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The End of an Era in Polymers-Part II: L'ESTOCADE

In the previous blog post, I pointed convincing Rheo-SANS experimental evidence challenging the predictions of the popular reptation model of polymer melt deformation and concluded that reptation was probably inadequate to describe the concept of
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  The End of an Era in Polymers- Part II: L’ESTOCADE ESTOCADE pronunciation: es.to.kad/ noun, feminine, plurial estocades the mortal impalement of a bull in bullfighting. In the previous blog post, I pointed convincing Rheo-SANS experimental evidence challenging the predictions of the popular reptation model of polymer melt deformation and concluded that reptation was probably inadequate to describe the concept of entanglements, often described as responsible for the large increase of the Newtonian viscosity as the molecular weight of the macromolecules exceeds a critical value Mc. In this post, I show that a new way to analyze classical data raises intriguing new questions regarding the effect of molecular weight on  viscosity, taking us far, very far, from a molecular dynamic explanation (thus the Estocade’s cartoon). Viscosity of polymers is key to their behavior in the molten state and thus to their processing. The well-known equations of rheology giving the temperature and molecular weight dependence of melt viscosity assume that these two variables separate in the expression of viscosity. It is admitted that (Newtonian) viscosity depends on the product of two parameters: a friction factor which is controlled solely by local features such as the free volume, and a structure factor which is controlled by the large scale i.e. the configuration of the chains. The friction factor depends on temperature only and not on the molecular weight characteristics (M w , M n ). It is best expressed as a function of (T-T g ), at least up to approximately T g +100 o C. The structure factor, on the other hand, depends on the number of chains per unit volume and on their molecular weight and dimensions. It is also admitted that the structure factor is largely the same regardless of the chemical nature of the repeating units which form the macromolecules. This universality is expressed by the constants entering the WLF equation. The above statements express the classical interpretation of the separation of the effect of molecular weight and temperature on the viscosity. For polymers of low molecular weight (M<M c ), it is admitted that the viscosity is quite reasonably well described by the Rouse model, with no adjustment for intermolecular interactions, which can be written: (1)  o  = K M (M<M c ) Where M is the molecular weight of the chains,  o  is the Newtonian viscosity at temperature T, and K is a constant which varies with (T-T g ). For longer chains (M>M c ), the well-known 3.4 power dependence reflects the strong influence of the entanglements on the viscosity: (2)  o  = K' M 3.4  (M>M e ) The critical molecular weight, M c ,   is obtained from intersecting the straight lines Log  o  vs Log M drawn in the two regions M<M c  and M>M c . Formula (1) and (2) above simply state that molecular weight and temperature effects separate in the expression of viscosity of polymers. The temperature dependence of K or K' in Eqs (1) and (2) is often written with the WLF expression, Eq. 3, which, admittedly, explains well, between T g  and  T g +100, the typical curvature observed in Arrhenius plots of Log(  o ) v s 1/T, (3) Log(  o ) = (-C 1g *(T-T g )) / (C 2g +(T-T g )) + Log(  og ) where  og  is the viscosity at T g,  C 1g  and C 2g  are adjustable constant, often admitted to have the universal value of 17.44 and 51.6 respectively. The 3.4 power dependence of molecular weight M for M> M c  has been extensively investigated and explained by several models of entanglements. The friction theory of Bueche determines a value 3.5, whereas the reptation model of de Gennes predicts a value of 3.0, short of the experimental value of 3.4, but later modified by Doi and Edwards to predict 3.4, invoking the effect of tube length fluctuation. There seems to be, at present, the consensus that the 3.4 exponent is a universal characteristic of entanglements in macromolecular chains, that it is constant, independent of temperature, pressure or stress. And it is clear that the continuous publication of theoretical improvement papers which have been published in the course of the 40 years following de Gennes’s srcinal contribution has succeeded in dogmatizing the concept of reptation as the mechanism of macromolecular deformation which explained entanglements in polymers. Yet, as I suggested in my last blog post, the End of an Era, and in many posts before that, there is now forceful experimental evidence to challenge the reptation interpretation of macromolecular deformational behavior, linked to the present understanding of entanglements and, therefore, of melt viscosity at increasing M. In this blog post, I report some of the findings of a new examination of the basic assumption behind the classical approach: the admitted separation of molecular weight (M) and temperature (T) in the formulation of viscosity. I re-analyzed published viscosity data on a series of monodispersed Polystyrene samples covering a wide range of molecular weight across Mc (from M=550 to M=1.2 million g/mole) tested over a broad temperature span in the melt. I reviewed the adequacy of the classical formula, and found a rather good agreement, overall, yet I also analyzed the data in the light of the predictions of the Dual-Phase and Cross-Dual-Phase models of polymer interactions that provide a new interpretation of entanglements, and, surprisingly, emerged a totally new understanding of the impact of the separation of the variables on the comprehension of the physics of (polymer)  interactions. I will show a few of the findings in this post. I showed that the classical expected behavior, both below and above the critical molecular weight for entanglement, M c , was pretty respected but actually only approximately valid on close examination; for instance that the 3.4 exponent for the variation of Newtonian viscosity with molecular weight for M>M c  was almost constant, yet varied slightly and moreover systematically with temperature, roughly validating the separation of M and T in the expression of viscosity, but not really so, only for certain values of the molecular weights which were multiples of Mc: 2Mc, 4Mc, 8Mc etc. (period doubling). Inspired by the Cross-Dual-Phase interpretation of melt entanglement which assumes that a split of the statistical system of interactions into two systems occurs at the critical molecular weight M c , I analyzed the influence of M and T on viscosity across M c  by new formulas which provided the Cross-Dual-Phase parameters for the same viscosity data obtained on Polystyrene already analyzed by the classical formulas (Eqs. 1-3 above). This formulation of the viscous flow behavior from the Cross-Dual-Phase perspective offered new arguments regarding the reptation model historical dilemma: while the srcinal de Gennes’s theory predicted a power exponent of 3 for the molecular weight dependence of Newtonian melt viscosity, several authors successfully tweaked the mathematics of the initial reptation model to explain the “reality”, i.e. that viscosity appeared to follow a 3.4 exponent behavior instead of 3. Were the improvements by Doi, Edwards, Wagner, Marrucci, McLeich and many others actually necessary? In the Cross-Dual-Phase treatment of viscosity, I find that viscosity is the product of two terms, each M and T dependent. One of the terms varies with M with an exponent 3, while the other term varies with M with an exponent 0.4. I assign this behavior to the existence of the interactive and coupled Cross-Dual-phases: the core phase and the entanglement phase. Is de Gennes’s molecular dynamic calculation correctly addressing the interactions occurring in the core phase of the cross-dual-phase model, only missing the elastic sweeping dissipative component srcinating from the second-dual phase (above M c ), or is it just a pure coincidence? Are the classical formulations of viscosity simply curve-fitting expressions misleading the comprehension of the physics behind polymer flow deformation and entanglements? For instance, when working at constant free volume instead of constant temperature, the exponent for M>  M c  is no longer in the 3.4 range but close to 5.3, which raises the question of truly understanding and quantifying the influence of free volume on the viscosity. TESTING THE SEPARATION OF THE VARIABLES USING CROSS-DUALITY PHYSICS Yet, there is more profound reasons for abandoning the molecular dynamic interpretation of the viscosity results and of entanglement. When comparing the molecular dynamic approach and the Cross-Dual-Phase approach, one can quantitatively formulate the M dependence of the expression of viscosity that uses the separation of the variables in terms of the parameters of the Cross-Dual-Phase viscosity formula. This allows to test the conditions which validate the separation of M and T. The results of this investigation may be quite significant, not just for polymer physics, but for physics, more generally speaking. To simplify (and without justification here): (4) Log  o = F(T)+ a 1 (M) from the classical approach a 1 (M) = a 11 (M,T) + a 12 (M,T) from comparing classical with the Cross-Dual-Phase approach. with (for PS monodispersed grades, M>Mc): (5) a 1 =a 11 +a 12 = b 1 +b 2   exp(  H 2R  /T)+b 4  exp(  H 2  /T)   with  H 2R  a constant and  H 2  an exponentially decreasing function of M, and: b 1  = b 1 (a 11 )+b 1 (a 12 ) with b 1 (a 12 )=0 b 2  = b 2 (a 11 )+b 2 (a 12 ) with b 2 (a 11)  independent of M b 4  = b 4 (a 11 )+b 4 (a 12 ) where the sub-index 11 refers to the entanglement dual-phase and 12 to the core dual-phase. When b 2  and b 4  = 0 or when {b 2  exp(  H 2R  /T) +b 4  exp(  H 2  /T)}=0 then
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