Polymers in Confined Geometry.pdf
Polymers in Confined Geometry.pdf
Polymers in Confined Geometry.pdf
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84 APPENDIX B. CALCULATION FOR THE CONFINED POLYMER<br />
the first <strong>in</strong>tegral is calculated here, the others are evaluated <strong>in</strong> the same way,<br />
s s ′<br />
s (τ−s ′ )/ld<br />
dτ<br />
dτ dx e −x <br />
s<strong>in</strong> x − π<br />
<br />
4<br />
s ′<br />
0<br />
dτ ′ K(τ − τ ′ ) = −ld<br />
s ′<br />
s<br />
= ld<br />
√2<br />
= l2 d √2<br />
= l2 d<br />
2<br />
dτ<br />
<br />
e<br />
τ/ld<br />
s ′<br />
(s−s ′ )/ld<br />
τ−s′<br />
− ld s<strong>in</strong><br />
0<br />
<br />
1<br />
√2 − e s−s′<br />
ld s<strong>in</strong><br />
s<br />
− l + e d s<strong>in</strong><br />
s<br />
ld<br />
τ − s′<br />
ld<br />
dy e −y s<strong>in</strong> y −<br />
τ<br />
− l − e d s<strong>in</strong> τ<br />
ld<br />
s ′ /ld<br />
s/ld<br />
′ s − s<br />
+<br />
ld<br />
π<br />
<br />
4<br />
′ s<br />
− π<br />
<br />
s′<br />
− l − e d s<strong>in</strong><br />
4<br />
ld<br />
<br />
dz e −z s<strong>in</strong> z<br />
<br />
− π<br />
<br />
. (B.9)<br />
4<br />
The trigonometric relations used can be found <strong>in</strong> [1]. Calculat<strong>in</strong>g the rema<strong>in</strong><strong>in</strong>g<br />
<strong>in</strong>tegrals and comb<strong>in</strong><strong>in</strong>g the results gives<br />
〈R(s) · R(s ′ <br />
)〉 = 1 − ld<br />
<br />
ss<br />
2 lp<br />
′ + l3 d<br />
2 √ <br />
e<br />
2 lp<br />
− |s−s′ <br />
|<br />
′ |s − s |<br />
ld s<strong>in</strong> +<br />
ld<br />
π<br />
<br />
4<br />
<br />
s<br />
− s<br />
l − e d s<strong>in</strong> +<br />
ld<br />
π<br />
<br />
s′ ′<br />
− s l − e d s<strong>in</strong> +<br />
4<br />
ld<br />
π<br />
<br />
+<br />
4<br />
1<br />
(B.10)<br />
√<br />
2<br />
valid for all s, s ′ .<br />
From this result the mean-square end-to-end distance is obta<strong>in</strong>ed <strong>in</strong> the l<strong>in</strong>es<br />
follow<strong>in</strong>g eq. (3.37).<br />
B.3 Parallel end-to-end distance correlation<br />
The mean-square end-to-end distance is calculated <strong>in</strong> detail, by <strong>in</strong>sert<strong>in</strong>g the<br />
derivatives of the average undulations eqs. (3.30) and (3.31) <strong>in</strong> eq. (3.40)<br />
˙R||(s) ˙ R||(s ′ ) = d2<br />
ds ds ′<br />
R||(s)R||(s ′ ) <br />
= 1 − ld<br />
+<br />
2 lp<br />
l2 d<br />
16 l2 p<br />
+ l2 d<br />
8 l2 e<br />
p<br />
−2 |s−s′ |<br />
ld s<strong>in</strong> 2<br />
′ |s − s |<br />
−<br />
ld<br />
π<br />
4<br />
<br />
=〈 ˙r ||(s) ˙r ||(s ′ )〉<br />
<br />
. (B.11)<br />
An <strong>in</strong>tegration for s ≥ s ′ yields the desired result for the projected length correlation<br />
<br />
R||(s)R||(s ′ ) s s ′<br />
= dτ dτ<br />
0 0<br />
′ R||(s) ˙ ˙ R||(s ′ ) <br />
<br />
= 1 − ld<br />
+<br />
2 lp<br />
l2 d<br />
16 l2 <br />
ss<br />
p<br />
′ + l2 d<br />
8 l2 s s ′<br />
dt dt<br />
p 0 0<br />
′ K(|τ − τ ′ |), (B.12)