Free Energy of Finite Length Rigid Rods

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MS

C

P

HYSICS AND

A

STRONOMY

TRACKTHEORETICALPHYSICS

M

ASTER

T

HESIS

Free Energy of Finite Length Rigid Rods

by

R

AMON

C

REYGHTON

5952948

July 2017

60 EC

Research carried out in the

THEORY OFBIOMOLECULARMATTERGROUP at AMOLF, September 2016 – July 2017

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Wetenschappelijk denken is persoonlijke ervaring (gelijk een artistieke aandoening) en persoonlijke creatie (gelijk het penseelen van een schilderij

of het componeeren van een muziekstuk).

De leek ziet de wetenschap als een soort causaal kraaltjesrijgen. Hij neemt kennis van het geschematiseerde eindproduct en roept uit

– is dit nu de beschrijving der natuurverschijnselen!

Is er niet meer tusschen hemel en aarde dan deze schamele producten!

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A general introduction to the topics

and questions in this thesis

I

For the layman

A common question in physics is: what parameters are relevant for the physical quantity I want to determine? Or, more precisely: how do I justify the presence of each of the terms I allowed in my expression? One of the central questions in this thesis belongs to this category. It is asked in the context of the theory of liquid crystals, which will be introduced below.

There are well-established ways to answer questions like these, and the main chapters of this thesis will do just that, in a formal and technical way. First, however, I will introduce the general principles that guide this enterprise.

An education in physics provides one with a pantry stocked with phenomena, quantities, variables, constants, and laws. Moreover, one has trained oneself a toolbox full of methods and intuitions that one can employ to piece together insight from scrambled formulas and data. This often leads to not just one answer, but a host of possible ways to tackle a problem. In any case, the items one selects from one’s pantry should give a non-negligible contri-bution to the quantity one is interested in, the rest of the supply can be left unused. Even so one may need to concede that one’s toolbox is insufficient to process al the relevant items, so one is forced to narrow down one’s question and study a special case first. This story can either be framed as a large-scale failure or as a detailed success, depending on one’s epistemological ambitions with physics. This thesis is no exception.

But there is another, much more interesting side to the decision about relevant terms in expressions, which originates in the difference between phenomenological (macroscopic) theories and fundamental (microscopic) theories. The former are informed by experimentally observed variables and parameters that appear to govern a system. While those are by no means trivial to determine — How to be certain one does not overlook a relevant parameter? How to be sure a variable is not composed of two more fundamental ones? — the task of the theorist who is asked to arrange the data in a meaningful way is in some sense straightforward. The theory that is most widely applicable and has the most predictive power while maintaining a degree of mathematical elegance is probably accepted. Phenomenological theories can do without a fundamental justification for every term they allow, as long as they work.

Microscopic theories, on the other hand, meet a higher burden of proof. As a consequence, many have a limited usefulness. (To find exceptions to this statement, one only needs to point to the most famous laws of physics. I am aware of that, but it is probably healthy not to aim for the level of e.g. Maxwell in the context of a Master’s Thesis.)

Obviously, it is every physicist mission to combine the best of both worlds. Luckily, many strategies to simplify expressions in physics are useful in both domains.

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achieved by the assumption of ceteris paribus, an ideal that is actually achievable in a physics lab. But physicists often do have to deal with a somewhat analogous problem: that of boundary effects. In the study of physical systems that are composed of a collection of things — a fluid of water molecules, a solid of silicon atoms, a gas of nitrogen molecules — the collection is in practice never infinite. It must come to an end. That happens at a surface or boundary of the system, that is: at the interface with the rest of the world. All kinds of interesting physical phenomena may take place there. Those are often much more complicated than the behavior on the inside of the system, in what is called the bulk. Stripping down the problem to the bare minimum often entails: pretending as if there is no boundary, focussing on the bulk only. As a consequence one chooses to ignore every term in the expressions that is relevant on the boundary only.

Second, parameters that are simply way too small to be relevant to the problem at hand are neglected. This is done by a combination of what is called dimensional analysis and a comparison of orders of magnitude, and it depends on a choice of working conditions. One could limit a study of the behavior of water molecules to the conditions in which they form a gas, that is: above some minimum temperature. In those conditions, one can be sure that certain types of forces that influence the molecules are not relevant, simply because they are many orders of magnitude smaller than those that drive the fast motion and collisions of the molecules. Moreover, other parameters may be excluded definitively using the subtle art of dimensional analysis, which is based on the requirement that every term in an expression about, say, energy must be a composition of parameters such that its unit is energy.

Third, there often are fundamental symmetries that can be exploited to simplify expres-sions. In this thesis, for instance, I will work with particles that have toothpick- or cigar-like shape: long and thin, with identical head and tail. A mathematical expression that aims to represent these particles’ orientations should acknowledge that it does not matter whether they point to, say, to the upper-left or the lower right. For these particles those directions are indistinguishable.

In this thesis, all of the above methods are used in an attempt to match a fundamental and a phenomenological description of a class of materials called liquid crystals. I study them without considering boundary effects. I limit myself to a subclass made up of rigid rods, and to conditions (densities, temperatures) in which they are in the nematic phase with a homogenous center of mass distribution. Also, I assume that those rods are much longer than that they are thick, that they are completely rigid nonetheless, and that they have both an axial and a head-tail symmetry.

In order to understand this jargon I just poured out over the reader, she or he is encour-aged to read the physicists’ introduction and the technical chapters. A simple summary of the results is provided at the end of this thesis.

II

For the biologists and the historically inclined

Ever since Otto Lehmann coined the term fliessende Krystalle for the milky substances he observed with his Kristallisationsmikroskop, physicists and chemists have studied these liquid crystals, these materialized contradictiones in adjecto.(2)The botanist Friedrich Reinitzer had asked Lehmann for his opinion on this strange class of substances that the former had encoun-tered during research on the liquids that make some carrots yellow. They showed double refraction of light, an effect associated with solid crystals since the days of Christiaan Huygens. Liquids were supposed to be highly isotropic, how could they provide the electric anisotropy needed for this peculiar optic response called birefringence?

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Almost half a century earlier, two German eye doctors had already noted similar optical properties of a brain fluid called myeline.(3)Lehmann brought the study of these materials into the realm of physics, as he ventured they could be seen as a phase. A liquid crystal is a mixture of various components, that will exhibit qualitatively different properties1_depending

on e.g. temperature or pressure. This new mesophase was crystalline in some aspects — as indicated by the birefringence — and fluid in others — indeed the stuff did definitively flow. Physical chemists in the early twentieth century argued over the homogeneity of liquid crystals. Were they mere emulsions of crystalline particles in some oily fluid, or did the crystalline nature emerge due to some as yet unknown process from a homogeneous fluid? Eventually, the debate was settled in favor of the latter hypothesis.2

In the meantime, many physicists were struck by the analogies they observed between liv-ing nature and the behavior of the many liquid crystals that were beliv-ing discovered. Lehmann wrote:

Die Theilung und Copulation von Krystalltropfen erinnern an die ebenso benannten Vorgange im Reiche der lebenden Wesen, womit nicht behauptet werden soll, dass die Aehnlichkeit eine tiefergreifende ist.(3)

Lehmann follows the opinion of the German biologist and Darwin defender Ernst Haeckel, that all matter can be seen as living matter, and that what is called ’living’ ordinarily, is merely a complex collection of basic units of lower order. Presumably, crystals could constitute such basic building blocks of life just as well as organic matter. Coincidentally, many then-known liquid crystals were composed of organic matter, and the way they seemed to move under the microscope resembled bacteria.

The view that liquid crystals were not that far below basic life forms never really stuck and was abandoned almost universally after World War II. But recent decades have proved yet again that in the history of scientific disciplines — just as in physical systems — many interesting things happen at the boundaries — not just in the bulk.

The models of liquid crystals, developed by physicists and chemists, appear to be useful for many biological systems. Cell membranes, for instance, are made of layered lipids, lengthy particles with a head and a tail. The long ends of the molecules are not rigidly stacked together, but act as fluid. The molecules’ positioning, however, is ordered in some respects and this gives the membrane some rigidity. Thus, the membrane is liquid and crystalline in nature and the theory of liquid crystals is suitable for modeling its properties, such as its flexibility.(4–6)Moreover, liquid crystals can be used in self-assembling complex layered structures in bio-inspired materials.(7)

Lastly, liquid crystal theory proves fruitful in understanding the interior structure of cells. That structure is formed by the cytoskeleton, which consist of long and stiff polymers such as actin. Ioanna Gârlea studied the ordering of actin filaments and their mechanical interaction with the cell membrane by modeling them as the components of a typical liquid crystal: strongly confined long and rigid rods.(8;9)It is the numerical results of this work that gave rise to the need to take a closer look at the microscopic theory of liquid crystals in the case of rods that are long — not just with respect to their thickness, but also with respect to the container in which they are confined. The simulations show that discontinuities in the LCs configuration can emerge purely as a function of their increasing length in the small container. That is one of the motivations for the work that is presented in this thesis.

1 _{That is, be polymorphous, allotropic or physically isomeric, to quote some jargon from that time.} 2 _{Notably, the first Dutch female Ph.D. in chemistry, Ada Prins, played a role in that.}

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III

For my fellow physicists

i Phenomenology

Phases of matter are characterized by their qualitatively different properties that every layman is familiar with. A gas is distinct from a liquid, and a solid is another thing altogether. Physicist know how to quantitatively describe these phases. We can study phase transitions, during which a material switches from one phase to another, as a function of for instance temperature or density. One of the most characteristic differences between solid and fluid materials is the ordering of the molecules. In many solids the positions of the constituent molecules are more or less fixed with respect to one another.

In crystalline materials this ordering is especially neat: there is a positional lattice that gives a repetitive pattern of equal distances between molecules. The ordering within a material can mathematically be described by a density-density correlation function.(10)For crystals, this is a periodic function that holds true over long distances: given the position of a molecule, one can accurately predict the locations of neighboring molecules, and those of distant neighbors just as well. This has many consequences for the macroscopic properties of the material. For instance, crystals are very rigid and their periodicity is reflected in the way they interact with light.

For most liquids, no such long microscopic order is present: they are homogeneous. Except at short distances, the density-density correlation function gives no more information then that there is an average density of molecules in the entire liquid. Random positioning of particles is only possible when they have enough space to wiggle around, hence the generally lower density of liquids. Also, it is stimulated by their thermal motion, which is positively related with temperature, hence the higher temperatures at which the liquid phase is usually found.

Various mesophases

Some materials can be in a liquid crystal (LC) phase: a mesophase between liquid and solid. It is an intermediate phase in the sense that it occurs at densities or temperatures between those at which the material is solid and those at which it is liquid. A liquid crystal material has more than one phase transition between these two. Molecules that constitute a material that can transition into a LC phase are called mesogens, and if the transition happens mainly as a function of temperature it is called thermotropic. LCs can be formed by mesogens in some liquid solutions as well. In that case the phase transition is also driven by the density –or rather: concentration– of the mesogens, and it is called lyotropic.(11)

The use of the word mesophase is justified by the observation that there is indeed more than one qualitatively distinct phase transition between liquid and solid, and by a specific set of macroscopic properties that is not found in either of those phases. Typically, this set consists of some properties that are usually associated to liquids — for instance their capability to flow — and others that are reminiscent of crystals — for instance an anisotropic magnetic or optical response. In fact, there is a classification of various LC phases on a scale between liquid and solid, and some materials can transition through many of them.

In order to appreciate this from a microscopic point of view, we need to include not just the positioning of particles, but also their orientation. Obviously, the orientation of particles need only be described for particles that themselves have some anisotropy. To describe a ball one usually does not need its orientation, as it is very symmetric, but for a stick it does matter in which direction it points. In going from liquids to crystals, the first mesophase one

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Figure 1– Figure reproduced from S. Savenko and M. Dijkstra, ‘Computer simulation of lyotropic LC from spherocylinders’, Phys. Rev. E, 70, 011705, 2004. The isotropic phase (left) and the nematic phase (right) for hard rods with L/D = 15, obtained using Monte Carlo simulations at two packing fractions. In the nematic phase, the center of mass distribution is still homogeneous, but the particle orientation is anisotropic.

encounters is one in which the orientation of the mesogens is no longer random, even though their centers of mass are still distributed homogeneously, like in the liquid phase. This is called the nematic phase, after the Greek word νήμα for ‘thread’. There is an average orientation of the mesogens: they tend to point in one direction, see Figure 1. This often has macroscopically observable consequences. For instance the optic and magnetic response may be sensitive to this direction. In fact, it is this property that led to the discovery of LCs.

The next phase of LCs is the smectic phase, after the Greek and Latin words for ‘soap’. Smectics may have soap-like properties, as the mesogens are ordered in layers or columns that can slide over each other. Within each layer or column, there is only a weak periodicity in the positioning of the centers of mass, on top of the orientational order that is inherited from the nematic phase. In fact, within the two dimensions of the layer or the single dimension of a column the system is homogeneous like a nematic. The distance between layers or columns is more regular. In that sense, a layered smectic is crystalline in one dimension of the three, and a columnar smectic is so in two of the three. Moreover, the orientation of the mesogens can change from layer to layer, for instance in a chiral way, and there are other complicated varieties of smectics. When even more order is added (by a decrease in temperature or an increase in density) the material will loose more of its anisotropy and become fully crystalline or at least solid.

Physics in the nematic phase

Phase transitions of LCs have been studied extensively and the methods used for that will be introduced briefly in the next section. This thesis will not elaborate on that further, but instead focus on the description of an LC phase itself, namely the simplest case of nematics. The combination of positional homogeneity and orientational order gives rise to a wealth of phenomena that are worthy of study. It has been the object of an entire field of research for decades, which has resulted in a fine-grained understanding of the ways in which nematics

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spontaneously organize, deform under mechanical stress, accommodate boundary conditions, respond to magnetic and electric fields and guide light.(10)So much so in fact, that this allowed for (and was stimulated by) the development of liquid crystal display (LCD) technology using twisted nematics.(12)

With ‘twisted’ it is meant that the orientation of the nematic — the local average ori-entation of the mesogens — slowly changes within the system. In LCDs the oriori-entation is twisted up to an angle of 90 degrees from one end to the other of the small cell that acts as one pixel in the display. On top of that, an electric field to which the mesogens couple further directs the nematic, allowing light to pass through. These twists in nematics are an example of distortions. In absence of external conditions or stimuli, the average nematic orientation will point in the same direction throughout the system. That is, the nematic system will return to a state that is homogenous in its anisotropy, for that is the lowest energy state.

In this context the word elasticity is central. In physics in general elasticity is a material’s response to stress, that forces it into a configuration that is entropically or energetically costly. A well known model for rubber elastic bands uses just an ideal chain of polymers. Stretching the rubber amounts to a decrease in the number of available microstates. So, in this picture, the resulting storage of elastic energy and the pull of the rubber band are entirely entropic in nature. In other systems a rise in elastic energy is a consequence of the less-than-ideal configuration into which the molecules in the system are forced, internally or with respect to one another. Much physical theory has been developed about the ways in which such materials deform and the energy cost associated to each of those ways. Typically, this can be cast into the form of elastic moduli that link some type of external stress to material deformations such as compression and shear.

In nematics, ‘elasticity’ is used in an analogous but subtly different way. It is not about deformations of the system as a whole, prompting configurational changes within, but about internal configurational changes, such as those in twisted nematics. These can for instance be achieved by external fields or by conflicting boundary conditions at opposite ends of a confinement. It turns out that there are three distinct ways in which the bulk of a nematic can deform and there are three independent elastic constants that account for the energy cost of these types of deformations. ‘Constants’ is a bit of a misnomer, as a microscopic theory might reveal that they are a function of parameters such as the density or the size of the mesogens. In the next subsection the details of this phenomenology are discussed and it is connected with the microscopic theories of nematics. The origin of the elastic response to deformations is usually sought in the interactions between the mesogens, as those are the driving factor for the emergence of a liquid crystalline phase in the first place. This thesis will investigate the possible contribution of entropy to the nematic elastic constants. There is no direct analogy between this idea and the entropic elasticity of rubber bands, but I will highlight other reasons to pursue it, especially when relatively long rods are used as mesogens.

ii Three main theoretical approaches

I will now briefly, and with a minimum of formulary, introduce the main theories that are used for understanding nematics, and discuss their strengths and weaknesses. One is fundamental in that it starts from a description of microscopic physics, but is fruitful for simple systems only. The other two are phenomenological, one tailored to model phase transitions and the other very successful in describing common types of distortions.

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Onsager

A new wave of microscopic theories of liquid crystals began with the seminal 1948 paper by the Norwegian-American theoretical chemists and physicist Lars Onsager.(13)The definition of liquid crystals had been made more precise, and Onsager showed that a nematic phase could emerge in a relatively simple system of long rods, at sufficiently high densities. With ‘rods’, here and later in this thesis, a specific type of particles that can act as mesogens are implied. Rods are stiff objects that are symmetric in all but one direction and that are much longer (L) than thick (D): L/D 1. Onsager required this ratio to be at least two orders of magnitude, which does not hold for most actual mesogens.

Onsager’s basic argument is that when enough of these rods are brought together, it is much more efficient to organize them like a ordered pack of mikado sticks: all neatly aligned. An unordered collection — such as the configuration that you get when you drop mikado sticks from some height into a bowl — typically needs more space. Under the right circumstances rigid rods in solution will indeed align, Onsager showed, even though entropy generally prefers an unordered state. Thus, symmetry is broken: the angular distribution of the rods is no longer fully random.

Let me emphasize the crucial insight at the core of Onsager’s result. He arrived at an expression for the free energy of the system, in which two terms compete. One is the orientational entropy term, which was derived with a relatively straightforward statistical mechanical analysis of the particles considered as an ideal gas. The other is the interaction term, which was found using interactions of a very basic type only. For high L/D rods, steric interactions — that do not involve repulsion or attraction, but just exclude particles out of each other’s volume — are sufficient for a system to provide a nematic state that is stable.

Onsager’s theory has as a central statistical object the single-particle distribution function ρ(1)_{(r, ˆ}_{ω) that depends on the position r and the rod direction ˆ}_{ω. Characteristic for the entropy} term is a factor ρ(1)_{(r, ˆ}_{ω) log ρ}(1)_{(r, ˆ}_{ω). The steric interactions are included up to the ‘second} virial’ contribution. All interactions are to be captured with an expression that just describes the connection (actually: exclusion) between two particles, given by two single-particle distri-butions. Hence the characteristic factor ρ2in the interaction term of the free energy. Maier and Saupe developed a method that is capable of handling more complex interactions through a mean-field approach.(14–16)The inclusion of long range forces between mesogens yielded quantitatively accurate predictions for the phase transitions for thermotropic LCs.(17)

Landau – De Gennes

In 1969 Pierre-Gilles de Gennes postulated that the transition from the isotropic to the nematic phase (the N-I transition) can be quantified with just a few coefficients that relate to the order in a system.(18)One of them is dependent on the temperature difference with the transition point, and on a exponent that is not known a priori. The other coefficients are unknown as well, which makes this a purely phenomenological theory. Together, the coefficients give the proportionality of the terms in De Gennes’s expression for the free energy, which is nothing more than an expansion in an order parameter (OP) and its first derivatives.

That classifies De Gennes proposal as a Landau theory of phase transitions.(19;20) In that very general and widely used framework, a parameter that is suitable to capture order during the transition is expanded around its equilibrium value. In writing down any Landau model some choices need to be made. The most important one is the selection of an order parameter. This can be guided by experimental observations of the system at hand, and by group theoretical considerations about the reduction of symmetries in the ordered phase.

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From the choice of order parameter usually some constraints on the possible coefficients in the expansion arise.

De Gennes elected a rank 2, symmetric and traceless tensor as the suitable order parame-ter for the N-I transition. Such a tensor, he claimed, is always proportional to the dielectric tensor αβ, the magnetic susceptibility tensor χαβ, or ‘any other [such] tensor, defining a local

property of the fluid’.(18)To promote this tensor to the order parameter ties in with all kinds of experimental studies of the properties of nematics. De Gennes called this tensor Q and ever since this Landau – De Gennes approach has been dubbed ‘Q-tensor theory’.(21)

This model is not juist applicable to phase transitions, but to some extent also capable of neatly describing the properties of the nematic phase itself. The Q-tensor can be made position dependent.(22)_{Then, it captures both the degree of ordering and the orientation of that ordering}

at every point in the system, given by a local average direction of the mesogens. Fundamentally this picture requires a local domain that is large enough to define a statistical average in a meaningful way, but small enough to constitute one spot in a macroscopic continuum of positions. Not all types of systems strictly allow for this, so in practice computational modeling of nematics that connects to Q-tensor theory uses some form of averaging as well. The average local order on small scales, even down to the magnitude of the particle size, can be obtained using for instance Monte Carlo simulations or Molecular Dynamics calculations.

In a 3 dimensional system that orientational order can be either uniaxial or biaxial. In the former case, the distribution of particle orientation around the local mean is isotropic: there are as much particles pointing slightly to the left or right of the the local average direction than there are pointing slightly upwards or downwards. In the biaxial case, the distribution around the mean is distorted: apart from a first direction that gives the mean orientation, there is a second direction in which the deviations from the mean dominantly occur. In the uniaxial case the Q-tensor has two equal eigenvalues, while all three of a biaxial Q are different.(21)

Important in the context of this thesis are the derivatives of Q: they describe the strength of the spatial variations in the order. For various reasons that I will elaborate on later, it is a sensible first approximation to allow in the expression for the free energy two distinct types of terms composed of derivatives of Q. Associated to them are two coefficients that De Gennes named the ‘elastic constants in the isotropic phase’. A persisting problem in liquid crystal theory is their connection with the three macroscopically observed elastic constants.

Oseen – Frank

In the Interbellum Carl Wilhelm Oseen worked on a the theory of distortion in the order of liquid crystals.(23;24) A host of other authors joined, culminating in a single edition of Transactions of the Faraday Society in 1933 with many papers on distortions in the nematic state.(25–27)Their crucial contribution was the ‘director’ field picture. In it the director, a single unit vector n(r), represents the local orientation of the nematic fluid.

Compared with the Q-tensor this object contains less information: it necessarily assumes a homogeneous density distribution and an orientational order that is homogenous in strength and devoid of biaxiality. It is only the direction of the order that varies. And smoothly so: just as De Gennes would do later, Oseen and others added only the first derivatives of the director field as distortion terms to an expression for the free energy. In reality, the local orientation may be forced to change suddenly, sometimes so discontinuously that the degree of ordering and even the density are affected as well. Such spots are called defects, and subtypes in include point defects of various topological strength, disclinations (orientational line defects) and dislocations (including density inhomogeneities).

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This vocabulary was introduced by Frederick Charles Frank, who recast Oseen’s ap-proach firmly in a pure continuum theory, in which defects are presented as mathematical singularities.(28)_{Moreover, he recognized that the director field should be viewed as a line}

field, not a vector field. Given their symmetry, most nematic mesogens can be flipped (or rotated by 180 degrees) without any change to the continuum physics. For all intents and purposes n =−n. Mathematically the nematic orientation is not a vector on the sphere S2_{, but}

a line through n and−n, living on the real projective plane RP2_.(29)_{With this realization Frank}

showed that some of the possible lowest order distortion terms are actually identical. He also identified a new type of terms, but that eventually turned out to be reducible to surface terms.(30) Finally, the consensus in the field is that only three distinct elastic constants are relevant in the Oseen – Frank theory for the bulk of an ideal uniaxial nematic.(10;31)

Various ways of relating the three theories

So far, we encountered three theoretical approaches to liquid crystals. A schematic overview of the landscape is given in Figure 2, which includes arrows that symbolize the connections between the theories that I will now introduce. Each of the theories has its limitations, for example in its range of applicability. Ideally, theories should agree on quantitative predictions wherever they overlap. Unfortunately, there is already quite a bit of qualitative difference between the three.

For practical purposes the theories are simply used in parallel, each to solve different problems. For instance, the Maier – Saupe theory has proven useful in finding suitable nematic mesogens for display applications, and Oseen – Frank theory was essential in modeling the elastic response in cells of those nematics. From a theoretical point of view, however, there is an abiding interest in rigorously connecting the dots. One central item on the agenda is the justification of phenemenological terms in the expression for the free energy from a funda-mental, microscopic point of view: the statistical mechanics of LCs. A second programme is about mapping the Landau – De Gennes tensor theory to the Oseen – Frank director picture. The latter project is in principle easily accomplished, or so it seems at first: one assumes (or takes the limit to) a uniaxial Q-tensor and a homogeneous density distribution. Then, it is matter of recasting variables to retrieve Frank’s expression. Alas, it appears there are a couple of problems, even though the algebra is often doable and informative. First, it has been suggested on mathematical grounds that nematic configurations at equilibrium must be either uniform, or show some biaxiality in order to allow for deformations, which could be necessary to accommodate boundary conditions.(32)The assumption of uniaxiality may thus be problematic. Even so, while biaxiality may be natural in the the Landau – De Gennes framework, it is rarely observed in nature — but it should be noted that such an experimental task is not easy. Second, the Q-tensor is much richer in information than the director, so going from one to the other may very well be not uniquely defined. In particular, in some system topologies there is a non-trivial choice of orientation of the director field.(29;33).

Third, and most prominently, there is the number of distinct elastic terms and associ-ated independent elastic constants. De Gennes identified two, whereas the Oseen – Frank perspective and experiments find three. This discrepancy stems from the different symmetries of Q and n and the way those play out with the assumption of smooth (long wavelength) distortions of the order. Both are phenomenological theories, and thus could in principle be supplemented with extra terms in cases of emergency like these, especially as experimental findings provide an occasion for that. A straightforward way to do so is to allow higher order terms in the expansion in Q. Indeed, to De Gennes’s two terms many more have been so

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Onsager

and Density Functional Theory tensor order parameter

Landau – De Gennes

Phenomenology

average orientation of molecules on mesoscale

free energy functional from statistical mechanics

Fundamentals

director (line) field

Oseen – Frank

Priest / Straley F&Y (2002) 

and this thesis

choose: uniaxial

homogeneous

- expansion in ! around phase transition - derivatives of ! for distortions

- 2 or many elastic constants

in nematics, smoothly changing " - derivatives of " for distortions -

3 elastic constants -

energy of system: competing ideal gas/entropy term, and a (second virial) interaction term

Figure 2– The three relevant theoretical approaches along with their central mathematical objects, main quantita-tive method and findings, and relations. The approach from fundamentals (bottom: Onsager and other Density Functional Theories) may be used to better justify the expressions in phenomenolgical theories (top: Landau – De Gennes and Oseen – Frank). The blue line represents the type of connection Fukuda & Yokoyama make and that will be used in this thesis as well: a DFT is supplemented with the Q-tensor and the resulting expressions are cast in terms of Frank elastic constants.

added in the plethora of articles that have been written about this infamous elastic constants problem. The problem, however, is that most of these attempts provide no clear justification for precisely which terms they choose to add. Meanwhile, the number of available, more or less equivalent higher order terms is rather large.(34–36)_{This prompted Vertogen & De Jeu to}

remark that all of these are discarded ‘for the simple reason that they are hard to deal with’.(31) This problem reinforces the need to link the microscopic theories to the macroscopic observables, not just for the occurrence of phase transition, but also for the static properties within a mesophase. Early contributions to this were made by Priest and Straley.(37;38)With the aim to find the three distinct elastic terms, they calculated higher moments of the density and orientational distribution function, based on expansion of the interaction term in the Onsager free energy of rigid rods. These attempts proved fruitful in the sense that from those higher moments reasonable expressions for all three elastic constants in the Oseen – Frank picture could be extracted. This connection does not, however, apply to the more general Landau – De Gennes theory. It cannot, in fact, as the Q-tensor is deliberately limited to the first non-zero moment of the orientational order.

The general framework in which Onsager’s and Maier & Saupe’s results can be expressed is density functional theory (DFT).(39) _{Its core idea is to encapsulate the configuration of a}

statistical system in the single particle distribution function, much like Onsager did. The re-sulting expression for the free energy is a function of the density and orientational distribution of the mesogens as usual, but it can now simply be constrained to a specific configuration, by adding as Lagrange multipliers (external) conditions. Minimization of this constrained expression then gives the desired equilibrium configuration of the system, complying with the constraints.(40)

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When the De Gennes order parameter is treated as a constraint to a density functional, the resulting expression for the equilibrium free energy will naturally contain terms propor-tional to it. This procedure thus offers a microscopic justification for the presence of each term in such an expression of Q-tensor terms. The foundation with a minimized density functional guarantee that they are not just phenomenologically suspected to play a role, but also fundamentally shown to be relevant for the physical equilibrium state.

This procedure has mostly been applied to strengthen the Landau – De Gennes ap-proach’s predictions for the N-I transition and helps to improve its quantitative predictions in the nematic phase. In addition, Ball and Majumdar considered static distortions (and thus: elastic constants), but added four elastic terms by hand.(41)_{This does not add justification or}

insight into the origin of the elastic terms, even though they found remarkable constraints on the equilibrium solutions that are possible with those terms.

Next to manually inserting the elastic terms or tracking them down in the interaction term, a final possibility is to search for them in the entropy term of the free energy density functional. This is what Fukuda and Yokoyama (F&Y) did.(42;43) In a series of papers they calculated the energy and entropy of semi-flexible polymers, using (i) DFT constrained with the tensor order parameter, and (ii) a diagrammatic expansion in the correlations in the polymers.(44;45)This implies that they included the length of these polymers and calculated how the entire objects contribute to the entropy. In contrast with virtually all earlier calculations of the entropy term of mesogens, this now no longer is a entirely local property. Thus, the particles’ length plays a role in the spatial variations of the orientational order. F&Y included this effect up to first derivatives of the Lagrange multiplier that couples the tensor order parameter to the free energy. At the end of this lengthy exercise, they considered the limit in which the polymers are very stiff, so that they are like rigid rods. When the resulting expression was evaluated for a 3 dimensional ideal uniaxial system, all three elastic terms and expressions for the their respective constants were found. In conclusion: F&Y discovered that the entropy term alone suffices to derive the macroscopic elastic terms from a microscopic description of a simple nematic system. For real systems it is expected that both entropy and interactions play a role of comparable magnitude in creating the elastic reponse.

iii Problems, questions, and an outline of this thesis

Fukuda and Yokoyama’s result acted as the main source of inspiration for this thesis. The work that will be presented in the second chapter has been guided by this question: Can the elastic constants of an ideal uniaxial nematic be derived via the De Gennes tensorial picture from the entropy term of a density functional that faithfully accounts for the finite length of the mesogens, if those are modeled as ideally rigid rods from the outset?

More or less rigid

This question has the potential to address some of the open problems that I discussed so far. But firstly, it is an attempt to clarify F&Y’s work by exploring the significance of its assumptions. Most importantly, they required the complex apparatus of diagrammatic expansions in order to calculate the energy of semi-flexible rods. This work avoids such complications by assuming rigidity from the very first moment that the length of the particles is introduced.

Even beforehand, there is ample reason to believe that this may not be equivalent to a semi-flexible approach. In 1981, Nico van Kampen showed that the statistical physics of a trimer in the limit towards rigidity is not equivalent to that of a purely rigid trimer.(46)The motions of the former take place in a somewhat smeared out track in phase space, while the

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Figure 3– Left: figures reproduced from I. Gârlea and B. Mulder.(9)_{Isotropic and nematic phase in a strongly} confined 2D hard rod system with low L/B. Right: pictorial explanation of non-local contributions to the entropy: rods with a center of mass (yellow dots) outside a local domain (green circle) can be relevant to it as well. latter follows an ideal line. This introduces a difference that does not average out. Van Kampen notes that ‘rigid rods do not exist in Nature’, but a choice of potential for a semi-flexible trimer is not easily justifiable either. He is also relieved to remark that the influence of this subtly and paradoxically different notions of rigidity most likely have little influence on the properties of larger systems, as other forces and interactions are dominant. For this thesis, however, the focus is precisely on this subtle notion of rigidity, as interactions will not be considered. Hence, Van Kampen’s article may be considered a warning.

In contrast with that, Jeff Chen’s work should be mentioned. In the appendices to two papers he developed an expression for the free energy of rods that includes their finite length, first in a way quite similar to that of F&Y, and then for rigid rods.(47;48)He showed a degree of equivalence between these two approaches. While these calculations boost confidence in the merit of the finite rod length approach, Chen’s result does not preemptively make the work of this thesis redundant with F&Y’s, as he mainly focussed on the interaction term and did not develop the expressions from the entropy term in detail.(49)

Non-local entropy contributions

A second set of questions that may be elucidated by this work has to do with the entropy term. As mentioned, this usually takes the form of ρ(1)_{(r, ˆ}_{ω) log ρ}(1)_{(r, ˆ}_{ω), where ρ}(1)_{(r, ˆ}_{ω) is a} function of the center of mass position r and the particle orientation ˆω. This is a correct and complete way of counting the possible states of a system, but a problem arises if the nematic orientation changes on length scales smaller than the length of the particles. Then, locally, an expression based on center of mass presence alone does not encapsulate the fact that at a mesogen may still be very much present at a small distance away from r, in the direction ˆω. As illustrated in the right part of Figure 3, non-local contributions can be relevant on scales that are small with respect to the rod length.

If nothing else this is a problem of fundamental interest. But there are at least two questions that may be connected to it; I already mentioned both in the above. First, there is the energy cost of defects that is poorly understood in the context of Oseen – Frank theory. At a point defect the director field can for instance be a divergence well. As a consequence, close to the defect no centers of mass are present and a local mean orientation cannot be defined. Meanwhile, it is clear that the ends of the rods that point towards the defect do influence the its configuration. Hence the need for to probe for non-local contributions to the entropy.

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Second, this problem may become more pressing in the case of long rods in small confinements (e.g. sides B). Numerical simulations suggest that such configurations induce discontinuities in the nematic orientation, see the left of Figure 3.(8;9;50)These defects are a consequence of the ratio L/B primarily. This stands in contrast with the nematic defects that are usually studied, that originate from a topological mismatch in boundary conditions that must be overcome somewhere in the nematic. Arguably, the need to account for non-local entropy contribution is particularly important for rods that are long with respect to the system size.

Outline

In conclusion, the aim of this thesis is to explore a non-local addition to the entropy term of the free energy of rigid rods, in the hope of shedding light on both the long standing problem of elastic constants and on the energy cost of defects that arise for long rods. In a more abstract sense, the objective is to rethink the justification of the terms that contribute to the distortion free energy of nematics.

The first chapter will be spend on formally introducing the theories I just outlined. I will give Onsager’s statistical mechanics of hard rod systems in some detail. Landau – De Gennes theory of the N-I transition will be discussed a bit more broadly, mostly focussing on the definition of the order parameter, but I will explain his two elastic constants and outline why adding more is problematic. Then, using the Oseen – Frank picture, I will derive the three elastic terms of that framework.

The second chapter contains the main contribution of this thesis, and consists entirely of the F&Y-inspired finite length rigid rod entropy calculation. Some time will be spent on the necessary expansions, rotations and integral calculations, even though the most technical material is placed in the appendix.

The third chapter is a discussion of the results. In it, I will revisit the motivations I outlined in this introduction. Finally, a brief summary of the conclusions and an outlook is given.

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Chapter 1

A technical introduction to the relevant

theories of liquid crystals

The most relevant parts of the existing theories of liquid crystals will be presented in this chapter. The derivations and explanations are not new, but follow some trusted sources closely, most notably René van Roij’s PhD thesis,(51)Bela Mulder’s notes on density functional theory,(39)De Gennes & Prost’s and Vertogen & De Jeu’s textbooks,(10;31)and many primary sources such De Gennes’s articles.

1.1 Statistical mechanics of LC phases

Following Van Roij, I will now introduce the statistical physics of simple liquid crystals and eventually arrive at the type of energy functional that (a) allowed Onsager to derive the nematic phase for rigid rods, and (b) I will use in the next chapter. The considered systems are simple in the sense that (i) finite size effects or surface influence are neglected, (ii) non-equilibrium or dynamical situations such as fluid flow are too, (iii) if interactions are included, they will be of the steric type, and (iv) the only LC-phase that will be pursued is the nematic case, so positional order remains highly random.

Thanks to the first two assumptions, the framework of statistical mechanics can be directly employed to relate microscopic states to macroscopic observables. Here, the former may be written as Γ(qN_{, p}N_{) = Γ(q}

1, . . . , qN, p1, . . . , pN) where the variables denote the

positions of the N particles in the fluid, and their generalized momenta. So, qN represents N 3-dimensional positions, not a to-the-power-N . One qiis a three dimensional vector, for

brevity I will save myselve the trouble of writing qior pi sometimes

To this phase space, particle orientation (and possibly, angular momentum) must be added when spatially extended particles are considered, as we will do later. The macrostate may be described by e.g. the Helmholtz free energy, F (N, V, T ), where N is the number of particles, V is the system’s volume, and T is its temperature.

1.1.1 Partition function and free energy

The proposition of statistical mechanics is to avoid calculating the time evolution of the Hamiltonian HN(Γ) for all particles, but instead find expectation values for macroscopical

observables by averaging some ensemble of microstates. I state the results for the canonical ensemble:

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• All microstates are weighed with the Boltzmann weight exp [−βHN(Γ)]. Here β =

(kBT )−1, which has units of inverse energy, such that βHN is dimensionless.

• The expectation value for an observableO is now given by hOi = R dΓ exp [−βHN(Γ)]O(Γ)

R dΓ exp [−βHN(Γ)]

(1.1) where the denominator takes care of normalization. Here, the differential of Γ is given by dΓ =QN i=1 dqidpi hc

, where c = 3 for particles without angular momentum, and c = 6 for those with, for dimensional reasons.

• Above, I already almost identified the canonical partition function Z = 1

N ! Z

dΓ exp [−βHN(Γ)] . (1.2)

From it, the free energy is found: F (N, V, T ) =−1 βlog Z

Crucial for the development of the next section is the assumption that the Hamiltonian can be split in a kinetic and a potential part: HN = UN(qN) + KN(pN), where the kinetic term KN

may be written simply as KN =Ppi

pi·pi

2m for point particles. Then, the partition function may

be simplified as follows: Z = 1 N ! Z dΓ exp " −β UN(qN + X pi pi· pi 2m !# = 1 N ! h3N Z N Y i (dqiexp−βUN(qN) × Z N Y i dpiexp h −βpi· pi 2m i

naming the first integralW, and performing the last one (Gaussian, see below)

= 1 N ! h3NW(N, V, T ) × r 2mπ β 3N = W N !VTN . (1.3)

The factorW is the configurational integral ; I will evaluate it later. In the last line, I made use of the thermal volume of a point particle (hence the third power of h), which we can also write as this power of the De Broglie wavelength Λ:

VT = h √ 2πmkBT 3 = Λ3. (1.4)

This thermal volume must be multiplied with an additional factor to incorporate the effect of angular momentum, if the particles are not point-like. Since LCs are necessarily made of anisotropic (non-spherical) particles, this is formally required, just as c = 6 is actually the appropriate choice.(52)_{However, this factor to}_V

T can be ignored without physical consequence

in most LC studies. What is more: while the thermal volume remains present as an overall factor in the free energy (together with the density ρ it is a dimensionless quantity that will appear in a logarithm), it entirely drops out of the expression for the distribution function that determines the configuration of the system at normalization.

The Gaussian integral of the point-particle kinetic part was performed as follows Z dp exp − β 2mp· p = s π β 2m =r 2πm β =p2πmkBT (1.5)

and one should note that this integral is done not just N but 3N times, since pi is actually a

vector of three dimensions.

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1.1.2 Virial Expansion

Solving the configurational integralW is hard in general, and doable in very specific cases. For zero potential UN we have the ideal gas case:

Wid= Z dqNe0 = Z dq N = ΩN. (1.6)

In the last equality we defined the one-particle configurational volume Ω. The orientation of rod-like particles is given by two angular variables, so with a solid angle we have Ω = 4πV .

The free energy can now be specified with respect to this ideal case: F =−1 β log Z =− 1 β log W N !_VTN =−1 β log Wid Wid W N !_VTN =−1 β log Wid N !VTN + log W Wid ,

whereWidis the configurational integral for the ideal gas case (UN = 0). The first term yields

an ideal gas contribution to the free energy, while the second term remains to be determined, as we will see later. To obtain a dimensionless expression we take β to the LHS and compute:

βF = + log N !VT N ΩN − log W Wid ≈ (N log N − N) + N log VT Ω − log W Wid

= Ω(ρ log N _{− ρ) + Ωρ(log V}T − log Ω) − log

W Wid = Ω (ρ log(ρ_VT)− ρ) | {z } − log W Wid = βFid − log W Wid . (1.7)

The final expression explicates that the free energy consists of an ideal gas part and a yet to be determined interaction part. The factor ρ log ρ is the well-known signature feature of the ideal gas part. In the derivation we used Stirling’s approximation for N ! in the second equality, assuming N is very large. In the third equality, we introduced a quantity ρ = N_Ω that is proportional to the number density. As we will see in the next subsection, it can be reinterpreted as the single-particle distribution function. It can only be defined under the assumption that the system is close to equilibrium and not subject to external forces. A second assumption we will make now is that the particles are neutral or only have interactions on ranges that are short compared to the inter-particle distances. This allows us to confine ourselves to pair-wise interactions. We thus define a potential as:

U =

N

X

i<j

u(qi, qj). (1.8)

This eases our case, but not sufficiently so. One approach that makes the integral ofW actually solvable is to make use of the Mayer-function

fM(qi, qj) = exp [−βu(qi, qj)]− 1, (1.9)

which is judiciously defined such that it, when introduced in the configurational integral, splits it in an ideal-gas part and an expansion in orders of the density ρ, like so:

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Free Energy of Finite Length Rigid Rods W = Z dqN_exp  −β N X i<j (u(qi, qj))   = Z dqN N Y i<j exp [_−βu(qi, qj)] = Z dqN N Y i<j (1 + fM(qi, qj)) ,

which, ignoring terms of higher order in fMcan be written as

= Z dqN  1 + N X i<j fM(qi, qj) +O(fM2)   = _Wid + Wid Wid Z dqN N X i<j fM(qi, qj) +O(fM2) = _Wid    1 + 1 R dqN N X i<j Z dqNfM(qi, qj) +_O(fM2)    . Now writing out dqN = dqidqj

QN k6=i,jdqk

and noting that k-factors are shared by numerator and denominator. = _Wid    1 + N X i<j " R QN k6=i,jdqkR dqidqjfM(qi, qj) R QN k6=i,jdqkR dqidqj # +_O(fM2)    = _Wid    1 + N X i<j 1 R dqidqj Z dqidqjfM(qi, qj) +_O(fM2)    = Wid    1 + N X i<j 1 ΩiΩj Z dqidqjfM(qi, qj) +O(fM2)   

The sum over i < j has N (N− 1)/2 terms, which we approximate as N2/2 for large N . Also, the integral over qi and qj is equal for all i, j, so just write as 1, 2.

= _Wid 1 +N (N− 1) 2Ω2 Z dq1dq2fM(q1, q2) +O(fM2) ≈ Wid 1 +1 2ρ 2 Z dq1dq2fM(q1, q2) +O(fM2) .

In the fourth line I neglected higher order terms of the Mayer function, that is: I calculated the second virial approximation only. The second order term captures the pair-wise interaction, so it is natural that it is proportional to the density squared. When this expression is used for W in (1.7), and it is recalled that log(1 + x) = x + O(x2_{), this expression can be obtained for}

the second virial series of the free energy: βF = βFid−

1 2ρ

2Z _dq

1dq2fM(q1, q2) +O(ρ3). (1.10)

The computation of higher order terms can be cast in the language of diagrams, where the Mayer function acts as bond between vertices defined by a single-particle distribution function ρ(q), that we will meet in the following section.(53)

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1.1.3 Density functional theory

Nowhere in the above discussion the particles were prescribed to position and orient them-selves in a certain fashion. In fact, the derivations hold only for a homogeneous and isotropic distribution of the particles. Given such a phase, equation (1.10) will provide you with the associated free energy. How to inject some order into the particles’ configuration? Or, even more useful, how to find out what order is energetically favorable? The answer is density functional theory.

Earlier we defined a quantity that was equal or at least proportional to the number density and called it ρ. Now, we will promote it to be the single-particle distribution function ρ(q). To stress that it is a single-particle function (SPDF) it is often denoted as ρ(1)_{(q). For hard} rods, q is usually a position r and an orientation vector ˆω. In such cases, ρ(1)_{(r, ˆ}_{ω) can be} defined as:(39) ρ(1)_{(r, ˆ}_{ω) =} * _N X i=1 δ(r− ri)δ(ˆω− ˆωi) + . (1.11)

The average and the sum over all particles make clear that it is a statistical property, not a function of one particular particle, even though its name might suggest that. The delta functions detect the presence of a single particle with position r and orientation ˆω, and this is done throughout space. The normalization is obvious:R dq ρ(1)_{(q) = N .}

Density functional theory works with a functionalG[ρ(q)] about which a few statements are true. They were found in specific cases at first, and later formally proven.(54)_{There exists a}

G[ρ] such that

• for a given pair-potential u(qi, qj), it is unique;

• it depends on the chemical potential µ, volume V , and temperature T ;

• it is minimized at equilibrium: G[ρeq]≤ G[ρ] for the equilibrium distribution ρeq;

• this minimum gives the equilibrium grand canonical potentialGeq.

Thus, the thermodynamics of a system can be obtained from (i) the pair potential u(qi, qj),

(ii) the variational principle, and (iii) the three grand canonical parameters. Often, µ can be traded for N with ease. Since ρ(q) is normalized to N ,

G[ρ] = F[ρ] − µ Z

dq ρ(q)

rephrasesG[ρ] in terms of a Helmholtz free energy functional. That now must be equal to an equilibrium Helmholtz free energy, like the one we met in the previous section. This transformation thus offers a solution to a question that is in principle wide open: what to choose as an initialG[ρ].

The minimization of the functional often involves multiple functional derivatives, which are denoted by δF [r] / δf (r) for a functional F [f ]. For future reference, I now state the defining equation, which uses a variation ∆f :

F [f (r) + ∆f (r)] = F [f (r)] + Z

dr δF

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1.1.4 Onsager’s result

One of Onsager’s crucial insights is that in the case of long hard rods, the third order virial term (kernel f3) and higher terms can be neglected, because

hf3i hf2 Mi =_O D L log L D .

This he derived by geometrically exploring the ways two or three long rods can be brought into contact. For cubes and plates, the third virial term is much more relevant. Onsager’s simplification can be interpreted as the analog for hard bodies with steric interactions of mean field theories for particles with long range interactions. It holds under the assumptions that (i) the rods are very long: L D, (ii) their interactions are purely steric, and (iii) their volume fraction in the solution is small: Φ =1_/₄_πρLD2 _{1. In practice, values for ΦL/D ∼ 4 are}

found around the N-I transition.(10)

Here I will mention some of the steps in Onsager’s approach, in order to illustrate the rather abstract treatment in the above. Onsager’s approach was a density functional theory avant la lettre, so recasting it in DFT language is justifiable.(52)

I start with a free energy functional informed by the Helmholtz free energy from equation (1.10): β_F[ρ(1)_] ₌ Z drdˆωρ(1)_{(r, ˆ}_{ω) log} VTρ(1)(r, ˆω) + 1 2 Z drdˆω Z dr0dˆω0ρ(1)_{(r, ˆ}_ω)ρ(1)_(r0_{, ˆ}_ω0₎_e−βu(r,r0_{, ˆ}_{ω, ˆ}_ω0₎ − 1. (1.13) The term on the first line is the ideal gas and orientational entropy contribution. On the second line stands the second virial term, which uses the Mayer function. For hard rod-like objects with steric interactions it may be defined as

fM(qi, qj) =

−1 particles i and j overlap

0 i and j do not overlap (1.14)

Given the definition (1.9), this may be accomplished by defining the pair-potential as u(qi, qj) =

∞ overlap

0 no overlap (1.15)

A first observation about the expression (1.13) of the energy is that the second virial term is independent of temperature. This implies that the emergence of a mesophase in hard body fluids — for which the Mayer function holds — is determined by entropy alone. This can be made explicit by calculating the internal energy:

E = ∂βF ∂β N,V = Ωρ 1 ρVT ρ∂VT ∂β = Ωρ(2πmkBT ) 3/2 h3 h3 (2πm)3/2 ∂ ∂β β+3/2 = Ωρβ−3/23 2β +1/2₌ 3 2ΩρkBT = 3 2N kBT.

This is linear in T . Using the identity F = E− ST we see that the entire free energy F is only connected linearly with T and that minimizing it simply amounts to maximizing the entropy S.(51)This is to say that LCs that use hard rods as mesogens are athermal. In fact, the

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packing fractions that can be determined from the Onsager model for nematic and isotropic configurations just around the phase transition are independent of temperature as well.(10)

The way Onsager’s theory of hard rods works, then, is that the distribution of particles is determined by minimizing the energy functional, including steric interaction in the second virial term that impacts the molecular freedom and packing of the system. The result is a configuration ρ(1)_{(r, ˆ}_{ω) with a certain entropy cost, which determines the Helmholtz free} energy of the system.

Now, let’s actually perform some calculations along this line. I assume an homogeneous distribution of rods’ center of masses in real space, such that I can rewrite the single-particle density as a constant times an axial orientation distribution Ψ via ρ(1)_{(r, ˆ}_{ω) = ρ Ψ(ˆ}_{ω). Then} I demand the stationary condition on this functional, while maintaining the normalization R dˆωΨ(ˆω) = 1 as a Lagrange multiplier. That is:

δ_G[Ψ] δΨ ! = 0, (1.16) where G[Ψ] = βF − λ Z dˆωΨ(ˆω)− 1 (1.17) = ρ Z drdˆωΨ(ˆω) log(Ψ(ˆω)) + ρ Z drdˆωΨ(ˆω) log(VTρ) − 1 2ρ 2 Z dr1dˆω1 Z dr2dˆω2Ψ(ˆω1)Ψ(ˆω2) e−βu(r1,r2, ˆω1, ˆω2)_{− 1} − λ Z dˆωΨ(ˆω)− 1 .

In the third line after the last equality, the integration is done over two rods, now numbered for clarity. The integrals over r each yield one volume factor V .

A functional derivative with respect to the multiplier λ gives back the normalization of Ψ; I will plug it in wherever appropriate. The stationary condition forG with respect to Ψ will give us some idea of what the latter will look like. To this end, note that the first line (with Ψ log Ψ) yields 1 + log Ψ, because the integrand of the functional has that as its Ψ-derivative (using the product rule). The second line simply yields the logarithm of the thermal volume and constant density. The third line is more complicated, and is best understood using the notion of excluded volume.

In order to make the step towards this notion more transparent, we make the following shifts of variables for this interaction term:

r1,2 = r2− r1 R = (r2+ r1)/2 =⇒ r1 → R − r1,2/2 r2 → r1+ r2 = R + r1,2/2 . (1.18)

This is indeed a simple shift with unit Jacobian which does not alter the integral. R is the position of a pair of rods, and r1,2is difference space vector between their centers of mass. The

conceptual step is now to assume that u is independent of R, which is valid for a pair-wise interaction. Thus, R can be integrated out, giving a volume factor. Let me recap the functional:

G[Ψ] = ρV Z dˆωΨ(ˆω) log(Ψ(ˆω)) + ρV log(VTρ) Z dˆωΨ(ˆω) − 1₂ρ2 Z dRdr1,2 Z dˆω1dˆω2Ψ(ˆω1)Ψ(ˆω2) e−βu(r1,2, ˆω1, ˆω2)_{− 1}

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Free Energy of Finite Length Rigid Rods − λ Z dˆωΨ(ˆω)_{− 1} .

Using ρV = N , the number of rods, and defining the excluded volume thus E(ˆω1, ˆω2) =− 1 V Z dR Z dr1,2Ψ(ˆω1)Ψ(ˆω2) e−βu(r1,2, ˆω1, ˆω2)_{− 1} =−1 VV Z dr1,2Ψ(ˆω1)Ψ(ˆω2)fM(r1,2, ˆω1, ˆω2), (1.19)

that is, pre-emptively pulling out one factor V to be able to collect a factor N of all terms ofF and, physically, to obtain a volume that is purely a property of this pair of rods, one can now write G[Ψ] N = Z dˆωΨ(ˆω) log(Ψ(ˆω)) + log(_VTρ) + ρ 2 Z dˆω1dˆω2Ψ(ˆω1)Ψ(ˆω2)E(ˆω1, ˆω2) − λ Z dˆωΨ(ˆω)_{− 1} . (1.20)

Onsager assumed rods of length L and width D with the former much larger than the latter, and showed that the excluded volume approaches E = 2L2D_{| sin γ|, where γ is the angle} between the two rods. The excluded volume can be understood as ‘the volume around a fixed rod A that the center of mass of another rod B cannot enter, given that the direction B points in differs from that of A by an angle γ’.

Now the angular distribution can be found by demanding 0 =! δG[Ψ] δΨ = (1 + log Ψ)− λ +ρ 2L 2_{D 2} Z dˆω0Ψ(ˆω0)_{| sin γ|, so:} βµ = log Ψ + 1 + ρL2D Z dˆω0Ψ(ˆω0)| sin γ| (1.21) where I plugged in the normalization in the last line and set λ = βµ to fix dimensions. It is natural to identify µ as the chemical potential, given the thermodynamic identity connecting the grand potential with the Helmholtz free energyGeq = F−µN. The factor 2 in the third line

originates from the symmetry of the functional derivative to the two versions of Ψ. Solutions to this equation need to be normalized, which allows for cancelling constant terms and the chemical potential.

There is always an isotropic solution to equation (1.21). It is a constant, and normalization requires this to be Ψ = _4π1 . For sufficiently high concentration, Onsager found additional, nematic solutions using a variational method with an ansantz Ψ = C cosh( cos θ), with θ the angle between the nematic axis and the particle orientation ˆω, C a normalization constant, and the variational parameter. This turns out to give a strongly ordered state.

Another approach was developed by Kayser & Ravaché, using bifurcation analysis, in which new solutions branch off from the constant one at specific densities.(55)They showed that the first few bifurcation equations are just eigenvalue problems for the excluded volume function E(ˆω1, ˆω2). For instance, the first bifurcation equation reads(56)

Ψ1(ˆω)∝ −ρ0

Z

dˆω0E(ˆω, ˆω0)Ψ1(ˆω0).

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Using the series of bifurcation equations one can obtain a diagram in which ordered solutions branch off from the isotropic one at a specific density.

In general, the minimization of free energy (1.13) as just performed entails a competition between the ideal entropy (∝ ρ log ρ) and the interaction term (∝ ρ2_and_{| sin γ|). The former}

favors isotropically oriented particles, while the latter prefers aligned particles: small γ. The latter dominates at higher densities, when particles get close together. Then it becomes important that they get well-aligned, such that they take up less space with respect to one another (smaller excluded volume) and have more room to wiggle around (larger phase space), which is entropically favored. Counterintuitive but true: a dense collection of rods obtains more freedom by aligning.

1.2 Phenomenological Theories

1.2.1 The Landau – De Gennes theory for the nematic phase

In a Landau theory of a phase transition an appropriate order parameter needs to be defined. It should be zero in the unordered, isotropic phase, and non-zero as soon as order emerges. It can be modeled with an expansion in the parameter around its equilibrium unordered value. Although the nematic-isotropic (N-I) phase transition is formally of first order, it is sufficiently smooth for a Landau theory to explore the nematic phase. This is what De Gennes did.(18;22) There are many excellent introductions to his theory, most importantly his book and a nice review by Gramsbergen et al.(10;19)Here I present the essential insights only.

The choice of order parameter is a delicate matter. It can be purely phenomenological, just informed by the symmetries of the system. Landau offered arguments for those types of reasoning. It can also be partly informed by a microscopic theory of the system. Here I start with the latter, assuming a nematic system of rigid rods.

Order parameter: microscopically informed

In the introduction, I mentioned the symmetry in the orientation ˆωof rods: it is best described as line through ˆωand−ˆωon RP2. What is needed, is a probability measure for the orien-tation dµ(ˆω) = Ψ(ˆω)dˆω on S2_{, where µ(ˆ}_{ω) = µ(}_−ˆ_{ω) and}R

S2dµ(ˆω) = 1 for symmetry and normalization.(29) Then, a function is desired that maps this distribution to a number that parametrizes the order in the system. A first choice would be: the average of the scalar product of the orientation —hˆω· ˆωi — but that is unity as ˆωis a unit vector. The next natural choice would be to consider higher moments of ˆω. The first moment, however, is zero:(29)

hˆωi = Z S2 ˆ ωdµ(ˆω) = 1 2 Z S2 ˆ ωdµ(ˆω) + Z S2−ˆ ωdµ(−ˆω) = 0

Hence the second moment is considered

O = Z S2 ˆ ω_{⊗ ˆ}ωdµ(ˆω), or, equivalently: Oστ = Z S2 ωσωτdµ(ˆω). (1.22)

Free Energy of Finite Length Rigid Rods

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Free Energy of Finite Length Rigid Rods

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Contents

A general introduction to the topics

and questions in this thesis

I

For the layman

II

For the biologists and the historically inclined

III

For my fellow physicists

Onsager

Landau – De Gennes

Phenomenology

average orientation of molecules on mesoscale

free energy functional from statistical mechanics

Fundamentals

Oseen – Frank

Chapter 1

A technical introduction to the relevant

theories of liquid crystals

1.1

Statistical mechanics of LC phases

1.2

Phenomenological Theories