J. Phys. Chem. 1996, 100, 15255-15260
15255
Size Dependence of Tracer Diffusion in Supercooled Liquids Gerhard Heuberger and Hans Sillescu* Institut fu¨ r Physikalische Chemie, Johannes Gutenberg-UniVersita¨ t, Mainz, D-55099 Mainz, Germany ReceiVed: April 2, 1996; In Final Form: June 17, 1996X
We have determined by forced Rayleigh scattering the diffusion coefficients D of several photochromic tracers with van der Waals radii between 0.38 and 8 nm (the largest ones being photolabeled polystyrene micronetworks) in 10 glass-forming liquids at temperatures between the glass temperature Tg and ∼1.2Tg. The results were analyzed in terms of power law plots, D(T) ∝ T/η(T)ξ, where η is the solvent shear viscosity, and temperature shifts, D(T) ∝ T/η(T + ∆T). The shift ∆T was related with the width of the rotational correlation time distribution via the time-temperature superposition principle.
I. Introduction Tracer diffusion in supercooled liquids and polymers close to the glass transition temperature Tg has been the subject of many recent studies, and some puzzling phenomena have been discovered, in particular, a pronounced enhancement of translational diffusion in comparison with rotational diffusion and shear viscosity.1-5 The fast diffusion of very small molecules (He, N2, etc.) in polymers is almost independent of the R-process determining the glass transition.1 For larger tracers, the influence of R-relaxation upon the translational diffusion coefficient D becomes observable and can be accounted for by a phenomenological relation, D ∝ T/ηξ, where η is the shear viscosity, and the exponent ξ increases with increasing tracer size.3,6 In free volume interpretations ξ has been related with the ratio of tracer and matrix molecular jumps sizes7 and shapes.8 However, this explanation is difficult to reconcile with our finding3 that ξ < 1 for self-diffusion in o-terphenyl (OTP), where tracer and matrix molecules are identical, which requires ξ ) 1 unless different influences of free volume redistribution upon D and η are assumed. There is also the surprising observation that the rms displacement of a tracer molecule during its average rotational correlation time can extend many times over its own size.5,6,9 This phenomenon can be explained by assuming a domain structure where translational diffusion occurs predominantly in “fluidized” (mobile, loosely packed, etc.) domains.10-13 One should expect that the enhancement of translation should not be observable if the tracer size exceeds the domain size and diffusion averages over the spatial heterogeneity. This has been confirmed in experiments with variable tracer size.5,6,14 In this paper, we present the results of a more extended study with tracers of variable size and flexibility and 10 glass-forming liquids, some being identical with those of a recent dielectric relaxation study15 providing the corresponding data on rotational diffusion. The analysis of our data will provide some new aspects that are hoped to help elucidate the complexities of molecular motion at the glass transition. II. Experimental Section Glass-Formers (Chart 1). Preparation of Glass-Formers 1-5. Glass-former 1 was synthesized by esterification of the phenolic hydroxyl groups of phenolphthalein (Aldrich) with benzoylchloride (Aldrich), and 2, 4, and 5 by etherification of cresolphthalein (Aldrich), 4,4′-bis(octahydro-4,7-methano-5HX
Abstract published in AdVance ACS Abstracts, August 15, 1996.
S0022-3654(96)00968-9 CCC: $12.00
inden-5-ylidene)bisphenol (Kodak), and phenolphthalein with dimethyl sulfate (Aldrich) in a standard procedure. 3 was prepared by condensation of 9-fluorene (Aldrich) under acidic conditions with phenol and etherification of the product with dimethylsulfate. Glass-formers 6, 7, 8, and 9 were obtained from Aldrich, and 10 from F. Stickel (MPI fu¨r Polymerforschung, Mainz). All glass-formers were purified by distillation and recrystallization. Viscosity measurements for glass-formers 1-5 were performed by T. Pakula (MPI fu¨r Polymerforschung, Mainz). Viscosity data of 6-10 were taken from literature: 6, refs 16, 17; 7, ref 18; 8, refs 16, 17, 19; 9, ref 18; 10, refs 20, 21. Tracers (Chart 2). The photochromic dye molecules 2,2′bis(4,4-dimethylthiolan-3-one) (TTI) and the cesium salt of 4′(N,N-dimethylamino)-2-nitrostilbene-4-carboxylic acid (ONS) were prepared as previously described (ref 2). The anthracene derivative A-ONS was the same sample as used in ref 6. Aberchrome 540 (ACR), a fulgide dye, was obtained through the courtesy of the BASF Ludwigshafen. The fulgide derivative N-ACR was obtained from A. Dietrich (Institute Organic Chemistry, Universitat Mainz). Polystyrene network spheres (P6, P8) were prepared by radical copolymerization of styrene, diisopropenylbenzene, and p-chloromethylstyrene (molar ratio 100:5:1) in microemulsion. Surfactants used were tetradecyldiethanolamine for P6 (ref 22) and tetradecyltriethylammonium chloride for P8 (ref 23). The hydrodynamic radii were measured by dynamic light scattering in toluene, yielding about 6 nm for P6 and 8 nm for P8. The network spheres were labeled with ONS as described elsewhere (ref 24). Sample Preparation. Solutions of the dye tracers were prepared by stirring for 1-2 days at 20 K above Tg of the glassformer. The samples had a concentration of about 100 ppm (by weight) for the dyes and 1% for the micronetwork tracers. Solutions of low viscosity were filled with a syringe into glass cuvettes of 1 mm optical length. For viscous liquids a droplet was placed between two glass plates separated by a 0.7 mm Teflon ring. Samples of salol, glycerol, and 1,2-propanediol were prepared under dried N2 or in a glovebox. It should be noted that TTI is insoluble in glycerol and 1,2-propanediol. Experimental Setup. The experimental setup for the forced Rayleigh scattering (FRS) experiments using the 488 nm line of an argon ion laser was essentially the same as in our previous experiments.2,6 Typically, holograms were formed by irradiating light intensities of about 100 mW/mm2 for 10 ms with a laser spot size of 1 mm2. The grating distance d ) λ/2 sin(θ/2) can be varied from 100 µm to about 150 nm. The FRS scattering intensity could generally be fitted by the exponential I(t) ) [A © 1996 American Chemical Society
15256 J. Phys. Chem., Vol. 100, No. 37, 1996
Heuberger and Sillescu
CHART 1. Glass-Formers
exp(-t/τ) + B]2 + C with τ-1 ) 4π2D/d2 and small background corrections B and C. The samples were in close contact with brass sample holders for heating or cooling, respectively, using commercial regulation units where the thermocouple is at a position outside the cuvette. The temperature difference to the hologram position was determined by a calibration curve that was recorded in a separate run with a second thermocouple inside the sample. We estimate the absolute accuracy of the T values given in Figures 1-6 to be better than (1 K . III. Results The size dependence of the tracer molecules (Chart 2) was probed in two similar glass-forming liquids (PDE, CDE, Chart 1) differing only by two additional methyl groups in CDE, thus reducing the internal flexibility of this molecule. The experimental diffusion coefficients are compared in Figure 1 with the inverse shear viscosity η of the solvent via the Stokes-Einstein relation
Dη ) kBT/nπηrvdw
(1)
rvdW are van der Waals sphere radii of the tracers (Chart 2), which have been calculated following the procedure of Edward25 from atomic radii. The rvdW values of the polymer network spheres were identified with their hydrodynamic radii determined by dynamic light scattering. We have chosen the “stick boundary” value n ) 6 for the largest tracers P6-ONS, P8-ONS, and A-ONS, whereas the “slip boundary” value n ) 4 was used for the other tracers and for PDE and CDE in the comparison of Dη with self-diffusion. This rather arbitrary procedure resulted in qualitative agreement between the experimental D values and Dη at the highest temperatures. At lower temperatures we find D > Dη for the small tracers, whereas D ≈ Dη was obtained at all temperatures for the polymer network spheres (P6-ONS, P8-ONS) as well as the large tracer A-ONS. The quantitative agreement for the latter may result from opposite influences of the molecular flexibility and the nonspherical shape upon the apparent rvdW. In Figure 2a, the dependence of TTI diffusion coefficients upon the size of weakly polar glass-forming liquids is shown. The best agreement between D and Dη is obtained for m-TCP, probably related with the internal flexibility of the m-TCP
Tracer Diffusion in Supercooled Liquids
Figure 1. Diffusion coefficients of different tracers. (a) solvent: PDE. Tracers: ACR (0), A-ONS ()), P6-ONS (O). (b) solvent: CDE. Tracers: TTI (9), N-ACR ([), P8-ONS (b). The self-diffusion coefficients (measured by NMR32 ), are denoted by *. Some curves are shifted upward by 1 or 2 decades as indicated by +1 and +2. The full lines denote scaled inverse viscosities (see text).
J. Phys. Chem., Vol. 100, No. 37, 1996 15257
Figure 2. Diffusion coefficients in different liquids. (a) tracer: TTI. Solvents: m-TCP (4), Salol ()), OTP (0),33 ODE (O), FDE (~), PDBE (3), PS (*).2 (See Figure1 for CDE. Note the shifts indicated by +1 and -1.). (b) Tracer: ACR. Solvents: propanediol ([), glycerol (2), PDE (9). The full lines denote scaled inverse viscosities (see text).
CHART 2. Tracersa
Figure 3. Diffusion of ACR in PDE: log-log plot of D/T versus η-1 normalized with respect to Tg and ηg ) η(Tg).
a 5 and 6 are assigned to P6-ONS and P8-ONS, the ONS-labeled spherical polystyrene micronetworks having hydrodynamic radii of 6 and 8 nm, respectively (see text).
molecules. In liquids of large rigid molecules (FDE, PDBE) the Dη curves appear to be shifted to higher temperatures with respect to the experimental D values. This T shift will be discussed further below. In polystyrene2 (PS) we have used monomeric friction factors taken from the literature26 for estimating Dη . In Figure 2b, ACR diffusion in PDE was compared with that in two hydrogen-bonding liquids, glycerol and 1,2-propenediol. The observation that D < Dη in glycerol will be discussed further below.
Figure 4. Diffusion of ACR in PDE: derivative plot (see text) with x ) D/T (O) and x ) ηTS-1 (*), respectively.
IV. Discussion In Figure 3, we give an example of a log-log plot providing the exponent ξ from the phenomenological relation D ∝ T/ηξ which was observed previously for diffusion in OTP.3,6 For comparison, we show in Figure 4 a “derivative plot” proposed by Stickel15 which yields a straight line if the Vogel-FulcherTammann (VFT) relation is valid and crosses the abscissa at the Vogel temperature T0 in this case. The full line in Figure 4 is a fit of the dielectric absorption maxima (taken from ref
15258 J. Phys. Chem., Vol. 100, No. 37, 1996
Figure 5. Temperature shift of diffusion coefficients. The D values of TTI (O) and N-ACR (0) in CDE shown in the upper left inset along with x ) T/η (full line) are shifted by ∆T ) 10 K and 5 K, respectively, resulting in the master plot, where all values coincide approximately. The dotted line is a VFT fit of T/η crossing the abscissa at T0 ) 323 K.
Figure 6. Horizontal and vertical shifts of diffusion coefficients. The symbols denote TTI diffusion coefficients D(T) in CDE. The full line is Dη(T). The dashed line is Dη(T + 10 K), which agrees with D(T). The dotted line is Dη(T), as defined in eq 1, however, with a van der Waals radius reduced from 0.38 nm to 0.19 nm.
15) in the regime T J 340 K. The shear viscosities of PDE (asterisks) have been shifted by 3 K in order to produce agreement with the full line in the high-T regime; that is, we have chosen x ) ηTS(T) ) η(T + 3 K). Whereas the dielectric relaxation data agree with those of ηTS over the whole T range,15 the diffusion data deviate at low temperatures, where ξ ) 0.71 was determined from Figure 3. The influence of T shifts upon the data (see Figure 10 in ref 15b) is explored further in Figures 5 and 6. For tracer diffusion we have chosen x ) D/T in Figure 5 and have applied the shifts DTS(T) ) D(T - ∆T) with ∆T ) 5 K and 10 K for N-ACR and TTI, respectively, in CDE. The full line corresponds to x ) η-1 of CDE. The unshifted data are shown in the upper left part of the figure for comparison. It is apparent that the horizontal shift by ∆T results in an excellent fit of the diffusion
Heuberger and Sillescu with the viscosity data (except for the lowest temperatures), which also agree with dielectric relaxation.27 In Figure 6 we demonstrate the effect of the horizontal ∆T shift in an Arrhenius plot of TTI diffusion coefficients and the “rescaled viscosity” Dη as defined in eq 1 via the Stokes-Einstein relation with n ) 4. Here, the full line, Dη(T), is considerably below the experimental diffusion coefficients, whereas the dashed line, Dη(T + 10 K), is in almost perfect agreement. On the other hand, a Vertical shift obtained by reducing the van der Waals radius from rvdW ) 0.38 nm to 0.19 nm yields the dotted line in Figure 6, which agrees with the experimental D values at the highest temperatures but differs by up to almost 2 decades at the lowest temperatures. A corresponding power law plot, D ∝ T/ηξ, yields ξ ) 0.85 and also differs by almost 2 decades from the ξ ) 1 line at the lowest temperatures.28 Here, the crossover between the ξ < 1 and a hypothetical ξ ) 1 regime is somewhat above the largest experimental D values shown in Figure 6. The observation that the T dependence of tracer diffusion coefficients can be forced (at least partly) into coincidence with that of T/η by a T shift is reminiscent of the time-temperature superposition principle, which is well-known in polymer rheology.26 Thus, one can assume that the spectrum determined at some temperature T1 can be brought into coincidence with that at a different temperature T2 by a logarithmic shift, ∆ log ω, of the frequency scale. This implies the shift of the average relaxation time determined from the maximum of the spectrum and of the whole “distribution of relaxation times”, which determines the shape of the spectrum and is assumed T independent in “thermorheologically simple” glass-formers.29 In recent attempts to explain the enhancement of translational diffusion over other relaxation processes (rotational diffusion, shear viscosity) one assumes that the relaxation time distribution originates from spatial heterogeneity with mobile domains dominating translational molecular displacement and less mobile domains determining structural relaxation.13,14 If we assume further that the correlation time distribution can be quantitatively mapped upon a corresponding glass transition temperture distribution by way of the t - T superposition principle,29,30 it is obvious that the relaxation behavior in the “fast” domains can be brought into coincidence with that in the “slow” domains by a simple temperature shift. This explains why D(T) dominated by the fast domains can be forced upon Dη(T) by the shift D(T - ∆T) ≈ Dη(T) since Dη(T) is dominated by relaxation processes within the slow domains. It should be noted that a shift of Tg by 10 K corresponds to a change of the average relaxation time by about 2 decades, which is also about the width of the correlation time distribution in typical glass-forming liquids close to Tg.3,6,15 The difference between the T shift and power law assumptions can be discussed if we assume a VFT relation for η(T). In this case we obtain (see eq 1)
(
)
-B T - T0
Dη(T) ) TD0 exp
(2)
(
DTS(T) ) Dη(T + ∆T) ) (T + ∆T)D0 exp
(
Dξ(T) ) TD0 exp
)
-ξB T - T0
)
-B T + ∆T - T0 (3) (4)
Thus, the T shift can be considered as a shift of the Vogel temperature T0,15 whereas the “power law” assumption changes B into ξB.2 By comparison of eqs 3 and 4 we find (neglecting
Tracer Diffusion in Supercooled Liquids
J. Phys. Chem., Vol. 100, No. 37, 1996 15259 >1 is the H bonding between the tracer and glycerol molecules, which may increase on lowering T relative to that between the glycerol molecules themselves. For self-diffusion in glycerol no trend toward ξ >1 is detectable at the lowest accessible temperatures (T > 256 K). If the tracer and solvent molecules are of comparable size (F ≈ 1), the ξ values can be rather small (e.g., ξ ) 0.71 for ACR in PDE), whereas ∆T ≈ 0 within experimental uncertainty. It should be noted that the numerical value of ∆T depends upon the choice of rvdW (see vertical shift in Figure 6) and the absolute T accuracy of the experimental setup . For example, a relative T error of 3 K between the D and η experiments would change ξ from 0.85 to 0.95 in salol.28 A disadvantage of most systems studied was that the crossover from ξ < 1 to ξ ) 1 could not be observed since the accessible data did not extend sufficiently into the high-T regime. Only in PDE, salol, and m-TCP could we observe the crossover and determine approximate values of Tco ) 343 K, 260 K and 245 K, respectively. These crossover temperatures are close to 1.2 Tg, which was found to characterize the crossover between the liquid and glass transition regimes in other experiments with glass-forming liquids.3,4,15 For example, the discrepancy from the high-temperature VFT fit shown in the derivative plot in Figure, 4 starts at approximately the temperature assigned to the crossover in the power law plot of Figure 3. V. Conclusions
Figure 7. Exponents ξ of power law plots (upper half) and temperature shifts ∆T (lower half, see text) plotted versus the ratio of van der Waals radii F ) rvdW(tracer)/rvdW(solvent). At the symbols, the first number corresponds to the solvent (Chart 1), the second number to the tracer (Chart 2). The estimated uncertainty of ξ and ∆T is indicated by two representative error bars.
the T shift of the prefactor)
∆T ) (T - T0)(ξ-1 - 1)
(5)
This relation, which predicts that ∆T decreases with decreasing T, is in contrast with our experimental finding (Figures 4 and 5) that ∆T increases on lowering T, whereas ξ is found T independent at low temperatures (Figure 3). The reason for this discrepancy is that the VFT relation is not valid in the T regime close to Tg. This is apparent from the derivative plots Figures 4 and 5, which only in the high-T regime show good agreement with the VFT relation given by the straight lines crossing the abscissa at T0 ) 278 K and 323 K, respectively. The large deviations at lower T were already emphasized by Stickel et al.15 and previous authors.16 Figure 7 summarizes our results of determining the exponents ξ from power law plots, D ∝ T/ηξ, and the shifts ∆T from fits of D(T) ) Dη(T + ∆T), where ∆T was used as a fit parameter in the attempt to find optimum agreement of the D(T) and Dη(T + ∆T) curves (see Figure 6). The number code in Figure 7 refers to Chart 1 (first number) and Chart 2 (second number). The abscissa denotes the ratio of the van der Waals radii, F ) rvdW (tracer)/rvdW (solvent), which are also listed in Chart 1 and Chart 2. Qualitatively, it is apparent from Figure 7 that the smallest ratios F correspond to the smallest ξ and largest ∆T values. Furthermore, we find ξ ≈ 1 and ∆T ≈ 0 for large tracer molecules (F > 1). Possibly, the diffusion of ACR in glycerol (7/1 in Figure 7) is exceptional in that ξ > 1 and ∆T < 0 seems to be just outside the experimental uncertainty, as can be inferred from Figure 2b and a corresponding power law plot.28 This is noteworthy since it has been discussed in relation with the free volume interpretation of ξ whether or not ξ >1 is possible.31 In glycerol and other H-bonding liquids a possible reason for ξ
The decoupling of tracer diffusion coefficients and shear viscosity in glass-forming liquids has been analyzed in terms of power law plots, D(T) ∝ T/η(T)ξ (Figure 3), and temperature shifts, D(T) ∝ T/η(T + ∆T) (Figure 4, Figure 6). For small and medium tracer sizes we observe ξ < 1 close to Tg and a crossover to ξ )1 at Tco ≈ 1.2 Tg. Within experimental uncertainty of ξ (about (0.05) we find ξ ) 1 for larger tracers that may, however, exceed the size of the solvent molecules by only about 30% (Figure 7). The hope that large spatial inhomogeneities of ∼1-2 nm29 can be detected from the size dependence of tracer diffusion3 was therefore not substantiated. This does not exclude the existence of large inhomogeneities, in particular, since A-ONS is rather flexible and the diffusion of ACR in glycerol and 1,2-propanol may be influenced by the H-bonding dynamics in these liquids. However, Cicerone and Ediger have also observed that the moderate size increase from tetracene to rubrene already suppresses the enhancement of translational diffusion in o-terphenyl close to Tg.14 In systems where the size of the tracers was smaller than that of the solvent molecules a temperature shift by ∆T was found to be an interesting alternative to the power law plot (Figure 7). Although the shift ∆T depends somewhat upon the choice of the tracer radius (vertical shift in Figure 6), it leads to coincidence of D(T) and Dη(T + ∆T) ∝ T/η(T + ∆T) over a large T range (see eq 1 and Figure 6). Only for some systems, where tracer and solvent molecules have about the same size, are deviations observed that would require an increase of ∆T at temperatures close to Tg (Figure 4). A possible explanation was suggested via a relation of time-temperature superposition26 with spatial inhomogeneity.29,30 Here, one can assume that translational diffusion is dominated by the motion in fluidized domains6,13 having a lower Tg than the less mobile domains determining structural relaxation and shear viscosity. In this scenario, the fast dynamics in the fluidized domains can be mapped upon the slow dynamics by a Tg shift which shifts D(T) onto Dη(T + ∆T). Finally, let us comment on the relation between translational and rotational diffusion. All glass-forming liquids listed in Chart 1 have been studied by dielectric relaxation15,27 except PDPE.
15260 J. Phys. Chem., Vol. 100, No. 37, 1996 The T dependence of the dielectric relaxation times was found to be in agreement with that of η if T shifts |∆T| e 3 K were permitted. From these results we expect that the rotational diffusion of the tracers should also scale with the solvent η, as was also observed by Ediger et al.5,14 A more extensive discussion of rotational diffusion will be published after completion of a current self-diffusion study.32 Acknowledgment. We thank Dr. F. Stickel for making available to us his dielectric relaxation results27 prior to publication and for interesting discussions. We are grateful to Dr. T. Pakula for the viscosity measurements (see section II), to Dr. A. Dietrich for supplying the fulgite derivative N-ACR, and to Dipl. Phys. I. Chang for the self-diffusion coefficients determined by NMR.32 Support by the Deutsche Forschungsgemeinschaft (SFB 262) is gratefully acknowledged. References and Notes (1) Crank, J., Park, G. S., Eds., Diffusion in Polymers; Academic Press: New York, 1968. (2) Ehlich, D.; Sillescu, H. Macromolecules 1990, 23, 1600. (3) Fujara, F.; Geil, B.; Sillescu, H.; Fleischer, G. Z. Phys. B. 1992, 88, 195. (4) Ro¨ssler, E.; Eiermann, P. J. Chem. Phys. 1994, 100, 5237. (5) Cicerone, M. T.; Blackburn, F. R.; Ediger, M. D. Macromolecules 1995, 28, 1600. (6) Chang, I.; Fujara, F.; Geil, B.; Heuberger, G.; Mangel, T.; Sillescu, H. J. Non-Cryst. Solids 1994, 172-174, 248. (7) Vrentas, J. S.; Vrentas, C. M. Macromolecules 1993, 26, 1277; 1994, 27, 4684; J. Polym. Sci., Polym. Phys. Ed., 1993, 31, 69 and references therein. (8) Coughlin, C. S.; Mauritz, K. A.; Storey, R. F. Macromolecules 1991, 24, 1526 and references therein. (9) Our display of this phenomenon in ref 6 has been misunderstood in ref 11. We have not suggested that the particles “diffuse translationally without rotating appreciably”.11 However, we discussed several ways of
Heuberger and Sillescu how to avoid this unphysical motion and favored a model where the translational displacement of a particle occurs in “fast” spatial regions, whereas the mean rotational correlation time is approximated by the residence time in “slow” regions.6 (10) Stillinger, F. H.; Hodgdon, J. A. Phys. ReV. E 1994, 50, 2064. (11) Tarjus, G.; Kivelson, D. J. Chem. Phys. 1995, 103, 3071. (12) Liu, C. Z. W.; Oppenheim, I. Phys. ReV. E 1996, 53, 799. (13) Sillescu, H. Phys. ReV. E 1996, 53, 2992. (14) Cicerone, M. T.; Ediger, M. D. J. Chem. Phys. 1996, 104, 7210. (15) Stickel, F.; Fischer, E. W.; Richert, R. J. Chem. Phys. 1995, 102, 6251; 1996, 104, 2043. (16) Laughlin, W. T.; Uhlmann, D. R. J. Phys. Chem. 1972, 76, 2317. (17) Cukierman, M.; Lane, J. W.; Uhlmann, D. R. J. Chem. 1973, 59, 3639. (18) Landolt-Bo¨rnstein Transportpha¨ nome 1, Bd. 2 5a, 6th ed.; Springer Verlag: Berlin, 1969. (19) Ja¨ntsch, O. Z. Kristallogr. 1957, 108, 190. (20) Barlow, A.; Erginsav, A. Proc. R. Soc. London 1972, A327, 175. (21) Riande, E.; Agawal, P. Unpublished data. (22) Antonietti, M.; Nestl, T. Macromol. Rapid.Commun. 1994, 15, 111. (23) Antonietti, M.; Bremser, W.; Mu¨schenborn, D.; Rosenauer, C.; Schupp, B.; Schmidt, M. Macromolecules 1991, 24, 6636. (24) Antonietti, M.; Coutandin, J.; Gruetter, R.; Sillescu, H. Macromolecules 1984, 17, 798. (25) Edward, J. T. J. Chem. Educ. 1970, 47, 261. (26) Ferry, J. D. Viscoelastic Properties of Polymers, 2nd ed.; Wiley: New York, 1980. (27) Stickel, F. Ph.D. Dissertation, Universita¨t Mainz, 1995. (28) Heuberger, G. Ph.D. Dissertation, Universita¨t Mainz, 1995. (29) Donth, E.-J. Acta Polym. 1979, 30, 481; Relaxation and Thermodynamics in Polymers; Akademie Verlag: Berlin, 1992. (30) Zetsche, A.; Fischer, E. W. Acta Polymer 1994, 45, 168. (31) Zielinski, J. M.; Sillescu, H.; Romdhane, I. H. J. Polym. Sci. (Polym. Phys. Ed. 1996, B34, 121 and references therein. (32) Chang, I. Ph.D. Dissertation, Universita¨t Mainz, in preparation. (33) Lohfink, M.; Sillescu, H. In AIP Conference Proceedings No 256; Kawasaki, K., Tokuyama, M., Kawakatsu, T., Eds.; American Institute of Physics: New York, 1992; pp 30-39.
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