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Polymers are met under a large variety of chemical achievements, from biopolymers like DNA, xanthan, cellulose, proteins, and actin filaments, to synthetized polymers like polyethylene (PE), polybutadiene (PB), polystyrene (PS), polymethylmetacrylate (PMMA), and so on, all of great industrial interest. They are employed under numerous forms, in solutions, as gels, rubbers, synthetic fabrics, moulded pieces, and so on. [ABSTRACT FROM AUTHOR]

Real samples of condensed matter phases are necessarily bounded. The most obvious effect of the confinement is local: Molecular interactions in the bulk and at the boundaries are different. These differences lead to local changes that normally occur over length scales of a few atomic radii. Because the volume-to-surface ratio grows with the system size, the local surface effects are often neglected when a large system is considered. However, there can also be global consequences of surface interactions. For example, bounded volumes of liquid crystals in equilibrium contain topologically stable defects that are stable exclusively because of the surface conditions, as in Fig. 11.13. In this chapter, we consider surface effects both for isotropic and for anisotropic media. Monolayers and bilayers of surfactants are reviewed in Chapter 14. [ABSTRACT FROM AUTHOR]

The microscopic stability of soft materials systems (liquid crystals, colloids, polymers) is a problem with many facets. [ABSTRACT FROM AUTHOR]

In Chapter 3, we have considered the notion of order parameter, its amplitude and phase. The order parameter is a continuous field (scalar, vector, tensor, etc.) describing the state of the system at each point. Generally, it is a function of coordinates, ψ(r). Distortions of ψ(r) can be of two types: those containing singularities and those without singularities. At singularities, ψ is not defined. For a 3D medium, the singular regions might be either zero-dimensional (points), one-dimensional (lines), or two-dimensional (walls). These are the defects. Whenever a nonhomogeneous state cannot be eliminated by continuous variations of the order parameter (i.e., one cannot arrive at the homogeneous state), it is called topologically stable, or simply, a topological defect. If the inhomogeneous state does not contain singularities, but nevertheless is not deformable continuously into a homogeneous state, one says that the system contains a topological configuration (or soliton). [ABSTRACT FROM AUTHOR]

Dislocations in crystalline solids are rather simple examples of topological singularities: The symmetry of translation is broken along a line, and the Burgers vectors of merging dislocations add up linearly; on the other hand, they are complex physical objects, whose most important properties relate to irreversible phenomena that attend plastic deformation. [ABSTRACT FROM AUTHOR]

Both smectic A and columnar liquid crystals possess quantized symmetry translations (one such translation in SmAs, two in columnar phases). This is the reason why dislocations, the defects that break translational symmetries, are found in these phases. Defects involving other kinds of symmetry breaking might exist, such as disclinations or topological singular points, which break rotations. [ABSTRACT FROM AUTHOR]

Because of their translational symmetries, 1D (smectic) and 2D (columnar) positionally ordered liquid crystals show dislocations of translation of various characteristics. Their elastic and plastic properties, which depend strongly on the Burgers vector, their relationship with disclinations, and so on, make them original with respect to their counterparts in 3D solid crystals. These dislocations are also worth studying in some detail because of their role in complex phases with thermodynamical defects, such as TGB phases (smectics with periodic twist sub-boundaries), which we have already alluded to, or Moiré phases (columnar phases with periodic twist sub-boundaries), whose existence has been recently predicted. [ABSTRACT FROM AUTHOR]

Dislocations are responsible for the plastic deformation of crystalline materials such as metals, and play a role in a number of other properties of crystals, such as crystal growth, electrical properties of semiconductors, radiation damage through their interaction with point defects, and so on. Their theoretical discovery dates back to the years just before the second World War, and their visualization by various techniques, essentially electron microscopy, to the 1950s. They are an essential ingredient of physical metallurgy. They carry internal stresses and are "topologically" related to the symmetries of the crystal. It is this double character that makes them act as sources of plastic deformation. Their topological properties were fully appreciated with the appearance of a general classification of topological defects in ordered media, which comprehends superfluids, magnetic systems, and liquid crystals. Although defects and their textures in liquid crystals were observed (by optical microscopy) long before defects and their textures in solids, it is only in the last 10 years that investigations on the role of structure and texture on the rheological properties have been developed. A good knowledge of the bases of the physics of defects in solids cannot but help to progress in the investigation of rheological properties and instabilities of mesomorphic materials. [ABSTRACT FROM AUTHOR]

Euclidian geometry describes the world as a pattern of simple shapes: spheres, triangles, lines, and so on, with an intuitively clear concept of dimension: 0 for a point, 1 for a line, 2 for a plane, and so on. However, this classic picture is not a complete image of Nature, for "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." B. Mandelbrot, who developed the new family of shapes and coined the term fractal, gives one of the possible definitions: "A fractal is a shape made of parts similar to the whole in some way." [ABSTRACT FROM AUTHOR]

Static continuum theory allows one to find the equilibrium configurations of the order parameter field by minimizing an appropriate free-energy functional. If the system is slightly disturbed, it usually relaxes back to this equilibrium. Most of the perturbations relax quickly, over a characteristic time of collisions between the molecules (which is of the order of 10-10-10-14 s for classic fluids). However some variables, called hydrodynamic variables, relax slowly. A classic example is the mass density: The frequency ω (inverse time) of the sound waves vanishes when their wavelength becomes infinite, L → ∞, according to the dispersion law ω Rt c/L, where c is the sound velocity. In general, densities of conserved variables (mass, linear momentum, and energy) are hydrodynamic variables. In media with continuous broken symmetries, such as nematic liquid crystals, there is a second class of hydrodynamic variables, namely, the degeneracy parameters (or phases of the order parameter), which is discussed in Chapter 3. [ABSTRACT FROM AUTHOR]

We describe the state of deformation of a nematic or a cholesteric phase, at a fixed temperature T, by the director field n(r). The free energy associated with the deformation depends necessarily on the gradient of the director, ∇n, whose components $$ \frac{{\partial n_i }} {{\partial x_j }} $$ will be noted ni,j. We assume that the distortions are small, ni,j« 1/a, where a is a typical molecular length. This assumption has some advantages, as follows: There is a well-defined "tangent" perfect (liquid) crystal at each point r, with orientation n(r), whose spatial extention is large enough to make a continuous description possible.Therefore, the order parameter is a locally well-defined constant s[T(r)], which depends on temperature uniquely.At any point r, the symmetry properties of the tangent liquid crystal should reflect in its free energy density f(r), which however depends not only on n(r), but also on its derivatives ∇n. It is stated that f must be invariant under any change of the orientation of n and of the values of its derivatives ∇n, which are allowed by the symmetries. This invariance is by no means trivial, and it should be considered as a principle, to which one could attach the name of Noll (principle of material invariance); it goes much farther than does the invariance of the Landau expansion, which does not depend on the derivative. [ABSTRACT FROM AUTHOR]

Physical matter might exist in different forms, or phases, that differ by the type of order, mass density, and so on. Transitions between the phases are described by the behavior of thermodynamic potentials, such as the Gibbs G or the Helmholz F free energies, and by the derivatives of these potentials. The phase transition is called first order if the first derivatives of the potentials (such as entropy, order parameter) exhibit a discontinuity, a jump at the transition point. These transitions involve nonzero latent heat. If the first derivatives are continuous but the second derivatives (such as heat capacities) diverge at the transition, it is a second-order phase transition. It is customary to lump into the second category of continuous phase transitions all transitions with no latent heat. Although the thermodynamic potentials in the second-order transitions change gradually, the symmetry of the system changes discontinuously. [ABSTRACT FROM AUTHOR]

The concept of an order parameter has appeared with the attempt to describe the orderdisorder transition of alloys, specifically, to define a degree of disorder (see Section 3.1.6). First elaborated by Gorsky and Bragg and Williams to describe order-disorder transitions in alloys, it has been developed in its modern form by Landau for the purpose of a phenomenological description of phase transitions. More specifically, it addresses the question of the description of the long-range order of the structural (crystallographic) or thermodynamic (magnetic, dielectric, etc) properties, which repeat uniformly in a given system. Related quantities are intensive thermodynamic variables. The opposite notion of shortrange order refers to spatial thermodynamic fluctuations; these are particularly important near a second-order phase transition A ↔ B (see Chapter 4). Fluctuations that develop in, say, the high temperature phase A, announce the order properties of B. An important related quantity is the spatial extent ξ of these fluctuations, the coherence length. The same term of coherence length (or correlation length) is also used to conote the range of distortions induced in the order parameter of phase B by a local perturbation (Chapter 5, section 5.6), e.g., the extent of the central region of a defect, where the order parameter is "broken" (singular), the core of the defect. [ABSTRACT FROM AUTHOR]

This is not the place to discuss in detail the crystalline arrangements of atoms. Most text books on condensed matter physics describe the 14 Bravais lattices (groups of translations) and the 230 Schönflies-Fedorov groups (translations and point symmetries). These groups exhaust all possible symmetry groups with 3D discrete translations. In classic crystallography, the elements that build the symmetry of the group are atoms; they can be generally represented by points or, when the picture requires more complex characters, by the density of matter. Because of the large (compared with kBT) binding energies, the atoms can be considered as fixed, and the symmetries can be easily visualized through rigid displacements (translations, rotations). These symmetries are not the main concerns of soft matter physics. In soft matter, the order is related to the specific peculiarities of molecular shapes, and fluidity, hence, entropy, are more relevant factors. We first describe how the concept of atomic order can be extended when taking into account the precise shapes of the atoms, no longer considered as points, and how these extensions enter naturally in the description of "complex" soft systems. [ABSTRACT FROM AUTHOR]

It is classic to distinguish three states of matter: gas, liquid, and solid. In the first approximation, the gas state is characterized by the absence of interactions between atoms or molecules, which therefore display statistical disorder. The requirement of maximum of entropy controls this state. Gases are outside the scope of this textbook. In contrast, the more condensed liquid and solid phases are controlled not only by entropy, but also by interparticle interactions. These states are stabilized by a complex interplay between attractive interactions, which are responsible for the condensation of chemical species, and repulsive interactions. The balance yields a local order, defined on some characteristic length. In crystals, this local order reproduces on distances much larger than the interparticle distances, so that one can speak of long-range order. Long-range order of crystalline materials is described in terms of the elements of the group of symmetry under which this order is invariant. Independently of the type of arrangement, all condensed matter media in liquid or solid states present some common features as follows: 1The energies of interaction that stabilize the local order vary in the rang 0.1eV ÷ 10eV per atom or molecule. The order of magnitude that is frquently met is 1 eV.2The molar volumes are all of the same order of magnitude; typical distances between atoms are of the order of 1 Å.3Vibration frequencies of atomic bonds are of the order of 1013-1014 Hz. [ABSTRACT FROM AUTHOR]