Polymers in Confined Geometry.pdf

82 APPENDIX B. CALCULATION FOR THE CONFINED POLYMER
(−1 + i)l −1
d
(−1 − i)l −1
d
Im k
(1 + i)l −1
d
(1 − i)l −1
d
path of integration
for s > s ′
Re k
path of integration
for s < s ′
Figure B.1: Complex plane of the function ˜ f(k) with indicated poles.
Inserting the residues, we obtain for s > s ′
〈x(s)x(s ′ )〉 = kBT l3 ⎧ ⎫
s − s′
s − s′
⎪⎨ exp i exp −i ⎪⎬
d ld
ld
s − s′
+
exp −
8κ ⎪⎩
1 + i
1 − i ⎪⎭
ld
= kBT l3 d
4 √ 2κ exp
′
s − s′ s − s
− cos −
ld
ld
π
(B.4)
4
which—by additionally closing the contour for s < s ′ in the lower half plane—
results in
〈x(s)x(s ′ )〉 = kBT l3 d
4 √ 2κ exp
− |s − s′ ′ | |s − s |
cos −
ld
ld
π
. (B.5)
4
This is the result in eq. (3.26) used for further calculations.
B.2 End-to-end distance correlation
Here we present the details of the calculation for the mean-square end-to-end
distance 〈R2 (L)〉. We start by calculating the distance between a point r(s) on
the contour and the origin of the polymer r(0), given by the integral
R(s) = r(s) − r(0) =
s
0
dτ t(τ). (B.6)