Spontaneous Flow Transitions in Active Polar Gels

Spontaneous Flow Transitions in Active Polar Gels

ABSTRACT

Active polar gels are a type of viscoelastic soft material formed by polar filaments 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 filaments in the cytoskeleton which plays an important role in many cellular processes.

Using diffferent boundary conditions, Fredericks-like flow transitions driven by the activity are expected. These transitions occur from homogeneously polarized static states in thin gel layers to flowing states with polarization tilts in larger layers.

In addition, a few notes on the limitations of this work and possibilities for additional work are briefly expounded on at the end.

TABLE OF CONTENT

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

INTRODUCTION

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 flow 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.

REFERENCES

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