the order along the z-axis is more uniform, more or less compatible with the Janus
cylinder-type morphology.
These conclusions have been strengthened by an analysis of suitable correlation
functions and structure factors [99]. These results show (Fig. 31) that a cylindrical
bottle brush is a quasi-1D object and, as expected for any kind of 1D system, from
basic principles of statistical thermodynamics, statistical fluctuations destroy any
kind of long-range order in one dimension [108]. Thus, for instance, in the lamellar
structure there cannot be a strict periodicity of local composition along the z-axis,
rather there are fluctuations in the size of the A-rich and B-rich domains; as one
proceeds along the z-axis, these fluctuations are expected to add up in a random
fashion. However, in the molecular dynamics simulations of Erukhimovich et al.
[99] no attempt could be made to study such effects quantitatively because the
backbone contour length L was not very large in comparison with the domain size l d
of an A-rich (or B-rich, respectively) domain.
Because a periodic boundary condition in the z-direction was used to eliminate
effects due to the free chain end of the backbone, axial correlations could only be
studied over distances z distinctly smaller than L/2, to avoid finite size effects.
Another complication was that the density distribution in radial direction is still
nonuniform for the rather short side chain lengths accessible in the simulations.
Also, very long-lived fluctuations occurred, which created strong deviations from
the average circular shape of the cross-section of the bottle brush in the xy-plane
perpendicular to the backbone. These fluctuation phenomena not only concern the
lamellar-like phase, where the translational symmetry along the z-axis is broken,
but they also affect the Janus cylinder structure or the phase where the cylinder
cross-section has a Janus dumbbell shape (Fig. 30). Here one expects that the
orientation of A–B interfaces randomly changes when one proceeds along the
z-axis [105, 109, 110]. This problem has been investigated by us with large scale
Monte Carlo simulations of a lattice model of bottlebrush polymers [105, 109, 110] ,
applying the pruned-enriched Rosenbluth method (PERM) [111, 112]. By this
method, precise results could be obtained for bottle brush polymers using three
choices of the side chain length (N s ¼ 6, 12, and 18) and three choices of
the backbone contour length (L ¼ 32, 48, and 64 lattice spacings, respectively),
studying various solvent conditions. Clearly, these side chain lengths are very
short, but they do correspond to the range that is relevant experimentally [93, 113].
When a phase separation into a Janus cylinder structure occurs, e.g., where the
upper half of the cylinder contains the B-rich phase and the lower half the A-rich
phase, we have a planar AB interface (Fig. 32a) and the quantity that we wish to
record is the vector normally oriented to this interface for any monomer of the
backbone. Studying the orientational correlations of this vector will yield the
desired information on possible fluctuations of interface orientation (Fig. 32b).
Since the AB interface at nonzero temperature is not a sharp dividing surface, but
rather has a finite width, a numerical characterization of the local orientation of this
interface normal is difficult. Therefore, an essentially equivalent but numerically
unambiguous characterization of this Janus cylinder-type ordering has been
Structure Formation of Polymeric Building Blocks: Complex Polymer Architectures
149
cylinder-type morphology.
These conclusions have been strengthened by an analysis of suitable correlation
functions and structure factors [99]. These results show (Fig. 31) that a cylindrical
bottle brush is a quasi-1D object and, as expected for any kind of 1D system, from
basic principles of statistical thermodynamics, statistical fluctuations destroy any
kind of long-range order in one dimension [108]. Thus, for instance, in the lamellar
structure there cannot be a strict periodicity of local composition along the z-axis,
rather there are fluctuations in the size of the A-rich and B-rich domains; as one
proceeds along the z-axis, these fluctuations are expected to add up in a random
fashion. However, in the molecular dynamics simulations of Erukhimovich et al.
[99] no attempt could be made to study such effects quantitatively because the
backbone contour length L was not very large in comparison with the domain size l d
of an A-rich (or B-rich, respectively) domain.
Because a periodic boundary condition in the z-direction was used to eliminate
effects due to the free chain end of the backbone, axial correlations could only be
studied over distances z distinctly smaller than L/2, to avoid finite size effects.
Another complication was that the density distribution in radial direction is still
nonuniform for the rather short side chain lengths accessible in the simulations.
Also, very long-lived fluctuations occurred, which created strong deviations from
the average circular shape of the cross-section of the bottle brush in the xy-plane
perpendicular to the backbone. These fluctuation phenomena not only concern the
lamellar-like phase, where the translational symmetry along the z-axis is broken,
but they also affect the Janus cylinder structure or the phase where the cylinder
cross-section has a Janus dumbbell shape (Fig. 30). Here one expects that the
orientation of A–B interfaces randomly changes when one proceeds along the
z-axis [105, 109, 110]. This problem has been investigated by us with large scale
Monte Carlo simulations of a lattice model of bottlebrush polymers [105, 109, 110] ,
applying the pruned-enriched Rosenbluth method (PERM) [111, 112]. By this
method, precise results could be obtained for bottle brush polymers using three
choices of the side chain length (N s ¼ 6, 12, and 18) and three choices of
the backbone contour length (L ¼ 32, 48, and 64 lattice spacings, respectively),
studying various solvent conditions. Clearly, these side chain lengths are very
short, but they do correspond to the range that is relevant experimentally [93, 113].
When a phase separation into a Janus cylinder structure occurs, e.g., where the
upper half of the cylinder contains the B-rich phase and the lower half the A-rich
phase, we have a planar AB interface (Fig. 32a) and the quantity that we wish to
record is the vector normally oriented to this interface for any monomer of the
backbone. Studying the orientational correlations of this vector will yield the
desired information on possible fluctuations of interface orientation (Fig. 32b).
Since the AB interface at nonzero temperature is not a sharp dividing surface, but
rather has a finite width, a numerical characterization of the local orientation of this
interface normal is difficult. Therefore, an essentially equivalent but numerically
unambiguous characterization of this Janus cylinder-type ordering has been
Structure Formation of Polymeric Building Blocks: Complex Polymer Architectures
149