Christos N. Likos

EFFECTIVE INTERACTIONS IN SOFT CONDENSED MATTER PHYSICS

Abstract 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 μm 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.

Phase Diagram of Star Polymer Solutions

The phase diagram of star polymer solutions in a good solvent is obtained over a wide range of densities and arm numbers by Monte Carlo simulations. The effective interaction between the stars is modeled by an ultrasoft pair potential which is logarithmic in the core-core distance. Among the stable phases are a fluid as well as body-centered cubic, face-centered cubic, body-centered orthogonal, and diamond crystals. In a limited range of arm numbers, reentrant melting and reentrant freezing transitions occur for increasing density.

Criterion for determining clustering versus reentrant melting behavior for bounded interaction potentials.

We examine in full generality the phase behavior of systems whose constituent particles interact by means of potentials that do not diverge at the origin, are free of attractive parts, and decay fast enough to zero as the interparticle separation r goes to infinity. By employing a mean field-density functional theory which is shown to become exact at high temperatures and/or densities, we establish a criterion that determines whether a given system will freeze at all temperatures or it will display reentrant melting and an upper freezing temperature.

Soft matter with soft particles

In this review we present a summary of recent progress achieved in examining the equilibrium and dynamical properties of concentrated solutions of two novel kinds of soft matter systems: and starburst molecules known as . The two systems share a host of interesting properties. The both consist of highly branched polymers, they allow for tuning of their properties through modification of the macromolecular architecture and they are both representatives of a quite novel class of colloidal particles, termed . On the other hand there are also important differences, reflecting the fundamental difference in their architecture. It will be shown that a combination of scattering techniques and rheology with computer simulations and analytical methods from the realm of theoretical physics can shed light on the unusual properties of such systems. In this fashion, new ways appear for the manipulation of soft matter systems under external influences and promising perspectives for the fabrication of new materials are opened up and the versatility in manipulating soft matter is underlined.

Why do ultrasoft repulsive particles cluster and crystallize? Analytical results from density-functional theory

We demonstrate the accuracy of the hypernetted chain closure and of the mean-field approximation for the calculation of the fluid-state properties of systems interacting by means of bounded and positive pair potentials with oscillating Fourier transforms. Subsequently, we prove the validity of a bilinear, random-phase density functional for arbitrary inhomogeneous phases of the same systems. On the basis of this functional, we calculate analytically the freezing parameters of the latter. We demonstrate explicitly that the stable crystals feature a lattice constant that is independent of density and whose value is dictated by the position of the negative minimum of the Fourier transform of the pair potential. This property is equivalent with the existence of clusters, whose population scales proportionally to the density. We establish that regardless of the form of the interaction potential and of the location on the freezing line, all cluster crystals have a universal Lindemann ratio Lf=0.189 at freezing. We further make an explicit link between the aforementioned density functional and the harmonic theory of crystals. This allows us to establish an equivalence between the emergence of clusters and the existence of negative Fourier components of the interaction potential. Finally, we make a connection between the class of models at hand and the system of infinite-dimensional hard spheres, when the limits of interaction steepness and space dimension are both taken to infinity in a particularly described fashion.

Gaussian effective interaction between flexible dendrimers of fourth generation: A theoretical and experimental study

We propose a theory for the effective interaction between soft dendritic molecules that is based on the shape of the monomer density profile of the macromolecules at infinite dilutions. By applying Flory-type arguments and making use of the experimentally measured density profiles, we derive a Gaussian effective interaction whose parameters are determined by the size and monomer number of the dendrimers that are derived from small-angle neutron scattering (SANS) measurements. By applying this theory to concentrated dendrimer solutions we calculate theoretical structure factors and compare them with experimental ones, derived from a detailed analysis of SANS-data. We find very good agreement between theory and experiment below the overlap concentration, where drastic shape deformations of the dendrimers are absent.

Asymmetric caging in soft colloidal mixtures

The long-standing observations that different amorphous materials exhibit a pronounced enhancement of viscosity and eventually vitrify on compression or cooling continue to fascinate and challenge scientists, on the ground of their physical origin and practical implications. Glass formation is a generic phenomenon, observed in physically quite distinct systems that encompass hard and soft particles. It is believed that a common underlying scenario, namely cage formation, drives dynamical arrest, especially at high concentrations. Here, we identify a novel, asymmetric glassy state in soft colloidal mixtures, which is characterized by strongly anisotropically distorted cages, bearing similarities to those of hard-sphere glasses under shear. The anisotropy is induced by the presence of soft additives. This phenomenon seems to be generic to soft colloids and its origins lie in the penetrability of the constituent particles. The resulting phase diagram for mixtures of soft particles is clearly distinct from that of hard-sphere mixtures and brings forward a rich variety of vitrified states that delineate an ergodic lake in the parameter space spanned by the size ratio between the two components and by the concentration of the additives. Thus, a new route opens for the rational design of soft particles with desired tunable rheological properties.

Structural Arrest in Dense Star-Polymer Solutions

The dynamics of star polymers is investigated via extensive molecular and Brownian dynamics simulations for a large range of functionality f and packing fraction eta. The calculated isodiffusivity curves display both minima and maxima as a function of eta and minima as a function of f. Simulation results are compared with theoretical predictions based on different approximations for the structure factor. In particular, the ideal glass transition line predicted by mode-coupling theory is shown to exactly track the isodiffusivity curves, offering a theoretical understanding for the observation of disordered arrested states in star-polymer solutions.

Counterion-induced entropic interactions in solutions of strongly stretched, osmotic polyelectrolyte stars

We examine the conformations and effective interactions of star-branched polyelectrolytes with and without added salt, by employing monomer-resolved molecular dynamics simulations and an analytical theory. The simulations take into account the excluded-volume and Coulomb interactions between the individual monomers, as well as the counter- and coions. The theory is based on a variational free energy that is written as a sum of electrostatic, polymer, and entropic contributions of the counter- and coions. For the conformations of isolated polyelectrolyte stars, we find strong stretching of the chains, resulting in a linear scaling of the star radius with the degree of polymerization, as well as trapping and condensation of a large fraction of counterions. The effective interactions at arbitrarily strong overlaps between the stars are shown to be dominated by the entropic contributions of the trapped counterions, with the electrostatic contribution playing only a minor role due to an almost complete neutral...

Effective interactions between star polymers and colloidal particles

Using monomer-resolved molecular dynamics simulations and theoretical arguments based on the radial dependence of the osmotic pressure in the interior of a star, we systematically investigate the effective interactions between hard, colloidal particles and star polymers in a good solvent. The relevant parameters are the size ratio q between the stars and the colloids, as well as the number of polymeric arms f (functionality) attached to the common centre of the star. By covering a wide range of qs ranging from zero (star against a flat wall) up to about 0.75, we establish analytical forms for the star-colloid interaction which are in excellent agreement with simulation results. A modified expression for the star-star interaction for low functionalities, f10 is also introduced.