Helical, angular and radial ordering in narrow capillaries

Abstract

Abstract. -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 dimensionless parameter q * R, where R is the cylinder radius and q * is the wave number 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.Introduction. -During the last years there has been large interest in the behavior of block copolymer (basically diblock copolymer) melts confined in narrow capillaries, which were studied both experimentally [1-6] and via computer simulation [7][8][9] as well as by numerical calculations within the self-consistent field theory [1,10,11]. The most characteristic of these morphologies are lamellae ordered normal to the cylinder axis z (slab morphology [10] ) and coaxial shells or multiwall morphologies [7,10] as well as various helices [1,8,9], the morphologies' symmetry being strongly depended on the ratio of the gyration radius R G of the diblock macromolecules under consideration and the radius R of the cylinder the macromolecules are confined to. However, some important issues have not been addressed yet. In particular, the various morphologies (up to 20 in ref [11] and up to 30 in ref [9]) are distinguished only visually by their snapshots and no attempts to relate them to a quantitative order parameter are made until now. Meanwhile, without a quantitative definition of an order parameter and, accordingly, finding the quantitative change of such an order parameter at the phase transitions

Supramolecular Organization of Polymeric Materials in Nanoporous Hard Templates

“…It is again remarkable that confinement-induced chiral structures are generated in systems containing only achiral components. As pointed out by Erukhimovich and Johner [218], the free energies of both enantiomeric morphologies are identical so that statistically equally frequent formation of right-handed and left-handed helices is to be expected.…”

“…This quantity was, for example, defined as D p /L 0 , where L 0 is the bulk period of the model BCP [213]. Erukhimovich and Johner, who attempted to put confined structure formation of BCPs into a more general context, suggested a dimensionless parameter that is the product of D p and the wave number of fluctuations of a critical order parameter that determines the ordering in the corresponding bulk system [218]. Whereas the critical order parameter in the case of an order-disorder transition in a BCP melt is related to the concentration of the repeat units of its constituents, the more general definition of a reduced pore diameter allows the treatment of ordering in cylindrically confined systems to be extended to a broader range of structure formation processes.…”

“…However, apart from visualizing morphology snapshots, only a few attempts have been made to develop classical phase diagrams based on the relation of the body of identified morphologies to a quantitative order parameter. Erukhimovich and Johner derived reduced phase diagrams in which, for example, phase boundaries were displayed in the reduced temperaturereduced D p plane for different surface field strengths [218]. Here, the reduced D p -value is, as mentioned above, the product of D p and the wave number of fluctuations of a critical order parameter determining the ordering in the corresponding bulk system, and the reduced temperature is related to the bulk order-disorder transition.…”

Mesophase formation in two-component cylindrical bottlebrush polymers

“…It is then also interesting to ask how such a situation compares to the case where such block copolymers fill a cylinder (at the same density) without constraint on the locations of the A-B junctions. Of course, the latter problem has already been studied in other contexts [62][63][64][65][66][67][68][69][70][71][72][73][74][75][76]. Here we generalize the treatment of Ref.…”

“…Here we generalize the treatment of Ref. [75] to take this constraint on the location of the A-B junctions into account to formulate a weak segregation theory [77,78] of ordering in binary bottle-brushes (Sec. 2).…”