Ion Dranca/Chem.J. Mold. 2008, 3 (1), 31-43The steps shift to higher temperature with increasing the heating rate. The application <strong>of</strong> the isoconversionalmethod to several systems demonstrated [18] that the variability in E correlates with the dynamic fragility. For maltitoland glucose the reported values <strong>of</strong> the dynamic fragilitry, respectively, are 75 [69] and 70 [70]. As these values are notvery large, we may expect a moderate variability in E throughout the glass transition. The actual dependencies <strong>of</strong> theeffective activation energy on conversion are shown in Fig. 10. As expected, we observe a moderate decrease for bothsystems. For maltitol the activation energy decreases from ~250 to ~150 kJ mol -1 whereas for glucose from ~220 to~170 kJ mol -1 .2.42.22.0C p/ J g -1 K -11.81.61.4C p,l1.2C p,g1.00 20 40 60 80Fig. 9. Temperature dependence <strong>of</strong> the heat capacity for maltitol (circles) and glucose (squares).C p . l . and C p . g . show the values extrapoled to the T gvalues (dashed lines)that are used in evaluating by eq. 7T / o CThe E vs plots (Fig. 10) were converted into E vs T plots by replacing with temperature which is estimatedas an average <strong>of</strong> the temperatures corresponding to this at different heating rates.260240E / kJ mol -12202001801601400.0 0.2 0.4 0.6 0.8 1.0Fig. 10. Variation <strong>of</strong> the activation energy with the extent <strong>of</strong> relaxationfor the glass transition in maltitol (circles) and glucose (squares)The resulting dependencies are shown in Fig.11 and 12. As mentioned earlier, a decrease in the effective activationenergy with temperature is typical for the glass transition (-relaxation). We observed similar effects for polystyreneand polystyrene composite [71] as well as for poly(ethylene terephthalate) and boron oxide [18]. For the most fragilesystem studied [18], poly(ethylene terephthalate) (m=166), the value <strong>of</strong> E decreased more than 2 times per 7 o C, whereasfor boron oxide (m=32) it decreased less than 1.5 times per 57 o C.240200E / kJ mol -1160120800-20-10400 10 20 30 40 50 60T / o CFig. 11. Variation <strong>of</strong> the activation energies for the sub-T g(solyd symbols) and T g(open symbols) relaxation <strong>of</strong> maltitolwith average temperature <strong>of</strong> the process. Numbers by the points represent annealing temperatures39
Ion Dranca/Chem.J. Mold. 2008, 3 (1), 31-43Maltitol and glucose have intermediate fragilities <strong>of</strong> about the same value so that they demonstrate similarvariability in E that amounts to 1.7 times 10 o C for maltitol and 1.3 times per 8 o C for glucose. Therefore, the correlation<strong>of</strong> the variability in E with the fragility appear to hold for these two systems as well.240200-1E / kJ mol16012080-25 -20-10400 10 20 30 40 50T / o CFig. 12. Variation <strong>of</strong> the activation energies for the sub-T g (solid symbols) and T g (open symbols)relaxation <strong>of</strong> glucose with average temperature <strong>of</strong> process. Numbers by the pointsrepresent annealing temperaturesThe Arrhenius activation energies reported for the -relaxation in maltitol and in glucose span a rather widerange which is not surprising as their values depend on the temperature and, therefore, on the temperature region usedtheir determination. For this reason, single value <strong>of</strong> E (440 [72] and 460 [69] kJ mol -1 ) reported for maltitol are difficultto compare with our variable values. On the other hand, our dependence fits well within the reported [73] decrease inE from 560 to 50 kJ mol -1 with increasing temperature. Single values <strong>of</strong> E for the -relaxation in glucose are 180, [74]320 [75], and 420 kJ mol -1 [70].It should be stressed that the effective activation energy landscapes presented in Figs. 11 and 12 are remarkablysimilar to the temperature dependence <strong>of</strong> the effective activation energy for stress relaxation as well as for viscous flowin polymers. These latter also show a significant increase in E on approaching the glass transition region from the glassystate, followed by a significant decrease in E while further relaxing toward the liquid state. This effect was originallypredicted by Fox and Flory [76] and observed experimentally by McLoughlin and Tobolsky [77]. Passing <strong>of</strong> the Evalue through a maximum can be understood in terme <strong>of</strong> molecular cooperativity that is very low significantly below T gwhere relaxation occurs via a noncooperative local process that has a small activation energy. As the temperature rises,the molecular mobility intensifies increasing cooperativity and the activation energy that reach their maximum in theregion <strong>of</strong> T g. Above T g, the free volume starts to quickly increase with temperature so that cooperativity decreases, andthe effective activation energy drops down to the values characteristic <strong>of</strong> the viscous flow.3.5. Sizes <strong>of</strong> Cooperatively Rearranging Regions in maltitol and glucose. Figure 9 displays C pvs T data formaltitol and glucose. The data agree well with the earlier measurements [69, 78]. By applying Eqs. 7, 8, 9, 10 to the C pdata (Fig. 9) we determined the values <strong>of</strong> V to be 30.6 (maltitol) and 36.4 (glucose) nm 3 . The values <strong>of</strong> the characteristiclength were determined as =(V ) 1/3 yelding, respectively, 3.1 and 3.3 nm. The values are very close to each other andcomparable to the value reported [62] for sorbitol, 3.6 nm. By its meaning the value <strong>of</strong> provides an estimate <strong>of</strong> theaverage distance between the mobility islands in the heterogeneous glassy system [48].The closeness <strong>of</strong> the valuessuggests similarity <strong>of</strong> the heterogeneous structures <strong>of</strong> the maltitol and glucose glasses. The C pdata can also be used toestimate the number <strong>of</strong> molecules involved in the cooperatively rearranging region as [48].The resulting values are ~90 for maltitol and 190 for glucose. The value for glucose is almost the same as forsorbitol (N =195 molecules [48]), whose molecular mass is similar to that <strong>of</strong> glucose.3.6. Sub-T gRegion. While reheating polyvinyl chloride samples annealed significantly below the glass transitiontemperature, T g, Illers [79], detected small endothermic DSC peaks that appears before the main glass transition step.The effect was later studied more extensively for metallic glasses by Chen [80] and for various polymers by Bershteinand Egorov [81]. An interpretation <strong>of</strong> the effect was given by Chen as a partial enthalpy relaxation and recovery thatoccur at the expense <strong>of</strong> the faster part <strong>of</strong> a broad relaxation spectrum <strong>of</strong> the glassy state. It was suggested [80, 81]that effective activation energy, E <strong>of</strong> the underlying process can be determined from the shift in the annealing peaktemperature, T pwith the heating rate, q asln qE R1(13)dT pwhere R is the gas constant. The E values determined this way [80, 81] were found to be several times smaller thanthe activation energies <strong>of</strong> the -relaxations in the respective glassy systems. The annealing peaks are easily produced40
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