Pressure Steps

In addition to temperature, pressure can also be introduced as a variable. Three types of volume-recovery experiments can be considered in which the pressure is dif-ferent from, or is not maintained at, 1 bar. In the first, the temperature is stepped up or down at constant pressure but for pressures exceeding 1 bar; in the second, the pressure is stepped at constant temperature; and in the third, volume recovery is examined for densified glasses formed by compressing the liquid, cooling it to below Tg, and then releasing the pressure [Robertson et al., 1985]. For the first two

174 VOLUME RELAXATION AND THE LATTICE–HOLE MODEL

FIGURE 4.8 Increase in T_gand glassy volume of PVAc as the cooling rate is increased over the range of 10⁴. (Adapted from from Vleeshouwers [1993] and Vleeshouwers and Nies [1994].)

experiments, polystyrene (PS) was used as the example material, allowing comparison with the experimental results [Rehage and Goldbach, 1966; Goldbach and Rehage, 1967; Rehage, 1970; Oels and Rehage, 1977].

To compute the recovery kinetics of PS under different histories of pressure and temperature, various material parameters are needed, those given in Table 4.3 and others described below. The first three quantities in Table 4.3 are the S-S characteristic parameters for the pressure–volume–temperature properties of liquid PS. From these, all of the desired PVT properties and the free-volume functions for the liquid can be derived. For the glass, however, further PVT properties are needed. These include Tg

as a function of pressure, Tg(P), the thermal expansivity of the glass as a function of pressure, αglass(P), and the bulk modulus of the glass as a function of pressure, Bglass(P). For use in the calculations, these data can be used in the form of empirical equations. (Further details can be found in the Appendix of Robertson et al. 1985).

Other quantities in Table 4.3 are the nominal glass transition temperature, Tg, the time-temperature shift parameter of the liquid as a function of time-temperature and pressure, aT,p, and the transition region size parameters Ns and ζ.

TABLE 4.3 Parameters for Recovery Kinetics of PS P*= 7453 bar^a S-S characteristic parameters V*= 959.8 mm³/g^a

T*= 12,680 K^a

c_l= 13.3^b time–temperature shift parameters c2= 47.5 K^b

Tg= 373 K

τg= 1 h (3600 s) nominal relaxation time at glass transition Ns= 40 number of segments in free-volume region ζ= 12 size ratio for region controlling free volume

R= 5.3 translation factor between macroscopic and microscopic processes

aQuach and Simha [1977].

bPlazek [1965].

We first show the comparison of theory with Rehage and Goldbach’s volume recovery data from temperature steps for PS at 1 bar. These data are a companion to volume-recovery data from pressure steps, described below. The computation of volume recovery for PS is the same as that described above for PVAc. The polymer is assumed to be in equilibrium at the initial temperature T0, and then at time t= 0, the temperature is suddenly stepped to the final temperature T1. The data of Rehage and Goldbach [Rehage and Goldbach, 1966; Goldbach and Rehage, 1967] are shown in Figure 4.9 for steps from various initial temperatures, T0, to the final temperature, T1, of 90.70^◦C. The thermally polymerized PS examined had a number-average molec-ular weight of 500 kg/mol. The results basically agree with those of Kovacs [1958], Hozumi et al. [1970], Uchidoi et al. [1978], and Adachi and Kotaka [1982]. The ordinate in Figure 4.9 is the relative deviation of the volume from equilibrium; the volume difference in the denominator, (V0− V1), is that existing immediately after the step in temperature. The curves computed, given by the solid lines, were moved along the time axis until a reasonable fit with the data was obtained. The fit to the data in Figure 4.9 is fairly good and yields the parameter of R in Table 4.3. [The large difference between the value of R for PS (5.3) and that used previously for PVAc (0.0022) is believed to arise largely from different relaxation times at the assumed glass transition temperatures.]

The computation of volume recovery following pressure steps at constant tem-perature is analogous to that described above for temtem-perature steps under constant

FIGURE 4.9 Relative change in volume versus time of PS specimens after temperature steps of various magnitude at 1-bar pressure to 90.70^◦C. (V0, initial volume; V1, volume at 90.70^◦C.

(Adapted from Robertson et al. [1985]; data from Rehage and Goldbach [1966] and Goldbach and Rehage [1967].)

176 VOLUME RELAXATION AND THE LATTICE–HOLE MODEL

pressure. The polymer is assumed to be in equilibrium at the initial pressure p0

and temperature T. Then at time t = 0, the pressure is suddenly stepped to the final value p1without change in temperature. Rehage and Goldbach have measured the volume recovery following pressure steps from several elevated pressures down to atmospheric pressure at the temperature of 91.84^◦C [Rehage and Goldbach, 1966;

Goldbach and Rehage, 1967]. Their data are shown in Figure 4.10 along with the predictions computed. The parameters in the computation were the same as those used for the volume recovery following temperature steps discussed above, including the value of R obtained from fitting those data.

The fit between experiment and computation in Figure 4.10 is only approximate.

One feature reproduced in the calculation is the degree of spread between the data in Figures 4.9 and 4.10. Rehage and Goldbach drew particular attention to the much larger spread in the temperature-step than in the pressure-step data. This difference is also predicted by the calculation. However, it is unclear why there is not better agreement between theory and experiment. In contrast to the assumptions of the computation, the data seem to suggest that recovery from pressure steps is basically different from recovery from temperature steps. Although recovery from the pressure steps occurred at a higher temperature (91.84^◦C) than recovery from the tempera-ture steps (90.7^◦C), the recovery from the pressure steps is slightly slower. Lacking the usual S-shape slowdown as equilibrium is reached, however, the recovery from the pressure steps appears to be complete at nearly the same time as that from the temperature steps.

FIGURE 4.10 Relative change in volume versus time of PS specimens after pressure steps of various magnitude at 91.84^◦C to 1 bar. V0, initial volume; V1, volume at 1 bar. (Adapted from Robertson et al. [1985]; data from Rehage and Goldbach [1966] and Goldbach and Rehage [1967].)

FIGURE 4.11 Paths for attaining equilibrium volume below the glass transition temperature at atmospheric pressure with densified glass. (Adapted from Robertson et al. [1985].)

Densified glasses allow further exhibitions of structural recovery. For example, one can consider the following question: Is it possible to reach rapidly an equilibrium liquid state below the glass transition temperature at atmospheric pressure by pres-surizing the liquid, cooling it below the glass transition, and then deprespres-surizing it?

This sequence of steps is shown schematically in Figure 4.11. The liquid at A is pres-surized to B, cooled through the glass transition at C to D, and then deprespres-surized to E, the equilibrium volume at that temperature at atmospheric pressure. An alternative path, indicated by the dashed line, would be to cool the pressurized glass to room temperature at D^before releasing the pressure and then heating the depressurized glass from D^ to E. It would seem that for either path, the equilibrium volume at E would be more quickly reached in this way than if the glass had been cooled at atmospheric pressure, along the upper, glass line to the temperature of D and E, and allowing it to recover to E.

Although the densified glass at E in Figure 4.11 has the equilibrium volume, would it be in equilibrium? Various indirect experiments suggest that it would not. For example, Oels and Rehage [1977] found that all of their densified glasses, produced under pressures up to 5000 bar, tended at 22^◦C to expand with time. Yet one of these densified glasses had a volume very close to equilibrium at 22^◦C immediately following depressurization.

Since we are generally assuming that the total volume can be divided into filled and unfilled space, specimens of the same material having the same volume at the same temperature and pressure are assumed to have the same free volume. However, there

178 VOLUME RELAXATION AND THE LATTICE–HOLE MODEL

FIGURE 4.12 Development of the volume in time at 86, 90, and 94^◦C for a glass having the equilibrium volume and free volume at 90^◦C and a mean-squared fluctuation in free volume assumed to be 15% smaller than equilibrium. (Adapted from Robertson et al. [1985].)

is a parameter other than volume and average free volume that is used by the kinetic theory to describe the structure of the glass, and that is the free-volume distribution.

Therefore, in the following, only the free-volume distribution will be assumed not to be in equilibrium on arrival at E. There is still the question of whether the distribution has the symmetry of the equilibrium distribution. We assume that it does and consider only two distortions of the equilibrium free-volume distribution. We suppose that the temperature at which the PS is depressurized, corresponding to point E, is 90^◦C and the breadth of the free-volume distributions for 40 monomer-size regions is roughly 8% narrower or 8% broader than the equilibrium distribution. These distributions have mean-square fluctuations smaller and larger by 15% than at equilibrium. The time developments of the volume for these distributions are shown in Figures 4.12 and 4.13.

If maintained at 90^◦C, the densified glass with the narrower free-volume distribu-tion is seen in Figure 4.12 to go through a maximum before returning to the initial (and equilibrium) volume. The reason for the maximum is that the higher free-volume half of the distribution moves up toward equilibrium before the lower free-volume half moves down. This causes the average free volume to go through a maximum, and hence so does the total volume. Later, the lower half of the initial distribution will move downward to bring the entire distribution into equilibrium. In contrast, the densified glass with the broader free volume is seen in Figure 4.13 to go through a minimum. This occurs in like manner because the upper half of the initial distribution moves downward toward the equilibrium distribution before the lower half of the

FIGURE 4.13 Development of the volume in time at 86, 90, and 94^◦C for a glass having the equilibrium volume and free volume at 90^◦C and a mean-squared fluctuation in free volume assumed to be 15% larger than equilibrium. (Adapted from Robertson et al. [1985].)

initial distribution moves upward. An equilibrium distribution, of course, would not have changed.

Two other curves are shown in Figures 4.12 and 4.13. These assume that after depressurization at 90^◦C, the temperature is moved up or down by 4^◦C before recovery begins. (The initial displacements of these curves from the curves for 90^◦C arise from the thermal expansion of the glass.) If the free-volume distribution of the densified glass is narrower than at equilibrium, a maximum is superimposed on the volume curve at the beginning or the end of the transition, whichever is at the higher volume.

But the height of the maximum decreases as the temperature differs from that at depressurization. In contrast, if the free-volume distribution of the densified glass is broader than at equilibrium, a minimum is superimposed on the volume curve at the beginning or end of the transition, whichever has the smaller volume, although the minimum has essentially disappeared for the downstep to 86^◦C in Figure 4.13.

Although either a narrower or a broader distribution of free volume than at equilibrium seems possible for densified glasses, the results of Oels and Rehage [1977], and of Kogowski and Filisko [1986], suggest that densified PS has a narrowed free-volume distribution.

In the computation of the volume–recovery curves in Figures 4.12 and 4.13, only thermal agitation, or Brownian motion, was assumed for the driving force for recovery.

Stress field effects were not taken into account. However, the release of pressure from the densified glass is equivalent to an application of a negative pressure (i.e., an

180 VOLUME RELAXATION AND THE LATTICE–HOLE MODEL

expansion) to the stable mechanical system that had existed under the pressure, and this stress application could be a further driving force for the expansion of the glass.

Vleeshouwers and Nies have extended this work somewhat to include both temper-ature and pressure in the formation of glasses [Vleeshouwers and Nies, 1992, 1994, 1996]. Their work follows along lines similar to the above. Vleeshouwers and Nies found that at higher pressures, the free volume alone, as in Eq. (4.4), is no longer sufficient to describe mobility. However, by adding temperature as a second indepen-dent variable, in addition to free volume, they were able to describe the results of experiment satisfactorily.

4.6 CONTINUUM LIMIT OF FREE-VOLUME STATES AND THE