Specific volume-temperature curves

Fig. 4. Specific volume-temperature curves at different pressures of the 1-l.c. (isobaric measurements) 37 38)...

Figure 4.3 Specific volume-temperature curves for a semicrystalline polymer. (A) Liquid region (B) viscous liquid with some elastic response (C) rubbery region (D) glassy region (E) crystallites in a rubbery matrix (F) crystallites in a glassy matrix. 

Solids absorb heat on melting and, with the notable exception of ice, expand. They evolve heat when they undergo polymorphic transformation to a more stable polymorphic and contract. Consequently, dilatometric (specific volume-temperature) curves bear a close resemblance to calorimetric (enthalpy-temperature) curves. The melting dilation corresponds to the heat of fusion, and the coefficient of cubical expansion, a, corresponds to the specific heat capacity, c. The ratio cja is virtually a constant independent of temperature. 

One of the most important conclusions associated with the Takayanagi model is that the distribution function of the free volume fraction Fj f) is evaluated by the following equation based on the specific volume-temperature curve ... 

Figure 5.4. Specific volume-temperature curves for polystyrene fractions [Adapted, by permission, from Fox TQ Flory P J, J. Appl Phys., 21, 581, 1950].

FIGURE 12.12. Specific volume - temperature curves for four epoxy resins with increasing crosslink density from 1007/DPS to 828/DPS. Measurements were made during cooling at 57min under a pressure of 5 MPa. 

Under some favorable circumstances, some polymers can align their molecules in a regular lattice shape. Such materials are said to be crystallizable. Often, if the cooling starts from the melt phase, the crystallization is only partial, and hence the materials possess both crystalline and amorphous structure and so are called semicrystalline. There is a jump of specific volume at a transition zone in the specific volume/temperature curve below the melting temperature (T ), and then there is a region of tough solid. The mechanism of the crystallization is still the object of intense discussions (Tanner and Qi 2005 Pantani et al. 2005). 

Fig. 3.1 Specific volume-temperature curves for pure poly(decamethylene adipate), for its mixtures with dimethyl formamide (ui = 0.60), O and for its mixtures with diphenyl ether (ui = 0.18), A.(4)...

Experimental results clearly indicate that stereo-irregular polymers do indeed crystalhze as though they were copolymers. For example, specific volume-temperature curves for isotactic poly(propylene) display all the characteristics expected for a random type copolymer. The results of such a study by Newman (44)... 

This phenomenon is illustrated by the PVT (pressure - specific volume - temperature) curves plotted for individual plastics (see Figure 3.4). The PVT graphs for an amorphous plastic (PS) and a semi-crystalline plastic (HOPE) show the ideal setting process for a plastic in a mould after filling at 80 MPa pressure. 

In Fig. 3 c the schematic volume-temperature curve of a non crystallizing polymer is shown. The bend in the V(T) curve at the glass transition indicates, that the extensive thermodynamic functions, like volume V, enthalpy H and entropy S show (in an idealized representation) a break. Consequently the first derivatives of these functions, i.e. the isobaric specific volume expansion coefficient a, the isothermal specific compressibility X, and the specific heat at constant pressure c, have a jump at this point, if the curves are drawn in an idealized form. This observation of breaks for the thermodynamic functions V, H and S in past led to the conclusion that there must be an internal phase transition, which could be a true thermodynamic transformation of the second or higher order. In contrast to this statement, most authors... 

Fig. 6. Specific volume pressure curves for the l.c. polymer shown in Fig. 5. Thin dashed lines pressure dependence of the phase transformation temperatures l.c. to isotropic, Tc, and the glass transition temperatures, T , full line specific volume-temperature cut at 2000 bar (isothermal measurements)...

As there exists a phase equilibrium both phases must have reached in the internal thermodynamic equilibrium with respect to the arrangement and distribution of the molecules the measuring time. Therefore, no time effects or path dependencies of the thermodynamic properties in the liquid crystalline phase should be expected. To check this point for the l.c. polymer, a cut through the measured V(P) curves at 2000 bar has been made (Fig. 6) and the volume values are inserted at different temperatures in Fig. 7, which represents the measured isobaric volume-temperature curve at 2000 bar 38). It can be seen from Fig. 7 that all specific volumes obtained by the cut through the isotherms in Fig. 6 he on the directly measured isobar. No path dependence can be detected in the l.c. phase. From these observations we can conclude that the volume as well as other properties of the polymers depend only on temperature and pressure. The liquid crystalline phase of the polymer is a homogeneous phase, which is in its internal thermodynamic equilibrium within the normal measuring time. 

The specific volume v also shows an anomalous temperature dependence near 7. The behavior of v is universal among all systems having a finite 7. When measured at a constant cooling rate q, tJ follows the behavior of the enthalpy, shown in Fig. 3, as it decreases linearly with T and changes shape at a temperature dependent on q but close to the 7 observed for Cp. Below this breakaway temperature, the system is not in equilibrium. If the system is annealed, not far below 7, the behavior shown in Fig. 5 is observed. Volume v decays toward a lower asymptotic value Jq, which can lie either on the extrapolated volume-temperature curve for the liquid or above it, if the annealing temperature is low enough. The latter observation suggests the existence for metastable equilibrium of a Oq versus T curve that breaks away from the extrapolated liquid curve, but no information is yet available on where or how it breaks away. There is also... 

It was established by the earlier authors that the specific volume of polymers diminishes linearly with the temperature until the Tg. Below this temperature the diminution continues but at a small rate, as shown in Figure 5.4 obtained by Fox and Floiy for fractions of polystyrene of different molecular weights. Moreover, it was observed " that all the volume-temperature curves of the liquid state above the transition temperature, if they are extrapolated, intersect each other practically at the same point, at absolute zero temperature (V(o)iiq = 0.7674 cm /g). This volume was considered as the remaining space between atoms and molecules when no movement is allowed. Kanig ° proposed that the difference between the volume observed at absolute zero temperature and the volume measured at the transition temperature was constant for all polymers and equal to 0.0646 cm /g. This volmne difference was considered the space which, in the amorphous solid, is available for oscillations. 

FIGURE 4 The specific volume-temperature cooling curves obtained on the fully cured 1007, 1004,1001, and 828/DDS resins. 

Fig. 2.2. The specific-volume-temperature cooling curves for the epoxy resin EPON lOOlF/DDS fully cured at four different rates of cooling as indicated. The corresponding TgS identified by the intersection point of the equilibrium and glass lines are listed.

FIGURE 3.9 (a) Specific volume versus temperature for A, an amorphous polymer, and B, a partly crystalline polymer, (b) Volume-temperature curve for pure poly(V,At -sebacoyl piperazine), for which is 82°C and is 180°C-181°C. (Data from Flory, P. J., and H. K. Hall, J.Am. Chem. Soc., 73, 2532, 1951.)... 

Humid volumes are given by the curves entitled Volume mVkg diy air. The volumes are plotted as func tions of absolute humidity and temperature. The difference between dry-air specific volume and humid-air volume at a given temperature is the volume of water vapor. 

As the temperature is decreased, free-volume is lost. If the molecular shape or cross-linking prevent crystallisation, then the liquid structure is retained, and free-volume is not all lost immediately (Fig. 22.8c). As with the melt, flow can still occur, though naturally it is more difficult, so the viscosity increases. As the polymer is cooled further, more free volume is lost. There comes a point at which the volume, though sufficient to contain the molecules, is too small to allow them to move and rearrange. All the free volume is gone, and the curve of specific volume flattens out (Fig. 22.8c). This is the glass transition temperature, T . Below this temperature the polymer is a glass. 

Figure 28 shows the key features of the humidity chart. The chart consists of the following four parameters plotted as ordinates against temperature on the abscissas (1) Humidity H, as pounds of water per pound of dry air, for air of various relative humidities (2) Specific volume, as cubic feet of dry air per pound of dry air (3) Saturated volume in units of cubic feet of saturated mixture per pound of dry air and (4) latent heat of vaporization (r) in units of Btu per pound of water vaporized. The chart also shows plotted hiunid heat (s) as abscissa versus the humidity (H) as ordinates, and adiabatic humidification curves (i.e., humidity versus temperature). Figure 28 represents mixtures of dry air and water vapor, whereby the total pressure of the mixture is taken as normal barometric. Defining the actual pressure of the water vapor in the mixture as p (in units of mm of mercury), the pressure of the dry air is simply 760 - p. The molal ratio of water vapor to air is p/(760-p), and hence the mass ratio is ... 

The specific volume curves intersect at a point where Ar = 0 the latent heat, on the contrary, changes only very slightly with the temperature, and its curve is either horizontal, or exhibits a maximum, falling off slightly at higher pressures, and probably approaching the p axis. Thus, when Ar = 0, L f has a considerable positive value, and when L/ = 0, Ar has (probably) a considerable negative value. 

Figure 3 Curves of specific volumes vs. temperature for poly(vinyl acetate) measured on cooling. Equilibrium values measured 0.02 h and 100 h after cooling, as indicated. Tg and Tg are glass transitions respectively at fast and slow cooling rate. Reproduced from Ref. [2] with permission of John Wiley Sons, Inc.

Figure 5 The upper panel shows the logarithm of the specific volume as a function of temperature for a cooling rate T = 52.083 10-6, with error bars determined from 55 independent cooling runs. The lines are fits with a constant expansion coefficient in the melt (continuous line) and glass phase (dashed line), respectively. The lower panel shows the common fit curve for all cooling rates in the melt and fit curves in the glass for four cooling rates given in the legend.

Applying this prediction to the cooling rate dependence of a break points in the specific volume curves, one obtains a Vogel-Fulcher temperature of To = 0.35 that agrees well with that determined from the temperature dependence of the diffusion constant in this model, which is T = 0.32. 

---