Thermodynamical Properties of a Model Liquid Crystal

Thermodynamical Properties Of A Model Liquid Crystal

vorgelegt von Diplom-Physiker Haiko Steuer aus Berlin

Von der Fakult¨at II - Mathematik und Naturwissenschaften der Technischen Universit¨at Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften – Dr.rer.nat. – genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr.rer.nat. Achim Hese Berichter: Prof. Dr. Siegfried Hess Berichter: Prof. Dr. Martin Schoen

Tag der wissenschaftlichen Aussprache: 6. Mai 2004

Berlin 2004 D 83 ii 1 Acknowledgments

First, I want to thank god, not only for the creation of all the matter with such interesting behavior, but also for his grace.

As high as the heavens are above the earth, so great is his love for those who fear him. The bible, Psalm 103, Verse 11

Further, I thank

Prof. Dr. Siegfried Hess for the opportunity to get insight in the rich world of • liquid crystals and the interesting methods of molecular simulations. His fruitful guidance and the warm atmosphere in the work group enabled this work.

Prof. Dr. Martin Schoen for making available a well-structured Monte Carlo • algorithm and for substantial help and advise.

Prof. Dr. Achim Hese for his willingness to head the doctorate committee. • Igor Stankovi´c for the structure analysis of several conﬁgurations. •

Dr. Patrick Ilg for the literature hint at the alternative K2 computation and for • proofreading this thesis.

Nana Sadowsky for her very detailed proofreading of this thesis. • Dr. Henry Bock for substantial help and discussions. • Financial support from the Deutsche Forschungsgemeinschaft for the Collaborative Re- search Center (SFB) 448 Mesoskopisch strukturierte Verbundsysteme is gratefully ac- knowledged. 2 3 Abstract

A simple model liquid crystal is studied via Monte Carlo computer simulations. The anisotropy of the particles enters the Lennard-Jones-like interaction potential in the attractive part such that side-by-side configurations are favored. The bulk system shows isotropic, nematic and solid phases. Comparisons of the phase behavior and pressures with analytical results show good agreement for high temperatures. The influence of confinement is studied by sandwiching the liquid crystal between two flat and smooth walls (slab geometry). The walls are modeled with homeotropic as well as twist alignment. The twist alignment is constructed with homogeneous planar anchoring at both walls with 90◦ different directors. The nematic phase in this geometry shows a linear decay of the azimuthal angle of the director along the cell. In both geometries the confinement shifts the phase transitions in support of higher order. The nematic phase is only slightly stabilized, but the enlargement of the solid phase regime is very pronounced. Therefore, the walls mainly increase the positional order of the system. The twist elasticity is studied by computing the corresponding Frank elastic constant K2 with a new method. The change of the free energy caused by a twist deformation is evaluated in order to obtain K2 as an average value even for moderate system sizes. 2 The relation K2 = c(nS2) is found, in agreement with experimental findings and other theories.

Zusammenfassung

Ein einfaches Modell fur¨ Flussigkristalle¨ wird mit Hilfe von Monte Carlo-Simulationen untersucht. Die Anisotropie steckt derart im anziehenden Anteil des Lennard Jones- ahnlic¨ hen Potentials, dass Seite-an-Seite-Konfigurationen energetisch bevorzugt werden. Das uneingeschränkte System zeigt isotrope, nematische und feste Phasen. Phasen- verhalten und Druck sind bei hohen Temperaturen in guter Ub¨ ereinstimmung mit an- alytischen Rechnungen. Der Einfluss von Einsperrungen des Fluids zwischen flache, glatte Wände wird untersucht (Plattengeometrie). Dabei werden die Wände so model- liert, dass sowohl eine homeotrope als auch eine verdrillte (twist) Ausrichtung ensteht. Die verdrillte Ausrichtung wird mit planarer Verankerung an beiden Wänden, aber mit 90◦ verdrillten Direktoren erstellt. In der nematischen Phase ergibt sich dann ein linearer Abfall des Azimutwinkels des Direktors entlang der Zelle. In beiden Geometrien werden die Phasenub¨ ergänge zugunsten höherer Ordnung verschoben. Während die nematische Ordnung nur leicht verstärkt wird, erhöht sich die Positionsordnung erheblich. Die Verdrillungselastizität wird untersucht durch Berechnung der zugehörigen Frank’- schen Konstante K2 mit einer neuen Methode. Dazu wir die Anderung¨ der freien En- ergie durch eine Verdrillung ausgewertet, um K2 als Mittelwert auch bei moderaten 2 Systemabmessungen zu bestimmen. Es ergibt sich K2 = c(nS2) , in Ub¨ ereinstimmung mit experimentellen Ergebnissen und anderen Theorien. 4 Contents

1 Introduction 7

2 A simple model for liquid crystals 11 2.1 Liquid crystals ...... 11 2.2 The model potential ...... 13 2.3 Analytic estimates: Pressure and phase behavior ...... 15 2.4 Range of interaction ...... 17 2.5 Boundaries ...... 18

3 Method 21 3.1 The Monte Carlo (MC) method ...... 22 3.2 The Hamiltonian ...... 23 3.3 NVT Monte Carlo ...... 24 3.4 NPT Monte Carlo ...... 25 3.5 Observables ...... 27 3.6 Consistency test NVT – NPT ...... 30

4 MC results in the bulk system 33 4.1 Isotherms ...... 33 4.2 Isochores ...... 35 4.3 Isobars ...... 41

5 MC results with conﬁning walls 45 5.1 Homeotropic alignment ...... 45 5.1.1 Isotherm ...... 45 5.1.2 Isochore, including structure analysis ...... 49 5.1.3 Isobars ...... 53 5.2 Twist alignment ...... 55 5.2.1 Isotherm ...... 55 5.2.2 Isochore ...... 57 5.2.3 Isobars ...... 61

6 Twist elastic coeﬃcient 63 6.1 First estimate ...... 64 6.2 Method 1: Alignment tensor in Fourier space ...... 67

5 6 CONTENTS

6.3 Method 2: Free energy distortion ...... 70 6.4 MC results ...... 76

7 Conclusions and outlook 83

A Rotational energy of a uniaxial particle 87

B The partition function 89

C Derivatives of the potential 91

D Proof of lemma 93

Bibliography 95 Chapter 1

Introduction

Many materials in daily life take the form of fluids, i.e. liquids or gases. Most of them are not simply atomic fluids but complex fluids, sometimes called soft matter. These are materials which are composed of macromolecules or form anisotropic fluids. Notable examples include polymer solutions, biological materials, gels, ferrofluids, sur- factant assemblies and liquid crystals. The study of soft matter and its behavior poses a significant challenge to modern science. The theory must account for qualitatively new effects which are present neither for simple fluids nor for solids. On the other hand, these materials have found many industrial applications. Polymers1, for example, are mainly used as elastomers (rubber) or plastics. This thesis will focus on liquid crystals and their equilibrium properties. Liquid crystals represent a special state of matter between solids and liquids – forming an anisotropic fluid. They are very useful due to this dual nature and their easy response to surface forces[1]. In 2002, liquid crystal displays (LCDs) became the most produced display type world wide. LCDs consist of a nematic2 liquid crystal sandwiched between two plates of glass (slab geometry). Therefore, the physics of confined liquid crystals is an important subject from the technological point of view. An understanding of the thermodynamical behavior of liquid crystals is vital for the development of new applications and for the improvement of existing ones. Two equilibrium properties of a model liquid crystal will be studied here.

1. The phase behavior of the bulk system and the influence of confining walls will be analyzed in a slab geometry with different anchoring mechanism, including a twisted nematic (TN) cell.

2. The twist elasticity of this model will be studied. The elasticity of a liquid crystal inﬂuences its response to external ﬁelds.

A LCD can be build up of TN cells, which are switched by external fields. Thus, the present thesis will contribute to the knowledge of the behavior of LCDs. Furthermore, the influence of confinement on the phase behavior of liquid crystals is of high academic

1Polymers consist of many small molecules that can be linked together to form long chains. 2The nematic phase is characterized by a common direction of the particle orientations . This is then called the director.

7 8 CHAPTER 1. INTRODUCTION interest. The interplay between bulk and surface forces gives rise to a complex phase behavior. In experiments the confinement is found to induce capillary condensation[2]. Fluids can be studied with the help of interaction potentials, which model the potential energy between two particles. The purpose of statistical physics is to predict macroscopic properties from such microscopic models. This can be done by computer simulations. The development of computer simulation techniques in statistical physics started in 1953, when Metropolis et al. showed how to obtain the equation of state for a given interaction potential[3]. They used the Monte Carlo (MC) integration scheme – integrating over a random sample of points. An alternative method, called Molecular Dynamics (MD) was presented in 1959[4]. In MD simulations the Newtonian equation of motion is numerically solved.. The success of these methods in predicting thermodynamical properties for model fluids composed of hard discs[5], Lennard-Jones molecules[6] and for many other fluids and solids in the following years have made computer simulations an important tool in theoretical physics. Computer experiments have two main advantages: First, theoretical model systems can be developed by testing and comparing its properties with experimental results. Second, simulations give insight into processes, which are not accessible to the experiment, e.g. the observation of the trace of a single particle. Limitations of molecular simulations are the capacity of computers and the inaccuracies of the theoretical models. During the last decades modifications and combinations of these methods were used – supported by the rapid development of computer hardware. Furthermore, computer simulations are nowadays not only used for physics, but also for chemistry, car traffic, meteorology, climate and forest fire mod- eling, networks of biology, economics, telecommunications, internet and society, and many others. The development of these research areas was highly inspired by physics. In 1972 – the year when the author of this thesis was born – the first computer simulation of liquid crystals were done[7, 8]. The first confined liquid crystal system was simulated in 1991[9]. Until now, there have been many other theoretical studies of liquid crystals in restricted geometries with help of both molecular simulations [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] and density functional theories[22, 23, 24, 25, 26, 27, 28] or combinations of both[29, 30, 31]. Compared with the great variety of chemical compounds which possess liquid crystalline phases, relatively few theoretical models have been conjectured and studied. In addition to the well established Lebwohl-Lasher[7, 8, 9, 15, 18, 19, 21] Gay-Berne[32, 33, 10, 34, 35, 11, 36, 12, 14, 17], ellipsoidal[37, 29, 38, 31] and spherocylinder[39, 22, 40, 41, 16] models, theoretical analysis of new model systems is helpful and desirable for computational studies of the phase behavior of liquid crystals[25], in particular in restricted geometries[9, 10, 13, 42], and of transport processes[43]. Recently[44], a simple interaction potential has been introduced and the phase behavior was determined by an augmented van-der-Waals theory[45]. The model potential describes the anisotropy of one particle in the attractive part of the Lennard Jones-like pair interaction. The pressure as function of the temperature and of the density was calculated. Estimates for the gas-liquid coexistence and for the critical temperature were given as well as for the isotropic-nematic transition temperature. Here, this model liquid crystal will be analyzed via MC simulations. The phase behavior with and without walls as well as the twist elasticity are determined. 9

The Monte Carlo data for the bulk system[46] agree well with the former analytical results for high temperatures. Higher ordered (solid) phases are also found. Simulations with confining flat walls[47] reveal the influence of different kinds of substrates on the phase behavior and alignment effects of this model liquid crystal. Two different kind of substrates are used. The first one is modeled with homeotropic3 alignment. The second is modeled such that in front of both walls homogeneous planar4 alignment is preferred, but with 90◦ difference between the azimuthal angles at the walls. The nematic phase is then characterized by an inhomogeneous director field. The physics is that of a TN cell without any orienting field. The performance of liquid crystal devices often depends on the values of the elastic constants K1, K2 and K3. These coefficients are instrumental in determining the ease of reorienting the nematic liquid crystal. Many methods have been developed to obtain them experimentally by observing Frederiks transitions5. The measurement of the twist elastic constant K2 is the most difficult[48, 49]. Theoretically, there are several ap- proaches to determine the elastic constants[50, 51, 52, 53, 54, 55, 56]. Some are based on density functional theories[50, 52, 56] or make use of fluctuations of the alignment tensor in Fourier space[57, 53, 54, 55], which only works in the long-wavelength limit. We will calculate the twist elastic constant K2 with a new method. It is based on an analytical expression derived from the change of the free energy caused by a twist deformation. The relevant quantity is given by an average to be evaluated in the undisturbed state and can be computed via Monte Carlo simulations even with moderate system sizes. This thesis proceeds as follows: In Chapter 2 the model and the analytical estimates are reviewed after a short introduction to liquid crystals. The model is specialized for the purpose of computer simulations, i.e. the range of interaction and boundary conditions are specified. Details of the MC simulations are outlined in Chapter 3. The method is described in the NV T as well as the NP T ensemble for anisotropic particles and the observables are introduced, whose average values will be computed in oder to analyze the phase behavior. Results for the bulk system are presented in Chapter 4. These provide a reference for the analysis of the behavior of the model system in restricted geometries, which is done in Chapter 5. The twist elastic constant K2 is calculated according to the new method and compared with another one[53] in Chapter 6.

3In homeotropic alignment the director is perpendicular to the wall. 4In homogeneous planar alignment the director is parallel to the wall. This is usually achieved by rubbing the wall surface. 5In Frederiks transitions a nematic liquid crystal is reoriented by an external ﬁeld. 10 CHAPTER 1. INTRODUCTION Chapter 2

A simple model for liquid crystals

2.1 Liquid crystals

A brief introduction to liquid crystals and their phase behavior is given. The thermophysical properties are of importance in technical applications. Materials which form liquid crystalline phases, are called liquid crystals. They typically consist of rodlike molecules like p-azoxyanisole (PAA, ﬁgure 2.1). Disc-shaped molecules may also form liquid crystalline phases. If the liquid crystalline phases only

Figure 2.1: Example for a rodlike molecule (PAA)[58]. occur in solution and the phase behavior depends on the solvent concentration we speak of a lyotropic liquid crystal. On the other hand, a thermotropic liquid crystal shows temperature dependent liquid crystalline phases. Here, we will pay attention only to thermotropic liquid crystals composed of rodlike molecules. A liquid crystalline phase is an intermediate phase lying between the isotropic liquid and the solid crystalline phase. The crystalline solid phase is characterized by a long- range positional order. For temperatures between the melting point and the clearing point, there may be one or more liquid crystal phases (see ﬁgure 2.2). Contrary to the isotropic liquid, where there is neither positional nor orientational long-range order, a liquid crystal phase has at least orientational order, but otherwise behaves as a viscous liquid. Liquid crystalline phases are birefringent. This is a visible manifestation of their long-range orientational order and can be observed between crossed polarizers. Above the clearing point, the liquid is isotropic. There are several types of liquid crystal phases. Several of them may be observed for one compound. The nematic phase shows a common orientation (director) of the particles, but no positional order. A smectic phase

11 12 CHAPTER 2. A SIMPLE MODEL FOR LIQUID CRYSTALS

crystalline solid smectic-A nematic isotropic liquid

T melting point clearing point

Figure 2.2: A typical history of liquid crystalline phases between solid and isotropic phases during heating. Each molecule is drawn as an ellipsoid. Between the melting point and the clearing point there is shown a smectic-A and a nematic phase.

has a long-range positional order in one direction, i.e. layers are formed. If there is no structure in the layer, one has either a smectic-A phase (if the layer normal equals the director) or a smectic-C phase (otherwise). These are two-dimensional liquids. Other types of smectic phases may occur, where additional order is found within the layers. Alternatively, the liquid crystalline phases may be chiral. In a chiral nematic phase, the director ﬁeld is not uniform. Instead, the local preferred direction is spontaneously twisted. The repetition period of the helical structure, which is responsible for the reﬂection color, is temperature dependent. Therefore, one application of chiral nematics is the thermography, i.e. they can be used as optical temperature indicators.

Today, the most important technical application of liquid crystals are liquid crystal displays (LCD’s). These electro-optical devices make use of two features, which influ- ences the director field and therefore the optical behavior. First, boundary effects can lead to a certain alignment near the well-prepared boundary. Second, the reorientation of the alignment caused by an electric field can be used to control the optical behavior of the device with low voltages. A twisted nematic (TN) cell (figure 2.3) is mainly composed of two planar walls with a nematic liquid crystal film in between. The walls are prepared such that the particles align planar at the wall with a definite direction (due to a rubbed surface). But the alignment directions due to the surfaces differ about 90◦, such that without any external field the director smoothly rotates from one wall to the other (twist deformation). If outer polarizers are arranged appropriately, light will be led through the cell by the twisted director field. By applying an electric field E parallel to the wall normal the director will align with that field. Therefore, the light will not find a way through the polarizers and the cell, which will appear dark.

In this thesis, it will be modeled both the situation in ﬁgure 2.3 (a), i.e. a twisted homogeneous planar alignment and a homeotropic alignment, where the director is parallel to the wall normal. 2.2. THE MODEL POTENTIAL 13

(a) (b) E

Figure 2.3: Nematic liquid crystal ﬁlm of a TN cell (a) without, and (b) with an applied electric ﬁeld E. 2.2 The model potential

We consider a fluid composed of (effectively) axisymmetric particles whose orientation is characterized by a unit vector uˆ parallel to the figure axis. A vector with a hat ˆ will always be normalized in this thesis. The interaction potential between two particles located at the positions r1 and r2 with orientations uˆ1 and uˆ2 depends on the three vectors r = r r , uˆ and uˆ as shown in figure 2.4. The interaction energy for two 2 − 1 1 2

r u u22 uu1

Figure 2.4: Interaction of two uniaxial particles.

ﬂuid particles is written as[44]

r −12 r −6 Φ (r, uˆ , uˆ ) = 4Φ 0 0 (1 + Ψ(ˆr, uˆ , uˆ )) , (2.1) ff 1 2 0 r − r 1 2 ·³ ´ ³ ´ ¸ where r = r ˆr and the quantities r0 and Φ0 set the characteristic length and energy scales. The anisotropy in the attractive term is described by

Ψ(ˆr, uˆ , uˆ ) = 5ε P (uˆ uˆ ) + 5ε [P (uˆ ˆr) + P (uˆ ˆr)] (2.2) 1 2 1 2 1 · 2 2 2 1 · 2 2 · Here 3x2 1 P (x) = − (2.3) 2 2 is the second Legendre polynomial. Note that uˆ is equivalent to uˆ , so the head-tail j − j symmetry is satisﬁed. In the calculations the potential was cut oﬀ at a distance rc = 3. 14 CHAPTER 2. A SIMPLE MODEL FOR LIQUID CRYSTALS

The terms 5P2(n1 n2) in the anisotropy function can be motivated with rotational invariants constructed· from tensors of the second rank (l = 2) 15 1 ϕ (n) := n n δ , (2.4) νµ 2 ν µ − 3 νµ r µ ¶ for which the orthonormality

1 2 ϕ ϕ 0 0 := ϕ (n)ϕ 0 0 (n) d n h νµ| ν µ i 4π νµ ν µ 1 Z 1 = (δ 0 δ 0 + δ 0 δ 0 ) δ δ 0 0 (2.5) 2 νν µµ νµ µν − 3 νµ ν µ is valid. Then, ϕ (n)ϕ (n) = 2l + 1 = 5 and ϕ (n )ϕ (n ) = 5P (n n ). νµ νµ νµ 1 νµ 2 2 1 · 2 The characteristic length r0 and energy Φ0 may be used to define a characteristic 3 temperature T0 = Φ0/kB and pressure P0 = Φ0/r0. In the following, all quantities are given in these characteristic units. In these standard Lennard-Jones (LJ) units the expression (2.1) is rewritten as Φ (r, uˆ , uˆ ) = 4 r−12 r−6(1 + Ψ(ˆr, uˆ , uˆ )) , (2.6) ff 1 2 − 1 2 i.e. the length r and the energy Φ are£ expressed in LJ ”diameter”,¤ and potential depth, respectively [59]. To allow comparisons with the analytical calculations in [44] the anisotropy coefficients εi are chosen to be ε1 = 0.04 and ε2 = 0.08. In this case the side-side configuration is energetically favored, since the side-side− interaction potential Φside−side(r) := Φ (eˆ r, eˆ , eˆ ) = 4 r−12 r−6(1 + 5ε 5ε ) ff ff x z z − 1 − 2 8 = 4 r−12 r−6 £ ¤ (2.7) − 5 · ¸ side−side side−side 2 has a much deeper minimum Φff (rmin ) = (1 + 5ε1 5ε2) = 2.56 than the end-end potential − − − Φend−end(r) := Φ (eˆ r, eˆ , eˆ ) = 4 r−12 r−6(1 + 5ε + 10ε ) ff ff z z z − 1 2 2 = 4 r−12 r−6 £ ¤ (2.8) − 5 · ¸ Φend−end(rend−end) = (1 + 5ε + 10ε )2 = 0.16 (see figure 2.5). The orthogonal ff min − 1 2 − unit vectors eˆν specify the basis according to the simulation box. The minima are side−side 1/6 end−end 1/6 rmin = (2/(1 + 5ε1 5ε2)) 1.04 and rmin = (2/(1 + 5ε1 + 10ε2)) 1.31 respectively. The effectiv− e length-to-width≈ ratio l/w for our particles can be defined≈ end−end side−side with help of the potentials Φff (r) and Φff (r) either through the ratio of the minima or through the ratio of the zeros of these functions. Both strategies lead to the same result end−end 1/6 rmin 1 + 5ε1 5ε2 l/w = side−side = − r 1 + 5ε1 + 10ε2 min µ ¶ = 41/6 1.26 (2.9) ≈ 2.3. ANALYTIC ESTIMATES: PRESSURE AND PHASE BEHAVIOR 15

Figure 2.5: Interaction potential Φff for end-end, T, X and side-side conﬁguration in comparison with the spherical LJ potential which is closer to one compared with other models. To get an further insight into the interaction of two particles, we put one with orientation uˆ1 = eˆz into the origin r1 = 0 of a coordinate system and have a look at another particle with the same orientation uˆ2 = eˆz, located at the variable position r2 = (x, 0, z). With r2 = (r sin ϕ, 0, r cos ϕ) the potential energy is then given by

Φparallel(r, ϕ) := Φ (r , eˆ , eˆ ) = 4 r−12 r−6(1 + 5ε + 10ε P (cos ϕ)) , (2.10) ff ff 2 z z − 1 2 2 £ end−end side−side ¤ which is a generalization of the potentials Φff (r) and Φff (r). The contour- parallel plot of Φff is shown in ﬁgure 2.6. For large distances the interaction becomes zero. The black minima show the regions which are favored as particle distances. The white region in the middle can be interpreted as an enlarged picture of one particle. For a comparison with a real substance our ”particles” should be identiﬁed with a nearly spherical agglomerate of strongly non-spherical particles, in the spirit of the ideas originally put forward by Maier and Saupe[60]. In contrast to model systems where mainly the shape of the particles and therefore repulsive forces are considered our model potential takes the anisotropy from the attractive intermolecular forces. Furthermore, the presented model potential is computationally less demanding than for example that of the Gay-Berne model.

2.3 Analytic estimates: Pressure and phase behavior

An overview of the analytic results obtained for this model[44] (without walls) will be given in order to compare with the results in the bulk system presented below. 16 CHAPTER 2. A SIMPLE MODEL FOR LIQUID CRYSTALS

2 ϕ r

1 x

parallel Figure 2.6: Potential energy Φff of two parallel particle. The white region in the parallel middle contains distances with Φff > 0.

First, the pressure in the fluid is estimated according to an augmented van der Waals-approximation[45] p nkT + pW CA + kT n2(B BW CA) , (2.11) ≈ − where pW CA is the pressure in a reference system (to be specified below). The distortion pressure pdis = p (nkT + pW CA) is approximated with pdis kT n2(B BW CA). Here, B and BW C−A are the second virial coefficients of the fluid≈ and the−reference system, respectively. The idea is, that this expression should be valid for small densities according to the virial expansion and that for high densities, the pressure is dominated by the reference system. Therefore, the reference system is chosen to be purely repulsive. Here, it is chosen the W CA potential[61]

4(r−12 r−6) + 1 (r < 21/6) φW CA(r) := − , (2.12) (0 otherwise for which the pressure can be approximated with a modified Carnahan-Starling[62] expression nBW CA(T ) (nv (T ))2 pW CA(n, T ) nkT + 2 eff . (2.13) ≈ (1 nv (T ))2 (1 nv (T ))3 µ − eff − eff ¶ 3 W CA ! The effective particle volume veff (T ) = πd /6 is defined through φ (d) = kT . The distortion pressure is split into an isotropic and an alignment part: ∂f pdis = n2kT (B BW CA) + n2 align (2.14) iso − ∂n 2.4. RANGE OF INTERACTION 17

The alignment part of the distortion pressure deﬁnes a corresponding alignment free energy falign, which can be written as ρ(uˆ) f = kT ρ(u) ln d2uˆ align ρ Z 0 nkT ρ2 c (r, uˆ , uˆ )χ(uˆ )χ(uˆ ) d2uˆ d2uˆ d3r (2.15) − 2 0 anis 1 2 1 2 1 2 Z Φ (r, uˆ , uˆ ) with c (r, uˆ , uˆ ) = k(r) + exp ff 1 2 (2.16) anis 1 2 − kT µ ¶

The orientational distribution function is written as ρ(uˆ) = ρ0(1 + χ(uˆ)), where ρ0 = 1/(4π) and canis(r, uˆ1, uˆ2) is the anisotropic part of the direct correlation function, which contains information from the potential. With help of the expansion χ(uˆ) = a ϕ + a ϕ we get coeﬃcients a = χ ϕ and a = χ ϕ , which νµ νµ νµλκ νµλκ νµ h | νµi νµλκ h | νµλκi turn out to be the alignment tensors of second and fourth rank. For the assumption of uniaxial alignment we get just two scalar numbers a2 and a4, by which the alignment free energy is expressed. falign(a2, a4) can then be minimized in order to obtain the correct coeﬃcients.

2.4 Range of interaction

To save CPU time the interaction range is limited to the cut-oﬀ radius rc. This allows to organize the particles in boxes and make neighbor lists for each particle[59]. Then many distances with minor contributions to the potential do not have to be calculated.

Φff (r, uˆ1, uˆ2) := 0 (r > rc) (2.17)

Here rc = 3 is chosen. The model then slightly differs from the original potential without cutoff. For the pressure the difference can be estimated as follows: For small densities n = N/V we have (neglecting the anisotropy of the potential)

n2 p r F d3r = p + p (2.18) ≈ 6 · MC err Z where ∂ F r = 4(r−12 r−6) r · −∂r − − · = 24(2r¡−12 r−6) ¢ (2.19) − and perr shall be the pressure difference n2 ∞ 16πn2 p = r F 4πr2 dr (2.20) err 6 · ≈ − 3r3 Zrc c due to cut off. The pressure without cut off (rc = ) would be smaller, because the cut off neglects negative pressure contributions. F∞or densities n 0.1 this gives ≈ 18 CHAPTER 2. A SIMPLE MODEL FOR LIQUID CRYSTALS

perr 0.006, which is small in comparison to the error obtained for typical pressure values.≈ −For large densities n 1 we expect a measurable effect. Therefore, the cut off ≈ distance rc is an important part to be known of the model studies here. The abrupt cut off can be done without problems in Monte Carlo simulations. How- ever, in molecular dynamics simulations the potential energy should be smooth. Oth- erwise, the calculation of forces can become difficult. Then it would be necessary to smoothly cut off the potential. A smooth cut off has been done for the short-range attractive potential SHRAT[63].

2.5 Boundaries

Except for any wall interactions, the boundaries of the simulation box should not influence the results. This can be achieved with periodic boundary conditions. If the simulation box has the edges dx, dy and dz, it is periodically repeated in all three directions, i.e. any particle with position r give rise to 33 1 = 26 mirror particles at positions r + m d eˆ + m d eˆ + m d eˆ , where m , m−, m 1, 0, 1 except x x x y y y z z z x y z ∈ {− } (mx, my, mz) = (0, 0, 0). This means, that a particle at the margin of the simulation box can interact with a particle at the opposite site (see figure 2.7). Theoretically, one may allow infinite mirror particles, i.e. m > 1. However, in practice the cut off radius | ν| rc and dimensions dν of the simulation box are chosen such that only the neighboring boxes (with m 1, 0, 1 ) are relevant, i.e. 2r < min d , d , d . Then, no further ν ∈ {− } c { x y z}

1 1 1

2 2 2

1 1 1 rc dy

2 2 2

1 1 1

2 2 2

Figure 2.7: Periodic boundary conditions in 2 dimensions. The middle cell is the simulation box. Mirror particles (in the 32 1 = 8 repeated boxes) are grey. For particle 1 the − interaction range is indicated by a circle with radius rc. Particle 1 at r1 interacts with two particles, one neighbor in the simulation box and the mirror r2 + dyeˆy of particle 2. mirror particles with mν = 2 have to be taken into account. The boundary conditions for our system without walls§are periodic in all three directions. 2.5. BOUNDARIES 19

In simulations with walls we have to change the boundary condition in one direction. We place two ﬂat walls at the planes z = z1/2 = dz/2 parallel to the xy-plane (see ﬁgure 2.8). Therefore we do not apply periodic boundary§ conditions in the z direction.

(a) (b) y

z (c)

Figure 2.8: (a) Walls conﬁning the simulation box at z1 = dz/2 and z2 = dz/2 (b) homogeneous planar alignment with g (uˆ) = (uˆ aˆ )2 (c) homeotropic− alignment with w · w gw(uˆ) = 1.

The walls consist of particles interacting with fluid particles with orientation uˆ at distant r according to a Lennard-Jones type potential φ (r, uˆ) = 4(r−12 r−6g (uˆ)) (2.21) fw − w with the anchoring function gw(uˆ). It is chosen to model the desired alignment. For g (uˆ) = (uˆ aˆ )2 we get homogeneous planar alignment at the wall with orientations w · w parallel to the unit vector aˆw, for gw(uˆ) = 1 we get homeotropic alignment. A twisted 2 2 nematic cell can be modeled using the anchoring functions g1(uˆ) = uˆy and g2(uˆ) = uˆx. The fluid-wall potential is cut off at distance rc = 3. Now one could put particles on a lattice in the solid wall (discrete walls) and let the fluid particles interact pairwise with them[14]. Another much easier possibility is to handle the walls continously[64, 10, 38]. We follow the latter way and assume a smooth wall with a particle density of ρw. By integrating the wall-fluid potential over the wall[64] w we get

2 Φfw(r, uˆ) = ρw φfw( r rw , uˆ) d rw w k − k Z ∞ = 2πρ 4 (r2 + (z z )2)−6 (r2 + (z z )2)−3g (uˆ) r dr w w − w − w − w w w w Z0 2 £ ¤ = 2πρ (z z )−10 (z z )−4g (uˆ) , (2.22) w 5 − w − − w w µ ¶ where r = (x, y, z) is the position of the ﬂuid particle. Similar choices for the wall anchoring[10, 14, 38] have been made in the literature. 20 CHAPTER 2. A SIMPLE MODEL FOR LIQUID CRYSTALS Chapter 3

Method

The goal to gain insight into the phase behavior of the liquid crystal model studied here can be achieved by observing thermodynamical properties of this complex ﬂuid. These are obtained by calculating average values A from some observables A like internal energy or order parameters (to be speciﬁed later).h i Average values can be calculated as integrals

A = A(Γ) f(Γ)dΓ (3.1) h i Z in the phase space. Here, Γ is a point in the phase space, that may contain positions and orientations of all particles and f(Γ) is the distribution function (probability density) in phase space. The goal of a Monte Carlo simulation will be the calculation of such average values A . Random numbers will be used for numerical integration. It is a h i stochastic simulation, in which time plays no role at all. Therefore only equilibrium properties can be extracted with Monte Carlo simulations. Alternatively A could be calculated with help of a time average (T large) h i

1 T A = A(Γ(t))dt (3.2) h itime T Z0 of a dynamical, deterministic simulation (molecular dynamics). Here, the trace Γ(t) of the phase space point is followed using Hamiltonian dynamics (may be restricted by constraints like constant temperature). It is necessary to calculate the forces between all pairs of particles in each discretized step of the dynamics, which makes this method time consuming. For an ergodic system one has A = A time. However, the Monte Carlo method is faster (forces are not necessary) ash wiell ash morei suitable to heat constraints (by choosing the appropriate ensemble distribution function f) and will be used here. For non-equilibrium properties, molecular dynamics simulations would be the method of choice.

21 22 CHAPTER 3. METHOD 3.1 The Monte Carlo (MC) method

In the canonical ensemble (constant temperature T ) f is given through e−βH(Γ) f(Γ) = with Z = e−βH(Γ)dΓ , (3.3) Z Z where H(Γ) is the Hamiltonian of our system (see section 3.2), β = 1/(kBT ) and Z is the partition function. Usually such integrals have to be calculated numerically. This could in principle be done with easy standard algorithms like the trapeze method. But here we have to take at least some hundred particles (typical N = 1000) to ensure thermodynamic limit. Therefore the phase space has a very high dimension N. A ∼ very eﬃcient method to numerically calculate high dimensional integrals like (3.1) is the Monte Carlo integration scheme, which works as follows: The idea is to generate a large set of random points Γ1, Γ2, . . . , ΓM which is distributed according to some probability density p(Γ). Then{ our integral can} be approximated by

A(Γ)f(Γ) 1 M A(Γ )f(Γ ) A = p(Γ) dΓ j j (3.4) h i p(Γ) ≈ M p(Γ ) j=1 j Z X This is both easy and efficient with the distribution p(Γ) = f(Γ). The generation a of a set of points distributed according to f(Γ) can be done with help of the Metropo- lis algorithm[3]. There, a numerical representation of a Markov process Γ1, Γ2, . . . is generated through the transition probability W (Γ Γ ) = ω(Γ Γ ) p(Γ Γ ) , (3.5) j → j+1 j → j+1 j → j+1 where ω(Γ Γ ) is the probability of trying the new phase space point Γ , if Γ j → j+1 j+1 j is currently given and p(Γj Γj+1) is the acceptance probability of this change. It is given by → f(Γ ) p(Γ Γ ) = min 1, j+1 , (3.6) j → j+1 f(Γ ) ½ j ¾ assuming a symmetric trial probability ω(Γj Γj+1) = ω(Γj+1 Γj) (otherwise the definition of p has to be slightly modified). Because→ of →

f(Γj+1) = exp β(H(Γj+1) (H(Γj)) (3.7) f(Γj) {− − } the acceptance probability can be calculated without knowing the partition function Z. It can be proven that the transition probabilities W (Γ Γ ) fulfill detailed balance j → j+1 W (Γ Γ ) f(Γ ) = W (Γ Γ ) f(Γ ), (3.8) j → j+1 j j+1 → j j+1 which is sufficient for the Marcov process to converge. The associated master equation d f(Γ , t) = (W (Γ Γ ) f(Γ , t) W (Γ Γ ) f(Γ , t)) (3.9) dt j − j → j+1 j − j+1 → j j+1 Γ Xj+1 3.2. THE HAMILTONIAN 23 then has the stationary solution f(Γ). Therefore, the Metropolis algorithm converges to the generation of points Γ1, Γ2, . . . distributed according to f(Γ). The details of the algorithm as well as its adaption to the NP T ensemble, where the pressure is fixed, will be explained in the following sections. Two main difficulties have to be considered. First, the system may be too small. The absence of the thermodynamic limit may lead to finite size effects. These can be avoided by repeating a simulation with another system size. If the results do not depend on the dimensions of the simulation box, it is sufficiently large. Large enough systems were achieved already with several hundred particles. However, some tolerable finite size effects like nonzero order parameter ( 0.1), which should be exactly zero, are still ≈ observed. Second the finite CPU time of the simulation (i.e. finite number of Γj) leads to uncertainties in the average values. Especially near phase transitions, the system need more time to be in equilibrium. The simulation may not lead to a stable but to a metastable phase space point. For the average values, fluctuations can give a hint whether the statistical average has enough input or not.

3.2 The Hamiltonian

A system consisting of N uniaxial particles, where the pair potential Φff (rij, uî, uˆj) is given by (2.6) is considered. The particles may feel the wall potential Φfw(rj, uˆj), see equation (2.22), due to flat smooth walls. The phase space is spanned by positions rj and momenta pj of the particles as well as the Eulerian angles Ωj (ϕj, ϑj, χj) and their conjugate momenta Λ (Λ , Λ , Λ ). The first two Eulerian ≡angles express the j ≡ ϕj ϑj χj orientation uˆ (ϕ, ϑ) of a particle (polar coordinates of the symmetry axis, see figure 3.1). The third≡Eulerian angle χ, describing a rotation around the symmetry axis, plays

z u

¡£¢¥¤ ¦¨§ ©£ ©

¡¤ © ©

¦¨§ © y ¡£¤ ϕ x

Figure 3.1: Eulerian angles ϕ and ϑ of a particle orientation uˆ. no role in the potential. However, this additional degree of freedom contributes to the kinetic energy since our particles are uniaxial but not linear. According to the rotational energy Trot of uniaxial particles (see appendix A), the 24 CHAPTER 3. METHOD

Hamiltonian H is given by

N 2 2 2 2 p Λϑ (Λϕ Λχ cos ϑj) Λχ H = Φ + j + j + j − j + j , (3.10) tot 2m 2J 2J sin2 ϑ 2J j=1 " ⊥ ⊥ j k # X with the total potential energy

N N 2

Φtot = Φff (rij, uˆi, uˆj) + Φfw(rj, uˆj) , (3.11) j=1 "i=j+1 w=1 # X X X where r = r r . m is the mass of a particle. The moments of inertia for rotations ij i − j about the symmetry axis and the axes perpendicular to it are Jk and J⊥. The angles χj are cyclic variables for the uniaxial particles studied here.

3.3 NVT Monte Carlo

In the simulations up to N = 1000 particles in a box of volume V , with periodic boundary conditions, were studied at a given temperature T . In this canonical (NV T ) ensemble the equilibrium average of a phase space function A(Γ) is given by 1 A = f (Γ)A(Γ) dΓ , (3.12) h iNV T h6N N! 2N (2π)N NV T Z where 1 H(Γ) f (Γ) = exp (3.13) NV T Z − k T NV T µ B ¶ is the probability density of the canonical ensemble. The factors 2N and (2π)N in the denominator occurring in (3.12) are associated with the head-tail symmetry and the rotational symmetry about the ﬁgure axis of each particle. The partition function is given by (see appendix B)

Φ (Γc) N Z (k T )3N exp tot d2uˆ d3r , (3.14) NV T ∼ B − k T j j B j=1 Z ½ ¾ Y c 2 where Γ (r1, uˆ1, . . . , rN , uˆN ) specifies a configuration and d uˆj = sin ϑj dϑj dϕj is the surface≡element on the sphere of orientations. For an observable A which depends on the configuration only, i.e. A = A(Γc), one has

N A = f c (Γc)A(Γc) d2uˆ d3r (3.15) h iNV T NV T j j j=1 Z Y with the conﬁgurational distribution function c c c exp Φtot(Γ )/(kBT ) fNV T (Γ ) = {− } . (3.16) Φtot N 2 3 exp d uˆj d rj − kBT j=1 R n o Q 3.4. NPT MONTE CARLO 25

An efficient way to calculate integrals like in (3.15) is to generate a numerical representation of a Markov process in configuration space corresponding to the distri- c c bution function fNV T (Γ ) in equation (3.16). This can be done through Metropolis’ c algorithm[3]: At the beginning we randomly choose a starting configuration Γ1. For a given particle we randomly change either the orientation or the position. Then we c c calculate the associated change in energy ∆Φtot = Φtot(Γ2) Φtot(Γ1) and accept the new configuration with probability − ∆Φ p(Γc Γc ) = min 1, exp tot . (3.17) 1 → 2 − k T ½ µ B ¶¾ This being done for all N particles corresponds to one Monte Carlo step (or sweep) c eventually leading to a new configuration Γ2. The range of the changes in positions is adjusted during the simulations in order to gain fast convergence to equilibrium. The rotation angle range for the change of the orientations is set to π/10. The particles are organized in boxes and neighbor lists to save time for calculating particle distances. Neighbor lists must be updated in certain intervals on account of particle diffusion. In the limit of a sufficiently large (but finite) number M1 of sweeps the algorithm serves c c to generate a sequence (or set) of further configurations ΓM1+1, . . . , ΓM2 distributed according to f c , so that NV T © ª M2 1 c A NV T A(Γi ) . (3.18) h i ≈ M2 M1 i=M1+1 − X Typical Monte Carlo runs consisted of M2 = 100 000 sweeps with an initial equilibration of M1 = 20 000 sweeps. The equilibration criterion was that the pressure fluctuates around a practically constant average value. The number M1 of sweeps necessary for achieving equilibrium is much larger in the vicinity of phase transitions.

3.4 NPT Monte Carlo

Sometimes it is desirable to achieve a homogeneous equilibrium with a constant pressure P . Then the two-phase region with inhomogeneities is automatically avoided, because the system easily reaches a fitting volume. In addition, a constant volume simulation often needs more time to achieve equilibrium, because the particles have to manage their positions in a finite size, which may be small. A constant pressure has the opportunity to first enlarge the system, giving all particles place to arrange each other. In the NP T -ensemble simulations up to N = 1000 particles were studied at a given temperature T and a given pressure P . In the case of confining walls, we keep the distance dz constant. Therefore, we can only fix the lateral pressure Pk. To be precise, the equilibrium ensemble studied keeps constant the particle number N, the temperature T , the distance dz and the lateral pressure Pk. One may therefore speak of the NdzPkT ensemble. If P is the pressure tensor, then 1 P = (P + P ) (3.19) k 2 xx yy 26 CHAPTER 3. METHOD takes into account forces parallel to the walls. In bulk simulations for isotropic phases, 1 the scalar pressure P = 3 (Pxx + Pyy + Pzz) equals the lateral pressure Pk. According to the isothermal-isobaric distribution function in configurational phase space

exp Φtot+PkV c c − kBT fNP T (Γ , V ) = (3.20) ∞ Φtot+PkVn 2 3 o 2 3 exp d uˆ1 d r1 . . . d uˆN d rN dV 0 − kBT R R n o we have to compare the potential Φtot energy of our system before and after the vol- c ume change. The space integration of the partition function in the denominator ZNP T depends implicitly on the other integration variable, namely the volume. To get rid of this, we use the substitution sjν := rjν/dν, where the box of our system has side lengths dν, that is V = dxdydz. Then, the volume change, which we have to perform in the simulation, can be handled by scaling of particle positions[65, 66]. The variables sj will 3 3 remain constant for a volume change. With d rj = V d sj the partition function can now be written as ∞ Φ + P V Zc = V N exp tot k d2uˆ d3s . . . d2uˆ d3s dV (3.21) NP T − k T 1 1 N N Z0 Z ½ B ¾ For an observable A which depends on the conﬁguration only, i.e. A = A(Γc), we can obtain average values in the NP T ensemble through

M2 1 c A NP T A(Γi ) . (3.22) h i ≈ M2 M1 i=M1+1 − X c The set of configurations Γi i=1,...,M2 is produced as follows according to Metropolis’ algorithm[3], which is adapted{ } for the constant pressure ensemble[67, 68, 59]: At the c beginning we randomly choose a starting configuration Γ1 in a box of volume V1 = dxdydz. The z-length must be dz = zw=2 zw=1. For a given particle we randomly change either the orientation or the position. Then− we calculate the associated change in energy ∆Φtot and accept the new configuration with probability min(1, exp( ∆Φtot/kBT )). When all particles have been taken into account we have done one NV −T Monte Carlo step. Now, we finally try to change the volume. Thereby we keep the distance dz between the walls constant and only change dx and dy with the same random factor new new c (0.995, 1.005), i.e., dx = cdx and dy = cdy The positions of the fluid particles are∈ scaled accordingly: xnew = cx and ynew = cy. The acceptance probability of this volume change ∆V = V V is given through new − old V N ∆Φ + P ∆V p(V V ) = min 1, new exp tot k old → new V − k T ( µ old ¶ µ B ¶) ∆Φ + P ∆V Nk T ln c2 = min 1, exp tot k − B , (3.23) − k T ½ µ B ¶¾ 2 because Vnew/Vold = c . This acceptance probability is also valid for the positional and orientational changes made before, where especially ∆V = 0 and c = 1. The change of 3.5. OBSERVABLES 27

potential energy ∆Φtot must be calculated with some caution. The cut off at distance rc should not lead to the artificial effect that one pair of particles may be considered for one volume but not for the other (bigger) one. Therefore, we cut off each pair interaction according to a common criterion

1 (r + r ) > r . (3.24) 2 old new c

c Finally we arrive at a new configuration Γ2 at volume V2. After M1 Monte Carlo steps of this kind we find that our observables fluctuate around some average value. Typical numbers in our simulations were M1 = 10 000 and M2 = 100 000 with higher numbers in the vicinity of phase transitions.

3.5 Observables

Quantities calculated are the internal energy E, the pressure P as well as the orientational order parameter S2 and the smectic order parameter %1. In the thermodynamic limit their average values should not depend on the ensemble in which they were calculated. In our simulation we could indeed conﬁrm, that

A = A . (3.25) h iNV T h iNP T Therefore, the ensemble index will be omitted in the following. Fluctuations of the quantities are calculated with help of the variance

∆A = A2 A 2 (3.26) h i − h i q which us used to obtain errorbars for the observables.

Internal energy

For the internal energy per particle E = H /N one has h i Φ E = 3k T + h toti , (3.27) B N

The ﬂuctuation of the internal energy is given by

2 2 Φtot Φtot ∆E = h i − h i , (3.28) q N because T is ﬁxed in our simulations. 28 CHAPTER 3. METHOD

Pressure The pressure P for a homogeneous system is calculated through Clausius’ virial expression

1 P V = Nk T + (r r ) F , (3.29) B 3 i − j · ij * i j

48 60 24 F = r + ε [(u ˆr)u + (u ˆr)u ] [1 + 5ε P (u u ) 5ε ] r ij r14 r7 2 i · i j · j − r8 1 2 i · j − 2 240 ε (u ˆr)2 + (u ˆr)2 r , (3.30) − r8 2 i · j · £ ¤ where r = r r . j − i Nematic order parameter The nematic phase may be characterized by the Maier-Saupe order parameter[69, 60, 58] MS S2 and by the alignment tensor of rank two[70], which is the average of the tensor

1 N Q = u u (3.31) N j j j=1 X for a homogeneous system. The symbol 1 1 a = (a + a ) a δ (3.32) νµ 2 νµ µν − 3 λλ νµ denotes the symmetric traceless part of a tensor aνµ. Here we have uνuµ = uνuµ 1 MS 3 − 3 δνµ. Now we can deﬁne S2 as the largest eigenvalue of the tensor S := 2 Q . MS h i With help of the corresponding normalized eigenvector n one has S2 = nνSνµnµ, and consequently[71]

3 1 N 1 1 N SMS = n n u u δ = P (n u ) . (3.33) 2 2 ν µ N j ν j µ − 3 νµ N 2 · j * j=1 + * j=1 + X µ ¶ X Note that at ﬁrst one has to calculate the alignment tensor Q in order to take the MS h i average value S2 , because the director n must be known. For isotropic phases S and MS consequently S2 vanish. In our Monte Carlo simulations an isotropic phase is typically characterized by a nematic order parameter below 0.1. The nematic order parameter does not completely vanish due to the ﬁnite size of the simulation system. Perfect MS alignment uj = n corresponds to S2 = 1. For the nematic phase we expect typical MS values of S2 & 0.4. 3.5. OBSERVABLES 29

The biaxiality of S = 3 Q is characterized by the parameter b, deﬁned[72] through 2 h i b2 = 1 6 (S S S )2 (S S )−3. (3.34) − µν νλ λµ µν νµ If two of the eigenvalues of S are equal (uniaxial tensor), b = 0. In this case we use MS uni S2 = S2 and write 3 3 S = Q = Suni nn . (3.35) 2 h i 2 2

uni By taking the norm a := √aνµaνµ of (3.35) we can compute S2 , with nn = 2/3 according to k k k k p

uni 3 S2 = Q . (3.36) r2 kh ik The biaxiality b was found to be small (less than 0.1) in the nematic phase. The order uni MS parameter S2 is therefore expected to agree with S2 . During the simulation of a system with orientational order, it is possible that the nematic mean orientation changes and Q mixes these orientations. In this case the mean eigenvector of Q cannot be interpretedh i as a director. It is therefore preferable h i c to calculate an order parameter S2 directly from Q(Γ ) as

3 S2 = Q . (3.37) *r2 k k+ Here, this quantity was used as nematic order parameter. Alternatively, we also used equation (3.33) taking n as the mean orientation of the current conﬁguration. The results agreed because of the small degree of biaxiality.

Smectic order parameter A Smectic order parameter should be sensitive to the formation of layers. Of course, layers may also occur in solid phases. Any periodicity of the particle density %(r) should lead to a high ”smectic order” parameter. Usually the smectic order parameter is chosen to be the ﬁrst Fourier coeﬃcient %1 of the density %(r) resolved according to

∞ 2πkn r %(r) = % + % cos · φ . (3.38) 0 k d − Xk=1 µ ¶ The average particle density is given by %0. The periodicity d, the layer normal n and the oﬀset φ are unknown. For φ = 0 one layer should lie in the origin of the coordinate system. For a smectic-A phase the layer normal should be equal the director and can therefore be extracted from the alignment tensor. But the dependency of %1 on n is very critical. A slightly wrong n due to statistical errors gives a too small order parameter. In the case of walls parallel to the xy plane (see section 2.5), however, we already know the expected layer normal n = ez. Next we can get rid of the oﬀset φ by writing 30 CHAPTER 3. METHOD cos(ϕ φ) = α cos(ϕ) + β sin(ϕ), where α + iβ = eiφ. So we have for z [0, D] the Fourier−series ∈

∞ 2πkz 2πkz %(z) = % + α% cos + β% sin (3.39) 0 k d k d Xk=1 · µ ¶ µ ¶¸ with coeﬃcients 1 D 2πz % α = %(z) cos dz and (3.40) 1 D d Z0 µ ¶ 1 D 2πz % iβ = %(z)i sin dz (3.41) 1 D d Z0 µ ¶ By summing equations (3.40) and (3.41) and taking the absolute value we have

1 D 2πiz % = %(z) exp dz (3.42) 1 D d ¯ Z0 µ ¶ ¯ ¯ ¯ ¯ ¯ Because we can not calculate the¯ particle density without discretization¯ errors the better way to calculate this quantity is the average value

N 1 2πizj %1 = exp (3.43) *¯N d ¯+ ¯ j=1 µ ¶¯ ¯ X ¯ ¯ ¯ This was also used in [73] as a positional¯ order parameter,¯ motivated through the static structure factor. The remaining problem of obtaining the periodicity d can be solved by just calculating %1 in the expected range of d and then take the maximum[74]. The range for the expected periodicity is d [0.7, 1.3]. ∈ 3.6 Consistency test NVT – NPT

To test our NP T algorithm for the bulk system we first simulate N = 800 particles at the temperature T = 1.4 in a cubic box of fixed volume V = 103 (NV T ensemble). The result is shown in figure 3.2(a). In the NV T simulation the scalar pressure P is found to be P 3.0168. Now we run a simulation with variable volume and fixed pressure P = 3.0168.≈ This results in a average volume of V 1000.4. The constant pressure k ≈ simulations with a desired lateral pressure Pk are expected to fulfill Pxx = Pyy = Pk. The component Pzz may differ in the inhomogeneous system with walls. For all our NP T calculations we found this expectation to be fulfilled in the limit of accuracy, which can be estimated from fluctuations of Pxx and Pyy. This means that the thermodynamic limit is reached in our calculations. In addition, the algorithms for NV T and NP T Monte Carlo simulations are shown to be consistent. However, there will still be problems with the thermodynamic limit by looking at local average values in inhomogeneous systems. The regions in which the local average values are calculated will be too small. 3.6. CONSISTENCY TEST NVT – NPT 31

(a)

(b)

Figure 3.2: N = 800 particle at temperature T = 1.4. (a) Average pressure for a NV T run at V=1000. (b) Average volume for a NP T run at P=3.0168 32 CHAPTER 3. METHOD Chapter 4

MC results in the bulk system

The analysis of the bulk state via Monte Carlo (MC) simulations will lead to an insight of the phase behavior of the model liquid crystal studied here. For some isotherms (T = 1.0, T = 1.2, T = 1.4) comparison with analytical results[44] will be made in order to check the accuracy of the estimates and validity of the approximations. The isotherms are calculated with the help of NV T -MC simulations. The isotherm T = 1 will also be analyzed in greater detail via NP T -MC simulations. Two isochores (for n = 0.7 and n = 1.0) will be studied in the NV T ensemble and ﬁve isobars (P = 0.1, P = 0.3, P = 0.5, P = 0.7, P = 1.0) in the NP T ensemble. The phase transitions isotropic-nematic (IN) and nematic-solid (NS) will provide a reference with which it is possible to analyze the impact of conﬁning walls on the phase behavior. Walls will be considered in Chapter 5.

4.1 Isotherms

In ﬁgure 4.1 the pressure is displayed as function of the number density. Points represent MC results connected by dashed lines to guide the eye. These were computed for N = nV particles in a box of volume V = 1000. The solid curves stand for corresponding analytical results[44], obtained via the augmented van der Waals-approximation (eq. (2.11)), where the minimization of the alignment free energy (2.15) is considered. For high densities and low temperatures there are two solutions, revealing an isotropic state (solid lines) and a nematic state (thin lines). The analytic curves agree well with the MC data for higher temperatures, at lower temperatures the agreement is not so good. In particular, the isotropic-nematic phase transition is shifted to lower densities, compared with the Monte Carlo calculations. The isotherm T = 1 was studied for comparison in the NP T ensemble (see ﬁg- ure(4.2)). The desired pressure was changed from p = 0.1 to p = 5.5. At the isotropic- nematic phase transition (p = 1.6) the particle density increases almost smoothly. If we decrease the pressure again we observe no hysteresis for pressure steps of ∆p = 0.1. The simulations were done with N = 1000 particles. In experiments the thermal hysteresis for the isotropic-nematic phase transition is small[75] or may also vanish[76]. The results of the constant pressure simulations agree with these of the constant volume

33 34 CHAPTER 4. MC RESULTS IN THE BULK SYSTEM

8 analytic (isotropic) 7 analytic (nematic)

6 MC (isotropic)

¢¤¦ 5 MC (nematic) 4

¨¢©¤ ¦ 3 2

1 § 0 ¢¡¤£¦¥ -1 0 0.2 0.4 0.6 0.8 1 n

Figure 4.1: Isotherms for T = 1.0, T = 1.2 and T = 1.4. Pressure (see eq. (3.29)) is plotted versus the number density n = N/V . Dots are NV T -MC results connected by dashed lines. Squares indicate a nematic phase. Solid lines show analytic results[44], the thin lines are branches for nematic phases.

n 1 S2

0.8

0.6 ∆ 0.16 E 0.4 0.12

0.2 0.08

0 1 2 3 4 5 6 0 0 1 2 3 4 5 6 p

Figure 4.2: Isotherm for T = 1.0 calculated with constant pressure Monte Carlo simulations. Density n, nematic order parameter S2 and energy ﬂuctuation ∆E (inlet) are shown. simulations shown above and below. An additional conﬁrmation of the functionality of the NP T algorithm is the observation, that the pressure calculated as average value equals the desired pressure used in the volume change probability (like in Chapter 3.6). Equilibrium was typically reached after 10 000 sweeps and average values are taken over 4.2. ISOCHORES 35

40 000 sweeps. In the vicinity of the phase transition we had to double these sweep numbers. We gain deeper insight into the phase behavior by studying the complete isotherm T = 1, including high pressures to enter the solid state and including the complete expansion process in order to study the hysteresis forced by the strong first-order phase transition nematic-solid. For this temperature we will present similar isotherms in a restricted geometry and compare them with the bulk results in order to gain insight into the influence of confinement on the phase behavior. Some results for the isotherm T = 1, obtained via constant pressure MC simulations, are shown in figure 4.3. We start with an isotropic fluid at low pressure Pk = 0.1. NP T runs for which Pk = 0 are notoriously difficult on account of a dramatic increase of volume fluctuations as P goes to zero. The nematic order parameter S2 as well as the smectic order parameter %1 are small in the isotropic phase, that is lower than 0.1 for the finite-size system we observe. The average values for small pressures Pk < 1 confirm the results from the isobaric calculations for T = 1 (to be presented in section 4.3). The isotropic-nematic phase transition takes place at PIN = 1.7, indicated through a peak in the energy fluctuation ∆E, which is proportional to the heat capacity. The phase transition is accompanied by a hardly visible jump in the number density around n = 0.85 and by a remarkable jump in the nematic order parameter S2 > 0.4. At PNS = 7.4 the system enters the solid state. This phase transition manifests itself clearly in the peak in ∆E and a discontinuity in n. The order parameters jump to higher values too. The smectic order parameter %1 0.3 indicates an increase in positional order. Snapshots of a configuration in the nematic≈ phase at Pk = 6, where S2 = 0.74 and a configuration in the solid state at Pk = 8, are shown in figure 4.4. The orientation of each particle is indicated by a cylinder. In the solid state configuration, there are no layers perpendicular to the z direction. Because %1 is calculated assuming formation of layers perpendicular to z, its value remains small compared to 1.0. At Pk = 12 compression terminates and pressure is subsequently released in small steps. The systems remains in the solid state even for small pressures. At Pk = 3 we observe a small increase in order where the pressure increases. This can be explained as a reorganization of crystal structure, which may happen in finite size and finite ”time” Monte Carlo runs. Indicative for this is also the density jump around Pk = 3. We observe a strong hysteresis due to solidification, reflecting a pronounced first-order phase transition. The solid-nematic and nematic-isotropic transitions occur almost at the same pressure around Pk = 1.4.

4.2 Isochores

To analyze the isotropic-nematic phase transition in greater detail, the system was studied for selected values of number density and the temperature was changed in small steps. This can be done in the NV T ensemble because we expect that the density jump during the isotropic-nematic phase transition is small. First the system was cooled, in small steps, from a high temperature, where it is in an isotropic phase, to a low temperature, where an ordered state is expected. Once this ﬁnal temperature is reached the system is heated up again in small temperature steps. For each temperature we 36 CHAPTER 4. MC RESULTS IN THE BULK SYSTEM

1.1

1 n 0.9

0.8

2 4 6 8 10 12

0.8

0.6

0.4 order parameter 0.2 S ρ2 1 2 4 6 8 10 12

0.2 E ∆

0.1

2 4 6 8 10 12

P||

Figure 4.3: Complete Isotherm T = 1 for the bulk system obtained via NP T -MC. For compression and expansion processes (indicated by the arrows) the number density n, the order parameters S2 and %1 and the energy ﬂuctuation ∆E are plotted as functions of the desired pressure Pk. 4.2. ISOCHORES 37

Figure 4.4: Snapshots of bulk configurations at Pk = 6 (top), where the system is in a nematic phase and at Pk = 8 (bottom), where the system is in a solid state. The configurations are taken from the isotherm T = 1 obtained via NP T -MC. wait several thousand sweeps for equilibrium (see section 3.3) and average the cartesian components (in the space fixed frame) of the pressure tensor

Nk T 1 P = B δ + (rν rν) F µ (4.1) νµ V νµ V i − j ij * i j

Results for n=0.7 Results for n = 0.7 are presented in ﬁgure 4.5. During the initial cooling period the temperature was lowered from T = 1 to T = 0.48 and eventually heated up again to the ﬁnal temperature T = 1. For high temperatures the pressure tensor is isotropic, that 38 CHAPTER 4. MC RESULTS IN THE BULK SYSTEM

(a) ¢¡£¡ 0

-0.5

p -1

-1.5 ¤¢¥¦¥

-2 §©¨ ¨

(b) 1

0.8

0.6 2 S 0.4

0.2

0 (c) 0.4

0.3

E 0.2 ∆

0.1

0 0.4 0.5 0.6 0.7 0.8 0.9 1 T

Figure 4.5: Isochore for n = 0.7 cooling and heating, obtained via NV T -MC: (a): pressure tensor components pνν, (b): order parameter S2, (c): fluctuation of internal energy ∆E is its diagonal components are equal. For smaller temperatures the pressure becomes negative indicating a mechanically unstable state. Negative pressures are avoided in a NP T ensemble (see above). A snapshot of a configuration for T = 0.9 is shown in figure 4.6(a). There does not seem to be any orientational order. The calculated order parameter is S2 < 0.1. At TIN = 0.82 the isotropic-nematic phase transition occurs, revealed by a peak in ∆E and leading to a small step of the pressure values towards 0, i.e. the pressure increases. The reason for this can be seen in a snapshot for T = 0.75 4.2. ISOCHORES 39 in figure 4.6(b). The box is not uniformly filled with particles. There are two different density regions indicating that we are in a two-phase state. Therefore the box volume V taken in equation (4.1) is above the real volume for the particles in the nematic phase and we cannot interprete our average values as the ”pressure”. In addition, the term isochore has to be used with caution for this series of simulations. For even smaller temperatures (T < 0.68) snapshots show a smectic-like phase (figure 4.6(c)). A more detailed analysis[77] however, indicates that it is a solid state where we have a three-dimensional ordered crystal. The lack of a smectic phase is not unusual for liquid crystals like for example PAA[78, 79]. For other anisotropy parameters ε1 and ε2 however our model may be able to show a smectic phase too. In the solid phase, the diagonal components of the pressure tensor have different values. This phase is metastable when the system is heated up again up to T = 0.84, so we have hysteresis. The origin of this hysteresis is the first-order phase transition nematic-solid in our finite size simulation system, observed in finite time. The detailed analysis of the solidification and melting transitions are outside the scope of this article. The nematic regime during heating is very narrow, from T = 0.84 to T = 0.87. Metastable nematic phases, which can be obtained only with decreasing temperature, called monotropic, are well known in experiments[80, 81, 82]. In the analytical calculation the isotropic-nematic phase transition temperature for particle density n = 0.7 was TIN = 1.14, that is larger than in the simulations, where this transition takes place at T = 0.82 (cooling) and T = 0.87 (heating).

Results for n=1.0

To prevent voids in our system we increased the density to n = 1. Data from simulations of cooling and heating procedures in the temperature range T [0.7, 1.6] are presented ∈ in figure 4.7. In these simulations the cubic box with the edges dx = dy = dz = 8 is filled with N = 512 particles. Snapshots of the nematic and the solid phase from the cooling period are presented in figure 4.8. We observe three phases: At high temperatures an isotropic liquid, at low temperatures a solid. The solid phase partly shows negative pressure values. Between these two states there is a nematic phase. Compared with the results for n = 0.7, the nematic phase is present in a considerably larger temperature interval. During the heating period the nematic phase is present only in a narrow temperature range from T = 1.38 to T = 1.46 with a strong hysteresis period. The first-order phase transitions fluid-solid occur at T = 0.92 for cooling and T = 1.38 for heating. Qualitatively, the hysteresis observed here is very similar to that of the purely spherical Lennard-Jones system with the same density n = 1, with slightly different transition temperatures T = 0.8 and T = 1.3. The nematic phase regime is again smaller in the heating period than in the cooling period. The isotropic-nematic transition temperatures observed in the simulations are T = 1.34 for cooling and T = 1.46 for heating. The corresponding analytical calculations give TIN = 1.42, between these values. 40 CHAPTER 4. MC RESULTS IN THE BULK SYSTEM

(a) (b)

(c)

(d) (e)

Figure 4.6: Snapshots of conﬁgurations at density n = 0.7 for diﬀerent temperatures: (a) T = 0.9, (b) T = 0.75 and (c) T = 0.6. (d) and (e) show two more T = 6 snapshots with rotated perspectives. The isochore is obtained via NV T -MC. 4.3. ISOBARS 41

(a) 12

6 p

3 ¤¦¥§¥

0 ¨ © © ¢¡£¡ -3 (b) 0.8

0.6 2 S 0.4

0.2

0 0.3 (c)

0.2 E ∆ 0.1

0 0.8 1 1.2 1.4 1.6 T

Figure 4.7: Isochore for n = 1 cooling and heating: (a): pressure tensor components pνν, (b): order parameter S2, (c): ﬂuctuation of internal energy ∆E. The isochore is obtained via NV T -MC.

4.3 Isobars

To complete our analysis of phase transitions in the bulk system, we observe some isobars. The isotropic-nematic (IN) phase transition is studied for diﬀerent isobars with pressure values Pk = 0.1, Pk = 0.3, Pk = 0.5, Pk = 0.7 and Pk = 1.0 using constant pressure MC simulations (NP T ensemble). For the bulk system it would not be necessary to restrict the NP T volume changes 42 CHAPTER 4. MC RESULTS IN THE BULK SYSTEM

(a) (b)

Figure 4.8: Snapshots of configurations at density n = 1 for different temperatures: (a) T = 1.1 and (b) T = 0.7 taken from the cooling procedure. The isochore is obtained via NV T -MC. to the x- and y- direction. However, we want to use the bulk behavior only as a reference for the restricted geometries. Therefore, the bulk simulations are performed in the same manner as the simulations with walls, namely, letting the volume breathe only in the x- and y- direction. Each simulation was done with N = 1000 particles in a box with length dz = 15 and variable dx and dy. In figure 4.9 we show the density and nematic order parameter as functions of temperature, which was lowered until one enters the regime of nematic phases. During a cooling series we start a new run (with a slightly lowered temperature) with an initial configuration and volume taken from the end of the former simulation. The particle densities n at T = 1 can be compared with these obtained for the pressures Pk < 1 of the isotherm T = 1 (figure 4.3). The comparison shows good agreement. The transition temperature TIN is found to be between 0.88 and 0.96 depending on the pressure, at which the cooling procedure was performed. For larger pressures Pk we find higher particle densities, smaller energies E, a higher nematic order parameter S2 and higher transition temperatures TIN . The isotropic-nematic phase transition are indicated by a peak of the energy fluctuation ∆E, a jump in the internal energy E, a slightly jump in the particle density n and a clear increase of the nematic order parameter S2. We also simulated the heating procedures back into the isotropic phase and were unable to observe any hysteresis within the accuracy of the temperature steps we chose, even with ∆T = 0.001. This results from the weakness of the first-order phase transition, which is typical for the IN transition[76]. 4.3. ISOBARS 43

0.9 n 0.8 P = 0.1 P = 0.3 P = 0.5 0.7 P = 0.7 P = 1.0 0.8 0.85 0.9 0.95 1 -5

-6 E -7

-8

0.8 0.85 0.9 0.95 1 0.8

0.6 2

S 0.4

0.2

0.8 0.85 0.9 0.95 1 T

Figure 4.9: Isotropic-nematic phase transition through cooling for diﬀerent isobars. The number density n, the energy E and the nematic order parameter S2 are plotted vs. the temperature T for diﬀerent pressure values Pk. The isobars are obtained via NP T -MC. 44 CHAPTER 4. MC RESULTS IN THE BULK SYSTEM Chapter 5

MC results with conﬁning walls

To study the effect of confinement on the phase transitions observed, we now restrict the geometry by two flat walls (slab geometry). Two different anchoring types will be used, leading to homeotropic and twisted homogeneous planar alignment, respectively. For both anchoring types the isotherm T = 1, the isochore n = 0.9 as well as some isobars are studied to allow comparison with the bulk simulations presented in Chapter 4. In addition a structure analysis of the isochore n = 1.0 according to [77] is made. These isochores are calculated via NV T MC simulations. All other simulations (isotherm, isobars) are obtained via NP T MC simulations.

5.1 Homeotropic alignment

We begin by focusing on particle-wall interactions favoring homeotropic alignment. This can be modeled by setting both anchoring functions g1(uˆ) = g2(uˆ) = 1, thereby elimi- nating the orientation dependent part in the interaction potential (2.22):

2 Φ (r) = 2πρ (z z )−10 (z z )−4 (5.1) fw w 5 − w − − w µ ¶ with z1 = 7.5 and z2 = 7.5. The wall particle density ρw is chosen to be ρw = 0.8. The lack of−the orientational dependence first leads to a pure layering of the center of masses of the particles near the walls without any orientating effects due to the wall interaction. However, the orientation dependent fluid-fluid interaction in the first layers favoring side-side configurations give rise to the alignment parallel to the layer normal and therfore to the wall normal.

5.1.1 Isotherm Figure 5.1 shows the particle density, order parameters and energy fluctuation for the isotherm T = 1. We start with compressing the system in the isotropic phase. The smectic order parameter increases already for small pressures, but only slightly, indicating the formation of layers (see figure 5.2). The IN phase transition is smoother than in the bulk system (compare figure 4.3) and takes place at PIN = 1.5, that is, for a slightly

45 46 CHAPTER 5. MC RESULTS WITH CONFINING WALLS

1.1

n 0.9

0.8

0.7

2 4 6 8

0.8

0.6

0.4 order parameter

0.2 S ρ2 1 2 4 6 8

0.2 E ∆

0.1

2 4 6 8

P||

Figure 5.1: Isotherm T = 1 for the homeotropic aligned system, obtained via NP T MC simulations. For compression and expansion processes(indicated by the arrows) the bulk number density n, the order parameters S2 and %1 and the energy ﬂuctuation ∆E are plotted as function of the pressure Pk. 5.1. HOMEOTROPIC ALIGNMENT 47

P||=2.4 P||=1.0 P||=0.1 2 local n 1

-8 -6 -4 -2 0 2 4 6 8 z

Figure 5.2: NP T MC Simulations with homeotropic alignment. Density proﬁles for Pk = 0.1, Pk = 1.0 and Pk = 2.4 from the isotherm T = 1. The box indicates the region, where the bulk particle density is measured.

lower pressure PIN compared with the bulk. The particle density, measured in the middle of the simulation cell, smoothly changes around n = 0.85, that is the same density as during the IN transition in the bulk. For the slab geometry it is easier to form a nematic phase, because at least one layer is formed very early (see figure 5.3). In this layer the particle-particle interaction favors a side-side configuration (all uˆj parallel), resulting in homeotropic alignment. A high nematic order in the first layer near the wall is known for Gay-Berne particles too[12]. Therefore, the anchoring mechanism is responsible for the shifted NI transition. During this transition we observe a weak density jump and large peak in the energy fluctuation. Solidification takes place at PNS = 2.5, which is much lower than in the bulk. Layering turns out to be quite pronounced and causes positional order. The mere presence of a surface may, under favorable geometric conditions (i.e., suitable choice of dz) support solidification if dz is close to such that an unstrained solid can actually form. The nematic-solid phase transition is accompanied by a jump of the particle density and the order parameters. The crystal structure can be seen from two different perspectives in the snapshots. In figure 5.3 (middle) the perspective is chosen such that the crystalline structure can be viewed in the middle of the cell. Rotations around the z-axis would reveal the long-ranged positional order in the regions near the walls. Thus, the crystal is not perfect, but has some dislocations. To analyze the crystal structure, we focus on the second and third layer near the substrate (z = 7.5) for the − final Pk = 6 configuration, see figure 5.3 (bottom). The layers are found to consist of a two-dimensional hexagonal structure. For Gay-Berne particles it is already known that the first layer forms a two-dimensional lattice even in the bulk nematic phase. It is of importance for the onset of orientational order[12]. A more detailed analysis of some crystal structures will be presented for the isochor n = 0.9 below. For pressures Pk > 5 we find that the smectic order parameter is changing its value 48 CHAPTER 5. MC RESULTS WITH CONFINING WALLS

y 0

-2

-4

-4 -2 0 2 4 x

Figure 5.3: Snapshots of the isotherm T = 1 for the homeotropic alignment, obtained via NP T MC simulations. Configurations at Pk = 1.5 (top), where the system enters the nematic phase and at Pk = 6 (middle), where the system is in a solid state, are shown. For Pk = 6 the bottom picture shows configurations of the second (empty circles) and third (filled circles) layer. 5.1. HOMEOTROPIC ALIGNMENT 49 from time to time. This is indicative of a nonequilibrium situation, where the crystal structure reorganizes spontaneously. At Pk = 8 we start expanding our system, observing again hysteresis. In the expansion process the nematic regime is quite narrow between Pk = 1.5 and Pk = 1.2, which is very similar to the bulk.

5.1.2 Isochore, including structure analysis As already known from the bulk isochores n = 0.7 and n = 1 obtained via NV T MC simulations (figures 4.5-4.8) we have to be careful with the term isochore, because the particle density may be inhomogeneous. For the high bulk density n = 1 the system remained homogeneous, so we chose a similar density for the simulations in the confined system, namely n = 0.9. For this global particle density n = N/V we expect a slightly higher density in the middle of the simulation cell, because there are no particles near the walls. Therefore, these MC simulations with walls are expected to be comparable with the bulk results already discussed. The NV T MC simulations are carried out with N = 1350 particles in a box with lengths dx = dy = 10 and dz = 15, resulting in the density n = N/V = 0.9. Here, only the cooling procedure will be presented. Figure 5.4 shows pressure p = tr(p)/3, nematic order parameter S2, smectic order parameter %1 and fluctuation of the energy ∆E versus the temperature T . The temperature was cooled down from T = 1.8 to T = 0.6. The bottom curve (figure 5.4(d)) shows the amounts of crystal structures for the confined system. The structures are obtained by common neighbor analysis[77]: For each particle, polyhedra formed by relevant neighbors are translated into planar graphs with polyhedra edges as branches and neighbors as nodes. The comparison with planar graphs of main ideal crystal structures (bcc1, fcc2 and hex3) reveals the structure, which surrounds each particle. The ratio between the number of particles which are found to belong to a structure and the total number of particles is taken as a measure of the amount of crystal structure. The sum of these amounts may be greater than 1, because a particle may be part of different structures. This structure analysis is stable against thermal displacements and reorientations. However, to gain a good statistical information the system should be very large (N > 10 000 instead of N = 1350). Therefore, the method was adapted to average over 10 different configurations. At high temperatures T > 1.4 the system is in an isotropic phase. The nematic and smectic order parameters as well as the amounts of crystal structures are low. At T = 1.28 the IN phase transition occurs, which is confirmed by a peak in the energy fluctuation and an increase of the nematic order parameter S2. This transition is much smoother than in the bulk system as already known from the comparison of the isotherms. The knowledge of the accurate transition temperatures is difficult to obtain because of the relatively large fluctuations of the energy fluctuation ∆E. This would be better with higher sweep numbers (50 000 for these simulations, thereof 45 000

1body centered cubic 2face centered cubic 3hexagonal close packed 50 CHAPTER 5. MC RESULTS WITH CONFINING WALLS

(a) 9

6 p 3

(b) S 0.8 ρ2 1 0.6 0.4 0.2 order parameter 0 (c) 0.08

0.06 E ∆ 0.04

0.02

(d) 0.6 amount of fcc amount of hex 0.4 structure analysis

structure 0.2

0 0.6 0.8 1 1.2 1.4 1.6 1.8 T

Figure 5.4: Cooling process for the isochore n = 0.9 of the homeotropic aligned system, obtained via NV T MC simulations: (a): pressure p, (b): order parameters S2 and ρ1, (c): fluctuation of internal energy ∆E. (d): amount of fcc/hex crystal structure obtained by common neighbor analysis[77]. The Simulations are done in the NV T ensemble. 5.1. HOMEOTROPIC ALIGNMENT 51 for averaging). From T = 1.2 to T = 0.74 there is a smooth increase of positional order. This can be seen in the smectic order parameter ρ1 as well as in the amounts of fcc and hcp crystal structures. The occurrence of fcc structures is typical for Lennard Jones-like model systems. In the nematic phase the fcc structure dominates the system. The nematic-solid phase transition at T = 0.74 is accompanied by a hardly visible peak in the energy fluctuation, a slightly jump in the pressure p and a jump of the smectic order parameter ρ1 to values greater then 0.8. In addition, the amount of hexagonal close packed structure exceeds that of the fcc structure. This confirms the observation of a hexagonal structure in the first layers done for the solid phase configuration obtained for the isotherm T = 1 (figure 5.3). The nematic phase configuration for T = 1.2 is shown in figure 5.5 together with the inhomogeneous distributions of particle density nlocal(z) and nematic order parameter S2(z). It can be seen, that near the walls the layering mechanism gives rise to particle

3 S2 nlocal 2.5 T = 1.2

1.5

0.5

0 -8 -6 -4 -2 0 2 4 6 8 z

Figure 5.5: Top: Snapshot of the isochore n = 0.9 for the homeotropic alignment, taken at T = 1.2. Bottom: Distribution of particle density and nematic order parameter for the same phase space point. The simulations are performed in the NV T ensemble. 52 CHAPTER 5. MC RESULTS WITH CONFINING WALLS density peaks. We clearly are already in the nematic phase, because the local nematic order parameter is almost constant S2(z) 0.6. The configuration shows that the director is indeed parallel to the wall normal.≈ The particle density fluctuates around n 1 in the middle of the cell, so comparison with the n = 1 isochore for the bulk local ≈ system is justified: There, the IN and NS phase transitions occurred at TIN = 1.34 and TNS = 0.92, compared with TIN = 1.28 and TNS = 0.74 for the confined system. This is surprising, because it is expected that the confined system reaches the ordered phases at higher temperatures instead of showing lower transition temperatures. This may be due to the faultiness of the transition temperatures in the confined system (high fluctuations). Better results will be achieved with higher sweep numbers and/or in the NP T ensemble, where the equilibrium may be reached faster. However, the main difficulty may be the smoothness of the transitions. A further structure analysis can be made by comparing the distribution of directions to the next neighbors (which is accessible via the common neighbor analysis[77]) with the distribution of particle orientations. The result can be seen in figure 5.6. First,

0 ϕ distribution of directions to distribution of next neighbors

orientations

¤ ¢ T = 1.0

0 ¥

£ T = 0.6 ©¨

0 ¦¨§ ¡ 0 0

Figure 5.6: Distribution comparison of the directions to the next neighbors[77] (left) and the orientations (right) for the temperatures T = 1.0 (top) and T = 0.6 (bottom). The data are obtained from NV T MC simulations of the isochore n = 0.9. have a look at the right column, where the distribution of orientations uˆj is shown. At T = 1.0 the director n = eˆz of this nematic phase conﬁguration can slightly be seen. This is more clear at T = 0.6, where the system is in a solid state. Both states are accompanied by neighbor distributions, which reveal the hcp structure (ﬁgure 5.7): 5.1. HOMEOTROPIC ALIGNMENT 53

Figure 5.7: Next neighbors for the hexagonal close packed (hcp) crystal structure.

The hexagonal layers with 6 next neighbors are parallel to the layers of the centers of masses of the particles. They can be seen as ϕ = π/2-and-3π/2-layer in the neighbor distribution. Another three neighbors each in direction of n are visible as circles around θ = π/2. Altogether the hcp structure is clearly conﬁrmed§ in the left column of ﬁgure 5.6. In addition to this the correlation with the director n reveals the impact of the geometry of the wall, because its normal dominates both the orientations of the particles and the orientation of the crystal structure.

5.1.3 Isobars

In analogy to section 4.3 we finally have a look at the family of isobars Pk = 0.1, Pk = 0.3, Pk = 0.5, Pk = 0.7 and Pk = 1.0, which are obtained via NP T MC simulations. Here, we combine the advantage of the studied isotherm (fast convergence and no voids in the system due to the NP T MC method) and the isochore (temperature can be used as parameter during one simulation series, therefore omitting high pressures). Here, all MC NP T simulations are carried out with N = 1000 particles with box lengths dz = 15 and variable d and d . During the simulations these lengths varied from d = d = 7.9 x y h xi h yi to d = d = 9.9. h xi h yi The main observables (bulk particle number density nbulk, energy E, smectic order parameter %1 and the nematic order parameter S2) are plotted versus the temperature for all isobars in figure 5.8. The bulk particle number density nbulk is the particle density measured in the middle of the simulation cell (compare figure 5.2). The isotropic- nematic phase transition is observed for all the isobars during the cooling procedure from T = 1.0 to T = 0.8. Qualitatively, the behavior of the observables is the same as in the bulk system (figure 4.9). The transition temperatures TIN (0.88, 0.95) are almost identical with the bulk behavior. They are difficult to see in figure∈ 5.8, because the strong nematic-solid phase transition dominates the picture. The great jumps of E and nbulk between T = 0.82 for Pk = 1.0 and T = 0.9 for Pk = 0.1 are found to mark the nematic-solid phase transition. This is confirmed by the jump of the smectic order parameter %1. In the solid state reorganizations of the crystal structure may lead to finite-time effects like jumps in the particle density. A high density jump is observed 54 CHAPTER 5. MC RESULTS WITH CONFINING WALLS

1.1 P = 0.1 P = 0.3 0.8 1 P = 0.5 P = 0.7 P = 1.0 0.6 0.9 1 bulk ρ n 0.4 0.8

0.2 0.7

0.8 0.85 0.9 0.95 1 0.8 0.85 0.9 0.95 1 -5 0.8

-6 0.6 2 E

-7 S 0.4 -8 0.2 -9

0.8 0.85 0.9 0.95 1 0.8 0.85 0.9 0.95 1 T T

Figure 5.8: The cooling process for diﬀerent isobars with homeotropic alignment. The bulk number density nbulk, the energy E and the order parameters %1 and S2 are plotted vs. the temperature T for diﬀerent pressure values Pk. The isobars are obtained via NP T -MC.

for the isobar Pk = 0.7 such that the isobar Pk = 1.0 reaches less high particle densities at small temperatures. The IN transitions occur at densities around 0.8 as in the bulk system. The increase of the nematic order parameter is very smooth, which leads to an additional hiding of the correct transition temperatures. In summary, the IN transition temperatures are slightly raised. The NS transition temperatures are large enough to be observed above T = 0.8 for all isobars. 5.2. TWIST ALIGNMENT 55 5.2 Twist alignment

Finally, we use the slab geometry again, but with different anchoring. The anchoring 2 functions are now chosen to be different for the two walls, namely g1(uˆ) = uˆy for the wall at z = 7.5 and g (uˆ) = uˆ2 for the wall at z = 7.5. 1 − 2 x 2 2 −10 −4 2 2πρw 5 (z zw) (z zw) uˆy (w = 1) Φfw(r, uˆ) = − − − (5.2) 2πρ 2 (z z )−10 (z z )−4uˆ2 (w = 2) ( w ¡ 5 − w − − w x¢ ¡ ¢ where again ρw = 0.8. This will favor homogeneous planar alignment at the walls with a director field, aligned along the y-axis at z = 7.5 and along the x-axis at z = 7.5. Therefore the director field is forced to be inhomogeneous.− In the nematic phase, the simulation scenario will be equivalent to a twisted nematic cell without external fields. For such optical applications anchoring effects are very important[30, 21].

5.2.1 Isotherm In figure 5.9, data for the isotherm T = 1 are given, where the system is compressed from Pk = 0.1 to Pk = 8 and then expanded back to Pk = 0.1. During this procedure, isotropic, nematic and solid phases are found. For low pressures Pk < PIN = 1.5 the system is in an isotropic phase. The order parameters increase with increasing pressure for the same reason as for the other geometries. At PIN = 1.5 the peak in the energy fluctuation indicates a transition in the nematic phase. The particle density measured in the middle of the simulation cell smoothly changes around n = 0.86. This seems to be a common value for the IN transition for the isotherm T = 1. Here, the nematic order parameter seems to remain quite small. The inhomogeneity of the director field artificially keeps this order parameter small, because it was defined for a homogeneous system. To find out the real value of the nematic order parameter, we consider local values of S2(z), as shown in figure 5.10. In the isotropic phase at Pk = 0.2 the nematic order parameter turns out to be quite large (almost 0.3). This is due to the finite extent of layers into which the box was cut in order to measure local values. On average, each nonempty layer contains about 15 particles. In the nematic phase at Pk = 3, where the global order parameter was lower than 0.5, we actually find local values of S2(z) 0.7. This number may be even larger for layers in the immediate vicinity of the walls.≈ Configurations in the nematic phase may be seen as examples for a twisted nematic cell without external fields. In Chapter 6 we will analyze the behavior of the twist deformation in more detail. There, the twist elastic coefficient K2 will be calculated. At PNS = 4.6 solidification takes place, accompanied by a peak in the energy fluctuation and jumps of the particle density and the smectic order parameter. Solidification pressure is between those observed in the bulk and in the homeotropically aligned cell. It is more difficult to achieve long-range positional order if the orientational order is only short-range due to the inhomogeneous director field. During some reorganization processes the crystal eventually reaches a configuration, where even the global nematic order parameter has values S2 > 0.6. A snapshot of such a configuration (at Pk = 6) together with one of the nematic phase is shown in 56 CHAPTER 5. MC RESULTS WITH CONFINING WALLS

0.9 n

0.8

0.7 2 4 6 8 0.8

0.6

0.4 order parameter 0.2 S ρ2 1 2 4 6 8

0.2 E ∆

0.1

2 4 6 8

P||

Figure 5.9: Isotherm T = 1 for the twisted homogeneous planar aligned system, obtained via NP T MC simulations. For compression and expansion processes (indicated by the arrows) the bulk number density n, the order parameters S2 and %1 and the energy ﬂuctuation ∆E are plotted as function of the pressure Pk. 5.2. TWIST ALIGNMENT 57

1 P|| = 3 P|| = 0.2 0.8

0.6 2 S 0.4

0.2

0 -8 -6 -4 -2 0 2 4 6 8 z

Figure 5.10: NP T MC Simulation with twisted homogeneous planar alignment. Local nematic order parameter S2(z) for the isotherm T = 1 with diﬀerent pressures. At Pk = 3 the system is in a nematic phase.

figure 5.11. The surprisingly high global nematic order parameter S2 = 0.7 can be rationalized as follows: In this crystal the right wall has gained control over almost the whole cell. Therefore, apart from the first three layers near the left wall, the director is homogeneous. Because of a spontaneous break of symmetry caused by the right wall, its anchoring mechanism dominates the system. This can also be seen in figure 5.12, where the local azimuthal angle ϕ(z) of the director n(z), defined through tan(ϕ) = ny/nx, is plotted for different pressures. A value of ϕ = 0◦ belongs to the left wall at z = 7.5, where the particles are anchored along the x-axis, and ϕ = 90◦ belongs to the anchoring− at the right wall. In nematic phases (see Pk = 2 or Pk = 4) we find a linear tilt, so the director changes smoothly from one side to the other. Linear director profiles are known to occur for a lattice model liquid crystal confined in a hybrid cell[19]. For Pk = 6 the director jumps near the left wall and remains homogeneous otherwise. Expanding the simulation box in small pressure steps back to the isotropic phase we find again a large hysteresis. This time, we observe a single solid-isotropic transition at PSI = 1.2. During this transition the peak of the energy fluctuation as well as the jump in the particle density are very sharp. The nematic phase only occurs during the compression process, such that we have a monotropic nematic phase.

5.2.2 Isochore The success of the calculation of the isochore n = 0.9 for the homeotropic aligned system motivates the calculation for the twisted homogeneous planar aligned system as well. In section 5.1.2 we found that the isotherm n = 0.9 calculated in the NV T ensemble is accompanied by an eﬀective bulk densities of n 1, where the system remains homogeneous at least in the nematic phase. ≈ 58 CHAPTER 5. MC RESULTS WITH CONFINING WALLS

Figure 5.11: Snapshots of the isotherm T = 1 for the twisted homogeneous planar alignment, obtained via NP T MC simulations. Conﬁgurations at Pk = 3 (top), where the system is in a nematic phase and at Pk = 6 (bottom), where the system is in a solid state, are shown.

ϕ -45

P|| = 6 P|| = 4 -90 P|| = 2 -6 -4 -2 0 2 4 6 z

Figure 5.12: Twist of azimuthal angle ϕ of the director n(z) along the cell. The con- ﬁgurations are taken from the isotherm T = 1 with 90◦ twisted homogeneous planar alignment, obtained via NP T MC simulations, for diﬀerent pressures. 5.2. TWIST ALIGNMENT 59

(a) 12

6 p

(b) 0.6

0.4 2 S

0.2

0 0.1 (c)

0.08 E ∆ 0.06

0.04 0.8 1 1.2 1.4 1.6 1.8 T

Figure 5.13: Isochore n = 0.9 for the twisted homogeneous planar aligned system, obtained via NV T MC simulations,: (a): pressure p, (b): order parameter S2, (c): ﬂuctuation of internal energy ∆E. The cooling procedure is calculated via NV T -MC.

Figure 5.13 shows the result of the NV T MC simulation of the isochore n = 0.9. As for the other geometries, the scalar pressure, the nematic order parameter S2 and the energy fluctuation ∆E are plotted versus the temperature, which was lowered from T = 1.8 to T = 0.8. Again, the three phases isotropic fluid, nematic fluid and crystalline solid are observed during the cooling process. The isotropic-nematic phase transition occurs at T = 1.26 indicated though a peak of ∆E as well as a smooth increase of the nematic order parameter S2. The pressure values give no hint for this transition. This is again 60 CHAPTER 5. MC RESULTS WITH CONFINING WALLS due to the weakness of the first-order IN phase transition. In the nematic phase, the global measured nematic order parameter S2 remains again quite small (S2 < 0.55) due to the inhomogeneous director field in the twisted anchored system. The solidification at T = 0.96 is accompanied by small jumps of the pressure P and the nematic order parameter S2 as well as a peak of the energy fluctuation ∆E.

4 S T = 0.9 2 nlocal

0 -8 -6 -4 -2 0 2 4 6 8 z

Figure 5.14: Top: Snapshot of the isochore n = 0.9 for the twisted homogeneous planar alignment, taken at T = 0.9. Bottom: Distribution of particle density and nematic order parameter for the same phase space point. The simulations are performed in the NV T ensemble.

A snapshot of a solid phase configuration at T = 0.9 is shown in figure 5.14 together with the distribution of the particle number density and local nematic order parameter along the cell. The peaks of the number density nlocal(z) show the layers which were formed one after the other because of the influence of the wall-particle interaction. In the left side of the simulation cell a homogeneous director field is achieved, which is dominated by the left wall. Therefore, the global nematic order parameter is larger than in the nematic phase (S2 > 0.6). The local nematic order parameter has values 5.2. TWIST ALIGNMENT 61 about S(z) 0.8. For z = 2 the solidification is not yet perfect. No crystalline structure can be seen≈by rotating the perspective.

5.2.3 Isobars To compare with the bulk and the homeotropic aligned system we ﬁnally simulate the series of isobars Pk = 0.1, Pk = 0.3, Pk = 0.5, Pk = 0.7 and Pk = 1.0. Again, these isobars are obtained via NP T MC simulations with N = 1000 particles and dz = 15. The results are shown in ﬁgure 5.15. In the conclusions (Chapter 7) the comparison with

0.9 0.2 1 bulk ρ

n 0.8 P = 0.1 P = 0.3 P = 0.5 0.1 P = 0.7 0.7 P = 1.0 0.8 0.85 0.9 0.95 1 0.8 0.85 0.9 0.95 1

-5 0.5

0.4 -6 2 E S 0.3

-7 0.2

0.1 -8 0.8 0.85 0.9 0.95 1 0.8 0.85 0.9 0.95 1 T T

Figure 5.15: Isotropic-nematic phase transition through cooling for different isobars for twisted homogeneous planar alignment. The bulk number density nbulk, the energy E and the order parameters %1 and S2 are plotted vs. the temperature T for different pressure values Pk. The isobars are obtained via NP T -MC. the bulk and the homeotropic aligned system will be shown in one picture. Here, the bulk number density nbulk, the energy E and the order parameters %1 and S2 are plotted versus the temperature T , which was lowered from T = 1 to T = 0.8. Here, the solid phase is not achieved for any isobar (% < 0.23), revealing that it is more difficult to freeze the twisted homogeneous planar aligned system. We already saw during comparison of the isotherms T = 1 for different anchoring mechanisms that the homeotropic aligned 62 CHAPTER 5. MC RESULTS WITH CONFINING WALLS system freezes more easy. Because of the lack of the solid phase, the number densities achieve not as high numbers as in the homeotropic aligned system. The IN phase transition takes place at TIN (0.88, 0.95), indicated through a high energy fluctuation (visualized via errorbars) and∈an increase of the nematic order parameter S2, which is not quite large because of the inhomogeneous director. The transition temperatures TIN are almost identical with the bulk results (see section 4.3) and the results for the homeotropic aligned system (section 5.1.3). Chapter 6

Twist elastic coeﬃcient

The switching behavior of the TN cell is influenced by the value of the elastic constants K1, K2 and K3. They play a critical role in describing how nematics respond to applied fields. Experimental, they are obtained by observing Frederiks transitions[48, 49], where an external field is used to reorient the director field. In order to introduce these coefficients the density of the free energy f = F/V has to be studied. The phenomenological continuum theory of liquid crystals due to Oseen and Frank[83] uses the director field n(r) to describe a liquid crystal. The free energy is expanded around the state of uniform alignment. After taking into account the head-tail symmetry n n the free energy density f is supposed to take the approximated form − ≡ K K K f = f + 1 ( n)2 + 2 [n ( n)]2 + 3 [n ( n)]2 , (6.1) 0 2 ∇ · 2 · ∇ × 2 × ∇ × splay twist bend where f0 is the free|energy{z densit} y|of the {zuniform director} | field.{zThe coefficien} ts K1, K2 and K3 are called Frank elastic constants. Distortions of the director field n(r) from the uniform distribution give rise to free energy changes. The first term describes a splay deformation, the third a bend deformation. Here, the interesting distortion is the second term, which occurs due to a twist deformation. In a twisted nematic cell the director field is assumed to be πz cos 2L n(r) = sin πz . (6.2)  2L  0   Then, z = 0 marks the left wall and has a director n = ex, where z = L marks the right wall with the director n = ey. In between there is a linear increase of the twist angle of π the director. For this director field we have n = 2L n and n = 0. Therefore, only the twist deformation part in the Frank∇free× energy− density surviv∇ · es. Because of K K π 2 f = 2 [n ( n)]2 = 2 (6.3) twist 2 · ∇ × 2 2L we expect in a cell with volume V a change of the free³energy´ of V K ∆γ 2 ∆F = 2 (6.4) 2 L µ ¶

63 64 CHAPTER 6. TWIST ELASTIC COEFFICIENT due to the twist deformation of ∆γ = π/2 along the distance L. The goal of this Chapter is to study the twist elastic coefficient K2 for the present liquid crystal model. For this we have a look at two different strategies. Both use MC simulation in a bulk nematic phase. The first is a well-known method in the literature[53, 54] for calculating all three elastic constants. It is based on the calculation of certain susceptibilities[57], which are related to the elastic constants[84]. These susceptibilities are given with help of the alignment tensor in Fourier space and will be used in section 6.2. The second method is derived in section 6.3, where the distortion of the equilibrium free energy due to a twist deformation is calculated in order to obtain the twist elastic constant. MC results of both methods will be presented and compared in section 6.4. A first estimate of the energy change due to a twist deformation can be done simply by applying the appropriate deformation and measure the potential energy in the system, neglecting entropic contributions to the free energy F = E T S. This will be done in section 6.1. −

6.1 First estimate

In order to apply a twist deformation we study the model liquid crystal at T = 1 and P = 3. Here, we will find a stable nematic phase. The simulations are carried out in the NP T ensemble with N = 1000 particles, a box length of L = dz = 15 and a desired lateral pressure of Pk = 3. Walls are needed to control the director field. In a first 2 simulation we use two identical anchoring functions g1(uˆ) = g2(uˆ) = uˆx to obtain a homogeneous director field n = eˆx. Then a series of simulations follows where the first wall is rotated step by step. For the whole series of simulations the anchoring functions are chosen to be

2 (uˆx cos φ + uˆy sin φ) (w = 1, left wall) gw(uˆ) = 2 (6.5) (uˆx (w = 2, right wall) with varying rotation angle φ [0, π/2]. During these simulations the twist deformation ∈ will increase. The results are shown in figure 6.1. The bulk density nbulk and the fluid- fluid potential energy E are shown versus the rotation angle φ. The bottom curves show beneath the nematic and smectic order parameter the biaxiality parameter b, defined through (3.34). In the accuracy of the errorbars the bulk particle density nbulk and the potential energy per particle E remain constant. However, one may find a little tendency of the energy to increase with φ. For the initial simulation φ = 0 we indeed get a homogeneous director, as can be seen in figure 6.2(a). The director n = eˆx points into the paper plane. The nematic order parameter is about S2 = 0.7, the smectic order parameter %1 < 0.1. The low biaxiality parameter b 0.2 confirms the uniaxiality of the alignment tensor. This will be lost for the inhomogeneous≈ director field for φ > 0. Therefore the parameter b continously increase with φ. At the same time the globally defined nematic order parameter S2 will decrease for the same reason. The smectic order parameter stays low (% 0.2) for all φ. The bulk density n is about n 0.94 for all rotation 1 ≈ bulk ≈ 6.1. FIRST ESTIMATE 65

0.95 bulk n 0.94

0.93 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

-7.2 E

-7.3

-7.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 S ρ2 0.8 1 b

0.6

order 0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 φ

2 Figure 6.1: Twist deformation caused by a rotated left wall g1 = (uˆx cos φ + uˆy sin φ) 2 and a ﬁxed right wall g2 = uˆx. The bulk density nbulk, potential energy E and order parameters S2 and %1 are shown vs. the rotation angle φ. In addition the biaxiality parameter b is plotted. The average values are obtained via NP T MC simulations at Pk = 3 and T = 1. 66 CHAPTER 6. TWIST ELASTIC COEFFICIENT

(a)

(b)

(c) 90 φ = 90 φ = 67 φ 75 = 45 φ = 22 φ = 0 60

45 phi 30

-6 -4 -2 0 2 4 6 z

Figure 6.2: (a) and (b): Snapshots of conﬁgurations obtained for φ = 0 and φ = π/2. (c): Twist of azimuthal angle phi=ϕ of the director n(z) along the cell for diﬀerent wall rotations φ, including φ = 0 and φ = π/2. The simulations are performed in the NP T ensemble with Pk = 3 and T = 1. 6.2. METHOD 1: ALIGNMENT TENSOR IN FOURIER SPACE 67

angles φ. This confirms former results for the isotherms T = 1, analyzed at Pk = 3 for different geometries. It can be concluded that all the simulations for different φ are done in a nematic phase. Finally the φ = 90◦ twisted cell is reached (see figure 6.2(b)). In figure 6.2(c) the distribution of the azimuthal angle ϕ(z) of the director n(z) is shown, revealing a linear relation for all φ [0, π/2]. The potential energy E(φ) per particle according to the fluid-fluid interaction∈ shows a little increase with φ. The errorbars are quite large but the effect is visible and may be estimated to an energy shift of E(π/2) E(0) 0.05. Neglecting the entropic influence we have a first guess for the − ≈ free energy change ∆F N(E(π/2) E(0)) 50 in the volume V = Nnbulk = 943 due to a π/2 twist deformation≈ along L =−15, whic≈h is

V K π 2 ∆F = 2 50 (6.6) 2 2L ≈ ³ ´ With this approximation we get K2 10 for the nematic phase space point at P = 3 and T = 1. ≈

6.2 Method 1: Alignment tensor in Fourier space

According to the literature[84, 57] the elastic constants K1, K2 and K3 (see (6.1)), are related to certain correlations of components of the alignment tensor in Fourier space

V N Qˆ (k) := uˆ uˆ exp(ik r ) . (6.7) N j j · j j=1 X

Assuming that the director is parallel to eˆz and k = (kx, 0, kz) lies in the xz-plane, the symmetry breaking variables Qxz and Qyz lead to susceptibilities

2 ˆ ˆ S2 V kBT Qxz(k)Qxz( k) = 2 2 and (6.8) − kxK1 + kz K3 D E 2 ˆ ˆ S2 V kBT Qyz(k)Qyz( k) = 2 2 , (6.9) − kxK2 + kz K3 D E which is valid for small k. The k vector components used are restricted by the length scales of the simulation. The minimum kx can be kmin = 2π/Lx. Therefore the simulation box has to be large in order to achieve good results. These equations were already used to calculate the elastic constants for other models like hard ellipsoids[53, 54] or a Gay-Berne ﬂuid[55]. Here, we extract only the twist elastic constant by setting kz = 0 in (6.9) and calculating

2 S2 V kBT ! 2 E(kx) := = K2kx (6.10) Qˆ (k )Qˆ ( k ) yz x yz − x D E for several kx 2π/Lx. The twist elastic coeﬃcient K2 can then be obtained by ﬁtting ! 2≥ E(kx) = K2kx for small k. 68 CHAPTER 6. TWIST ELASTIC COEFFICIENT

The calculation of Qˆyz(k) has to be done in a coordinate system, where n = eˆz. However, in the space ﬁxed system used during a simulation, this is not the case. Instead, the director may be written as

nx sin ϑ cos ϕ n = n = sin ϑ sin ϕ . (6.11)  y    nz cos ϑ     These angles ϕ and ϑ can be obtained for any conﬁguration during the MC simulation either by calculating the greatest eigenvalue of the alignment tensor with the corresponding eigenvector n or by calculating the mean orientation n statistically. Two other unit vectors, which form an orthogonal basis together with n, are ωˆ cos ϑ cos ϕ αˆ sin ϕ x x − ωˆ = ωˆ = cos ϑ sin ϕ and αˆ = αˆ = cos ϕ . (6.12)  y     y    ωˆ sin ϑ αˆ 0 z − z         With help of the transformation n ωˆ ωˆ ωˆ n ωˆ n 0 x x y z x · n αˆ αˆ αˆ n = αˆ n = 0 (6.13)  y   x y z   y      n → n n n n n · n 1 z x y z z ·           we get the correct components of the director. Corresponding other transformations give

x ωˆ r , uˆ αˆ uˆ and uˆ n uˆ , (6.14) j → · j jy → · j jz → · j which are needed to calculate

V N V N Qˆ (k ) = uˆ uˆ eikxxj (αˆ uˆ )(n uˆ ) exp(ik (ωˆ r )) (6.15) yz x N jy jz → N · j · j x · j j=1 j=1 X X Note, that the unit vectors ωˆ and αˆ may be interchanged by each other. This would transform into another valid basis. Finally, with help of the complex relations eikxxj = cos(k x ) + i sin cos(k x ) and z 2 = a2 + b2 for z = a + ib and real a, b, we have x j x j | | N 2 2 V Qˆ (k )Qˆ ( k ) = Qˆ (k ) = (αˆ uˆ )(n uˆ ) exp(ik (ωˆ r )) yz x yz − x yz x N · j · j x · j ¯ j=1 ¯ ¯ ¯ ¯ X ¯ ¯ ¯ ¯ 2 ¯ ¯ V N ¯ ¯ ¯ = (αˆ uˆ¯ )(n uˆ ) cos(k (ωˆ r )) + ¯ N · j · j x · j Ã j=1 ! X 2 V N (αˆ uˆ )(n uˆ ) sin(k (ωˆ r )) . (6.16) N · j · j x · j Ã j=1 ! X

The freedom of basis choice allows us to calculate a second term Qˆyz(kx)Qˆyz( kx) with exchanged vectors ωˆ αˆ . The exchange of these basis vectors means an exchange− of x- ↔ 6.2. METHOD 1: ALIGNMENT TENSOR IN FOURIER SPACE 69

and y-axis in the coordinate system with n = eˆz. For large systems this should give the same result. For our ﬁnite size system we get a smaller statistical error by taking the middle value of both results. This result can be averaged in a MC simulation in order to obtain the function E(k ). This is done for a set of M k -values k , k , . . . , k , x x { x1 x2 xM } where the smallest value is given through 2π/Lx. The resulting function should be of ! 2 the form E(kxn) = K2kxn. To get the best K2 that ﬁts this function, we minimize the error

M 1 1 2 e(K ) = . (6.17) 2 E(k ) − K k2 n=1 xn 2 xn X µ ¶ The reciprocal function is taken to calculate the error, because in this way the small kx-values are weighted more than the large ones. Minimizing (6.17) with respect to K2 leads to

de 2 M K 1 1/k4 = 2 = 0 K = n nx . (6.18) dK K3 E(k )k2 − k4 ⇒ 2 1/(E(k )k2 ) 2 2 n=1 xn xn xn n xn xn X µ ¶ P For simple checks of the method a large bulk conﬁgurationP was produced. This was achieved with help of a NV T MC simulation with N = 7440 particles in a cubic box of volume V = 203 at temperature T = 1. At equilibrium the nematic order parameter was S2 = 0.6. The pressure obtained is P = 3 such that comparison with the rough estimate at (T, P ) = (1, 3) is possible. Starting from this conﬁguration we averaged over 20 000 steps to get M = 20 values E(k ) in the range k 2π/L , . . . , 4π/L . These can xn xn ∈ { x x}

1.5 ) x E(k 1

0.5

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

kx/kmin

2 Figure 6.3: Fit of E(kx) = K2kx versus kx/kmin, where kmin = 2π/Lx. The points are average values E(kx) from a NV T MC simulation with n = 0.94 and T = 1. The line is the fit with K2 = 3.35. The points in brackets are not considered. be seen in figure 6.3 together with the fit function K k2 with K = 3.35 0.07. The error 2 x 2 § 70 CHAPTER 6. TWIST ELASTIC COEFFICIENT

2 ∆K2 = 0.07 was calculated as the mean displacement of E(kx)/kx from K2. Points with kx > 1.74kmin were not considered, because they clearly are too large for the parabolic behavior. So, only 14 values are taken into account. Taking even fewer points does not improve the accuracy of the fit because of an offset E0 in the parable. The alternative 2 approach taking into account this offset and fitting to the function E(kx) = E0 + K2kx leads to unsatisfactory results with very different K2- and E0- values for different system sizes. Better fitting results are expected to be produced with kz = 0, where we would obtain the other elastic constants too. The value of K = 3.35 0.076 is a factor 3 smaller 2 § than the rough estimate K 10 of section 6.1. 2 ≈

6.3 Method 2: Free energy distortion

Another possibility to obtain the twist elastic constant K2 will be based on the calculation of the free energy distortion due to a twist deformation in analogy to elastic coefficients for shear deformations[85]. This will enable us to calculate K2 as an average value of an observable. We start with the assumption of the equilibrium free energy, which is guaranteed by the MC method. Let Z be the canonical partition function. Because of F = kT ln Z the total free energy in a non-disturbed nematic phase shall be in its minimum− Φ (Γc) F [Φ ] = kT ln exp tot dΓc , (6.19) tot − − kT Z µ ¶ where Γc (r , uˆ , . . . , r , uˆ ) are configurations in the nematic phase. The orien- ≡ 1 1 N N tations uˆj fluctuate around a common director. Any orientational deformation uˆj c → uˆj + δuˆj leads to an new configuration Γdef (r1, uˆ1 + δuˆ1, . . . , rN , uˆN + δuˆN ) with a shifted total energy, which may be estimated≡through the Taylor expansion

N ∂Φ 1 N N ∂2Φ Φ (Γc ) Φ (Γc) δuˆ tot + δuˆ δuˆ : tot . (6.20) tot def − tot ≈ j · ∂uˆ 2 j k ∂uˆ ∂uˆ j=1 j j=1 j k X X Xk=1 2 δΦtot δ Φtot

These deformations would enhance| the free{z energy} by| {z }

∆F = F [Φ (Γc )] F [Φ (Γc)] (6.21) tot def − tot δF 1 δ2F δΦ (Γc) dΓc + δΦ (Γc )δΦ (Γc) dΓc dΓc ≈ δΦ (Γc) tot 2 δΦ (Γc )δΦ (Γc) tot 1 tot 1 Z tot ZZ tot 1 tot δF δ2F with the functional| deriv{z atives deﬁned} |through {z }

1 Φtot c δF = exp δΦtot dΓ exp Φtot dΓc − kT − kT Z µ ¶ = δΦ (6.22) hR toti¡ ¢ 6.3. METHOD 2: FREE ENERGY DISTORTION 71 and

Φtot c 2 1/(kT ) δΦtot exp kT dΓ Φtot c δ F = − 2 exp δΦtot dΓ + exp Φtot dΓc − kT R − kT ¡ ¢ Z µ ¶ 1 Φtot 2 c ¡R ¡ ¢ exp¢ δ Φtot δΦtotδΦtot/(kT ) dΓ exp Φtot dΓc − kT − kT Z µ ¶ − ¡ ¢ R ¡ δ¢Φ δΦ δΦ 2 = δ2Φ h tot toti − h toti . (6.23) tot − kT 2 δ FBG 2 ® δ Ffluct | {z } | {z } The first derivative δF should vanish in equilibrium, because the free energy is in its 2 2 minimum. The second derivative δ F is split in a Born-Green part δ FBG and a fluc- 2 tuation part δ Ffluct. The fluctuation part equals the square of the fluctuation of δΦtot, which may be small, because the average value of δΦtot itself should vanish. This will be checked during simulations. The remaining problem is to calculate the average value 2 of the observable δ Φtot, which can be obtained according to (6.20) for any given orientational deformation δuˆj. Then the twist distortion of the free energy is given through

1 ∆F δ2Φ . (6.24) ≈ 2 tot ® We switch to the special case of the potential (compare equations (2.6)-(2.3))

15 1 Φ = 4 r−12 r−6 1 + ε (uˆ uˆ )2 tot jk − jk 2 1 j · k − 3 Xj

∆γ δuˆ = ω uˆ = (r ωˆ )ωˆ uˆ . (6.26) i i × i L i · × i

The twist angle ∆γ (r ωˆ ) increases along the direction ωˆ , which should be chosen to be L i · i perpendicular to the director n. In the case of the twisted nematic cell, where ωˆ = eˆz and n = eˆ , it follows δuˆ = ∆γ z eˆ uˆ . At the left wall z = 0 the particles do x i L i z × i i not feel any orientational deformation and at the right wall zi = L they feel the full twist δuî = ∆γ eˆz uî, where usually ∆γ = π/2. However, in a bulk simulation the director may have an×y direction n. The twist axis ωˆ may be chosen as the basis vector introduced in equation (6.12), such that ωˆ n = 0 is fulfilled. For any configuration the twist axis is calculated in this way. The· orientational dependent part of the total 72 CHAPTER 6. TWIST ELASTIC COEFFICIENT potential energy yields a derivative

∂Φtot −6 ∂ 2 2 2 = 30 rik ε1(uˆi uˆk) + ε2 (uˆi ˆrik) + (uˆk ˆrik) ∂uˆj − ∂uˆj · · · i

The ﬁrst order energy distortion can be calculated according to

N ∂Φ N ∂Φ N ∂Φ δΦ = δuˆ tot = (ω uˆ ) tot = ω uˆ tot tot j · ∂uˆ j × j · ∂uˆ j · j × ∂uˆ j=1 j j=1 j j=1 j X X X µ ¶ N ∂ = ω Φ with := uˆ (6.28) j · Lj tot Lj j × ∂uˆ j=1 j X With this twist operator applied to the total potential energy Lj

∂Φtot jΦtot = uˆj (6.29) L × ∂uˆj N = 60 r−6 [ε (uˆ uˆ )(uˆ uˆ ) + ε (uˆ ˆr )(uˆ ˆr )] − jk 1 j · k j × k 2 j · jk j × jk k=1 Xk6=j it follows

N δΦ = ω Φ (6.30) tot j · Lj tot j=1 X = 60 r−6 [ε (uˆ uˆ ) ω (uˆ uˆ ) + ε (uˆ ˆr ) ω (uˆ ˆr )] − jk 1 j · k j · j × k 2 j · jk j · j × jk Xj6=k = 60 r−6 [ε (uˆ uˆ ) ω (uˆ uˆ ) + ε (uˆ ˆr ) ω (uˆ ˆr )] − jk 1 j · k j · j × k 2 j · jk j · j × jk Xj

ω ω ω ω ∆γ ω ω ω with jk := j k. Note that according to j = L (rj ˆ ) ˆ all the j have the same direction perpendicular− to the director n. · The second order energy distortion is

N δ2Φ = ω (δΦ ) = 60 r−6 (6.31) tot i · Li tot − jk i=1 X Xj

∂ ω u (u b) a (u b) = (ω a) (ω b)(b a) (ω u)(u a) · × ∂u · · × · − · · − · · (6.32) µ ¶ ¡ +(u b) 2(ω b¢)(u a) 2(ω a)(u b) + (ω u)(a b) . · · · − · · · · With help of this lemma it£ follows for example ¤

ω (uˆ uˆ ) ω (uˆ uˆ ) = (ω ω ) (ω uˆ )(uˆ ω ) (ω uˆ )(uˆ ω ) j · Lj j · k jk · j × k j · jk − j · k k · jk − j · j j · jk +(uˆ uˆ ) 2(ω uˆ )(uˆ ω ) 2(ω ω )(uˆ uˆ ) + (ω uˆ )(ω uˆ ) (6.33) j · k j · k j · jk − j · jk j · k j · j jk · k £ ¤ The whole ε1-term can be calculated to

ω (uˆ uˆ ) ω (uˆ uˆ ) ω (uˆ uˆ ) ω (uˆ uˆ ) = (6.34) j · Lj j · k jk · j × k − k · Lk j · k jk · k × j = ω2 (ω uˆ )2 (ω uˆ )2 + jk − jk · k − jk · j (uˆ uˆ ) 4(ω uˆ )(ω uˆ ) 2ω2 (uˆ uˆ ) + (ω uˆ )(ω uˆ ) (ω uˆ )(ω uˆ ) j · k jk · k jk · j − jk j · k k · k jk · j − j · j jk · k £ ∗ ¤ The -term can be rewritten as | {z } ∗ = (ω uˆ )(ω uˆ ) (ω uˆ )(ω uˆ ) ∗ k · k jk · j − j · j jk · k = ω uˆ (ω uˆ ) uˆ (ω uˆ ) + uˆ (ω uˆ ) uˆ (ω uˆ ) jk · j k · k − k k · j k k · j − k j · j £ added 0 ¤ = ω ω (uˆ uˆ ) uˆ (ω uˆ ) , compare (D.8) jk · k × j ×| k − k {zjk · j } = (ω uˆ ) (ω uˆ ) , (6.35) − j£k · k jk · j ¤ because ω ω . Therefore the ε -term can be reduced to jk k k 1 ω (uˆ uˆ ) ω (uˆ uˆ ) ω (uˆ uˆ ) ω (uˆ uˆ ) j · Lj j · k jk · j × k − k · Lk j · k jk · k × j = ω2 (ω uˆ )2 (ω uˆ )2 + (uˆ uˆ ) 3(ω uˆ )(ω uˆ ) 2ω2 (uˆ uˆ ) jk − jk · k − jk · j j · k jk · k jk · j − jk j · k = ω2 (1 2(uˆ uˆ )2) + ω 3(uˆ uˆ )uˆ uˆ uˆ uˆ uˆ uˆ ω . (6.36) jk − j · k jk · j · £ k j k − j j − k k · jk ¤ £ ¤ 74 CHAPTER 6. TWIST ELASTIC COEFFICIENT

Note, that this equals

ω2 (1 2(uˆ uˆ )2) + ω 3(uˆ uˆ )uˆ uˆ uˆ uˆ uˆ uˆ ω jk − j · k jk · j · k j k − j j − k k · jk ω ω 2 = 3 jk uˆjuˆj uˆkuˆ£k jk 2ωjk uˆjuˆk : uˆjuˆk . ¤ (6.37) · · · −

The ε2-term in (6.31) gives with help of the special form of the lemma (D.4) ω (uˆ ˆr ) ω (uˆ ˆr ) + ω (uˆ ˆr ) ω (uˆ ˆr ) j · Lj j · jk j · j × jk k · Lk k · jk k · k × jk = ω2(1 2(uˆ ˆr )2) + ω 3(uˆ ˆr )uˆ ˆr uˆ uˆ ˆr ˆr ω j − j · jk j · j · jk j jk − j j − jk jk · j + ω2(1 2(uˆ ˆr )2) + ω 3(uˆ ˆr )uˆ ˆr uˆ uˆ ˆr ˆr ω . (6.38) k − k · jk k ·£ k · jk k jk − k k − jk jk¤ · k The second order energy distortion can£now be rewritten by taking into¤ account the special form of ω = ∆γ/L (r ωˆ )ωˆ as j j · ∆γ 2 δ2Φ = 60 r−6 (6.39) tot − L jk µ ¶ Xj

V K ∆γ 2 1 ∆F = 2 δ2Φ (6.40) 2 L ≈ 2 tot µ ¶ ® we conclude that the twist elastic constant K2 can be derived from the average value 2 δ Φtot . This result is not satisfactory, because it takes into account global deformations.h i However, the free energy distortion is approximated for small deformations only. In the ε2-term the absolute positions of the particles inﬂuence the angle ωj = ∆γ/L (r ωˆ ). It follows, that the result for K depends on the system size. Therefore j · 2 we change the orientational deformations to local ones: We have to translate the positions of each pair of particles rj and rk such that their center of mass is in the origin (relative coordinates), that is about (r + r )/2: − j k r r ω ω r jk , r − jk ω jk , ω − jk (6.41) j → 2 k → 2 ⇒ j → 2 k → 2 2 With this the results for δΦtot and δ Φtot are rewritten as ∆γ δΦ = 60 r−6 (r ωˆ ) ωˆ ε (uˆ uˆ ) (uˆ uˆ ) tot − L jk jk · · 1 j · k j × k Xj

2 and δ Φtot accordingly such that the twist elastic coeﬃcient can be obtained as an average value K Kobs with the observable 2 ≈ 2 60 ® ε ε Kobs := r−6(r ωˆ)2 ε + 2 2 (uˆ ˆr )2 + (uˆ ˆr )2 2 − V jk jk · 1 2 − 2 j · jk k · jk j

Here, the second part, which is proportional to (ωj +ωk) is omitted according to (6.41). For the ε1-term this makes no diﬀerence, because

( + )(uˆ uˆ )2 = uˆ 2uˆ (uˆ uˆ ) + uˆ 2uˆ (uˆ uˆ ) = 0 . (6.45) Lj Lk j · k j × k j · k k × j j · k

Note, that the observable K2 (6.43) is strongly dependent on the range of interaction −4 rcut, because it decreases only with r . The above mentioned equilibrium configuration 3 obs at (N, V, T ) = (7440, 20 , 1) is now used to obtain the value of K2 for different cut off radii rcut, see figure 6.4. Indeed, the cut off radius has an crucial influence on the value of K . From r = 3 to r = it increases by about 20%. However, with r = 3 2 cut cut ∞ cut we are confident to get results which are in accordance with the model potential, which used the same cut off radius. The result Kobs(r = 3) 2.1 is smaller than the 2 cut ≈ rough estimate K2 10. An even better result is obtained after averaging over 20 000 ≈ obs configurations for the cut off radius rcut = 3 used. It follows K2 = K2 = 1.97 0.04. In order to check the consistency of the method we calculate δΦ . For the simulation§ tot ® above we find h i δΦ h toti = 0.2 40 (6.46) ∆γ/L − §

Therefore, the first order term indeed vanishes. This confirms that the free energy is in a minimum due to an equilibrium state. So far we have just calculated the Born-Green obs part K2 = K2 of the twist elastic coefficient. The fluctuation part of the free energy change δ2F in equation (6.23) can now be calculated through the fluctuation of the fluct ® 76 CHAPTER 6. TWIST ELASTIC COEFFICIENT

2.4

2.2 obs 2 K 2

1.8

2 3 4 5 6 7 8 9 10

rcut

obs Figure 6.4: Twist elastic coefficient K2 versus cut off radius rcut. One single configuration of a NV T MC simulation with n = 0.94 and T = 1 ( P = 3) is used. → observable δΦtot, here ∆δΦtot = 40 ∆γ/L . It yields a corrected value of the twist elastic constant of Kcorr = K Kfluct with 2 2 − 2 1 ∆δΦ 2 Kfluct = tot , (6.47) 2 V k T ∆γ/L B µ ¶ fluct 2 3 that is K2 = 40 /20 = 0.2 in the example above. Here, the corrected value is Kcorr = 2.0 0.2 = 1.8. 2 − 6.4 MC results

Here, results of NVT MC simulations are presented for the isotherms T = 0.8, T = 0.85, T = 0.9, T = 0.95 and T = 1.0 in the nematic phase regime. The particle numbers are between N = 1200 and N = 2300 in a cubic volume of V = 123. Both methods are used to obtain values for the twist elastic coefficient K2. The results of method 1 (section Q 6.2) are indicated by K2 , reminding of the alignment tensor Q, whose Fourier space Q 2 representation was used to get E(kx) = K2 kx. Values from method 2 (section 6.3) are labeled and calculated as Kcorr = Kobs Kfluct. 2 2 − 2 Figure 6.5 shows some results for the isotherms studied. The average values in the ® MC runs were averaged over 50 000 sweeps after another 50 000 sweeps for equilibration. Q corr The nematic order parameter S2 is shown beneath the values of K2 and K2 . Most of the simulations are performed in the nematic phase, where S2 > 0.4. Only four simulations are done in the isotropic phase (top of figure 6.5). The two MC runs at T = 0.8 with densities n = 0.72 and n = 0.75 have not yet been equilibrated because of 6.4. MC RESULTS 77

0.9

0.8

0.7

0.6

2 0.5 S 0.4

0.3 T=1.00 T=0.95 0.2 T=0.90 T=0.85 0.1 T=0.80 0.7 0.8 0.9 1 1.1 1.2 1.3

12 11 10 9 8 7 Q 2

K 6 5 4 3 2 1 0 0.7 0.8 0.9 1 1.1 1.2 1.3 T=1.00 7 T=0.95 T=0.90 6 T=0.85 T=0.80 5

4 corr 2 K 3

0 0.7 0.8 0.9 1 1.1 1.2 1.3 n

Figure 6.5: Isotherms T = 0.8, T = 0.85, T = 0.9, T = 0.95 and T = 1.0 obtained via NVT MC simulations. Shown are the nematic order parameter and the twist elastic Q corr coeﬃcient calculated as K2 and K2 versus the particle density n. 78 CHAPTER 6. TWIST ELASTIC COEFFICIENT

14 E(kx) Q 2 12 K2 kx

) 8 n = 1.04 x

E(k 6

2 n = 0.81

0 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

kx/kmin

Figure 6.6: Fit of the parables E(kx) for T = 0.8 at n = 0.81 and n = 1.04, used to Q Q 2 obtain K2 and the fit functions K2 kx. the vicinity of the phase transition isotropic-nematic, which seems to be strong for this isotherm. The middle and bottom plots of figure 6.5 show the twist elastic coefficient Q corr Q corr calculated as K2 and K2 . The values of K2 are less accurate as those of K2 . Especially at densities n > 1 there is a high fluctuation, where method 2 gives highly corr ordered results with increasing K2 as temperature is decreased or density increased. Figure 6.6 shows the data points E(kx) for T = 0.8 at two densities n = 0.81 and Q Q 2 n = 1.04, which were used to obtain K2 , together with the parable fits K2 kx. For Q the low density n = 0.81 the fit works quite well with K2 = 2.04 0.03, which is corr §corr larger than the result K2 = 1.27 of method 2. For n = 1.04, where K2 = 4.22, the data points E(kx) show no good parable, such that the fit gives the imprecise result KQ = 12.4 0.3. Again, an offset to the parable would improve the fit. Therefore, 2 § the better way to use this method will probably be with kz = 0, where all the elastic constants can be obtained. Here, we will focus now on the results6 of the second method corr K2 . From experimental measurements it is known[58] that the elastic coefficients Kλ are 2 corr proportional to the square of the nematic order parameter S2 . The results K2 of corr method 2 are now collected to check this relation. In figure 6.7 the coefficient K2 is 2 plotted versus (nS2) . Obviously, there is a linear relation, such that one has Kcorr(n, T ) c (nS (n, T ))2 (6.48) 2 ≈ 2 with c 5.1, which is a global quantity for the present liquid crystal model. Only one point la≈y beside the line, which comes from the (n, T ) = (0.72, 0.8)-simulation. This one was not in equilibrium. For constant particle density, which is according to the experiments, we confirm the relation K S2. The comparison of the corrected value 2 ∼ 2 6.4. MC RESULTS 79

2 4 K 3

2 corr 1 K2 0 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 2 (n S2)

corr obs fluct Figure 6.7: Twist elastic coefficient K2 = K2 K2 versus the square of the prod- − 2 uct of particle density and nematic order parameter (nS2) for different temperatures ® obs and densities. The empty circles are the Born-Green contributions K2 . ® corr obs fluct obs K2 = K2 K2 with the uncorrected Born-Green part K2 shows that the fluctuation part−particularly is important for small densities. A closer look at the data ® corr ® points reveals that one has K2 < 0 for small values of nS2 and that there is a higher 4 order term (nS2) for large nS2. The negative values are due to too large fluctuation contributions. In good accordance with the data points it is

Kcorr = 0.05 + 5.1 (nS )2 + 0.4 (nS )4 (6.49) 2 − 2 2 for all temperatures and particle densities which are in the nematic or isotropic region. Higher order terms were also predicted by a more general theory according to Par- dowitz and Hess[51, 86]. This molecular theory made use of expansions of thermody- 2 2 3 namic functions with respect to S2. It gives K2 = n (cS2 + dS2 + . . .). The coeﬃ- cients c, d, . . . can be calculated as integrals of the interaction potential Φff (r, uˆ1, uˆ2) (2) and the pair-correlation function g (r, uˆ1, uˆ2). For small S2, this theory yields the 2 same relation K2 = c(nS2) as obtained above by MC results. To calculate c for the present model, we assume that the pair-correlation function may be approximated with (2) g exp( Φff / kT ). This approximation makes sense for a dilute system. Then it follows≈ [86] −

15 r2 ∂Φ Φ d2uˆ d2uˆ c = γ(ˆr, uˆ , uˆ ) − r ff exp ff 1 2 d3r (6.50) 28 1 2 3 · ∂r −k T (4π)2 Z µ ¶ µ B ¶ 80 CHAPTER 6. TWIST ELASTIC COEFFICIENT

8 c

6 0 10 20 30 40 50 T

Figure 6.8: The coeﬃcient c according to Pardowitz [51, 86] versus temperature. where the angle dependent function

γ(ˆr, uˆ , uˆ ) = 12 tr uˆ uˆ uˆ uˆ ˆrˆr 7 uˆ uˆ : uˆ uˆ (6.51) 1 2 1 1 · 2 2 · − 1 1 2 2 ³ ´ can be calculated as 2 γ = 12(uˆ1 uˆ2)(uˆ1 ˆr)(uˆ2 ˆr) 11(uˆ1 uˆ2) + 5 · · · − · (6.52) 4 (uˆ ˆr)2 + (uˆ ˆr)2 − { 1 · 2 · } and the model specific part is given by (see appendix C) r2 ∂Φ − r ff = 16r−10 8r−4(1 + Ψ(ˆr, uˆ , uˆ )) . (6.53) 3 · ∂r − 1 2 Note, that c is temperature dependent. Because of the approximation for the pair- correlation function we have to integrate with high temperatures to ensure a dilute system. The integral over r has to be cut off at rc = 3. We show c(T ) in Fig. 6.8. For high temperatures, the temperature dependence is not very strong. At T = 10, c = 6.5 is near the Monte Carlo result of c = 5.1. Therefore, Both theories give similar results. With the Pardowitz theory it is possible to calculate c very fast, because only two-particle integrals have to be calculated We apply it to obtain the dependence from the anisotropy parameters ε1 and ε2. The integral c at T = 10 for different anisotropy parameters ε1 and ε2 is shown in Fig. 6.9. For zero anisotropy parameters c = 0 is found. This confirms that an isotropic fluid has no elasticity for orientational deformations. In addition, it can be seen that c mainly depends on ε1 and that c is proportional to ε1. The precise result due to these integrations is c 150 ε A test MC run with double ε , ≈ 1 1 resulting in c = 10.2 instead of c = 5.1, confirms this behavior. Thus, both theories for K show the common result c ε . The more accurate MC computations give 2 ∼ 1 c 130 ε . (6.54) ≈ 1 6.4. MC RESULTS 81

Figure 6.9: The coeﬃcient c for T = 10 according to Pardowitz [51, 86] versus the anisotropy parameters. The x marks the anisotropy parameters, for which all other results of this thesis are performed.

Finally, we conclude that the twist elastic constant for our model liquid crystal can be approximated with

K 130 ε (nS (n, T ))2 (6.55) 2 ≈ 1 2 for temperature T and particle density n. 82 CHAPTER 6. TWIST ELASTIC COEFFICIENT Chapter 7

Conclusions and outlook

As an example for a complex fluid we studied a simple model liquid crystal[44]. Two thermodynamical properties were mainly focused. First the phase behavior of the bulk system and the influence of confining walls were analyzed in a slab geometry with different anchoring mechanism, including a twisted nematic (TN) cell. Second the twist elasticity of this model was studied by calculating the twist elastic constant. The computations were done with Monte Carlo (MC) simulations in the NV T (constant volume) as well as in the NP T (constant lateral pressure) ensemble. Important average values obtained were the pressure tensor, internal energy and its fluctuation, order parameters for the nematic and layered phase as well as the twist elastic constant according to the new method introduced in section 6.3. The MC results for the phase behavior in the bulk system agree qualitatively with analytical calculations[44], especially for high temperatures. A nematic phase was observed. For smaller temperatures solidification takes place without revealing any smectic phase. A detailed analysis was done for the isotherm T = 1 as well as for the isochore n = 1. For n = 0.7 we observed with NV T MC an inhomogeneous particle density. Any solidification leads to a strong hysteresis due to the stability of the crystal structure. Several isobars were also studied as a reference for the confined system with walls. In the slab geometry two different anchoring mechanisms at the two flat solid walls were used. The first one favors a homeotropic alignment. The second one produces a twist alignment, where both walls model a homogeneous planar anchoring with a 90◦ difference between the walls. For our model liquid crystal the phases for the isotherm T = 1, the isochore n = 0.9 (where nbulk 1) as well as several isobars were studied with both anchoring mechanisms. This allo≈ws a comparison with the bulk system phase behavior. A structure analysis in the solid phase reveals a hcp crystal, where the hexagonal layers are perpendicular to the director. The influence of the confinement may be reviewed at the example of the phase transitions observed for the isotherm T = 1. Here, compression and expansion processes were studied in constant pressure Monte Carlo simulations. For a summary, we compile in table 7.1 pressures at which phase transitions occur. In all three cases the pressure regime for the nematic phase is very narrow in the expansion process or even vanishes (see figures 4.3, 5.1 and 5.9). This is because of a large hysteresis, which is typical for first-order phase transitions. The solid state here is very stable. The confinement of flat walls forces the phase transition

83 84 CHAPTER 7. CONCLUSIONS AND OUTLOOK

Bulk system Homeotropic alignment Twist alignment PIN 1.7 1.5 1.5

PNS 7.4 2.5 4.6

PSN 1.5 1.5

PNI 1.3 1.2

PSI 1.2

Table 7.1: Comparison of phase transition pressures of the isotherm T = 1 for different kind of geometries. The system with twist alignment showed no nematic phase in the expansion process. to be shifted compared with the bulk, especially for solidification and melting. This can be understood because of the positional order, that the wall brings into the system. A first layer beside the wall is formed very early and in this layer it is easy for the system to achieve highly ordered states. Figure 7.1 shows the nematic and smectic order parameter for the compression process. The isotropic-nematic transition is only slightly shifted towards smaller pressures by the walls, the nematic-solid transition is clearly shifted, especially for the homeotropic alignment. In the case of twisted homogeneous planar alignment we found a linear relationship between the azimuthal angle of the director and the z coordinate in the simulation box (figure 5.12). The twist elasticity was also studied. A new method was used to compute the twist elastic constant K2 as an average value of an observable, obtained with the help of the distortion of the free energy due to a twist deformation. The results are compared with another method based on director fluctuations, which uses the alignment tensor in Fourier space Qˆ (k). Computing this only for ky = kz = 0 gives less good results, especially for high densities. The new method works very good already for moderate 2 system sizes and reveals the relationship K2 = 130ε1 (nS2) . This is in accordance with another molecular theory and with the experimental finding, that K2 is proportional to the square of the nematic order parameter S2 at constant particle density n.

Summary In summary the following conclusions can be drawn for the model liquid crystal studied:

The bulk system shows the phases isotropic ﬂuid, nematic and solid. The phase • behavior agrees qualitatively with analytical calculations.

Conﬁnement causes an increase of positional order. Phase transitions are shifted • accordingly. The nematic phase is only slightly stabilized.

A twist conﬁguration shows a linear twist angle decay in the nematic phase. • The twist elastic coeﬃcient can be calculated even for moderate system sizes and • 2 the relation K2 = 130ε1(nS2) is found. 85

0.8

0.6 2 S 0.4

0.2 bulk twisted homogeneous planar homeotropic 0 0 2 4 6 8

0.8

0.6 1 ρ 0.4

0.2

0 0 2 4 6 8

P||

Figure 7.1: Comparison of nematic (S2) and smectic (%1) order parameters as function of the pressure for the isotherm T = 1 and diﬀerent kind of geometries. Only the compression process, obtained via NPT-MC simulations, is shown.

Outlook The knowledge of the static properties of the bulk fluid and the behavior of the liquid crystal in the vicinity of walls as presented here is a prerequisite for the analysis of dynamic phenomena in confined and in mesoscopically structured systems. It is desirable to study via Molecular Dynamics (MD) simulations the influence of walls on the translational and rotational diffusion and to compare it with results inferred from NMR experiments[87]. Non-Equilibrium Molecular Dynamics (NEMD) simulations of the viscous properties as calculated previously for ellipsoids and Gay-Berne particles[88] should be performed for the model liquid crystal used here. Furthermore, an extension of the present study to fluids of Janus particles[89] is desirable. 86 CHAPTER 7. CONCLUSIONS AND OUTLOOK Appendix A

Rotational energy of a uniaxial particle

In this section we need to distinguish vectors from columns and bilinear forms from matrices. We will use the following notation: For a vector a we write its components with respect to the basis = ex, ey, ez as a column C( , a). The components of a bilinear form A willBbe written{ as }a matrix M( , A,B), if it operates like A(a, b) = C( , a)tM( , A, )C( , b) as a matrices product.B AB transformation into a new basis ˜B= e˜ , Be˜ , e˜ B canBthan be done using the transformation matrix B { x y z} M( , ˜) = (C( , e˜x), C( , e˜y), C( , e˜z)) containing the components of the new basis vectorsB Bwith respBect to theB old basisB as columns.

C( ˜, a) = M( ˜, )C( , a) (A.1) B B B B M( ˜, A, ˜) = M( ˜, )M( , A, )M( , ˜) (A.2) B B B B B B B B

We need the components of the inertial tensor Jνµ and the angular velocity ων in the same basis. Then we could derive the bilinear form 1 T = ω J ω (A.3) rot 2 ν νµ µ in this manner. In the body-ﬁxed basis it is obviously K

J⊥ 0 0 M( , J, ) = 0 J 0 (A.4) K K  ⊥  0 0 Jk   if the 3-axis is the symmetry axis. The first two Eulerian angles (ϑ, ϕ) can be used to express the orientation u of this 3-axis in the space-fixed external basis = ex, ey, ez . We use the convention of [90] for the Eulerian angles and find that B { }

cos ϕ sin ϑ C( , u) = sin ϕ sin ϑ . (A.5)   B cos ϑ   87 88 APPENDIX A. ROTATIONAL ENERGY OF A UNIAXIAL PARTICLE

The angular velocity for changing Eulerian angles (ϑ, ϕ, χ) is

0 sin ϕ cos ϕ sin ϑ − C( , ω) = 0 ϕ˙ + cos ϕ ϑ˙ + sin ϕ sin ϑ χ˙ (A.6)       B 1 0 cos ϑ       In order to derive C( , ω) we need the transformation matrix K cos ϕ cos ϑ sin ϕ cos ϕ sin ϑ − M( , ) = sin ϕ cos ϑ cos ϕ sin ϕ sin ϑ . (A.7)   B K sin ϑ 0 cos ϑ −   which we get from a rotation around ey with ϑ followed with a rotation around ez with ϕ. Note that M( , ) = M( , )t. It can now easilyK Bbe shownB thatK

sin ϑ ϕ˙ − C( , ω) = M( , )C( , ω) = ϑ˙ (A.8) K K B B   cos ϑ ϕ˙ + χ˙   and from equations (A.4) and (A.8) we get (see also [91]) 1 T = C( , ω)t M( , J, ) C( , ω) rot 2 K K K K J J = ⊥ ϕ˙ 2 sin2 ϑ + ϑ˙2 + k (ϕ˙ cos ϑ + χ˙)2 (A.9) 2 2 ³ ´ Now we want to express Trot with help of the conjugate momenta ∂T Λ = rot = J sin2 ϑ + J cos2 ϑ ϕ˙ + J cos ϑ χ˙ (A.10) ϕ ∂ϕ˙ ⊥ k k ∂Trot ¡ ¢ Λϑ = = J⊥ϑ˙ (A.11) ∂ϑ˙ ∂T Λ = rot = J cos ϑ ϕ˙ + J χ˙ (A.12) χ ∂χ˙ k k and ﬁnally arrive at the phase space function

2 2 2 Λϑ (Λϕ Λχ cos ϑ) Λχ Trot = + − 2 + (A.13) 2J⊥ 2J⊥ sin ϑ 2Jk leading to the Hamiltonian in equation (3.10) Appendix B

The partition function

The partition function in the canonical ensemble is

1 H(Γ) N Z = exp d3r d3p d3Ω d3Λ (B.1) NV T h6N N! 2N (2π)N − k T j j j j B j=1 Z · ¸ Y with d3Ω = dϑ dϕ dχ. We can factorize ZNV T into a product of a kinetic and a conﬁgurational integral, if we transform the conjugate momenta Λ of the Eulerian angles Ω into a real angular momentum l, so that Trot does not longer explicitely depend on Ω. With the substitution (see [90]) Λ = l cos ϑ l sin ϑ (B.2) ϕ z − x Λϑ = ly (B.3)

Λχ = lz (B.4) we get from equation (A.13)

2 2 2 lx + ly lz Trot(l) = + . (B.5) 2J⊥ 2Jk Note that d3Λ = sin ϑ d3l (B.6) Now we can write 1 Z = QN Z (B.7) NV T h6N N! 2N (2π)N c where p2 + Trot(l) Q := exp 2m d3p d3l (B.8) − k T Z ( B ) Φ N and Z := exp tot sin ϑ d3Ω d3r . (B.9) c −k T j j j B j=1 Z ½ ¾ Y

89 90 APPENDIX B. THE PARTITION FUNCTION

3 3 2 The integral Q can be calculated to yield (2πkBT ) m J⊥Jk. The integrals over the cyclic angles χ in Z lead to a factor (2π)N . In summary we get j c p N 1 4π3(k T )3 m3J 2 J Φ N Z = B ⊥ k exp tot d2u d3r (B.10) NV T N! h6 −k T j j Ã p ! B j=1 Z ½ ¾ Y with the surface element d2u = sin ϑ dϑ dϕ of the orientation u. Appendix C

Derivatives of the potential

In section 3.5, the force between two particles (3.30) is needed in order to express the pressure observable. In section 6.4 the scalar product of this force with the distance vector has to be known in equation (6.53) for calculating the integral expression of the twist elastic constant. Here, both quantities will be derived, starting from the interaction potential (2.6)

Φ (r, uˆ , uˆ ) = 4 r−12 r−6(1 + Ψ(ˆr, uˆ , uˆ )) , (C.1) ff 1 2 − 1 2 and the anisotropy function (2.2) £ ¤ 15 1 15 2 Ψ(ˆr, uˆ , uˆ ) = ε (uˆ uˆ )2 + ε (uˆ ˆr)2 + (uˆ ˆr)2 . (C.2) 1 2 2 1 1 · 2 − 3 2 2 1 · 2 · − 3 · ¸ · ¸ First, have a look at the derivative ∂Φ ∂Ψ ∂rˆ ff = 48r−13ˆr + 24r−7ˆr(1 + Ψ) 4r−6 ν . (C.3) ∂r − − ∂rˆν ∂r Because of rˆ = r (r2 + r2 + r2)−1/2 for ν 1, 2, 3 and its derivative ν ν 1 2 3 ∈ { } ∂rˆ 1 ν = (eˆ rˆ ˆr) (C.4) ∂r r ν − ν we have ∂Φ ∂Ψ ∂Ψ ff = 48r−13ˆr + 24r−7ˆr(1 + Ψ) + 4r−7 ˆr ˆr . (C.5) ∂r − ∂ˆr · − ∂ˆr ·µ ¶ ¸ The scalar product of the last term [. . .] with ˆr vanishes, such that (compare eq. (6.53)) r2 ∂Φ − r ff = 16r−10 r−4(1 + Ψ) (C.6) 3 · ∂r − is valid, independent from the form of the function Ψ(ˆr, uˆ1, uˆ2). In order to calculate the force F = ∂Φ /∂r we have to take the derivative − ff ∂Ψ = 15ε [(uˆ ˆr)uˆ + (uˆ ˆr)uˆ ] , (C.7) ∂ˆr 2 1 · 1 2 · 2

91 92 APPENDIX C. DERIVATIVES OF THE POTENTIAL which leads to ∂Ψ ∂Ψ ˆr ˆr = 15ε (uˆ ˆr)2ˆr + (uˆ ˆr)2ˆr (uˆ ˆr)uˆ (uˆ ˆr)uˆ . (C.8) ∂ˆr · − ∂ˆr 2 1 · 2 · − 1 · 1 − 2 · 2 µ ¶ £ ¤ Finally, the expression

∂Φ ff = 48r−13ˆr + r−7 60ε (uˆ ˆr)uˆ + (uˆ ˆr)uˆ + ∂r − − 2 1 · 1 2 · 2 ½ (C.9) 1 £ 1 ¤ 24 + 180ε (uˆ uˆ )2 + 240ε (uˆ ˆr)2 + (uˆ ˆr)2 ˆr 1 1 · 2 − 3 2 1 · 2 · − 2 · µ ¶ µ ¶¸ ¾ is found, which equals the negative force in equation (3.30). Appendix D

Proof of lemma

2 In section 6.3 the following expression was used to obtain the second derivative δ Φtot of the total potential energy (equation (6.32): Lemma: If a, ω and b are constant vectors and b2 = 1, it follows for u with u2 = 1

∂ ω u (u b) a (u b) = (ω a) (ω b)(b a) (ω u)(u a) · × ∂u · · × · − · · − · · (D.1) µ ¶ ¡ +(u b) 2(ω b¢)(u a) 2(ω a)(u b) + (ω u)(a b) . · · · − · · · · Note, that one can write £ ¤

∂ (u b)2 (u b) a (u b) = a u · (D.2) · · × · × ∂u 2 µ ¶ Therefore, for a = ω the expression to be calculated equals

∂ 2 (u b)2 ∂ ω u · = ω u (u b) ω (u b) (D.3) · × ∂u 2 · × ∂u · · × · µ ¶¸ µ ¶ µ ¶ which reveals, that these terms come from the second derivative of the (u b)2-like potential. In this case (a = ω) the lemma (D.2) shortens to ·

∂ 2 (u b)2 ω u · = ω2(1 2(u b)2) + ω 3(u b)ub uu bb ω .(D.4) · × ∂u 2 − · · · − − · · µ ¶¸ £ ¤ The vectors may have cartesian components (denoted by Greek subscripts) in the or- thonormal basis (eˆx, eˆy, eˆz), such that a = aνeˆν. The Einstein sum convention is used. But now, let us prove the lemma: The starting point is the calculation of the derivative

∂ (u b) a (u b) = a (u b) b + (u b) (b a) (D.5) ∂u · · × · × · × £ ¤ Applying ω (u ) gives · × ◦ ω (u b) a (u b) = ω (u b) a (u b) + ω u (b a) (u b) . (D.6) · L · · × · × · × · × × · £ ¤ £ ¤ 93 94 APPENDIX D. PROOF OF LEMMA

The cross products can be written as b a = eˆλελµνbµaν, where ελµν is the totally antisymmetric tensor of third rank (Levi-Civita× tensor)

1 for (λ, µ, ν) (1, 2, 3), (2, 3, 1), (3, 1, 2) ∈ { } ελµν := 1 for (λ, µ, ν) (3, 2, 1), (2, 1, 3), (1, 3, 2) . (D.7) − ∈ { } 0 else

Because of 

δµα δνα ελµν ελαβ = = δµαδνβ δµβδνα (D.8) δµβ δνβ − ¯ ¯ ¯ ¯ ¯ ¯ it follows ¯ ¯ ω u (b a) = ω ε u ε b a · × × µ λµν ν λαβ α β = (ω b)(u a) (ω a)(u b) . (D.9) £ ¤ · · − · · On the other hand the relation

δλα δµα δνα ε ε = δ δ δ (D.10) λµν αβγ ¯ λβ µβ νβ ¯ ¯ δ δ δ ¯ ¯ λγ µγ νγ ¯ ¯ ¯ = δ¯ λα δµβδνγ δµγ¯δνβ + δλβ δµγδνα δµαδνγ + δλγ δµαδνβ δµβδνα ¯ − ¯ − − is needed to show that¡ ¢ ¡ ¢ ¡ ¢

2 ω (u b) a (u b) = ελµνωλuµbν εαβγaαuβbγ = (ω a) 1 (u b) + · × · × · − · (D.11) (ω u) (u b)(b a) (u a) + (ω b) (u a)(b u) (b a) · · · − · · · ¡ · − · ¢ holds. Here, b2 = u2 ¡= 1 is used. In summary¢ it is found¡ from (D.9) and (D.11¢ )

ω (u b) a (u b) + ω u (b a) (u b) = · × · × · × × · = (ω a) (ω b)(b a) (ω u)(u a) (D.12) · − · £ · − ¤· · +(u b) 2(ω b)(u a) 2(ω a)(u b) + (ω u)(a b) , · · · − · · · · which together with (£D.6) proves the lemma (D.1). ¤ Bibliography

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