Langevin behaviour

C. Gilles, P. Bonville, K.K.W. Wong, S. Mann, Non-Langevin behaviour of the uncompensated magnetization in nanoparticles of artificial ferritin. Eur. Phys. J. B 17, 417-427 (2000)... 

Equations (28) and (29) are derived from the statistical theory based on the Gaussian statistics which describes the network behaviour if the network is not deformed beyond the limit of the applicability of the Gaussian approximation33). For long chains, this limit is close to 30 % of the maximum chain extension. For values of r, which are comparable with rmax, the force-strain dependence is usually expressed using the inverse Langevin function 33,34)... 

Local motions which occur in macromolecular systems can be probed from the diffusion process of small molecules in concentrated polymeric solutions. The translational diffusion is detected from NMR over a time scale which may vary from about 1 to 100 ms. Such a time interval corresponds to a very large number of elementary collisions and a long random path consequently, details about mechanisms of molecular jump are not disclosed from this NMR approach. However, the dynamical behaviour of small solvent molecules, immersed in a polymer melt and observed over a long time interval, permits the determination of characteristic parameters of the diffusion process. Applying the Langevin s equation, the self-diffusion coefficient Ds is defined as... 

Onsager model with generalized mechanical behaviour. It is clear from the last four equations that if the true y obeys a modified Langevin equation with memory function 2D(t), the essential transform... 

As we see, the parameter 1 results from Langevin reorientation of the polarizability ellipsoid and is always positive. The second of the above parameters, 2, corresponds to Bom s term in the Kerr effect and can be positive or negative, depending on the electric structure of the molecule. The third, the Debye parameter 3, has no counterpart in other phenomena of molecular orientation, and is specific to the non-linear dielectric behaviour of dipolar substances. 

Recently, reliable relaxation experiments at liquid/liquid have been performed by different techniques. Bonfillon Langevin (1993) measured the dilational elasticity of different surfactants (Triton X-100, SDS in water and in 0.1 M NaCl) at water/air as well as alkane/water interfaces by a modified longitudinal wave damping method (cf. Section 6.3.1). While at the water/air interface, the behaviour of Triton X-100 could be described by a... 

This phenomenon can be explained allowing for the usual presence of dodecanol in SDS solutions. At first the salt leads to an increase in surface activity (shift of about one order of magnitude of the adsorption isotherm to lower concentration), and secondly the potential impurity dodecanol, which strongly adsorbs at the interface water/air, will more or less transfer to the dodecane phase after it has been adsorbed at the water/dodecane interface. Thus, no different mechanism is needed to describe the relaxation behaviour, as done by Bonfillon Langevin (1993). 

It should be emphasized that the essence of the Rouse model is in the universal nature of the modelling of the dynamics of a connected object. The central assumption in the Rouse model is that the dynamics is governed by the interactions localized along the diain. In fact, if one assumes a linear Langevin equation for R with localized interaction, one ends up with the Rouse model in the long time-scale behaviour. To see this, consider the general form of the linearized Langevin equation... 

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. 

In an earlier development, Haward and Thackray [55] had proposed a very similar representation to describe the yield behaviour of polymers. Their model is shown schematically in Figure 12.23. The initial part of the stress-strain curve is modelled by the Hookean spring E and the yield point and subsequent strain hardening by the Eyring dashpot and the Langevin spring. Haward and Thackray relate the total strain e and the plastic strain 6a from the activated dashpot to the nominal stress a (load applied divided by initial cross-sectional area). We have... 

---