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

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

Fluctuation Theory of Random Copolymers

We investigate the behavior of symmetric AB random copolymer melt near the point of microphase separation in the framework of a one-loop (Hartree-Fock) approximation, that exactly solves the problem. We have shown that composition fluctuations result to complete elimination of the microphase separation predicted in the framework a mean-field approximation [1-5]. Using variational techniques the temperature dependence of the density-density correlation function and its length scales are calculated. The diagonal in the replica space correlation function is stable with respect to the off-diagonal perturbation (glass-like order parameter). We discuss the general relation between microphase separation in random copolymers and spinodal decomposition in a mixture of A and B unconnected monomer units below their critical demixing point

Microphase separation in multiblock copolymer melts: Nonconventional morphologies and two-length-scale switching

The phase behavior of AfmN(BN/2AN/2)B(1−f)mN multiblock copolymer melts is studied within the weak segregation theory. The interplay between ordering on different length scales is shown to cause dramatic changes both in the ordered phase symmetry and periodicity upon small variation of the architectural parameters of the macromolecules. Phase diagrams are presented in the (f,χN) plane (χ is the Flory-Huggins parameter) for various values of the architecture parameters n and m. Near the critical surface, i.e., for (f−0.5)2«1, such nonconventional cubic phases as the face-centered cubic (FCC), simple cubic (SC), (double) gyroid, and the so-called BCC2 (single gyroid) are found to be stable. The lamellar morphology is shown to be replaced by BCC2, FCC, or SC (depending on the structural parameters) as the most stable low-temperature phase.

Nonconventional morphologies in two-length scale block copolymer systems beyond the weak segregation theory

The order-disorder and order-order transitions (ODT and OOT) in the linear multiblock copolymers with two-length scale architecture AfmN(BN/2AN/2)nB(1−f)mN are studied under intermediate cooling below the ODT critical point where a nonconventional sequence of the OOTs was predicted previously within the weak segregation theory (WST). To describe the ordered morphologies appearing in block copolymers (BCs) under cooling, we use the pseudospectral version of the self-consistent field theory (SCFT) with some modifications providing a good convergence speed and a high precision of the solution due to using the Ng iterations and a reasonable choice of the predefined symmetries of the computation cell as well as initial guess for the iterations. The WST predicted sequence of the phase transitions is found to hold if the tails of the BCs under consideration are symmetric enough (|0.5−f|≤0.05); the quantitative agreement between the WST and SCFT phase diagrams is reasonable in a narrow (both in f and χ~ = χN) region close to the critical point, though. For |0.5−f|>0.05, a large region of the face-centered cubic phase stability is found (up to our knowledge, first within the SCFT framework) inside of the body-centered cubic phase stability region. Occurrence of the two-dimensional and three-dimensional phases with the micelles formed, unlike the conventional diblock copolymers, by the longer (rather than shorter) tails, and its relationship to the BC architecture is first described in detail. The calculated spectra of the ordered phases show that nonmonotonous temperature dependence of the secondary peak scattering intensities accompanied by their vanishing and reappearance is rather a rule than exception.

Microphase separation of diblock copolymers with amphiphilic segment

We present a statistical mechanical approach for predicting the self-assembled morphologies of amphiphilic diblock copolymers in the melt. We introduce two conformationally asymmetric linear copolymer models with a local structural asymmetry, one of a ‘‘comb-tail’’ type and another that we call ‘‘continuous jackknife model.’’ The copolymers consist of amphiphilic and ‘‘monophilic’’ (non-amphiphilic) blocks, which have different segmental volume and tend to segregate into subphases. Using a self-consistent field theory (SCFT) framework, we explore the phase diagrams for these copolymers and compare them with that known for conventional, conformationally symmetric diblock copolymers. To determine the impact of structural effects on the self-assembly of copolymer melts, copolymers with a variation in both molecular architecture and chemical composition, f, are studied for different values of the Flory–Huggins parameter, χ. The composition dependence of the phase diagrams is shown to be basically determined by the conformational asymmetry. Remarkably, the stable lamellar structures exist even in the very compositionally asymmetric case, f < ¼. An interesting geometric distinction of the ‘‘direct’’ and ‘‘inverse’’ morphologies is introduced. The presence of an internal structure is found to influence the high χ behavior, where a stable two-scale (structure-in-structure) hexagonal morphology is found to be formed for some compositions. Therefore, the local chemical structure of monomer units can dictate the global morphology of copolymer melts.

Nonmonotonic incommensurability effects in lamellar-in-lamellar self-assembled multiblock copolymers

Using the self-consistent-field theory numerical procedure we find that the period D of the lamellar-in-lamellar morphology formed in symmetric multiblock copolymer melts AmN/2(BN/2AN/2)nBmN/2 at intermediate segregations changes nonmonotonically with an increase in the relative tail length m. Therewith D reveals, as a function of the Flory χ-parameter, a drastic change in the vicinity of the internal structure formation, which can be both a drop and a rise, depending on the value of m. It is argued that the unusual behavior found is a particular case of a rather general effect of the incommensurability between the two length scales that characterize the system under consideration.