Statistics of randomly branched polymers in a semi-space

We investigate the statistical properties of a randomly branched 3--functional $N$--link polymer chain without excluded volume, whose one point is fixed at the distance $d$ from the impenetrable surface in a 3--dimensional space. Exactly solving the Dyson-type equation for the partition function $Z(N,d)=N^{-\theta} e^{\gamma N}$ in 3D, we find the "surface" critical exponent $\theta={5/2}$, as well as the density profiles of 3--functional units and of dead ends. Our approach enables to compute also the pairwise correlation function of a randomly branched polymer in a 3D semi-space.Comment: 15 pages 7 figsures; section VII is slightly reorganized, discussion is revise

Mesophase formation in two-component cylindrical bottle-brush polymers

When two types of side chains (A,B) are densely grafted to a (stiff) backbone and the resulting bottle-brush polymer is in a solution under poor solvent conditions, an incompatibility between A and B leads to microphase separation in the resulting cylindrical brush. The possible types of ordering are reminiscent of the ordering of block copolymers in cylindrical confinement. Starting from this analogy, Leibler's theory of microphase separation in block copolymer melts is generalized to derive a description of the system in the weak segregation limit. Also molecular dynamics simulation results of a corresponding coarse-grained bead-spring model are presented. Using side chain lengths up to N = 50 effective monomers, the ratio of the Lennard-Jones energy parameter between unlike monomers $(\epsilon_{AB})$ and monomers of the same kind $(\epsilon _{AA} = \epsilon_{BB})$ is varied. Various correlation functions are analyzed to study the conditions when (local) Janus cylinder-type ordering and when (local) microphase separation in the direction along the cylinder axis occurs. Both the analytical theory and the simulations give evidence for short range order due to a tendency towards microphase separation in the axial direction, with a wavelength proportional to the side chain gyration radius, irrespective of temperature and grafting density, for a wide range of these parameters.Comment: 26 pages, 19 figure

Frequency Dispersion of Sound Propagation in Rouse Polymer Melts via Generalized Dynamic Random Phase Approximation

An extended generalization of the dynamic random phase approximation (DRPA) for L-component polymer systems is presented. Unlike the original version of the DRPA, which relates the (LxL) matrices of the collective density-density time correlation fumctions and the corresponding susceptibilities of polymer concentrated systems to those of the tracer macromolecules and so-called broken links system (BLS), our generalized DRPA solves this problem for (5xL)x(5xL) matrices of the coupled susceptibilities and time correlation functions of the component number, kinetic energy and flux densities. The presented technique is used to study propagation of sound and dynamic form-factor in disentangled (Rouse) monodisperse homopolymer melt. The calculated sound velocity and absorption coefficient reveal substantial frequency dispersion. The relaxation time is found to be N times less than the Rouse time (N is the degree of polymerization), which evidences strong dynamic screening because of interchain interaction. We discuss also some peculiarities of the Brillouin scattering in polymer melts. Besides, a new convenient expression for the dynamic structural function of the Rouse chain in (q,p)-representation is found.Comment: 37 pages, 2 appendices, 48 references, 1 figur

The Hartree approximation in dynamics of polymeric manifolds in the melt

The Martin-Siggia-Rose (MSR) functional integral technique is applied to the dynamics of a D - dimensional manifold in a melt of similar manifolds. The integration over the collective variables of the melt can be simply implemented in the framework of the dynamical random phase approximation (RPA). The resulting effective action functional of the test manifold is treated by making use of the selfconsistent Hartree approximation. As an outcome the generalized Rouse equation (GRE) of the test manifold is derived and its static and dynamic properties are studied. It was found that the static upper critical dimension, $d_{\rm uc}=2D/(2-D)$, discriminates between Gaussian (or screened) and non-Gaussian regimes, whereas its dynamical counterpart, ${\tilde d}_{uc}=2d_{\rm uc}$, distinguishes between the simple Rouse and the renormalized Rouse behavior. We have argued that the Rouse mode correlation function has a stretched exponential form. The subdiffusional exponents for this regime are calculated explicitly. The special case of linear chains, D=1, shows good agreement with MD- and MC-simulations.Comment: 35 pages,3 figures, accepted by J.Chem.Phy

Necklace-Cloverleaf Transition in Associating RNA-like Diblock Copolymers

We consider a ${\rm A}_m{\rm B}_n$ diblock copolymer, whose links are capable of forming local reversible bonds with each other. We assume that the resulting structure of the bonds is RNA--like, i.e. topologically isomorphic to a tree. We show that, depending on the relative strengths of A--A, A--B and B--B contacts, such a polymer can be in one of two different states. Namely, if a self--association is preferable (i.e., A--A and B--B bonds are comparatively stronger than A--B contacts) then the polymer forms a typical randomly branched cloverleaf structure. On the contrary, if alternating association is preferable (i.e. A--B bonds are stronger than A--A and B--B contacts) then the polymer tends to form a generally linear necklace structure (with, probably, some rear side branches and loops, which do not influence the overall characteristics of the chain). The transition between cloverleaf and necklace states is studied in details and it is shown that it is a 2nd order phase transition.Comment: 17 pages, 9 figure

Polyelectrolytes in the presence of multivalent ions: gelation versus segregation

We analyze solutions of strongly charged chains bridged by linkers such as multivalent ions. The gelation induced by the strong short range electrostatic attractions is dramatically suppressed by the long range electrostatic correlations due to the charge along the uncrosslinked monomers and ions. A modified Debye-Huckel approach of crosslinked clusters of charged chains is used to determined the mean field gelation transition self-consistently. Highly dilute polyelectrolyte solutions tend to segregate macroscopically. Semidilute solutions can form gels if the Bjerrum length $l_B$ and the distance between neighboring charged monomers along the chain $b$ are both greater than the ion size $a$

Localization transition of random copolymers at interfaces

We consider adsorption of random copolymer chains onto an interface within the model of Garel et al. Europhysics Letters 8, 9 (1989). By using the replica method the adsorption of the copolymer at the interface is mapped onto the problem of finding the ground state of a quantum mechanical Hamiltonian. To study this ground state we introduce a novel variational principle for the Green's function, which generalizes the well-known Rayleigh-Ritz method of Quantum Mechanics to nonstationary states. Minimization with an appropriate trial Green's function enables us to find the phase diagram for the localization-delocalization transition for an ideal random copolymer at the interface.Comment: 5 page

A Modified Random Phase Approximation of Polyelectrolyte Solutions

We compute the phase diagram of salt-free polyelectrolyte solutions using a modified Debye-Huckel Approach. We introduce the chain connectivity via the Random Phase Approximation with two important modifications. We modify the electrostatic potential at short distances to include a bound on the electrostatic attractions at the distance of closest approach between charges. This modification is shown to act as a hard core in the phase diagram of electrolyte solutions. We also introduce a cut-off on the integration of the modes of wave length smaller than the size over which the chains are strongly perturbed by the electrostatic interactions. This cut-off is shown to be essential to predict physical phase diagram in long chain solutions

Fluctuation effects in the theory of weak supercrystallization in block copolymer systems of complicated chemical structure

To investigate the influence of fluctuation effects on conditions of microphase separation in two-component incompressible molten block copolymers of complicated chemical structure an alternative and more general formulation of the weak crystallization theory is developed using a new variational principle for the calculation of the free energy. Unlike the Fredrickson-Helfand theory, ours accounts for a given angular dependence of vertex functions more rigorously and enables us to evaluate correlation functions in both disordered and supercrystal states. A comparison of phase diagrams constructed by means of both our and the FH theories is made for the case of star copolymers $({\rm A}_n)_k ({\rm B}_m)_k$. The numerical minimization of free energies corresponding to supercrystal and homogeneous states was carried out with account of exact expressions for higher correlators of these systems calculated by us early. It is shown that both theories are in good agreement within the limit of strong fluctuation effects whereas the FH theory underestimates moderate fluctuation effects in crossover region