Polymers in Confined Geometry.pdf

Appendix B
Calculation for the confined
polymer
B.1 Undulation correlations
The Fourier back transform of the undulation correlation function, eq. (3.25),
into real space is the integral
〈x(s)x(s ′
dk dk′
)〉 = eiks
2π 2π e−ik′ s ′
〈˜x(k)˜x ∗ (k ′ )〉
dk kBT e
=
2π κ
ik(s−s′ )
k4 + 4l −4 , (B.1)
d
integrating out the δ-function in eq. (3.25).
For evaluation of the remaining integral, we can use the complex analysis. We
have to sum over the residues of the integral kernel, hence we need to find the
poles of ˜ f(k). Expanding the denominator, the zeros are found
˜f(k)
k 4 + 4l −4
d = k 2 + i2l −2
2 −2
d k − i2ld
= k − (1 − i)l −1
−1
d k + (1 − i)ld · k + (1 + i)l −1
−1
d k − (1 + i)ld . (B.2)
These poles, as well as the path of integration, are shown in figure B.1. The
integration over the real axis is done, for s > s ′ , by closing the contour in the
upper half complex plane, where the exponential suppresses any contribution of
the closed path.
The Fourier transformation is then given by
〈x(s)x(s ′ )〉 = kBT
2πκ 2πi
81
Res ˜ f(k) + Res
k=−1+i
˜ f(k)
k=1+i
. (B.3)