Relaxation isothermal

En remplagant P par AP dans les expressions (81), et en tenant compte de (79), (80) et de (82), on obtient finalement l equation difteren-tielle qui controle la relaxation isotherme de P ... 

Horizontal shift distances at different points between the relaxation isotherms were measured. For PC (7), PST (8), and the 25/75 blend, the shift factors are independent of time, and complete master curves can be obtained by simple tj nf g laxation isotherms along... 

Deduction of Shift Factors. Time-temperature shift factors for the blends were obtained by shifting the experimental relaxation isotherms to the calculated master curves (10). The temperature and time dependence of the shift factors of the 75/25 and 50/50 blends are represented in Figures 10a and 10b at t = 10 sec and t = 1000 sec for a reference temperature of 140°C. The empirically determined shift factors of the pure components are given in these figures by dotted lines their temperature dependence is of the WLF type. 

Using the general expression for the relaxed (isothermal) modulus one can write... 

Volume Relaxation. Isothermal volume relaxation was assessed for the a-PS/PPO blend system using a precision mercury dilatometer up to a final aging time of proximately 2 days. Four aging temperatures of Tg-lS C, T,-30°C, Tg-4S C, and Tg-60 C were used, and the resulting relaxation rates ( Pv = (l/V)dV/dlog(ti) ) (7P) are presented in Figure 3 (initial curvature in volume rdaxation plots was not used in... 

Fig. 11. Stress relaxation isotherms for a polyisobutylene amorphous polymer. After To-bolsky (25).

If a glass is allowed to relax isothermally, it is difficult to be sure that its final volume represents structural equilibrium, rather than being an unstable state in which it is trapped by its high viscosity. Since glasses can be prepared with volumes above or below the equilibrium value, it is possible to establish that they approach the same state from both directions, as illustrated in... 

The first term on the right is the common inverse cube law, the second is taken to be the empirically more important form for moderate film thickness (and also conforms to the polarization model, Section XVII-7C), and the last term allows for structural perturbation in the adsorbed film relative to bulk liquid adsorbate. In effect, the vapor pressure of a thin multilayer film is taken to be P and to relax toward P as the film thickens. The equation has been useful in relating adsorption isotherms to contact angle behavior (see Section X-7). Roy and Halsey [73] have used a similar equation earlier, Halsey [74] allowed for surface heterogeneity by assuming a distribution of Uq values in Eq. XVII-79. Dubinin s equation (Eq. XVII-75) has been mentioned another variant has been used by Bonnetain and co-workers [7S]. 

If the coiiplin g parameter (the Bath relaxation constan t in IlyperChem), t, is loo Tight" (<0.1 ps), an isokinetic energy ensemble results rather than an isothermal (microcan on leal) ensemble. The trajectory is then neither canonical or microcan on-ical. You cannot calculate true time-dependent properties or ensemble averages for this trajectory. You can use small values of T for Ih CSC sim ii lalion s ... 

The isothermal curves of mechanical properties in Chap. 3 are actually master curves constructed on the basis of the principles described here. Note that the manipulations are formally similar to the superpositioning of isotherms for crystallization in Fig. 4.8b, except that the objective here is to connect rather than superimpose the segments. Figure 4.17 shows a set of stress relaxation moduli measured on polystyrene of molecular weight 1.83 X 10 . These moduli were measured over a relatively narrow range of readily accessible times and over the range of temperatures shown in Fig. 4.17. We shall leave as an assignment the construction of a master curve from these data (Problem 10). 

Shown in Fig. 4a is the temperature dependence of the relaxation time obtained from the isothermal electrical resistivity measurement for Ni Pt performed by Dahmani et al [31. A prominent feature is the appearance of slowing down phenomenon near transition temperature. As is shown in Fig. 4b [32], our PPM calculation is able to reproduce similar phenomenon, although the present study is attempted to LIq ordered phase for which the transition temperature, T]., is 1.89. One can confirm that the relaxation time, r, increases as approaching to l/T). 0.52. This has been explained as the insufficiency of the thermodynamic driving force near the transition temperature in the following manner. 

The calculated results are demonstrated in Figs. 7a and 7b [25, 33). The system maintained at T—1.85 is subject to down quenching with a temperature step AT = 0.05 and isothermal aging is followed at each temperature to equilibrate the system. After reaching the equilibrium at T =1.5, the operation is reversed up to T =1.85. In Fig. 7a, one sees that the resultant relaxation time is shortened with decreasing temperature and... 

Additional isothermal treatments at neighbouring temperatures small step annealing) yield plateau values of resistivity corresponding to equilibrium values at certain temperatures which reflect the order parameter in thermal equilibrium as a function of temperature ( equilibrium curve , curve 4 in Figure 1). This study can be used for an analysis of the kinetics of order-order relaxations (see Figure 3 below). 

For isothermal measurements, it is advisable to use a furnace of low thermal capacity unless suitable arrangements can be made to transport the sample into a preheated zone. The Curie point method [132] of temperature calibration is ideally suited for microbalance studies with a small furnace. A unijunction transistor relaxation oscillator, with a thermistor as the resistive part with completion of the circuit through the balance suspension, has been suggested for temperature measurements within the limited range 298—433 K [133]. 

The feed is charged all at once to a batch reactor, and the products are removed together, with the mass in the system being held constant during the reaction step. Such reactors usually operate at nearly constant volume. The reason for this is that most batch reactors are liquid-phase reactors, and liquid densities tend to be insensitive to composition. The ideal batch reactor considered so far is perfectly mixed, isothermal, and operates at constant density. We now relax the assumption of constant density but retain the other simplifying assumptions of perfect mixing and isothermal operation. 

As shown in Figure 3.5.3, the relaxation time versus pressure curves are dramatically different from those obtained using CF4 at a temperature well above its critical point. Indeed, while the overall form of the Tx curves for CF4 in fumed silica was similar to that of the bulk gas, the shape of the Ti plots for c-C4F8 in Vycor more closely resembles that of an adsorption isotherm (Ta of CF4 in Vycor is largely invariant with pressure, as gas-wall collisions in this material are more frequent than gas-gas collisions). This is not surprising given that we expect the behavior of this gas at 291 K to be shifted towards the adsorbed phase. The highest pressure... 

The measured NMR signal amplitude is directly proportional to the mass of adsorbate present, and the NMR signal versus pressure (measured at a fixed temperature) is then equivalent to the adsorption isotherm (mass of adsorbate versus pressure) [24-25]. As in conventional BET measurements, this assumes that the proportion of fluid in the adsorbed phase is significantly higher than the gaseous phase. It is therefore possible to correlate each relaxation time measurement with the calculated number of molecular layers of adsorbate, N (where N = 1 is monolayer coverage), also known as fractional surface coverage. 

Figure 8. Relaxation strength versus crystallinity in isotactic polypropylenes of Figure 7. Unrelaxed low temperature modulus Q)> relaxed y modulus (A), relaxed 0 modulus (0), relaxed a modulus ( >). Filled symbols are for the isothermally crystallized (68%) specimen. 

In observing the time dependent changes in birefringence and stress-optical coefficient, for elongated samples at 25 C, it was found that the rate of crystallization of high trans SBR s was very much faster, some 10 times more rapid, than that for NR (8). This is consistent with the reported rates of isothermal crystallization for NR (2.5 hours at -26°C) and for 807. trans-1,4 polybutadiene (0.3 hours at -3°C) in the relaxed state (12). 

The difference between the static or equilibrium and dynamic surface tension is often observed in the compression/expansion hysteresis present in most monolayer Yl/A isotherms (Fig. 8). In such cases, the compression isotherm is not coincident with the expansion one. For an insoluble monolayer, hysteresis may result from very rapid compression, collapse of the film to a surfactant bulk phase during compression, or compression of the film through a first or second order monolayer phase transition. In addition, any combination of these effects may be responsible for the observed hysteresis. Perhaps understandably, there has been no firm quantitative model for time-dependent relaxation effects in monolayers. However, if the basic monolayer properties such as ESP, stability limit, and composition are known, a qualitative description of the dynamic surface tension, or hysteresis, may be obtained. 

Hysteresis was generally observed in the compression-expansion cycles of the force-area isotherms, indicating that the timescale for relaxation of the fully compressed film back to its expanded state was slower than the movement of the barrier of the Langmuir trough. Our studies, like many others, imply that monolayers are metastable and that reversible thermodynamics can only be applied to their analysis with caution. 

The fitting of space relaxation data using Eq. (1) to this mechanistic scheme (space relaxation data are always isothermal, because transient temperature effects are not relevant for the amplitude change of a concentration disturbance this is just an advantage of wavefront analysis of reaction kinetics), reported in (3, 5, 12), supposing a Langmuir type chemisorption for (CO) and (I O) has confirmed that (see Figure 10, 11) ... 

Poisoning is caused by chemisorption of compounds in the process stream these compounds block or modify active sites on the catalyst. The poison may cause changes in the surface morphology of the catalyst, either by surface reconstruction or surface relaxation, or may modify the bond between the metal catalyst and the support. The toxicity of a poison (P) depends upon the enthalpy of adsorption for the poison, and the free energy for the adsorption process, which controls the equilibrium constant for chemisorption of the poison (KP). The fraction of sites blocked by a reversibly adsorbed poison (0P) can be calculated using a Langmuir isotherm (equation 8.4-23a) ... 

---