Equivalent weight (EW) is the number of grams of dry Nafion per mole of sulfonic acid groups when the material is in the acid form. This is an average EW in the sense that the comonomer sequence distribution (that is usually unknown to the investigator and largely unreported) gives a distribution in m in this formula. EW can be ascertained by acid-base titration, by analysis of atomic sulfur, and by FT-IR spectroscopy. The relationship between EW and m is EW ) 100m + 446 so that, for example, the side chains are separated by around 14 CF2 units in a membrane of 1100 EW. Common at the time of this writing are Nafion 117 films. The designation “117” refers to a film having 1100 EW and a nominal thickness of 0.007 in., although 115 and 112 films have also been available. Early-reported studies involved 1200 EW samples as well as special experimental varieties, some being rather thin. The equivalent weight is related to the property more often seen in the field of conventional ion exchange resins, namely the ion exchange capacity (IEC), by the equation IEC ) 1000/EW. The mention of the molecular weight of high equivalent weight (EW > 1000 g‚mol-1) Nafion is almost absent in the literature, although the range 105-106 Da has been mentioned. As this polymer does not form true solutions, the common methods of light scattering and gel permeation chromatography cannot be used to determine molecular weight as well as the size and shape of isolated, truly dissolved molecules. Studies of the structure of this polymer in solvent (albeit not a true solution) will be mentioned in the scattering section of this review. It should be noted that Curtin et al. performed size exclusion chromatography determinations of the molecular weight distribution in Nafion aqueous dispersions after they were heated to high temperatures (230, 250, and 270 °C).1 Before heating, there was a high molecular weight shoulder on a bimodal distribution, due to molecular aggregates, but this shoulder disappeared upon heating, which indicated that the aggregates were disrupted. The peaks for the monomodal distribution for the heated samples were all located at molecular weights slightly higher than 105 g‚mol-1. Also, light scattering experiments revealed that the radius of gyration had a linear dependence on the molar mass of the aggregates, which suggests that the particles are in the form of rods or ribbons, or at least some elongated structure. Nafion ionomers are usually derived from the thermoplastic -SO2F precursor form that can be extruded into sheets of required thickness. Strong interactions between the ionic groups are an obstacle to melt processing. This precursor does not possess the clustered morphology that will be of great concern in this article but does possess Teflon-like crystallinity which persists when the sulfonyl fluoride form is converted to, for example, the K+ form by reacting it with KOH in water and DMSO. Thereafter, the -SO3H form is achieved by soaking the film in a sufficiently concentrated aqueous acid solution. Extrusion of the sulfonyl fluoride precursor can cause microstructural orientation in the machine direction, * Address correspondence to either author. Phone: 601-266-5595/ 4480. Fax: 601-266-5635. E-mail: Kenneth.Mauritz@usm.edu; RBMoore@usm.edu. 4535 Chem. Rev. 2004, 104, 4535−4585

State of Understanding of Nafion

The entanglement concept in polymer rheology

1. Introduction to colloid science and rheology 2. Hydrodynamic effects 3. Brownian hard spheres 4. Stable colloidal suspensions 5. Non-spherical particles 6. Weakly flocculated suspensions 7. Thixotropy 8. Shear thickening 9. Rheometry of suspensions 10. Suspensions in viscoelastic media 11. Advanced topics.

Colloidal Suspension Rheology

The field of microrheology is concerned with how materials store and dissipate mechanical energy as a function of length scale. Recent developments in the theory and instrumentation of the microrheology of complex fluids are reviewed. Equal emphasis is given to the physical phenomena probed, advances in instrumentation, and specific experimental systems in which this field has already had an impact. The inversion of the compliance data, measurement of sample heterogeneity, high frequency viscoelasticity, effects of shear flow, single molecule experiments, surface viscoelasticity and time evolution studies are considered. The techniques highlighted include particle tracking microrheology, diffusing wave spectroscopy, laser tracking, magnetic tweezers and atomic force microscopy. Specific examples of complex fluid systems are chosen from the fields of polymers, colloids and biological assemblies.

https://iopscience.iop.org/article/10.1088/0034-4885/68/3/R04/pdf

Microrheology of complex fluids

The linear viscoelastic and viscometric functions have been determined for solutions of wellcharacterized monodisperse linear and star-branched polystyrenes and for commercial, polydisperse polystyrene. The value of the product c\(\bar M_w \) for these solutions was large and was obtained by using both high and low\(\bar M_w \)

Shear flow properties of concentrated solutions of linear and star branched polystyrenes

The shear‐wave field generated in a linear viscoelastic medium confined between parallel plates (one fixed, the other oscillating sinusoidally in its own plane) has been evaluated and presented in a graphical form convenient for determination of the role of wave propagation effects in dynamic rheological measurements. The transition from closely spaced planes (“gap loading”) to the freely propagating plane wave (“surface loading”) limit is examined; the importance of deviations in magnitude and phase of the gradient profile from the gap loading limit is discussed in terms of high precision dynamic rheological experiments; significant deviations occur for shear wavelength to gap width ratios of 30 or less.

Deviation of Velocity Gradient Profiles from the “Gap Loading” and “Surface Loading” Limits in Dynamic Simple Shear Experiments

A new multiple lumped resonator apparatus for the measurement of the viscoelastic properties of very dilute macromolecular solutions is described. The instrument may be used to study the viscoelastic behavior appropriate to infinite dilution for comparison with current theories. It has a frequency range from 100 to 8300 Hz (discrete frequencies) and may be used with solution viscosities from 0.005 to 0.5 P. Values of G′ less than 0.5 dyne/cm2 have been measured with reasonable accuracy when used in conjunction with a computerized data acquisition system.

Application of the Birnboim Multiple Lumped Resonator Principle to Viscoelastic Measurements of Dilute Macromolecular Solutions

Theoretical predictions are presented for the operating characteristics of a quartz crystal microbalance (QCM) system consisting of the crystal, an attached viscoelastic sample layer, and a surrounding viscoelastic bath medium. Predictions are given for the spatial variation of “particle” velocity and velocity gradient throughout the sample layer, for the characteristic mechanical impedance acting on the crystal surface due to the viscoelastic sample layer plus surrounding bath medium, and for the resultant changes, Δfr, in the resonance frequency of the QCM. Errors introduced by employing the usual simple Sauerbrey-type Δfr/sample-layer–mass relationship, together with a constant frequency offset to approximate the effect of the viscoelastic bath medium, are explored. In general, the viscoelastic properties of both the sample and bath media can substantially affect the apparent thickness (mass) of the sample layer if it is obtained by employing the typical Sauerbrey-type approximation noted above.

Theoretical predictions for the mechanical response of a model quartz crystal microbalance to two viscoelastic media: A thin sample layer and surrounding bath medium

Dynamic Viscoelastic Properties of Polystyrene in High-Viscosity Solvents. Extrapolation to Infinite Dilution and High-Frequency Behavior

The storage and loss shear moduli, G, and G˝, have been measured for solutions of three samples of poly-γ-benzyl-L-glutamate with molecular weights from 16 to 57 × 104, by use of the Birnboim-Schrag multiple-lumped resonator. The frequency range was 106 to 6060 Hz, the concentration range 0.0015–0.005 g/ml, and the temperature 25°C. Two helicogenic solvents with widely different viscosities, dimethylformamide and m-cresol, were used to provide a broader effective frequency range. The intrinsic moduli, extrapolated to infinite dilution, were compared with the predictions of the theory of Ullman for rigid rods; agreement was rather good at the lowest frequencies, but unsatisfactory at high frequencies. The data over the entire frequency range of three of logarithmic decades could be described closely by a relaxation spectrum consisting of one terminal relaxation time separated by a gap from a sequence of relaxtion times spaced as in the Zimm theory. The terminal time agrees approximately with that calculated for end-over-end rotation of a rigid rod. The additional relaxation mechanisms are tentatively attributed to modes of flexural deformation of the helix.

John L. Schrag

Papers

Deviation of Velocity Gradient Profiles from the “Gap Loading” and “Surface Loading” Limits in Dynamic Simple Shear Experiments

Application of the Birnboim Multiple Lumped Resonator Principle to Viscoelastic Measurements of Dilute Macromolecular Solutions

Theoretical predictions for the mechanical response of a model quartz crystal microbalance to two viscoelastic media: A thin sample layer and surrounding bath medium

Dynamic Viscoelastic Properties of Polystyrene in High-Viscosity Solvents. Extrapolation to Infinite Dilution and High-Frequency Behavior

Infinite-dilution viscoelastic properties of poly-gamma-benzyl-L-glutamate in helicogenic solvents.