Molecular Theory of Nematic Liquid

Crystals

Paul van der SchootDepartment of Applied Physics, Eindhoven University of Technology

and

Institute for Theoretical Physics, Utrecht University

The Netherlands

Lecture Notes Postgraduate AIO/OIO School

Statistical Physics and Theory of Condensed Matter

March 12, 2018

Abstract

Nematic liquid crystals are symmetry-broken, uniaxial fluids in which the consti-

tutive particles are aligned along a preferred axis known as the director. There are

two kinds of nematic liquid crystal commonly referred to as thermotropic and lyotropic

nematics. Thermotropic nematics are fluid phases of low molecular weight compounds

whereas lyotropic nematics are dispersions of colloidal particles in a host fluid. The

driving force for spontaneous molecular alignment in thermotropic nematics is enthalpy,

and that in lyotropic nematics is not enthalpy but entropy. Molecular theories describing

the isotropic-to-nematic phase transition can be formulated for both types of system.

They are rooted in the same theoretical formalism. In these Lecture Notes I describe

how, from a general density functional theoretical description, the Maier-Saupe theory

for thermotropic nematics and the Onsager theory for lyotropic nematics can be derived.

I will discuss the ins and outs of both theories in some detail.

1

Contents

1 Introduction 2

2 Density functional theory 9

3 Onsager theory 15

4 Maier-Saupe theory 21

5 Concluding remarks 26

6 Bibliograpy 27

1 Introduction

The three familiar aggregated states of matter are the gas, the liquid and the solid.

The gas phase has a low density and does not keep its volume or its shape, that is,

it expands to fill its container. The liquid and solid are both dense phases, where the

former keeps its volume but not its shape, whilst the latter keeps both in any shape of

container. Gases and liquids flow when a force is exerted onto them and exhibit viscous

flow behaviour. Solids that deform under loading but do not flow, and (in principle) only

exhibit elastic deformation behaviour. Upon removal of the external force they relax to

their initial shape. Solids are crystals and exhibit long-range positional order, whereas

gases and liquids do not exhibit any kind of long-range order. The main difference

between a gas and a liquid is, in essence, the density, which differs typically by a factor

of one thousand.

At fixed pressure p but increasing temperature T , the typical sequence of the aggre-

gated state of matter goes from a crystalline phase, via a liquid phase, to a gas phase.

Figure 1 (left) shows a generic phase diagram of atoms, small molecules and of spherical

colloids that sometimes are seen as large model atoms. The drawn lines demarcate first

order, discontinuous transitions between the different states of matter. The lines repre-

sent a continuum of conditions where two phases co-exist. Thermodynamics stipulates

that, under conditions of co-existence, the phases have equal temperatures, chemical

potentials and pressures. Indicated in the phase diagram are also the triple point (TP)

at which a solid, a liquid and a gas simultaneously co-exist, and the critical point (CP)

at the end of the liquid-gas transition line. Beyond it, in the super critical fluid region,

there is no liquid-gas transition. Not surprisingly, the densities of the co-existing gas

and liquid phases approach each other when approaching the critical point.

2

Phase diagrams such as shown in Figure 1 (left) can be understood on the basis

of relatively simple pair interaction potentials U , schematically represented in Figure 1

(right). The interaction potential U(r) is presumed to be spherically symmetric and only

depends on the distance r between the centres of mass of the two particles. The potential

ignores any atomic detail of the molecules and is characterised by a short-range (steric)

repulsion, and by a longer-ranged Van der Waals attraction. An often-used model pair

potential is the Lennard-Jones potential ULJ ,

ULJ = ε

[(σr

)12−(σr

)6], (1)

where σ is a measure for the size of the molecules and ε a measure for the strength of

the Van der Waals interaction between them. These two model parameters have been

determined experimentally for a plethora of atoms and small molecules.

Figure 1: Left: Phase diagram of atoms and small molecules (and colloids). X: crystal,

L: liquid, G: gas. Indicated are also the triple point TP and the critical point CP. Right:

Schematic spherically symmetric molecular potential U(r), where r is the distance between

the particles and σ is their size.

Molecules that are not approximately spherical in shape may have additional states of

matter at (intermediate) temperatures where neiter the liquid nor the crystalline phase

are thermodynamically stable. These phases are for that reason called mesophases.

There are two types of mesophase, referred to as liquid crystals and plastic crystals.

Plastic crystals are crystalline solids and exhibit positional order, but they lack orienta-

tional order. Liquid crystals are symmetry-broken, ordered fluids in which one or more

angular and/or positional degrees of freedom are frozen in. As liquid crystals are by

definition not crystalline, this implies that not all three positional degrees of freedom

can be frozen in.

As a consequence, liquid crystals possess properties in between those of common

3

liquids and solids. That is, they flow like fluids but are also able to respond elastically

to certain types of mechanical deformation, and in that sense resemble solids. The

simplest and most comprehensively investigated liquid crystal is the uniaxial nematic

liquid crystalline phase, or nematic. A widely used compound that exhibits this phase

is 4-cyano-4’-pentylbiphenyl also referred to as 5CB, shown in Figure 2. Just like other

compounds that form nematic phases, so-called nematogens, the molecular structure of

5CB is characterised by an rigid (aromatic) core attached to which is a flexible tail, often

a short aliphatic (hydrocarbon) chain. The chemical structure of the molecule can be

represented by a rigid, cigar-shaped particle with a flexible tail. The rigid body defines

a unit main body axis vector ~u and is readily crystallisable. The flexible tail introduces

just enough disorder in the molecular structure to allow for the crystal phase to be

destabilised in favour of the nematic phase, although this only happens for a certain

temperature range.

Figure 2: Representation of a 5CB molecule focusing on the chemistry (left) and on the

physics (right). The aromatic moiety on the right of the chemical structure is rigid while the

aliphatic tail on the left is flexible. The unit vector ~u is the main body axis vector of the

rod-like molecule.

For 5CB the sequence of phase transformations under atmospheric pressure is (i)

crystal → nematic liquid crystal at 18 [◦C], and (ii) nematic liquid crystal → isotropic

fluid at 35 [◦C]. The isotropic fluid phase is equivalent to the usual liquid state. In

the nematic phase, the main body axis vectors are aligned along a common axis called

the director, represented by the unit vector ~n. See Figure 3. The nematic phase has

cylindrical and inversion symmetry. This means the main body axis vectors of the

particles, the unit vector ~u, are more or less aligned along the director ~n, which because

of inversion symmetry is a “double-pointed” vector. The thermodynamics of nematics

does not change by inverting the director ~n → −~n. As a consequence, particles with

orientations ~u and −~u are equally probable. Nematics for which this is not true are

called ferronematics. The ferro-nematic state is much less common than the “common”

4

nematic state.

Figure 3: Isotropic liquid phase (left) and nematic liquid crystalline phase (right). The

common axis is given by the director ~n. Because of inversion symmetry, we have ~n = −~n.

Nematic liquid crystals have unusual physical properties because of their uniaxial

symmetry. For instance, nematics are birefringent: the optical properties along the

director are different from those perpendicular to that, which is why they find appli-

cations in opto-electronic devices. Not surprisingly, nematics also have a very complex

flow behaviour depending on the flow direction and type, involving no fewer than five

different “viscosities” (so-called Leslie co-efficients). Nematics are also able to respond

elastically to distortions of the director and have no fewer than five different elastic

constants depending on the type of deformation. Finally, there are (in principle) three

types of interfacial tension, depending on the attack angle of the director field relative

to an interface.

All of this is outside the scope of these lecture notes, which focus entirely on molecular

theories of why nematic phases arise in the first place. Molecular theories of nematics

explicitly incorporate the notion of particles and interactions between them into the

description. For this to work for the two types of liquid crystal that we distinguish

and that are commonly referred to as lyotropic nematics and thermotropic nematics, we

need to consider what kind of physical system these comprise of. Parenthetically, the

nematogens of both types of nematic are often elongated particles and “rod-like”, but

they can also be flat and “disk-like”.

Thermotropic nematics are typically one-component, pure fluids of relatively low

molecular weight materials, such as 5CB. They can also be main-chain and side-chain

liquid crystal polymers comprised of chemically linked nematogen building blocks. Ly-

otropic nematics are rod- or plate-like particles finely dispersed in a host fluid. The

particles are in the size range from tens of nanometers to a few microns. They include

chromonics, which are self-assembled, supramolecular polymers, biopolymers such as

DNA and f-actin, surfactant micelles, and colloidal particles such as carbon nanotubes,

graphene flakes and clay particles. Of particular interest are rod-like viruses, such as

tobacco mosaic virus (TMV) and fd virus. Both act as model systems to study lyotropic

nematics on account of their monodisperse character and relative ease to produce in

5

significant quantities.

Molecular field theories that explain why molecules or colloidal particles sponta-

neously align along a common axis to form a nematic liquid crystal rely on the concept

of an orientational distribution function. The orientational distribution function P (~u)

describes the probability density that a particle has orientations between ~u and ~u+ d~u.

Because nematics obey cylindrical symmetry, we have P (~u) = P (θ, φ) = P (θ), where

the director is along the z-axis of a cartesian co-ordinate system and ~u can be de-

scribed in the usual polar angles θ and φ. Because of inversion symmetry, we also have

P (θ) = P (π− θ). See Figure 4. For the isotropic phase of the material, P (θ, φ) = 1/4π.

Figure 4: Left: The main body axis vector ~u of a nematogen makes an angle θ with the

director ~n. Right: The orientational distribution function, P (~u) = P (θ), of a nematic.

Because the nematic has cylindrical and inversion symmetry, the distribution function must

be a symmetric function of θ − π/2.

It turns out useful for our discussion to consider not the orientational distribution

function P (~n · ~u) but rather only one of its moments known as the nematic scalar order

parameter. The nematic scalar order parameter is defined as1

S ≡ 〈32

(~n · ~u)2 − 1

2〉 = 〈3

2cos2 θ − 1

2〉 = 〈P2(cos θ)〉 (2)

where 〈· · · 〉 ≡∫ 1

−1d cos θ

∫ 2π

0dφ(· · · )P (θ, φ) is an orientational average, and P2(x) ≡

3x2/2−1/2 the second Legendre polynomial. For perfectly aligned particles we have S =

1, while for randomly oriented particles S = 0. Typical values obtained experimentally

for the nematic order parameter are S ≈ 0.3 − 0.4 for thermotropic nematics and S ≈

0.6−0.8 for lyotropic nematics, at least for conditions near the transition to the isotropic

phase.

1Note that the first moment 〈~n · ~u〉 = 0 because of inversion symmetry.

6

Figure 5: Nematic order parameter S of a thermotropic liquid crystal as a function of the

temperature T . TIN is the isotropic-nematic transition temperature at which co-existence

between isotropic and nematic phases is possible.

Thermotropic nematic phase transitions are driven by enthalpy,2 i.e., attractive in-

teractions, which have to be strong enough to overcome the loss of orientational entropy

upon alignment. This implies that for thermotropics the temperature is the relevant

thermodynamic control variable. Indeed, if we plot the order parameter S as a function

of temperature T for a thermotropic nematic, we find that at the clearing or isotropic-

nematic transition temperature, TIN , the order parameter jumps from S = 0 to a

non-zero value ≈ 0.3−0.4. In other words, the isotropic-nematic transition is discontin-

uous, i.e., of first order. See Figure 5. Cooling an isotropic fluid of nematogens to the

clearing temperature TIN allows for the co-existence of isotropic and nematic phases,

provided that not of all of the latent heat has been extracted from the fluid.

Lyotropic nematics are driven, in essence, by the competition between two different

“kinds” of entropy: translational entropy and orientational entropy. Because of this,

temperature does not have a large impact on the properties of lyotropic systems and is

not an effective thermodynamic control variable. Indeed, lyotropic nematics turn out

to behave approximately athermally and only weakly respond to temperature changes.

Translational entropy depends on the availability of free volume, i.e., the volume not

taken up by the presence of the particles and this depends on how many particles there

are in a given volume. Hence, concentration must be the relevant experimental control

variable. See Figure 6, showing schematically how the order parameter S depends on

the particle concentration and temperature.

For sufficiently high temperatures, enthalpy is unimportant and the nematic transi-

tion is driven entirely by entropy and therefore athermal. With increasing concentration

2For incompressible systems such as fluids the distinction between enthalpy and energy is irrelevant.

7

Figure 6: Order parameter of lyotropic liquid crystals as a function of the concentration c and

the inverse temperature T−1. In the athermal region the energy plays a minor role. In the

thermal region, energy plays an important role and may involve gelation and solubility-limit

effects that widen the phase gap. The phase gap between the concentration in the co-existing

isotropic and nematic phases, cI and cN , in the athermal regime varies from about 5% to a

few 100% relative to cI , depending on the type of particle (flexibility, polydispersity)! The

volume fraction of particles at the transition scales as the inverse aspect ratio D/L, which

can become very low for long particles.

the order parameter jumps from zero to some value between, say, 0.5 to 0.8 depending

on the type of particle. Of course, for sufficiently low temperatures, enthalpy becomes

important again. Notice that under conditions of co-existence, not only the order pa-

rameters in the isotropic and nematic phases are different, but also the concentrations

of particles. The so-called phase gap, which is the difference in the concentrations in the

co-existing isotropic and nematic phases, can range from a few per cent to hundreds of

per cents, relative the concentration of particles in the isotropic phase. This depends

on the architecture of the particles, e.g., their aspect ratio and bending flexibility, the

solution conditions such as pH and ionic strength, but also on the polydispersity of the

particles: the distribution of lengths or widths or the nematogens present in the fluid.

In both lyotropic (L) and thermotropic (T) liquid crystals, the transition from the

isotropic (I) to the nematic (N) phase is of first order. A host of theories have been put

forward to describe the I-N transition. These include Landau theory (T and L), Flory

lattice theory (T and L), generalised van der Waals theory (T), Maier-Saupe theory (T),

Onsager theory (L), Khokhlov-Semenov theory (L), integral equation theory (T and L)

8

and polymer reference interaction site or PRISM theory (L). My aim is to focus on the

main theories in this context, being Maier-Saupe theory and Onsager theory, and derive

these from a common description, being density functional theory.

2 Density functional theory

We start by defining a Helmholtz free energy, F [J], and presume it to be some function

F [ρ(~u)] of the number density ρ(~u) [m−3] of particles with orientation ~u. Because

nematics are spatially uniform fluids, we have ρ(~u) = ρP (~u) with ρ = N/V [m−3] the

mean number of particles N per unit volume V [m3], and P (~u) the earlier-introduced

angular distribution function. Of course, this quantity is normalised,∫d~uP (~u) ≡

∫ π

0

dθ sin θ

∫ 2π

0

dφP (θ, φ) ≡ 1. (3)

We call F [P ] a functional rather than a function, as it is a function of the probability

distribution function P , and indicate this with the square brackets.

To set up the theory, we use the following ingredients: First, we identify the ideal

gas or ideal solution free energy, Fid. Second, we invoke the Gibbs entropy to formulate

an expression for the contribution of the orientational entropy to the free energy, For.

Third, we account for interactions between the particles through an excess free energy,

denoted Fexc. The total free energy is the sum of these three contributions:

F = Fid + For + Fexc. (4)

To start with the first contribution, for a closed system,3 i.e., one defined by the

number of particles N , volume V and temperature T , we know that the pressure p

[N m−2] obeys the phenomenological ideal gas law in the appropriate low-density limit,

limρ→0 p = −(∂F/∂V )N,T = ρkBT . Here, kB [J K−1] is Boltzmann’s constant and T [K]

the absolute temperature. The ideal gas is the relevant reference point for thermotropic

nematics. It allows us, by isothermal integration over the volume, to obtain the ideal

gas contribution to the free energy.

In lyotropic nematics we have a carrier fluid and, dispersed in it, colloidal particles.

In this case, the relevant quantity is not the pressure p, but a quantity known as the

osmotic pressure Π. It is defined as p = pS + Π, where pS is the pressure of the pure

solvent in a hypothetical reservoir that the solvent in the dispersion is in chemical and

thermal equilibrium with. For the semi-grand (N,V, T, µS) ensemble of interest, where

3Not to be confused with an isolated system for which the number of particles N , the volume V and U

are fixed.

9

Figure 7: Left: A closed, (N,V, T ), system in thermal equilibrium with a heat reservoir at

temperature T . Right: semi-grand, (N,V, T, µS), system in thermal and chemical equilibrium

with a heat and solvent reservoir, at temperature T and solvent chemical potential µS . Heat

transfer between system and reservoir is indicated with dQ and particle transfer with dN .

µS [J] is the chemical potential of the solvent in the reservoir, the osmotic pressure

obeys limρ→0 Π = −(∂F/∂V )N,T,µS = ρkBT , according to the phenomenological Van’t

Hoff law for ideal solutions. See also Figure 7 for an explanation of the fundamental

difference between a closed and a semi-grand ensemble.

Our aim is to derive, from these phenomenological laws, an “ideal” free energy. This

is possible, as already advertised, by a volume integration of the ideal gas law and the

equivalent Van ’t Hoff’s ideal solution law. The procedure requires some care because

naive integration does not necessarily produce a free energy that is extensive, as it

should. For an ideal solution we obtain

Fid = −∫dVΠ|N,T,µS = NkBT ln (ρω/e), (5)

where e is Euler’s constant and ω = ω(T, µS) [m3] a microscopic volume scale that has

its origin in the integration constant.4 The integration constant is a constant only of the

volume V , not the number of particles in that volume, N . If we choose it proportional

to N , this restores the extensivity of the free energy as required by thermodynamics.

Euler’s constant comes in for historical reasons.

For an ideal gas, we obtain the same expression, eq (5), except that the volume scale

can be calculated exactly by taking the classical limit in the quantum- statistical theory

of ideal point particles.5 This gives ω = ω(T ) = Λ3 with Λ =√h2/2πmkBT [m] the

thermal de Broglie wave length. Here, m [kg] denotes the mass of the particle and h

[J s−1] Planck’s constant. The thermal wave length is a measure for how spread out

4Note the integration is indefinite.5Euler’s constant emerges naturally from that treatment.

10

the wave packet of the point particle is. For an ideal solution, this calculation is not so

straightforward because of the presence of solvent molecules, but Λ is thought to be of

the order of the size of a solvent molecule.

The second contribution to the free energy is that due to the Gibbs entropy, asso-

ciated with the alignment of the particles. From statistical mechanics we know that

For = kBT∑ν Pν lnPν , where Pν is the probability of a microstate ν of the system. In

our case,

Pν = P (~u1, ..., ~uN ) =

N∏i=1

P (~ui) (6)

at least within a classical description of the particles, where they can be numbered

i = 1, ..., N . Invoked is also a mean-field approximation, where the orientations ~ui of

the individual particles are independent of each other. We next realise that∑ν =∏N

i=1

∫d~ui, and make use of (i) the realisation that lnPν =

∑Ni=1 lnP (~ui), (ii) the

independence of P (~ui) from P (~uj 6=i), and (iii) of the property of normalisation of all

distribution functions P (~ui). This gives

For = kBTN

∫d~uP (~u) lnP (~u), (7)

where P (~u) denotes the orientational distribution function of an arbitrary particle, all

particles being identical. Notice that we could also have written Eq (7) straight away,

because of the additivity of the entropy of N independent particles.

Finally, we address the issue of the excess free energy. This free energy represents all

relevant additional contributions not taken into account by the ideal free energy and the

orientational free energy. The main contribution comes from interactions between the

nematogens. As these particles are chemically complex and in the case of a lyotropic also

involve contributions from the presence of a solvent, we need to simplify the description

by relying on the concept of potentials of mean force. In potentials of mean force,

irrelevant degrees of freedom are, in a way, “integrated out”.6 Atomic details are lost in

the actual interaction potential even for quite small distances due to self averaging: the

interaction potential is the sum of the interaction potentials between all atoms of both

particles.

In practice, it means that we presume the details of the interaction between nemato-

gens to be hidden in the phenomenological potentials that we invoke. For simplicity,

we further presume that the only relevant degrees of freedom are the orientations of

the two particles ~u and ~u′, and the position vector ~r connecting the centre of mass of

one particle to that of the other. See Figure 8 (left). So, we expect the pair potential

6Strictly speaking, these degrees of freedom are literally integrated out in the partition function, leaving

only the degrees of freedom of the interesting particles.

11

Figure 8: Left: the pair potential U(~r, ~u, ~u′) is a complicated function of the relative position

~r and the orientations ~u and ~u′ of two test particles. Right: for a given angle γ(~u, ~u′) between

the main body axis vectors of two test particles we expect a Lennard-Jones type of interaction

potential, characterised by a short-range repulsion and a long-range van der Waals attraction.

The strength of the attraction, ε, depends on the relative orientation of the particles.

U = U(~r, ~u, ~u′) to be a function only of ~r, ~u , and ~u′. For any given angle γ = γ(~u, ~u′)

between the two main body axis vectors ~u and ~u′, we expect the interaction potential

to be Lennard-Jones-like, as shown in Figure 8 (right). Note that, because the particles

are not necessarily inversion symmetric in reality, the interaction potential need not be

symmetric for angles 0 ≤ γ < π/2 and π/2 < γ ≤ π. See Figure 9.

The overall potential energy Upot of a collection of N nematogens depends on the

instantaneous positions and orientations of all the particles. To calculate the expectation

value of this quantity, which is the thermodynamic value of the potential energy, we

invoke two approximations. First, we presume that the interactions between the particles

are pair-wise additive. Because in reality this is generally not the case, we need to (in

a way) renormalise the interaction in order to get a phenomenological description that

accounts for this inconvenience. As a result, the interaction potential can no longer

be predicted, e.g., from quantum mechanics, and has to be seen as a phenomenological

potential. Second, we invoke the mean-field approximation, which in fact we already

did in formulating the orientational free energy.

Within these approximations, we write

Upot =1

2N

∫d~uP (~u)Umol(~u), (8)

where the factor 12

corrects for double counting, and Umol is the “molecular field” each

particle experiences from the presence of all the others. The molecular field obeys

Umol =

∫d~r

∫d~u′ρ(~r, ~u′|~0, ~u)U(~r, ~u, ~u′), (9)

12

Figure 9: Absolute value of the interaction potential |U | between two nematogens need not

be symmetrical for parallel configurations (with angle γ0) and for antiparallel configurations

(with angle γ = π). See also Figure 8.

where the integrals count the number of particles at position ~r with orientation ~u′, given

that there is a test particle with orientation ~u at the origin ~0 of the co-ordinate system.

Here, ρ(~r, ~u′|~0, ~u) denotes a conditional density.

The conditional density ρ(~r, ~u′|~0, ~u) is usually converted to a pair correlation func-

tion, defined by

ρ(~r, ~u′|~0, ~u) ≡ ρ(~u′)g(~r, ~u, ~u′) = ρP (~u′)g(~r, ~u, ~u′), (10)

where we have made use of the fact that far from the test particle located at the origin,

we expect that ρ(~r, ~u′|~o, ~u) → ρ(~u′) = ρP (~u′). The pair correlation function g(~r, ~u, ~u′)

measures the deviation from this expectation at closer separations. Indeed, the central

particle influences the positions and orientations of nearby particles because it interacts

with them. The pair correlation function is, in essence, the ratio between the actual

number density of particles near a test particle and the hypothetical value it would have

if none of the particles interacted.

At very short separations, we expect g → 0 because of the Pauli exclusion. For

large separations, the influence of the central test particle becomes weak and g → 1. At

intermediate distances there are peaks and valleys, associated with solvation layers of

particles around the test particle. See Figure 10. We naively expect the first (primary)

peak to correspond to the minimum of the interaction potential. The reason is that

g, a probability, must be Boltzmann distributed. Hence, for low densities, g should be

equal to exp(−U/kBT ). In reality, that is, at non-zero densities, the relevant potential

is not the pair potential but the position-dependent, effective many-body interaction

experienced by the test particle of given orientation.

13

Figure 10: The pair correlation g as a function of the distance r between the centres of mass of

two particles. For short separations g → 0 due to Pauli exclusion. For large separations g →

1, whilst for intermediate separations there are peaks and valleys associated with solvation

layers. Onsager theory for lyotropic nematics focuses on the effects of Pauli exclusion, whilst

Maier-Saupe theory for thermotropics mainly deals with the effects of the attractive part of

the interaction.

For a uniform isotropic or nematic fluid, the molecular field any particle experiences

from all other particles becomes

Umol(~u) = ρ

∫d~u′P (~u′)

∫d~rg(~r, ~u, ~u′)U(~r, ~u, ~u′). (11)

As we have seen, the potential energy Upot of the collection of particles follows from

Upot = 12N∫d~uP (~u)Umol(~u). To calculate the excess free energy from this, we make

use of the thermodynamic identity Upot = ∂βFexc/∂β, where β ≡ 1/kBT [J−1] is the

reciprocal thermal energy.7 Hence,

βFexc =FexckBT

=

∫dβUpot. (12)

This expresses the excess energy of the system, Fexc, in terms of the orientational

distribution function P (~u) and the pair correlation function g(~r, ~u, ~u′), both of which at

this point are unknown functions.

Our purpose will be to calculate the orientational distribution function, given two

plausible educated guesses for the pair distribution function. One of these focuses on the

role of Van der Waals interactions and, in essence, the first peak in the pair distribution

7Note that the internal energy U of a system obeys ∂βF/∂β, with F the Helmholtz free energy and β =

1/kBT the usual reciprocal thermal energy kBT . This follows from applying the chain rule for differentiation

and inserting the identities S = −∂F/∂T and F = U − TS.

14

Figure 11: Cylindrical particles of length L and width D, and orientations ~u and ~u′. We

presume the interaction potential to depend steeply on the shortest separation, r, between

the main-body-axis vectors.

function. This will turn out to be relevant to thermotropic nematics. See Figure 10.

The other ignores the role of attractive interactions completely, and presumes that the

Pauli exclusion dominates the physics of liquid crystals. This turns out to be accurate

in the context of lyotropic liquid crystals. The first gives rise to the Maier-Saupe theory

of thermotropic nematics, the second to the Onsager theory. Let us first get to grips

with Onsager theory.

3 Onsager theory

As already alluded to, lyotropic nematics are dispersions of colloidal particles or macro-

molecules dispersed in a fluid medium. Onsager announced in a brief abstract in 1942

that he could explain the observations on the co-existence of isotropic and nematic

dispersion of both rod-like and plate-like particles by solely accounting for their “co-

volumes”, that is, their effective excluded volumes, in a thermodynamic description. In

1949 he published the full seminal paper, showing that entropy alone can drive phase

transitions. The account of Onsager’s theory I give here is different from the original

derivation, which is based on a virial expansion of the free energy. For those not familiar

with the virial expansion, it is slightly more intuitive and, more importantly, it allows

me to connect Onsager theory and Maier-Saupe theory.

I shall presume that the particles are infinitely rigid cylinders of length L and width

D. Pairs of cylinder interact solely via a steeply repulsive potential as illustrated in

Figure 11. The pair correlation function g(~r, ~u, ~u′) depends on the centre-to-centre

distance ~r as well as the orientations of the two particles, ~u and ~u′. To lowest order in the

15

Figure 12: Pair correlation function for an ideal gas of “hard” particles, i.e., particles that

interact via a harshly repulsive potential.

density, as already advertised in the preceding section, we expect that g ≈ exp(−βU).

If U is a steeply repulsive potential like the one shown in Figure 11, that is, if U is zero

for non-overlapping particles and infinity for overlapping ones, then the pair correlation

becomes a step function, schematically represented in Figure 12.

If we take g = exp(−βU) for our estimate of the pair correlation function, the

potential energy of our collection of rod-like particles becomes

Upot ≈1

2Nρ

∫d~uP (~u)

∫d~u′P (~u′)

∫d~rU(~r, ~u, ~u′) exp

(−βU(~r, ~u, ~u′)

). (13)

Notice that U has to drop sufficiently fast to zero with inter-particle distance for the

spatial integral not to diverge in the thermodynamic limit V →∞.

Integrating this expression over β now gives our excess free energy,

βFexc =1

2Nρ

∫d~u

∫d~u′P (~u)P (~u′)

∫d~r [1− exp(−βU)] . (14)

Here, we used the integration constant in β to make certain that for U = 0, we retrieve

the expected Upot = 0. Eq. (14) can be rewritten as

βFexc = NρB, (15)

where B [m3] is the second virial coefficient and we recall that ρ is a number density,

so Bρ is dimensionless as it should. For cylindrical particles it is defined in terms of

orientational averages 〈· · · 〉 ≡∫d~uP (~u)(· · · ) with a similar prescription for the primed

variable, as

B ≡ 〈〈B(~u, ~u′)〉〉′. (16)

Here, the second virial co-efficient of two cylindrical particles with fixed orientations ~u

and ~u′ is defined as

B(~u, ~u′) ≡ 1

2

∫d~r [1− exp(−βU)] . (17)

16

From this definition of B(~u, ~u′), and realising that the integrand is unity for overlap-

ping configurations of pairs of hard cylinder and zero in all other cases, we find (using

a cunningly chosen co-ordinate system) that

B(~u, ~u′) =1

2L2| sin γ(~u, ~u′)|2D +O(LD2), (18)

with γ(~u, ~u′) = arccos(~u · ~u′) the angle between the main body axis vectors of two

cylinders, and ~u · ~u′ = cos θ cos θ′ + sin θ sin θ′ cos(φ − φ′). Here, θ and θ′ denote the

polar angles and φ and φ′ the azimuthal angles of the two particles. Eq. (18) is actually

quite intuitive, if we keep in mind Fig. 11. It suggest that the volume excluded by

two rods that are skewed at an angle γ must approximately be equal to the area of a

parallelogram with equal sides at an angle γ and of length L, times twice the width D

of the rods.

Note thatB(~u, ~u′) is half the volume of a parallelepiped with base area L2| sin γ(~u, ~u′)|

and height 2D. In fact, it is, literally, half of the excluded volume of two rods of ori-

entations ~u and ~u′. Aligning rods reduces B, and hence we conclude that Fexc must in

that case decrease. This implies that the free volume accessible to all particles increases

if the particles align. Obviously, this happens at the cost of orientational entropy, and

hence leads to an increase of the orientational free energy For.

To formulate Onsager theory, we add all ingredients to our free energy functional

F = Fid + For + Fexc, to obtain

βF [P ]

N= ln ρω − 1 +

∫d~uP (~u) lnP (~u) + ρ

∫d~u

∫d~u′P (~u)P (~u′)B(γ), (19)

which may also be written in short-hand notation as

βF [P ]

N= ln ρω − 1 + 〈lnP 〉+ ρ〈〈B(γ)〉〉′. (20)

Because this free energy only accounts for the contribution of the second virial co-efficient

and higher order virials involving three and higher order body contacts are ignored, the

theory is an example of a second virial approximation. Interestingly, there are strong

indications that the second virial approximation is exact in the limit of infinite aspect

ratios L/D →∞. This makes Onsager theory one of the few exact theories in condensed

matter physics.

In thermodynamic equilibrium, the most probable distribution P minimises F , given

the normalisation constraint∫d~uP (~u) ≡ 1. To find the most probable distribution, we

need to functionally minimise the free energy

δ

δP

[F − λ

(∫d~uP (~u)− 1

)]= 0 (21)

17

where λ is a Lagrange multiplier introduced to ensure proper normalisation of the dis-

tribution function. A pedestrian recipe for obtaining the functional derivative is as

follows. First, insert P (~u) = Peq(~u) + δP (~u) into the expression for the free energy,

where Peq(~u) is the as yet unknown, most probable distribution function and δP (~u)

a perturbation away from it. Second, Taylor expand the integrand in the free energy

functional to linear order in δP (~u). Finally, we rearrange the expressions to produce an

expression of the form F [Peq + δP ] = F [Peq] +∫d~uδP (~u) [δF/δP ], defining the func-

tional derivative δF/δP as all terms between the square brackets. For more information

how to do functional integrals, refer to the textbook of Hansen and McDonald cited in

the bibliography.

This now produces what is known as the Onsager equation,

lnPeq(~u) = λ− 1− 2ρ

∫d~u′Peq(~u

′)B(~u, ~u′), (22)

which needs to be solved. We fix the value of λ by normalising P . So far, no-one has

succeeded in solving this non-linear integral equation analytically, except for the trivial

solution with P = 1/4π, which is the isotropic solution. We shall have to take recourse

to numerical methods to find non-isotropic solutions.

From the Onsager equation we deduce that Peq obeys, as expected, a Boltzmann

distribution,

Peq(~u) ∝ exp[−2ρ〈B(γ)〉′

]≡ exp [−βUmol(~u)] (23)

where βUmol(~u) is the earlier-introduced molecular field. This molecular field apparently

depends on the distribution function itself. Hence, equations of this type are known as

self-consistent field equations, and any theory that produces such an equation is a self-

consistent field theory. Self-consistent field theories are mean-field theories.

Another useful observation we can make from the Onsager equation is that the

product ρ〈B(γ)〉′ ∼ L2Dρ〈| sin γ|〉′ suggests a natural concentration scale c ≡ πL2Dρ/4,

because in the isotropic phase 〈| sin γ|〉′ = π/4. In the limit L/D � 1 this becomes equal

to c = φL/D, where φ is the volume fraction of the particles.8 If we do a bifurcation

analysis of the Onsager equation, we find that solutions to the Onsager equation branch

off from the isotropic solution P = 1/4π at c = 4. This implies that we expect the

dimensionless concentrations of the isotropic and co-existing nematic phases to be close

to this value. Hence, the transition to the nematic phase occurs around a volume fraction

of 4D/L → 0 if L/D → ∞, explaining why nematic phases can present themselves at

very low particle concentrations.

8The precise correspondence depends on whether the rods are cylinders or sphero-cylinders, i.e., capped

cylinders. In the limit L/D � 1 the distinction becomes insignificant.

18

Figure 13: Schematic of the order parameter S as a function of the dimensionless concentra-

tion c for a dispersion of hard rods.

A schematic of the numerical solution to the Onsager equation is given in Figure

13. Plotted is the order parameter S = 〈P2(cos θ)〉 as a function of the dimensionless

concentration c. For c < 3.5 there is only one solution to the Onsager equation. For

c ≥ 3.5 there are three. Two of these are free energy minima, one is a free energy

maximum. The free energy maxima are indicated with dashed lines. Stable branches

end in the isotropic and nematic spinodals, the limits of thermodynamic stability, also

indicated in the figure. The isotropic spinodal is at c = 4 and the nematic spinodal

at c = 3.5. The branch with negative order parameter represents a local (metastable)

minimum.

To find out where the binodal is, that is, where the concentrations of the co-existing

isotropic (I) and nematic (N) phases are located, we need to invoke the second law

of thermodynamics. According to the second law, any two phases in thermodynamic

equilibrium must be in chemical, mechanical and thermal equilibrium. This means equal

chemical potentials, µI = µN where µ = ∂F/∂N |V,T , equal osmotic pressures, ΠI = ΠN

with Π = −∂F/∂V |N,T , and equal temperatures, TI = TN . Since the theory describes

the balancing of two types of entropy and has no energy component, it is athermal.

This means that there is no meaningful temperature in this model, and we only need to

equate chemical potentials and osmotic pressures.

For the binodal we find for the concentration in the isotropic phase cI = 3.3 and

a corresponding order parameter S = 0, while for the concentration in the nematic

phase we obtain cN = 4.2 with order parameter S = 0.8. This means that not only are

the particles aligned in the nematic phase, the concentration of particles is about 30%

larger in the nematic than that in the isotropic phase. These values agree favourably

with results from computer simulations. In practice, rod-like colloidal particles do ex-

19

hibit some degree of bending flexibility, which makes direct comparison with experiment

not straightforward. Theories that incorporate a bending flexibility into the Onsager

theory agree well with experiments. As already advertised, the effects of electrostatic

interactions, residual van der Waals interactions, polydispersity, supramolecular poly-

merisation, and so on, have also been dealt with. All of this is outside the scope of my

lectures.

The question arises how to read Figure 13. At low concentrations, that is, for c ≤ 3.3,

the isotropic phase is the stable phase. On the other hand, for c ≥ 4.2 the nematic phase

is the thermodynamically stable state of the dispersion. The nematic order parameter

S ≥ 0.8 increases with increasing concentration and approaches the value of unity only

at very high densities. Preparing the dispersion at an overall concentration of particles

in the region 3.3 < c < 4.2 produces two co-existing phases: one isotropic phase with

concentration c = cI = 3.3 and order parameter S = 0, and one nematic phase with

concentration c = cN = 4.2 and order parameter S = 0.8. The fractional volume VN/V

of the nematic phase is fixed by the overall concentration: VN/V = (c− cI)/(cN − cI).

It rises from VN/V = 0 if c ≤ cI with increasing concentration to VN/V = 1 if c ≥ cN .

See Figure 14. The branch with negative order parameter is metastable with a free

energy that is always larger than that of the corresponding nematic with positive order

parameter.

Figure 14: Schematic representation how isotropic and nematic phases present themselves

in a lyotropic system. Left: For particle concentrations below where the nematic becomes

stable we observe only one phase, which is the isotropic phase. Right: For concentrations

larger than the minimum concentration needed to stabilise the ordered phase, again only

one phase (the nematic phase) is seen. Middle: For intermediate concentrations we have

co-existence of an isotropic phase and a nematic phase. Their relative volumes depend on

the overall concentration of the particles in the dispersion.

This brings us to the Maier-Saupe theory for thermotropic nematics, which in essence

has a very similar structure to the Onsager theory.

20

4 Maier-Saupe theory

Onsager theory is at the heart of our understanding of lyotropic nematics. In principle,

it should be accurate for infinitely rigid, infinitely long rod-like particle interacting via

an infinitely harsh repulsive potential. Temperature plays no role in Onsager theory.

Not surprisingly, it has not quite caught on in the field of thermotropic liquid crystals,

where the molecules have an aspect ratio of, say, five, and where temperature does

play an important role. Indeed, in thermotropics the main driving force for spontaneous

alignment of the molecules is not entropy but energy, or, rather, enthalpy as experiments

are usually done at fixed (atmospheric) pressure. Setting up a theory for thermotropic

nematics of relatively low molecular weight molecules is no trivial affair because these

are dense systems. This implies that a virial expansion is of no use and we need to

resort to physical insight to make headway.

To see how we can make headway, let us consider the nematogen 5CB that we

encountered in the Introduction. The molecular weight of 5CB is Mw ' 250 [g mol−1],

and under atmospheric pressure the isotropic-nematic transition occurs at a temperature

of TIN = 35 [◦C]. The relative density difference of the co-existing isotropic and nematic

phases is exceedingly small: (ρN − ρI)/ρI = 0.002, where ρI = 1 [g cm−3]. The

latent heat associated with the transition is surprisingly small too: ∆HIN ' 1.6 [J g−1]

≈ 0.15 kBTIN , where TIN is the isotropic-nematic transition temperature, also called

the clearing temperature.9 Finally, like any kind of liquid, the (linear) thermal expansion

co-efficient of 5CB is very small and in the range α ' 10−3 − 10−4 [K−1] depending on

the temperature.

All of this implies that temperature has very little influence on the mean separation

between the molecules, only on their orientations. Presuming this to be the case, we may

conclude that the potential energy Upot is only very weakly dependent on temperature.

Consequently,

βFexc =

∫dβUpot ≈ βUpot, (24)

where the integration constant is chosen such that for T = 0 K we retrieve the thermo-

dynamic identity F = U − TS = U with S now the entropy, not to be confused with

the nematic order parameter.

Next, we need to establish the potential energy of the system. Starting off from the

familiar expression that we also used to estimate the excess free energy of a lyotropic

9The nematic phase is turbid. This is caused by director field fluctions that scatter light on account of

the difference between the refractive indices parallel and perpendicular to the main optical axis, usually the

director. The isotropic phase, where there are no director field fluctuations, is not turbid. Hence, melting a

nematic leads to the emergence of a relatively “clear”, isotropic fluid.

21

Figure 15: Schematic of the second Legendre polynomial P2(cos θ) as a function of the polar

angle θ.

system, we have

Upot =1

2N

∫d~uP (~u)Umol(~u), (25)

where the factor of one-half corrects for double counting, and where

Umol(~u) = ρ

∫d~u′P (~u′)

∫d~rg(~r, ~u, ~u′)U(~r, ~u, ~u′). (26)

As before, U(~r, ~u, ~u′) denotes the pair potential.

The problem that we face is that g(~r, ~u, ~u′) is difficult to estimate for dense liquids

and that our earlier estimate in terms of a Boltzmann weight of the interaction potential

U fails miserably. Hence, we shall not attempt to estimate the pair correlation function

itself but focus instead on the molecular field Umol(~u). Within mean-field theory, we

expect the orientational distribution function P (~u) to be Boltzmann weighted, so pro-

portional to exp(−βUmol(~u)). This means that Umol(~u) must obey the same cylindrical

and inversion symmetry that P obeys in the nematic. In the isotropic phase we expect

Umol(~u) = Uiso to be independent of the orientation ~u of a particle.

A plausible form of the molecular field must therefore be

Umol(~u) = Uiso −∆εSP2(~n · ~u) (27)

where ~n denotes as before the director, the second Legendre polynomial P2(x) = 3x2/2−

1/2 ensures the correct symmetry and S = 〈P2(~n·~u)〉 =∫d~uP (~u)P2(~n·~u) is the nematic

order parameter. See Figure 15. Recall that the nematic order parameter S is zero in

the isotropic phase. Finally, ∆ε ≥ 0 is a measure of the anisotropy of the attractive

van der Waals interaction between the nematogens, favouring parallel orientations. See

Figure 16. We expect ∆ε to be some function of the density ρ of the material that sets

the average distance between the molecules.

22

Figure 16: Schematic of the interaction potential U between two nematogens as a function

of their distance r. Left: for two particles at right angles the potential well is −ε. Right: for

parallel particles the well is an amount −∆ε deeper, favouring parallel alignment.

From this we conclude that for thermotropic nematics,

Fexc = Upot =1

2NUiso −

1

2N∆ε〈P2(~n · ~u)〉2, (28)

giving for our estimate of the free energy functional

βF [P ] = N ln ρω −N +N〈lnP (~u)〉+1

2NUiso −

1

2N∆ε〈P2(~n · ~u)〉2. (29)

It turns out convenient to subtract from this free energy the free energy of the corre-

sponding isotropic phase, where P = 1/4π. This leaves us with an excess free energy

β∆F [P ] ≡ βF [P ]− βF [1

4π] = N〈ln 4πP (~u)〉 − 1

2Nβ∆ε〈P2(~n · ~u)〉2. (30)

Notice that we have constructed this free energy in such a way that in the isotropic

phase ∆F ≡ 0.

The unknown quantity in this expression is the orientational distribution function

P (~u). We follow that same recipe as before, and functionally minimise ∆F with respect

to P (~u) while keeping P (~u) normalised. Hence,

δ

δP (~u)

[β∆F − λN

(∫d~uP (~u)− 1

)]= 0, (31)

where λ is again a Lagrange multiplier. The factor N is arbitrary and for convenience

- it just renormalises the Lagrange multiplier.

Performing the functional minimisation according to the recipe of the preceding

section, and realising that 〈P2(~n · ~u)〉2 =∫d~u∫d~u′P (~u)P (~u′)P2(~n · ~u)P2(~n · ~u′), gives

ln 4πP (~u) = λ− 1 + β∆εSP2(~n · ~u), (32)

23

which again is a self-consistent field equation because S = 〈P2(~n·~u)〉 =∫d~uP (~u)P2(~n·~u)

is a function of the orientational distribution function P that we set out to calculate.

Not surprisingly, its structure closely resembles that of the Onsager equation.

We need to solve this equation to find the orientational distribution function P as a

function of the control parameter β∆ε. Noting that

P (~u) =1

4πexp (λ− 1 + β∆εSP2(~n · ~u)) (33)

we fix λ by insisting that∫d~uP (~u) ≡ 1. We immediately notice that −∆εSP2(~n · ~u) ≡

Umol(~u) now acts as the molecular field that in the isotropic phase by definition is zero.

Hence,

P (~u) =exp(β∆εSP2(~n · ~u))∫d~u exp(β∆εSP2(~n · ~u))

(34)

The order parameter S can now be determined self-consistently, because if we insert Eq

(34) into the definition, we deduce that

S =

∫d~uP2(~n · ~u) exp (β∆εSP2(~n · ~u))∫

d~u exp (β∆εSP2(~n · ~u)), (35)

must hold. Actually, it is not necessary to explicitly calculate the integrals in both

the numerator and denominator of Eq (35). Indeed, from Eq (35), we conclude that

S = ∂ lnZ/∂β∆εS, with Z =∫d~u exp (β∆εSP2(~n · ~u)) the normalisation constant of the

distribution function, i.e., the one-particle partition function. Therefore, it is sufficient

to calculate Z and derive from that the self-consistent field equation.

Analytical integration of the self-consistent field equation is not trivial, so we take

recourse to a numerical evaluation of it. Figure 17 shows the schematic of the order

parameter S as a function of the dimensionless reciprocal temperature β∆ε = ∆ε/kBT .

For β∆ε < 4.48 only the isotropic solution with S = 0 is stable. For β∆ε ≥ 4.48

there are two stable branches, with ∂2∆F/∂S2 > 0, and one unstable branch, for which

∂2∆F/∂S2 < 0. There are two spinodal points: one at β∆ε = 4.48 with order parameter

value S = 0.32, which is the nematic spinodal, and one at ∆ε = 5 with order parameter

S = 0, being the isotropic spinodal.

To calculate the binodal and pinpoint the isotropic-nematic transition temperature,

we again invoke the second law of thermodynamics, telling us that we must have equal

chemical potentials in the co-existing isotropic (I) and nematic (N) phases, µI = µN ,

equal temperatures, TI = TN , and equal pressures PI = PN . Unfortunately, we cannot

calculate those from our theory, as we do not know how ∆ε depends on the density of

the fluid. Fortunately, we can make use of the incompressiblity property of fluids that

we used to formulate the theory to work around this problem.

From thermodynamics we know that the Helmholtz free energy, F , describes a closed

system with a fixed number of particles N , volume V and temperature T . The Gibbs

24

Figure 17: Schematic of the nematic order parameter S as a function of the reciprocal

dimensionless temperature β∆ε for a thermotropic nematic according to Maier-Saupe theory.

Dashed lines indicate thermodynamically unstable branches. The spinodal points are also

indicated, as are the binodals. The sign of the second derivative of the free energy FSS with

respect to the order parameter is also indicated.

free energy, on the other hand, describes a closed system in mechanical equilibrium with

a reservoir. This is an (N,P, T ) ensemble. These two thermodynamic potentials are

connected via the Legendre transform G = F + PV . In equibrium, thermodynamics

stipulates that G = Nµ, with µ the chemical potential of the particles. Hence, µρ =

P + F/V with ρ = N/V their number density. From this, we conclude that equal

densities, chemical potentials and pressures translate into equal free energy densities!

So, for incompressible systems, the conditions µI = µN , TI = TN , and PI = PN , may

be replaced by ρI = ρN , TI = TN , and FI/V = FN/V , that is, ∆F/V = 0.

It suffices to find the reciprocal temperature βIN∆ε for which β∆F/N = 0. We find

numerically that βIN∆ε = 4.55, and the corresponding order parameter SIN = 0.43.

Extracting energy from the isotropic phase reduces its temperature until we reach the

clearing temperature for βIN∆ε = 4.55. Extracting more energy does not lead to further

decrease in temperature. What happens is that isotropic phase is converted into nematic

phase in proportion to how much of the latent heat has been extracted from the fluid.

Once all the latent heat has been extracted, the temperature decreases again and the

order parameter increases. Notice that a nematic with negative order parameter is

metastable, that is, its free energy is larger than that with positive order parameter.

Even if we could somehow prepare a nematic fluid with negative order parameter, it

would not be long-lived.

The latent heat per particle we obtain from Maier-Saupe theory equals ∆UIN/N =

∂β∆F/∂βN = 0.42kBTIN . It tells us that, according to Maier-Saupe theory, each

25

Figure 18: Schematic representation how isotropic and nematic phases present themselves

in a thermotropic system. Right: For temperatures above where the nematic phase emerges

we observe only one phase, the isotropic phase. Left: For temperatures below the clearing

temperature again only one phase is seen, which is the nematic phase. Middle: If not all of

the latent heat is removed from the isotropic fluid, we have co-existence of an isotropic phase

and a nematic phase.

particle gains just shy of half a thermal energy through collective alignment at the I-

N transition, which exactly compensates for the entropy loss. Notice that this value

is slightly larger than the measured value of 0.15 kBTIN for 5CB. The temperature

dependence of measured order parameters seems to reasonably accurately follow the

prediction of Maier-Saupe theory, presuming that ∆ε does not depend on temperature.

In general, Maier-Saupe theory is considered as successful despite its simplicity and

obvious drawbacks.

One drawback, the lack of a density jump at the transition, can be dealt with rela-

tively straightforwardly. This is outside the scope of these lecture notes. Another draw-

back, namely the mean-field approximation, is more severe. According to the theory,

the isotropic spinodal temperature is only 1.5% lower than the clearing temperature,

which corresponds to approximately 5 K for 5CB. In reality, the difference is much

smaller, about 1 K. This means that near the I-N transition critical fluctuations due to

the nearby spinodal cannot be ignored.10 These fluctuations make the transition weakly

first order, as may in fact also be deduced from the small latent heat.

5 Concluding remarks

In these lecture notes we have derived Onsager theory and Maier-Saupe theory from

the same density functional theory, making appropriate approximations that suit the

differences in the physics underlying lyotropic and thermotropic nematics. Both theories

10At the spinodal the free energy has an extremum and because of that the free energy cost of excitations

near it approach zero. This causes the amplitude of fluctuations to diverge approaching the spinodal.

26

are, despite their drawbacks, considered as central to our understanding of nematic

liquid crystals. Indeed, much more sophisticated density functional theories have been

put forward that attempt to deal more accurately with (aspects of) experimental reality.

Still, a tractable and accurate theory that deals with hard-core repulsion and soft-core

attraction on an equal footing, e.g., for rod-like polymers, remains elusive. An important

reason for this is that a simple van der Waals-type of approximation, in which the

impact of van der Waals attraction on the thermodynamics of the system is treated at

the two-body level, cannot be accurate due to translation-rotation coupling. Decoupling

of translational and rotational degrees of freedom is the central to almost all density

functional theories to go beyond the second virial approximation.

Acknowledgement

I thank Mariana Oshima Menegon and Shari Patricia Finner for a critical reading of

the manuscript, corrections and suggesting improvements. I thank Mariana Oshima

Menegon for the rendering of all the figures. This project has received funding from

the European Union Horizon 2020 research and innovation programme under the Marie

Sk lodowska-Curie grant agreement No. 641839.

6 Bibliograpy

For a quick reminder of the basic principles of statistical mechanics:

• Ben Widom, Statistical Mechanics - A Concise Introduction for Chemists (CUP,

Cambridge, 2002);

• David Chandler, Introduction to Modern Statistical Mechanics (OUP, Oxford,

1987).

A primer in soft matter physics and density functional theory:

• Jean-Luis Barrat and Jean-Pierre Hansen, Basic Concepts for Simple and Complex

Fluids (CUP, Cambridge, 2003).

More advanced discussions of virial expansions and density functional theory can be

found in:

• Jean-Pierre Hansen and I.R. McDonald, Theory of Simple Liquid - with Applica-

tions to Soft Matter, 4th edition (AP, Amsterdam, 2013).

The standard tome on liquid crystals:

27

• Pierre-Gilles de Gennes and Jacques Prost, The Physics of Liquid Crystals, second

edition (Clarendon, Oxford, 1995).

A more general discussion of the theory of solids, liquids and liquid crystals:

• Paul Chaikin and Tom Lubensky, Principles of Condensed Matter Physics, paper-

back edition (CUP, Cambridge, 2000).

Mathematical details may be obtained from:

• Michael Stone and Paul Goldbart, Mathematics for Physics (CUP, Cambridge,

2000).

28