Hartman equation

Coefficient of linear thermal expansion measurements can also be used to estimate some important characteristic parameters of polymers, such as free volume (Vf), hard-core volume (v ), and magnitude of intermolecular interactions (T ) in the materials (Brostow and Szymanski 1986). As temperature increases, polymers expand and their conformational entropy, a measure of disorder in polymer chains, increases. Theoretically, at a sufficiently high temperature, volume expansion allows each polymer chain to relax individually without interacting with its neighbors. This temperature is defined as T. In polymers, T is an extrapolated value above the degradation temperature, and for many commercial polymers T is above 500°C (Matsuoka 1997). Variables such as Vt, v, and T can be evaluated by the application of an equation of state such as the Hartman equation shown below (where V = v/v, T = TIT, and F = PIP ). (Hartman and Haque 1985a, b) ... 

It has been reported (Akinay et al. 2001,2002) that the Hartman equation works well for both polymeric solids and melts, including amorphous and crystalline polymers and their blends. Moreover, it also works well in more complicated multiphase systems, such as liquid crystalline polymers (LCPs). Use of the Hartman equation is relatively simple, and its variables can be measured experimentally. Several other equations of state are cited in the Polymer Handbook (Brandrup 1999). Some of these do not permit direct measurement of parameters or contain universal constants that complicate determination of the required parameters. Another advantage of the Hartman... 

Har] P. Hartman (1964), Ordinary Differential Equations. New York Wiley. 

Hartman, R, Ordinary Differential Equations, 2nd ed., Boston Birkhauser (1982). 

Certain reaction rate forms can lead to unusual bdiavior, that is not u ll represented by the general modulus approach, over the entire range of modulus values. For isothermal systems, these are assocuOed with rate equations that can exhibit empirical or approximate negative order bdiavior. For example, Satterfield, Roberts, and Hartman [87,88] have shown that rate equations erf the form... 

Similar types of equations were also proposed by Hartman et al. (1994) for nonspherical particles. They introduced the equations... 

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