Reptation, test

Experimentally, the maximum achieved value for t is only for = 3 10 chains, which is too short for the reptation test. For = 10 and 7 -10, it is T /10. The reduced squared radius variation appears close to an exponential in all cases, and the decay times are close to the estimated (and also measured for M = 7 -10 ) stress relaxation time. But this is not a key test of the reptation theory for example, the Rouse model predicts also the same time variation of both quantities at long times (within a factor two for the terminal time). 

We present here a simple experiment, conceived to test both the reptation model and the minor chain model, by Welp et al. [50] and Agrawal et al. [51-53]. Consider the HDH/DHD interface formed with two layers of polystyrene with chain architectures shown in Fig. 5. In one of the layers, the central 50% of the chain is deuterated. This constitutes a triblock copolymer of labeled and normal polystyrene, which is, denoted HDH. In the second layer, the labeling has been reversed so that the two end fractions of the chain are deuterated, denoted by DHD. At temperatures above the glass transition temperature of the polystyrene ( 100°C), the polymer chains begin to interdiffuse across the... 

The reptation model tests through diffusion measurements in linear polymer melts... 

For 100 < P < 1000, the measured diffusion coefficients for N = P no longer follow the N 2 reptation prediction. In the same range of N values, D remains proportional to N 2 if P N, i.e. if the motion of the chains surrounding the test chain are frozen down during the diffusion time of the test chain. The comparison of the data obtained with N = P and with N P clearly puts into evidence the acceleration of the dynamics associated with the matrix chains, similarly to what has yet been observed with other polymers [11, 12, 42 to 44] or in solutions [10]. This acceleration, by a factor close to three, can be attributed to the constraint release mechanism [7, 8, 13], the effects of fluctuations of the test chain inside its tube [9] being a priori the same in the two situations P = N and P N. 

From the theoretical point of view, the first refinement of the reptation approach has been to introduce the collective dynamics of the chains in terms of the constraint release process[7, 8, 13]. Due to the motions of the surrounding chains, some constraints which constitute the tube may disappear during one reptation lime, and thus give more freedom to the test chain. Quantitative attempts have been made to take into account these additional... 

Problems 3.7 through 3.11 test your ability to work with reptation ideas and the Doi-Edwards equation. 

While these functions have been adjusted to describe shear and uniaxial extensional flows, they seem to work poorly for planar extension of LDPE (Samurkas et al. 1989). Planar extensional flow represents a particularly difficult test for K-BKZ-type constitutive equations, since fits to shear data fix all the model parameters required for planar extension, and there is therefore no wiggle room left to obtain a fit to the latter. (This is because I = I2 in both shear and planar extension.) A recent non-K-BKZ molecular constitutive equation derived from reptation-related ideas shows improved qualitative agreement with planar extensional data (McLeish and Larson 1998). 

Although motional averaging might occur in ways other than that envisioned by Cates, temperature-jump experiments have yielded values of Tbr that indicate Tbr < in the region where the relaxation is nearly monoexponential, in agreement with Cates theory. In addition, Cates theory offers distinctive predictions for the concentration-dependencies of the viscoelastic behavior these allow the theory to be tested rather stringently. To obtain these predictions, we note that in the semi-dilute regime, the mean-field reptation time is L 4>, where 0 is the volume fraction of surfactant. Hence, from Eqs. (12-31) and... 

The stress relaxation modulus then decays exponentially at the reptation time [Eq. (9.22)]. The terminal relaxation time can be measured quite precisely in linear viscoelastic experiments. Hence, Eq. (9.82) provides the simplest direct means of testing the Doi fluctuation model and evaluating... 

In trying to answer these questions it is certainly instructive to examine previous fundamental experimental works in which such thin films have been employed. In particular, we have to mention studies on polymer interdiffusion in which thin polymer films were used for testing theoretical concepts of polymer physics [102-106]. Spin-coating was typically used to prepare the films because this process presents an easy way of obtaining smooth film of precisely controllable thickness, even in the nanometer range. This is an essential criterion for investigating, for example, polymer diffusion across an interface between two films. While these experiments successfully supported the model of chain reptation [102-106], they also indicated some deviations that hinted, for example, at the enrichment of chain ends at surfaces [104, 106]. 

Obviously, if found experimentally, the exponent -2 is only a sign of reptation motion but not a sufficient condition for it. Thus, in order to prove that reptation is a really dominant mode of chain motion in polymer concentrates, we have to test it not only with self-diffusion but also with other physical properties which reflect the local motion of polymer chains. One such property is the (coherent) dynamic structure factor 5(fc, r) (see Section 3.2 of Chapter 4 for its definition). In fact, it was predicted theoretically [45-47] that the k dependence of its decay with r in the range of k defined by... 

At present, no reported data on ring self-diffusion in polymer concentrates are available other than those of Mills et al. and no theory of this subject exists other than Klein s. Thus we see a virgin field of research open before us. What seems most needed is experimental data for self-diffusion in the melt and concentrated solutions of rings. Diffusion of linear chains in ring chain matrices should also be instructive, as pointed out by Mills et al. The reptation idea now dominating the study of polymer self-diffusion will face crucial tests when accurate and systematic diffusion data on these systems become available. 

Figure 4.19. The tube model. The effect of the surrounding chains on any test chain (shown bold) is to confine its motion to a tube (shown shaded). The test chain can move only by wriggling along the tube - this motion is called reptation .

Reptation assumes that the mobility of the matrix polymer plays no role in the relaxation of the tube constraint felt by a test molecule. However, if the matrix chains were very much more mobile than the test chain, additional lateral motion of the chain might be permitted by virtue of the constraining chains themselves moving away. This type of motion is called constraint release or tube renewal , and may be operative if some of the matrix chains are significantly shorter or intrinsically more mobile than the test chain (Green 1991, Composto etal. 1992). 

In the previous chapter, we discussed the dynamics of a polymer in a fixed network. We shall now discuss the polymer dynamics in concentrated solutions and melts. In these systems, though aU polymers are moving simultaneously it can be argu that the reptation picture will also hold. Consider the motion of a certain test polymer arbitrarily chosen in melts. If the test polymer moves perpendicularly to its own contour, it drags many other chains surrounother hand the movement of the test polymer along its contour will be much easier. It will be thus plausible to assume that the polymer is confined in a tube-like region, and the major mode of the dynamics is reptation. 

Yu et al. (2011) studied rheology and phase separation of polymer blends with weak dynamic asymmetry ((poly(Me methacrylate)/poly(styrene-co-maleic anhydride)). They showed that the failure of methods, such as the time-temperature superposition principle in isothermal experiments or the deviation of the storage modulus from the apparent extrapolation of modulus in the miscible regime in non-isothermal tests, to predict the binodal temperature is not always applicable in systems with weak dynamic asymmetry. Therefore, they proposed a rheological model, which is an integration of the double reptation model and the selfconcentration model to describe the linear viscoelasticity of miscible blends. Then, the deviatirMi of experimental data from the model predictions for miscible... 

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... 

Figure 8 Reptation of a chain within a tube. (A test chain [black] reptates through a tube with walls formed by segments of the chains [gray]).

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