Polymer self-diffusion

Phillies, GDJ, The Hydrodynamic Scaling Model for Polymer Self-Diffusion, Journal of Physical Chemistry 93, 5029, 1989. 

Previously, the assumption has been made that the polymer self-diffusion coefficient is proportional to l/f21. The assumptions involved in the derivation of this relationship have been shown experimentally to be invalid 8). 

Phillies, G., The hydrodynamic scaling model for polymer self-diffusion. Journal of Physical Chemistry, 1989, 93, 5029-5039. 

This book deals almost exclusively with equilibrium and steady transport properties. Time-dependent or dynamic processes are discussed only fragmen-tarily in connection with polymer self-diffusion (Chapter 8) and spinodal decomposition (Chapter 10). It far exceeds the ability and experiences of the author to write about them. However, we have to remark thcit the center of gravity in polymer solution study is rapidly moving toward dynamic processes and also surface phenomena. 

Figure 7-3. Trajectory of polymer self-diffusion. Each circle marks the center of mass of a migrating polymer chain at a certain instant.

In an excellent review article, Tirrell [2] summarized and discussed most theoretical and experimental contributions made up to 1984 to polymer self-diffusion in concentrated solutions and melts. Although his conclusion seemed to lean toward the reptation theory, the data then available were apparently not sufficient to support it with sheer certainty. Over the past few years further data on self-diffusion and tracer diffusion coefficients (see Section 1.3 for the latter) have become available and various ideas for interpreting them have been set out. Nonetheless, there is yet no established agreement as to the long timescale Brownian motion of polymer chains in concentrated systems. Some prefer reptation and others advocate essentially isotropic motion. Unfortunately, we are unable to see the chain motion directly. In what follows, we review current challenges to this controversial problem by referring to the experimental data which the author believes are of basic importance. 

Usually, C is assumed to depend on polymer mass concentration c and temperature T, unless the chain is too short. It can be adequately formulated by free volume theory [3]. On the other hand, Fg is assumed to be a function of c and N (or molecular weight M). The central theme in polymer self-diffusion studies is to evaluate the latter by theory or experiment for a variety of chain architectures and solvent conditions. 

Hess [13] neglected the hydrodynamic interactions among chain beads and treated the global motions of different chains as uncorrelated (this is to assume a small number of chain-chain contacts and thus to focus on the semi-dilute regime). He deduced that polymer self-diffusion consists of both lateral and longitudinal modes of chain motion until the entanglement parameter t/>(c, N) reaches unity, but it is dominated by the latter (i.e., chains move reptatively)... 

The basic weakness of Phillies work is that eq 3.5 does not yet have a reasonable theoretical background. Thus, despite its apparent success, this equation is of little interest to those who want to know the mechanism responsible for polymer self-diffusion. The reader is advised to see Wheeler et al. [43] for more comments on it. 

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. 

The following papers had experimental foci different from those examined above. Several provide very interesting information on polymer self-diffusion in solution. Others are not amenable to the above analysis. 

The above sections summarize a detjuled examination [1] of nearly the entirety of the published literature on polymer self-diffusion and probe diffusion in polymer solutions. Dependences of D, and Dp on polymer concentration, probe molecular weight, and matrix molecular weight were determined. We now attempt to extract systematic behaviors from the above particular results, asking What features are common to self- and probe-diffusion of all polymers in solution ... 

In the above, virtually the entirety of the published literature on polymer self-diffusion and on the diffusion of chain probes in polymer solutions has been reviewed. Without exception the concentration dependences of Dg and Dp are described by stretched exponentials in polymer concentration. The measured molecular weight dependences compare favorably with the elaborated stretched exponential, eq. 16, except that, when P M or M P, there is a deviation from eq. 16, that deviation referring only to the molecular weight dependences. The deviation uniformly has the same form The elaborated stretched exponential overestimates the concentration dependence of Dp, so that at elevated c the predicted Dp/Do is less than the measured Dp/Dp. Contrarywise, almost without exception the experimental data on solutions is inconsistent with models that... 

The power law exponent, n, is unity for a Newtonian fluid and less than unity for a shear-thinning fluid. One of the central questions of polymer physics concerns the molecular basis for the constitutive equations. Because NMR is so sensitive to molecular dynamical parameters, the simultaneous mapping of velocity profiles and molecular properties such as the polymer self-diffusion coefficient by means of the dynamic NMR microscopy technique offers an effective test of much molecular models. 

Figure 9.10 (a) Normalised velocity profiles for different concentration solutions of polyfethylene oxide) in water obtained using dynamic NMR microscopy. The concentrations increase in equal steps from 0.5% (w/v) ( ) to 4.5% (w/v) ( ). (b) The polymer self-diffusion profile for the highest concentration solution in units of 10 m s" Note that this was obtained in a separate experiment so that the capillary wall does not fall at precisely the same pixel as in (a), (c) Water solvent velocity and (d) diffusion maps for the 4.5% (w/v) poly(ethylene oxide) solution. (From Y. Xia and P.T. Callaghan [18] and reproduced by permission of the American Chemical Society.)... 

Combination of equations 18 and 19 and use of the definition of mobility leads to the following expression for the polymer self-diffusion coefficient, when... 

M. Tirrell, Polymer self-diffusion in entangled systems. Rubber Chem. Technol. 57, 523 (1984). 

JOA Joabsson, F., Nyden, M., and Thuresson, K., Temperature-induced fractionation of a quasi-binaiy self-associating polymer solutioa A phase behavior and polymer self-diffusion investigation. Macromolecules, 33,6772,2000. 

J.E. Martin, "Polymer Self-Diffusion Dynamic Light Scattering Studies of Isorefractive Ternary Solutions," Macromolecules. 17. 

M. Tirrell, "Polymer Self-diffusion in Entangled Systems," Rubber Chem, Technol, 51, 523-556 (1984). 

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