# DISLOCATIONS IN UNIAXIAL LAMELLAR PHASES OF LIQUID CRYSTALS, POLYMERS AND AMPHIPHILIC SYSTEMS

## R. Hołyst and P. Oswald

## International Journal of Modern Physics B 1995, Vol. 09, No. 13n14, pp. 1515-1573

Dislocations in soft condensed matter systems such as lamellar systems of polymers, liquid crystals and ternary mixtures of oil, water and surfactant (amphiphilic systems) are described in the framework of continuum elastic theory. These systems are the subject of studies of physics, chemistry and biology. They also find applications in industry. Here we will discuss in detail the influence of dislocations on the bulk and surface properties of these lamellar phases. Especially the latter properties have only been recently studied in detail. We will present the experimental evidence of the existence of screw and edge dislocations in the systems and study their static properties such as: energy, line tension and core structure. Next we will show how the surface influences the equilibrium position of dislocations in the system. We will give the theoretical predictions and present the experimental results on thin copolymer films, free standing films of liquid crystals and smectic droplets shapes. In semi-infinite lamellar systems characterized by small surface tension the dislocation is stabilized at a finite distance, h_{eq}, from the surface, due to the surface bending elastic constant, K_{s} (for zero surface tension h_{eq}≈K_{s}/2K, where K is the bulk bending elastic constant). For large surface tension the edge dislocations are strongly repelled by the surface and the equilibrium location for finite symmetric systems such as free standing liquid crystal films shifts towards the center of the system. The surface is deformed by dislocations. These deformations are known as edge profiles. They will be discussed for finite systems with small and large surface tension. Surface deformations induce elastic interactions between edge dislocations, which decay exponentially with distance with decay length proportional to where D is the size of the system normal to lamellas. Two screw dislocations in finite system interact with the logarithmic potential, which is proportional to the surface tension and inversly proportional to D. The surface induced elastic interactions will be compared to, well-known, bulk deformation induced interactions. A new phenonenon discussed in our paper is the fluctuations induced interactions between edge dislocations, which follows from the Helfrich mechanism for flexible objects. At suitable conditions, edge dislocations can undergo an unbinding transition. Also a single dislocation loop can undergo an unbinding transition. We will calculate the properties of the loop inside finite system and discuss in particular the unbinding transition in freely suspended smectic films. We shall also compute the equilibrium size of the loop contained between two hard walls. Finally we will discuss the dynamical bulk properties of dislocations such as: mobility (climb and glide), permeation, and helical instability of screw dislocations. Lubrication theory will also be discussed.