Volume recovery isothermal

Figure 5.16 Isothermal volume recovery curves of glucose by contraction and expansion at T = 30°C after quenching the samples to 25 °C and 40°C, equilibration of the samples at those temperatures and reheating to 30°C. The equilibrium volume at 30°C is denoted Uqo. Drawn after data from Kovacs (1963).

Importantly for the structural recovery of glasses, the model predicts an equilibrium decay function which is of KWW form (jS= 1 —n), see equation (81), for even a single primitive species. Thus the requirement of a non-exponential decay function is fulfilled by the model. Although the other models use a broad relaxation function to describe behavior, they neither make the prediction of equation (91), nor can the general equation (89) result from them. In general, n and t can be functions of Tand d however, we treat only the case where t is a function of 8, i.e. t = T (r, 5(t )). Then, rewriting equation (86) in terms of t and identifying the macroscopic variable 0 with the departure from equilibrium 5, we find that, for isothermal volume recovery, equation (89) becomes... 

As pointed out earlier, most aging studies of blends have employed enthalpy relaxation and very few measurements have been made using volume relaxation. Notably, the volume changes during isothermal volume recovery are small, typically of the order of 1% or less and require high-precision measurements. 

The simplest volume recovery experiment performed is the down-jump. In this experiment, a material initially above Tg and at equilibrium is subjected to a temperature down-jump to an aging temperature Ta below Tg. The isothermal evolution of volume at T, as indicated by the downward arrow in Figure 1, is monitored with time via length or volume dilatometry. Figure 2 shows tjq)i-cal data replotted from Kovacs data (9) for a series of aging temperatures for poly(vinyl acetate). These curves, called intrinsic isotherms, are plotted as the relative departure from equilibrium 5 versus the logarithm of time, with S defined as... 

For the case of isobaric temperature jumps or isothermal pressure jumps, these equations revert to the KAHR equations for volume recovery using (dS/dP)r = -Ak and (dS/dT)p = Aa. Ak is the difference in the compressibilities of the liquid and glass at Tg. For enthalpy recovery, (d5/dP)r = VTAa and (dS/dT)p = ACp. 

One of the (apparently) most complicated features of the kinetic behavior of glasses arises when the material is allowed to recover isothermally for a period of time t insufficient to reach equilibrium, and then heated to a higher temperature and allowed to recover. As shown in Figure 31, the volume departure from equilibrium can cross over the actual equilibrium and exhibits a maximum which depends upon the actual thermal history applied to the sample. These have been referred to as crossover or memory effects. As will be seen, they arise from the fact that the response function (e.g. for volume recovery) exhibits behavior equivalent to a multiplicity of retardation mechanisms. 

There are then three important facets to equation (92). First, at equilibrium, the decay function is non-exponential (equation 90). Second, the observed retardation time depends upon S and T. Third, both n and t appear inside the integral, which implies that the history-dependent change of both these parameters is inherent in the model and so they affect the volume-recovery response differently than in the Narayanaswamy-KAHR-Moynihan-type models. For the same isothermal recovery after a T-jump, we can compare equation (92) with the response for the KAHR model... 

For describing thermally stimulated current associated with transitions in amorphous polymers, the multiple order parameter concept has been used. This concept has been widely discussed in past years and a good fit of volume recovery under isothermal and isobaric conditions through the glass transition has been presented by Kovacs et al. The relative volume departure of the system from equilibrium, 5, has been assumed to have a rate of change t,- of... 

The decrease in volume that accompanies physical aging is known as volume recovery or volume relaxation. Dilatometry (Dil) can be used to follow the volume relaxation in glasses by monitoring the time-dependence of the volume change on aging. The material is either cooled from above Tg to the aging temperature 7], (down-jump) and the isothermal volume contraction is measured or the sample is heated in the glassy state (up-jump), in which case an expansion follows. 

Early studies using resins for isolation and analysis of trace organics, such as pesticides, PCBs, and organic acids, from small volumes of water showed excellent recovery and the potential of easy application to environmental samples. Isotherm studies in distilled water were used to define the sampling parameters for quantitative analysis of these compounds. Later, studies using resin samplers for large-volume environmental samples were extrapolated from the early low-volume resin work of Junk et al. (5,14) and Thurman et al. (27) (see Table I). 

Owing to the fact that the glassy state is a nonequilibrium or metastable state, the thermodynamic properties of a glassy system (volume, enthalpy, etc.) in isothermal conditions will evolve toward thermodynamic equilibrium. The evolution of the volume is usually expressed in terms of = p — Ve)/Ve, where v and are, respectively, the specific volume at time t and at equilibrium. After a T-jump cooling experiment, the variation of 8 with time, in isothermal conditions, follows trends similar to those shown (18) in Figure 12.16. This process is known as structural recovery. 

The key impurities present in a typical ROG for the recovery of H2 by a PSA process are bulk Ci and C2 and dilute C3 and C4 hydrocarbons. Figures 10.11 and 10.12 describe the pure gas adsorption isotherms of the components of ROG at 30 °C on the BPL activated carbon and a silica gel sample (Sorbead H produced by Engelhard Corp.), respectively.31 These data were also measured in Air Products and Chemicals, Inc. laboratories. It may be seen that the carbon adsorbs C3+ hydrocarbons very strongly. Consequently, desorption of these hydrocarbons from the carbon by H2 purge becomes rather impractical requiring a large volume of purge gas. 

A similar equation can be derived for the enthalpy. It turned out that this modification to give the retardation time a free volume dependence, was successful in describing one-step isothermal recovery but unsuccessful in describing memory effects. Kovacs, Aklonis, Hutchinson and Ramos (1979) attributed the latter to the contributions of at least two independent relaxation mechanisms involving two or more retardation times. These authors proposed a multiparameter approach, the so-called KAHR (Kovacs-Aklonis-Hutchinson-Ramos) model. The recovery process is divided into N subprocesses, which in the case of volumetric recovery may be expressed as ... 

In experiment B, for example, a -7.5°C temperature decrease is used to perturb the system from equilibrium. Recovery is allowed to take place until the point is reached at which the application of a 2.5°C temperature increase results in the sample obtaining its equilibrium volume at the temperature T -5°C. Thus, immediately after the second temperature jump, "the sample is apparently at equilibrium, having been brought to its equilibrium volume at the temperature T -5°C. If one monitors the volume after the attainment of this "equilibrium", one finds a spontaneous isothermal increase in volume followed by a subsequent decrease in volume back to the now real equilibrium condition. Different thermal pathways will result in different extents of memory as shown in the figure. If more complex temperature treatments are used, even more complex memory behavior will result. 

Work by Kovacs [75] on the volume relaxation of poly(vinyl acetate) highlighted the nonlinearity in the kinetics of isothermal recovery processes, and that the distribution of relaxation times, t, was necessary if the memory effects in glasses were to be explained. These ideas are incorporated in some of the phenomenological models developed to describe the aging of a glass and met in earlier sections. 

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