Neutron Scattering

15.3 .1 Entropie forces - the Rouse model
As the simplest model for chair relaxation, the Rouse model considers a Gaussian chair in a
heat bath. The building blocks of such a Gaussian chair are segments consisting of several
monomers, so that their end to end distance follows a Gaussian distribution. Their
conformations are described by vectors ami , = r, - r,+1 along the chair. Thereby r is the
position vector of the segment "n" . The chair is described by a succession of freely
connected segments of length f . We are interested in the motion ofthese segments on a length
scale
f < r < R e, where R e"2 = n P Z is the end to end distance ofthe chair. The motion is described by
a Langevin equation
~o dn =V
"F (rn) + J n(t) ,
where ;o is the monomerc friction coefficient . For the stochastic force f(t) we have
(fn(t»=0 and (fna(t)fmß( 0» =2kBT a, and /3 denote the
Cartesian components of r . F(r,d is the free energy of the polymer chair . The force terni in
Eq .[15 .10] is dominated by the conformational entropy of the chair
S=k,~n W({rn })
where W (lLnl) is the probability for a chair conformation lrn} of a Gaussian chair of n-
segments .
3/2
expp
{117-7 }
j -3 (Li
2 Ë Z
With the boundary conditions offorce free ends Eq .[15 .10] is readily solved by cosine Fourier
transformation, resulting in a spectrum of normal modes . These solutions are similar to e.g .
the transverse vibrational modes of a linear chair except that relaxational motions are
involved instead of periodic vibrations . The dispersion ofthe relaxation rates 11r is qua dratic
in the number ofknots p along the chair.
15-13