Rouse equation of motion

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... 

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. 

We can take the Rouse term l/ ke 02rm/0m2 (ke = 3kBT//2) entropic spring constant) into consideration formally, if we define the element Tnm of the Oseen tensor as Tnm = E/ . The equation of motion (13) thus becomes... 

The equations of motion (75) can also be solved for polymers in good solvents. Averaging the Oseen tensor over the equilibrium segment distribution then gives = l/ n — m Y t 1 = p3v/rz and Dz kBT/r sNY are obtained for the relaxation times and the diffusion constant. The same relations as (80) and (82) follow as a function of the end-to-end distance with slightly altered numerical factors. In the same way, a solution of equations of motion (75), without any orientational averaging of the hydrodynamic field, merely leads to slightly modified numerical factors [35], In conclusion, Table 4 summarizes the essential assertions for the Zimm and Rouse model and compares them. 

For the Rouse model shown in Fig. 2.1, the equations of motion of frictional elements are given as... 

Similar to the elastic dumbbell case, the equation of motion for any bead in the Rouse chain is written as... 

In the equations of motion for the beads in Eq. (11.4), it seems reasonable to assume that the bead-acceleration terms on the left side (which have the general appearance of a substantial derivative ) are negligibly small compared to the individual terms on the right side. This assumption has been discussed in several publications [15-19]. When the acceleration terms are omitted, the bead equation of motion reduces to a simple force balance used in the publications of Kramers [1], Kirkwood [2], Rouse [3], Zimm [4], and others ... 

Polymer melt relaxation on these length and time scales and for chains below the entanglement molecular weight, as in our case, is typically analyzed within the Rouse model. The equation of motion for this chain of phantom beads... 

Note that v is the number density of Rouse segments in chains of N segments in length and that are monodisperse. The indexp is the eigenmode from the solution to the equation of motion. Furthermore, equation 65 is the equation for a special form of the generalized Maxwell model having constant coefficients Gj = vkBT (see eq. 18). The relaxation times are given by Xr g/ p. ... 

The Rouse model is defined by the equation of motion (eqn [13] or [14]) without inertia with the simplest possible potential ... 

Using Eq. 59 and the equation of motion for Qi(f), it is a simple matter to compute the mean-square displacement of the center of mass of the Rouse chain at time t, averaged over the fluctuating forces ... 

In the Zimm model (see Fig. 2A) the hydrodynamic interactions are included by employing the Oseen tensor Him the tensor describes how the mth bead affects the motion of the /th bead. This leads to equations of motion that are not Unear anymore and that require numerical methods for their solution. In order to simplify the picture, the Oseen tensor is often used in its preaveraged form, in which one replaces the operator by its equiUb-rium average value [5]. For chains in -solvents, this leads for the normal modes to equations similar to the Rouse ones, the only difference residing in the values of the relaxation times. An important change in behavior concerns the maximum relaxation time Tchain> which in the Zimm model depends on N as and implies a speed-up in relaxation compared to the Rouse model. Accordingly, the zero shear viscosity decreases in the Zimm model and scales as Also, in the Zimm model the diffusion coefficient... 

Formally, the expression for the eigenvalues has the same form as that for the linear Rouse chain. Now, however, the intrachain phase shift rjr has to be coupled to the triple k = (fci, 2. ki). The latter can be easily determined by inserting the equations of motion for non-junction chain beads, Eq. 79, into the periodic boundary conditions, Eqs. 81 to 83. It brings us again to Eq. 75 for ki, k2, and k. ... 

To summarize, the relaxation times (or eigenvalues) of a rather complex system such as a 3-D topologically-regular network end-Unked from Rouse chains were determined analytically. In fact, one can do even better it is possible to construct all of the eigenfunctions of the network analytically (which amounts to the transformation from Cartesian coordinates to normal coordinates). Briefly, to construct the normal mode transformation, see Eqs. 84 and 85, one has to combine the Langevin equations of motion of a network jimction, Eq. 80, and the boundary conditions in the network junctions, Eqs. 87 to 92. After some algebra one finds [25,66] ... 

In the case of the Rouse model, the calculation first yields an equation of evolution for <(r, — jj) > (t), then leading to S,(q). This is similar in the case of the reptation model using the disengagement equation of motion, but this latter concerns only the third process of the Doi-Edwards model this requires to take into account ... 

Equation of Motion for the Normal Coordinates in the Rouse Model... 

The MD simulation eoupled with the Langevin thermostat simulates Rouse dynamics of a polymer chain. The Rouse relaxation time scales with the number of monomers on a chain as and it is necessary to perform at least cN (where constant c depends on the value of the integration time step At) integrations of the equation of motion for a ehain to completely renew its eonfiguration. During each time step, At, N N — l)/2 calculations of forces between monomers are performed. The CPU time required to do cN integrations of the equations of motion will grow with the number of monomers on a chain as xmd — N - Thus, the computational efficiency of MD simulation has the same N dependence as a MC simulation with only local moves. 

The Rouse modes of motion of a molecule whose ends are fixed were examined by Mooney, who found that the relaxation spectrum was the same as that corresponding to equations 1 to 4 with v substituted for p/M except for an additional contribution to the modulus with infinite relaxation time and magnitude vkT. Thus, the relaxation and dynamic moduli are given by... 

In the Rouse model N monomers (beads) are coupled to each other via harmonic springs [16, 17, 20], As is well-known, the forces are of entropic origin. It is customary to revert to a continuous picture in which n, the bead s running number, takes real values. For a detailed discussion see Doi and Edwards [17]. The Langevin equations of motion for such a polymer in the MdM flow field are... 

The equations of motion of the Rouse-chain are formulated while neglecting all inertial effects. Then the velocity dr//dt of the bead I is given by... 

We finish this section with the schematic drawing displayed in Fig. 6.4, meant to indicate how the time dependent fluctuations of the amplitude of a Rouse-mode could look-like. The interaction of a chain with its surroundings leads to excitations of this mode at random times. In-between, the mode amplitude decreases exponentially with a characteristic relaxation time as described by the equation of motion. These are the only parts in the time dependent curve which show a well-defined specific behavior the excitations occur irregularly during much shorter times. We may therefore anticipate that the shape of the time correlation function is solely determined by the repeated periods of exponential decay. Regarding the results of this section, we thus may formulate directly the time correlation function for the normal coordinate... 

The coupled equations of motion can be decoupled by going to normal modes. These variables, the Rouse modes, are defined by... 

The important difference between the Rouse and Zimm equations of motion comes from the nonlocality of the mobility tensor... 

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. 

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