POLYMER

In this review we study the behavior of a single labelled polymer chain in various polymer systems: polymer blends, diblock copolymers, gradient copolymers, ring copolymers, polyelectrolytes, grafted homopolymers, rigid nematogenic polymers, polymers in bad and good solvents, fractal polymers and polymers in fractal environments. We discuss many phenomena related to the single chain behavior, such as: collapse of polymers in bad solvents, protein folding, stretching of polymer brushes, coil¯rod transition in nematogenic main-chain polymers, knot formation in homopolymer melts, and shrinking and swelling of polymers at temperatures close to the bulk transition temperatures. Our description is mesoscopic, based on two models of polymer systems: the Edwards model with Fixman delta interactions, and the Landau¯Ginzburg model of phase transitions applied to polymers. In particular, we show the derivation of the Landau¯Ginzburg model from the Edwards model in the case of homopolymer blends and diblock copolymer melts. In both models, we calculate the radius of gyration and relate them to the correlation function for a single polymer chain. We discuss theoretical results as well as computer simulations and experiments.

We present a scaling Ansatz for the distribution function of the shortest paths connecting any two points on a percolating cluster which accounts for (i) the e ect of the nite size of the system, and (ii) the dependence of this distribution on the site occupancy probability p. We present evidence supporting the scaling Ansatz for the case of two-dimensional percolation.

In this work, we present a review of recently achieved progress in the field of soft condensed matter physics, and in particular on the study of the static properties of solutions or suspensions of colloidal particles. The latter are macromolecular entities with typical sizes ranging from 1 nm to 1 small mu, Greekm and their suspension typically contain, in addition to the solvent, smaller components such as salt ions or free polymer chains. The theoretical tool introduced is the effective Hamiltonian which formally results by a canonical trace over the smaller degrees of freedom for a fixed, frozen configuration of the large ones. After presenting the formal definitions of this effective Hamiltonian, we proceed with the applications to some common soft matter systems having a variable softness and ranging from free polymer chains to hard colloidal particles. We begin from the extreme case of nondiverging effective interactions between ultrasoft polymer chains and derive an exact criterion to determine the topology of the phase diagrams of such systems. We use star polymers with a variable arm number f as a hybrid system in order to interpolate between these two extremes. By deriving an effective interaction between stars we can monitor the change in the phase behavior of a system as the steepness of the repulsion between its constituent particles increases. We also review recent results on the nature and the effects of short-range attractions on the phase diagrams of spherical, nonoverlapping colloidal particles.

We review results concerning the critical behavior of spin systems at equilibrium. We consider the Ising and the general O(N)-symmetric universality classes, including the Nrightwards arrow0 limit that describes the critical behavior of self-avoiding walks. For each of them, we review the estimates of the critical exponents, of the equation of state, of several amplitude ratios, and of the two-point function of the order parameter. We report results in three and two dimensions. We discuss the crossover phenomena that are observed in this class of systems. In particular, we review the field-theoretical and numerical studies of systems with medium-range interactions. Moreover, we consider several examples of magnetic and structural phase transitions, which are described by more complex Landau¯Ginzburg¯Wilson Hamiltonians, such as N-component systems with cubic anisotropy, O(N)-symmetric systems in the presence of quenched disorder, frustrated spin systems with noncollinear or canted order, and finally, a class of systems described by the tetragonal Landau¯Ginzburg¯Wilson Hamiltonian with three quartic couplings. The results for the tetragonal Hamiltonian are original, in particular we present the six-loop perturbative series for the small beta, Greek-functions. Finally, we consider a Hamiltonian with symmetry O(n1)plus sign in circleO(n2) that is relevant for the description of multicritical phenomena.