Bruce Hartmann

Loss factor height and width limits for polymer relaxations

Polymer relaxations at the glass transition are often used in damping applications. Questions arise whether there is any limit to the height and width of the damping peak that can be achieved. A related question is whether there are any limits on the combination of these two properties that are achievable. This later question arises because of the experimental observation that the height of the peak is inversely related to the width of the peak. In this paper, these questions are addressed using various analytical models of polymer behavior. Starting with the single relaxation time model and progressing to the Cole–Cole model, the Davidson–Cole model, and finally the Havriliak–Negami (HN) model, height and width predictions are obtained. It is found that the HN model predicts a band of physically possible height and width combinations when using reasonable values of the model parameters. High peaks are narrow and broad peaks are low. For a loss factor peak height of 2, the half‐width must be less than 3 d...

Polyurethane networks with controlled architecture of dangling chains

Network formation from A x + B y precursors (A + B → A - B) with functionality and molecular weight distributions is described by the statistical theory of branching processes. Network formation is described in terms of sol and gel fractions, dangling chains, elastically active network chains (EANC), elastically active crosslinks, free chain ends and branch points partitioned between sol and gel. Various definitions of an EANC are considered. The general relations are applied to a special case of A1 + A2 + A3/B2 system. It is show how the size and weight fractions of dangling chains can be varied indepently by varying the functionality or molecular weight distributions. This is demonstrated experimentally analyzing corresponding polyetherurethane networks prepared from mixtures of polyoxypropylene polyols. The width of the main transition region correlates with the fraction of material in dangling chains. The dependence of the equilibrium shear modulus on the concentration of EANCs indicates much weaker intermolecular interactions for networks with many short dangling chains compared with systems having few longer dangling chains.

Dynamic mechanical relaxation in some polyurethanes

Abstract Dynamic mechanical measurements of complex shear modulus versus frequency and temperature were made on a series of polyurethanes of varying soft-segment molecular weight. The time-temperature shifted data (master curves) mapped out the soft-segment glass transition. Shift factor curves were fitted to the Williams-Landel-Ferry (WLF) equation to obtain shift constants. The fractional free volume and coefficient of thermal expansion at the glass transition, determined from the shift constants, decrease as the molecular weight increases. Master curves were fitted to the modified Havriliak-Negami equation and the fitting parameters related to molecular structure. The limiting low-frequency modulus is dependent on the soft-segment molecular weight and percent crystallinity, but the limiting high-frequency modulus is about the same for all these polymers. The average relaxation time decreases with increasing soft-segment molecular weight and is correlated with glass transition temperature.

Network structure dependence of volume and glass transition temperature

A series of polyurethanes was used to determine the molar contributions of chain ends (CE) and branch points (BP) to free volume and glass transition temperature Tg. The polyurethanes were copolymers of diphenylmethane diisocyanate and poly(propylene oxide) (PPO) with hydroxyl functionalities of one, two, and three. The equivalent weights of all the PPOs were equal, such that the chemical composition of the chain segments was essentially identical. Therefore, the only distinctions among polymers were differences in CE and BP concentration. Theory of branching processes computer simulations were used to determine the concentration of CE due to imperfect network formation. Other CE contributions were from the monofunctional PPO. Polymer volumes and Tgs were correlated to CE and BP concentrations, and the contributions of these species were determined from least squares fits. The molar volume and Tg contributions were then used to determine free volume thermal expansion coefficients. These values were compar...

Calculation of B/A for n-alkane liquids using the Tait equation

The B/A parameter of acoustic nonlinearity was calculated for a series of n-alkane liquids using the Tait PVT equation of state supplemented with specific heat data. The calculations of sound speed, sound speed derivatives, the two components of B/A, and the value of B/A itself were compared with experimental data taken from the literature and with earlier calculations using a different equation of state. In addition, a comparison of the results with Ballous rule (linear relation of B/A and reciprocal sound speed) was made. It is concluded that B/A can be calculated from the Tait equation of state with about the same accuracy as direct measurements of sound speed versus pressure and temperature, though the the temperature derivatives of the sound speed are calculated with much lower accuracy than pressure derivatives. The calculations made using the Tait equation are about the same accuracy as calculations made using our equation of state. Also, Ballous rule does not hold for these liquids.

Sound absorption height and width limits for polymer relaxations

An analysis was performed to relate the height and width of the peak in shear sound absorption per wavelength as a function of frequency for the glass transition of a polymer. For the single relaxation time model, the width is 1.14 decades only at small values of the ratio of relaxed to unrelaxed sound speed. For the larger values of sound-speed ratio observed in polymer glass transitions, the width increases as the height increases. For the Havriliak–Negami model, the width increases as the height decreases in a manner similar to that for the complex modulus, though in this case there is a cutoff below the absolute maximum. The curve can be described by the relation that height times width is 1.5 decades of frequency. These predictions are in good agreement with experimental data for 21 polyurethanes.

Calculation of Relaxation Time in Polyurethanes Using Additive Group Contributions

SYNOPSIS The method of additive properties was used to calculate the dynamic mechanical relaxation time for a series of polyurethanes. Calculations were also made of density and glass transition temperature. Group contributions for nine component groups were determined. With these group values, the densities of the 12 polymers used to determine the groups were calculated and found to agree with measured values within an average of 0.2%. Calculated glass transition temperatures also agreed with measured values within 0.2%. The relaxation time, defined as a parameter in the Havriliak-Negami equation, was shown to be correlated with the glass transition temperature, allowing relaxation time to also be expressed as an additive property. Calculated logarithms of relaxation times agree with measured values to within 7% over a range,of relaxation times covering many decades. 0 1996 John Wiley & Sons, Inc.

Pressure and Temperature Dependence of Electrical Conduction in Polymers

Experimental data from the literature on the pressure and temperature dependence of electrical conductivity in a poly(epoxide co sulfide) is analyzed. From the increase in resistivity (reciprocal of conductivity) with pressure, it is shown that electrical conduction, in this case, proceeds via an ionic diffusion process and should therefore be correlated with free volume. Using a recently proposed PVT equation of state to calculate free volume with the usual definition, fair correlation is found between log resistivity and reciprocal free volume fraction. The deviations from perfect correlation are not random, but vary in a systematic manner, showing that pressure dependence is not adequately described. Based on an observation by Utracki, much better correlation is found using a pressure‐reducing parameter in the equation of state which is larger than that used to fit the PVT data. This change is equivalent to changing the definition of free volume.

Review of the method of reduced variables for the dynamic mechanical properties of viscoelastic materials

Under certain conditions, a change in temperature is equivalent to a change in frequency for dynamic mechanical measurements in viscoelastic materials. This property allows one to extend the effective frequency range of the measurements by shifting isotherms along the log frequency axis using the method of reduced variables, also known as time–temperature superposition. The most commonly used analytical equations describing the required shift are the WLF equation and the Arrhenius equation, but shifting can be implemented without the necessity of an analytical representation. This review will consider the physical basis for this property, known as thermorheological simplicity, the limitations of its applicability, and the advantages and disadvantages of various shifting procedures. Techniques that have been used include shifting the real part of the modulus, the imaginary part of the modulus, the loss factor, and the use of complex compliance rather than complex modulus. Other important issues are the cho...

Acoustic nonlinearity calculations using the Tait equation of state

Beyer’s B/A parameter of acoustic nonlinearity was calculated using the Tait equation of state supplemented with specific heat results, in the same manner as done earlier with our equation of state [J. Acoust. Soc. Am. 82, 614 (1987)]. Calculations were made of sound speed, sound‐speed temperature and pressure derivatives, the two components of B/A, and the value of B/A for a series of six n‐alkane liquids as well as the limiting case of molten polyethylene. All these results were compared with experimental data from the literature. In addition, a comparison of the results with Ballou’s rule (linear relation of B/A versus reciprocal sound speed) was made. Overall, values calculated using the Tait equation are found to agree with experiment about as well as our equation of state, though the pressure derivatives of the sound speed and the final value of B/A are generally not as accurate using the Tait equation. [Work supported by the Naval Surface Warfare Center’s In‐house Laboratory Independent Research Pr...