**2.2.2 Temperature-pressure-volume control**

The design of devices based on the control of pressure requires breakthrough technologies. The major difficulty is to generate high pressure.

In polymer solidification, the effects of pressure can be studied through pressure–volume– temperature phase diagrams obtained during cooling at constant pressure. The effect of hydrostatic (or inert) pressure on phase transitions is to shift the equilibrium temperature to higher values, *e.g.*, the isotropic phase changes of complex compounds as illustrated in the works of Maeda et al. (2005) by high-pressure differential thermal analyzer and of Boyer et al. (2006a) by high-pressure scanning transitiometry, or the melting temperature in polymer crystallization as illustrated for polypropylene in the work of Fulchiron et al. (2001) by highpressure dilatometry. However, classical dilatometers cannot be operated at high cooling rate without preventing the occurrence of a thermal gradient within the sample. This problem can be solved by modelling the dilatometry experiment (Fulchiron et al., 2001) or by using a miniaturized dilatometer (Van der Beek et al., 2005). Alternatively, other promising technological developments propose to couple the pressure and cooling rates as shown with an apparatus for solidification based on the confining fluid technique as described by Sorrentino et al. (2005). The coupling of pressure and shear is possible with the shear flow pressure–volume–temperature measurement system developed by Watanabe et

Thermodynamics and Thermokinetics to Model Phase Transitions of Polymers

Δ*Vpol* of the polymer due to the sorption.

pressure, respectively. And *L*, *R,*

of the wire. *VC* is the volume of the polymer container.

Beret & Prausnitz, 1975; Behme et al., 1999).

fraction of the solvent (index 1,

binary adjustable parameter *k*12.

calculated with **eq. (5)**.

The mass fraction of solvent (the permeant),

( <sup>2</sup> 

over Extended Temperature and Pressure Ranges Under Various Hydrostatic Fluids 647

with a high pressure decay technique, the whole setup being operated under a fine control of the temperature. The limits and performances of this mechanical setup under extreme conditions, *i.e.*, pressure and environment of fluid, were theoretically assessed (Boyer et al., 2007). In the working equation of the vibrating-wire sensor (VW) **(eq. (1))**, unknowns are both the mass *msol* of solvent absorbed in the polymer and the associated change in volume

> 2 2 0

solvent. The other parameters are the physical characteristics of the wire, namely,

 

The volume of the degassed polymer is represented by *Vpol* and

\* , <sup>2</sup> *p* \* , 2 *T* \* ) being the characteristic parameters of pure compounds.

*sol g pol B g C pol L R m V V V*

which represent the natural (angular) frequencies of the wire in vacuum and under

The thermodynamics of solvent-polymer interactions can be theoretically expressed with a small number of adjustable parameters. The currently used models are the 'dual-mode' model (Vieth et al., 1976), the cubic equation of state (EOS) as Peng-Robinson (Zhong & Masuoka, 1998) or Soave-Redlich-Kwong (Orbey et al., 1998) EOSs, the lattice-fluid model of Sanchez–Lacombe equation of state (SL-EOS) (Lacombe & Sanchez, 1976; Sanchez & Lacombe, 1976) with the extended equation of Doghieri-Sarti (Doghieri & Sarti, 1996; Sarti & Doghieri, 1998), and the Statistical Associating Fluid Theory (SAFT) (Prigogine et al., 1957;

From the state of the art, the thermodynamic SL-EOS was preferably selected to theoretically estimate the change in volume of the polymer *versus* pressures and temperatures found in **eq. (1)**. In this model, phase equilibria of pure components or solutions are determined by equating chemical potentials of a component in coexisting phases. It is based on a welldefined statistical mechanical model, which extends the basic Flory-Huggins theory (Panayiotou & Sanchez, 1991). Only one binary adjustable interaction parameter *k*12 has to be calculated by fitting the sorption data **eqs. (2-4)**. In the mixing rule appears the volume

11 22 1 2 *ppp p* \* \*\* \*

1 1 22 1 2

\* \*

\* \*

 

*p p T T*

\* \* \* \*

The parameter *p*\* characterizes the interactions in the mixture. It is correlated with the

 

*<sup>p</sup> <sup>T</sup>*

1) in the polymer (index 2,

 

 2 2

*<sup>s</sup>* are, respectively, the length, the radius and the density

2), ( <sup>1</sup> 

12 1 2 *p* \* *k p p* (4)

1, at the thermodynamical equilibrium is

(2)

(1)

*<sup>g</sup>* is the density of the

0 and *B*

\* , <sup>1</sup> *p* \* , <sup>1</sup> *T* \* ) and

(3)

4 *<sup>S</sup>*

*g* 

al. (2003). Presently, performing of *in-situ* observations of phase changes based on the optical properties of polymers (Magill, 1961, 2001) under pressure is the object of a research project developed by Boyer (Boyer et al., 2011a).

To estimate the solubility of penetrating agents in polymers, four main approaches are currently generating various techniques and methods, namely: gravimetric techniques, oscillating techniques, pressure decay methods, and flow methods. However, with many existing experimental devices, the gain in weight of the polymer is measured whereas the associated volume change is either estimated or sometimes neglected (Hilic et al., 2000; Nalawade et al., 2006; Li et al., 2008).

The determination of key thermo-mechanical parameters coupled with diffusion and chemical effects together with temperature and pressure control is not yet well established. Approaches addressing the prediction of the multifaceted thermo-diffuso-chemomechanical (TDCM) behaviour are being suggested. Constitutive equations are built within a thermomechanical framework, like the relation based on a rigorous thermodynamic approach (Boyer et al., 2007), and the proposed formalism based on as well rigorous mechanical approach (Rambert et al., 2006; Baudet et al., 2009).
