On the Incidence of Tg and Ta on the Formulation of Rheological Equations J.P. Ibar New School Polymer Physics .Dynamic Frequency sweep data are obtained at constant temperature T: G’ (w, T), G”(w, T) , h*(w, T) For a given melt, with molecular weight M (monodispersed): Tg = Tgo -A /M Ta is also a function of frequency w: Ta (w) = Ta∞ + B /(Log w – C) When comparing melts at different temperatures, with different molecular weights, we propose to compare them at the same (T-Ta(M,w)). We will generalize to include other parameters that Influence Tg, such as pressure, cooling rate and elongational ratio l ObjectivesData analysed: -from Susuki & Pearson (University of Strasbourg) -from Marin and Graessley POLYSTYRENE Monodispersed fractions. M< Mc : 2,000-37,000 Susuki, Majeste data Ph-D thesis M> Mc : 37,500-680,000 Marin (W11, W16, W39, W68) Mc ~ 35,000 Also, PC data 3 grades Mw=18,000, 23,000, 33,000 I=2.2Newtonian viscosity: h*o (T) h*o (T) = lim (G”/w) w -->0 Rouse theory predicts that h*o is proportional to M h*o (T) = K(T) M where K is function of temperature through the molecular friction coefficient. It is admitted that Rouse model applies well for M < McData (Suzuki) are obtained for all molecular weights at T= 210 oCConclusions: The best line passing through the data has slope 1.49, not 1.0 The best fit is curved, not linear. Data from Majeste, Montfort, Allal confirm this result (slope is 1.25) Data from Pearson also confirm the same results (non-linearity and slope 1.34). The Rouse model applied to comparing molecular fractions at the same temperature seems to badly fail explaining the data for the M < Mc fractions.Let’s re-calculate all Newtonian viscosity values for the fractions to compare them at the same temperature with respect to their respective Tg knowing that Tg(M).CONCLUSION: After correction of the effect of Tg(Mn), the Rouse model is validated for the M < Mc fractions: Newtonian viscosity h*o is proportional to M Note that at low X= (T-Tg), 30 oC and below, the slope of Log (h*o)/M vs Log M is no longer 0, but negative. This is indicative that another correction might be needed: the influence of w on Tg?. This is what is considered next.Does Tg really vary with Frequency? The mechanical manifestation of Tg is called TaTemperature sweep (@1o C/min) with DMA measuring E’(T), E”(T) for Grade 2 PC. w=10 rad/s E’, E” Dyne/cm20.16 Hz 16 Hz b transition a transition c transitionFor various grades of PC (OQL=Grade 2, 141=Grade 1, PK= Grade 3)Transition a “Frequency-plot” From McCrum, Reads, Williams Symbols apply to various techniques to determine TaFrequency Map showing Ta and Tb transitions frequency dependenceIso PMMA mastercurve at T=180 oC (After A. Allal) a Transition a Transition determined by rheology: assumes superposition worksPartial Conclusions: 1. From our study of Newtonian viscosity vs M, for M Mc the exponent remains ~3.4 if Tg(w) is limited to Tgo,K at low w , but appears to be closer to 3 if Tg(w) is not bottomed out (but this result needs verification with other data sets before it can be definitely validated).What about the effect of strain rate on Tg ? Does the Cox-Mertz rule apply (equivalence strain rate & w)? What else can make Tg vary to influence rheological data?This plot indicates the kinetic nature of Tg and validates the assumption that the lowest value of Ta should be T2, the thermodynamic value of Tg at infinite viscosity.Annealing /QuenchingEffect of Extensional Stress on Tg : “thickening”CONCLUSIONS Incorporating the expression of Tg(w, q, P, M. l) into the rheological equations, e.g. the expression of viscosity, offers many advantages: •It ties up the parameters used in rheology with the well studied Tg (or Ta) transition by spectroscopists. Such a universality and variable reduction was attempted by Williams, Landel and Ferry with the WLF equation but they used the Tg (dilatometric value) instead of using Ta which varies with cooling rate q, frequency of oscillation w, elongation draw ratio l, pressure P, and molecular weight M. •Integrating Tg into the equations of rheology as a FUNCTION and not as a fixed parameter may provide a simple scale reduction pivot, allowing to reduce analytically the influence of many parameters to the variation of that Ta function. • Tg and Ta dynamics are different, although interrelated in the same sense that local interactions are synchronized in the dynamics of the entanglement network which modulates the global interactionThere is the need for a fundamental understanding of the properties of local interactions embedded in and modulated by the global interactions, and for the integration, in the field of rheology, of the knowledge about molecular motions gained by spectroscopists. We suggest that this scaling approach should be done by separating the effect of orientation from the effect of free volume. When this is done, the dual aspect of viscoelasticity becomes more apparent.Correlation of motions below Tg and above Tg through free volume structuring evolution.Deconvolution of interactive motions fused in a peak of relaxation. The same thing must be done with rheological data.Analysis of the deconvoluted elementary relaxations composing a transitional peak (TTg). “Compensation” describes the interactive mechanisms. The stability of the modulating network can be determined (104.5 sec). Perhaps because of the lack of integration into the field of rheology of spectroscopists’s knowledge about molecular motion interactions, linear viscoelasticity remains only partially understood (see the Great Myths of Rheology, part I (downloadable in the Content of WIZIQ). Furthermore, extrapolation to non-linear rheology (at higher strain) is rendered more difficult to quantify because parameters are presently defined and scaled from linear rheology (see the Great Myths of Rheology, part II). The proposed analytical method to separate out the effect of stress field on conformation molecular motions from those due to a reorganiizatio of free volume should contribute to a better understanding of the molecular mechanisms responsible for the rheological response. Equations can simply be re-framed by incorporating (T-Ta(M,w,P,q,l)) in the expression of the moduli. The first thing, we suggest, that should be popularized is the use of temperature sweep at constant w, instead of the classical frequency sweep. In such an experiment Ta remains constant, so (T-Ta) decreases linearly, i.e. controllably but without the need to know the Ta value (an example is shown in the last slide). Comparison with WLF (Newtonian) relates to shear-thinning. Conclusion