Unified description of transportation of polymer chains with different topologies through a small cylindrical pore

Instead of using free energy, we directly balanced confinement and hydrodynamic forces (fc = kBT/ξ and fh = 3πηule) on individual “blobs” to obtain a unified description of how polymer chains with different topologies (linear, star and branched) pass through a cylindrical pore with a diameter of D, much smaller than its size, under a flow rate (q), where kB, T, η, ξ, u (=q/D2), and le are the Boltzmann constant, absolute temperature, viscosity, “blob” diameter, flow velocity, and the blobʹs effective length along the flow direction, respectively; and each “blob” is defined as a maximum portion of the confined chain whose confinement free energy becomes of order thermal energy (kBT). Namely, using fc = fh, we easily locate at which minimum (critical) flow rate (qc) polymer chains with different topologies are able to pass through the pore without priori consideration of chain topology, i.e., a general description, qc/qc,linear = (D/ξ)2, where qc,linear equals [kBT/(3πη)](ξ/le). The only thing left here is to find ξ for each topology. Obviously, for a confined linear chain, ξlinear = D. For a confined star chain, ξstar = [2/(f + |f−2fin|)]1/2D, where f is arm number and fin is the number of arms first inserted into the pore; and for a branched chain, ξbranch = (D/a)α’Nt,Kuhnβ’Nb,Kuhnγ’, where a is the size of one Kuhn segment, Nt,Kuhn and Nb,Kuhn are respectively the numbers of Kuhn segments of the entire branched chain and the subchain between two neighboring branching points; and the three constant exponents (α′, β′ and γ′) are directly related to the well-known Floryʹs scaling exponents between the chain size and both Nt,Kuhn and Nt,Kuhn.