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

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).    

The 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 ^oC. 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 (Mc

), 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)    m_o = K M         (M < M_c)

Where M is the molecular weight of the chains, m_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)   m_o = K' M^3.4          (M>Mc)

The critical molecular weight, M_c, is obtained from intersecting the straight lines Log m_o vs Log M drawn in the two regions Mc

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(m_o) v s 1/T,

(3)

Log(m_o) = (-C_1g*(T-T_g)) / (C_2g+(T-T_g))  + Log(m_og)

where m_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 original 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 original 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 originating 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 m_o= F(T)+ a₁(M)  from the classical approach

 a₁(M) = a₁₁(M,T) + a₁₂(M,T) 

from comparing classical with the Cross-Dual-Phase approach. 

with (for PS monodispersed grades, M>Mc):

(5)

 a₁=a₁₁+a₁₂

    = b₁+b₂ exp(DH_2R/T)+b₄ exp(DH₂/T)

with DH_2R a constant and DH₂ an exponentially decreasing function of M, and:

b₁ = b₁(a₁₁)+b₁(a₁₂) with b₁(a₁₂)=0

b₂ = b₂(a₁₁)+b₂(a₁₂) with b₂(a₁₁₎ independent of M

b₄ = b₄(a₁₁)+b₄(a₁₂)

where the sub-index 11 refers to the entanglement dual-phase and 12 to the core dual-phase.

When b₂ and b₄ = 0 or when {b₂ exp(DH_2R/T) +b₄ exp(DH₂/T)}=0 then a₁(M)=b₁ and there is true separation of the variables M and T.

 We find that b₂ and b₄ become zero periodically for values of M which are multiples of Mc (~36,000), with period doubling: 2Mc,4Mc,8Mc, 16Mc. For all other values of M, the values of b₂ and b₄ are quite small but not zero, yet small enough to make the sum {b₂ exp(DH_2R/T) +b₄ exp(DH₂/T)} give the appearance that the separation of the variables is justified, although it is not. Furthermore, we observe that the sum {b₂ exp(DH_2R/T) +b₄ exp(DH₂/T)}, although small, structures with M in a way which makes the periodic molecular weights for the zeros the compensation points for the other M values, when M is expressed as log M.  

These results are illustrated in the Figures below (and detailed in the upcoming paper “Cross-Dual-Phase Inspired New Formulations of The Temperature and Molecular Weight Dependence of Newtonian Viscosity.  Explanation of the 3.4 Power Exponent for the Dependence of Molecular Weight on The Newtonian Viscosity”.

A SPECTACULAR ARRAY OF NEW HIDDEN CORRELATIONS 

Fig. 1

For M=134,000, in Fig. 1, plots of a₁₁(T) and a₁₂(T) can be perfectly fitted by a set of dual exponential functions with pre-exponential constants b₂(a₁₁), b₄(a₁₁), b₂(a₁₂), b₄(a₁₂) and constants b₁(a₁₁) and b₁(a₁₂) as defined in Eq. 5. Their sum a₁=a₁₁+a₁₂ of Eq. 4 would be constant (and the blue triangles line horizontal) should the separation of M and T be valid for this particular value of M. Fig. 1 shows that the horizontality is almost true in the higher T region, but that, actually, a₁(M) slightly varies with T. For certain discrete M values, such as 2Mc, 4Mc, 8Mc etc. the blue triangles line is horizontal. For the other values of M, a “structure of a₁(M) appears visible, fragmenting a₁₁(M,T) and a₁₂(M,T) into ranges of sequential M, for which compensations occur. The compensation points coincide with the values of M for which the blue lines are horizontal. 

Fig. 2a

In Fig. 2a, the values of a₁₂(M,T)  for Mc₁₂ vs log M reaches an asymptotic value which lines up this range of data with the data located above it, at higher M (Figs. 2b and c).

Fig. 2b

Fig. 2b explores range-2 of the values of the molecular weight showing a compensation point at M~4Mc. The value of a₁₂ at the compensation point is zero, like a₁₁(not shown), indicating that a₁=a₁₁+a₁₂=0 is a solution for the separation of M and T.

Fig. 2c

For the values of M located in range 3 (shown in Fig. 2c), the a₁₂(T,M) compensate at  M~2Mc. Interestingly, comparing the 3 molecular ranges fragmentation in Figs. 2a to 2c,  for the high M range (Fig. 2c) the compensation is “negative”, i.e sourcing from the lower M value, while it is  “positive” for range 1 (Fig. 2a), i.e. converging to a higher M value. The situation in range 2 is intermediate with the combination of a positive and negative compensation using the same compensation point. The description of a₁₂(M,T) can thus be described as an interlocked network of positive and negative compensations, a very interesting web new structure threaded by the interdependence of the effect of M  and T on viscosity.

The variation of a₁₁(M,T) is similar to a₁₂(M,T) and in many ways symmetrically “opposite”, as illustrated in Figs. 3a to 3c, for which T is kept constant- here limited to only 2 temperatures: T=403 ^oK (130 ^oC) and 478 ^oK (205 ^oC).

Fig. 3a

There is a symmetry of the behavior of a₁₁(M) and a₁₂(M), here shown for T=403 (Fig. 3a). For such low temperature, and up to T=448 ^oK, the axis of symmetry is slightly but noticeably different for the values of M superior or inferior to 4Mc (at 4Mc, both a₁₁ and a₁₂ are equal to 0).  

Fig. 3b

Split of a₁ into a₁₁(M) and a₁₂(M) at high T, 478 ^oK in Fig. 3c, showing the average value, a₁/2, represented by the dotted line, to be the symmetrical axis across the 4Mc compensation point. The green triangles display the behavior of a₁(M) vs log M which can be fitted by a straight line with slope ~ 3.4, the classical result.

Fig. 3c

Fig. 3c is a “clean” version of Fig. 3b, focusing on the variation of a₁₁(M) , the black squares with slope 0.414, and of a₁₂(M), the red dots with slope 2.93 representing the entanglement and the core-Dual-phase, respectively. Note that for T= 483 ^oK, the respective slopes are 0.328 and 3.0 for the Cross-dual-Phases, giving an exponent 3.328 for the variation of a₁(M) in Eq. 5.

Fig. 4a

Fig. 4a explains well how the sub-structure of a₁(M), which is so rich and diversified, as shown in the Figs. 2-3 above, in particular, varies with T, becomes temperature independent when combined as a₁=a₁₁+a₁₂. It is like the neutrality of the electrical charge of atoms which, we know, sub-structures nevertheless into negative and positive charges in electrons and protons, respectively. Fig. 4a shows the average value between a₁₁ and a₁₂, which is twice the value of a₁. This is a log-log plot since a₁ and a₁₁, a₁₂ are scaled like log m_o, and the M-axis is also scaled logarithmically. The slope of a₁ vs logM is, therefore, 2*1.686= 3.372; one can see that there is a single straight line with that same slope, and independent of T, as long as M remains above ~2 Mc. Below that value, we observe a compensation of lines as T varies instead of the unique straight line: T and M no longer separate. The details for M< 2Mc is blown up in Fig. 4b. We can clearly see that the slope of each isotherm increases with T up to T~ 448 ^oK (mentioned already above) beyond which the slope asymptotically converges to the 3.372 slope of the T independent region. 

Fig. 4b

I have added two graphs, Figs. 5a and b to summarize the results of this post (b1(a11) is defined in Eq. 5, it is equal to a₁ of log m_o at T constant):

Fig. 5a is the classical view on viscosity, in the light of the concepts of the Cross-Dual-Phase view on viscosity, showing that 2 straight lines (red and blue lines) add up (cyan line) to describe the experimental behavior (black squares), explaining the physical irrelevance of the 3.4 power exponent, and,

Fig. 5b  provides the same data as in Fig.5a plotted against M instead of log M.  The two cross-dual-phases can each be fitted by a simple pair of exponential terms of M, with the same exp(-M/M₁) and exp(-M/M₂) for both dual-phases, indicating their linear correlation for all M > Mc.

In essence, the reason for the extra 0.4 on top of the power exponent 3.0 determined by de Gennes is the presence of a dissipative elastic dual-phase (the entanglement dual-phase) amounting to approximately 10% of the core-phase values (expressed as b₁, b₂, b₄ in Eq. 4). This represents the new view on the influence of M on the Newtonian viscosity.

The requirement to have two cross-dual-phases for only M>Mc, and simply one dual-phase for MDH(M) and DS(M) of the system.   The split of the system at Mc into 2 cross-dual-phase interactive systems is induced by the instability of a single system solution beyond a critical value of M.

Fig. 5a

The cyan line (passing through the dark squares, the data) is the sum of the red and blue straight lines.

Fig. 5b

The best fitting equation for these two cuves is actually not the power laws shown in  Fig. 5a, but a pair of exp decay functions. A plot of the red values against the blue ones is a straight line with slope ~0.109.

CONCLUSIONS

So, it is reasonable to conclude that for T above approximately 448 ^oK (175 ^oC), the Newtonian viscosity of the PS melts will demonstrate a satisfactory classical behavior for all M > Mc.

The only difference with the classical view is that we have learned that this positive correlation is only an appearance, an approximation which becomes poor at lower T, say as T_g is approached. Moreover, we know that this illusion of a success is hiding one of the most interesting aspect of polymer physics, the duality and cross-duality of the interactions which give rise to the behavior depicted in Figs. 2a to 2c, for instance.

As a matter of fact, if one were to zoom in on each point of the straight line in Fig. 4a, in the range where there is a good apparent correlation with the classical separation of M and T leading to the 3.4 correlation between log m_o and logM, the structure of a₁(M) and the slight but systematic T dependence (with period doubling for M ), a₁(M,T), would reveal itself. But one needs to apply the Cross-Duality physics to the separation of the variables in order to reveal this new aspect of the influence of weight on the dynamics of the interactions.   

In conclusion, the results of the new analysis of the Newtonian data by the Cross-Duality-viscosity formula appear so rich in the discovery of new correlations that many scientists may decide to give reptation the Estocade to embrace an emerging new Era of Polymer Physics.  Eh, you never know!