Dynamik von Disklinationen in anisotropen Fluiden

125 Pages·2002·3.104 MB·German

by  Reichenstein M.

Dynamik von Disklinationen
in anisotropen Fluiden
Von der Fakult(cid:127)at Physik der Universit(cid:127)at Stuttgart
zur Erlangung der Wu(cid:127)rde eines
Doktors der Naturwissenschaften (Dr. rer. nat.)
genehmigte Abhandlung
von
Michael Reichenstein
aus Waiblingen
Hauptberichter: Prof. Dr. H.-R. Trebin
Mitberichter: Prof. Dr. H. Herrmann
Tag der mu(cid:127)ndlichen Pru(cid:127)fung: 18. Juli 2002
Institut fu(cid:127)r Theoretische und Angewandte Physik
Universit(cid:127)at Stuttgart
2002

Inhaltsverzeichnis
Symbolverzeichnis v
Summary vii
Einleitung 1
1 Anisotrope Fluide 5
1.1 Der nematische Ordnungsparameter . . . . . . . . . . . 6
1.2 R(cid:127)aumliche Variationen des Ordnungsparameters . . . . 13
1.3 Defekte und Disklinationen . . . . . . . . . . . . . . . . 16
1.4 Hydrodynamik des Direktors . . . . . . . . . . . . . . . 20
2 Hydrodynamik des Ausrichtungstensors 23
2.1 Die dynamischen Gleichungen . . . . . . . . . . . . . . . 23
2.2 Die verschiedenen Anteile der freien Energie . . . . . . . 28
2.3 Vergleich mit den Leslie-Ericksen-Gleichungen . . . . . . 30
2.4 Vergleich mit den Gleichungen einer isotropen Flu(cid:127)ssigkeit 32
2.5 Zusammenfassung der dynamischen Gleichungen . . . . 33
3 Skalierung und numerische Aspekte 35
3.1 Skalierung der freien Energie . . . . . . . . . . . . . . . 36
3.2 Einfu(cid:127)hrung der Reynoldszahl . . . . . . . . . . . . . . . 39
3.3 L(cid:127)osung des Druckproblems . . . . . . . . . . . . . . . . 42
3.4 Das diskretisierte Gleichungssystem. . . . . . . . . . . . 44
3.5 Randbedingungen . . . . . . . . . . . . . . . . . . . . . 45
4 Disklinationsdynamik im Direktorbild 49
4.1 Die Multidom(cid:127)anenzelle . . . . . . . . . . . . . . . . . . . 49
4.2 Die Multidom(cid:127)anenzelle als Displaypixel . . . . . . . . . 56
4.3 Die Multidom(cid:127)anenzelle als Mini-Defektlabor . . . . . . . 59
5 U(cid:127)bergang zum Ausrichtungstensor 67
5.1 Du(cid:127)nne Filme . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 Der Eigenwertu(cid:127)bergang . . . . . . . . . . . . . . . . . . 69
iv Inhaltsverzeichnis
6 Disklinationsdynamik im Ausrichtungstensorbild 77
6.1 Annihilation von Defekten . . . . . . . . . . . . . . . . . 77
6.2 Disklinationen im (cid:0) -Tensorbild . . . . . . . . . . . . . . 79
6.3 Die homogene Str(cid:127)omung . . . . . . . . . . . . . . . . . . 83
6.4 Anstr(cid:127)omen des Disklinationskerns. . . . . . . . . . . . . 86
Zusammenfassung und Ausblick 95
Anhang 97
A Vollst(cid:127)andige zeitliche Ableitung der freien Energie . . . 97
B Der Standardparametersatz . . . . . . . . . . . . . . . . 101
C Visualisierung . . . . . . . . . . . . . . . . . . . . . . . . 103
Literaturverzeichnis 105
Verzeichnis der Abbildungen, Boxen und Tabellen 112
Symbolverzeichnis
DiesisteineZusammenstellungeinigerh(cid:127)au(cid:12)ggebrauchterSymbole.Fu(cid:127)r
eine genauere Erl(cid:127)auterung sei auf die jeweiligen Abschnitte verwiesen,
in denen die Gr(cid:127)o(cid:25)en eingefu(cid:127)hrt werden.
Falls nicht in Komponenten angegeben, werden Vektoren fett gesetzt:
n,v.Tensorenwerdenebenfallsfett gesetzt,jedochohneSerifen: (cid:0) , .
PhysikalischeEinheitenwerdenohneSerifengesetzt:m, g .Vollst(cid:127)and(cid:0) i-
cms
ge zeitliche Ableitungen werden als d(cid:30)=(cid:30)_ geschrieben. Bei partiellen
dt
Ableitungen von skalaren Gr(cid:127)o(cid:25)en (cid:30) nach der Raumrichtung x schrei-
(cid:11)
ben wir @ (cid:30) = @ (cid:30) = (cid:30) und verwenden durchgehend griechische
@x(cid:11) (cid:11) ;(cid:11)
Indizesfu(cid:127)rdieKoordinatenrichtungen.HandeltessichumVektoren,so
gilt z.B. @ v =@ v =v .
@x(cid:11) (cid:12) (cid:11) (cid:12) (cid:12);(cid:11)
Variablen und Parameter
(cid:1) Einheitsmatrix
b Kantenl(cid:127)ange der Simulationsbox
c Elastische Konstanten der Landau-Energie
i
d Plattenabstand
d Kritischer Plattenabstand
krit
F Freie Energie
f Gesamte freie Energiedichte
ges
f Freie Energiedichte, Volumenanteil
bulk
f Freie Energiedichte, elastischer Anteil
elast
f Freie Energiedichte, elektrischer Anteil
elek
f Freie Energiedichte, chiraler Anteil
chiral
H Molekularfeld
(cid:11)(cid:12)
K corotational derivative von Q
(cid:11)(cid:12) (cid:11)(cid:12)
k Elastische Konstanten der Frank-Oseen-Energie
ij
Mbiax Biaxialit(cid:127)atsma(cid:25)
n Direktor
(cid:11)
p Hydrostatischer Druck
p Intrinsischer cholesterischerPitch
0
vi Symbolverzeichnis
Q Tensorieller Orientierungsordnungsparameter
(cid:11)(cid:12)
Re Reynoldszahl
St Stelzerzahl
1
St Starkzahl
2
s Defektst(cid:127)arke
t Reduzierte Temperatur in der Landau Energie
V Geschwindigkeitsgradient
(cid:11)(cid:12)
(cid:11) Viskosit(cid:127)aten in der Leslie-Ericksen-Theorie
i
(cid:14) Kronecker-Symbol
(cid:11)(cid:12)
(cid:15) Dielektrizit(cid:127)atstensor
(cid:11)(cid:12)
(cid:15) Vollst(cid:127)andig antisymmetrischer Tensor
(cid:11)(cid:12)(cid:13)
(cid:13),(cid:13)~ Rotationsviskosit(cid:127)aten in der Leslie-Ericksen-Theorie
(cid:26) Dichte
(cid:27) Spannungstensor
(cid:11)(cid:12)
(cid:27)elast Elastischer Anteil des Spannungstensors
(cid:11)(cid:12)
(cid:27)irr Viskoser Anteil des Spannungstensors
(cid:11)(cid:12)
(cid:30) Elektrische Spannung
’ Pretilt der MD-Zelle
p
(cid:18) Entropiedichte
# Pretwist der MD-Zelle
p
#D Kritischer Pretwist der MD-Zelle
p
(cid:22) Skalierter Tensor-Ordnungsparameter
(cid:11)(cid:12)
(cid:23) ;(cid:23) Viskosit(cid:127)aten in der Tensordynamik
1 2
(cid:23) Kinetischer Koe(cid:14)zient in der Tensordynamik
3
(cid:24) Koh(cid:127)arenzl(cid:127)ange
R
Summary
Liquid crystals are very particular materials. Although being (cid:13)uids,
theyrevealananisotropyofphysicalproperties,causedbyalong-range
orientationalorderofthemolecules.Someliquidcrystallinephasesaddi-
tionallyexhibitacertaindegreeofpositionalorder,thusahugenumber
of phase transitions can occur1.
Manyliquidcrystallinephasesarestableatroomtemperatureandtheir
characteristic length scales are in the wavelength range of visible light.
Thesepropertiesallowtheconstructionofliquidcrystaldisplays,where
thetransmissionoflightiscontrolledbyapplyinganelectricvoltageand
thusin(cid:13)uencingtheorientationalorderviaitscouplingtothedielectric
anisotropy.
In the orientational order, singularities can occur. Linear singularities
(linedefects),arecalleddisclinationsinanalogyofthenamedislocations
for defects in crystals. Disclinations in nematic liquid crystals are very
similar to vortices in super(cid:13)uids and to cosmic strings in quantum-
theoretical vacuum (cid:12)elds, but they are much more easily accessible to
the experiment.
In this workwe investigatethe dynamic properties of disclinations: the
motion, creation and annihilation of entire line defects as well as the
friction force on the defect core. For this purpose we set up the hy-
drodynamic equations for a tensorial order parameter which allows us
to correctly represent the core of the disclinations. In addition to the
well-knownNavier-Stokes-Equationof the liquid, we obtain an angular
momentum equation for the orientationaldegree of freedom.
Theory of the Nematic Order Parameter (Chapter 1)
Theorientationalorderofnematicliquidcrystalsisdescribedbyasym-
metricandtracelesstensorof secondrank,the orderparameter (cid:0) .The
expansion of the free energy in terms of (cid:0) up to fourth order yields a
bulk energyF , which canbe used tostudy phase transitions.At the
bulk
minimum of the bulk energy,the orderparameterisdegeneratedtothe
projective plane and (cid:0) can be written as (cid:0) = 3S(n n 1(cid:1) ). The
2 (cid:10) (cid:0) 3
moleculesalignalonganeasyaxisn,theso-calleddirector. 3Sdescribes
2
the degree of disalignment: S =0 corresponds to the isotropic, 3S =1
2
to the ideal nematic phase, i.e., a parallel alignment of all molecules.
1To maintain the readability of this small summary, we do not repeat the refe-
rencestoliterature,expressionsand(cid:12)gures,whichcanbefoundinchapter 1{6.
viii Summary
Spatial variations of the order parameter are taken into account by
the elastic energy F . For the typical length scale of liquid crystal
elast
displays, the elastic energy is very small compared to the bulk energy
and can be considered as a perturbation.
Thisisnottrueanymoreinthecoreofdisclinationlines.Thevariations
of the order parameter take place on very short distances ( 10nm),
(cid:25)
thus the elastic energy is no longer small and becomes comparable to
the bulk energy: the (cid:0) tensor exhibits biaxial states, the alignment
along a secondary axis perpendicular to the main director n becomes
important.
To quantitatively register the biaxiality, we introduce a measure of
biaxiality M , which is zero for prolate and oblate uniaxial order pa-
biax
rameters and tends towards one in the limit of biaxial states.
Hydrodynamics of the Alignment Tensor (Chapter 2)
The hydrodynamic equations for the director n are known as Leslie-
Ericksen theory. In this workwe study the correspondingequations for
the tensorial order parameter. We derive them in three steps: (cid:12)rst the
entropyproductionrateissetup.Thenweidentifytheconjugateforces
(the velocity gradients and the molecular (cid:12)eld) and (cid:13)uxes (the stress
tensor and the corotationalderivative of (cid:0) ).
Viscositiesareintroducedascoe(cid:14)cientsintheexpansionoftheforcesin
termsofthe(cid:13)uxes.TheresultisaNavier-Stokesequationandadynamic
equation for the order parameter (angular momentum balance). The
elastic and viscous parts of the order parameter (cid:12)eld couple to the
velocities via the stress tensor. The angular momentum equation, on
the other hand, contains the gradients of the velocity as well.
Scaling and Numeric Aspects (Chapter 3)
Toworkoutthebasicpropertiesofthe setofequationspreviouslyderi-
ved,wecombinethe conventionalscalingof the Navier-Stokesequation
(yielding aReynolds number)with the scalingof the free energyaccor-
ding to Hornreich.
The scaling of the free energy density is based on the characteristic
lengthb(e.g.thedimensionofadisplaypixel).Weexpressbintermsof
the coherencelength(cid:24) ,i.e., theintrinsiclengthscale,wheretheliquid
R
crystalline order decays. Thus we obtain
Summary ix
(cid:24)2
f =f + R f :
ges bulk b2 elast
Theratiobetweenthe volumeandthe elasticenergy,whichdependson
the length scale b, is now clearly visible and will be investigated in the
vicinity of thin (cid:12)lms (Chapter 5).
The properties of the hydrodynamic equations can be reduced to four
characteristicnumbers. These are:
the Reynolds number Re (the ratio between the kinetic and vis-
(cid:15)
cous (cid:13)ow energy),
theratioSt betweentheviscousrotationalenergyandtheelastic
2
(cid:15)
energy, both of them being related to the liquid crystallineorder,
a mixed ratioSt between the viscous(cid:13)owenergyand the elastic
1
(cid:15)
energy of the liquid crystalline order, and
the kinetic coe(cid:14)cient (cid:23) .
3
(cid:15)
Therescaledsetofequationsissolvednumericallyby(cid:12)niteelementand
(cid:12)nite di(cid:11)erence methods. As boundary conditions for the (cid:0) tensor we
employtheuniaxialbulkvalue,whichiscalculatedfromacon(cid:12)guration
of minimum bulk energy at a given temperature.
The Multidomain Cell (Chapter 4)
The multidomain cell provides a simple solution for one of the major
problems of liquid crystalline displays (LCD): the contrast of conven-
tional LCDs based on the twisted nematic (TN) cell strongly depends
onthe viewingangle.InthisTNcellthe moleculesatthe twocon(cid:12)ning
cover glasses are anchored parallel to the surface and enclose an angle
of 90 degrees. Therefore, the director forms a helix structure.
Byslightlytiltingthemoleculesatthesurface,thetwistingsenseofthe
helix can explicitly be chosen to be left or right handed. In a multido-
main cell (we use the version of Schadt et al.) left and right handed
helicesalternatelikeonacheckerboardpattern.Sincehelicesofopposite
twisting sense do not match, line defects appear.
Static investigations allow the derivation of a con(cid:12)guration phase dia-
gramasfunctionofthetiltangle# ofthedirectoratthesurface(mea-
p
sured from the surface normal) and the applied voltage U . For small
0
x Summary
values of # a defect-free con(cid:12)guration occurs since two helices reverse
p
their twisting sense due to a process denoted splay canceling.
DuetothecomplexgeometryoftheMDcell,wereduceoursetofequa-
tions toapurerotationaldynamicsbydiscardingthe velocity(cid:12)eld. We
simulate two di(cid:11)erent switching processes of the MD cell. One of them
mimics the application of the MD cell in displays, where disclinations
are always present. The other switching process can be used to study
the creation and annihilation of disclinations.
Wecalculatethetransmissionoflightbya2 2Jonesmatrixformalism.
(cid:2)
To achieve more detailed information on the disclination lines, a small
Burgerscircuitisperformedoneachplaneformedbyfourgridpoints.If
aclosurede(cid:12)ciencyremains,theplaneismarkedinitscenterbyasmall
sphere. By carrying out this procedure for each time step, we produce
a movie of the disclination line motion in space.
For the switching process in display applications we (cid:12)nd that the swit-
ching time is somewhat faster close to the disclination lines. At some
distance from the line defects, there is no di(cid:11)erence in switching time
compared to a conventionalTN cell. The process of switching on is the
reverse process of switching o(cid:11).
For investigatingthe creationandannihilation of disclinationswe start
with a defect-free con(cid:12)guration. During the switching process, two of
the four domains reverse their twisting sense. This is possible by the
nucleation of ring disclinations. First the liquid crystal reacts to the
applied electric voltage by aligning the directors parallel to the (cid:12)eld.
Ring disclinations open up where boundary conditions of opposite di-
rectionmeet.When theyarelargeenoughtomeet,theyrecombineand
twodisclinationlinesremainclosetothesurfaces,whichdistinguishthe
helices of left and right handed twisting sense.
When the voltage is turned o(cid:11) again, the order parameter (cid:12)eld rela-
xes according to the elastic torques. The disclinations move from the
surfaces to the center of the cell. There, again, a recombination takes
place. In the following, two of the domains turn their twisting sense,
thus yielding a homogeneouscon(cid:12)guration.
Properties of Thin Films (Chapter 5)
To achieve insight into the biaxial degree of freedom, we investigate
the static con(cid:12)gurations in thin (cid:12)lms. The liquid crystal is con(cid:12)ned by
two parallel surfaces at z = 0 and z = d. The order parameter at the