Malcolm L. Williams

Extensions of the Rouse Theory of Viscoelastic Properties to Undiluted Linear Polymers

The Rouse theory for viscoelastic properties of very dilute solutions is modified for application to undiluted linear polymers. With the effective segment mobility expressed in terms of steady‐flow viscosity, the theory is applied to polymers of rather low molecular weight essentially without further change. In high molecular weight polymers, it is assumed that for modes of motion with relaxation times above a critical value the effective segment mobility drops abruptly, in accordance with the effect of entanglement coupling on steadyflow viscosity as described by Bueche. Properties in both the transition region between glasslike and rubberlike consistency and the rubberlike or plateau region are predicted semiquantitatively with no arbitrary parameters. In an alternative application to the transition region, the average effective friction coefficient per monomer unit can be calculated for both linear and lightly cross‐linked systems.

Second approximation methods for determining the relaxation time spectrum of a viscoelastic material

Abstract Second-approximation methods are given for deriving the distribution function of relaxation times of a viscoelastic material from experimental measurements of stress relaxation after sudden strain, stress relaxation after cessation of steady-state flow, the real and imaginary parts of the complex dynamic rigidity, and the real part of the complex dynamic viscosity. The numerical coefficients required for the calculations are tabulated, and the methods are illustrated by application to experimental data on solutions of polyvinyl acetate.

Dynamic mechanical properties of polyethyl methacrylate

Abstract The real ( J ′) and imaginary ( J ″) components of the complex compliance have been measured between 24 and 2400 cycles/sec. in the temperature range from 75° to 155°C. for two fractions of polyethyl methacrylate of weight-average molecular weights 1.73 × 10 6 and 0.164 × 10 6 . The data were practically identical for the two fractions except that J ″ was slightly higher at the highest temperatures and lowest frequencies for the lower fraction. The method of reduced variables did not give superposed curves for J ′ and J ″ over the entire temperature range. However, the data could be analyzed in terms of two additive mechanisms; for the α mechanism, J ′ and J ″ superposed with reduction factors a T whose temperature dependence followed an equation of the WLF form, and for the β mechanism they also superposed with reduction factors a Tβ whose temperature dependence corresponded to a constant activation energy of 31 kcal. The coefficients of the WLF equation, reduced to the glass transition temperature, were similar in magnitude to those observed in other polymers. Relaxation and retardation spectra have been calculated for both mechanisms. Those for the α mechanism, attributed to the usual chain backbone motions, resemble in form those for other disubstituted vinyl chains; the friction coefficient per monomer unit in backbone motion has been calculated from the extended Rouse theory. The spectra for the β mechanism, attributed to side chain motions, are relatively broad. The relative locations of the two mechanisms on the time scale change rapidly with temperature.

Dynamic mechanical properties of polyvinyl acetate

Abstract The real and imaginary components of the complex compliance have been measured between 30 and 5100 cycles/sec. at nine temperatures between 50°C. and 90°C. on an unfractionated sample of polyvinyl acetate (Mn = 140,000, Mw = 420,000). The results superpose by the method of reduced variables to give composite curves covering the transition from rubberlike to glasslike consistency as a function of frequency at 75°C. From these, the reduced curves for the real and imaginary parts of the complex shear modulus, and the real part of the complex viscosity, have been calculated. Earlier data of Mead and Fuoss on dielectric dispersion and dielectric loss of polyvinyl acetate have been reduced to 75°C. in the same manner, and the mechanical and electrical reduction factors are found to be identical. The apparent activation energy for mechanical and electrical relaxation increases sharply with decreasing temperature, as observed for other polymers. Distribution functions of mechanical relaxation and retardation times, and of electrical relaxation times, have been calculated from both real and imaginary components of modulus, compliance, and dielectric constant, respectively. The results of the approximation formulas of Williams and Ferry and of Schwarzl and Staverman are found to be closely similar. The electrical relaxation spectrum is somewhat flatter, and lies at shorter times, than the mechanical retardation spectrum.

Free Volume Approach to Polystyrene Melt Viscosity

The Doolittle approach to Newtonian viscosity assumes η=A expB/f, where A and B are constants independent of temperature and f, the relative free volume, is (v−v0)/v0. Here v is the specific volume and v0 is the occupied volume. Using machine computations, this equation was applied to published data on polystyrene, replacing v by its equivalent vg+(dv/dT)(T−Tg), where vg is the specific volume at the glass temperature Tg. The agreement between calculated and observed viscosities is at least as good, and perhaps better, than that afforded by other existing empirical expressions. The constant B is independent of molecular weight and temperature while v0 depends slightly on molecular weight. A log‐log plot of A against molecular weight M produces two straight lines intersecting at M=35 000. At higher values of M, the slope is 3.4; at lower values, it is slightly greater than one. This treatment separates the melt viscosity of polystyrene into two factors, A, dependent only on structural parameters, and f, wh...

Temperature Dependence of Dynamic Properties of Elastomers. Relaxation Distributions

Abstract By the use of reduced variables, the temperature dependence and frequency dependence of dynamic mechanical properties of rubberlike materials can be interrelated without any arbitrary assumptions about the functional form of either The definitions of the reduced variables are based on some simple assumptions regarding the nature of relaxation processes. The real part of the reduced dynamic rigidity, plotted against the reduced frequency, gives a single composite curve for data over wide ranges of frequency and temperature; this is true also for the imaginary part of the rigidity or the dynamic viscosity. The real and imaginary parts of the rigidity, although independent measurements, are interrelated through the distribution function of relaxation times, and this relation provides a check on experimental results. First and second approximation methods of calculating the distribution function from dynamic data are given. The use of the distribution function to predict various types of time-depende...

Dynamic mechanical properties of polymethyl acrylate

Abstract The real and imaginary components of the complex compliance have been measured between 30 and 3000 cycles/sec. in the temperature range 25° to 89°C. for a sample of polymethyl acrylate of weight-average molecular weight 2.2 × 106. Results at all temperatures and frequencies superpose by the method of reduced variables to give the components at 25°C. over 7.5 decades of frequency, corresponding to the transition from rubberlike to glasslike consistency as a function of frequency at this temperature. Relaxation and retardation distribution functions have been calculated, and the friction coefficient per monomer unit has been estimated from an extension of the Rouse theory. The temperature dependence of the relaxation processes agrees with that derived from dielectric measurements, and it has been analyzed in terms of the free volume and its thermal expansion coefficient.

Dynamic mechanical properties of plasticized polyvinyl acetate

Abstract The real and imaginary components of the complex compliance have been measured between 30 and 4500 cycles per second in the temperature range −11°C. to 40°C. for a 50% (by volume) solution of polyvinyl acetate in tri-m-cresyl phosphate. Results at all temperatures and frequencies superpose by the method of reduced variables to give the components at 25°C. over 7.5 decades of frequency, corresponding to the transition from rubberlike to glasslike consistency at this temperature. Values of the steady-flow viscosity have been measured between 0° and 41°C. and found to give nearly the same temperature reduction factors as found empirically from the dynamic measurements. Relaxation and retardation distribution functions have been calculated. By reducing these data to a common temperature for both plasticized and undiluted polymer, it is concluded that the presence of 50% diluent reduces the monomeric friction coefficient by a factor of 107. By reducing the data to a hypothetical reference state of unit density and unit steady-flow viscosity and combining them with previous data on solutions of the same polymer in 1,2,3-trichloropropane, the distribution function of mechanical relaxation times can be obtained at 25°C. over 13.5 decades of time. The distribution function is found to be similar in shape and location on the time scale to that obtained for polystyrene in decalin previously studied in this laboratory.

Viscoelasticity and Flatspotting

Abstract Flatspotting involves delayed elasticity, so conventional principles of polymer viscoelasticity should be applicable. The strain behavior of the tire cord material during the complex stress and thermal history of the tire is idealized in a model, assuming flatspotting is caused mainly by the tire cord material and that the strains in the cord in the footprint area are less than the strains in the cords elsewhere in the tread. A flatspot index based on this model is empirically correlated with tire performance. It is shown here that this commonly used empirical index can be quantitatively interpreted in terms of a viscoelastic property of the tire cord material, in this case, creep compliance of nylon 66. Flatspotting depends on the shape of the viscoelasticity function, which is governed by chemical composition, stress differences in the tire, and the effect of environmental variables on the time scale. The most important environmental variables are temperature and moisture content. Certain quali...