Langevin chain

For fast flow deformations of polymer fluids, a non-linear theory of chain deformation and orientation is considered. To account for non-hnear effects and finite chain extensibility, inverse Langevin chain statistics is assumed. Time evolution of chain distribution function in the systems with inverse Langevin chain statistics has been discussed in earlier papers [12,13] providing physically sensible stress-orientation behaviour in the entire range of the deformation rates and chain deformations. 

Fig. 1.5 (a) Tension-elongation curve of the Langevin chain (solid line) and its Gaussian approximation... 

Fig. 4.7 Tension-elongation curve of a cross-linked rubber. Experimental data (circles), affine network theory by Gaussian chain (broken line), affine network theory (4.107) by Langevin chain (solid line).

The assumption of Gaussian chains in the affine network theory can be removed by using nonlinear chains, such as the RF model (Langevin chain), stiff chain model (KP chain), etc. These models show enhanced stress in the high-stretching region. The effect of nonlinear stretching will be detailed in Section 4.6. 

The elastic free energy of an affine network made up of such Langevin chains is written as... 

Due to the nonlinear property of a Langevin chain, the S-shaped tension-elongation profile as shown in Figure 4.7 (solid line with data points) is well reproduced by this theory. 

The earliest and simplest approach in this direction starts from Langevin equations with solutions comprising a spectrum of relaxation modes [1-4], Special features are the incorporation of entropic forces (Rouse model, [6]) which relax fluctuations of reduced entropy, and of hydrodynamic interactions (Zimm model, [7]) which couple segmental motions via long-range backflow fields in polymer solutions, and the inclusion of topological constraints or entanglements (reptation or tube model, [8-10]) which are mutually imposed within a dense ensemble of chains. 

In summary, the chain dynamics for short times, where entanglement effects do not yet play a role, are excellently described by the picture of Langevin dynamics with entropic restoring forces. The Rouse model quantitatively describes (1) the Q-dependence of the characteristic relaxation rate, (2) the spectral form of both the self- and the pair correlation, and (3) it establishes the correct relation to the macroscopic viscosity. 

Some years ago, on the basis of the excluded-volume interaction of chains, Hess [49] presented a generalized Rouse model in order to treat consistently the dynamics of entangled polymeric liquids. The theory treats a generalized Langevin equation where the entanglement friction function appears as a kernel... 

As reported in [22] and [29], Fig. 2 shows a typical sequence of images depicting nucleation of a lamella by a single chain of N = 700 beads as obtained from Langevin dynamics simulations. The chain is quenched to T = 9.0... 

The equation of motion for the position vector Ra,- of the ith segment of the chain a at time t is given by the Langevin equation,... 

Since l is proportional to and q is proportional to 1/L, i is proportional to. Substitution of Eq. (67) into Eq.(62) gives the Langevin equation for the Rouse modes of the chain within the approximations of preaveraging for hydrodynamic interactions and mode-mode decoupling for intersegment potential interactions. Equation (62) yields the following results for relaxation times and various dynamical correlation functions. 

Analogous to the derivation of the effective Langevin equation for the chain in dilute solutions, we get in semidilute solutions... 

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