Reptation model diffusion coefficient

The scaling dependence of the diffusion coefficient on N and Cobs Iso poses a number of questions. While the original scaling predictions, based on reptation dynamics [26,38], oc N, have been verified by some measurements [91,98], significant discrepancies have been reported too [95,96]. Attempts to interpret existing data in terms of alternative models, e.g., by the so-called hydrodynamic scaling model [96], fail to describe observations [100,101]. 

Diffusion of flexible macromolecules in solutions and gel media has also been studied extensively [35,97]. The Zimm model for diffusion of flexible chains in polymer melts predicts that the diffusion coefficient of a flexible polymer in solution depends on polymer length to the 1/2 power, D N. This theoretical result has also been confirmed by experimental data [97,122]. The reptation theory for diffusion of flexible polymers in highly restricted environments predicts a dependence D [97,122,127]. Results of various... 

The reptation model predicts that the viscosity of a melt scales with the chain length to the third power while the diffusion coefficient decreases with the second power of the chain length. 

Leibler et al. (1991) have developed a model for this process, which they call sticky reptation. For long chains with many stickers, the self-diffusion coefficient of a sticky reptating chain turns out to be... 

The primitive chain reptates along itself with a diffusion constant that can be identified as the diffusion coefficient of the Rouse model. Under the action of a force /, the velocity of the polymer in the tube is v =f /, where is the overall friction coefficient of the chain. It is expected that C is related to the friction coefficient of the individual segments, Q, by the expression... 

The above butyl-branched alkane was studied by solid-state 13C NMR, alongside its linear analogue C198H398, to establish the solid-state diffusion coefficient.150 Both alkanes were in the once-folded form. The progressive saturation experiments have shown that the longitudinal relaxation of magnetization is consistent with a solid state chain diffusion process. Reptation and one-dimensional diffusion models were demonstrated to satisfactorily represent the data. The addition of the branch to the alkane chain was shown to result in a decrease in the diffusion coefficient, which ranged from 0.0918 nm2 s 1 for the linear chain to 0.016 nm2 s 1 for the branched chain. These diffusion coefficients are consistent with those of polyethylene. 

The simple reptation model does not properly account for all the relaxation modes of a chain confined in a tube. This manifests itself in all measures of terminal dynamics, as the longest relaxation time, diffusion coefficient and viscosity all have stronger molar mass dependences than the reptation model predicts. Tn Sections 9.4.5 and 9.6.2, more accurate ana-... 

Recall that Fig. 9.3 showed the linear viscoelastic response of a polybutadiene melt with MjM = 68. The squared term in brackets in Eq. (9.82) is the tube length fluctuation correction to the reptation time. With /i = 1.0 and NjN = 68, this correction is is 0.77. Hence, the Doi fluctuation model makes a very subtle correction to the terminal relaxation time of a typical linear polymer melt. However, this subtle correction imparts stronger molar mass dependences for relaxation time, diffusion coefficient, and viscosity. 

Both diffusion coefficient and relaxation time obey stronger power laws in chain length than predicted by the simple reptation model [Eqs (9.8) and (9.12)]. 

The three-dimensional diffusion coefficient is predicted by the simple reptation model to be reciprocally proportional to the square of polymer molar mass ... 

Experimental values of the molar mass exponent close to 2 have been obtained. For example, for poly (methyl methacrylate), a value of 2.45 has found (see P. Prentice, Polymer, 1983, 24, 344-350). As with values of self-diffusion coefficient, this has been regarded as close enough to 2 for reptation to be considered a good model of the molecular motion occurring at the crack tip. 

Brochard and de Gennes [53] proposed a relaxation-controlled model for the dissolution of polymer droplets. When a droplet of polymer solution, of concentration Oo, is immersed in a solvent, two processes control the dissolution. The first step relates to the swelling of the polymer network by the solvent. This was assumed to be controlled by the cooperative diffusion coefficient, Dcoop- The second step corresponds to the viscous yield of the network and is controlled by the reptation time of the polymer, t,ep. The expression for the net solvent flux in such a system is given by... 

Thus, the reptation model predicts that Dg decreases as N as the number N of monomers in the chain grows. When N is quite large, the diffusion coefficient is very low. As a result, if you bring two polymer melts together, they will tend to intermingle very slowly, even if the thermodynamics suggests that the mixed state is the most favorable one (i.e., if the two polymers are miscible). 

The reptation model is more powerful than you might think. You can get much more out of it than just the simplest basic laws for the viscosity, the longest relaxation time, and the diffusion coefficient of a chain in a polymer melt. This model allows you to describe, for instance, the relaxation of a pol mier after a stress has been released, or the response to a periodic force. As a result, you gain a fairly complete picture of the dynamics of polymer liquids, and of their viscoelasticity in particular. 

The objectives of the present research were (i) to develop a solvent transport model accounting for diffusional and relaxational mechanisms, in addition to effects of the viscoelastic properties of the polymer on the dissolution behavior (ii) to perform a molecular analysis of the polymer chain disentanglement mechanism, and study the influence of various molecular parameters like the reptation diffusion coefficient, the disentanglement rate and die gel layer thickness on the phenomenon and (iii) to experimentally characterize the dissolution phenomenon by measuring the temporal evolution of the various fronts in the problem. 

This experiment thus affords two independent ways to test the reptation, or any other, model of polymer diffusion. At long times we can extract the center-of-mass diffusion coefficient (from a plot of I vs. t ) and determine, for... 

Relaxation 67,70,96,99,111,155 Reptation model 1,24,42 Resolution 14 Resonance NSE 20 Rheology 35,55 Rotational isomeric state 118 Rotational transitions 117 Rouse diffusion coefficient 28,42, 175 Rouse model 24-26,30-35,38, 117, 119, 142, 193,200 —, generalized 47 Rouse time 27 RPA 162, 163, 199... 

Both models can thus explain the low exponent of 1.3 for the scaling law for and for both cases reptation is no longer necessary for the relaxation of stress. The mechanisms could probably be distinguished by the concentration dependence on the self-diffusion coefficients A of the surfactant molecules. In a solution with a connected network, a surfactant molecule should be in the same situation as in a bicontinuous L3 phase, where is independent of the surfactant concentration. However, Kato and co-workers have shown that the values increase with concentration for a solution with rod-like micelles thus, a diffusion-limited bond-interchange mechanism is more likely for the explanation of the scaling law of the structural relaxation time than the assumption of connected networks consisting of thread-like micelles. 

The process of disentangling, as it is envisaged in the reptation model, is sketched in Fig. 6.11. The motion of the primitive chain , the name given to the dynamic object associated with the primitive path, is described as a diffusion along its contour, that is to say, a reptation . The associated curvilinear diffusion coefficient can be derived from the Einstein relation, which holds generally, independent of the dimension or the topology. Denoting it D, we have... 

Hence, according to the reptation model, the transition from a non-entangled to an entangled polymer melt should be accompanied by a change in the exponent of the power law for the diffusion coefficient, D M, from i/ = —1 to = —2. 

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