Next: Appendix A Up: The Gaussian chain Previous: Green's function method

# One Gaussian chain in a box

As an example of the use of the Green's function method, we calculate the pressure exerted by a Gaussian chain on the walls of a confining box .

The potential is zero everywhere inside the box, and infinite everywhere outside the box. Then at the walls of the box. Solving Eq. (3.18) yields

 (3.21)

 (3.22)

where

 (3.23)

 (3.24)

 Z = (3.25) Zi = (3.26)

Introducing the eigenfunctions and eigenvalues from Eqs. (3.23) and (3.24) we get

 (3.27)

where the prime at the summation sign indicates that only odd n should occur in the sum.

Now look at two limits

i.

The polymer is much smaller than the box
 Zi = (3.28) Z = (3.29)

The pressure on wall one is

 (3.30)

i.e. independent of the wall number, and equal to the ideal gas result.

ii.

The polymer is very much constrained by the box

 (3.31)

 P1 = (3.32) = (3.33)

In this case the pressure on the different walls depends on the size of the box orthogonal to the wall.

Next: Appendix A Up: The Gaussian chain Previous: Green's function method
W.J. Briels