Stretched exponential form

Phillies [296] observed that several authors [87,270] using different underlying argnments have obtained stretched exponential form for diffusion in presence of polymer matrix ... 

Johnson et al. [186] measured diffusion of fluorescein-labeled macromolecules in agarose gels. Their data agreed well with Eq. (85), which combined the hydrodynamic effects with the steric hindrance factors. Gibbs and Johnson [131] measured diffusion of proteins and smaller molecules in polyacrylamide gels using pulsed-field gradient NMR methods and found their data to fit the stretched exponential form... 

The relation between collective and self-motion in simple monoatomic liquids was theoretically deduced by de Gennes [233] applying the second sum rule to a simple diffusive process. Phenomenological approaches like those proposed by Vineyard [ 194] and Skbld [234] also relate pair and single particle motions and may be applied to non-exponential functions. The first clearly fails to describe the PIB results since it considers the same time dependence for both correlators. Taking into account the stretched exponential forms for Spair(Q.t) (Eq. 4.21) and Sseif(Q>0 (Eq 4.9), the Skold approximation ... 

It is found empirically that a stretched exponential form,... 

D = D° exp(-ac ), where D is the diffusion, D represents the zero-concentration limit, c is the concentration, a and v are parameters, fits the data from a wide variety of probes and matrix polymers ( ). Several theoretical justifications for this behavior have been presented (97-1011. but it is not possible to tell yet which, if any, is uniquely correct. The treatments range from simple physical considerations (98) to treatments of hydrodynsumical interaction of probe and matrix (97,991. Other more complex and general treatments (1001 do not explicitly arrive at the stretched exponential form, but do closely fit the available data. Much more work needs to be done on probe diffusion in such transient networks. Beyond enhancing the arsenal of gel characterization, the problem is quite fundamental to a number of other important processes. 

H NMR study of I2 doped poly[2-buthoxy-5-methoxy phenylene vinylene] (PBMPV) has been carried out by Lee et al.114 They found magnetization recovery to follow a stretched-exponential form M(f) =Mo[l — exp(f/Ti) ] with n = 0.8 0.03, which is characteristic of a system with distribution of the correlation time. 1/Tj versus/ 172 in samples with different conductivity follows/ 172 dependence over 20 40 MPIz range indicating the main SLR is ID spin diffusion. A correlation between 1/T1 and dc conductivity has been made which is found follow a simple power law. 

The Mittag-Leffler function interpolates between the initial stretched exponential form [7,31]... 

A supercooled hquid or amorphous sohd can be driven out of (restricted) equihbrium in a wide variety of ways, and the kinetics of relaxation back to (restricted) equihbrium subsequently followed. This variety includes sudden temperature or pressure changes, mechanical working, application of a polarizing electric field, and irradiation with energetic particles. As mentioned in Section I, the linear-response relaxation kinetics of supercooled liquids as monitored by any of several properties is observed to follow a Kohlrausch-Wilhams-Watts (KWW) stretched exponential form [see Eq. (2)],... 

Here Qa is the mean value of property Q averaged over basin a (at energy ), and (X) is the spectral weight in the continuum limit of the modes with exponential decay constant X. If 2(0 in fact has the stretched exponential form, then (X) will be proportional to the Laplace transform F(X), for which both numerical (Lindsey and Patterson, 1980) and analytical (Helfand, 1983) studies are available. In the simple exponential decay limit= 1, F(X) reduces to an infinitely narrow Dirac delta function but it broadens as p decreases toward the lower limit to involve a wide range of simple exponential relaxation rates. 

Much attention has been given to the relaxation of various properties toward their equilibrium values, especially in the glasses literature. Kohlrausch first proposed a stretched exponential form as a description of viscoelasticity, while Williams and Watts suggested the same form for dielectric relaxation exp[-(t/r) ], where 0 < 9 < 1 and 0 = 1 corresponds to the Debye limit. The master equation solution, Eq. (1.41), has a decaying multiexponential form that could lead to a wide variety of behavior depending upon the system. 

Below Tg (4,5) co-operative molecular processes are usually assumed to be inactive. However, physical ageing implies that conformational changes may be still able to occur if rather infrequently. Structural relaxation processes are observed to be non-exponential and are represented by a continuous distribution or stretched exponential form (d). Thermorheologically simplicity (TRS) implies that the molecular relaxation process has the same form at different temperatures (7) and the validity of this assumption is addressed in this paper. Isobaric volume recovery (8,9) has been described by a single parameter mc el, however all fi ee volume models (10,11) have limitations and a distribution of hole sizes and relaxation times leading to a pseudo-linear theory is a more realistic model(72). Comparison of data fi om various techniques should throw light on the molecular nature of physical agdng. 

Several authors report data on rodlike polymers. Jamil, et aJ. [144] report QELSS measurements on random-coil polystyrene probes in solutions of the rigid-rod polymer poly-(7-stearyl-Q-L-glutamate) in its isorefractive solvent toluene, fitting the D, they found to a stretched-exponential form for its concentration dependence. They found V 1.3, and interpret > 1 as arising from end-to-end aggregation of the polymer polymers. Russo, et al. [145] measured D, for the rodlike polymer poly(7-benzyl-a,L-glutamate) in pyridine at concentrations above and below the isotropic cholesteric phase transition. Dp changes its concentration dependence at the transition. 

The stretched-exponential form exp(-91 ) can also be parameterized exp(- (t/r) ), thereby introducing relaxation times and Tr0f as replacements for the... 

In exponential models, the concentration dependence is an exponential or stretched exponential in concentration(2,3). For self-diffusion, the stretched exponential form is... 

The following two sections treat sedimentation of polymers in homogeneous solution, and sedimentation of probe chains and spheres through polymer solutions. A final section offers a systematic discussion of these results. Throughout, the analysis matches the experimental concentration dependence against the simple stretched-exponential form... 

From limited results, s(c) of sedimenting probes in a polymer matrix generally follows a stretched exponential in c. With a small matrix polymer, s(c) of a probe chain simply tracks the solution viscosity. In solutions of large matrix polymers, s and rj do not show the same concentration dependence. With probe spheres, 5(c) may track the solution viscosity or may show re-entrant behavior. The agreement of 5(c) with a stretched-exponential form is less outstanding when re-entrance is observed. The literature describes too few probe polymer pairs to be able to say if re-entrant behavior is common or rare. 

In conclusion, s(c) almost always follows the stretched-exponential form of Eq. 2.6. When there are measurements on enough members of a homologous series of polymers, a and v can show clear dependences on probe and matrix molecular weights. It appears that the dominant variable determining a is the polymer s extent, not the polymer s molecular weight, so that transferring a polymer from a good to a Theta solvent substantially reduces a. 

In contrast to these stretched-exponential forms, a simple scaling relationship for II would be... 

Tao, et al. used forward recoil spectroscopy and PFGNMR to determine Ds of hydrogenated polybutadienes in alkane solvents(15). Measurements of the dynamic shear moduli were used to determine the low-shear viscosity of the same solutions. Tao, et al. s measurements of Ds appear in Figure 8.7, together with fits of Ds to scaUng-law and stretched-exponential dependences on polymer concentration and molecular weight. The root-mean-square fractional errors in the displayed fits to the scaling-law and stretched-exponential forms are both 20%. For the... 

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