Polymer brushes in cylindrical pores: Simulation versus scaling theory

The structure of flexible polymers endgrafted in cylindrical pores of diameter D is studied as a function of chain length N and grafting density sigma, assuming good solvent conditions. A phenomenological scaling theory, describing the variation of the linear dimensions of the chains with sigma, is developed and tested by molecular dynamics simulations of a bead-spring model. Different regimes are identified, depending on the ratio of D to the size of a free polymer N-3/5. For D > N-3/5 a crossover occurs for sigma=sigma(*)=N-6/5 from the "mushroom" behavior (R-gx=R-gy=R-gz=N-3/5) to the behavior of a flat brush (R-gz=sigma N-1/3,R-gx=R-gy=sigma N--1/12(1/2)), until at sigma(**)=(D/N)(3) a crossover to a compressed state of the brush, [R-gz=D,R-gx=R-gy=((ND)-D-3/4 sigma)(1/8)< D], occurs. Here coordinates are chosen so that the y axis is parallel to the tube axis, and the z direction normal to the wall of the pore at the grafting site. For D < N-3/5, the coil structure in the dilute regime is a cigar of length R-gy=ND-2/3 along the tube axis. At sigma(*)=(ND1/3)(-1) the structure crosses over to "compressed cigars," of size R-gy=(sigma D)(-1). While for ultrathin cylinders (D < N-1/4) this regime extends up to the regime where the pore is filled densely (sigma=D/N), for N-1/4 < D < N-1/2 a further crossover occurs at sigma(***)=D-9/7N-3/7 to a semidilute regime where R-gy=((ND)-D-3/4 sigma)(1/8) still exceeds D. For moderately wide tubes (N-1/2 < D < N-3/5) a further crossover occurs at sigma(****)=(ND-7)-D-3, where all chain linear dimensions are equal, to the regime of compressed brush. These predictions are compared to the computer simulations. From the latter, extensive results on monomer density and free chain end distributions are also obtained, and a discussion of pertinent theories is given. In particular, it is shown that for large D the brush height is an increasing function of D-1.