Spontaneous Flow Transitions in Active Polar Gels

ABSTRACT

Active polar gels are a type of viscoelastic soft material formed by polar laments which are constantly driven out of equilibrium by the consumption of chemical fuel such as ATP. Following the approach of Voituriez et al, a generic hydrodynamic theory relying on symmetry arguments only was used to study the effects of quasi one-dimensional confinement on active polar gels. The phenomenological theory based on liquid crystal hydrodynamics is motivated by the dynamics of actin laments in the cytoskeleton which plays an important role in many cellular processes. Using different boundary conditions, Fredericks-like ow transitions driven by the activity are expected.

These transitions occur from homogeneously polarized static states in thin gel layers to owing states with polarization tilts in larger layers. In addition, a few notes on the limitations of this work and possibilities for additional work are brie y expounded on at the end.

TABLE OF CONTENTS

1 Introduction 2
1.1 Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Types of Liquid Crystals . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Nematic Liquid Crystals . . . . . . . . . . . . . . . . . 3
1.2.2 Chiral Nematic Liquid Crystals . . . . . . . . . . . . . 4
1.2.3 Smectic Liquid Crystals . . . . . . . . . . . . . . . . . 5
1.3 Biological Liquid Crystals . . . . . . . . . . . . . . . . . . . . 6
1.4 The Distortion Free Energy . . . . . . . . . . . . . . . . . . . 6
1.5 Active Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Active behaviour of the Cytoskeleton 11
2.1 The Cytoskeleton . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Actin Filaments . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 Intermediate Filaments . . . . . . . . . . . . . . . . . . 12
2.1.3 Microtubules . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Constitutive hydrodynamic equations of active polar gels . . . 13
2.2.1 Polar order, Fluxes and Forces in active gels . . . . . . 14
2.2.2 The Maxwell model . . . . . . . . . . . . . . . . . . . . 16
2.2.3 Dynamic equations . . . . . . . . . . . . . . . . . . . . 16
2.2.4 Boundary Conditions and Anchoring . . . . . . . . . . 17
3 Spontaneous Flow Transitions in 2-Dimensions 19
3.1 Linear Approximation . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Non-equilibrium steady states . . . . . . . . . . . . . . . . . . 20
3.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 22
3.3.1 Hydrodynamic free boundary conditions . . . . . . . . 22
3.3.2 Hydrodynamic no-slip boundary conditions . . . . . . . 25
3.3.3 Hydrodynamic mixed boundary conditions . . . . . . . 27
3.3.4 Active boundary conditions . . . . . . . . . . . . . . . 27
4 Endnotes 28
4.1 Scope of work and Recommendations for further study . . . . 28
4.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

CHAPTER ONE

Introduction

1.1 Liquid Crystals

It is widely believed that Matter exists in three distinct and interconvertible phases: The Solid, Liquid and Gaseous phases; with the major distinction between them given by the degree of order (or disorder) in their molecular arrangements. In actual fact however, this is not strictly true for all materials because a wide range of natural and synthetic substances exist that do not display a simple transition from the solid phase to the liquid phase,
but rather go through a series of intermediate phases (called mesophases or mesomorphic phases). These mesophases display properties that are intermediate between the solid and liquid states; and are therefore referred to as Liquid Crystals. A liquid Crystal for example may ow like a liquid but does not possess the isotropy an ordinary liquid displays, it may also have interesting optical properties similar to a crystalline material but will not display long-ranged three dimensional order like most crystalline materials.

From a microscopic point of view, the major difference between a liquid crystalline phase, the crystalline phase and the liquid phase stems from their molecular arrangement. In a crystalline solid there is long-ranged three dimensional order with a regular arrangement of atoms on a lattice and in a liquid the molecules are oriented in random directions with weak intermolecular forces in all directions. However, in a liquid crystal the intermolecular forces in the crystalline solid are not the same in all directions; in some directions the forces are weaker than in other directions. As such a material is heated, the increased molecular motion overcomes the weaker forces rst, but its molecules remain bound by the stronger forces. This produces a molecular arrangement that is random in some directions and regular in others thus leading to liquid-like order in at least one direction and variation in physical properties along others (i.e anisotropy)

1.2 Types of Liquid Crystals

Many Liquid Crystal materials are organic compounds that exhibit liquid crystalline phases either as a function of temperature or concentration of solute molecules in a solvent. The former class of materials are termed Thermotropic and the latter, Lyotropic. Thermotropic phases usually occur in a certain temperature range, above which the material behaves as an isotropic liquid and below which it forms a regular crystal[1]. Lytotropic phases can be formed in amphiphilic compounds by varying the volume balance between its constituents. There are also certain low-melting inorganic materials such as Zinc Chloride which when mixed with long chain soap-like molecules form mesophases with liquid crystalline behaviour as a function of both temperature and organic-inorganic composition ratio. Such liquid crystals are termed met-allotropic. Liquid crystals are examples of mesogenic materials because they form mesophases under appropriate conditions; this also implies that not all liquid crystal materials will be in liquid crystal phases under all conditions.

The major building blocks for liquid crystalline materials have been identified as either small rod-like or disk-like organic molecules, long helical rods in a liquid substrate, polymers and other associated structures such as the amphiphilic molecules above[2].

Various classifications for the different types of liquid crystals have been proposed, the most widely used are as follows:

1.2.1 Nematic Liquid Crystals

Nematics are the most commonly encountered liquid crystalline mesophase.

In nematics, the long axis of the rod-like constituent molecules tend to align themselves parallel to each other along a common direction called the anisotropic axis. This imposes long ranged directional order even though there is no positional order; implying that molecules are free to ow and the centre of mass positions remain randomly distributed as in an isotropic liquid.

In many cases, a unit vector called the director p is used to describe the average local direction of molecular alignment; it is often dened without polarity such that its sign has no physical significance i.e the states p and 􀀀p are indistinguishable. Nematic liquid crystals get their name from the threadlike topological defects (formally called disclinations) corresponding to lines of singularity in the director alignment. Examples of extensively studied nematic liquid crystals are p􀀀azoxyanisole (PAA) and 4-methoxybenzylidene- 40-butylaniline (MBBA). Mixtures of Nematic liquid crystals have interesting optical properties and are often used in liquid crystal displays (LCDs) because they can be easily aligned by external magnetic and electric elds.

Figure 1.1: Schematic representation of Molecular arrangement in a nematic liquid crystal, the short bold lines represent the molecules

1.2.2 Chiral Nematic Liquid Crystals

Chiral nematics exhibit a property called chirality or handedness meaning they exhibit a twisting of the constituent molecules perpendicular to the director, with the molecular axis parallel to the director giving rise to a helical structure. The structure of chiral nematics is a result of the chiral nature of the constituent molecules. Chiral molecules are different from their mirror images and therefore have either a right-handed or left-handed sense, such molecules are also called enantiomorphs. The helical axis of a chiral nematic is often assumed to be in the horizontal direction and the helix itself may be either right-handed or left-handed. At a given temperature, a sample of a cholesterol liquid crystal always produces helices of the same sense but there are cholesterics whose handedness of helix can be changed by varying the temperature [1]. The variation in the director alignment in a chiral nematic
sample has a periodicity L where L is the helical pitch dened as the distance measured along the helical axis over which the director rotates through a full 2 radians, the periodicity is half of this distance because the states described by p and 􀀀p are identical.

The chiral pitch is also aected by the concentration of chiral dopants in the liquid crystal host material i.e an a chiral liquid crystal host material will form a chiral nematic if doped with chiral molecules.

Figure 1.2: Schematic representation of Molecular arrangement in a chiral nematic liquid crystal

1.2.3 Smectic Liquid Crystals

Figure 1.3: Schematic representation of Molecular arrangement in (a) smectic A and (b) smectic C liquid crystal phases Smectics also derive their name from a greek word meaning \soap”; this is due to the fact that they have mechanical properties similar to those of soaps.

They have well dened layered structures (with axed interlayer spacing) that can slide over one another similar to soap. Smectics are thus more ordered than nematics and for a given material the smectic phase occurs at temperatures below that for which a nematic phase is observed [1]. Several types of smectics have been identified but the two most common are smectics A and C. In a smectic A phase, the molecules are arranged in layers where the director is on an average perpendicular to the layers and parallel to the layer normal.

In a smectic C phase (labelled b), the director is tilted at an angle relative to the layer normal; in both cases the director has no polarity i.e p and 􀀀p are indistinguishable.