Director dynamics in nematics 
The aim of this Chapter presenting the early stages of our research is to provide 
the machinery for the hydrodynamic description of liquid crystal dynamics and to 
get acquainted with the basic principles of 
ow generation and its in
uence on 
the nematic director. Moreover, we present numeric solutions to two-dimensional 
and quasi-three-dimensional switching processes, which are scarce in literature. We 
demonstrate that in conned systems the back
ow can lead to drastic eects far 
from mere perturbations, for which it is usually recognized. 
4.1 Introduction 
Problems involving hydrodynamic motion of the nematic liquid crystal due to the 
director reorientation have been studied mainly in terms of the Ericksen-Leslie continuum 
theory of the nematic liquid crystal [22{24]. For one-dimensional geometry, 
Clark and Leslie [25] have given a thorough approximative analysis of nematic relaxation 
upon removal of electric or magnetic eld; a complete numerical treatment 
of the problem has been contributed by van Doorn [26]. One-dimensional back- 

ow dynamics in the twist cell has been studied by Berreman [27,28]. Recently, an 
optical observation of the back
ow in the twist cell was reported [29]. Pieranski, 
Brochard and Guyon [30,31] have studied, both theoretically and experimentally, 
one-dimensional dynamic behavior in magnetic eld for three geometries (twisted, 
planar to homeotropic, homeotropic to planar), limited to small deformations (applying 
near-critical elds). They give the distortion wave vector and eective viscosity 
dependence on the magnetic eld strength. The instability against periodic 
distortion in the case of the Freedericksz transition (rst observed by Carr [32]) has 
been studied by Guyon et al. [33] for the two-dimensional case, and by Hurd et al. 
[34] for three dimensions. The pattern formation in a rotating magnetic eld has 
been observed experimentally and accounted for by a numerical study based on the 
Ericksen-Leslie equations [35,36]. An experiment measuring the rotational viscosity 
is presented by Bajc et al. [37], together with a full hydrodynamic numerical treatment 
in cylindrical geometry (one-dimension), yielding an exact expression for the 
eective viscosity, depending on the director eld conguration. Lately, the interest 
in hydrodynamic description of pattern formation in 
uids has been increasing, 
23

2. 
Directordynamicsinnematic. 


amongstothersinvolvingalsonematicandnematicpolymer
uids[38{44]. 


InthisChapterafulltwo-dimensionalhydrodynamicstudyofanematicsam. 
pleinmagneticeldispresented,producingnontrivialback
oweldseveninth. 
simplestgeometrieslikeasquareorarectangle.Firstashortreviewofnemato. 
dynamicequationsisgiven,followedbyanintroductionofcharacteristicscaleso. 
theproblem.Inthesecondpartthe
oweldsaretentativelyinterpretedbystric. 
analyticaswellaslessstrictarguments.Also,thein
uenceoftheback
owonth. 
directorreorientationisdiscussed.TheideapursuedthroughouttheChapterist. 
giveenoughqualitativephysicalunderstandingoftheback
owgenerationandit. 
eectonthedirectoreldtobeabletoexplainorevenforeseetheglobaltimepat. 
ofrelaxationprocesses[45].Therelaxationwiththeback
owisthencomparedt. 
thesimpliedcasewhereback
owisnottakenintoaccount.Theissuesinques. 
tionherearethechangeintheswitchingtimeofthecellcausedbytheback
ow. 
andthelocaldepartureofdirectororientationfromtheorientationinthesimpl. 
case,pursuedalongthewholepathofrelaxation.Thedrasticback
oweectasth. 
consequenceofaspecialmagneticeldswitchingisdemonstratedinthe2Dan. 
quasi-3Dexamples.The3Dgeometryisclosertoapossibleexperiment. 


4.2Ericksen-Leslietheor. 


HerewearegoingtopresenttheEricksen-Leslietheory[22,23],whichisthedynami 
theoryforthenematicdirectorcoupledtohydrodynamic
ow.Itcanbederive. 
followingthetheoreticalbasissetinChapter2.Atthispoint,wewillskipth. 
derivationandinvitethereadertovisitChapter5,wheretheEricksen-Leslietheor. 
inageneralizedformisderivedthoroughly. 


Threebasicequationsareinvolvedintheproblemofnematodynamics;thesear. 
theequationofmotionofthedirectoreld(2.40),thegeneralizedNavier-Stoke. 
equation(2.41),andtheequationofcontinuity.Thelatterissimplyreducedtoth. 
equationofincompressibility(2.42),whereastheformertwoarerelativelyextensiv. 
duetothe(uniaxial)anisotropyofthenematic
uidaswellastothecouplin. 
betweenthedirectorreorientationand
ow. 


Thetimeevolutionequationforthedirectoreldisabalancebetweengeneral. 
izedelastic,electric,magnetic,andviscousforces.Inprinciple,bothelectrican. 
magneticeldscanbeusedtomanipulatethenematicdirector.However,theus. 
oftheelectriceld,thoughmoreeAcientduetolargersusceptibilityanisotropies. 
bringsaboutsomediAcultiestodealwith,i.e.,thedielectricproblemhastob. 
solvedexactly,andtheconvectionofionsshouldbetakenintoaccount.Asaresul. 
ofthis,thetheoreticalstudytobepresentedinthisChapterusesamagneticeld. 


Toobtaintheelasticandmagneticpart,theFrankelasticfreeenergydensit. 
[46{48],[49,pp.102,119]isused. 


1. 
2
. 
2
. 


f=

.
K
11
(r
n)
2
+

.
K
22
[n
(rn)]+

.
K
33
[n(rn)]..


a

0
(n
H)
2
;(4.1. 


222. 



wherenisaunitvectorrepresentingthedirector,K
11
,K
22
,andK
33
arethesplay. 
twist,andbendelasticconstants,respectively,Histhemagneticeld,and
a
isth. 



Directordynamicsinnematic. 
2. 


magneticsusceptibilityanisotropy,i.e.,thedierencebetweenthesusceptibilitie. 
parallelandperpendiculartothedirector.Thecaseof
a
>0willbeconsidere. 
here,asthisisthesituationpresentinmostnematicsubstances.Intheoneelasti 
constantapproximation,theelasticpartofEq.(4.1)reducest. 


. 


f
one

=


K(rn)
2
. 
(4.2. 
. 


ThesurfacetermshavebeendroppedinEqs.(4.1)and(4.2)onaccountofxe. 
boundaryconditions,correspondingtoinnitelystronganchoring. 
TheEuler-Lagrangeequationsforthefreeenergyfunctiona. 


. 


. 


F=dVf(n;rn)..(r)n
. 
(4.3. 


n
2

withtheconstraint=1andfgivenby(4.1),givethegeneralizedelastican. 


magneticforce. 


 . 


@f@. 


h
e. 
. 
=..+@
. 
. 
(4.4. 


@n
. 
@(@
j
n
i
. 


Theequilibriumconditionread. 


h
em

=..(r)n. 
(4.5. 


h
em

whereistheLagrangemultiplier,i.e.,theforcemustbeparalleltonevery. 
where.Onegetsridoftheredundantdirectordegreeoffreedomandthemultiplie. 
byprojectingEq.(4.5)ontotheplaneperpendiculartothedirector. 


Thegeneralizedviscousforceisobtainedfromthedissipationfunction,Eq.(2.19. 
containingscalarinvariantsformedwithn,.
_n,andrv,beingbilinearinthelatte. 
two[50,p.142]. 


..h
v
=

1
N+

2
A
n. 
(4.6. 


wheretherotationalviscosity

1
and

2
areexpressedintermsoftheLeslieviscosit. 
coeAcients
i
[49,p.206],

1
=
3
..
2
,

2
=
3
+
2
.With.
_nbeingthemateria. 
timederivativeofthedirector. 


. 
N=.
_n..(rv)n=.
_n+W
. 
(4.7. 


. 


isthevectoroftherelativedirectorrotationwithrespecttotherotationofthe
uid. 
Thesymmetricandantisymmetricpartsofthevelocitygradient,A
ij
andW
ij
,hav. 
beendenedinEq.(2.34).Theviscousforcealsoneedstobeprojectedtothe

h
v

planeperpendiculartothedirector.Theequationofmotionofthedirectorread. 
brie
. 


. 
h
em
+h
. 
=0. 
(4.8. 
?. 
orinmoredetai. 


. 
@. 
@. 
. 
. 
h
em
..

2
A
n..

1
[W
n+(v
r)n. 
. 
?. 
:(4.9. 



2. 
Directordynamicsinnematic. 


ThegeneralizedNavier-Stokesequation. 


". 


@. 


+(v
r)v=..rp+r
(
v
+
e
);(4.10. 


@. 


whereisthedensity,pisthepressure,andthedivergenceofatensordeneda. 
(r
)
i
=@
j

ji
,involvestwostresstensorcontributions.Theviscouspartisobtaine. 
fromthesamedissipationfunctionasthegeneralizedforce(4.6)[50,p.142]. 



. 


=
1
n
n(n
A
n)+
2
n
N+
3
N
n. 

4
A+
5
n
(A
n)+
6
(A
n)
n;(4.11. 


withtheLesliecoeAcientsobeyingtheParodirelation[51. 



6
..
5
=
3
+
. 
(4.12. 


duetotheOnsager'sreciprocityprinciple.Forfuturepurposes,letussplitth. 
viscousstresstensorintothesymmetricandantisymmetricparts. 



. 


=
1
n
n(n
A
n)+
4
A. 


1. 


2
(N
n+n
N)+


(
5
+
6
)[(A
n)
n+n
(A
n)]. 
2. 
1. 



. 


=



1
(N
n..n
N)+



2
[(A
n)
n..n
(A
n)]:(4.13. 


2. 



Theelasticpartofthestresstensor(Eq.(2.26))isaconsequenceofdeformation. 
changingthedirectoreldgradients,Eq.(2.26),[49,p.152]. 


@. 



. 


ij
=..@
j
n
k
. 
(4.14. 
@(@
i
n
k
. 


ThepressureeldinEq.(4.10)issetbytheincompressibilitycondition(2.42),i.e.. 
ithastobedeterminedinsuchawaythatEq.(2.42)issatised. 


4.3Characteristicscale. 


TheproblemstobeaddressedinthisChapterinvolvetwolengthscales:thecon. 
tainersizeorthethicknessofthecapillaryLandthemagneticcoherencelength[49. 
p.123. 


. 


1K
1. 

m
=. 
(4.15. 


H
0
j
a
. 


Inorderthateldeectsbeprominent,
m
mustbesmallcomparedwithL.There. 
fore,
m
isthelengthrelevantforthedynamicsofthesystem. 
Thedirectorequationofmotion(4.9)andthegeneralizedNavier-Stokesequatio. 


(4.10)introduceacharacteristictimescaleeach.Typicalrelaxationtimeofth. 
directoreldi. 
. 


1

. 
m
. 
(4.16. 
K
1. 



Directordynamicsinnematic. 
2. 


if
m
isthecharacteristiclengthofthedirectorvariation.Sincethedynamicsi. 
governedbythedirectoreldrelaxation,isthecharacteristictimeoftheswitchin. 
process. 


Theothertimescaleisgivenbyatypicaltransitiontimeduringwhichth. 
velocityeldisequilibratedtoitsstationaryvalueduetoviscousforces. 


L
. 



0
=. 
(4.17. 


.

. 


TheisotropicviscositycoeAcient
4
(seeEq.(4.11))isofthesameorderofmagni. 
tudeastherotationalviscosity

1
,soitisconvenienttousethelatterintheestimat. 
(4.18).Typically,theratioofthetwotimescales|theunsteadinessparameter. 
isoftheordero. 



L
2

..
K
1. 


L
2


0
=. 
m

=
2

10
..6
. 
(4.18. 


.

. 


. 


. 
. 


ThismeansthatunlessthecontainersizeLismuchlargerthanthecoherencelength. 
thevelocityeldisadaptedquicklytoagivendirectoreldanditstimederivative. 
sothatduringthereorientationprocessitbehavesquasi-stationary|thepartia. 
timederivativein(4.10)canbedropped. 


Thecharacteristicmagnitudeofthevelocitycanbeestimatedbyequatingth. 
viscousforce(the
4
termin(4.11))andtheviscousforceexertedbythedirecto. 
rotation,whichdrivesthe
ow(the
2
and
3
termsin(4.11)),yieldin. 


.
K
11

v
0
==
m
=. 
(4.19. 




1

. 


Asindicatedin(4.19),thesameestimatecanbeobtainedinasimplerfashion. 
althoughitmightnotseemaslucid.Againitwasassumedthat
4

2


1
. 
ThecharacteristiclengthofthevelocityvariationisestimatedtoL.Tobemor. 
precise,bothlengthscales,Land
m
,areintertwinedhere,butLisusedinorde. 
tooverestimatetheReynoldsnumber. 



LK
11

Re=

L=
m

10
..6
. 
(4.20. 




2

.

. 


. 


UnlessthemagneticcoherencelengthistinyincomparisonwithL,theReynold. 
numberismuchsmallerthanunity,andthenonlinearadvectivederivativetermi. 
(4.10)canbedropped.Inaddition,iftheratio(4.18)issmallaswell,alsoth. 
partialtimederivativein(4.10)canbeomitted,asmentionedabove.Thereade. 
shouldnotethatusuallytheReynoldsnumberismorethananorderofmagnitud. 
smallerthantheunsteadinessparameter.Thenalequationtosolveisthu. 


0=..rp+r
(
v
+
e
). 
(4.21. 


MaterialparameterssuchastheviscosityandtheelasticcoeAcientswillcorre. 
spondtothoseforMBBA,listedin[49,pp.105,231].Itisconvenienttogivesom. 
typicalmagnitudes.Amagneticeldwithstrength0.1Tgivesacoherencelengtho. 
about10mandthecharacteristictime(4.16)ofaround2s.Extrapolationlength. 
[49,p.113]assmallas100nmorevensmallerarereadilyobserved,sothatth. 
stronganchoringlimitisrealistic. 



2. 
Directordynamicsinnematic. 


4.3.1Commentonheatdiusio. 


ThroughouttheThesisweassumethatthetemperatureisconstant.Oneshoul. 
notproceedfurtherwithouthavingjustiedthislimit.Inparticular,thedynamic. 
ofdefectsstudiedinChapters7and8mightbeaectedbytemperaturegradients. 
sincethemotionofdefectcores|regionsofhighfreeenergydensity|isgenerall. 
accompaniedbyatransportofheat.Theheatdiusionequatio. 


@T. 


..

r
(
rT)=. 
(4.22. 



@tc
. 


introducesyetanothertimescale. 


c
p
l
. 

Q
=. 
(4.23. 


. 


wherelisthecharacteristiclengthofthetemperaturevariation,c
p
isthespecichea. 
capacity,andisatypicalcomponentoftheheatconductivitytensor.Comparin. 

Q
andthecharacteristicdynamictimeoftheorderparameter(4.16),bothatth. 
samelengthl,oneget. 


Kc
p


Q
==. 
(4.24. 


1
. 


whichisoftheorderof5
10
..4
fortypicalmaterialparametersofliquidcrystals. 
Theratio(4.24)isthesamealsoifdefectsarepresent(Eq.(7.19)).Thismeanstha. 
theheatdiusionisfastcomparedtotheorderparameterdiusion,sothatther. 
cannotexistanysubstantialinhomogeneitiesintemperatureinducedbytheorde. 
parameterdynamics.Theconstanttemperaturelimitisrealisticinfact. 


4.4Descriptionoftheproblemandnumericalim. 


plementatio. 


Therelaxationofaconnednematicsampleuponswitchingamagneticeldwil. 
bestudied.Acontainerofsquareorrectangularcrosssection(thexyplane)i. 
adopted,extendingtoinnityinthezdirection.Innitelystronganchoringi. 
assumed,whichxesthedirectoreldattheboundaries.Inpractice,thismean. 
thattheextrapolationlength[49,p.113],,mustbemuchsmallercomparedbot. 
withthesamplesizeandthemagneticcoherencelength,i.e.,Land
m
. 
Standardno-slipboundaryconditionsareprescribedforthe
ow,settingthevelocit. 
tozeroattheboundaries. 


Thepartialdierentialequations(4.9)and(4.10)arecastindimensionlessfor. 
usingcharacteristicscalesintroducedabove.Theyaresolvedusingnitedierenc. 
discretisation.Theoutlineofthemethodisasfollows.Atagivendirectorel. 
anditstimederivative,thegeneralizedNavier-Stokesequation(4.10)withoutth. 
advectivederivativetermisexplicitlyiteratedintime.Afterthat,knowingth. 
velocityeld,thedirectorequation(4.9)isexplicitlyiteratedintimetoyieldth. 
newdirectoreld.Thenthevelocityisupdatedagain,andsoforth.Accordingt. 



Directordynamicsinnematic. 
2. 


thebigdierenceincharacteristictimescales(4.17)and(4.16),onemakesman. 
iterationsofEq.(4.10)beforeupdatingthedirectoreld.Foragenericseto. 
materialparameterstheunsteadinessparameter(4.18)issmallenough,soonecoul. 
dropthetimederivativetermin(4.10)intherstplace.However,withtheexplici. 
iterativenumericalschemedescribedabove,thishaslittlesense.Ontheotherhand. 
Eq.(4.10)withoutthenonlineartermandEq.(2.42)togetherresultinalargese. 
oflinearequationsforthediscretizedvelocityandpressurevariables,whichca. 
besolveddirectlyforthestationaryvelocityeld.Inpracticeitturnedouttha. 
theiterativemethodisfarmoreeAcient.Thevelocityandpressurevariablesar. 
discretizedonastaggeredgrid[52,p.331]inordertopreventtheoccurrenceo. 
thewell-knownoscillatorypressuresolution[53].Theincompressibilityconditioni. 
satisedinastandardwaybysolvingaPoissonequationforpressurecorrectionsa. 
everyvelocityiterationstep[52,p.340]usingtheSORmethod[54,p.655].Atth. 
boundaries,normalpressurecorrectionderivativesarespeciedinordertomeetth. 
incompressibilityconditionthere.Thecalculationsweretypicallydoneonasquar. 
meshofsize60x60. 


4.52Dproble. 


Therstproblemconsideredisfullytwo-dimensional(Fig.4.1):thequantitiesin. 
volveddependonxandy,whilethedirectorandthe
owvelocityarelyinginthex. 
plane.Theorientingmagneticeldpointsalongtheyaxis.Toavoidanyfrustration. 
thealignmentdictatedbythestronganchoringisparallelforhorizontalsides,whil. 
forverticalonesitishomeotropic.Whereconvenient,theangleparametrizationo. 
thedirectorandtheangle-conjugatedgeneralizedforce(thetorque)hwillused. 


n=(cos';sin';0). 
(4.25. 
h=(0;0;h)n. 
(4.26. 


ThroughouttheSection4.5,thefollowingdimensionlessquantitiesforlength,time. 
velocity,andmagneticeldwillbeused. 


r r=L;t t=
L
;v v
L
=L;H H=H
0
;(4.27. 


wher. 




1
L
. 
L
. 



L
==. 
(4.28. 


.

2

.
K
1. 
. 


isthecharacteristicrelaxationtimeforthedirectorelddeformationonthescal. 


ofthecontainersizeLan. 


. 


1K
11
H
0
. 
(4.29. 


L
0
j
a
. 


isthemagneticeldwiththecoherencelengthofL. 



3. 
Directordynamicsinnematic. 


(a) 
B Field ON 

Field OFF 


(b) 
y 


x 

Figure4.1Thecalculationsareperformedinasquareorrectangulargeome. 
try.Twotypesofrelaxationarestudied:(a)startingwithauniformdirecto. 
eld,themagneticeldisswitchedon,or(b)themagneticeldisswitche. 
otodisorientaeld-alignedsample.Thedirectorisxedattheboundarie. 
asshown. 



Director dynamics in nematics 31 
4.5.1 Mechanisms governing the problem 
Back
ow generation 
As indicated by calculations, the elastic stress tensor contribution (4.14) alters the 
velocity eld by up to 10%. Thus, while it is of some importance when studying 
the 
ow elds, in rst approximation it can be neglected when the in
uence of the 
back
ow on the director eld is in question, this in
uence itself also being small. 
Our interpretation of the velocity elds will be based solely on the viscous coupling 
given by Eq. (4.11). What is more, it turns out that the anisotropy of the 

uid viscosity, described by the 1, 5, and 6 terms in (4.11) has no qualitative 
importance. The driving force of all the interesting 
ow phenomena observed is the 
anisotropy of the coupling to the director rotation given by the 2 and 3 terms in 
(4.11). Since 2=3  70, only the 2 term needs to be taken into account when 
trying to interpret the results. There are two contributions to the force exerted on 
the 
uid described by this term, one depending on the gradient of director rotation, 
and the other on the director eld gradient. Putting '_ = ! and ' = 0 one obtains 
f1 = 2  0 ; 
@! 
@x! (4.30) 
for the rotation gradient dependent force. Generally, this force is always perpendicular 
to the director, while its magnitude depends on the ! derivative along the 
director, n 
 r!. The second contribution is best seen if we put r' = ('x ; 0): 
f2 = ..2!'x (cos 2' ; sin 2') : (4.31) 
Thus, the magnitude of this force depends only on !jr'j, whereas its direction is 
such that it makes twice the angle with r' as the director. The reader should bear 
in mind that 2 is negative and that the direction of the force just described depends 
on the sign of !. 
In
uence of back
ow on the director rotation 
Let us discuss the torque on the director exerted by the 
ow. A 
ow eld corresponding 
to a pure rotation (W 6= 0, A = 0) imposes the same rotation on the 
director, as Eq. (4.9) suggests putting hem to zero. Conversely, pure extensional 

ow (W = 0, A 6= 0) aligns the director along the axis of extension. For shear 
ow, 
which is a sum of the 
ows just mentioned, Eq. (4.9) gives 
'_ = ..
1
2
  
2 

1 
cos 2' + 1!; (4.32) 
where ' measures the angle relative to the velocity direction and  is the shear rate 
(see Fig. 4.2 for the sign convention). Equation (4.32) has a stationary solution only 
if j
2=
1j > 1, i.e., if 3 < 0, which is the condition for the 
ow-aligning nematic, as 
opposed to the 
ow-tumbling nematic, where 3 > 0 (see the end of Section 4.5.2 for

3. 
Directordynamicsinnematic. 



Figure4.2Thedirectorangle
'
inEq.(4.32),ismeasuredrelativetoth. 


shearasshown. 


ashortdiscussionontheback
oweectin
ow-tumblingnematics).Thestationar. 
solutionof(4.32)give. 


j'
0
j1. 
(4.33. 


since

2
=

1
..1.Thedirectoristhusrotatedtowardsthevelocitydirection.Th. 
solutionwith'
0
>0isstable,whereasthatwith'
0
<0isnot.ForMBBAth. 
alignmentangleisapproximately'
0
7A.Notethatwhenoutofequilibrium,th. 
directorisrotatedanticlockwiseonlyforj'j<j'
0
j,whereasforanyotherorientatio. 
thestationarystateisapproachedbyaclockwiserotation(forthesituationa. 
depictedinFig.4.2). 


4.5.2Resultsandinterpretatio. 


Field-orelaxatio. 


Inthisexamplethesampleisinitiallyalignedwithamagneticeld.Theeldisthe. 
switchedoinstantaneously,andthesystemrelaxesbacktotheundeformedcong. 
uration.Inthesquarecell,theearlystagesofthisrelaxationshowananticlockwis. 
centralvortex,accompaniedbytwoclockwiseeddiesatthebottomandtop,wherea. 
therearenoeddiesneartheverticalboundaries(Fig.4.3(a)).Equations(4.30)an. 
(4.31)canexplainthisasymmetry,forbothforcecontributionsdependontherel. 
ativeorientationofthedirectorwithrespecttor!andr',respectively,whic. 
isdierentforthehorizontalasitisfortheverticalboundaries.Thesizeofth. 
eddiesisoftheorderofthemagneticcoherencelength.Theircentersarelocate. 
neartheregioninwhichdirectoreldcurvatureisamaximum('==4).Inth. 
courseoftime,themaximumcurvatureregionmovestothemiddleofthecell,an. 
sodotheclockwiseeddies(Fig.4.3(b)).Finally,thecentralanticlockwisecurren. 
iscompletelyeliminated.Figure4.4showsthetotaldeformationofthe
uidduet. 
theback
owaftertherelaxationprocesshasstopped. 


Letusnowtakeaqualitativelookattheforcesneartheboundaries,considerin. 
regionsinthemiddleoftheedgeswherelateralderivativescanbeneglectedto. 
rstapproximation.Onlythecomponentsparalleltotheboundariesareconsidered. 
sincethismustbethedirectionofthevelocitythere.Onehastorealizethatth. 
initialrelaxationratej!jhasamaximumat'==4,sincethemagneticforcei. 
thestrongestthere,implyingamaximumelasticforcetobebalancedwith.Not. 
that!isnegative.Foroutermostpartsofthesample,where'<=4,j!jincrease. 



Director dynamics in nematics 33 
-190 
-161 
-133 
-104 
-75 
-46 
-18 
0
20 
40 
-33 
-28 
-24 
-20 
-16 
-12 
-8 
-4 
0 
0 1 
0
1 
t = 0.02 
-70 
-60 
-50 
-40 
-30 
-20 
-10 
0
10 
0 1 
0
1 
t = 0.001 
0 1 
0
1 
t = 0.006 
(c) vmax =1.17 
(b) vmax =1.49 
(a) vmax =7.34 
Figure 4.3 Field o in the square cell: subsequent snapshots of the velocity 
elds (left column), director elds and director angular velocity elds (rep- 
resented by levels of gray) show a typical two-step relaxation. (a) In the 
beginning, three vortices are present, while the director in the middle rotates 
in the opposite sense; the line of the stationary director eld is indicated by 
the dashed contour. (b) Clockwise vortices are becoming dominant, note the 
reverse director deformation at the center. (c) The clockwise vortices have 
joined to form a single vortex, the sense of rotation now being opposite to 
that at the beginning. The interior starts rotating in the right sense at a high 
rate (compare with the rotation in (a)).

3. 
Directordynamicsinnematic. 



Figure4.4Thetotaldeformationofthe
uidaftertheeld-orelaxationi. 


thesquarecell.Intheeld-oncase,thedeformationissimilar(thoughno. 


thesame)butopposite. 


onmovingawayfromtheboundary.Hence,Eq.(4.30)givesforcestryingtostart. 
clockwisecurrentroundthecell.However,theforcesdescribedbyEq.(4.31)oppos. 
therstonesfortheverticalboundaries,whereasthoseforthehorizontalboundarie. 
addconstructively.Thisisoneofthereasonsforthemissingeddiesnearthevertica. 
boundaries.Movingfurtherawayfromtheboundaries,'becomesessentially=. 
andtherearenoforcesparalleltotheboundariesgivenbyEq.(4.31).Ontheothe. 
hand,Eq.(4.30)doesyieldforcesforthehorizontalboundaries(andnoneforth. 
verticalones),startingananticlockwisecurrent. 


Ofcourse,thesituationistoocomplicatedforthesquarecelltobeinterprete. 
completelyduetothefactthatthecornersplayanimportantrole,theproble. 
therebeingfullytwo-dimensional.Thereforeweaimtomakeamoreexactanalysi. 
forone-dimensionalcases,obtainedbyextendingeitherthehorizontalorthevertica. 
boundariestoinnity.InthiswaywearriveatthesystemsstudiedbyPieranski. 
BrochardandGuyon[30,31]inthelimitofweakdeformations.Figure4.5(an. 
Figure4.10innextsection)representingacellwitha5:1sideratio,shouldservea. 
anillustrationfortheone-dimensionalcases. 


CalculationsforL
x
L
y
clearlygivethreevortices(Fig.4.5(a)),whereasthos. 
forL
y
L
x
resultinasinglevortexonly(L
x
andL
y
aredimensionsofthecellin. 
andydirection,respectively).BeingabletoexplainthisonthebasisofEqs.(4.30. 
and(4.31)wouldyieldevidenceforthenontrivial
oweldsbeingaconsequenceo. 


the
2
termin(4.11).Intheoneelasticconstantapproximation(4.2). 
h
em
=r
2
'. 
. 
H
2
sin2'. 
(4.34. 
. 
theinitialdirectorcongurationforlargeenougheldsi. 
'=2arcta. 
. 
e
. 
. 
. 
. 
. 
. 
(4.35. 
. 


whereHisthemagneticeldusedtoalignthesample(alwayslyinginthevertica. 
direction),andristhedistancefromtheboundary.Thissolutionisobtaine. 



Director dynamics in nematics 35 
-160 
-133 
-105 
-78 
-50 
-23 
0
33 
60 
0 5 
0
1 
t = 0.001 
-65 
-57 
-49 
-41 
-33 
-24 
-16 
-8 
0 
0 5 
0
1 
t = 0.006 
(c) vmax =2.36 
(b) vmax =1.99 
(a) vmax =11.1 
-33 
-28 
-24 
-20 
-16 
-12 
-8 
-4 
0 
0 5 
0
1 
t = 0.02 
Figure 4.5 Field o in a horizontal cell with Lx=Ly = 5 (note the scale): 
two-step relaxation. See Fig. 4.3 for the key to the gures. (a) In the beginning 
three vortices are formed (Fig. 4.7), the director eld undergoes a reverse 
rotation in the middle (note the zero rotation contour, shown dashed). (b) At 
the moment of 
ow reversal, note the deformation of the central director eld. 
(c) A clockwise current results, while the director in the center is rotating in 
the right sense at maximum rate (compare with the rotation in (a)).

3. 
Directordynamicsinnematic. 


f 


0 0.1 0.2 0.3 0.4 0.5
r 


Figure4.6Totalinitialforces(sumof(4.30),(4.31))forthehorizontalan. 


vertical(dashed)cellasfunctionsofthedistancefromtheboundary(
r
=0
:
. 


inthecenter). 


byusingtheboundarycondition'
. 
=0for'==2,wheretheprimedenote. 
dierentiationwithrespecttor.Thecorrectconditionrequires=0forr=1=2

'
. 


(centerofthecell),which,however,introducesellipticintegrals.Thus,solutio. 
(4.35)isvalidiftheeldislargeenoughsuchthatforr=1=2thedirectoriswel. 
aligned('=2). 


'
0. 
'
000
,!
0

Putting!=and=with'givenbythesolution(4.35),oneobtain. 
theforces(4.30)and(4.31).Thecomponentsparalleltotheboundaryareshowni. 
Fig.4.6.Theybehaveasexpectedfromtheprecedingdiscussion. 


Nowletuscalculateapproximateinitialvelocityprolesforboththehorizonta. 
andtheverticalone-dimensionalcell.Inonedimensionthepressuretermandth. 
elasticstressterminEq.(4.21)canbedroppedsincetheycanonlyyieldtransversa. 
forces.Theremainingequatio. 



v

r
=. 
(4.36. 


issolvedeasilywhenretainingonlythe
2
and
4
termsin
v
(Eq.(4.11)).Togethe. 
withEq.(4.35)thesolutionread. 


"(. 
. 




.

. 
..3=2. 


v(r)=..H
2
cos'


cos3'+Clncos
..1
'+tan'+D. 
.

. 
..1=2. 


withtheconstant. 


"(. 
(). 


13=214=. 


C. 
cos'
0



cos3'
0
... 
ln(cos
..1
'
0
+tan'
0
)1=262=. 


(. 


4=. 


D=. 


2=. 



Directordynamicsinnematic. 
3. 


v 


2 
0 r 
-2 
-4 
-6 
0 0.1 0.2 0.3 0.4 0.5 

Figure4.7Initialvelocityprole(Eq.(4.36))forthehorizontalandthever. 


tical(dashed)cell(
H
=20).Inthehorizontalcase,twoopposingcurrentsar. 


predicted.Thewidthofthecounter-currentneartheboundaryisoftheorde. 


ofthemagneticcoherencelength.Ontheotherhand,theverticalcellgive. 


asinglecurrentonly,althoughthenetforcechangessignneartheboundar. 


(Fig.4.6).Bothanalyticalpredictionsbasedonthesimpliedviscousstres. 


tensorareconsistentwithnumericalresults. 


where'
0
representsthedirectorangleinthemiddleofthecell.Theupperexpres. 
sionsstandforthehorizontalcell,theloweronesfortheverticalcell.Thesolution. 
areplottedinFig.4.7,wherethedierencebetweenthehorizontalandverticalcase. 
isclearlyseen.Itisworthpointingoutagainthat,qualitatively,itisalsopossiblet. 
usetheseresultsfortheregionsclosetothemiddleoftheboundariesofthesquar. 
cell,wherethesituationresemblestheone-dimensionalcase(compareFigs.4.3(a. 
and4.5(a)). 


Sofar,onlythegenerationoftheback
owhasbeenstudied.Wenowconside. 
theinversephenomenonofhowtheback
owin
uencesthedirectorrotation.Inbot. 
one-dimensionalcasesweareconfrontedwithpureshear
ow,theeectsofwhichar. 
describedbyEq.(4.32).Thesituationisnowcompletelydierentforthehorizonta. 
andverticalcases.Inthehorizontalcase,theshearingback
owslowsdownth. 
directorrotationinthemiddleofthecell(Fig.4.5(a)),andismoreoveractuall. 
abletoreversethedirectionofrotationduetoweakelasticforcesthere(Figs.4.5(b. 
and4.8).Thisphenomenonisknownasthekickbackeect[55,p.167].Nearth. 
boundaries,however,theshearisopposite,thusacceleratingtherelaxation.As. 
result,thegradientof!becomeslarger,whichinducesanevenstrongerback
ow(. 
positivefeedbackeect).Onehastokeepinmindthattheback
oweectisonl. 
aperturbationtotherelaxationwhich,inthiscase,isgovernedbyelasticforces. 
Thus,whentalkingaboutthepositivefeedback,onemustrememberthatthisi. 
onlyasmallcontributiontothedirectorand
owelds,whereasglobalbehavio. 
isstillsetbytheelasticity.Eventually,elasticforcesbecomedominanteveninth. 



3. 
Directordynamicsinnematic. 


middleofthecell(Fig.4.8),reversingtherotationoftheinnerpartandacceleratin. 
it,whileslowingtherotationoftheouterpart.Wearethenleftwiththecente. 
reorientingfasterthantheedges(Figs.4.5(c)and4.8),asituationoppositetoth. 
initialone.Asaresult,theback
owalsochangesdirection,transformingfromth. 
anticlockwisecentralcurrentwithtwooppositeeddiesatthetopandbottomto. 
singleclockwisecurrent.Thelatteristhenstable;acceleratingtherelaxationi. 
themiddleofthecellitgrowsstrongerinitially(apositivefeedbackagain),the. 
slowlyfadesastheelasticforcesvanish.Thistypeofprocesswhere,inaregiono. 
thesample,thedirectorisrotatedbackwardsforsomeperiodoftimeandthenth. 

owdirectionissuddenlyreversed,willbereferredtoasatwo-steprelaxation.On. 
cannoticethatinthesquarecelltherelaxationpathisquitesimilar.Thekickbac. 
eectisobserved(Fig.4.3(a),(b)),aswellasthereversalofthe
ow(Fig.4.3(c)). 


Fortheverticalcase,however,inthemiddleofthecellthedirectorisnotrotate. 
backwardsbytheback
ow,butmerelyheldintheeld-alignedorientation.B. 
contrast,neartheboundariesthereorientationisacceleratedbytheback
owasfo. 
thehorizontalcase.Asaresult,therelaxationproceedsfromtheedgesandthe. 
graduallybitestothecenter,muchlikethecaseofasimplerelaxationwithouttakin. 
intoaccounttheback
ow(Fig.4.8,lowerdiagram).Thisconservesthedirectio. 
ofthecurrent.Onecancallaprocesslikethisaone-steprelaxation.Eventually. 
weendupwithareversecurrentinthiscasealso,butthishappensmuchlateran. 
whenthesamplehasalmostrelaxed.Besides,thereversalofthecurrentproceed. 
veryslowlyandsteadilyfromtheoutside;itdoesnothappensuddenlyinthewhol. 
cellasinthehorizontalcase. 


Field-onrelaxatio. 


Nowweconsiderthecaseinwhichtheinitialdirectoreldisundeformed,and. 
verticalmagneticeldisappliedatasmallangle(2A)othenormaltothedirector. 
whichdoesnotsignicantlyalterthecriticalbehaviorofthetransition[55,p.105]. 
Duetotheincreasingrateofrotation!towardthecenter,ananticlockwisecurren. 
ispredictedbyEq.(4.30),whereastheforce(4.31)isinitiallyunimportant,sinceth. 
directorisuniform.Calculationsforthesquarecellshowthatinthecourseoftim. 
thecurrentissplitintotwovortices,whileinbetweenthesetwoclockwisevortice. 
arealsoformed,andwenishwith4majorvortices(Fig.4.9(b)). 


Againitisusefulifonerstfocusesontheone-dimensionalcases.Forth. 
horizontalcellinitiallythereisnocurrent,becausethedirectorisperpendiculart. 
thegradientof!(Eq.4.30).Ifthecellisnotinnite,butjustextendedinth. 
horizontaldirection,the
owislimitedtotheshortboundaries,wherer!point. 
alongthedirector.Foralongcell,however,thisisunimportant.Whenthedirecto. 
turnsalittle,theforce(4.30)becomeslargerinitiatingtheanticlockwisecurrent. 
Withthedeformeddirectoreldtheforce(4.31)alsostartscontributingtoth. 
anticlockwisecurrent.Afterthedirectorhasreachedtheangleofinstability(4.33. 
inthecenter,thereorientationisacceleratedbythecurrent,resultinginapositiv. 
feedbackstrengtheningtheback
ow.Thepositivefeedbackisgreatlyamplie. 
bythemagnetictorque,whichinthiscaseisthedrivingforceoftherelaxation. 



Directordynamicsinnematic. 
3. 


j 
0.0 
0.4 
0.8 
1.2 
1.6 
j 
0.0 
0.4 
0.8 
1.2 
1.6 
0.0 0.2 0.4 0.6 
x 
0.8 1.0 

0.0 0.2 0.4 0.6 0.8 1.0 
t = 0 
t = 0.004 
t = 0.008 
t = 0.012x 
t = 0 
t = 0.004 
t = 0.008 
t = 0.012 
Figure4.8Verticalcross-sectionsthroughthecenterofthehorizontalcel. 
(eld-o):directorangleasafunctionoftransversecoordinateisshownfo. 
subsequentmomentsintime(earlystageonly).Theback
owaltersthepro. 
cessinthecenter(above),comparedwiththesimplerelaxationwithoutth. 

ow(below). 



40 Director dynamics in nematics 
0
33 
65 
98 
130 
163 
195 
228 
260 
0 1 
0
1 
t = 0.01 
t = 0.008 
0
25 
50 
75 
100 
115 
125 
150 
175 
188 
200 
0 1 
0
1 
0
8
15 
23 
30 
38 
45 
53 
60 
0 1 
0
1 
t = 0.015 
(c) vmax =0.83 
(b) vmax =5.29 
(a) vmax =9.84 
Figure 4.9 Field on in the square cell. See Fig. 4.3 for the key to the gures. 
(a) The director rotation is maximum in the middle; the asymmetry of the 
corners is due to the back
ow, not to dierence in the splay and bend elastic 
constants. (b) Now the center is relaxing slower, as are the NWand SE corners 
(which were faster in (a)). An interesting fourfold 
ow pattern is observed. (c) 
Late stages of relaxation; the current has not changed direction completely.

Director dynamics in nematic. 
4. 


Itactslikeafusetriggeringthemagnetictorquebyrotatingthedirectoroth. 
eldnormal(Eq.(4.34)).Ontheotherhand,neartheboundariestherelaxatio. 
isslowedbythesheartherehavingoppositesign(thedirectortendstostay
ow. 
aligned).Consequently,wheneventuallytherelaxationceasesinthemiddleofth. 
cell,theouterpartsarestillrelaxing.Nowtheforce(4.30)changesdirection,tryin. 
tostartaclockwisecurrent.Itisopposedbyther'-dependentforce(4.31),whic. 
isstrongestneartheboundaries.Asaresult,the
owdiesoutinthemiddleofth. 
cellrst,followedbyacessationattheboundaries. 


Againthesituationisquitedierentfortheverticalcase(Fig.4.10).However. 
unliketheeld-orelaxation,nowitwillbethemoreinterestingone.Hereth. 
anticlockwise
owduetotheforce(4.30)isinitiatedimmediately,sincenowr. 
pointsalongthedirector.Moreover,inthemiddleofthecelltheshearstartst. 
rotatethedirectoratamaximumrateasaresultofbeingnormaltothedirecto. 
(Eq.(4.32)).Ontheotherhand,neartheboundariestheshearisopposite,thu. 
turningthedirectorinthereversesense,againatamaximumrate(Fig.4.10(a)). 
Herewearedealingwithtwoeectivemechanismsthatturnonthemagnetictorque. 
oneinthemiddleofthecellgeneratinganacceleratedanticlockwiserotation,th. 
otherattheboundariescausingclockwiserotation,whichisalsoampliedbyth. 
magneticeld.Hence,thisgivesrisetoastronganticlockwisecurrentrst(positiv. 
feedbackagain),asconrmedbyFig.4.10(a).However,theforce(4.31),becomin. 
moreimportantasthedeformationincreases,opposestheclockwiserotation.Th. 

owisthussubjecttoakindoffrustration.Asaresultofthis,itisabletochang. 
directionveryrapidly,astheforce(4.30)ceases.Onecanmakeanestimateo. 
thedirectorangle,atwhichthenetforcechangessign,using!=
1
H
2
sin2'wit. 


. 


Eqs.(4.30)and(4.31),yieldin. 





1
2

f=



2
H
2
2cos2'+cos2'..1'
0
. 
. 


Thisgives'=30A 
fortheangleofforcereversal.Sincetheangleismaximuminth. 
centerofthesample,thisiswherethe
owreversalisinitiated.Aftersometime. 
theinteriorhasalmostrelaxedwhentheouterdirectoreldjustbeginstorotatei. 
therightsense(Fig.4.10(b)).Thegradientof!isreversed,theforce(4.30)change. 
direction,nowactinginaccordancewiththeforce(4.31).Theclockwisecurren. 
acceleratestherelaxationoftheouterpart(positivefeedback)whileadditionall. 
slowingtherotationinthecenter(Fig.4.10(c)).Interestingly,thiscurrentislarge. 
thantheinitialanticlockwisecurrent. 


Thus,fortheeld-onrelaxation,theinterestingscenarioalsoinvolvestwosteps. 
inthebeginningaweakmagnetictorquegeneratesback
owduetonon-unifor. 
rotation.Insomeregions,theback
owacceleratestherotationwhichhasamajo. 
eectbecauseinthiswaythemagnetictorquebecomesprogressivelylarger.Con. 
sequently,theseregionsarerelaxedmuchmorequicklythanthoseforwhichth. 
back
owhasaretardingeect.Asaresult,theyarealreadyfarabovetheangleo. 
maximummagnetictorquewhenthedelayedregionsarriveatit.Nowthelatterre. 
gionsrelaxfaster,whichcausestheback
owalsotochangedirection.Thetwo-ste. 
scenarioisfullydevelopedintheverticalcase,whereasthehorizontalcellrelaxe. 
moreorlessinonestep,asdescribedpreviously. 



42 Director dynamics in nematics 
-125 
-72 
-19 
0
34 
88 
141 
194 
247 
300 
0 1 
0
5 
t = 0.006 
-20 
0
13 
45 
78 
110 
143 
175 
208 
240 
0 1 
0
5 
t = 0.008 
0
44 
88 
131 
175 
219 
263 
306 
350 
0 1 
0
5 
t = 0.0105 
(c) vmax =25.8 
(b) vmax =6.68 
(a) vmax =19.8 
Figure 4.10 Field on in the vertical cell with Ly=Lx = 5 (note the scale): 
two-step relaxation. See Fig. 4.3 for the key to the gures. (a) Flow-induced 
backward rotation near vertical boundaries is amplied by the eld (contour 
of zero rotation is shown dashed). (b) Back
ow is changing direction, note the 
director deformation near the vertical boundaries. (c) The 
ow has changed 
direction, as well as the gradient of '_ . The situation is similar to that in (a), 
but reversed. Note that the velocities are large in magnitude.

Director dynamics in nematic. 
4. 


0.6 

0.4

<sin2.> 


0.2 

0.0 

Field ONField OFFwith backflow 
without backflow 
0.00 0.05 0.10 0.15
t

Figure4.11Timedependenceofsin
. 
'
(squarecell),averagedoverthecell. 


forcaseswithandwithouttakingintoaccounttheback
ow.Magneticel. 


ofstrength
H
=10isturnedonando. 


Thesuddenformationoffourvorticesinthesquarecellcanalsobeexplainedb. 
thismechanism,onlythatnowallfourboundariesareimportant,resultinginamor. 
complexfourfold
owpattern(Fig.4.9(b)).Clearlythe
owreversingmechanismi. 
presentinthiscasealso,yettowardtheendtheoriginal
owdirectionisrestored. 
However,calculationsforL
y
=L
x
=2alreadyyieldadenite
owreversion. 


Finally,itshouldbementionedthattheprocessesjuststudieddonotdepen. 
criticallyonthesignoftheLesliecoeAcient
3
,whichsetsthe
ow-aligningor
ow. 
tumblingpropertiesofthenematic,subjecttothesimpleshear
ow.If
3
isse. 
tozeroorevenitssignisreversed(keepingitsvaluesmall),noradicalchangesar. 
observed.Forourdiscussiontobevalid,onlytheconditionj
3
=
2
j1mustb. 
satised. 


4.5.3Comparisonwiththesimpliedtreatmen. 


Therelaxationwiththeback
owistobecomparednowwiththesimpliedre. 
laxationwithouttakingintoaccounttheback
ow.Figure4.11showsthetim. 
dependenceofsin
2
',averagedoverthesquarecell,forboththefullandsimpl. 
treatments.Heretheeectoftheback
owismainlytospeed-uptherelaxatio. 
process,asituationthatismostfrequentlyobserved.Insomecases,however,th. 
in
uenceoftheback
owismorecomplicatedandonecannotspeaksimplyabou. 
changesoftherelaxationrate.Thisoccursintheverticalcellifastrongenoughel. 
isappliedsothatthetwo-stepprocessbecomesverydistinct(Fig.4.12).Atalowe. 



4. 
Directordynamicsinnematic. 


(sin 2 
) 

F 


0.8 

0.4 

0.0 

-0.4 

 F(0,0) with flow 
F(0,0) without flow 
F(2,0) with flow 
F(2,0) without flow 
0.00 0.01 0.02
t

Figure4.12TimedependenceofthelargestcosineFouriercomponentso. 
sin
. 
'
,forthefullandsimpliedtreatments(verticalcell,magneticeldo. 
strength
H
=20isturnedon).TheaverageisdenotedbyF(0
;
0),wherea. 
F(2
;
0)standsforthecomponentbelongingtocos(2
x=L
x
).Itisshow. 
thattheaveragebehaviorcanbemorecomplicatedthannormal(compar. 
Figs.4.11,4.13,or4.14).Notethatthedierenceismuchlargerforth. 
F(2
;
0)componentsthanfortheaverages,indicatingthatduetotheback
o. 
therotationisindeedfasterinthemiddleofthecellandslowernearth. 
boundaries(seealsoFig.4.10). 


eldstrength,ontheotherhand,thesimpleregimeisagainrestored(Fig.4.13). 


TheimportanceofthegeometryisclearlyseenifonecomparesFigures4.12an. 
4.14,showingthetimedependenceoftheFouriercomponentsofsin
2
'inthevertica. 
andhorizontalcells,respectively.Theeldsarerescaledrelativetothecriticalel. 


.
K
33
22

H
2

=(=L+(=L. 
(4.37)

c
x
)
y
)

.
K
1. 


sothattheratioH=H
c
isthesameinbothcases,allowingonetomakeadirectcom. 
parison.Evidently,whenturningontheeld,theback
oweectismuchstrongeri. 
theverticalcell(two-stepprocess)thaninthehorizontalcell(single-stepprocess). 
Asfarastheeld-orelaxationisconcerned,thesameconclusionholds.Thereth. 
eectisstrongerinthehorizontalgeometry,whichinthiscaseyieldsthetwo-ste. 
scenario. 



Directordynamicsinnematic. 
4. 


-0.4 
0.0 
0.4 
0.8 
(sin 2 .) F 
F(0,0) with flow 
F(0,0) without flow 
F(2,0) with flow 
F(2,0) without flow 
0.00 0.02 0.04 0.06 0.08 
t 

Figure4.13TimedependenceofthecosineFouriercomponentsofsin
. 
'
,de. 
nedinFig.4.12,forthefullandsimpliedtreatments(verticalcell,magneti 
eldofstrength
H
=10isturnedon).Notethatduetotheweakereldth. 
averagebehaviorislesscomplex,whencomparedwithFig.4.12. 


-0.4 
0.0 
0.4 
0.8 
(sin 2 .) F 
F(0,0) with flow 
F(0,0) without flow 
F(0,2) with flow 
F(0,2) without flow 
0.00 0.01 0.02 0.03 
t 

Figure4.14TimedependenceofthecosineFouriercomponentsofsin
. 
'
. 
denedinFig.4.12,forthefullandsimpliedtreatments(horizontalcell. 
magneticeldofstrength
H
=20isturnedon).Thegureshouldservea. 
acontrasttoFig.4.12,showingthattheback
oweectislesspronouncedi. 
thehorizontalcell. 



4. 
Directordynamicsinnematic. 


4.6Amplifyingthekickbackeec. 


4.6.12Dproble. 


Ifonetakestheeld-alignedconguration(theinitialcongurationinFig.4.3. 
andappliesasecondarymagneticeldinthexdirectionimmediatelyafterthe. 
eldhasbeenturnedof,thekickbackproducedbytheback
owisampliedb. 
thesecondarymagneticeld.Adomainofoppositedirectorrotationisformed. 
Fig.4.15,separatedfromtherestofthesamplebyadomainwallwiththicknes. 
oftheorderofthemagneticcoherencelength
m
,Eq.(4.15).Thestrengthsofth. 
primaryandsecondarymagneticeldsare20and40units(4.29),respectively.Onc. 
thedomainisformeditbeginstoshrink,whichisaslowprocesscomparedwit. 
theformationandisdrivenbythedomainwallcurvature.Thenalconguratio. 
isundeformed,justlikeintheeld-ocasedepictedinFig.4.3. 


Letusestimatetheshrinkingtimeofthedomain.Neglectingthe
owandth. 
elasticanisotropy,theshrinkingrateofathincirculardomainwallwithradiusRi. 
[56,p.213. 


dR. 
=..

. 
(4.38. 


dt. 
wherelengthandtimearescaledbyLand
L
(Eq.4.28),respectively,yieldin. 


R
2

(0)..R
2
(t)=2t. 
(4.39. 


Notethattheshrinkingratedoesnotdependonthemagneticeld.Assumin. 
R(0)1=4,theshrinkingtimeisestimatedtot1=32,whichissupportedb. 
Fig.4.16. 


Thepresenceofthedomainwallisre
ectedinanyintegralpropertyofthecel. 
A(t),e.g.,inthecellaveragehsin
2
'i,Fig.4.16.Inthecaseofalineardependenc. 
ofAonthewalllength(theaboveaverageissuchacase),onecanwrit. 


A(t)=A(0)+C[R(t)..R(0)]. 
(4.40. 


whereCisaconstant.ThenitfollowsfromEq.(4.39)tha. 


2

t=C
1
[A(t)..A(0)]+C
2
[A(t)..A(0)]:(4.41. 


TheconstantsC
1
andC
2
aredeterminedbyttingthepolynomial(4.41)toth. 
slowlyfallingregioninFig.4.16.ThenumericdataandthetareplotedinFig.4.1. 
showingexcellentagreement. 


4.6.2Quasi-3Dproble. 


Letusnowconsideramoregeneralexample,whichisclosertoexperimentalset-up. 
thanthe2Dproblem.Bycontrast,thedirectorandthe
owvelocitycanpoin. 
outoftheplanethistime,whilethereisstillnodependenceonthezcoordinate. 
Twoswitchingexampleshavebeenchosen.Thenematicsampleisconnedtoa. 




Director dynamics in nematics 47 
(a) vmax = 94.4 
(b) vmax = 13.3 
t = 0.0025 
(c) vmax = 15.1 
t = 0.001 
t = 0.015 
Figure 4.15 Amplication of the kickback: 
ow (left column) and director 
elds in selected moments of time. (a) The kickback caused by the anticlock- 
wise 
ow gets amplied by the horizontal magnetic eld. (b) The domain has 
formed, the 
ow changes direction. (c) Shrinkage of the domain, now the 
ow 
is clockwise.

4. 
Directordynamicsinnematic. 


<sin2.> 


0.8 

0.6 

0.4 

0.2 

0.0 

with backflow 
without backflow 
0.00 0.01 0.02 0.03
t

Figure4.16Comparisonofswitchingprocesseswithandwithoutthe
ow. 
timedependenceofthe
y
directorcomponentsquared,sin
. 
'
,averagedove. 
thecell.Theback
ow-aectedswitchingproceedsinthreesteps.Firstalmos. 
thewholesamplealignswiththe
x
eld,exceptforthedomainwall.Th. 


22

characteristictimeofthisprocessis

=1
=H
=1
=
900.Thenthecentra. 


m

domainisslowlyshrinking,drivenbythedomainwallcurvature.Intheend. 
whenthesizeofthedomainiscomparabletoitsthickness,thecharacteristi 


2

timeoftherelaxationisagain

.TimeisscaledbyEq.(4.28).

m

L
,

0.0125 

0.0100 

0.0075 

t-t 

1 

0.0050 

0.0025 

numerical calculation 
fit with (x,x 2)

0.0000 

-0.03 -0.02 -0.01 0.00 


<sin2.>-<sin2.> 


1

Figure4.17Thetofthepolynomial(4.41)tothenumericdataforhsin
. 
'
. 
inthetimeintervalbetween
t
1
=0
:
0075and
t
2
=0
:
02,whenthedomainwal. 
iswelldened. 



Directordynamicsinnematic. 
4. 



1 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 
0 
y 

x 


L 

Figure4.18Theinitialdirectoreldconguration.Shading(althoughredun. 


dantinthecaseofthedirectoreld)representstheout-of-plane(
z
)compo. 


nent,whilethein-planecomponentsaredrawnasnailsascustomary.Magneti 


eldwithcoherencelengthof
L=
30isappliedinthe
x
direction. 


innitelylongcapillaryofsquarecrosssection.Thedomainofcalculationcorre. 
spondstothecross-section(asquareinthexyplane)andisthustwo-dimensional. 
Theanchoringisinnitelystrong,itsdirectionisparalleltothetubeaxis(zaxis). 
Initialconditionsareidenticalinbothcases:thesampleisalignedbyanin-plan. 
magneticeldwiththecoherencelength(4.15)of1=30ofthetubethicknessL. 
pointingalongthexaxis(Fig.4.18).FollowingthediscussioninSection4.3,for. 
100mthickcapillarythecorrespondingeldisaround0.3T,andthecharacteristi 
time(4.16)isabout0.2s. 


Theswitchingprocessisstartedbysuddenlyrotatingthemagneticeldto. 
perpendiculardirection.Theswitchingtimeoftheeldmustbeshortcompare. 
withthecharacteristictime(4.16).Intherstexample,thenaleldisparallelt. 
thezaxis,whereasinthesecondexample,itliesintheyzplaneatanangle70A 
withrespecttothezaxis.Itwillbeshownthat,duetotheback
ow,theswitchin. 
processisagainalteredcompletely. 


4.6.3Axialmagneticel. 


Afterthemagneticeldhasbeenswitchedtothezdirection,neartheboundar. 
thedirectorisrotatedoutoftheplanebyelasticandeldtorques.Ontheothe. 
hand,theelasticandeldtorquesarealmostzerointhecenter.Ifonedisregard. 
theback
ow,thedirectorwillalignwiththeneweldbyaclockwiserotationabou. 
theyaxis,proceedingfromtheboundarytowardthecenter. 


Theback
ow,however,canproducealargeeectinthiscase,sincethedirecto. 



50 Director dynamics in nematics 
-1.83 
-1.53 
-1.22 
-0.92 
-0.61 
-0.31 
0
0.31 
0.61 
0.92 
1.22 
1.53 
1.83 
-0.1 
0
0.1 
0.2 
0.3 
0.4 
0.5 
0.6 
0.7 
0.8 
0.9 
1 
-0.23 
-0.19 
-0.16 
-0.12 
-0.08 
-0.04 
0
0.04 
0.08 
0.12 
0.16 
0.19 
0.23 
-1 
-0.8 
-0.6 
-0.4 
-0.2 
0
0.2 
0.4 
0.6 
0.8 
1 
-0.40 
-0.33 
-0.27 
-0.20 
-0.13 
-0.07 
0
0.07 
0.13 
0.20 
0.27 
0.33 
0.40 
-1 
-0.8 
-0.6 
-0.4 
-0.2 
0
0.2 
0.4 
0.6 
0.8 
1 
v max = 0.096 
vmax = 0.14 
vmax = 0.089 
t = 54 
t = 4.05 
t = 0.45 
Figure 4.19 Axial magnetic eld: velocity (left column) and director elds in 
selected moments of time (in units of  as dened in Eq. (4.16)). The number 
of mesh points displayed has been reduced by a factor of two in each direction 
to gain clarity. The key to the gures is given with Fig. 4.18. Blue and 
red levels represent components up and down, respectively. The maximum 
in-plane velocity is denoted vmax. The velocity unit is v0 as dened in (4.19). 
The reader should not confuse the heads of nails with the nails themselves, 
where the directors point almost in the z direction. (a) The kickback caused 
by the out-of-plane 
ow. (b) The kickback is amplied by the magnetic eld, 
a domain is formed. (c) As the domain shrinks, a major part of the domain 
wall becomes a twist wall due to the elastic anisotropy: K11 and K33 are about 
twice as large as K22.

Director dynamics in nematic. 
5. 


orientationinthecentralregionislabilewithrespecttotheeld,andthusquit. 
sensibletoperturbations.Therateofdirectorrotationhastwominima|atth. 
boundaryandinthecenter,andamaximuminbetween.Asaresult,anout. 
of-plane
uid
owisgenerated(Fig.4.19,(a)),wherethevelocitychangessig. 
three-timesaswemovealongxdirection,whereasonpassingalongyitissingle. 
signed.ThemechanismsgoverningthisphenomenonareexplainedinSection4.5.1. 
The
owrotatesthedirectorinthecenterintheoppositedirectionthanexpecte. 
(thekickbackeect).Thisrotationisampliedbythemagneticeld.Thus,w. 
areleftwiththecenterrotatingintheoppositedirectionastheouterpartofth. 
sample,whichleadstotheformationofadomain(Fig.4.19,(b)).Theshrinkin. 
timet=R
2
(0)=2(Eq.(4.39)),whereR(0)L=3
m
=10istheinitialradiusofth. 
domain,isestimatedtot50inunitsof,Eq.(4.16).Ontheotherhand,th. 
characteristictimefortheformationofthedomainisjust. 


4.6.4Obliquemagneticel. 


Inordertoshowthattheeectisnotlimitedonlytoacertaindirectionofthena. 
magneticeld,asecondexampleistobedemonstrated.Theinitialconguratio. 
isthesameasinthepreviouscase(Fig.4.18),butnowthenaleldisinthey. 
planeatanangle70A 
withrespecttothezaxis.Itisimportantthattheinitialan. 
naleldsarestillperpendiculartoeachother,orcloseenoughtothis,i.e.,afe. 
degrees.Ifnotso,theeldtorqueoutweighstheback
ow-generatedoneeveni. 
thebeginning,andtheback
owcangiveonlyquantitativeeects.Thesituationi. 
similarasinthepreviousexample,onlythatnowthe
owisbothout-of-planean. 
in-planeandthedomainwallismorecomplex(Fig.4.20). 


Itisworthmentioningthat,ifthenaleldistooclosetothecross-sectionplan. 
(afewdegrees),evenintheabsenceofthe
owthedirectorrotationiscomplicate. 
bytheelasticanisotropy.Namely,duetotheanisotropy,thedirectordeviate. 
slightlyfromthexdirectioninitially,whichcausesthedirectortorotateinopposit. 
directionsindierentpartsofthesample.Despitetheback
owisimportantinthi. 
casealso,wedonotaimtogiveexamples,sincetheyaretoocomplicatedandthu. 
notparticularlyinstructive. 


4.7Summar. 


InthisChapter,nematodynamicproblemshavebeenstudiedintheirfullform. 
makingnoapproximationotherthanalowReynoldsnumber,whichispracticall. 
exactfortheproblemsconcerned.First,onemustnotetheremarkablenon-trivialit. 
ofthegeneratedvelocityelds,i.e.,theformationofseveralvortices,despiteth. 
simplegeometryandstrictlylaminar
ow.Itisaconsequenceofthedelicateinter. 
connectionbetweenthedirectorandthe
oweld.Inaddition,ithasbeenshow. 
thattheformoftheback
ow,aswellasitstimeevolution,dependverymucho. 
thegeometryofthesystem,asdoesthein
uenceoftheback
owonthedirecto. 
reorientation.Itisstrongerforthetwo-stepprocesses,whicharecharacterizedb. 



52 Director dynamics in nematics 
t = 67.5 
t = 9.9 
t = 0.45 
-0.83 
-0.69 
-0.56 
-0.42 
-0.28 
-0.14 
0
0.14 
0.28 
0.42 
0.56 
0.69 
0.83 
-0.1 
0
0.1 
0.2 
0.3 
0.4 
0.5 
0.6 
0.7 
0.8 
0.9 
1 
-0.075 
-0.063 
-0.050 
-0.037 
-0.025 
-0.013 
0
0.013 
0.025 
0.037 
0.050 
0.063 
0.075 
-0.6 
-0.4 
-0.2 
0
0.2 
0.4 
0.6 
0.8 
1 
-0.150 
-0.125 
-0.100 
-0.075 
-0.050 
-0.025 
0
0.025 
0.050 
0.075 
0.100 
0.125 
0.150 
-0.8 
-0.2 
0
0.2 
0.4 
0.6 
0.8 
1 
v max = 0.025 
vmax = 0.079 
vmax = 0.51 
Figure 4.20 Oblique magnetic eld: velocity (left column) and director elds 
in selected moments of time. The number of mesh points displayed has been 
reduced by a factor of two in each direction to gain clarity. The key to the 
gures is given with Figs. 4.18 and 4.19. (a) Both the out-of-plane and in-plane 

ow velocities are comparable in magnitude, resulting in the kickback around 
an oblique axis. (b) A part of the splay-bend domain wall rst created (not 
shown) is transformed to a twist wall due to the elastic anisotropy, resulting 
in a complex wall structure. (c) As the domain shrinks, the structure of the 
wall becomes simpler.

Director dynamics in nematic. 
5. 


aglobalreversalofthe
owdirection. 


Besidesnumericalsolutions,thequalitativepictureoftheback
owproblemi. 
alsohighlightedinthisChapter.Thus,followingthediscussioninSection4.5.2,on. 
isabletoforeseetheglobal
owdynamicsinthecell,withouthavingtoperforman. 
extensivenumericalcalculations.Inparticular,itisworthpointingoutagaintha. 
theback
owscenariodependscruciallyontherelativeorientationofthemagneti 
eldwithrespecttothelongaxisofthecell.Ithasbeenassumedthatj
3
=
2
j. 
holdsforthetwoLesliecoeAcients,whereasthesignof
3
hasprovednottob. 
signicant. 


Ithasbeenshownthattherearecaseswheretheback
oweectiscrucial.I. 
inapartofthesamplethedirectorisnearanunstableequilibriumwithrespec. 
totheeld,theback
owusuallyproducesaperturbationstrongenoughtochang. 
thedirectorrotationinthatpart.Ofcourse,onlythosecasesareinteresting,wher. 
theback
owhasafrustratingin
uenceonthedirectoreld,i.e.,itcreatesregion. 
ofoppositedirectorrotation.Suchrelaxationprocesseshavebeenreferredtoa. 
thetwo-stepprocesses,asopposedtotheone-stepprocesses,whereback
owisles. 
important. 


InnitelystronganchoringhasbeenassumedthroughouttheChapter4.I. 
ordertoobserveanyrelevantback
oweectsinpractice,theanchoringshouldb. 
strongenough(=
m
1),whichcanbereadilyachievedinexperiments.Numerica. 
evidencefortheceasingback
oweectinthecaseofniteanchoringhasbeengive. 
towardtheendof[57].Disregardinganysurfaceviscosityeects,onecanmake. 
simpleestimateforthecasewhentheanchoringbecomesnite,>0,butremain. 
strong,=
m
1.Comparingthedirectorprolesneartheboundaryfortheinnit. 
andtheniteanchoringinthepresenceofthemagneticeld,onendsthedirecto. 
gradienttodecreaseas
m
=(
m
+)1..=
m
.Uponremovaloftheeld(or,i. 
the3Dexamples,rotatingtheeldintoaperpendiculardirection),thedirectori. 
rotatedbyelasticforces(Eq.(4.9)),decreasingas
m
1..2=
m
.Itthen


2
=(
m
+)
2
followsthatthedivergenceofthe
2
and
3
termsinEq.(4.11),whichrepresent. 
thesourcedrivingthe
owinEq.(4.10),decreasesas1..3=
m
astheanchorin. 
getsweaker.Hence,themagnitudeoftheback
owanditstorqueexertedonth. 
director(Eq.(4.9))decreasethesameway. 
Asindicatedbyadditionalcalculationsnotpresented,thequalitativepictur. 
doesnotdependontheexactgeometryofthetubecrosssection,i.e.,thesquar. 
couldbereplacedbyacircleorarectangle,etc.,providedthattheaspectrati. 
staysroughlythesame.Also,thedetailedstructureofthedomainwallappearst. 
dependontheratiooftheelasticconstants,asmentionedincaptionstotheFigure. 
4.19and4.20. 
ThereorientationproblemsdescribedinthisChaptermayberegardedasa. 
overturetotheprimarychallenge|thein
uenceofhydrodynamic
owondefec. 
dynamics.Fromtheearlydaysofourresearchwehavespeculatedthatthesear. 
theprocesseswherethe
owshouldreallycomeintoplay.Richerinexperienceand. 
inparticular,equippedwiththenumericalmethod,wearenowreadytomeetth. 
challenge.Ofcourse,thereisstillacrucialstepthatyethastobedone|w. 
mustgobeyondthedirectordescription.Therearetwopathsleadingfromherean. 



5. 
Directordynamicsinnematic. 


wewilltakeboth,oneafteranother.Fornematicdefects,onehasnochoiceothe. 
thanresortingtothedynamictheoryofthetensororderparameter.Ontheothe. 
hand,defectsinavectororderparametereldlikethatofthesmectic-Cphasear. 
masteredbyadirectgeneralizationoftheEricksen-Leslietheory. 



. 
Dynamicsofavecto. 
orderparamete. 


InthisChapter,thedynamictheoryforageneralvectororderparameterwillb. 
derived.ItrepresentsageneralizationofthestandardEricksen-Leslietheoryt. 
includethenonhydrodynamicdegreeoffreedom|themagnitudeofthevector. 
OneshouldstressthattheEricksen-Leslietheoryisavectorialtheory,andtherefor. 
aconsistentgeneralizationistheonetothefullvectororderparameter,nottoth. 
tensor.Moreover,theEricksen-Leslietheoryisnotatallconnectedwiththetenso. 
orderparameter,exceptforthemissingpolareects(e.g.,electricpolarization). 
theelasticfreeenergyandthecouplingtothe
owareofthevectorialnature.A. 
wehavenotcomeacrossthisstatementintheliterature,wedecidedtosupporti. 
inthisChapter.Thegeneralthree-dimensionalcasewillbeconsidered.Althoug. 
vectorialquantitiesliketheelectricpolarizationappearinthiscase,wewillomi. 
theelectricandmagneticdegreesoffreedomandlimitthediscussiontotheelasti 
distortionsand
owcouplingonly.Atwo-dimensionalversionofthetheorywillb. 
usedinChapter8forthedescriptionoftheSmCthinlmsystem. 


5.1Free-energydensit. 


Wewillfocusonthehomogeneousandelasticpartsofthefree-energydensit. 
f(c;rc),wherecisthevectororderparameter. 


11. 
f=Ac
2
+Cc
4
+

.
L
ijkl
(@
i
c
j
)(@
k
c
l
):(5.1. 
24. 
Thehomogeneousterms(thersttwoterms)describethephasetransition,A. 


. 


A
0
(T..T
0
),A
. 
>0,C>0,andgivetheequilibriummodulusofc,c
0
=..A=C. 
whichisthecondensedquantity.Inprinciple,theelasticpartcouldcontainalsosec. 
ondderivativetermsL
ijk
@
i
@
j
c
k
,buttheseyield,besidesthetermsalreadycontaine. 
in(5.1),onlysurfacecontributions[50,p.77]). 


ThedemandingtaskistondtheelasticcoeAcientsL
ijkl
thatareallowedb. 
symmetry.Onerequiresthatfisinvarianttotheinversion(r!..r,c!..c). 
whichimpliesthatL
ijkl
mustbeinvariantaswell.Furthermore,thepermutatio. 
symmetry(2.10),L
ijkl
=L
klij
,mustbeobeyed.Itturnsoutthatthesetofscala. 


5. 



.
L
5. 
Dynamicsofavectororderparamete. 
ijk. 
elasticter. 
commen. 


A
A
A
A
A
A
ik
A
j. 
ij
A
k. 
il
A
j. 
c
i
c
k
A
j. 
c
j
c
l
A
i. 
(c
k
c
l
A
ij
+c
i
c
j
A
kl
.
(c
j
c
k
A
il
+c
i
c
l
A
jk
.
c
i
c
j
c
k
c
. 
c
p
c
r

pij

rk. 


L
1
(@
i
c
j
)
. 
L
2
(@
i
c
i
)
. 
2L
3
(@
i
c
j
)(@
j
c
i
)
L
4
c
i
c
k
(@
i
c
j
)(@
k
c
j
)
L
5
c
j
c
l
(@
i
c
j
)(@
i
c
l
)
2L
6
c
j
c
k
(@
j
c
k
)(@
i
c
i
)
2L
7
c
j
c
k
(@
i
c
j
)(@
k
c
i
)
L
8
c
i
c
j
c
k
c
l
(@
i
c
j
)(@
k
c
l
)
L
9
(c
p

pij
@
i
c
j
)
. 


Table5.1Elastictermsforthevector

isotropi 
spla. 
@
i
(c
j
@
j
c
i
..c
i
@
j
c
j
)+(@
i
c
i
)
2
,surfac. 


2

[c
(rc)],derivativeparallelto 


2

[(rc)
c],Acparallelto 
[c
(rc)
c](r
c),length-spla. 
[(rc)
c]
[c
(rc)],length-ben. 


2

[c
(rc)
c],deriv.andAcparall.to 


twis. 


orderparameter,Eq.(5.1),withou. 


theprefactor1/2denedinEq.(5.1).Inthenonquadratictermsthefactor. 
hasbeenincludedforconvenience. 



(a. 
(b. 


Figure5.1The(a)length-splayand(b)length-benddistortions,accounte. 


forbythe
L
6
and
L
7
terms,respectively. 


invariantstoformtheelasticcontributionisnotunique,i.e.,someinvariantsforme. 
with
ijk
canbeexpressedbythoseformedwithA
ik
.Wedecidetointroduceth. 
minimumnumberofinvariantscontaining
ijk
.Amongthesequenceofinvariant. 
lik. 


(@
i
c
i
)
2
;c
2
(@
i
c
i
)
2
;::. 
(5.2)

j

wechoosetheonethatislowestorderinc.Thisdoesnotmean,ofcourse,thatterm. 
oforderandhigheraredroppedeverywhereinL
ijkl
.Thenonzerocomponent. 


.
L
c
2
ijkl
andthecorrespondingscalarinvariantsarelistedinTable5.1.Elasticconstant. 
L
i
,independentofthelengthofc,havebeenintroduced.TheL
1
termisanisotropi 
term,i.e.,ittreatsalldistortionsequally.TheL
2
andL
9
termsarethesplayan. 
twistterms,respectively.TheL
3
termisessentiallyasurfaceterm.TheL
4
termi. 
relatedtothebenddistortion,i.e.,itisnonzeroifthederivativeofcinthedirectio. 
ofcisnonzero.ThetermswithL
5
,L
6
,L
7
,andL
8
aredierentfromzeroonlyifth. 
lengthofcvaries.TheL
6
andL
7
termsareparticularlyinteresting:theycorrespon. 
todistortionsinvolvingthevariationofthelengthofcandasimultaneousspla. 
orbenddeformation(Fig.5.1).Thereisnosuchtermconnectedwiththetwis. 
distortion. 



Dynamicsofavectororderparamete. 
5. 


AcomparisonwiththeFrankdistortionterms(4.1)canbemadetoconnec. 
theFrankelasticconstantswiththefundamentalelasticconstantsL
i
.Thespla. 
andtwisttermsarealreadyamongtheinvariantsinTable5.1.Thebendtermi. 
expressedasfollows. 


2

[c(rc)]=c
i
c
k
(@
i
c
j
)(@
k
c
j
)+c
j
c
l
(@
i
c
j
)(@
i
c
l
)..c
j
c
k
(@
i
c
j
)(@
k
c
i
):(5.3. 


c
2
mustbeequalcontributionsfromtheisotropic(L
1
)termtothesplay,twist,an. 
benddistortion.Thus. 


Onlytherstterm(theL
4
term)contributesifisconstant.Inaddition,ther. 


c
2
+c
2

.
K
11
=L
1
L
2
. 
(5.4. 
c
2
+c
4

.
K
22
=L
1
L
9
. 
(5.5. 
c
2
+c
4

.
K
33
=L
1
L
4
. 
(5.6. 


wherealsotheleadingorderdependenceoftheFrankelasticconstantsonthelengt. 
ofchasbeenobtained(recallthattheFranktermsinvolvetheunitvector).Tose. 
thecontributionfromtheL
1
termexplicitly,oneexpressesthebendterminanothe. 
way. 


h. 


[c(rc)]
2
=c
2
(@
i
c
j
)
2
..(@
i
c
j
)(@
j
c
i
)..[c
(rc)]
2
:(5.7)

. 


c
2

Hence,ifisconstant,Eq.(5.7)representsarelationbetweentheinvariants,s. 


thatoneofthemisredundant.IntheFrankexpression,onegetsridoftheisotropi 
term(@
i
c
j
)
2
.Furthermore,thesecondterminthebracketcannowbewrittenas. 
divergence,asindicatedinTable5.1.Thisleadst. 


1
2
. 
2
(@
i
c
j
)
2
=(r
c)
2
+[c
(rc)]+[c(rc)]+@
i
(c
j
@
j
c
i
..c
i
@
j
c
j
);(5.8. 



c
. 
c
. 


showingthatinfacttheL
1
termmerelyrenormalizestheconstantsforsplay,twist. 
andbend. 


AsmentionedinSection2.2,theelasticfreeenergydensitycanbewrittenas. 
sumofsquaretermsonly.AllelastictermsinTable5.1arealreadyinthesquar. 
form,excepttheL
3
,L
6
,andL
7
terms.Aftertakingcareofthese,onegets. 


f
e. 
111

=

(L
1
..2L
3
)(@
i
c
j
)
2
+

(L
2
..L
6
)(@
i
c
i
)
2
+

L
3
(@
i
c
j
+@
j
c
i
)
2
. 


22. 


111

(L
4
..L
7
)(c
j
@
j
c
i
)
2
+

(L
5
..L
7
)(c
j
@
i
c
j
)
2
+

L
6
(@
i
c
i
+c
j
c
k
@
j
c
k
)
2
. 


22. 


111

L
7
(c
j
@
i
c
j
+c
k
@
k
c
i
)
2
+

(L
8
..L
6
)(c
i
c
j
@
i
c
j
)
2
+

L
9
(c
k

kij
@
i
c
j
)
2
:(5.9. 


22. 


ForthefreeenergydensitytobepositivedenitethecoeAcientsofthesquar. 
termsmustallbepositive,whichintroducesinequalityrelationsbetweentheelasti 
parametersL
i
. 


5.2Couplingtothe
o. 


Todescribethecouplingtothe
ow,wemustndtheexpressionsfortheviscou. 
stresstensorandthegeneralizedviscousforceonthevectorc,i.e.,wemustdetermin. 




5. 
Dynamicsofavectororderparamete. 



dissipationterm

dissipationter. 


.
S
ijk. 


.
R
ijk. 



A
A
ij
A
k. 
(A
ii
)
2
,compressibl. 


11

(A
ik
A
jl
+A
jk
A
il
)
0
(A
ij
)
. 
(A
ik
A
jl
..A
jk
A
il
)(W
ij
)
2
,forbidde. 


2. 


Table5.2PartsofthematricesSandR,notdependingonc.Thesear. 
thetermsthatremaininthecaseoftheisotropic
uid(c=0).Thereisn. 
c-independentcontributiontoM.Thedissipationgivenbythesetermsmus. 
vanishforarigidrotation,whichinterdictstheterm(W
ij
)
2
. 


.

thematricesS,M,R,C,D,andBinEqs.(2.35)-(2.37)withtheproperties(2.39). 
Inaddition,theentropysourcedensity(2.38)mustbeinvarianttotheinversion. 
implyingthatthematricesS,M,R,andBareeveninc,whereasCandDareodd. 
Again,thesetofscalarinvariantstoformtheentropysourcedensityisnotunique. 
i.e.,someinvariantsformedwith
ijk
canbeexpressedbythoseformedwithA
ik
. 
Oncemore,wedecidetointroducetheminimumnumberofinvariantscontainin. 
ijk
.Thetermsdescribingthedissipationintheisotropic
uidarecollectedi. 
Table5.2,thoseappearingadditionallyintheanisotropiccaseinTable5.3.Viscou. 
parameters
i
independentofthelengthofchavebeenintroduced. 


Theremustbenodissipation(2.38)andnoforces(2.35)-(2.37)forarigidrota. 
tion,v=!r,_=!c,forwhic. 


c

. 


A=0;!
k
=

.

ijk
W
ij
;c_
i
=..W
ij
c
j
:(5.10. 


. 


Requiringthisfortheisotropicpart(Table5.2)forbidsthe(W
ij
)
2
term.Inth. 
anisotropicpart(Table5.3)itforbidsthetermc
p
c
r

pij

rkl
W
ij
W
k. 
andintroduce. 
thefollowingrelationsbetweenthematerialparameters. 



8
=
7
=
3
. 

5
=
4
. 
(5.11. 


Finally,thesymmetricandantisymmetricpartsoftheviscousstresstensor. 
Eqs.(2.35)and(2.36),rea. 



. 
=+
. 
+Ac
k
c
i
)+
2
Ac
k
c
l
c
i
c
j
+

i. 

0
A
ij

1
(A
ik
c
k
c
jjkkl

. 


1

4
(N
i
c
j
+N
j
c
i
)+
6
c
k
c_
k
c
i
c
j
. 


. 



. 
1. 


i. 
=
3
(N
i
c
j
..N
j
c
i
)+
4
(A
ik
c
k
c
j
..A
jk
c
k
c
i
);(5.12. 


2. 


andthegeneralizedviscousforceonthevectorc,Eq.(2.37),read. 


..h
v
=_(5.13)

i

3
N
i
+
4
A
ij
c
j
+
6
A
jk
c
j
c
k
c
i
+
9
c
j
c
j
c
i
;

wher. 
N
i
=c_
i
+W
ij
c
. 
(5.14. 


isthevectoroftherelativerotationofcwithrespecttotherotationofthe
uid.On. 
cancompareEqs.(5.12)and(5.13)withtheEricksen-Leslieexpressions(4.13)an. 




Dynamicsofavectororderparamete. 
5. 



matri. 


matrixter. 


dissipationter. 


1

.
S
ijk. 
(c
i
c
j
A
kl
+c
k
c
l
A
ij
. 
c
i
c
j
A
ij
A
kk
,compressibl. 


. 


1

(c
i
c
k
A
jl
+c
i
c
l
A
j. 
+c
j
c
k
A
il
+c
j
c
l
A
ik
)
1
c
i
c
k
A
ij
A
kj

. 


c
i
c
j
c
k
c
. 

2
c
i
c
j
c
k
c
l
A
ij
A
k. 



1

.
R
ijk. 
(c
i
c
k
A
jl
+c
j
c
l
A
ik
..c
j
c
k
A
il
..c
i
c
l
A
jk
)
3
c
i
c
k
W
ij
W
kj

. 


c
p
c
r

pij

rk. 
c
p
c
r

pij

rkl
W
ij
W
kl
,forbidde. 


1

.
M
ijk. 
(..c
i
c
k
A
jl
..c
j
c
k
A
il
+c
i
c
l
A
j. 
+c
j
c
l
A
ik
)
4
c
i
c
k
A
ji
W
jk

. 



C
ij. 
c
k
A
i. 
c
k
_c
k
A
ii
,compressibl. 
. 
. 
(c
i
A
j. 
+c
j
A
ik
. 

5
c
i
_c
j
A
i. 
c
i
c
j
c
. 

6
c
i
c
j
c
k
_c
k
A
i. 



1

.
D
ij. 
(..c
i
A
j. 
+c
j
A
ik
. 

7
c
i
c_
j
W
ji

. 


c
2

.
B
i. 
A
i. 

8
_
. 


c
i
c
. 

9
(c
i
c_
i
)
. 


Table5.3PartsofthematricesSandR,dependingonc,andtheremainin. 


matrices.Thesetermsappearadditionallyinthecaseoftheanisotropic
uid. 


Thetermcontainingisforbiddenduetotherequirementofvanishing


ijk

dissipationatarigidrotation. 


(4.6),relatingtheveindependentLeslieviscouscoeAcients
i
tothefundamenta. 
coeAcients
i
. 



4
=
0
. 
c
2


5
+
6
=
1
. 
c
4


1
=
2
. 
(5.15. 
c
2



1
=
3
. 
c
2



2
=
4
. 


with

1
=
3
..
2
and

2
=
3
+
2
ascustomary.Theleadingorderdependenceo. 
theLesliecoeAcientsonthelengthofchasbeenobtained.Inthegeneralcase,ther. 
aretwoadditionalviscousparameters
6
and
9
,connectedwiththedissipationwhe. 
c_isparalleltoc.Theydescribethecomponentofthegeneralizedforceparallelt. 
candintroducethetimederivativeofctothesymmetricpartoftheviscousstres. 
tensor.Furthermore,the
4
contributiontothegeneralizedforceisnotrestrictedt. 
thedirectionperpendiculartocanylonger. 


Finally,letuswritedownthedissipation,i.e.,theentropysourcedensity. 


T.
_s
. 
=
0
(A
ij
)
2
+
1
(A
ij
c
j
)
2
+
2
(A
ij
c
i
c
j
)
2
+
3
N
2
+

. 


2
4
N
i
A
ij
c
j
+2
6
A
ij
c
i
c
j
c
k
c_
k
+
9
(c
i
c_
i
)
2
:(5.16. 


AsmentionedinSection2.3,theentropysourcedensity(5.16)canbediagonalize. 



6. 
Dynamicsofavectororderparamete. 


tosquaretermsonly. 


 ! . 



. 

. 
46

T.
_s
. 
=
0
(A
ij
)
2
+
1
..(A
ij
c
j
)
2
+
2
..(A
ij
c
i
c
j
)
2
. 


.

. 

. 



3
h
2
+
9
g
2
. 
(5.17)

i

wit. 


.

4

.
h
. 
=N
i
+A
ij
c
j
. 
(5.18. 


.

. 


.

6

g=c
i
c_
i
+A
ij
c
i
c
j
. 
(5.19. 


.

. 


Hence,thereareonlyvesquaretermsdeterminingthedissipation,withtheco. 
eAcientsthathavetobepositive.Thetwoadditionalparametersare
4
=
3
(th. 
analogueofthereactionparameter

2
=

1
intheEricksen-Leslietheory)and
6
=
9
. 
Theydenetherotationinthe
uxspacetoyieldthe
uxeschosenoriginally,i.e.. 
N
i
insteadof(5.18)andc
i
c_
i
insteadof(5.19). 


Wearenowabletowritedownthedynamicequations(2.40)-(2.42).Thereisn. 
pointinreallywritingthemdownhere. 



6 
Defects 
In this Chapter, enough background on the defects will be covered to understand the 
annihilation processes studied in next Chapters. It is not our aim to give a thorough 
perspective of the subject here. In particular, we do not go into the group-theoretical 
considerations of defect topology | the homotopy theory [58,59]. Nevertheless, we 
do use some of its results relevant in our context. We will restrict the discussion 
to point defects in a two-dimensional system or line defects in a three-dimensional 
system, without discussing the point defects in the 3D system, as they will not be 
studied in the Thesis. The discussion will apply both to defects of the director and 
vector order parameters | the latter are merely a subset of the former. 
Defects are a \registration mark" of systems with broken symmetry. Let us rst 
try to give a useful denition of the defect. Naively we can say that the defect is 
an irregularity in the order parameter eld, i.e., a discontinuity. It can take place 
in a single point, a line, or a plane, resulting in zero-, one-, and two-dimensional 
defects. Their fundamental properties depend on the order parameter, or more 
precisely, on its symmetry. Of course, physically it is hardly possible to speak 
about any discontinuities, so there may be a problem with our naive denition of 
the defect. In fact, in the nematic the discontinuity is present only as long as the 
director description with a xed degree of order is considered. If this restriction is 
abandoned and changes of the degree of order and/or the biaxiality are allowed, a 
continuous solution is obtained [60{64]. Therefore, a more general denition of the 
defect must be searched for. Fig. 6.1 shows a point defect in two dimensions or a 
cross-section of a line defect in three dimensions. Orientational defects of the order 
parameter that carries information on a direction are called disclinations. There are 
two types of disclinations, wedge (Fig. 6.1) and twist disclinations [55, p. 126]. We 
will concentrate on the wedge disclinations only, as they are easier to envisage. 
If a loop is imagined around the defect and then traversed counterclockwise so as 
to return to the starting point, a winding number s can be dened as a measure 
of the total angle  the director/vector is rotated by on this trip, s = =2. Since 
the loop passes over defectless structure only, the continuity of the director/vector 
eld imposes that the angle of rotation must be an integer multiple of  or 2, 
61

6. 
Defect. 



(a)s. 
1=. 
(b)s. 
. 
Figure6.1Examplesofwedgedisclinations.Aloopisplacedaroundeac. 
defecttodeneitswindingnumber
s
,(a)
s
=1
=
2,(b)
s
=1.Inthevecto. 


case,(a)doesnotrepresentapossibleconguration. 



Figure6.2Fingerprint\defects"resembledisclinationsinthenematic.. 
defect-antidefectpairwithstrengths1
=
2and..1
=
2appearsonmyforenger. 



respectively. 


Defect. 
6. 


. 



. 

3
0;

;1;
;:::;directo. 
s=
2. 
:(6.1. 


0;1;2;3:::;vecto. 


Thewindingnumberdoesnotdependonthesizeoractualshapeoftheloopbu. 
solelyonthetypeofthedefectencircled.Therefore,itidentiesthedefectcom. 
pletelyandisalsoreferredtoasthestrengthofthedefect.Evenifthesingularityi. 
thecenter(thecoreofthedefect)issomehowsmeared(e.g.,bymelting|restorin. 
theisotropicphase,tobethecasebelow),thewindingnumberdoesnotchange.I. 
fact,todeterminethestrengthsofthedefectwedonotneedanyinformationo. 
whatthecentralcongurationis.Therefore,thecoreregionisnotofimportancefo. 
themacroscopicdescriptionoftheso-calledtopologicaldefects,i.e.,defectsthatcan. 
notbeconvertedtoadefectlessstructurebymeansofanycontinuoustransformatio. 
ofthedirector/vectoreld[58].Topologicallyspeaking,alldefectstransformabl. 
intoeachotherbycontinuoustransformationsareidentical.Thismeansthatthos. 
whichcanbetransformedinthiswaytoadefectlessstructure,arenotdefectsi. 
thetopologicalsense.Atthispointoneshouldmentionthatthewedgeandtwis. 
disclinationlinesaretopologicallyidentical. 


Usually,topologicaldefectsareenergeticallystable,althoughtheydonotcor. 
respondtostatesofthelowestfreeenergy[65].Indeed,whentryingtotransfor. 
themtoadefectlessstructure,ahighenergybarrieroccursduetodiscontinuitie. 
thatinevitablytakeplaceatsuchatransformation. 


6.1Disclinationsofa2Ddirector/vecto. 


Atwo-dimensionalorderparameterisaconvenientexampletostartwith.Th. 
systemmaybetwo-orthree-dimensional,itisonlythedirector/vectorwhichi. 
restrictedtoaplane.Hence,thediscussionwillapplytopointdisclinationsinth. 
2Dsystemorlinedisclinations(disclinationlines)inthe3Dsystem.Inthelatte. 
casethefreeenergies(6.5),(6.7),(6.11),(6.9),and(6.12)mustbeinterpretedasth. 
energiesperunitlengthofthedisclinationline.Theclassicationofdefectsisquit. 
illustrativehere,andintheoneconstantapproximation(Eq.(4.2))thecalculatio. 
ofstructuresissimple[66,p.147].Anexampleofthephysicalsystempossessin. 
the2Dvectororderparameteristhesmectic-CliquidcrystalstudiedinChapter. 
asarepresentativeoftheXY-model. 


Apartfromthesurfaceterms,thereisnodierenceinthelowestorderdirecto. 
andvectorelasticfreeenergies,i.e.,theFrankexpressionappliestobothcases.Inth. 
oneconstantapproximation(4.2),theequilibriumconditionforapointdisclinatio. 
locatedatthecenterofthecoordinatesystemreads[67. 


r
2

=0. 
(6.2. 


whereisthepolarangleofthedirector/vector. 


n=(cos;sin). 
(6.3. 



6. 
Defect. 


ThesolutionofEq.(6.2)candependonlyonthepolarangleandmustsatisfyth. 
continuityconditionforthedirector/vector.Thus. 


. 


=s+
0
=sarctg


+
0
;s=0;(
1
);1;(
3
);:::;(6.4. 
x
2. 


where
0
isafreeparameter.The(half-)integralnumbersisthestrengthofth. 
defectexactlyasdenedabove.Theelasticfreeenergyofthedefectstructurei. 
obtainedbyintegrationofEq.(4.2)forthesolution(6.4). 


2. 


 !
. 
 !
2

ZZ

K
. 
2. 
@@
2
. 


45

F
d
=rdrd+=Ksln;(6.5. 


2r. 
. 
@x@yr
. 


whereRisatypicalsizeofthesample,andr
0
isamicroscopiccut-orequiredfo. 
thefreeenergynottodiverge.Physically,thismeansthatatdistancesnearr
0
th. 
director/vectorconguration(6.4)cannotpossiblybecorrect.Inthisregionth. 
deformationbecomeslarge,sothattheFrankelastictheoryceasestobevalidan. 
changesinthescalarinvariantsoftheorderparametermustbetakenintoaccount. 
Inrstapproximation,acoreofradiusr
0
withthesystemintheisotropic(melted. 
stateisinvented,havingthe. 


. 


0;r<r
0

S. 
. 
(6.6. 


S
0
=const;r>r
. 


whereSisthescalarorderparameterofthenematicorthelengthofthevectororde. 
parameter.Ofcourse,thisisnotthecongurationwiththeminimalfreeenergy. 
ThecorefreeenergyF
c
duetothepresenceoftheisotropicphasei. 


F
c
=r
2

f. 
(6.7)

0

where
fisthedierenceinfreeenergydensitiesoftheisotropicandtheordere. 
phase.ByminimizingthetotalfreeenergyF
d
+F
c
,theradiusofthecoreissett. 


. 


Kn
. 


r
0
=. 
(6.8. 


2
. 


soitincreaseslinearlywiththestrengthofthedefects.Thesizeofthecoreiso. 
theorderofthecorrelationlength,Eqs.(7.18)or(8.8),whichisinthenanomete. 
range.Itisverysmallifcomparedtowavelengthoflightsoonecanconcludetha. 
opticscannotbeusedforinvestigationofdefectcores.Finally,thetotalfreeenerg. 
ofthedefectstructureinthisapproximationi. 


. 


1. 


F=F
c
+F
d
=s
2
K


+ln. 
(6.9. 
2r
. 


Incasemultipledefectswithstrengthss
i
arepresent,duetothelinearityo. 
Eq.(6.2)theequilibriumcongurationisobtainedsimplybysummingthesolution. 
(6.4)forasingledefect. 


XX

y..y
i

=(s
i

i
+
0i
)=s
i
arctg+
0
:(6.10)

0

i. 
x..x
. 



Defect. 
6. 


Thefreeenergyoftwodefectsis[65,p.529. 


. 


F=F
1
+F
2
+2Ks
1
s
2
ln. 
(6.11. 


. 
whereristhedistancebetweenthecentersofthedefects.Thersttwotermsstan. 
forthefreeenergies(6.9)ofsingledefects,whereasthethirdtermsrepresentsthei. 
interactionfreeenergy.Evidently,defectswithequallysignedstrengthsrepeleac. 
other,whilethosewithoppositesignsofthestrengthareattracted.Explicitly,th. 
freeenergyofadefectpairwiths
2
=..s
1
read. 


2
. 
1. 
. 


F=2sK


+ln;s=js
1
j;r
1
=r
2
=r
0
;(6.12. 
2r
. 


i.e.,thelogarithmicdivergencewithsystemsizeiseliminatedinthiscase. 


Letusnowdiscussbasicpropertiesofthedisclinationsofthetwo-dimensiona. 
director/vectororderparameter.Ourmaininterestwillbeinhowtopologicallydif. 
ferentdefectsarecharacterizedandthen,howtheymaycombine.Thedisclination. 
havealreadybeencharacterizedbytheirwindingnumberorstrengths.Forth. 
two-dimensionaldirector/vectoritisfoundthatdisclinationsofdierentstrength. 
aretopologicallydierent[58],i.e.,theycannotbecontinuouslytransformedint. 
eachother.Furthermore,twodefectswithstrengthss
1
ands
2
cancombinein. 
continuoustransformationtoformadefectwithstrengths
1
+s
2
,i.e.,incombin. 
ingthewindingnumbersaresimplysummed[58].Particularly,itfollowsthattw. 
defectswithoppositewindingnumberscancombinetoformadefectlessstructur. 
withzerowindingnumbers=0.Evenifthepairremainsunannihilated,Eq.(6.12. 
showsthatthelogarithmicdivergenceiseliminatedifs
1
+s
2
=0,sothefreeenerg. 
ofthedefectsissmall,providedthattheyarenotveryfarapart.Thiscanbeun. 
derstoodapriori,sinceusingaloopthatencirclesbothdefectsawindingnumbe. 
s=s
1
+s
2
=0isdetected,whichre
ectsadefectlessstructurewithlowdistortio. 
energyoutsidetheloop. 


Generally,defectswithoppositelysigned(notnecessarilyequalinmagnitude. 
strengthswillcombineinordertoreducethedistortionenergy.Ontheotherhand. 
itisenergeticallyfavorableforadefectwithalargestrengthtodecayintodefect. 
withlowerstrengths,whichcanthenmoveapartreducingthedistortionenergy.I. 
principle,oneshouldbemorepreciseandapplyEq.(6.12)tothiscases,accountin. 
forthecoreenergiesandcut-osr
0
thatdependonthewindingnumber.However. 
thesearesmallenergycorrections,toosubtletobedescribedinthecurrentapprox. 
imation.Nevertheless,Eq.(6.12)shows,thatchangingtheseparationofthedefect. 
foraslittleasonlyafewcoresizesalreadypredominatestheotherfreeenerg. 
contributions. 


6.2Disclinationsofa3Ddirector/vecto. 


Allowingforathree-dimensionalorderparameter,thetopologicalpictureischange. 
dramatically.Continuoustransformationschangingthewindingnumberbyanin. 
tegerarenowpossible[58].Thisimpliesthattopologicallyalldefectswithintege. 



6. 
Defect. 


0.60 

0.60 

0.45 

0.45 

0.30 

0.30 

0.15 

0.15 

0.00 

0.00 

-0.15 

-0.15 

-0.30 

-0.30 

-10 -5 0 510 


-10 -5 0 510 


rr 

(a)s=1=. 
(b)s=. 


Figure6.3RadialdependenceoftheQ-tensoreigenvaluesfor(a)the1
=
. 


and(b)1disclinations.Lengthisscaledbythecorrelationlength(7.18). 


Analyticexpansionsforsmall
r
aregiveninEq.(9.8). 


strengthsarenodefectsatall,sincetheycanbecontinuouslytransformedtoade. 
fectlessstructure.ForthedefectinFig.6.1(b)suchatransformation(aso-calle. 
escapeinthethirddimension[68,69])isachievedbyaprogressiverotationofdirec. 
torsaroundtheperpendicularaxeslyingintheplanewhengoingfromtheboundar. 
towardsthecenter.Hence,inthecaseofthevectororderparameterthereexistn. 
topologicaldisclinationlines(orpointsinthe2Dsystem).Similarly,half-intege. 
defectsarecontinuouslytransformableintoeachother,thusbeingtopologicall. 
identical.Hence,inthecaseofthedirectororderparameterthereexistsonly. 
singletopologicallinedefect(orpointdefectinthe2Dsystem);wechooseittob. 
thedisclinationlinewithwindingnumbers=1=2,Fig.6.1(a).Thecombinatio. 
lawisagainsimplytheadditionofwindingnumbers[58].Withonlyonetopologica. 
defect,thereislittlepossibilityleft:twodisclinationswithstrengthss=1=2ca. 
combinetoformadefectlessstructurewiths=0. 


Letusjustmentionthatthereexistanexactanalogybetweenthedisclinatio. 
linesandmagneticsystems[65,p.530]. 


6.3Structureofdisclinationcore. 


Inthecaseofthevectororderparametercthestructureofthedisclinationcorei. 
verysimple,i.e.,themagnitudecofthevectorvanishesinthecenter.Intheon. 
elasticconstantapproximation,thegeneralsolutionforsmallri. 


r
jsj

c/. 
(6.13. 


wherethevectorcisexpresseda. 


c=c(r)(.
^e
r
cos +.
^e

sin ); =(s..1)+ 
0
:(6.14. 


Inthecaseofthedirector,thefulltensororderparametermustbesolvedfo. 
[60{62].Intheoneelasticconstantapproximation(7.1),thesolutionforsmallri. 



Defect. 
6. 


giveninEq.(9.8).Thecrosssectionthroughthestrength1=2and1disclination. 
isdepictedinFig.6.3.Farfromthecoretheorderingisuniaxial,butalsointh. 
center,wherethelargestinabsoluteeigenvalueisnegative,i.e.,thedistributio. 
ofmoleculesisplanar.ThiscanbebestseeninFig.6.4containingthecomplet. 
informationontheQ-tensoreldinthedisclinationcores. 



6. 
Defect. 



(a)s=1=. 



(b)s=. 


Figure6.4Crosssectionsthroughthedisclinationlineswith
s
=1
=
2an. 
s
=1.TheQ-tensoreigensystemisrepresentedbythebox,thelengthso. 
theedgescorrespondtotheeigenvalues(aconstantisaddedtomakethe. 
non-negative). 



7 
Pair-annihilation of disclination lines in nematics 
7.1 Introduction 
The research of defects in order parameter elds corresponding to various condensed 
matter systems is driven by many aspects of motivation. Defects can be readily observed, 
either directly (e.g., by optical methods) or through other physical properties 
of the system, which are crucially modied in the presence of defects. In many cases 
of application defect-free structures are required, while in the others (e.g., in some 
liquid crystal displays) structures containing defects might be essential. In the latter 
case, one must know something about static or dynamic properties of defects. Theoretically, 
defects oer a rich playground for mathematically oriented excursions. 
Their topological properties can be very interesting and nontrivial, if only the order 
parameter has enough degrees of freedom. Defects play a decisive role in any 
phase transition, since in the late stages the ordering is governed exclusively by the 
dynamics of the defects created at the transition. An important part of the motivation 
arises from the universality of defects, i.e., they can occur in any system 
with a rich enough order parameter. Their major properties are independent of 
the underlying physics, determined solely by symmetries and dimensionalities of the 
order parameter, the defect, and the system. Lately the aim towards the exploitation 
of this universality has been experienced in the area, motivating the research 
of laboratory-friendly condensed matter systems such as liquid crystals in order to 
yield knowledge in completely dierent realms of physics (e.g., the physics of the 
universe, elementary particles, and elds) [70{74]. 
In order to study the statics or dynamics of defects in nematic liquid crystals, 
the full tensorial description of the nematic ordering must be considered. Nevertheless, 
there have been some attempts using the director description. The monopoleantimonopole 
annihilation of point defects has been studied in the scaling regime 
by Pargellis et al. [75]. The annihilation of a wedge disclination pair in a hybrid 
nematic cell and the annihilation of straight disclination lines have been studied 
by Minoura et al. [76] and Denniston [77], respectively. Peroli, Bajc, et al. [78{81] 
have studied the annhilation of point defects in a capillary. The dynamics of loop 
disclinations has been modelled by Sonnet and Virga [82]. Stark and Ventzki [83] 
have calculated the Stokes drag of spherical particles in a nematic solvent. 
69

7. 
Pair-annihilationofdisclinationlinesinnematic. 


Therearetworeasonsforthenecessityofthetensordescription.First,thehalf. 
integerdefectsareonlypossiblewhentheorderparameterpossessesthetensoria. 
symmetry.Onecanevadethisproblembyusingadirectordescriptionthatpreserve. 
thissymmetry[84].However,theapproachisstillinadequateforthesecondreason. 
whichistheinabilityofdescribingthedefectcore.Thecoremustnecessarilyb. 
included,foritisonlyinthiswaythatthelengthscaleisdenedtowhichothe. 
lengthscanbecompared,i.e.,theinterdefectdistance!Also,intheabsenceo. 
thecoreproblemsregardingthediscretisationarise,e.g.,theartifacturalpinningo. 
defects.Puncturingaholeatthespotofthedefectcoreandshrinkingittoapoin. 
assuggestedbyGartlandetal.[85]cannotbeofbenet,asthelengthscaleisstil. 
notintroduced,nottomentiontheobscureboundaryconditionsemergingatth. 
cuttingsurface.Auniaxialtensordescriptionwithavaryinglengthofthedirecto. 
hasbeenusedbyPismenandRubinstein[86].Thecompletetensorapproachha. 
beentakenbyKilian[87].Anicetensorialcalculationusinganadaptivemes. 
renementapproachhasbeenperformedbyFukudaetal.[88]. 


Ifonewantstoincludehydrodynamiceects,normallydescribedbytheEricksen. 
Leslietheory[22,23],Chapter4,ageneralizationofthelatterisrequiredtodescrib. 
thecouplingofthetensorialdynamicsandthe
ow[89{92].Stillkeepingthedirec. 
tordescription,onemightexpecttoremedytheproblemjustbyallowingavariatio. 
ofthedegreeoforder.Itturnsout,however,thatinthedefectcentertheequation. 
soobtainedareill-conditionedandincapableofaccuratelydescribingthehydrody. 
namicpartoftheproblem. 


Theeectofhydrodynamic
owonkineticsofnematic-isotropictransitionha. 
beenstudiedbyFukuda[93],asimilartopic,howeverwithadierentmethod|th. 
latticeBoltzmannalgorithm[94],hasbeenstudiedbyDennistonetal.[95].Recentl. 
aworkonhydrodynamicsoftopologicaldefectswaspublishedbyToth,Denniston. 
andYeomans[96].Theystudiedtheeectofback
owandelasticanisotropyonth. 
pair-annihilationofstraightlinedefectswithstrengths1/2,againusingthelattic. 
Boltzmannalgorithm.Theirtreatment,however,isnotbasedontheEricksen-Lesli. 
theoryandinvolvesonlytwoviscouscoeAcients.Thereisalsoasignicantamoun. 
ofexperimentalworkonthedynamicsofdefectsinnematics[97,76,98]. 


TheaimofthisChapteristopresentthesolutiontothepair-annihilationo. 
straightdisclinationlineswithstrengths1/2,startingfromthedynamictheor. 
forthetensororderparameter[92].Weconsideranunconnedbulksystem.I. 
thetheory[92],onlythosedissipationtermsareincludedthatreducetotheLesli. 
termsintheuniaxiallimitwithaconstantdegreeoforder.Therefore,thetensoria. 
theoryinvolvesthesamenumberofviscousparametersastheEricksen-Leslietheory. 
expressedassimplelinearcombinationsoftheLeslieviscositycoeAcients. 


Symmetrypropertiesofthestresstensorwithrespecttochangingthesigno. 
thewindingnumberwillbediscussed,resultinginasimpleidenticationofstres. 
tensorterms,responsiblefortheobserved
owasymmetryandtheaccelerationo. 
theannihilationprocess.Further,itistobeshownthatthehydrodynamiceec. 
dependsonthedirectorphaseangle,i.e.,unliketheorderparameterdynamicsi. 
caseofelasticisotropyconsideredhere,itisnotinvariantunderthehomogeneou. 
rotationofdirectors.Againthecorrespondingstresstensortermwillbepointe. 



Pair-annihilationofdisclinationlinesinnematics7. 


out. 


Itshouldbestressedthatalthoughthetensorialapproachworksverywella. 
smalldefectseparations,thepassageto>1mlengthscalesthatcanberesolve. 
experimentallyishinderedbyenormouscomputationalcomplexityoftheproble. 
andthelarge(severalordersofmagnitude)ratioofthedefectseparationtothesiz. 
ofthedefectcore. 


7.2Dynamicequation. 


ThestartingpointisthebulkfreeenergydensityexpressionintermsofQ[10,p. 
156]. 
. 
f=(Q)+L(@
i
Q
jk
)(@
i
Q
jk
). 
(7.1. 
. 
wherethehomogeneouspartisgivenb. 


11. 



)
2

(Q)=

AQ
ij
Q
j. 
+BQ
ij
Q
jk
Q
ki
+C(Q
ij
Q
ji
:(7.2. 
23. 
ItwastakenintoaccountthatC
1
(Q
ij
Q
ji
)
2
+C
2
Q
ij
Q
jk
Q
kl
Q
l. 
=(C
1
+1=2C
2
)(Q
ij
Q
ji
)
. 
andanewconstantC=C
1
+C
2
=2wasintroduced.Intheelasticpartof(7.1),onl. 
thetermwithL
1
Lisretained,resultinginisotropicelasticity.Termsofthir. 
orderinQareneededtoreachthesplay-bendelasticanisotropy[99],theeectso. 
whichhavebeenstudiedin[96]. 


RequiringtheQtensorbetracelessandsymmetric,theEuler-Lagrangeequatio. 
forthefreeenergyfunctiona. 


. 


F=dV[f(Q;rQ)..Q
ii
..
i

ijk
Q
jk
. 
(7.3. 


givesthehomogeneousandelasticpartofthegeneralizedforceonthetensororde. 
parameterQ. 


@. 


h
h. 
. 


i. 
=L@
k
Q
ij
..+A
ij
+
k

kij
. 
(7.4. 
@Q
i. 


TheLagrange-multipliertermsmerelystatethattheisotropicandantisymmetri 
componentsof(7.4)arenotspeciedandhavetobedeterminedbytheconstraints. 


h
he

i.e.,theisotropicandantisymmetricpartsmustbesubtractedfromtheforce
i. 
h
he

(projectiontothetracelesssymmetricsubspace).Toputitinanotherway,
mustbeprojectedontothesymmetricandtracelesssubspaceofQ.Theelasti 
stresstensorisobtainedinastandardmanner,Eq.(2.26),a. 


@. 



. 
ij
=..@
j
Q
kl
. 
(7.5. 


@(@
i
Q
kl
. 


TheviscousstresstensorandtheviscousgeneralizedforceontheQtensorare[92. 


. 



. 
ij
=
1
Q
ij
Q
kl
A
kl
+
4
A
ij
+
5
Q
ik
A
k. 
+
6
Q
jk
A
k. 
+
2
N
ij
..
1
Q
ik
N
k. 
+
1
Q
jk
N
ki
. 


. 
(7.6. 



7. 
Pair-annihilationofdisclinationlinesinnematic. 


. 


..h
. 
=


2
A
ij
+
1
N
ij
. 
(7.7)

ij
. 



wher. 


dQ
ij

.
N
ij
=+W
ik
Q
k. 
..Q
ik
W
kj
. 
(7.8. 


d. 
withthematerialtimederivativedQ
ij
=dt=@Q
ij
=@t+(v
r)Q
ij
andthesymmetri 
andantisymmetricpartsofthevelocitygradientAandW,Eq.(2.34).Onlythos. 
termshavebeenincludedthatintheuniaxiallimitwithaconstantdegreeoforde. 
reducetothestandardLeslieviscousterms
i
.Thus,theviscouscoeAcientsi. 
(7.6)and(7.7),linkedbytherelation
2
=
6
..
5
,canbeexpressedintermso. 
theLesliecoeAcientsandtheconstantvalueofthescalarorderparameter[92]. 
Finally,theequationofmotionfortheQ-tensoristhetracelesssymmetricpar. 
(denotedby?)ofthegeneralizedforcebalance. 


n. 
h
he
+h
. 
=0. 
(7.9. 
. 


.
Q
withtheconstraint. 
ii
=0;
ijk
Q
j. 
=0. 
(7.10. 


ThegeneralizedNavier-Stokesequationwithinthelow-Reynolds-numberapprox. 
imation(omittingthenonlinearadvectivederivativeterm(v
r)v),regularlyusedt. 
describetheorderparameterelasticitydrivendynamicsinliquidcrystals(Chapte. 
4),read. 



@v
i
=..@
i
p+@
j
(
v
+
e
). 
(7.11)

j. 
ji

@. 
withthedensityandtheviscousandelasticstresstensorsgivenin(7.6)and(7.5). 
Usually,alsothesteadystateapproximationismade,omittingthetimederivativ. 
term(Chapter4).Thepressureeldpmustbesuchthattheincompressibilit. 
conditio. 
@
i
v
i
=. 
(7.12. 


issatised. 


7.3Characteristicscale. 


Letusrewritethefreeenergydensityusingtheuniaxialansatz(3.11. 


.
Q
. 
ij
=S(3n
i
n
j
..A
ij
). 
(7.13. 
. 


3
2
1
3
9
4
. 
9

f=

AS+

BS+CS+

L(rS)
2
+

LS
2
(rn)
2
:(7.14. 


4416. 
. 



TheEuler-LagrangeequationforS,puttingrntozero,read. 



. 
@. 


Lr
2
S..=0. 
(7.15. 


. 
@. 



Pair-annihilationofdisclinationlinesinnematics7. 


Forahomogeneoussystem,thesecondtermmustvanishinequilibrium,fromwher. 
thebulkequilibriumvalueofSisobtained. 


. 


q

. 


S
0
=


..B=3C+(B=3C)
2
..8A=3C:(7.16. 
. 


LinearizingEq.(7.15)forsmalldeviationsfromequilibrium,S(r)=S
0
+
S(r). 
oneget. 




2@
2
. 




r
2


S..


S=0. 
(7.17. 
3L@S
2
S. 
fromwhereacharacteristiclengthscalecanbeextracted|thenematiccorrelatio. 
lengt. 


. 


3. 


=


. 
(7.18. 
2f
00
j
S. 


Usingthecorrelationlength(coupleofnanometersusually)andacorrespondin. 
characteristictime(Eq.(4.16). 


=

1

2
=K=
1

2
=L. 
(7.19. 


where

1
isthedirectorrotationalviscosityandKisthedirectorelasticconstant. 
Eq.(7.9)andthestationaryEq.(7.11)arebothputtoadimensionlessform.Th. 
timeisthecharacteristicrelaxationtimeoftheorderparameterdeformationo. 
thelengthscaleof,whichistypicallytensofnanoseconds.Inthefollowing. 
dimensionlessquantitieswillbeused,i.e.r r=forlength,t t=fortim. 
andv v=forthevelocity.Afterdoingso,thematerialparametersenterth. 
equationsonlythroughcombinationsgivenin(7.22)and(7.23). 


LetusestimatetheReynoldsnumberandtheunsteadinessparameterofth. 

ow,i.e.,theratioofcharacteristicdynamictimesofthe
oweldandtheorde. 
parametereld.TheestimatediersfromthosemadeinChapter4,inthatno. 
thereisnosimplerelationbetweencharacteristicdeformationlength(7.18)ofth. 
orderparametereldanditsrelaxationtime.Instead,onecanempiricallyidentif. 
thelatterwiththeannihilationtime.ThisyieldstheReynoldsnumberandth. 
unsteadinessparametero. 


.. 
K
R
. 
R
. 
0.


Re=10
..6

. 
(7.20. 




. 
. 
t. 


R
2

where
0
istheinitialdefectseparationandtistheannihilationtime.Theisotropi 
viscositywasputequaltoforbrevity.Thevalueof=t,obtainedempirically,is

.


R
. 
1
0

oftheorderofafewunits.Whatismore,followingthephenomenologicalequatio. 
ofmotiongivenbyPleiner[100,101,96]. 


 . 


dR1. 


/ln
... 
. 
(7.21. 


.

0

dt. 



. 
R
2

whereR(t)istheactualdefectseparationandscaleswith,thevalueof
0
=. 
exhibitsonlyaweaklogarithmicdependenceonR.Thus,forlargeenoughdefec. 
separationscomparedwith,theempiricalestimateisquitegeneralinvalidity.I. 
conclusion,theReynoldsnumberandtheunsteadinessparameteraretinyindeed. 
sothatinEq.(7.11)boththeadvectiveandpartialtimederivativescanbeomitted. 



7. 
Pair-annihilationofdisclinationlinesinnematic. 


7.4Technicalitiesandmaterialparameter. 


ThenumericalmethodusedisdescribedinChapter4.Thecalculationsweredon. 
onaninhomogeneoussquaremesh,consistingofanemeshof160x160pointsinth. 
centercontainingbothdefects,andacoarserinhomogeneousgridwithincreasin. 
spacingaroundittoyieldthetotalof280x280points.Thepositionofthedefect. 
wasdeterminedbyndingalocalminimumofthetraceQ
ij
Q
ij
. 


Thevelocitywassettozeroattheboundary.Inordertomeetthesituatio. 
presentinabulksystem,thedefectseparationwassmallcomparedtothesizeofth. 
computationalarea(theratioofthetwowas3/20)andthederivativesoftheorde. 
parameternormaltotheboundaryweresettozero.Initially,theQtensorwasse. 


. 


P

S. 
. 
y..yk

toQ
ij
=(3n
i
n
j
..A
ij
),wheren=(cos;sin)and=m
k
arctan. 


. 
k=1
x..x. 


whichistheoneelasticconstantequilibriumdirectorcongurationwithtwodefect. 
ofstrengthm
k
positionedat(x
k
;y
k
),Eq.(6.10).Afterwards,enoughcomputin. 
stepswithoutthehydrodynamicswereperformedtorstestablishthefulltensoria. 
conguration.Theinitialdefectseparationwasabove70correlationlengths(7.18). 
inordertoreachthefarregimeofmotion,wherethedefectsarewellisolated.Ason. 
realizes,therearethreelengthscalesinthesystem,whichshouldbewellenoug. 
separated:thecorrelationlengthandthedefectspacingastherelevantphysica. 
scales,plusthecontainersizeasthetechnicalone. 


TheviscositycoeAcientsin(7.6)and(7.7)wereobtainedfromthestandar. 
LesliecoeAcientscorrespondingtoMBBA[49,p.231]asdescribedin[92].Numer. 
icalvaluesoftherelevantratiosar. 



2
=
1
..1:92;
1
=
1
0:17;
4
=
1
1:99;
5
=
1
0:70;
6
=
1
..0:79. 


(7.22. 
TheLandaucoeAcientsA,B,CandtheelasticconstantLin(7.1)and(7.2)wer. 
takenfrom[102].Numericalvaluesoftherelevantratiosar. 


A
2
=L..0:064;B
2
=L..1:57;C
2
=L1:29;(7.23. 


withthecorrelationlength(7.18)2:11nm.Thecharacteristictime(7.19. 
32:6nscompletesthesetofmaterialparameters. 


7.5Resultsanddiscussio. 


Theresultsforthepairannihilationof1=2defects(Fig.7.1)arepresentedi. 
Fig.7.2.Itshouldbepointedoutthatduetothehighcomputationalcomplexityo. 
theproblemandthebroadrangeoflengthscalesinvolved,onlydefectseparationso. 
lessthan1mandannihilationtimesoflessthan1mscanbereached.Thismean. 
thatforthetimebeingtherestillexistsalargegapbetweennumericcapabilitie. 
andpossibleexperimentalobservations. 


InFig.7.2onenoticestwodistinctfeatures:duetothehydrodynamic
owth. 
annihilationisfasterandasymmetric.Fig.7.3showsthatitisparticularlythe+1=. 
defectwhosemotionisaectedbythe
ow.AlsoclearlydemonstratedbyFig.7.. 



Pair-annihilation of disclination lines in nematics 75 
(a) 
(b) 
Figure 7.1 A schematic representation of a pair of 1=2 defect lines: the 
eigenvectors corresponding to the largest absolute eigenvalue of Q (directors) 
are depicted in the cross-sectional plane, perpendicular to the disclination 
lines. Two isomorphs (a) and (b) are shown, diering only in a homogeneous 
rotation of the directors. For clarity, the number of mesh points has been 
reduced by a factor of 4 in each dimension and the correlation length has been 
increased by a factor of 2 (only the central homogeneous region of the mesh is 
shown).

7. 
Pair-annihilationofdisclinationlinesinnematic. 


30 

y 0 

-30 

0 1000 2000 
(b) 
(c) 
(a) 
+1/2 
-1/2 
t 

Figure7.2Positionofthedefectsasafunctionoftime,measuredfromth. 
initialmiddlepointbetweenthedefects.Threesituationsaredisplayed:th. 
twoisomorphs(a)and(b)(seeFig.7.1)andthecasewithoutthe
ow(c). 
wheretheisomorphsbecomedegenerate.Recallthatlengthismeasuredrel. 
ativeto

2
:
1nm,andtimeismeasuredrelativeto

33ns. 


0.3 
+1/2 
-1/2 
no flow 
0.2 

v 

0.1 

0.0 

r

Figure7.3Velocityofthedefectsasafunctionoftheinterdefectdistanc. 
(isomorph(a)).Forcomparison,thesameisshownforthecasewithoutth. 
hydrodynamic
ow.Thevelocityofthe+1
=
2defectisstronglyincrease. 
bythe
ow.Notethenonmonotonicbehavioratearlystagesoftheprocess. 
wheretheinitialequilibriumQ-tensorcongurationisadaptingtoadynami 
one.Thedistanceandthevelocityaremeasuredrelativeto

2
:
1nman. 
=
65nm
=
s,respectively. 


0 20 40 60 



Pair-annihilationofdisclinationlinesinnematics7. 


0.12 

0.08 

0.04 

v 

0.00 
-0.04 
-0.08 
+1/2 
-1/2 isomorph (a) 
isomorph (b) 
no flow 
1020304050 6070


r 

Figure7.4Velocityofthedefectswithoutthecontributionofadvectiona. 


afunctionoftheinterdefectdistance.Withoutthehydrodynamic
ow,bot. 


defectsmovesymmetrically.Notethatthepartofthevelocitycomingfro. 


theorderparameterdynamicsislargerforthe..1
=
2defect.Alsonoteth. 


dierencebetweentheisomorphsoriginatingfromthedierentcouplingt. 



owanddierent
owelditself,bothofwhicharemostlyduetothe

. 


viscousterm. 


(seealsoFigs.7.4and7.5)isthenonmonotonicbehaviorofthedefectvelocitie. 
atearlystagesoftheannihilation[86],[72,p.58].Itisaconsequenceofstartin. 
withtheequilibriumcongurationofxeddefectsratherthanwithadynamicone. 
whichisbeingapproachedbythesysteminthecourseofannihilation.Sinceou. 
simulationsrepresentonlytheverylatestageofanactualannihilationprocess,thi. 
nonmonotonicbehaviorshouldbeviewedasanunphysicalartifactoftheinitia. 
condition.Laterwewillshowthatitcanbeeliminatedbystartingwithaprope. 
dynamicconguration,evenwithoutthrowingawaycomputationalresourcesfo. 
simulatinglargerdefectseparations.Alternatively,itcanberegardedasaninertia. 
eectduetoaneectivemassthatcanbeattributedtothedefect[100].Asth. 
defectmoveswithaspeedv,itdistortstheorderparameteraroundit,thedistortio. 
dependingonv.Thepartofthedistortionenergyquadraticinvcanberegarde. 
asaneectivekineticenergy,fromwheretheeectivemasscanbeextracted. 


First,letusconcentrateonqualitativefeaturesofthe
ow-drivingmechanis. 
byinspectingthestresstensors(7.5)and(7.6).Oneistemptedtoexplaintheeasil. 
perceivedcharacteristicofthe
oweld(Fig.7.6(a)):duetoadvectionthe+1=. 
defectisspedup,whilethe
owismuchweakeraroundthe..1=2defect. 


AsestimatedinChapter4andveriednumerically,the\passive"
1
,
5
,an. 

6
termsintheviscousstresstensor(7.6)(ortheircounterpartsinthestandar. 
Ericksen-Leslietheory,
1
,
5
,and
6
),describingthedependenceofthe
uidvis. 



7. 
Pair-annihilationofdisclinationlinesinnematic. 


0.00 

-0.02 

v 

-0.04 

-0.06 

-1/2 
+1/2 
isomorph (a) 
isomorph (b) 
10 20 3040 50 60 70


r 

Figure7.5Theadvectivecontributiontothevelocityofthedefectsforth. 


twoisomorphiccases.Thesurprisinglylargedierencebetweenthevelocitie. 


ofthe+1
=
2defectismainlyduetothe

2
viscousterm.Atsmallseparation. 


notshown,themotiondrivenbytheorderparameterdynamics(Fig.7.4. 


becomesdominant. 


cosityontheorderparameter,giveonlyminorquantitativeeects.Thereforeon. 
canignoretheminstrivingtogainaqualitativepicture.Ontheotherhand,th. 


_

remaining
1
and
2
terms,whichcontaintheorderparametertimederivativeQ. 
andalsotheelasticstresstensor(7.5),representthesourcedrivingthe
owan. 
thereforehavetobeanalyzedcarefully. 


7.5.1The
owasymmetr. 


Atthisstage,weareinterestedonlyinsymmetries,i.e.,thebehaviorofthestres. 
tensortermsconsidereduponchangingtheorderparametereldlocallyastotrans. 
formthe+1=2and..1=2defectsoneintotheother.Inoneelasticconstantap. 
proximation,thiscanbeachievedbymirroringtheQtensorontheaxisjoiningth. 
defects(theyaxis,Fig.7.1)[95]:Q
xy
!..Q
xy
,sincethefreeenergydensity(7.1. 
isleftunchangedbythisprocedure.Anystresstensorterms,invariantwithrespec. 
tothistransformation,treatbothdefectsequallyandclearlydonotcontributet. 
the
owasymmetry.Ontheotherhand,anynoninvarianttermsmustbeidentie. 
asthe
owsymmetry-breakingcomponents. 


Bydenition(7.5)theelasticstresstensorisinvariant,whichisadirectconse. 
quenceoftheelasticisotropy.Asaresult,the
oweldisthesameforbothdefect. 
(Fig.7.6(b)).Inaddition,itsdirectionissuchastoreducetheinterdefectsepara. 
tionandtherebythefreeenergyofthesystem.Thisfollowsimmediatelyfromth. 
denitionofanystresstensor,Eq.(2.25). 


Theviscoustermswillbeanalyzedforthecasev=0,i.e.,onlythedriving(Q_
. 



Pair-annihilation of disclination lines in nematics 79 
(a) complete stress tensor, vmax = 0:011 
(b) elastic terms, vmax = 0:0057 
(c) 1 term, vmax = 0:0085 
(d) 2 term, vmax = 0:0041 
Figure 7.6 Flow elds resulting from dierent driving stress tensor terms: (a) 
the complete stress tensor, (b) elastic stress, (c) the 1 viscous term, and (d) 
the 2 viscous term. In all cases also the isotropic 4 viscous term is included. 
For clarity, the number of mesh points has been reduced by a factor of 4 in 
each dimension; only the central homogeneous region of the mesh is shown. 
The approximate positions of defects are marked with circles, the radius of 
the defect core is roughly four grid points. The maximum velocity magnitude 
vmax corresponding to the longest velocity vector is given for each 
ow eld 
(relative to =).

8. 
Pair-annihilationofdisclinationlinesinnematic. 


dependent)partin(7.8)willbeconsidered.The
2
termhasnodenitesymmetr. 
forsomeofitscomponentstransformsymmetricallyandsomeantisymmetrically. 
Atthedefectspotsthe
owdrivenbythistermisratherweakcomparedtoth. 


_

contributionfromtheothertermsinquestion,becauseQisextremalthereyieldin. 
avanishingdivergence.Hence,the
2
termdoesnotgiveadominantcontributio. 
totheadvectivemotionofthedefects. 


Ontheotherhand,the
1
termisfullyantisymmetricwithrespecttothetrans. 
formation,yieldingexactlytheopposite
owforthe..1=2defectascomparedwit. 
thatnearthe+1=2defect(Fig.7.6(c)).Onenoticesthatthe
owisthestronges. 
atthedefectpositionsinthiscase.Thus,duetoadvectionthistermalonecangiv. 
risetothe
owasymmetryobserved.OnecanverifybyinspectingEqs.(7.6)an. 
(7.7)thattherelativemagnitudeofthisantisymmetriccontributiontotheadvectiv. 
derivativeterm(v
r)Qin(7.9)isapproximatelyproportionalto
1
,providedtha. 
allothermaterialparametersarekeptxed.Ontheotherhand,scalingallvis. 
cositiesequallywithrespecttotheelasticconstantleavesthedynamicsunchange. 
completelyandmerelyaltersthecharacteristictime(7.19),astatementbasedpurel. 
ondimensionalgrounds(seeSection7.2). 


Inadditiontothe
owasymmetry,theannihilationprocessisalsosignicantl. 
spedupwhencomparedtotheannihilationwithoutthe
ow.Followingthepreviou. 
discussion,thiseectiscausedmostlybytheelasticstressdriven
ow.Thus,th. 
annihilationdynamicsoersaniceexampleshowingtheimportanceoftheelasti 
stressinliquidcrystals,whichisusuallyconsideredlesssignicant,e.g.inLCcells. 
Additionally,theelasticand
1
viscoustermsactinconcordnearthe+1=2defect. 
whereasforthe..1=2defecttheycombinedestructively.Thisexplainsthedieren. 
velocitymagnitudesinthevicinityofthedefects(Fig.7.6(a)). 


7.5.2Reorientation-drivendefectmotionvs
owadvectio. 


Itisalsoofone'sinteresttoquantifytheratioofdefectmotionduetoadvectiona. 
opposedtothemotionpropelledbytheorderparameterdynamics.Figures7.4an. 
7.5showthatthevelocitiesinquestionarequitecomparableinmagnitude.More. 
over,inChapter9,wheretherepulsivemotionoftwo1/2disclinationisstudied,w. 
ndthatthecontributionoftheadvectivetransporttothetotalmotionincrease. 
withtheincreasinginterdefectseparation(Fig.9.10).Weexpectthistobethecas. 
alsofortheattraction.Onceagainthisre
ectstheimportanceofthe
owindefec. 
dynamicsascomparedwiththelimitedperturbingeectsitnormallyhas,e.g.i. 
LCcells,Chapter4.Furthermore,onemustrealizethatalsoasecondary
oweec. 
besidesadvectionisimportant,namelythein
uenceofthe
owontheorderpa. 
rameterdynamics.ItisclearfromFig.7.4thattheorderparameterdynamicsitsel. 
isfasterbecauseofthecouplingtothe
ow.ComparingFigs.7.4and7.5oneca. 
statethatthecontributionofthiscouplingtothe
owasymmetryislessimportan. 
thanthatoftheadvection,whereasitsacceleratingeectisjustasimportant. 



Pair-annihilationofdisclinationlinesinnematics8. 


7.5.3In
uenceofthedirectororientationangleonthe
o. 


Inoneelasticconstantapproximation,thefreeenergydensity(7.1)andthusth. 
orderparameterdynamicsareinvariantwithrespecttoahomogeneousrotationo. 
theeigensystemoftheQ-tensorineveryspacepoint.Consequently,defectpair. 
dieringonlyinthisconstantphaseangleofdirectorrotation|letuscallthe. 
isomorphs(Fig.7.1)|behaveexactlyinthesameway(e.g.,forthecaseofa+. 
defectsuchisomorphsaretheradialandtangentialdefects,aswellasanyothe. 
formbetweenthetwo).Withthe
owpresent,however,thissymmetryisbroke. 
(Fig.7.2).Itisquiteinstructivetostudythedependenceoftheimportantstres. 
tensortermsuponsucharotation.Besidestheelasticterm(7.5),the
1
pairo. 
termsisalsoleftunchangedbytherotation.Thisiswhytheeectofadvectio. 
shouldberoughlysimilarforallisomorphs.Itisworthmentioningthatalsoth. 
in
uenceofthe
owontheQ-tensorgivenbythe
1
termin(7.7)isnotaectedb. 
therotation. 


Ontheotherhand,the
2
stresstensortermisnotinvariant.Onecanse. 
inFigure7.5thatitintroducessignicantdierencesevenasfarastheadvectio. 
ofthedefectsisconcerned.Forgeneralisomorphsthe
2
termyieldsa
owel. 
lackingthesymmetryofre
ectionontheaxisjoiningthedefects.Additionally. 
the
2
termintheviscousforce(7.7)isdierentfordierentisomorphs.Itisdu. 
bothtothedierentcouplingofthe
owtotheorderparameterdynamicsandt. 
thedierencesinadvectionthattheisomorphsarenotequivalentdynamically.A. 
veriednumerically,the
1
,
5
,and
6
termsagainbringonlyaverysmalldierence. 


7.6Summar. 


Wehavestudiedtheattractionandannihilationofstraightlinedefectswithstrengt. 
1=2inbulknematics.OurapproachisbasedontheEricksen-Leslie-likedynami 
theoryforthetensororderparameterofthenematicliquidcrystals.Ithasbee. 
shownthatduetothehydrodynamic
ow,theannihilationisfasterandasymmetric. 
Further,wehaveidentiedthegoverningstresstensorterms:the
1
and
2
viscou. 
termsandtheelasticstress.Symmetriesofthetermsuponinvertingthesigno. 
thewindingnumberandperformingahomogeneousin-planerotationoftheQ. 
tensoreigensystemhavebeendiscussed.Boththe
1
termandtheelasticstressar. 
invariantupontherotationandhenceidenticalforallisomorphs.The
1
termi. 
antisymmetricwithrespecttochangingthesignofthedefects,therebycontributin. 
dominantlytotheannihilationasymmetry.Ontheotherhand,theelasticstressi. 
symmetric,sothatitcausestheannihilationprocesstogoonfaster.Theonlyterm. 
distinguishingbetweendierentisomorphsarethe
2
termsin(7.6)and(7.7)(the. 
alsodistinguishbetweenthe+1=2and..1=2defect).Thus,onecanconcludetha. 
thedierenceindynamicsbetweentheisomorphsisgovernedbytheratio
2
=
1
. 
Theremaining
1
,
5
,and
6
termsintheviscousstresstensor(7.6)introduceonl. 
inferiorcorrectionstothe
oweld. 


Oneshouldemphasizeoncemorethatduetolengthscalesseveralorderso. 
magnitudeapartandenormouscomputationalcomplexityoftheproblem,withth. 



8. 
Pair-annihilationofdisclinationlinesinnematic. 


presentmethodoneisunabletoreachthe>1mrangeofinterdefectdistances. 
whichcanberesolvedinexperiments.Nevertheless,itisquitereasonabletobeliev. 
thatthehydrodynamiceectsdescribedinthisChapter,i.e.,the
owasymmetr. 
andthereductionoftheannihilationtime,willbepresentandevenstrongera. 
largerdefectseparations. 



8 
Pair-annihilation of vortices in SmC lms 
Our research of defect dynamics in SmC lms has been motivated by a preliminary 
experiment on the free-standing SmC thin lm system by Link et al. [103{106], 
showing an unexpected behavior of a pair of annihilating vortices. This triggered 
speculations on the 
ow eects being responsible for it. Apart from an approximate 
analytical study of forces on a single defect by Pleiner [100], there have been no hydrodynamic 
studies of defects in SmC lms reported so far. For a nonhydrodynamic 
treatment see the work by Pargellis et al. [107]. 
In this Chapter, we dene the SmC order parameter and set the scene for an 
adequate description of defect dynamics in SmC lms. The SmC thin lm system is 
reduced to the XY -model. Then we show that the pair-annihilation of vortices with 
winding numbers 1 (Fig. 8.2) is accompanied by strong hydrodynamic 
ow, which 
speeds up the process as compared with the model situation without the 
ow, and, 
assisted by the elastic anisotropy, gives rise to asymmetry in defect speeds. 
8.1 SmC order parameter 
In the phase sequence I | N | SmA | SmC, the SmC phase occurs, if present, 
at the lowest temperature. In the SmA phase, the director n is normal to the 
smectic layers, whereas in the SmC phase, this symmetry is broken and the director 
is tilted. Experimentally most convenient liquid crystal system for the study of 
defects is the free-standing SmC thin lm, only a few smectic layers in thickness. 
The projection of the director onto the smectic plane is a two-dimensional vector, 
called the c-director, which is the order parameter of the SmC phase. One has to 
point out immediately, that the c-director, despite its name, is in fact a vector, i.e., 
c 6= ..c. The length of the vector (also called the tilt or amplitude) is the condensed 
quantity that becomes nonzero at the transition, while its angle (or phase) is the 
hydrodynamic quantity with a Goldstone mode. Originally, the c-director has been 
considered a unit vector, representing only the hydrodynamic degree of freedom. 
For description of defects, however, one must include also the variation of its length. 
In general, the c-director is coupled to the smectic order parameter and possesses 
also the nematic tensorial structure. To rst approximation, both will be neglected, 
assuming a uniaxial nematic ordering with a constant degree of order and straight 
83

8. 
Pair-annihilationofvorticesinSmClm. 


andxedsmecticlayerswithaconstantthickness,irrespectiveofthec-director. 


Topologicaldefectsofthec-directoraredisclinationlineswithintegerwindin. 
numbers,i.e.,vortices.Halfintegerstrengthsarenotallowedduetoc=6..c.I. 
ordertoavoiddiscontinuitiesintheceld,thetiltmustbeallowedtovary.Wit. 
that,inthedefectcorethesystemrevertslocallytotheSmAconguration. 


ItmustbestressedthattheorderparametersoftheSmCandnematicphasesar. 
fundamentallydierent,anditistheSmCthinlmsystem|withintherestriction. 
givenbelow|ratherthanthenematicthatisarepresentativeoftheXY-model. 
Therefore,itisbelievedthattheSmCdynamicshasawiderrangeofapplicability. 
whichessentiallymotivatesitsanalysis.Moreover,the1vortexisunstablei. 
thenematiccase(Chapter9)|itisdecomposedintoarepellingpairof1=. 
disclinations,sothatitsdynamicscannotbefollowed.Thus,onehastoresortt. 
thevectororderparametertostudyvortices. 


8.2Dynamicequation. 


ThestartingpointisthehydrodynamictheoryofSmCliquidcrystalsproposedb. 
Carlsson,Leslie,Stewart,andClark[108,109].Itassumesaconstantsmecticlaye. 
thicknessandaconstantaveragetiltofthemolecules.Inordertodescribeth. 
structureofthevortices,however,atleasttheconstraintofconstanttilthastob. 
relaxed,i.e.,aslightgeneralizationofthetheoryisnecessary.Atthesametime. 
asubstantialsimplicationwillbemade,thatis,asystemwithvariationsonlyi. 
twodimensionsandwithstraightsmecticlayerswillbeassumed,eliminatingth. 
layernormaldegreeoffreedomcompletelyandxingitto.
^e
z
.Experimentally,Sm. 
thinlmsaremuchclosertothetwo-dimensionaltheoreticaldescriptionthanth. 
nematicsinChapter7,wherethedisclinationlinescancurveand
uctuate.Duet. 
thethinlmgeometry,twospatialvariablesxandy,withr=.
^e
x
@
x
+.
^e
y
@
y
,an. 
aplanar
ow,v=v
x
.
^e
x
+v
y
.
^e
y
,areassumed.Hence,wehavereducedtheSm. 
thinlmsystemtotheXY-model.Itcanbeshownthatundertheseassumptions. 
theconstant-tiltSmCtheory[108,109]reducestotheEricksen-Leslie(EL)theor. 
ofthenematicliquidcrystalexactly.Inaddition,themodulusofc,correspondin. 
tothesineofthetilt,willbeallowedtovary[100].Wehavethusarrivedatatwo. 
dimensionalversionofthevectororderparameterdynamics,consideredinChapte. 
5.Nothingchangesinthe2Dcase,exceptforthemissingtwistterminEq.(5.1). 
Thereis,however,aprincipaldierenceintheinversionsymmetryofthegenera. 
3Dvectorandc-directorsystems,stemmingfromthefactthatinrealitythe2. 
c-directorisembeddedinthe3Dspaceandthatn=..nstillholds:cisinvarian. 
toaglobalinversionc!..c.Asaresult,thefreeenergyandthedissipationo. 
theSmCsystemareinvarianttoseparateinversionsofthecoordinateandtheorde. 
parameter,i.e.,theyareinvarianttor!..r,c!c,aswellastor!r,c!..c,a. 
opposedtothe3Dsystem.Forexample,thedistortionsdepictedinFig.8.1areno. 
equivalentinthegeneralvectorcase,whereasinthecaseofthec-directordene. 
onstraightsmecticlayerstheyareidentical.Onejustneedsto
ipoverthesmecti 
layerstoseethis.Nevertheless,thisdistinctiondoesnotplayarolewiththefre. 



Pair-annihilationofvorticesinSmClm. 
8. 



(a. 
(b. 


Figure8.1Thedistortions(a)and(b)ofageneralvectoreldarenotequiv. 


alent,whereasintheSmCsystemmodelledtheyare. 


energyoftheform(5.1)andthedissipationoftheform(2.38),astheyarealread. 
invarianttotheseparateinversions. 


Inspiteofthethinlmgeometry,weignorethesurfacetermsinthefreeenerg. 
density.Moreover,inordertokeepthenumberofmaterialparametersaslowa. 
possible,wewillneglectagreatnumberoftheelastictermsinTable5.1.However. 
wedowanttoaccountforthesplay-bendelasticanisotropy,whichinSmCliqui. 
crystalscanbelargeduetospontaneouspolarizationeects[110{112].Thenth. 
freeenergydensitycanbeconvenientlyexpresseda. 


1
2
1
4
1. 


f=

Ac+

Cc+

B
1
(rc)
2
+

B
2
(r
c)
2
:(8.1. 


242. 



BycomparisonwiththeelastictermsinTable5.1,onecanshowthatuptoth. 
surfacetermstherelation. 


B
1
=L
1
;B
2
=L
1
+L
. 
(8.2. 


hold.Thus,disregardingthesurfaceterms,thetermswithL
4
-L
8
havebeencon. 
sistentlyomittedfromEq.(8.1).Theirrelevanceislimitedtoregions,whereth. 
modulusofcvaries,i.e.,tothedefectcores. 


Intheoriginalconstant-tiltdescription[108],theelasticpartinvolves9coeA. 
cients,whichforxedandstraightsmecticlayersreducetoabendandsplayter. 
only.Inourcase,allowingforavariationofthelengthofcthebendandspla. 
elasticconstantsB
1
andB
2
aretilt-independent,Eq.(8.2).Ifonewantedtous. 
adirectorofunitlengthinsteadofc,Eq.(8.1)wouldimplytheelasticconstant. 


~
c
2

todependonthetiltasB
i
/=sin
2
,whichisinaccordwiththesymmetr. 
considerationsin[108]. 


. 


TheEuler-LagrangeequationforthefreeenergyfunctionalF=dVf(c;rc. 
givesthehomogeneousandelasticpartofthegeneralizedforceactingonthevecto. 
c. 


2
@
2

h
i
=..(A+Cc)c
i
+B
1
j
c
i
+(B
2
..B
1
)@
i
@
j
c
j
:(8.3. 


Theelasticstresstensorisobtainedfrom(8.1)usingEq.(2.26). 


@. 



. 


ij
=..@
j
c
k
. 
(8.4. 
@(@
i
c
k
. 


Originally,thetheory[109]involves20viscousterms,ofwhichonlythestandar. 
Leslietermsareleftinthepresentlimitoflateral
ow,straightsmecticlayersandn. 
gradientsinthedirectionofthelayernormal.Accountinginadditionforthevariabl. 



8. 
Pair-annihilationofvorticesinSmClm. 


lengthofc,Eqs.(5.12)and(5.13)representtheproperdescriptionofthedissipativ. 
forcesinoursystem.Nevertheless,toresorttotheknownmaterialparameters,w. 
wewillreducethenumberofviscoustermstotheLeslietermsonly,omittingth. 
termswiththecoeAcients
6
and
9
inEqs.(5.12)and(5.13).Again,theomitte. 
termsaectonlythecoreregionofthedefect.Hence,theviscousstresstensori. 



. 
. 
i. 
=
0
A
ij
+
2
c
k
c
l
A
kl
c
i
c
j
+
3
(N
i
c
j
..c
i
N
j
). 


. 


111



4
(N
i
c
j
+c
i
N
j
)+(
1
..
4
)c
i
A
jk
c
k
+(
1
+
4
)A
ik
c
k
c
j
;(8.5. 
22. 


andthegeneralizedviscousforceonthevectorci. 


..h
. 


i
=
3
N
i
+
4
A
ij
c
j
;N
i
=c_
i
+W
ij
c
j
;(8.6. 


withthematerialtimederivative_=@c=@t+(v
r)candthesymmetricandanti. 


c
symmetricpartsofthevelocitygradientAandW,Eq.(2.34).AlthoughEqs.(8.5. 
and(8.6)areexactlythoseoftheEricksen-Leslietheory(Eqs.(4.11)and(4.6)). 
thereisafundamentaldistinctioninthatheretheviscousparametersdonotde. 
pendonthecondensedquantity|thetilt,whereastheoriginalLesliecoeAcient. 
do(Eq.(5.15)). 


Onemustemphasize,thatbyallowingthemodulusofctovaryinEqs.(8.1). 
(8.5),and(8.6),wemakeanaturalgeneralizationoftheELdescriptiontoth. 
caseofthenon-unitvectororderparameter.Herebyweautomaticallyrecoverals. 
thecorrecttiltdependenceoftheviscousforces,whichin[109]mustberegulate. 
bytilt-dependentcoeAcients,suggestedbysymmetryarguments.Fornematics. 
incontrast,onlythetensorialdescriptionwillprovidetheproperdependenceo. 
thematerialparametersonthedegreeoforder/biaxiality.Hence,itisexactlyth. 
modelledSmClmsystemratherthanthenematictowhichtheELtheoryapplie. 
rigorously(withintherestrictionsconsidered). 


Theequationofmotionforthevectorcreadsbrie
. 


h+h
v
=0. 
(8.7. 


TogetherwiththegeneralizedNavier-Stokesequation(2.41)andtheincompressibil. 
itycondition(2.42)itformsthesetofthreepartialdierentialequationsgovernin. 
thedynamicsoftheSmClmsystem. 


Theequationsarecastindimensionlessformbyintroducingacharacteristi 
length,i.e.,thetiltcorrelationlengt. 


. 


.
B
0

. 
. 
(8.8. 


2

(A+3Cc
0
. 


coupleofnanometersusually,andacharacteristictim. 



3

. 


=. 
(8.9. 


.
B
. 


whereB
0
=(B
1
+B
2
)=2.Thetimeisthecharacteristicrelaxationtimeofth. 
cdeformationsonthelengthscaleof,orequivalently,thedynamictimeofth. 



Pair-annihilationofvorticesinSmClm. 
8. 


modulusofc,typicallytensofnanoseconds.Inthefollowing,dimensionlessquan. 
titieswillbeused,i.e.r r=forlength,t t=fortimeandv v=forth. 
velocity.Doingso,thematerialparametersentertheequationsonlythroughratio. 
givenbelow. 


8.3Technicalitiesandmaterialparameter. 


ThenumericalmethodisthesameasinChapter7.Thecalculationsweredoneo. 
asquaremesh,consistingofanehomogeneousmeshof80x80pointsinthecente. 
containingbothdefects,andaninhomogeneousgridwithincreasingspacingaroun. 
ittoyieldthetotalof140x140points.Thevelocitywassettozeroattheboundary. 
Inordertosimulateabulksystem,thedefectseparationwassmallcomparedtoth. 
sizeofthecomputationalarea(theratioofthetwowas3/20)andthederivative. 
oftheorderparameternormaltotheboundaryweresettozero. 


Agenericsetofviscousparametersin(8.5)and(8.6)wasused,correspondin. 
totheLesliecoeAcientsofthenematicsubstanceMBBA[49,p.231],withth. 
relevantratios
4
=
3
..1:0,
2
=
3
0:085,
0
=
3
1:1,
1
=
3
0:19,TheLanda. 
coeAcientsA,C,andtheelasticconstantsin(8.1)wereintheratiosofA
2
=B
0
. 
..0:50,C
2
=B
0
2:0(yieldingequilibriumtiltvalueof30A),withthecorrelatio. 
length2:4nm.Thecharacteristictime88nscompletesthesetofmateria. 
parameters. 


Earlystagesoftheannihilationprocessexhibitadependenceontheinitialcon. 
guration(Fig.8.4).Startingwiththeequilibriumstructurecontainingtwoxe. 
defects,atransitionperiodexists,duringwhichtheequilibriumcongurationi. 
changingtoadynamicone[72,86],Chapter7.Itcanbealsoregardedastheinertia. 
eectduetotheeectivemass[100],asmentionedinChapter7.Astherelaxatio. 
timeoftheceldonthelengthscaleoftheinterdefectdistanceRisproportiona. 
toandsoistheannihilationtimeinthelimitofR=1,thetransitionperio. 


R
2

makesuproughlyaconstantfractionoftheannihilationtime,whichisunpleasan. 
asitwastesthecomputationalresources.Therefore,ascalingtechniquewasuse. 
toobtainamoresuitablestartingconguration,basedontheself-similarityofth. 
celd,attainedwhenfarawayfromthestartbutstillinthelimitR.Th. 
defectswerelefttoannihilatetohalftheinitialseparation,followedbyarescalingo. 
theceldtotheinitialdefectseparationandashortsimulationruntoequilibrat. 
thetilt.Thisstartingcongurationisconsideredasausefulapproximation|i. 
reality,thescalingregime,whereR/(t
0
wouldhold,isapproachedonlyat

..t)
1=2

verylargedistancesduetologarithmiccorrections[101]. 


8.4Resultsanddiscussio. 


Firstwefocusonthehydrodynamiceectsonthepair-annihilationintheoneelasti 
constantapproximation,B
1
=B
2
.Signicantasymmetryinthedefectmotionan. 
reductionoftheannihilationtimeascomparedtothenonhydrodynamictreatmen. 
isobserved(Fig.8.3),verysimilartothecaseof1=2defectsinnematics(Chapte. 



88 Pair-annihilation of vortices in SmC lms 
Figure 8.2 Left: radial-hyperbolic pair of annihilating 1 disclinations in c 
eld (the grid has been coarsened for clarity, only the central region of the 
mesh is shown). Vector heads have been omitted for legibility; yet there is no 
ambiguity, since in the absence of any external elds the system is invariant 
under a global transformation c ! ..c. Right: the central part of the corre- 
sponding 
ow eld. The +1 defect is subject to strong advection, as opposed 
to the ..1 one. 
7). It is possible to understand this easily by inspecting the 
ow-driving terms in the 
stress tensor (those containing _c): the elastic terms (8.4) and the 3 and 4 terms 
in the viscous stress (8.5). The main 
ow eect appears to be the advection, i.e., 
the hydrodynamic mass transport, which is to be discussed in the following. The 
viscous in
uence on the c vector comes second, though it is not negligible. 
If one performs a re
ection of the c vectors in the line joining the defects, the 
winding number of the defects is reversed, but the order parameter dynamics stays 
the same in the one elastic constant approximation. To recover the original conguration 
(up to an irrelevant global minus sign), a  rotation of the sample around 
an axis perpendicular to the lm through the middle point between the defects is 
required. Performing the re
ection on the stress tensor terms mentioned, one can 
verify that the 3 term is antisymmetric (provided that the 
ow is generated by this 
term only and that the 1 and 2 terms are neglected), the 4 term has no denite 
symmetry, while the elastic stress is symmetric by denition. This means that the 

ow generated by the elastic stress is symmetric with respect to the rotation about 
the perpendicular axis, while the one driven by the 3 viscous term is antisymmetric 
(Fig. 8.5). With other words, the elastic stress driven 
ow carries the defects symmetrically 
toward each other (Fig. 8.5(b)), as in this way the free energy is reduced. 
Thereby it contributes to the speedup of the process. On the other hand, the 
ow 
driven by the 3 term carries both defects with equal speeds and in the same direction, 
downward in Fig. 8.5(c). This is the main reason for the +1 defect moving

Pair-annihilationofvorticesinSmClm. 
8. 


12

8 

4 

y 0 

-4 

-8 

-12 

t

0 150 300 450 
-1 
(f) 
(e)
(d) 
(a) 
(b) 
+1 
(c) 
Figure8.3Positionofthedefectsvs.time,measuredfromtheinitialmid. 
dlepointbetweenthedefects.Thecaseswithoneelasticconstant:(a)R. 
(Fig.8.2)and(b)THdefectpair,(c)thecasewithouthydrodynamics,(d. 
RHpairwiththe

3
coeAcientdoubled.Combinedeectof
owandelasti 
anisotropy,withtheaverageelasticconstant
B
0
xed:(e)
B
1
=B
2
=5(RH). 
(f)
B
2
=B
1
=5(TH).Lengthandtimearemeasuredrelativeto

2
:
4n. 
and

88ns,respectively. 


0.6 

+1

0.5 
-1

 no flow

 equilibr. struct. initially

0.4 
v 0.3 
0.2 
0.1 
0.0 
r

Figure8.4Velocityofthedefects(relativeto
=
27nm/

s)vs.inte. 
defectdistance,oneelasticconstant.The+1defectisstronglyspedupb. 
the
ow,the..1isslightlysloweddown(therethe
owisoppositetoit. 
motion,Fig.8.2).Startingwiththeequilibriumcongurationofxeddefects. 
anonmonotonicbehaviorisobserved(dashed). 


4 8 12162024 



9. 
Pair-annihilationofvorticesinSmClm. 


faster.Whatismore,onthisbasisitcanbeunderstoodwhythe
ownearthe+. 
defectismuchstrongerascomparedwiththatnearthe..1defect(Fig.8.5(a)):i. 
therstcasethe
oweldsfromthetwosourcesareadded,whileinthesecondthe. 
combinedestructively.Finally,thevelocitymagnitudeofboth
owsrelativetoth. 
speedofdefectmotionjustduetoreorientationofcisproportionalto
3
=
0
. 


Theasymmetric
4
viscoustermcomplicatesthesituation(Fig.8.5(d)).Itisthi. 
termthatismainlyresponsibleforthedierent
oweectincaseofdierentdefec. 
pairs(e.g.radial-hyperbolic(RH)vs.tangential-hyperbolic(TH,ahomogeneou. 
=2rotationofcvectorsontheRHstructure),Fig.8.3(a),(b)),sincethe
3
an. 
theelastictermsareleftunchangedbythehomogeneousrotation,andsoisth. 
viscoustorqueonthecvector,givenbythe
3
partoftheviscousforce(8.6).Th. 
passiveviscousterms(
1
,
2
,
4
)neednotbediscussedinthequalitativepicture. 


Theasymmetryofdefectmotionisgivenrisetoalsobyelasticanisotropy.I. 
SmCchiralsystems,thiscanbelargeduetoc-vectordeformationinducedgradient. 
ofpolarizationPandthusappearanceofanelectriccharge,e=..r
P.Thepolar. 
izationvectorliesinthesmecticplane,usuallyperpendicularlytoc,thusincreasin. 
thebendelasticconstant[110].However,inverythinsystemssurfacepolarizatio. 
mightdominate[112],strengtheningtheresistancetothesplaydeformation[111]. 
Theannihilationprocessesatdierentratiosoftheelasticconstantsarepresentedi. 
Fig.8.6,combinedeectofthe
owandtheanisotropyisdemonstratedinFig.8.3. 
curves(e)and(f).Withoutthe
ow,invertingtheratioandcorrespondinglychang. 
ingthestructure,RH$TH,doesnotchangethedynamics,soitisenought. 
consideronetypeofelasticanisotropyonly. 


8.5Summar. 


UndertherestrictionsmappingtheSmCthinlmsystemtotheXY-model,w. 
havereducedtheSmCdynamictheory[108,109]totheELtheory.Wehavedemon. 
strated,thatthelatter,naturallygeneralizedtothevariablemodulusofthevecto. 
orderparameter,exactlydescribesthemodelsystemcontainingvortices.Numeri. 
cally,wehaveshownthatthein
uenceofhydrodynamicsdependsprimarilyonth. 
ratioofrotationalandtranslationalviscosity
3
=
0
,controllingthehydrodynami 
accelerationoftheprocessandthedefectspeedasymmetry,andontheratio
4
=
0
. 
breakinginvarianceuponcongurations,dieringonlybyahomogeneousrotatio. 
ofthevectorsc.Toalesserextent,themotionasymmetryiscontributedtoals. 
bytheelasticanisotropy.Ontheotherhand,rescalingtheelasticconstantswit. 
respecttotheviscositieshasnoeectonthedynamicsotherthanchangingth. 
characteristictime. 



Pair-annihilation of vortices in SmC lms 91 
(a) complete stress tensor, vmax = 0:01 (b) elastic terms, vmax = 0:0049 
(c) 3 term, vmax = 0:0043 (d) 4 term, vmax = 0:0019 
Figure 8.5 Flow elds resulting from dierent driving stress tensor terms: (a) 
the complete stress tensor, (b) elastic stress, (c) the 3 viscous term, and (d) 
the 4 viscous term. In all cases also the isotropic 0 viscous term is included. 
For clarity, the number of mesh points has been reduced by a factor of 2 in 
each dimension; only the central homogeneous region of the mesh is shown. In 
(a) the approximate positions of defects are marked with circles, the radius of 
the defect core is roughly two grid points. The maximum velocity magnitude 
vmax corresponding to the longest velocity vector is given for each 
ow eld 
(relative to = as dened in Eqs. (8.8) and (8.9)).

9. 
Pair-annihilationofvorticesinSmClm. 


-12 
-8 
-4 
0 
4 
8 
12 
(f) 
(e) 
(d)
(c)
(b)(a) 
y 
-1 
+1 
0 300 600 900

t 

Figure8.6Theeectofelasticanisotropy(withoutthehydrodynamics).Th. 
ratioofelasticconstantsis(a)1,(b)2,(c)4,(d)8,and(e)16;
B
0
iskep. 
constant.(f)Forcomparison,thehydrodynamiconeelasticconstantcasewit. 

3
doubledisshown. 



9 
Decay of integer disclinations in nematics 
In this Chapter, we study in some respects an opposite phenomenon to the annihilation 
| the decay of disclinations with integer strengths and the subsequent motion 
of the resulting disclinations. The study has been motivated by a numerically observed 
instability of the strength 1 disclination line, leading to the decomposition 
into a pair of repelling strength 1/2 disclinations. 
Like in Chapter 7, straight and innitely long disclination lines are assumed. 
First we focus on the early stage of the process, performing a stability analysis of 
the integer strength disclination. We want to check whether there exists any local 
stability. Should it not, it will be interesting to see what is the nature of the 
uctuations 
responsible for the instability, i.e., which components of the order parameter 
are coupled. With this intention we study the complete Q-tensor dynamics without 
the 
ow for small deviations from the initial structure. 
The 
uctuation problem of the strength 1 disclination line has been studied in the 
director description with and without variable degree of order by Ziherl and Zumer 
[113,114]. In the former case, a model radial prole for the scalar order parameter 
was assumed, but with an incorrect behavior near the origin. In the spherical geometry, 
the stability of a radial point defect (hedgehog) has been studied in [115] by 
constructing a specic perturbation without solving the eigenmode problem, and in 
[116] by assuming the Lyuksyutov's constraint [117] and a restriction to a subspace 
of perturbations. We solve a general linearized Q-tensor 
uctuation problem for a 
disclination line with any integer winding number. This enables us to nd the growing 

uctuations, which are responsible for the instability of the integer disclination. 
For the strength 1 disclination, we determine the correction to the growth rate of the 
critical 
uctuations due to the 
ow. In the nonlinear regime, we study numerically 
the in
uence of the 
ow on the repulsive motion of the 1=2 disclinations, created 
after the decay of the strength 1 disclination. 
9.1 Fluctuation problem 
In this Section, we study the dynamics of perturbations of a long and straight 
nematic disclination line with a general integer winding number. Cylindrical coordi- 
93

94 Decay of integer disclinations in nematics 
nates (r; ; z) with corresponding orthonormal base vectors (^er; ^e; ^ez) will be used. 
The disclination line coincides with the z axis. 
In the one elastic constant approximation (9.3), the free energy is invariant upon 
a homogeneous rotation of the Q-tensor. This implies that the Q-eigensystem will 
rotate as   =  0+(s..1) with respect to the above base vectors when we encircle a 
defect of strength s located at the origin;   is the angle between the director and ^er 
and  0 is the free parameter of the defect conguration, corresponding to the angle 
between the director at  = 0 and the x axis (e.g., for +1 defects  0 = 0 represents 
the radial defect, while the circular one has  0 = =2). There is no dependence on 
 other than this rotation, i.e., the scalar invariants of Q (the degree of order and 
biaxiality) are independent of . It is due to this generalized cylindrical symmetry 
of the unperturbed defect structure that the eigenmode problem is tractable. 
Let us dene another orthonormal triad (^e1; ^e2; ^ez), 
^e1 
^e2  =  cos   sin   
..sin   cos   ^er 
^e : (9.1) 
With this, in the unperturbed conguration (or ground state, as referred to below) 
the Q-tensor eigensystem coincides with the triad everywhere. Further, we dene 
the ve orthonormal symmetric traceless base tensors [118{120], Fig. 9.1, 
T0 = 1=p6(3^ez 
 ^ez .. I); 
T1 = 1=p2(^e1 
 ^e1 .. ^e2 
 ^e2); 
T
..1 = 1=p2(^e1 
 ^e2 + ^e2 
 ^e1); (9.2) 
T2 = 1=p2(^ez 
 ^e1 + ^e1 
 ^ez); 
T
..2 = 1=p2(^ez 
 ^e2 + ^e2 
 ^ez); 
with Tr(TiTj) = Aij . By virtue of the denition (9.1), the resulting eigenmode 
equations will be independent of  0, but will depend on the winding number s 
through spatial derivatives of the base tensors (9.2). 
In one elastic constant approximation, Eq. (7.1), the standard free energy density 
in terms of Q reads 
f = 1
2ATrQ2 + 1
3B TrQ3 + 1
4C (TrQ2)2 + 1
2LTr(rQ 
 rQ); (9.3) 
where in the last term the contraction over the gradient components is denoted by 
the dot. Expressing the Q-tensor as 
Q(r; t) = ai(r; t)Ti(r); i = ..2;..1; 0; 1; 2; (9.4) 
and inserting it into Eq. (9.3) while being careful with the gradient r = ^er@=@r + 
^e@=r@ of the base tensors, f is expressed in terms of the tensor components ai. 
The balance of generalized forces leads to the equations of motion for the components 
ai, in a dimensionless form: 
1 _ai = r 
 
@f 
@rai .. 
@f 
@ai 
= 
@ 
@r 
@f 
@ @ai 
@r 
+ 
1
r 
@f 
@ @ai 
@r 
+ 
@ 
r@ 
@f 
@ @ai 
r@ .. 
@f 
@ai 
: (9.5)

Decay of integer disclinations in nematics 95 
Figure 9.1 Schematic representation of the perturbations described by the 
base tensors (9.2) for a uniaxial distribution with a positive degree of order 
(dashed). The Q-tensor eigensystem is represented by the box, the length of 
the edges corresponds to the eigenvalue (plus a constant). The long axis of the 
box (usually called the director) is parallel to ez. T0 describes a perturbation 
of the degree of order, T1 describes a biaxial perturbation, T
..1, T2, T
..2 
represent rotations of the eigensystem. The interpretation of the perturbations 
varies according to which of the axes has been identied with the director. 
Irrespective of this, the perturbations given by T
..1, T2, and T
..2 possess 
Goldstone modes, while those given by T0 and T1 are massive. 
By symmetry, the ground state consists solely of the components a0 and a1, 
as opposed to perturbations, where all the components are allowed. According to 
Eq. (9.5), the ground state components, q0 = a0, q1 = a1, satisfy 
@2q0 
@r2 
+ 
1
r 
@q0 
@r .. 
A
L 
q0 .. 
1 
p6 
B
L 
(q2
0 .. q2
1) .. 
C
L
(q2
0 + q2
1)q0 = 0; (9.6) 
@2q1 
@r2 
+ 
1
r 
@q1 
@r .. 
4s2 
r2 
q1 .. 
A
L 
q1 + s2
3 
B
L 
q0q1 .. 
C
L
(q2
0 + q2
1)q1 = 0; (9.7) 
and in the vicinity of r = 0 behave as 
q0  c0 + c2 r2; q1  b rj2sj; (9.8) 
with c2 = c0(A + Bc0=p6 + Cc2
0)=4 and c0, b extracted from the numerical solution 
if needed. Putting 
ai(r; t) = ( qi(r) + xi(r; t) ; i = 0; 1 
xi(r; t) ; i = ..1; 2;..2 
; (9.9) 
where q0;1 are the ground state components and xi are the perturbations, xi  
q0;1, and linearizing the equations (9.5), one obtains two groups of coupled linear 
equations for the perturbations xi: 
x_ 0 = r2x0 .. f0(r) x0 + f01(r) x1; (9.10) 
x_ 1 = r2x1 .. 
4s2 
r2 
x1 .. 
4s 
r2 
@x..1 
@ .. f1(r) x1 + f01(r) x0; (9.11) 
x_..1 = r2x..1 .. 
4s2 
r2 
x..1 + 
4s 
r2 
@x1 
@ .. f..1(r) x..1; (9.12)

96 Decay of integer disclinations in nematics 
and 
x_ 2 = r2x2 .. 
s2 
r2 
x2 .. 2s
@x..2 
@ .. f2(r) x2; (9.13) 
x_..2 = r2x..2 .. 
s2 
r2 
x..2 + 2s
@x2 
@ .. f..2(r) x..2; (9.14) 
where r2x = @2x 
@r2 + 1
r 
@x 
@r + @2x 
r2@2 is the Laplacian in cylindrical coordinates and 
f0(r) = A + q2=3Bq0 + C(3q2
0 + q2
1); 
f1(r) = A .. q2=3Bq0 + C(q2
0 + 3q2
1); 
f..1(r) = A .. q2=3Bq0 + C(q2
0 + q2
1); (9.15) 
f01(r) = (q2=3B .. 2Cq0)q1; 
f2(r) = A + B=p6(q0  
p3q1) + C(q2
0 + q2
1): 
In Eqs. (9.10)-(9.15), dimensionless quantities have been introduced: r   r=, 
t   t= , (A;B;C)   (A;B;C)2=L, with the correlation length of the degree of 
order (7.18) and the characteristic time (7.19). It is worth pointing out that there 
is no dierence between defects with strengths s and ..s except for two minus signs, 
which can be absorbed in the base tensors, i.e., s ! ..s and T
..1;..2 ! ..T
..1;..2 
conserves the equations (9.10)-(9.15). 
9.1.1 Fluctuation eigenmodes 
The eigensolutions of the systems (9.10)-(9.12) and (9.13)-(9.14) are sought by separation 
of variables using the ansatze (we write cos(m) instead of C1 cos(m) + 
C2 sin(m), m is an integer) 
8> 
<> 
: 
x0 
x1 
x..1 
9> 
=> 
; 
= 8> 
<> : 
R0(r) cos(m) 
R1(r) cos(m) 
R..1(r) sin(m)
9> 
=> 
;
exp(..t); (9.16) 
 x2 
x..2  =  R2(r) cos(m) 
R..2(r) sin(m) exp(..t): (9.17) 
Proper combinations of the angular functions have been chosen in (9.16) and (9.17) 
to satisfy the equations. One could also include a factor cos(kz) and add the dependence 
on z to Eqs. (9.10)-(9.14), which is only through @2=@z2 in the Laplacian, so 
that the z coordinate is easily separated. Then the eigenvalue  (the inverse time 
constant) would be 
 = r + k2; (9.18) 
where r is the eigenvalue of the radial and angular part. This time we are not 
interested in the z dependence and have omitted it from the equations for brevity.

Decay of integer disclinations in nematic. 
9. 


Ineithercase,onlyeigenvaluesystemsfortheradialfunctionsR
i
(r)remain,wher. 

r
istheeigenvalue(wewriteinsteadof
r
). 


 . 
r
2
R
0
+..f
0
(r). 
m
. 
r
. 
R
0
+f
01
(r)R
1
=0. 
(9.19. 
. 
. 
r
2
R
1
+..f
1
(r). 
m
2
+4s
. 
r
. 
R
1
. 
4s. 
r
. 
R
..1
+f
01
(r)R
0
=0;(9.20. 
. 
. 
r
2
R
..1
+..f
..1
(r). 
m
2
+4s
. 
r
. 
R
..1
. 
4s. 
r
. 
R
1
=0;(9.21. 
an. 
. 
. 
r
2
R
2
+..f
2
(r). 
m
2
+s
. 
r
. 
R
2
. 
2s. 
r
. 
R
..2
=0;(9.22. 
. 
. 
r
2
R
..2
+..f
..2
(r). 
m
2
+s
. 
r
. 
R
..2
. 
2s. 
r
. 
R
2
=0:(9.23. 


Onenoticesthatthethree-andtwo-functionoperators(9.10)-(9.12)and(9.13). 
(9.14)areself-adjoint,implyingrealeigenvaluesandorthogonaleigenmodes.Thes. 
mustbesolvedfornumerically,eitherbydiscretization,Bessel-functionexpansion. 
relaxation,orshooting.Althoughforlinearsystemsthersttwomethodswoul. 
usuallybepreferred[121],wedecidetousetheshootingprocedure[54,p.582]a. 
itissimpleandquiteeAcientwhenonlyafeweigenfunctionsaresearchedfor.I. 
whatfollows,wewillfocusourattentiontopossiblegrowingmodes,i.e.,thosefo. 
which<0. 


Duetothesingularityofthecylindricalcoordinates,thebehavioroftheradia. 
eigenfunctionsneartheoriginmustbedeterminedanalyticallypriortonumericall. 
solvingtheeigensystems(9.19)-(9.23).Onemakesuseofthegroundstateexpan. 
sion(9.8),whichentersthefunctionsf
i
.TheunknowncoeAcientsoftheradia. 
eigenfunctionexpansionhavetobedeterminedtogetherwiththeeigenvalue.Fo. 
numericalreasons,werestricttheeigenmodestovanishatanarbitrary,butnotto. 
largeavalueofr=r
0
.Themodesconcernedarelocalizedandhenceremainun. 
aectedbytherestriction,ifonlyr
0
islargeenoughcomparedtothecharacteristi 
decaylengthofthemode.Fornonlocalizedmodestherestrictioncorrespondst. 
aphysicalcylindricconnementofthedefectwithstronganchoring;inthiscase. 
however,nonumericaldiAcultiespreventr
0
fromlargervalues.Thus,startingwit. 
theanalyticexpansionoftheradialfunctionsandatrialvalueof,theequationsar. 
integratedtor
0
byaRunge-Kuttamethodwithanadaptivestepsize[54],wherei. 
isrequiredthatR
i
(r
0
)=0.Withnradialfunctionscoupled,therearenunknow. 
coeAcientsofexpansion,oneofwhichisarbitrary.Togetherwiththisgives. 
freeparameters,whichintheshootingprocedurearedeterminedbythenendin. 
conditions. 


9.1.2Eigenmodesleadingtodeca. 


Themodesresponsibleforthedecaydonotinvolvethecomponentsx
2
orx
..2
,sinc. 
bysymmetrytheQ-tensoreigensystemdoesnotgetrotatedoutofthexyplanei. 



9. 
Decayofintegerdisclinationsinnematic. 


0 5 10 15 
-1.0 
-0.5 
0.0 
0.5 
1.0 
r 
R0 
R1 
R-1 
Figure9.2Radialeigenfunctionsofthegrowing
uctuation,
s
=1,
m
=2. 



..0
:
22.Thelengthunitis

=2
:
11nm,thetimeunitis

=32
:
6ns.Th. 


fastestgrowingmodeleadingtotheescapeofthedefecthas

..0
:
0042. 


thisprocess.Therefore,thesystem(9.10)-(9.12)mustbeexamined.Thelowest. 
orderexpansionofthesystem(9.19)-(9.21)aroundtheorigindoesnotinvolveth. 
groundstatecoeAcients(9.8),butrequire. 


. 


r
jm..2sj

a
1

r
m

R
0
;R
1
. 
. 
(9.24. 


r
jm+2sj

a
2

wherethetwosolutionsforR
1
andR
..1
areindependentandthecoeAcientsa
1
,a
. 
mustbedeterminedtogetherwiththeeigenvalue. 


Westudyindetailthesimplestcase,i.e.,thedecayofthe1defect.The. 
wemakeageneralizationtodefectsofhigherintegerstrengths.Inthecaseofth. 
decayofthe1defecttotwo1=2defects,themodesinquestionmustexhibit. 
quadrupolarsymmetry,whichsetsm=2intheangularpartofEq.(9.16).Asingl. 
growingmode(..0:22)isfound(Figs.9.2and9.3),whichislocalizedwithin. 
fewcorrelationlengths,whilealltheothers(includingthosewithdierentm)ar. 
decayingandnonlocalized.Duetothelocalization,thegrowingmodecannotb. 
aectedbyanyconnementunlessitcomesdowntothescale|itisanintrinsi 
featureofthedefectstructure.Itisnosoonerthanataconnementofr
0
3:5. 
thatthemodebecomesdecaying. 


Moreprecise,andapplyingtoanyintegerstrength,itisthefunctionR
..1
tha. 
islocalizedifandonlyiftheeigenmodeisofthegrowingtype,whichcanbesee. 
fromitsasymptoticbehavio. 


. 


r
..1=2..r

e


R
..1
. 
(9.25. 



Decayofintegerdisclinationsinnematic. 
9. 



Figure9.3Crosssectionthroughthedisclinationlinewith
s
=1:thegroun. 
state(gray)isperturbedbythegrowingmode(red,shownexaggerated),lead. 
ingtotwo1/2disclinationsonthe
x
axis. 



10. 
Decayofintegerdisclinationsinnematic. 


ThefunctionsR
0
andR
1
arefoundtobelocalizedalsoforthedecayingmodesi. 
theregionoflowenough.Signicantly,themodeswith<0haveadiscret. 
spectrum,whereasthespectrumofthosewith>0iscontinuous.Thus,th. 
growingmodescanbecounted. 


Onecantestthetimeevolutionofthegrowingmodeinapurenumericsim. 
ulation.Regardlessoftheinitialperturbation,i.e.,ifonlyitcontainsanonzer. 
projectionontothegrowingmode,anexponentialgrowthofanyquantitylinearl. 
dependingonthemodeamplitudeisobservedafterashorttransient(decayofothe. 
modes),withatimeconstantverycloseto. 


Itisinstructivetostudythein
uenceofthehydrodynamic
owgeneratedb. 
theorderparameterdynamicsonthegrowthrateofthemode.Thisisperforme. 
numerically,wherethecouplingofthe
owandQ-tensoreldsisdescribedbyth. 
tensorialversionoftheEricksen-Leslietheory[92],Chapter7.Theoneelasti 
constantapproximationisused,thenumericalmethodandthematerialparameter. 
arethesameasinChapter7.Itisfoundthatthehydrodynamiccorrectiontoth. 
growthrateissmall,i.e.,lessthan5%,speedingupthemodes.Thecorrectio. 
isexpectedtobesmall,sincethevelocityofthe
owgenerateddecreaseswit. 
decreasingQ(thatis,decreasingTrQ
2
),asdoesitsin
uenceonQ(Eqs.(7.6)an. 
(7.7)).Atthesametime,oneshouldbequitereserved,sincethedescriptiono. 
the
ow-to-Q-tensorcoupling[92]isnotcomplete[122],andthemissingterms[123. 
couldplayanimportantroleinthedynamicsofthedefectcore.Besides,on. 
mustalsorealizethattheapplicabilityofhydrodynamicequationsisquestionabl. 
atlengthandtimescalesthatsmall(1nm,10ns). 


Inthecaseofdefectswithhigherstrengthsthereisanincreasingnumbero. 
growingmodes,astherearemoreandmorewaysthedefectcandecay.Itturn. 
outthatforeverydecompositionallowedtopologically,onecanndatleaston. 
growingmode,providedthatnoneoftheresultingwindingnumbersistoohigh. 
Eachofthesemodesexhibitsadistinctiveangularsymmetry,setbyitsvalueofm. 
Generally,adefectofstrengthsdecaystomsymmetricallyplaced1=2defect. 
surroundingasm=2defect,whichremainsinthecenter(Fig.9.4).Forexample. 
apossibledecaychannelofadefectwithstrength2is:2!..1+61=2,whereth. 
..1defectstaysinthemiddle,surroundedbythe1=2defects.Thecorrespondin. 
modehasasixfoldsymmetry,m=6.Ontheotherhand,the+2defectisstabl. 
withrespecttothedecay2!..3=2+71=2andhigher.Allpossibledecaypath. 
ofdefectswithstrengths2and3aregiveninTables9.1and9.2.Thedecayto1=. 
defectsonlyisalwaysthefastest. 


Inprinciple,thereisnolimitationtothewindingnumbers.Thereare,however. 
twotechnicalpoints.Firstly,thecoresizeofthedefectincreasesproportionall. 
toitsstrengths,whichgetstimeandspaceconsumingwithprogressingwindin. 
numbers.Secondly,moretermsshouldbeaddedtotheexpansion(9.24)toreac. 
betteraccuracy,requiredparticularlyforthelocalizedmodeswithhighervalueso. 
m.Theseriesismorecomplicatedinthiscase,asitcontainsboththegroundstat. 
coeAcients(9.8)andtheeigenvalue.Itisbeyondouraimtopursueaccurac. 
issueshere. 


Instead,onemustemphasizeanimportantpointregardingthenumericsimu. 



Decay of integer disclinations in nematics 101 
Figure 9.4 A ninefold (m = 9) decay of the s = 3 defect. On the lower 
diagram we can see a -3/2 defect in the center and nine 1/2 defects around it.

10. 
Decayofintegerdisclinationsinnematic. 



decay

m

... 



1!21=2

2

0.2. 



2!1+21=2

2

0.06. 


2!1=2+31=2

3

0.2. 


2!41=2

4

0.3. 


2!..1=2+51=2

5

0.2. 


2!..1+61=2

6

0.04. 


Table9.1Fastestdecaymodeswithagiven
m
ofthe
s
=1
;
2disclinatio. 
line. 



decay

m

... 



3!2+21=2

2

0.04. 


3!3=2+31=2

3

0.1. 


3!1+41=2

4

0.2. 


3!1=2+51=2

5

0.3. 


3!61=2

6

0.4. 


3!..1=2+71=2

7

0.3. 


3!..1+81=2

8

0.2. 


3!..3=2+91=2

9

0.1. 


3!..2+101=2

10

0.01. 


Table9.2Fastestdecaymodeswithagiven
m
ofthe
s
=3disclinationlin. 


lationofdefectswithwindingnumbershigherthan1.Thesquaregridstandardl. 
usedinsimulationsreducesthesymmetryofthespacetoafourfoldsymmetryonly. 
i.e.,C
. 
!C
4
.IfagrowingmodeisinvariantunderC
4
(m=4;8;12;:::),itwil. 
experienceanarticialboostfromthegrid,suchthatitsgrowthratewillnotvanis. 
withvanishingamplitude!Now,assoonasthespectrumcontainsasinglesuc. 
mode,itwilloverwhelmotherpossiblyfastermodes,leadingtoanarticialfourfol. 
decayofthedefect.AnexampleofsuchdecayisshowninFig.s9.5and9.6.T. 
envisagetheoriginoftheboostitisappropriatetoaskwhythereisnotanywit. 
anonfourfold-symmetricmode.Inthiscase,therotationsofC
4
generateatleas. 
twodierentmodes,asopposedtotheidentityrepresentationinthepreviouscase. 
Thesearedegeneratedwithrespecttotheirplacementonthegrid,sincethegridi. 
invarianttoC
4
.Therefore,inapuregroundstateneithermustbefavoredandhenc. 
theirgrowthratemustvanishatzeroamplitude.Sincealldefectstructuresexcep. 
the1onepossessfourfold-symmetricgrowingmodes,thenumericalsimulationo. 
themodedynamicsisonlypossibleforthe1defect. 


9.1.3Eigenmodesleadingtoescap. 


Inanunconnedsystem,planardefectsofintegerstrengthscanescapetotheun. 
deformedcongurationwithazerodeformationfreeenergy(escapeinthethir. 
dimension)[68,69].Theissuesconcerningthe(meta)stabilityofdefectcoreshav. 



Decay of integer disclinations in nematics 103 
0 5 10 15 20 
-1.0 
-0.5 
0.0 
0.5 
1.0 
r 
R0 
R1 
R-1 
(a) 
0 5 10 15 20 
-1.0 
-0.5 
0.0 
0.5 
1.0 
r 
R0 
R1 
R-1 
(b) 
Figure 9.5 Radial eigenfunctions of the growing modes with m = 4 for the 
s = 2 defect: (a)   ..0:39, (b)   ..0:035 
Figure 9.6 Decay of the s = 2 defect to four 1/2 defects, m = 4; the 
corresponding radial eigenfunctions are depicted in Fig. 9.5. The four- 
fold-symmetric (m = 4) modes are numerically boosted due to the fourfold 
(m = 4) symmetry of the computational grid.

10. 
Decayofintegerdisclinationsinnematic. 


beenaddressedbyR.Meyer[69]in1973.Atthattimethetensorialdefectstructur. 
hadnotbeenpresentedyet,soadirectanswercouldnothavebeengiven.Equippe. 
withthepresentformalism,oneshouldlookforanothertypeofpossiblygrowin. 
modesleadingtotheescape.AsheretheQ-tensoreigensystemisrotatedouto. 
thexyplane,thesystem(9.13)-(9.14)mustbeexaminedthistime.Inparticular. 
oneexpectstheperturbationx
2
tobecrucial,asitcorrespondstoarotationofth. 
directoroutoftheplane.Thelowest-orderexpansionofthesystem(9.22)-(9.23. 
aroundtheorigini. 


. 


r
jm..sj

a
1

R
2
. 
. 
(9.26. 


r
jm+sj

a
2

theratioofa
1
anda
2
isdeterminedtogetherwiththeeigenvalueintheshootin. 
procedureasbefore.Form=0theequations(9.22)and(9.23)aredecoupled.Iti. 
indeedinthiscasethatonends<0andthediscretespectrumforallthemode. 
x
2
.Themodesx
..2
aredecaying,asarealltheothermodeswithm=60. 


Notingthatf
2
(r)!0andf
..2
(r)!f
..2
(1)>0,thegenera. 
asymptoticbehaviori. 


. 
. 
r
..1=2
e
. 
r
..1=2
e
. 
f
..
2(1).


..rr

R
2
;R
..2
;(9.27. 


i.e.,R
2
islocalizedfor<0.Itturnsoutthatherethelocalizedmodesaremuc. 
moreextensivethanthoseofthedecay. 


Onemaythinkthereislittlepointinstudyingthedecayifthedefectsarealway. 
unstabletotheescape.Thereexist,however,alargedierenceingrowthrateso. 
thetwotypesofunstablemodes,connectedwiththelargedierenceinlocalization. 
Inthecaseofthestrength1defectthedecayisapproximately53-timesfastertha. 
theescape.Whence,providedwepreparedthecongurationwiththestrength1. 
itwouldalwaysdecaybeforeitcouldevenstartescaping.Ofcourse,itisjustdu. 
tothefastdecaythattheinitialcongurationisveryhardtoprepare. 


9.1.4Remarksonthe
uctuationproble. 


.
Q
Theeigenmodeproblemhasbeensolvedintheoneelasticconstantapproximation. 
Beyondthisapproximation,onewouldhavetoincludemoreelastictermsinthefre. 
energydensity(9.3).ThereisonlyoneadditionalbulktermquadraticinQan. 
intherstderivative:(@
i
Q
ik
)(@
j
Q
jk
).Itdistinguishesbetweenthesplayandben. 
distortions(relevantforthedecay)onlyifthescalarinvariantsofQvary,whichdoe. 
takeplaceinourcase.Todistinguishbetweenthesplayandbendintheuniaxia. 
limitwithconstantdegreeoforder,however,onehastoincludethethird-orderter. 
ij
(@
i
Q
kl
)(@
j
Q
kl
).Inbothcasesthegeneralizedcylindricalsymmetryofthegroun. 
stateislost,makingtheeigenmodeproblemtwo-dimensional,i.e.,thevariables. 
andcannotbeseparatedanylonger.Itisonlyinthecaseofthe+1disclinatio. 
with 
0
=0or 
0
==2(radialandcirculardisclinations)thatthecylindrica. 
symmetryisretained,sothatwithoutthethird-ordertermtheseparationwoul. 
stillbepossible.Thethird-orderterm,however,bringsaboutmixedderivatives. 
whichinevitablypreventtheseparation. 



Decayofintegerdisclinationsinnematic. 
10. 


40 
20 
y 0 
-20 
-40 

(c) 
(b)
(a) 
0 500 1000 1500 2000 2500


t 

Figure9.7Positionoftherepellingdefectsasafunctionoftime,afterth. 


strength(a)1and(b)..1defectshasdecayed:(a)1
=
2defects,(b)..1
=
. 


defects,and(c)thedegeneratecasewithoutthe
ow.Recallthatthelengt. 


unitis

=2
:
11nmandthetimeunitis

=32
:
6ns. 


Asmentioned,thedependenceoftheeigenmodesonthecoordinatezissimpl. 
cos(kz)orsin(kz),withkenteringEq.(9.18)togivetheeigenvalue.Furthermore. 
itisalsopossibletostudythedisclinationswiththehalf-integerwindingnumbers. 
inparticularthe1/2disclinationline.Tosatisfythecontinuityoftheeigenso. 
lutionsinthiscase,intheangularpartoftheansatz(9.17)itisrequiredtha. 
m=1=2;3=2;5=2;:::,whiletheansatz(9.16)remainsthesame.Thus,withth. 
approachtakeninthisChapter,oneisabletosolvethefull
uctuationproblemo. 
astraightandinnitelylong1/2disclinationline. 


9.2Hydrodynamicspeedu. 


Nowletusstudytherepulsivemotionofthetwo1=2disclinations,createdafte. 
thestrength1disclinationhasdecayed.Weshallfocusonthein
uenceofth. 
hydrodynamic
ow.Duetothesymmetryoftheproblem,thereisno
ow-induce. 
asymmetry(Fig9.7)liketheoneencounteredinChapter7.However,thespeedu. 
causedbythe
owislargerthistime,becauseboththeelasticstress(7.5)an. 
theviscousstressgivenbythe
1
term(Eq.(7.6))drivethedefectsapart(seeth. 
discussionofChapter7).Theeectdependsontheratio
1
=
4
,Fig.9.8. 


Thereisalsonosubtletywiththeinitialcondition(thenalstateinthiscase. 
likethatinChapter7.Thisenablesustostudythebehaviorofthedefectvelocit. 
atlargerseparations.Figure9.9(a)showsthevelocityofthedefectvasafunctio. 
ofthedefectseparationr.Afterthedefectshavebeenwellisolatedfromeachother. 



10. 
Decayofintegerdisclinationsinnematic. 


8060 
r 40 
20 
0 

1 
2 
4 
0.5 
0 100 200 300 400


t 

(a. 


0.20

0.15 
v 0.10 
0.05 

0.00 

1 
2 
4 
0.5 
0 20 40 60 80


r 

(b. 


Figure9.8(a)Defectseparation
r
asafunctionoftimeand(b)defectve. 
locity
v
asafunctionoftheseparationfordierentratios

1
=
4
.Duetoth. 
questionablehydrodynamicsituationandill-deneddefectpositionatsmal. 
separations,in(b)thecurvefortheratio0.5isabovethatfortheratio. 
initially.Thekinkofthecurvesin(b)isanartifactoftheinhomogeneou. 
grid. 



0.4 
with flow3.0 with flow 
without flow without flow 
0.3 
2.0 
v 0.2 v r 
1.0 
0.1 
0.0 0.0 
0 20 40 60 80 0 20 40 60 80 
rr 
(a. 
(b. 
Figure9.9(a)Defectvelocity
v
and(b)thequantity
vr
asfunctionsofth. 
separation
r
.Thebehavioratsmall
r
isnotparticularlyinformativeasth. 
defectpositionisill-denedthere. 



Decayofintegerdisclinationsinnematic. 
10. 


thevelocitydecreaseswithr.Beforethat,thevelocitycannotbeproperlydened. 
Figure9.9(b)showsther-dependenceofthequantityvr,whichatleastforth. 
nonhydrodynamiccaseisconstantinthelimitr=!1(scaleinvariance). 


Figure9.10isparticularlysignicant.Itdisplaystheratiooftheadvectiv. 
velocity(transportbythe
ow)andthetotalvelocityofthedefect.Onecanse. 
thattheratioincreaseswithr,andreachesavaluewellabove0:5ashintedb. 
arougheye-madeextrapolationtothedatapoints.Thismeansthatatlarge. 
separationsthecontributionofthetranslationalmotion(advection)tothedefec. 
speedismoreimportantthanthatofthedirectorreorientation,especiallyifon. 
recallsthattheseparationsreachedinourcalculationsarestillquitesmall|onl. 
80or0.17m. 



10. 
Decayofintegerdisclinationsinnematic. 


0.50 

v 

adv 

v 0.25 

tot 

0.00 

0 20 40 60 80


r 

Figure9.10Ratiooftheadvectiveandtotalvelocitiesofthedefectas. 
functionoftheinterdefectseparation.Byaroughextrapolation,therati. 
iswellabove0
:
5atlargeseparations,indicatingtheimportanceofthe
o. 
transport. 



10 
Conclusion 
It is now time to summarize what has been presented in the Thesis, reviewing 
the concepts and results and putting them into context of the research going on 
in the eld. The aim of the Thesis has been to study selected problems coupling 
hydrodynamics and order parameter dynamics in liquid crystals. They were chosen 
according to theoretical and experimental interest, and, of course, our capability of 
solving them. 
The back
ow switching problems of Chapter 4 have been studied in terms of the 
well-established Ericksen-Leslie theory for the nematic director. Primarily they were 
intended to serve as an introduction to the research area, setting up the numerical 
approach to be used in the work to follow. Conceptually, they do not represent 
any novelty. Regarding their computational complexity, however, they are quite 
advanced compared to the one-dimensional examples that had been studied previously. 
More conned geometries increase the number of spatial variables, but at 
the same time oer more possibility for manipulation with external elds. Having 
managed to cope with the increasing complexity, we were able to point out special 
cases, where the hydrodynamic 
ow causes the perturbation enough to completely 
alter the time evolution of the system. Eects of this kind had only been studied 
in context of spinodal instabilities of liquid crystals, mostly in the linearized form, 
and in various pattern-forming systems involving electro- and thermal convection. 
The derivation of the dynamic theory for the complete vector order parameter 
presented in Chapter 5 features two viewpoints of importance. The vectorial theory 
is required if one wants to study defects in a medium with the vector order parameter 
| the SmC system in our case. On the other hand, by demonstrating that the 
Ericksen-Leslie theory can be naturally extended to yield the vectorial theory, or, 
in other words, the latter can be consistently reduced to the former, we have shown 
that the Ericksen-Leslie theory is exactly the reduced version of the vectorial theory, 
restricted to the unit vector. Despite the Ericksen-Leslie theory has been derived 
and used for nematics, it does not have any connection with the nematic tensor 
order parameter. 
To study the dynamics of defects in nematics, which has been our primary objective, 
one has to start over and construct the tensorial theory. It can be reduced 
to the Ericksen-Leslie theory in the limit of constant scalar invariants of the ten- 
109

11. 
Conclusio. 


sororderparameter.Theconversepathisnotpossible,i.e.,tryingtoupgradeth. 
Ericksen-Leslietheory,whichislinearinthedirector,byaddingavariationofth. 
scalarorderparameterresultsinahybridwithseveredrawbacks.InChapter7,w. 
havesolvedthehydrodynamicpair-annihilationproblemofstraightnematicdiscli. 
nationlinesusinganabridgedformofthetensorialtheory.Wewereabletoaccoun. 
forthespeculatedvelocitydierenceofthetwodisclinations,whicharisesmainl. 
duetothe
oweects.Unfortunately,coupleofmonthsbeforeourstheworko. 
anothergroupwaspublished[96],demonstratingessentiallyidentical
oweects. 
Thefactthattheirtreatmentisnotbasedonthecustomaryphenomenologicalde. 
scriptionofnematicsbutrestsonasomewhatleanermicroscopicmodelwithfewe. 
materialparametershasbeenlessimportantsofar.Thesituationmightchangei. 
thefuture. 


DespiteexperimentalconvenienceoftheSmClmsystem,therehavebeenn. 
numericalstudiesofdefectdynamicsreportedintheliterature.Apossiblereasonfo. 
thedeciencycouldbetheenormouscomplexityofthegoverningequationsandth. 
greatnumberofmostlyunknownmaterialparameters.Inaddition,thegeneraliza. 
tionoftheoriginalequationstothevariablelengthofthesmecticc-directorneede. 
forthedescriptionofdefectswouldbeabackbreakingtask.Instead,inChapter. 
wehavebenetedfromtheobservationthatundertherestrictingassumptionsth. 
equationscanbereducedtotheEricksen-Leslieequationsexactly.Atthislevel. 
thepassagetothevariablelengthofthec-directorisalsoquitecomfortable.Th. 
modelledSmClmsystemcorrespondstotheclassoftheXY-model.Itsdynam. 
icsisgovernedbythedynamicequationsforthevectororderparameterderivedi. 
Chapter5.Wehaveshownthatthe
oweectaccompanyingthepair-annihilatio. 
ofvorticesisqualitativelyequaltotheoneinnematics,i.e.,thedisclinationwit. 
thepositivewindingnumberisfaster. 


Vorticesofstrength1areunstableinthenematiccase,i.e.,theyspontaneousl. 
decaytoapairofidentical1=2disclinations,whichwerstobservednumerically. 
InChapter9,thismotivatedustostudythedynamicbehaviorofaperturbe. 
strength1disclinationline.Inoneelasticconstantapproximationwewereabl. 
tosolvethecompletetensor
uctuationproblemforthestraightdisclinationlin. 
withageneralintegerwindingnumber.Wefoundtwotypesofgrowing
uctuatio. 
eigenmodes,leadingtothedecaytodisclinationswithlowerwindingnumberso. 
totheescapeinthethirddimension,respectively.Theyarelocalizedandexhibi. 
discretespectra.Inaccordwiththestrongerlocalization,thecharacteristictimeo. 
themodesleadingtothedecayismorethananorderofmagnitudesmallerthantha. 
oftheescapingtype,suggestingthatthedecayingscenarioiswhattakesplace,e.g.. 
afteratemperaturequench,ratherthantheescapingone.Wehavealsostudie. 
thehydrodynamiccorrectiontothegrowthrateofthe
uctuationleadingtoth. 
decayofthestrength1disclination,andfoundittobeunder5%,increasingth. 
growthrate.Thevalidityofthisresultisquestionableduetotheincompletenes. 
ofthetensorialapproachandshort(nonhydrodynamic)lengthandtimescales.I. 
Chapter9wealsostudiedthein
uenceofthe
owontherepulsivemotionofth. 
two1=2or..1=2disclinations,createdafterthestrength1or..1disclinationha. 
decayed.Astrongspeed-uphasbeenobserved. 



Conclusio. 
11. 


IfthereisonethingoneshouldlearnfromtheThesis,itistheimportanceo. 
thehydrodynamic
owaccompanyingthedynamicsofdefectsinliquidcrystals.W. 
haveshownthatthecontributionofthe
owtransport(advection)tothemotio. 
ofthedefectisquitecomparabletothecontributionduetotheorderparamete. 
reorientation.Forthespecicchoiceofviscousmaterialparameters,correspondin. 
tothenematicsubstanceMBBA,theadvectivecontributionevendominatesa. 
relevantdefectseparations. 


10.1Futureperspective. 


Thereisstillalotofworktodo,eitherimprovingthemethodsused,orchallengin. 
newproblemsrelatedtothosethathavebeensolved. 


Inmyopinion,byfarthemostimportantimprovementtobedoneistoreplac. 
thesomewhatcumbersomemethodofsolvingtheNavier-Stokesequationwith. 
moreeAcientmethod,e.g.,aFourier-basedsolver,orafunctionexpansionmethod. 
ThemaindiAcultyindefectsimulationsisthelargediscrepancybetweenthesiz. 
ofthedefectcoreandtheinterdefectdistancerelevantforexperiments,whichar. 
readilyinaratioof1=10
5
.ThereplacementoftheNavier-Stokessolverisabsolutel. 
inevitableifonewantstoreachdefectseparationsoftheorderof100m,sotha. 
numericalresultscouldbedirectlycomparedwiththemeasurements. 


Oneoftheproblemstobeattackedinthefutureisthedynamicsofnematicpoin. 
defects,e.g.,theannihilationofahedgehog-antihedgehogpair,eitherinacapillar. 
wheremostofthemeasurementshavebeendone,orinbulk.Thisisa3Dproblem. 
Onecouldgetridoftheextraspatialcoordinatebyassumingcylindricalsymmetr. 
abouttheaxisjoiningthedefects.However,numericalindicesexistthattheactua. 
congurationdoesnotpossessthissymmetry. 


Anotheropenproblem,whichhasbeenalreadysolvedtosomeextentinChapte. 
9,arethe
uctuationsofthestrength1=2nematicdisclinationline.Assuggeste. 
bytheequations,apartofthe
uctuationspectrumcorrespondstothesimpleone. 
dimensionaldiusionspectrumwiththedispersion=k
2
,whereistherelaxatio. 
rateandkisthewavevectorofthemode,whichisparalleltothedisclinationlinei. 
thiscase.Inotherwords,asfarasthese
uctuationsareconsidered,thedisclinatio. 
linebehaveslikeadampedmasslessstringsubjecttolinetension.Inaddition,i. 
wouldbeinterestingandquitenontrivialtostudythein
uenceofhydrodynamicso. 
therelaxationratesofthe
uctuations,particularlythosejustmentioned.Accordin. 
towhatwehavelearnedaboutthe
oweectsondefectdynamicsweshouldexpec. 
signicantcorrections,especiallyforlongwavelength
uctuations. 


Similarcalculations(abitsimpler)asinChapter9canbeperformedalsoforth. 
disclinationsofthevectororderparameter.Inthesesystems,onecouldthusstud. 
the
uctuationsofthestrength1disclinationandthedecayofhigherstrengt. 



11. 
Conclusio. 


disclinations. 


??. 


Isitinthehumannatureorjustinminetheimpressionthataftertheproblem. 
havebeensolvedtheyinescapablystarttoseemtrivial?Afterafortnightorso.I. 
Iwastostartoverfromscratchagain,Iwouldhavedonemanythingsdierently. 
Muchbetter,Ibelieve.Nevertheless,afteralltheuncertaintyoftheearlydays. 
months(years?!),Iamprettysatisedwiththeoutcome.AndIcanhardlyconside. 
myselfbeingsomeonewhogetssatisedeasily.Still,Iwouldhavedonemanything. 
dierently...Butifsomeonehadforetoldmeaboutthepresentsituationwhen. 
wasjuststartingwithmyresearch,Iwouldprobablyneverhavehadthenervet. 
competewithit.Timeisakiller.Especiallyifonemakeseort. 



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ERRATU. 


WhileEqs.(5.9)and(5.17)(pp.57and60)themselvesareintact,theydono. 


f
el

representthediagonalizedquadraticsandT.
_s
i
asreferredtointhetext,bu. 
merelyexpressthemasthesumsofsquareterms.Thus,Eqs.(5.9)and(5.17)d. 
notstipulatethepositivityofthecoeAcients,asfalselystatedinthetext.Th. 
quadraticscanbediagonalizedinprinciple.However,thediagonalizedformsar. 
tooconvolutedtobeuseful.