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We review the theoretical research on the influence of disorder on structure and phase behavior of condensed matter system exhibiting continuous symmetry breaking focusing on liquid crystal phase transitions. We discuss the main properties of liquid crystals as adequate systems in which several open questions with respect to the impact of disorder on universal phase and structural behavior could be explored. Main advantages of liquid crystalline materials and different experimental realizations of random field-type disorder imposed on liquid crystal phases are described.

Understanding of phase and structural behavior of various systems which are randomly perturbed by some static origin of disorder is of considerable interest for different branches of physics [

The pioneering studies have been mostly carried out in various randomly perturbed magnetic materials [

It was shown that the 1st order phase transitions become gradual as disorder exceeds the critical value [_{d}_{d} ∝ w^{2/(d–4)} is predicted, where

In recent years it has been shown that randomly perturbed liquid crystal (LC) phases and structures [

LCs have several extraordinary (mechanical, electrical and optical) properties which make them indispensable in biological systems and in our daily life [

In order to study influence of random field-type disorder on phase and structural behavior one commonly enforces disorder to LCs either by mixing them with “impurities” or confining LCs to various porous matrices [

The paper is organized as follows. In Section II, we present a brief overview on liquid crystals, containing the principal characteristics and properties of the three main classes and the definition of the nematic orientational order parameter. In Section III, the basic properties of principal models of nematic-isotropic phase transition are shown. In Section IV, the influence of the random field on structural and phase transition properties is theoretically described in the framework of a mesoscopic phenomenological model. The experimental observations which support the theoretical model are also described. In the final Section V, we draw some conclusions.

Liquid crystals were discovered in 1861 by Planer, confirmed in 1888 by Reinitzer [

Liquid crystals or mesophases constitute a state of matter whose physical properties are intermediate between those of an isotropic liquid and a crystalline solid [

Liquid crystals possess many of the mechanical properties of a liquid,

The constituent organic molecules of liquid crystalline materials could be in general very different, but it is essential that they are anisotropic and rigid for at least some portion of the molecule length (since it must maintain an elongated shape in order to produce interactions that favor alignment). The liquid crystals composed from rod-like molecules are called “calamitics”, while those formed by disk-shape molecules are known as “discotic” [

Two types of liquid crystals must be differentiated: (i) thermotropic and (ii) lyotropic. The transitions involving thermotropic liquid crystals are effected by changing temperature. Lyotropic liquid crystals are found in solutions and concentration is the important controllable parameter. Usually a lyotropic liquid crystal molecule combines a hydrophobic group at one end with a hydrophilic group at the other end. These amphiphilic molecules forms ordered structures (micelles, vesicle, lamellar phases or reversed phases) in both polar and non-polar solvents as in the case of soaps and various phospholipids [

Another important class of liquid crystals is derived from certain macromolecules, usually in solution but sometimes even in the pure state. These liquid crystals are known as “liquid crystal polymers” [

Based upon their symmetry, Friedel distinguishes three major classes of liquid crystals—nematics, cholesterics, and smectics.

An isotropic liquid possesses full translational and orientational symmetry _{∞}_{∞ h}

In this phase there exists no long-range positional order (between the centers of mass of the molecules). The molecules tend to be parallel to some common axis, labeled by a unit vector

The biaxial nematic phase results due to further breaking of the rotational symmetry and is characterized by three orthogonal directors, the Goldstone modes
_{2 h} point group symmetry and the corresponding second rank tensor properties have three different principal components.

If the liquid crystal molecules are chiral (lacking inversion symmetry), the uniaxial nematic phase is replaced by the chiral nematic phase or twisted nematic phase or cholesteric phase. Locally, a cholesteric is very similar to a nematic. Again, the centers of mass have no long-range order and the molecular orientation shows a preferred axis labeled by a director

When the crystalline order is lost in two dimensions, one obtains the two-dimensional liquids, called smectics. Smectics liquid crystals have layered structures with a well-defined interlayer spacing that can be measured by X-ray diffraction. In addition to the orientational order, the smectic molecules exhibit some correlations in their position. In most smectics the molecules are mobile in two directions and can rotate around one axis. The interlayer attractions are weak compared with the lateral forces between molecules and the layers are able to slide over one another relatively easily which gives rise to fluid property of the system with higher viscosity than nematics. When smectic and nematic phases are found in one compound, the nematic phase is almost always found at higher temperatures with the exception of reentrant nematic phase.

The observed smectic phases differ from each other in the way of layer formation and the existing order inside the layer [_{∞ h}_{∞ h}_{2 h} symmetry.

As an example of compound that can be in both nematic and smectic A phases we cite the very much used octyl-4-cyanobiphenyl (8CB) with chemical formula shown in

The fundamental characteristics of a liquid crystal is the presence of long-range orientational order while the positional order is either absent (nematic phase) or limited (smectic phases). One phase differs from another by its symmetry. The transition between phases of different symmetry can be described in terms of an order parameter

The macroscopic orientational nematic order parameter is a traceless, symmetric, second rank tensor with components given by [

A uniaxial nematic state is described by the condition that two eigenvalues of

The microscopic ordered parameters

The uniaxial nematic state state is specified by only three parameters: the magnitude _{1}_{2}, ..._{N}_{B}T_{B}_{N}_{i}_{i}_{i}

If the mesogenic molecules possess cylindrical symmetry, _{l}_{2}〉, _{4}〉, etc., can be calculated. The most important one is usually called the scalar order parameter

The experimental observations [_{NI}_{NI}

In the microscopic models we deal with the system at the molecular level and we start the calculations on the basis of partition function or the exact density functionals of the free energy. Accordingly, for a complete and satisfactory microscopic theory a knowledge of the intermolecular interaction is necessary. Unfortunately such a knowledge is almost entirely lacking. Even if the essentials of the intermolecular interaction are known, the successful application of the theory is very difficult due to enormous calculation problems. Owing to these difficulties the model potentials have been introduced,

The first extremely successful molecular model (of mean field type) of liquid crystal ordering was developed by Onsager [

The configurational free energy (per rod) of a system of hard rods can be expressed as
_{0} is an additive constant and _{exc}_{i}_{j}_{exc}_{ij}_{i}

The singlet orientational distribution function can be determined by minimizing the free energy subject to the constraint ∫

Onsager obtained an approximate variational solution which is based on a trial function of the form _{iso}, nematic phase _{nem}, and order parameter

The Onsager model of nematic-isotropic phase transition is exact in the limit of the infinite length to width ratio

The mean-field model of Maier and Saupe [

The standard Maier–Saupe free energy difference between nematic and isotropic phases can be written as

The trial singlet orientational distribution function is chosen to be given by
_{2}(cos ^{2}

The entropy change associated with the nematic orientational order is given by

The order parameter is calculated by minimizing Δ

The possible extensions of the Maier–Saupe theory have been presented

A complete molecular theory of the mesophases must include both anisotropic short range repulsive and long-range attractive forces. The vdW types theories [

The classical version and development in the present form have been discussed at length in other reviews [

The Landau–de Gennes theory of phase transitions in liquid crystals [

The most general form of the free energy

Near the nematic-isotropic transition point, in a mesoscopic approach the phenomenological Landau–de Gennes form of the homogeneous free energy density _{h}^{4} J/K cm^{3}, ^{5} J/cm^{3}, and ^{5} J/cm^{3} [_{NI}

There is no term linear in

If

To ensure stability of the nematic phase,

Using the representation

The uniaxial nematic state (with the choice of
_{NI}^{2}/24_{nem} = _{iso} = 0) coexist in equilibrium. The minimum corresponding to the nematic phase becomes metastable for _{NI} < T < T^{+} = ^{2}/64^{+}, the superheated limit temperature of the nematic phase. The minimum corresponding to the isotropic phase is metastable for _{NI}_{NI}^{2}_{NI}^{2}.

The elastic free energy density (_{e}_{e}

The most essential surface contribution (anchoring term) of the free energy is given by

For the uniaxial ordering, using the representation

To obtain an insight into the structure and origin of the phenomenological parameters _{2n}(cos _{2} is by far the most important both from theoretical and experimental point of view, and is referred to as the nematic scalar order parameter

Using _{B}

Liquid crystals, by virtue of their fluidity, intrinsically soft elasticity and experimental accessibility, offer opportunities for the study of the structural and dynamical effects of quenched disorder, which can be introduced, for example, by confinement within appropriate random porous media [

Numerous studies [

In the following we analyze structural and phase behavior of a nematic liquid crystal experiencing quenched random anisotropy field (RAN) disorder in the simplest possible mean-field type approach. We first consider a typical domain structure in the nematic director field. We show that a domain structure is practically always expected, at least temporarily, if a lower symmetry (

The main part of the paper is concerned with phase and structural behavior of randomly perturbed systems in which domain-type patterns are assumed. For this purpose we list evidences supporting this description in phases or structures exhibiting continuous symmetry breaking, where the lower symmetry phase possesses long range order (or at least quasi long range order).

In general, at the mesoscopic continuum level the degree of ordering in the broken symmetry configuration is determined by an order parameter field (OPF). On the other hand a selected symmetry breaking state is described by a symmetry breaking field (SBF), also referred to as a gauge field [

An example of a continuous symmetry breaking in orientational ordering represents the isotropic to nematic liquid crystal phase transition. In the nematic phase ordering can be well described by the nematic director field

For illustration purposes we consider orientational ordering in the nematic phase in the remaining part of this Section. We first show that a domain-type pattern in the SBF is inevitably formed at least temporally in the lower symmetry configuration after a quench from the higher symmetry phase even in absence of disorder. Then we give evidences that such domain pattern could be stabilized by impurities, which impose a quenched random field-type of disorder. In both cases the domain patterns are well described by a single characteristic length. We determine regimes where such patterns are expected.

We consider the onset of orientational ordering in the thermotropic nematic phase which is reached via the temperature quench from the isotropic phase in absence of static disorder. Immediately after the quench in causally disconnected parts of the system, a random value of SBF is chosen depending on a local preference mediated by a fluctuation [

The coarsening dynamics following the quench evolves via three qualitatively different stages. Immediately after the quench the _{eq}_{eq}_{d}_{d}^{γ}

The above described mechanism of domain formation is universal and is referred to as the Kibble–Zurek mechanism [_{c}_{Q}_{0} |^{−υ}_{0} ^{−η}_{c}_{c}_{0} and _{0} are bare relaxation length and relaxation time estimating size of _{G}

Next, we estimate characteristic size _{d}_{d}

We express the elastic free energy and the random anisotropy field density [_{d}_{RA}_{r}_{d}_{RA}_{e}

The Imry–Ma estimation of _{d}

For this purpose [

_{0}. Local orientational ordering of a particle at the i-th site is given by an unit vector

The interaction energy

The short range interaction _{ij}_{i}_{i}

In simulations one originates either from spatially randomly distributed orientations of spins, or from a homogeneously aligned sample along a single direction. We refer to these system histories as the

A system configuration is obtained by minimizing

From obtained configurations the orientational correlation function

In order to obtain structural details from calculated dependencies _{d}_{d}_{d}

Representative

To check the validity of the Imry–Ma scaling the obtained _{d}

Results show that in the strong anchoring limit (_{d}_{d}^{−1±0.15} and _{d}^{−2±0.3}, respectively. On the contrary, for the homogeneous initial configurations significant departures from the Imry–Ma behavior are observed. In this case SRO is not obtained for weak enough anchoring conditions, although systems still reveal a characteristic length as already suggested by Giamarchi and Doussal [_{d}^{−1.6±0.1} for 2D and _{d}^{−3.2±0.25} for 3

It was further analyzed [

Here
_{ij}

The system is enclosed within a cube of volume
_{0} is the characteristic size of the unit cubic cell of the lattice. At the systems’ boundary the periodic boundary conditions are imposed. Impurities of concentration _{0}. It is assumed, that the

The orientations of unit vectors
_{int}_{0},

The positions

The orientation of the _{B}

In the simulations the average domain size was monitored which is estimated from a pattern as follows. One calculates an average volume _{d}

In _{d}^{0.49}. Simulations show negligible influence on _{d}_{d}_{d}_{d}_{d}

In the following we assume that the random anisotropy field gives rise to a domain type pattern. We analyze phase expected behavior for perturbed nematic LC and nematic-non nematogen mixture.

The random anisotropy nematic type disordering field enters the expression for the free energy _{s}_{d}_{r}_{d}_{r}^{3}. Note that _{d}_{n}

For numerical purposes we introduce instead the length
_{d}_{r}

Taking into account these previous considerations, the free energy density can now be written as

For convenience we introduce nondimensional quantities. The temperature is replaced by the reduced temperature _{NI}_{NI}^{2}/24_{NI}^{+} = 9/8. The nematic order parameter is normalized with respect to its value at the nematic-isotropic phase transition of the homogeneous system ^{2}^{3}^{4}. The nondimensional length, elastic constant and surface anchoring are given by _{r}^{2}_{d}B^{3}.

Omitting the bar notation, the dimensionless free energy density is written in the following form:

Taking the limit

Minimizing _{c}

The (Λ

For a weak enough disorder (Λ _{c}_{P}_{N}_{c}_{c}

The behavior of the correlation length

For low values of the order parameter _{c}_{r}

The profiles of the order parameter for two values of Λ shown in

In this section we present some results concerning the influence of a random anisotropy type disorder on the phase separation of the nematogen–non-nematogen mixture. We assume that the impurities (

The mixture is characterized by the volume fractions of the two components:
_{i}_{i}_{1} = _{2} = _{2} = Φ (due to the conservation of the number of particles Φ_{1} = 1 _{αβ}

The free energy

The first term is the free energy density of the isotropic mixing for the two components [_{B}_{0}/_{B}T

The second term in _{αβ}

The elastic free energy density _{e}_{s}

Using the non-dimensional variables defined in section (4.2.) the dimensionless free energy density is given by
^{2}_{B}TC^{3}/^{4}.

Note that the local ordering tendency of impurities induces a finite value of _{p}, S_{p}

_{s}_{e}

In this case the paranematic phase is replaced by the isotropic one and _{p}_{i}_{i}_{NI}

To show how the onset of nematic ordering triggers phase separation in conventional liquid crystal, we rewrite _{eff}_{eff}^{2}/Γ stands for the effective Flory–Huggins interaction parameter. We see that the nematic ordering effectively increases the Flory–Huggins interaction parameter, that can potentially lead to order-induced phase separation.

Minimizing _{c}

The subscripts

To calculate the phase diagram we use the equilibrium conditions given by the _{p}_{p}_{s}_{s}

The phase diagram (_{NI}

In the vicinity of _{NI}_{NI}_{p} < S_{c}

The described phenomenological RAN model roughly describes behavior of nematic LC experiencing a quenched random anisotropy field disorder. In the following we show some experimental evidences supporting this approach and discuss in which studied physical properties the influence of disorder might be significant for a temperature interval covering isotropic and nematic ordering in bulk LC samples.

In experimental studies the disorder is typically introduced geometrically by different perturbers via a varying LC–perturber interface [

All these perturbers introduce into systems a new characteristic length scale _{s}_{s}

Most studies have been performed in mixtures of LC phases and aerosil nanoparticles. The spherular aerosil particles have diameter 2_{s}^{2}/^{2}/_{s}_{s}^{3} the aerosil particles are more or less randomly distributed in the system. For _{s}^{3} the aerosils form a gel-like thixotropic network structure. In the so called _{s}^{3} and _{s}^{3}, the aerosil network is relatively responsive. It can rearrange in order to partially anneal the elastic stress imposed by the surrounding LC phase. In the _{s}^{3}, the aerosil network becomes a rigid-like, imposing a quenched type of disorder to surrounding LC molecules.

Several investigations have been also carried out in LC confined to aerogel matrices. Aerogels [_{a}

X-rays studies [_{s} ∼ ρ _{a}φ_{p}_{p}

Much work focusing on the influence of disorder on LC behavior has also been carried out in Controlled-pore glasses (CPG) [

In all these samples the orientational anchoring condition at the perturber–LC interface depends on interface treatment and also on the type of LC used. However as a rule, LC–perturber interfaces tend to increase degree of LC ordering locally [_{IN}

Regarding the range of ordering of the disordered phase, most studies confirm the prediction that the broken phase exhibits a domain-type short range order (SRO) in LC ordering. SRO was reported in samples using aerogels [

In randomly perturbed LCs on commonly observes a suppressed value of the paranematic-speronematic phase transition temperature

Furthermore, for small enough values of _{c}_{IN}_{c}

Note that the transition from the critical to the noncritical LC temperature behavior on decreasing _{c}

We have presented typical phase and structural behavior of randomly perturbed systems exhibiting continuous symmetry breaking using nematic liquid crystal phase as a testing ground. In liquid crystalline materials several features can be experimentally probed due to relatively good experimental accessibility, which can be exposed to different strengths and types of disorder. The impact of disorder on liquid crystal ordering is pronounced due to the softness of liquid crystal phases.

We have reviewed behavior of a nematic liquid crystal experiencing quenched random anisotropy field disorder assuming domain-type orientational ordering. Typical ordering of such systems has been demonstrated using relatively simple numerical simulations based on Lebwohl–Lasher lattice type model. It has been shown that a domain-type pattern characterized by a single characteristic length in the symmetry breaking field is inevitably formed at least temporally in the lower symmetry state (nematic) after a quench from the higher symmetry phase (isotropic) even in the absence of disorder. Such domain pattern can be stabilized by impurities. If impurities impose a random field-type disorder a domain pattern again characterized by a single characteristic length obeying the Imry–Ma scaling can be formed. We have presented cases in which the Imry–Ma scaling is expected and have shown that memory effects could be pronounced.

Assuming the domain-type nematic structure, we have used the Random Anisotropy Nematic (RAN) phenomenological model to study the phase behavior of nematic (speronematic)-isotropic (paranematic) phase transition. It has been shown that the main features which such a simple approach predicts are indeed realized in several experimental systems,

The authors acknowledge financial support from Ministries of Research and Education from Slovenia and Romania.

The arrangement of molecules in the nematic phase.

The arrangement of molecules in the cholesteric phase; the plans have been drawn for convenience, but do not have any specific physical meaning.

The arrangement of molecules in the smectic A phase.

The chemical formula of octyl-4-cyanobiphenyl (8CB).

_{d}

Value of

The time evolution of the characteristic domain length _{d}_{d}^{0.49} is obeyed, which is plotted with the dotted line. ^{3},

The domain growth for different concentrations of impurities after sudden quench from isotropic phases or homogeneously aligned structures. ^{3},

Saturated average domain length values as a function of ^{3},

The phase diagram (Λ_{c}_{c}

Phase diagram for nematic – non-nematic binary mixture in the absence of disorder for Γ =

The influence of disorder on the phase separation for nematogen – non-nematogen binary mixture (see the main text).