Rouse equation

It is worth noting that for the limiting case of a=l, the X0-Mw-c relationship exhibits the same proportionalities as obtained for the Rouse equation [Eq. (18)]. On the basis of Eq. (25) it is possible to obtain a standardisation of the relaxation time if the product i0 c1+1/a is plotted against the overlap parameter, c-[r ],as shown in Fig. 12. 

Equations (35) and (36) define the entanglement friction function in the generalized Rouse equation (34) which now can be solved by Fourier transformation, yielding the frequency-dependent correlators . In order to calculate the dynamic structure factor following Eq. (32), the time-dependent correlators are needed. 

In order to calculate the effects of CLF we have to ask how the fraction of monomers that is released through CLF at the chain ends grows with time. It has been recently shown that for Ktr the effect of reptation on escaping from the tube is negligible in comparison to CLF [90]. It is the first passage of a chain end that is assumed to relax the constraint of a tube segment on a chain. From the scale invariance of the Rouse equation (Eq. 3.7) an exact asymptotic result... 

The Fourier transformed position coordinate x q) thereby is taken from Eq. 3.8. Having evaluated the force we now can reformulate the Rouse equation (Eq. 3.6) introducing the new force term of Eq. 5.5. The Fourier-transformed Rouse equation then reads ... 

Thereby, U is a numerical constant in the order of 1. The sign describes the direction of the net force, which is dictated by the direction of strain propagation along the chain. After Fourier transformation Eq. 5.12 may be introduced into the modified Rouse equation (Eq. 5.6) yielding ... 

The combination of enhancement terms for the molecular friction coefficient from Eqs.(6.22) and (6.26) with the Rouse equation for viscosity yields... 

However tempting it may be, further physical exploitation of the above results must be tempered by the realization that both the Rouse and the freely jointed chain models are in some sense artificial. We have nevertheless extended, somewhat beyond the ball-and-spring concept, the validity of the Rouse equations, and the prospect of developing the special case a = 0 for nonlinear phenomena is not without possible phenomenological interest. 

Demonstrate that the Rouse equations for different normal modes are decoupled from each other with the form... 

Therefore, we obtain the so-called Rouse equation in the continuous limit. 

On the whole, the three assumptions lead us to a very simple Rouse equation (VI.S) describing the relaxation of an elongated state of the chain. Because it is a linear equation, the solutions can be analyzed in terms of eigenmodes... 

On the whole, we are not even sure that the very concept of modes retains a fundamental validity when the three corrections are included. On the original Rouse equation [eq. (VI.5)], modes emerged naturally because the equation was linear. However, if we incorporate the backflow corrections properly, the mobility matrix /u. becomes a function of the distance r - r the equation is then nonlinear, and all modes get mixed. The result is seen more easily on what experts in mechanics call a spectrum of relaxation times (or rates).This is shown in Fig. VI.6. 

The above equation is known as the Rouse equation (Rouse 1953), when there are no intersegment potential interactions (Gaussian chain). Now the dynamics is equivalent to that of a vibrating string. By adopting the continuous chain representation, il = s,0 < s < L, whereL = Nl is the contour length of the chain, the Rouse equation for a Gaussian chain becomes... 

This velocity at the ith segment is converted to the local force by multiplying by Adding this additional hydrodynamic force to the Rouse equation (Equation 7.24), we get... 

Consider a chain of N monomers whose configuration is described by the position of monomer i. The Rouse equation of motion is a classical Langevin equation... 

Within this preaveraging approximation, the Zimm equation of motion becomes linear and can be studied by the same methods as the Rouse equation. The main results are summarized here. 

Another excursion to chain dynamics provides this final model. The dynamics of a single chain is described by the Rouse equation, and we use the Rouse theory in its continuous version where the chain variables can be brought in indirectly. The Rouse equation for the probability is given by the functional differential equation... 

In the Rouse equation there are now different length scales via the q"s and, since the differential equation in terms of the RgS is diagonal, they can be studied separately. Now we can use the above equations to divide the q"s into two classes of length scales. For this purpose consider the physical meaning of R s) in terms of its Fourier expansion. i (s) defines the contour of the polymer in real space. Since all terms in Q are summed up to infinity over all modes, this is a function which is capable of being arbitrarily abrupt in its changes of direction. If we now delete the range of q values, and consider... 

The Rouse equation of motion can be treated as a special case of Eq. 100. To show this, the memory function given in Eq. 106 is subjected to the following approximations ... 

Accounting for this and Eq. 103, Eq. 100) can be shown to coincide with the Rouse equation of motion given by Eq. 45. 

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