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

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

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

Ordering Lamellar-Forming Copolymer Thin Films in 3D Bicontinuous Morphologies via Lamellar Patterned Substrate

The formation of ordered morphologies in thin films of symmetric diblock copolymer melts is considered theoretically. Somewhat surprisingly, under proper boundary conditions the presence of a lamellar chemical pattern on the substrate, being sufficiently pronounced and with the right period, is found to induce the formation of diamond-like morphologies. The phase diagram of the most stable phases on the plane (the substrate period Lx the film width H) is built within the self-consistent field theory numerical procedure. We also discuss the behavior of the order parameter Fourier spectrum at the transitions between the various morphologies

Microphase separation in correlated random copolymers

In this paper we present the results of a calculation of the phase diagram of a highly polydisperse multiblock copolymer in the weak segregation limit. The theory for polydisperse systems developed by Erukhimovich and Dobrynin [Erukhimovich, I.; Dobrynin, A.V. Macromol.Symp, 81, 253 (1994)] has been used. The model of the copolymer has the following characteristics: the blocklengths, as well as the molecule lengths are highly polydisperse (M(w)/M(n) = 2). The average number of blocks per molecule is very large and the effects of the finiteness of the blocklengths (the fluctuation corrections) are ignored. The resulting phase diagram shows some remarkable differences with the phase diagram of a regular monodisperse multiblock. Known differences are e.g. the order of the transition from the homogeneous state, and the variation of the period of the microstructure with the chi-parameter. Ln addition, we found a peculiar feature at the critical point: the phase boundaries have discontinuous derivatives

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

Helical, Angular and Radial Ordering in Narrow Capillaries

To enlighten the nature of the order-disorder and order-order transitions in block copolymer melts confined in narrow capillaries we analyze peculiarities of the conventional Landau weak crystallization theory of systems confined to cylindrical geometry. This phenomenological approach provides a quantitative classification of the cylindrical ordered morphologies by expansion of the order parameter spatial distribution into the eigenfunctions of the Laplace operator. The symmetry of the resulting ordered morphologies is shown to strongly depend both on the boundary conditions (wall preference) and the ratio of the cylinder radius and the wave length of the critical order parameter fluctuations, which determine the bulk ordering of the system under consideration. In particular, occurrence of the helical morphologies is a rather general consequence of the imposed cylindrical symmetry for narrow enough capillaries. We discuss also the ODT and OOT involving some other simplest morphologies. The presented results are relevant also to other ordering systems as charge-density waves appearing under addition of an ionic solute to a solvent in its critical region, weakly charged polyelectrolyte solutions in poor solvent, microemulsions etc.Comment: 6 pages, 3 figure