Polymers in Confined Geometry.pdf

54 CHAPTER 4. SIMULATION METHODS
For stiff and/or confined polymers this is also a quantity, which can be
experimentally measured on fluorescently dyed polymers (e.q. [44, 38]).
mean (square) parallel end-to-end distance
2
R|| , R|| :
The parallel end-to-end distance is the component of the end-to-end distance
along the confining ‘tube’ (here z-axis). So this only makes sense in
connection with the simulation in confined geometries. In the unconfined
case the simple relation R2
2
|| = 〈R 〉 /3 must hold.
The parallel end-to-end distance is given by
R|| =
N
i=1
ˆti . (4.34)
z
This quantity should coincide with the end-to-end distance in the very
strong confined limit, due to the orientation along the axis. It is more of
theoretical interest, since it can be used to explain some interesting global
features (cf. section 5.2.4).
apparent end-to-end distances 〈Ra〉, 〈R2 a〉,
2
Ra|| , Ra|| :
Another end-to-end distance is important, when experiments of flexible
polymers in confined environments are investigated (e.g. [44, 38]). Since on
fluorescently dyed polymers, the real ends are difficult to discern from the
rest of the chain. Only a blurry picture is seen as depicted in figure 1.2.
Therefore the apparent end-to-end distance is introduced, which is the maximal
distance between two beads along the chain. The distance between
two beads i, j at positions Ri, Rj is defined as
dij = Rj − Ri
(4.35)
so that in the same spirit as before, apparent and the apparent parallel
end-to-end distance are defined as
mean-square (parallel) radius of gyration R 2 g
Ra = max |dij| , (4.36a)
i,j
Ra|| = max
i,j (dij)z. (4.36b)
A scaling analysis and comparison to experimental data is provided in section
5.2.3.
2 , Rg|| :
The radius of gyration can be calculated directly as given in eq. (2.7).
Where the form with the center of mass Rcm is preferred since it only
involves order 2N calculations not order N 2 .