First-order kinetics, glass transition

The decoloration deviates from first-order kinetics below the glass transition temperature (Tg) although the reaction obeys first-order kinetics above T... 

There are several well known examples of photochemical reactions which exhibit a discontinuous change in rate constants and quantum yields when they are carried out in solid matrices and not in solution (see Table 2.5). The deviation from first-order kinetics for a reaction carried out in a matrix below its glass transition temperature is a characteristic feature of solid-state reactions. 

Brinker and coworkers also monitored the absorbance (A) at 355 nm (it it transition) in situ to ealeulate the kinetics of the ds trans isomerization of the azobenzene ligands contained in the nanocomposite film deposited on the ITO/ glass substrate in buffer solution (Liu, 2004). The first-order plot o ln Aao, Aq)/ (Aao—At)) vs. t was not perfectly linear in the entire region. The slope (rate constant, k) gradually decreases (t< 250 min) and remains constant (t> 250 min). The deviation from first-order kinetics is common for azobenzene ligands confined in sol-gel matrices or polymers (Bohm et al., 1996 Ueda et al., 1992). These data imply that the kinetics is composed of two parts, a fast one (ki) and a slow one ( 2). When t is small, the fast proeess is predominant. On the contrary, the slow process becomes predominant as time (i) increases. A double exponential equation was used to fit the data and evaluate ki and k2-... 

Glass transitions involve mainly the onset or freezing of cooperative, large-amplitude motion and can be studied using thermal analysis. Temperature-modulated calorimetry, TMC, is a new technique that permits to measure the apparent, fiequency-dependent heat capacity. The method is described and a quasi-isodiermal measurement method is used to derive kinetic parameters of the glass transitions of poly(ethylene terephthalate) and polystyrene. A first-order kinetics expression can describe the approach to equilibrium and points to the limits caused by asymmetry and cooperativity of the kinetics. Activation energies vary from 75 to 350 kJ/mol, dependent on thermal pretreatment. The preexponential factor is, however, correlated with the activation energy. 

In the glass transition region the approach to equilibrium may be approximated by first order kinetics as long as the distance fix)m equilibrium is small. The instantaneous number of high-energy configurations is represented by and the relaxation time by x ... 

Creation, motion, and destraction of holes are cooperative kinetic processes and may be slow. This leads to deviations from Eq. (1) if the measurement is carried out faster than the kinetics allows. Applied to the glass transition, one can write and solve a simple, first-order kinetics expression [40] based on the equilibrium expression, above ... 

However, when the dye is dispersed in a polymer matrix, the decay of the colour follows more complex kinetics. Initially, the decay appears to be first order and is essentially the same as that observed in the liquid. This is because some of the molecules are located in parts of the matrix where there is sufficient free volume for the rotation process to occur, which is why first-order kinetics, as in a liquid, are observed. After a very short period of time, the decay slows down and becomes non-exponential. This slower decay is associated with molecules which are constrained in the matrix and do not have sufficient free volume to move. So at this stage the colour change depends on the rate at which the molecule can gain enough volume to move. In other words, the decay process depends on the rate at which the matrix can create the necessary free volume. This creation of free volume is essentially the same process as that involved in the glass transition temperature discussed in Chapter 4. 

To get a kinetic expression, one makes use of the first-order kinetics of Fig. 2.8 and writes Eq. (3), with N (T) being the equilibrium number of holes, and N its instantaneous value. The rate constant, which was called k in Fig. 2.8, is replaced by 1/r, the relaxation time. By introduction of the heating rate q - (dr/dt), the change of time dependence to temperature dependence at constant heating rate is accomplished. Equation (3) describes the nottiso-thermal kinetics of the glass transition. A full solution of the equation is... 

Also, analysis of the kinetics and the glass transition temperature suggested greater interaction between aramid fibres and elastomer matrix. In particular, the degradation of PU or PU composites reinforced with aromatic PA or short carbon fibres followed first-order kinetics. 

By plotting/as a function of Ar, one may check the ten degree mle predictions validity. The predictions are valid if the values of Eqs. (9) and (15) superimpose onto a single curve. The temperature must be below a value which initiates other chemical processes or physical transitions such as glass transition, melting or other phenomena not associated with normal aging processes [73, 74]. Initiations of any of such processes would also not validate the first order kinetic assumption. 

X 10 min i and E = 5.29 x 10" J/mol, in which the solid line is the best fit of data to Eq. (14.6) with the values EJE = 0.337 and FJE = 0.194. Using these values of EJE and FJF, g( g( calculated from Eq. (14.6), and then the time to reach vitrification, is estimated from Eq. (14.4). Table 14.1 gives a summary of estimated for the first-order (n = 1) kinetics at various cure temperatures together with experimental results, where the glass transition temperature is assumed to be equal to the cure temperature = Tg) It is seen in Table 14.1 that the higher the cure temperature, the sooner vitrification is reached. The differences between the estimated and experimentally determined may be attributable to the assumption of first-order kinetics in the calculation of (see Eq. (14.4)). 

Although crystallization is controlled by kinetic factors, it is a phenomenon that is thermodynamic in origin (a first-order phase transition). On the other hand, the glass transition may be purely kinetic. We ll say more about that shortly, but one manifestation of this is the dependence of the T on the rate of cooling. It is observed at a somewhat lower temperature if a sample is cooled slowly than if it is cooled quickly (Figure 10-16). 

Several cautions are, however, in order. Polymers are notorious for their time dependent behavior. Slow but persistent relaxation processes can result in glass transition type behavior (under stress) at temperatures well below the commonly quoted dilatometric or DTA glass transition temperature. Under such a condition the polymer is ductile, not brittle. Thus, the question of a brittle-ductile transition arises, a subject which this writer has discussed on occasion. It is then necessary to compare the propensity of a sample to fail by brittle crack propagation versus its tendency to fail (in service) by excessive creep. The use of linear elastic fracture mechanics addresses the first failure mode and not the second. If the brittle-ductile transition is kinetic in origin then at some stress a time always exists at which large strains will develop, provided that brittle failure does not intervene. 

First-order solid-state amorphization occurs due to an entropy catastrophe [39] causing melting of superheated graphite and decompressed diamond below Pg when the entropy of the ordered crystal would exceed the entropy of the disordered liquid. This condition is resolved with the occurrence of a kinetic transition to a (supercooled) glass whereby the exact kinetic conditions during carbon transformation will be critically Pg-depen-dent [39]. It is important to consider the crystal to liquid transition and the effect of a superheated crystal whereof the ultimate stability is determined by the equality of crystal and liquid entropies [40]. When this condition is met, a solid below its Pg will melt to an amorphous solid, particularly... 

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