Critical free volume

The constant B is not an activation barrier but is related to the critical free volume for transport, V, and to expansivity. In polymer electrolytes, V is generally taken as fixed by the size of the polymer segment rather than the motion of the ion since the polymer strands must move before either cations or anions can be transported. 

The propagation kinetic constant remains relatively constant at low to moderate conversions where diffusion of the smaller monomer molecules is unhindered. As the critical free volume for the diffusion control of propagation is reached, kp begins to decrease. Diffusion of monomer molecules is now the rate controlling step for propagation. As polymerization continues and conversion increases (free volume decreases), the diffusion of the monomer and kp drastically decrease. 

The probability distribution, Eq. 10.11, can be used in Eq. 10.2 as an estimate for evaluating DL. Above the critical free volume D(V) is probably nearly constant therefore,... 

Vfcr critical free volume of the system when kt becomes diffusion controlled... 

Efcrp critical free volume when kp becomes diffusion controlled free volume of species j at its glass transition temperature Xn number-average degree of polymerization... 

Fig. 24. The fractional free volume / at the glass temperature T, versus the fractional critical free volume B required for a segmental motion for various polymers assuming the expansion factor a for the occupied state is zero, O, or equal to that for the glass, see also Table 6...

Cohen and Turnbull s critical free-volume fluctuations picture of selfdiffusion in dense liquids is similar to the vacancy model of self-diffusion in crystals. However, in crystals individual vacancies exist and retain their identity over long periods of time, whereas in liquids the corresponding voids are ephemeral. The free volume is distributed statistically so that at any given instance there is a certain concentration of molecule-sized voids in the liquid. However, each such void is short-lived, being created and dying in continual free-volume fluctuations. The Frenkel hole theory of liquids ignores this ephemeral, statistical character of the free volume. 

In eq. (5.28) /g is the critical free-volume fraction at Tg and a is the volumetric (dilatational) thermal coefficient of expansion of the sub-cooled liquid. Then, the shift factor discussed in Section 5.3.1 is... 

Critical free volume at the onset of the gel effect (m ) Polymerization rate (mol/m s)... 

Fig. 4. Sdiematic tempoature dependence of reactivity in solid polynia s (A) Polyiner-polynier leaaion (difTnsion controlled). (B) Polymer side diain-anall molecule reaction (difibaon controlled). (Q Intramolecular main-chain sdsaon. (D) Intramolecular reaOion in polymer side dmin. (E) Intramolecular reaction of sm molecule with small critical free volume...

When photochromic reactions of spirobenzopyran, azobenzene, and fulgide in polymer films are compared, it is found that the critical free volume n ded fw a reaction is relatively large for spirobenzopyran, but very small for azobenzene and fulgide. The thermal decoloration reaction of spirobenzopyran in a solid film below was much slower than in solutions and was considered diffusion-controlled. Thermal cis to trans isomerization of azobenzene in polycarbonate at room temperature proceeded at a rate comparable to the rate in solutions, except for an anomaly at the initial stage tjjg most part of this reaction can be considered chemically-... 

Two types of cteviation from the first-orda kinetics are noted for photo-aialthamal isomerization reactions in polymer films. The first is the normal type, in which the reaction rate is the same as or smaller than that in solutions at the initial stage and then progressively becomes smaller. Typical examples are thermal decoloration of the photocolored merocyanine form of spirobenzopyran molecularly dispersed in or chemically bound to a polymer matrix > and photoisomerization of the transazobenzene residue incorporated in polymer main chains The first interpretation for the decoloration of the merocyanine form assumed the existence of different isomers, each of which fades independently following first-order kinetics On the other hand, Kryszewski et al. proposed the kinetic matrix effect, which means that the distribution of free volume may lead to the deviation from first-order kinetics. His idea was based on the finding that deviations from first-order kinetics can be observol even in simple molecules such as azobenzene which has only one trans or cis isomeric form. The effect of free volume distribution on reactivity was further demonstrated by studies of annealed polymer films The distribution function of free volume as well as the critical free volume v were estimated for the merocyanine form of spiropyran in poly(methyl methacrylate) derivatives of azobenzene in polystyrene and azobenzene in polycarbonate The deviation from first-order kinetics was also observed in cyclizing imidization of model poly(amic acid) in a polyamide matrix... 

The time scale over which an event occurs determines wlm molecular motitms and interactions influence a particular event In the case of photoisomerization, the reaction proceeds only with the molecules which lie at the sites where the reactant is mobile dining the exdted lifetime. Consequently a very small quantum yield or an apparent rate coefficient is often observed for the photoreaction whidi requires a large critical free volume, e.g. for the photodecoloration of the merocyanine form of spirobenzo-pyran in polycarbonate film... 

In a photophysical process with some critical free volume, e.g., excimer formation of oligostyrene diastereomers in PMMA and l,3-di-(l-pyrenyl)propane in polystyrene the steady-state fluorescence intensity ratio of the excimer to the monomer... 

The photodecoloration of the cyclized form of fulgide in PMMA film proceeds following first-order kinetics, and its quantum yield (4> = 0.06) is the same as that in toluene solution This fact suggests a very small critical free volume v, and also suggests that the ring-opening decoloration reaction is not restricted by the mobility of Ae matrix polymer at room temperature. The apparent rate of photodecoloration... 

Unfortunately, there is no way to verify experimentally the validity of these two theories. Although the zero conformational entropy can be calculated, the second-order transition temperature, Ti, is experimentally nonverifiable because of the foregoing kinetic freeze-in phenomenon. On the other hand, the free volume , responsible for the experimentally observed glass transition, cannot be quantified because, depending on the definition used, the estimated values of the critical free volumes differ by several orders of magnitude [18]. 

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