The Great Myths of Polymer Rheology-Part I

The Great Myths of Polymer Melt Rheology. Part I Jean-Pierre Ibar New School Polymer Physics http:/newschoolpolymerphysics.blog.com -Abstract….p. 2 -Introduction…p.3 -Shear-Thinning : non-Newtonian behavior…p.5 -Description of the Data Sources… p. 7 -Analysis Protocole… p.10 -Accuracy Consideration… p.11 -Critical Analysis of the Equations of Rheology…p. 14 -Universality of WLF constants at Tg… p. 14 -Validity of the 3.4 exponent for M> Mc… p. 18 -For Mpolynomials or Fourier series that describe well any type of curves). In that context, it is shown that models that take roots in the spectrum of relaxation concept, such as the Rouse model, de Genne’s, Doi & Edwards’, and all their improved versions (for instance by Klein, Montfort, Graessley, Larson, Wagner, Marrucci, Allal, MacLeich), that describe well the molecular weight and temperature dependence of relaxation times, are necessarily limited in their description of melt deformation to the linear regime, where the curvefitting power of such mathematical tools is perhaps the reason for their success. The non-linear regime (high strain rates, high strain) is the only regime important to” real life”, i.e. to processors of plastic melts. In particular, the present understanding of shear-thinning, normal stresses and strainharddenin of polymer melts in terms of “chain disentanglement” deserves critical attention. Forty years ago, the polymer field was dominated by chemists and physical chemists who understood linear visco-elasticity in terms of networks of dashpots and springs, but were puzzled by large amplitude strain rates and strain behavior, especially by the effect of strain. Their interest and success in describing “molecularly” rubber elasticity and swelling (in equilibrium conditions only) by Gaussian chains whose length could affinely be related to macroscopic strain, can be viewed as the birth mark of modern physics, a la de Gennes, but also, perhaps, the source of the mis-characterization and mis-understanding of what entanglements are, and how their existence affects melt deformation, in particular with respect to the entropic melt orientation. We critically review the present classical understanding of the influence of entanglements on melt deformation, and expose its limitations. After critical discussion of the implications and limitations of the classical views, another model of melt deformation and of the influence of entanglements will be presented in part II. This model elaborates a profound different understanding of the source of visco-elastic behavior and of rubber elasticity. Introduction The knowledge of viscoelastic properties and non-Newtonian flow of polymer melts is of paramount importance to the understanding of their processing behavior. The viscosity of polymeric melts has received considerable attention over the last sixty years [1], and is admitted to depend 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 structure and configuration of the chains [1,2]. The friction factor depends on temperature only and not on the molecular weight characteristics (Mw, Mn). It is best expressed as a function of (T-Tg), at least up to approximately Tg+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 [1]. 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. For polymers of low molecular weight (Mw Mc), the well-known 3.4 power dependence reflects the strong influence of the entanglements on the viscosity: (2) ho = K' Mw 3.4 The critical molecular weight Mc is obtained from intersecting the straight lines Log ho vs Log Mw drawn in the two regions (Mw Mc). 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, describes well, between Tg and Tg+100, the typical curvature observed in Arrhenius plots of Log(ho) v s 1/T: (3) log ho = log hog -(C2g + (T -Tg)) C1g (T -Tg) where C1g and C2g are adjustable constants, often admitted to have the universal value of 17.44 and 51.6 respectively [16], hog is the viscosity at Tg, generally considered to be close to 1013 poises. Note that (3) is sometimes re-written as a Vogel-Fulcher equation: (3 bis): with To = Tg -C2g The 3.4 power dependence of molecular weight Mw has been extensively investigated and explained by several models of entanglements [3-5]. The friction theory of Bueche [3] determines a value 3.5, whereas the reptation model of de Gennes [4] and Edwards [5] predicts a value of 3.0, short of the experimental value of 3.4, but later modified, by way of tube length fluctuation, by de Gennes reptation model’s protagonists to predict 3.4 [6]. Although it has been reported [7, 8] that in some polymers the 3.4 dependence increases at higher molecular weight, it is admitted, on the contrary, that the power exponent tends towards its uncorrected theoretical reptation value of 3 at high (M/Me) [9]. In general, there seems to be a good consensus, at present, that the 3.4 exponent is a universal characteristic of entanglements in macromolecular chains, and, furthermore, that this exponent is constant, independent of temperature, or molecular weight. This is, therefore, the expression that theorists try to understand. log ho = A + T -To BShear-Thinning: non-Newtonian viscous behavior. More than a decade ago, Hieber and Chiang [10] assessed the relative merits of the classical equations describing the shear-rate dependence of viscosity, a phenomenon described as “shear-thinning”, based upon a compilation of steady-shear and dynamicshhea viscosity data from the literature for polystyrene melts. Their analysis still holds the admitted understanding today. These authors generalize the popular models of Cross [11] and Carreau [12] and include an additional term to describe the second-Newtonian limit of polymer melt behavior at high strain rate and high frequency: (4) m = (1-s). mc + s. mo where s is a constant, the ratio of the second-Newtonian viscosity at high rate to the low rate Newtonian viscosity (mo), and mc is the viscosity according to the generalized Cross-Carreau’s formula: (5) mc = mo /[1+(a1. mo w)a2] (1-a3)/a2 with a1, a2, and a3 temperature independent curvefitting parameters, w, frequency for shear-dynamic data, or shear rate, dg/dt, in the case of steady-shear flow. In particular, Eq. 5 reduces to the Cross model if a2 = (1-a3), and to the Carreau model if a2= 2. Equations 4 and 5 incorporate the admitted fundamental “universal” aspect of the rheology of polymer melts, stating both the principle of time-temperature superposition and the uniqueness of the viscosity -strain rate (or frequency) curves when the variables are rescaled as reduced viscosity, m/mo, plotted against a reduced time scale, mo.w , a socallle Vinogradov's plot [13]. The Cox-Merz ‘s rule [14] stipulates that curves of m vs dg/dt, at constant T, are identical to curves of m* vs w obtained in dynamic shear conditions at the same temperature. m* is the dynamic viscosity, calculated from the dynamic modulus G* and the frequency of of oscillation of the melt, w, expressed in rad/s. The Cox-Merz ‘s rule seems to still be a debatable proposition among scientists[15, 1]. Yet, very carefully conducted experiments on Polystyrene, both by capillarity and dynamic methods [17], seem to confirm the rule, which is generally admitted to be true for Polystyrene and Polycarbonate melts [16-18]. To study shear-thinning, capillary data are more difficult to handle and require careful corrections for the effect of pseudo-plasticity on the expression of shear rate (Rabinovitch’s correction [19]), and for the end of die effect on the value of the stress (Bagley’s correction [20]). In contrast, dynamic shear data do not require the application of any correction. The other main advantage of generating dynamic data at various frequency, various temperature, is that G’, the storage modulus, and G”, the loss modulus, are provided, from which m* is obtained, both calculated from the value of the phase angle q between strain and stress: (6) G’ /G* = cos q G” /G* = sin q (7) G* = (G’2 + G”2)0.5 m* = G* /w The viscoelastic aspect of a melt is entirely characterized by the value of G’(w,T) and G”(w,T), at least in the linear viscoelastic region. The amount of elasticity in the melt increases with increasing frequency at constant T, as shown in Fig. 1 for a PE melt, and with decreasing temperature at constant frequency, as also shown in Fig. 1 by the relative position of the 4 curves obtained at 4 different temperatures. Theoretically, G’/G* can take all values between 0 and 1. All existing theoretical models of viscoelasticity predict that G’/G* = 1 is the theoretical limit, corresponding to a totally elastic melt. For instance, for a generalized Voigt model, increasing frequency w to infinity corresponds to such an elastic state [21]. This is also the prediction of the Leonov’s model [22] and of the HN’s model [23]. We must critically examine this proposition, as well as determine the origins of the second Newtonian regime obtained at high frequencies. We will show that the data do not display such a 2nd Newtonian plateau when properly filtered for the influence of the transition relaxation terms, reflected by Tg (w, g, dg/dt). We also examine the low frequency region, where the behavior becomes Newtonian, and determine the influence of extrapolating from higher frequencies, using various linearizing functions, such as Eq. (4) and (5), on the value of the Newtonian viscosity. Description Of the Data Sources. Newtonian viscosity data obtained at various temperatures from Tg to Tg+100 oC, for a series of monodispersed PS of various molecular weight Mw , are collected from the published thesis of Pierson [24] and Susuki [25]. We also examine dynamic results published by Marin and Graessley [34} and by Majeste [35]. The data, covering a very large molecular weight range (Mw = 800 to 1,200,000), spread on both sides of the critical molecular weight (Mc=33,000), and are tabulated as a function of temperature and molecular weight. Fig. 2 reproduces the Susuki's data [14]. These studies [24, 25] regroup the work of several authors, along with Pierson's and Susuki's own experimental data. Very good agreement with previously published work is claimed by both Pierson and Susuki. The dispersity ratio is close to 1.02 for all 23 molecular weight studied, so the PS chosen can be considered monodispersed.. Dynamic shear data on polystyrene published by Pfandl et Al [17] are obtained between 124 oC and 290 oC, at 21 temperatures, for 16 frequency values w between 0.00314 and 31.4 rad/sec. The molecular weight is Mw = 391,000 (obtained by GPC), and the dispersity ratio is 2.24. These data are used to analyze the non-Newtonian behavior (shear-thinning), and to determine the variation of Newtonian viscosity with temperature from dynamic data (the Newtonian value is obtained by extrapolation at low frequency from curvefitted expressions). We also analyze new dynamic data performed on three GE grades of polycarbonate with various molecular weight characteristics. The melt flow rates for the three grades is respectively: 65 for Grade 1, 10.5 for Grade 2, and 2.5 for Grade 3. Grade 1 and 2 are two linear polymers with Tg =136 oC and 145.4 oC respectively (measured from PVT data extrapolated to P=0). Grade 3 is a branched polymer, with Tg = 151 oC. Both capillary and dynamic shear viscosity data are obtained at different temperature, strain rate and frequency. For both capillary and dynamic shear data, the Newtonian viscosity mo (or mo*) is obtained by curvefitting the non-Newtonian range, using several equations, including equations (4) and (5). The capillary data are collected with a Goettfert Rheograph 2001, and corrected with Rabinowisch’s formula [19]. Bagley’s plots were obtained and the correction for end effects found negligible. Fig. 3 is a plot of log (m) vs Log c (strain rate) at 24 temperatures ranging from 195 oC to 305 oC, in 5 degree steps, across 2 decades of strain rates, for Grade 1, a high flow polycarbonate resin with relatively few entanglements. The number of data points for each temperature is very large, as can be seen by the density of points in Fig. 3. Fig. 3 clearly demonstrates shear-thinning, e.g. the decrease of viscosity, at each temperature, as strain rate is increased. Fig. 3 The rheometer used to obtain the dynamic shear data for the 3 PC grades is a Rheometrics RDA-700, configured with a parallel plate cell, using 2" wide (in diameter), 2.0 mm thick specimens. Fig. 4 shows the dynamic viscosity versus frequency for Grade 1, a high flow grade PC resin used to inject mold compact disks, here analyzed under oscillatory shear conditions at 5% strain, between 168 oC and 283 oC, for 23 frequencies per isotherm (from 0.1 to 500 rad/s). Fig. 4 clearly demonstrates that applying a shear deformation at faster frequency causes the material to shear-thin, similarly to the effect of strain rate. The dynamic data are obtained from Tg+25 oC to Tg+140 oC for the three grades, with a frequency varying between 0.1 and 500 rad/s. The viscosity data obtained on polycarbonate are, we believe, of high reliability because they have been obtained by a unique operator, on a single instrument (Rheometrics RDA-700 or the Goettfert Rheograph 2001), using the same experimental set up and protocol for all samples, the same method to mold them (by injection molding) and with an exceptionally wide range of frequency, temperature and step intervals. Fig.4 Analysis Protocole For the purpose of this critical review of classical equations, we conform to the traditional procedure to analyse the data: the data are entered in a spreadsheet and analyzed in two ways: At constant Mw, the temperature dependence of mo according to WLF formula is tested, Eq. 3, as well as the universal aspect of the free volume constants C1, C2 and mog. Plots of Log(mo) vs Log(Mw), either at constant T or at constant (T-Tg), are constructed at each T to determine the constancy of the 3.4 exponent. For Non-Newtonian behavior, the principle of superposition is tested in several ways: 1. plots of (G’/G*) versus Log(w) at different temperatures are superposed by horizontal shifting; no determination of the accuracy of the fit is done, except “visually”. 2. plots of Log (m) vs Log(w) are superposed analytically to determine the best horizontal and vertical shift factors . One curve is chosen as reference and fitted with a mathematical expression which minimizes the rms deviation. The curvefitting constants found for the reference curve are then used to fit the other isotherms, only allowing the constants which correspond to a translation of the curve to vary. This technique [26] provides the horizontal and vertical shift factors by regression analysis and calculates the accuracy of the fit in terms of a r2 and a standard deviation, thus allowing a quantitative evaluation of the validity of the procedure. 3. generalized Cross-Carreau equation, Eq. (5), is curvefitted by non-linear regression at each T. Other equations are also tested. The constancy of the exponents with temperature is checked to determine if these popular equations are merely good regressional fits or if they have any other merit. In particular, we address the issue of the accuracy in the determination of the Newtonian viscosity, mo, when it is obtained by extrapolation from the non-Newtonian range. Accuracy Consideration. Hieber and Chiang [10] compile viscosity data on polystyrene from a large variety of publications and claim apparent success in normalizing all these data into a single equation, Eq. (5), incorporating the classical formula (Eqs 2 and 3) for molecular weight and temperature dependence. However, upon re-examination of the accuracy of the fits found by these authors, it seems appropriate to investigate the kind of accuracy which one should expect to get from viscous results. Hieber and Chiang produce a Vinogradov masterplot, i.e. a plot of reduced variables, Log (m /mo) vs Log( mo . w), for 67 data sets from the literature (1175 data points) and, based on Eq. (5) , find a rms deviation of 17.7% (Fig. 4 of ref. [10]). The WLF equation, Eq. (3), is said to be verified with a rms deviation of 14% (Table 3 of [10]), and Eq. (2), which predicts the effect of molecular weight, is also verified, for 102 data points, with a total rms deviation of 55%. Are the rms deviations quoted by Hieber and Chiang so large that many types of formulation tested could be validated equally well? When the variables used to verify an equation or a fit are on the logarithm scale, there is a real contraction of the scale, compare to a linear representation (which is precisely the reason for its use), and one does not really visualize that a 17.7% rms deviation is, in fact, a very poor fit. Likewise, the predicting power of a formula, such as Eq. (2), which is shown to be correct with a 55% rms deviation, on a log scale basis, needs serious reconsideration. A slight variation of a few percents in the value of the exponent in Eq. (2) causes a drastic change in the value calculated for viscosity. If it is admitted that all the laws of rheology are correct within that accuracy, we need to verify that experimental results are, indeed, supporting such a poor performance. For the viscous data obtained on polycarbonate, for both the capillary and dynamic experiments, we conducted a series of tests to check the repeatability of the data and to determine the expected accuracy. Fig. 5 plots Viscosity (Pa-s) against Stress (Pa) for two independent capillary runs done at the same temperature, various strain rates, for Grade 1. The curves are not visually distinct. Fig. 5 Fig. 6 shows that the difference between the two curves of Fig. 5 is random (except for the first points at low rates) and around 1-2%. The accuracy for the dynamic data is even better: for each temperature, three runs were systematically performed, each run with a new sample put in the cell. We could not distinguish between curves obtained at the same temperature, (except at the lowest frequency, which presented some particularity, as we shall explain, and the rms deviation between the curves was better than 0.5%. When we did observe some deviation, we could explain it and find what the reason was: either the temperature was not identical, or the preparation of the sample had been slightly different. Fig.6 Take the curve of Fig. 7, for instance, which is a plot of m* (Poises) vs G* (dyne/cm2) obtained for Grade 1 at various frequency, constant temperature, 223 oC. If we rerun the same experiment 3 times, we obtain exactly the same curve, except for the portion at the beginning which is shown to drop from 38,394 to 26,753 Poises in this instance. In another run, the initial value was 31,440 and in still another run, 37,224, but it always dropped down to the same value 26,753 after the first 6 frequencies, and continued on with the same set of value of (m*, G*) at the other frequencies. Fig. 7 Fig. 8 gives the cell temperature near the specimen tested at each measurement done. One sees that, indeed, there is a slight increase of temperature, here 1.85 oC (which was the worse case we observed), between the first measurement (obtained at the lowest frequency) and the sixth measurement, and that, perhaps, this error could account for the sharp drop of viscosity observed at the beginning of Fig. 7. However, for this Grade, the effect of temperature on the Newtonian viscosity (which is the case here since this is the low frequency range), can only justify 1/3 of the viscosity drop observed at the initiation of the curve. The rest can be related to the sample preparation technique which creates a thermal history on the specimen, slightly variable from specimen to specimen. The difference between specimens is quickly relaxed out at the beginning of the test, which takes place at the lowest frequencies. Once the small influence of thermal history has been erased, the viscosity curves become independent of any other parameter other than temperature, yielding excellent reproducibility and accuracy. In fact, it is probably significant to observe that the initial drop off observed at low frequency in Fig. 7 decreases in magnitude as temperature increases (because thermal history relaxes faster), and even disappears for temperatures higher than 260 oC. Fig.8 In summary, it seems in order to warn against the use of “uncontrolled” data from various sources to test the validity of sensitive equations. “Randomness” of the residuals for alleged good fits might be artificially created by mixing data from several workers, and mistaken with the effect of thermal history, which is known to create scatter and an unacceptable level of accuracy compare to experimental errors. As already said before, it seems more appropriate to generate the data carefully, using the same protocol of sample preparation, the same procedure to test the specimen, the same instrument to determine molecular weight characteristics, the same instrument to measure the viscoelastic data, by the same operator, and especially using the same lot for the resin. CRITICAL ANALYSIS OF THE EQUATIONS OF RHEOLOGY Universality of WLF constants at Tg. The simplest myth to knock down, because it is already largely admitted, is the myth that universal constants C1g and C2g enter the WLF formulation of Newtonian viscosity, Eq (3). Fig. 9 is a plot of Log(mo/mog) vs (T-Tg) for the three Polycarbonate grades. The value of Tg for the respective polymers is known from PVT analysis, by intercepting the rubbery and glassy volume-temperature behavior, at atmospheric pressure. The Newtonian viscosity mo at each temperature is obtained by fitting the non-linear behavior with the generalized Cross-Carreau equation, Eq. (5). For each grade, we verify that Log(mo) vs T can be fitted rather well (with r2 better than 0.999) by an hyperbolic function, which can, indeed, be rewritten as a WLF equation, Eq. 3 [16], from which the two constants C1g and C2g are drived. The value of mog in Fig. 9 and Eq. 3 is computed by extrapolation from the hyperbolic fit, knowing the value of Tg. See Table 1. Fig.9 Fig. 9 demonstrates that plots of Log(mo/mog) vs (T-Tg) for the 3 Polycarbonate grades do not resume to a single curve, as predicted if the WLF constants were universal [16]. The values obtained for C1g and C2g are convincingly different from the universal constants proposed by William, Landel and Ferry [16], respectively 17.44 and 51.6. Also, it is observed that Log(mog) for the 3 grades is not equal to 13, and does not stay constant with molecular weight, even for the two linear polymers Grade 1 and 2. Fig. 10 is a similar Figure than Fig. 9, but for Polystyrene. Six monodispersed Polystyrene melts, of various Mw on both sides of Mc, seem to validate well the universality concept. Fig.10 In Fig. 10, Tg is a function of molecular weight: Tg (Mw) is provided in Fig. 11, showing the leveling of Tg at high Mw. From a visual inspection of the data, Fig.10 seems to show that superposition of Log(mo/mog) vs (T-Tg) is quite good for PS, with the exception of the lowest molecular weight (M=1,670), which is singled out at the bottom of the graph. However, by changing the scale to examine the data better, as done in Fig. 12, a systematic deviation from a unique curve is clearly observed. For instance, if one compares the curves for M=19,300 and M=1.2 million at Log(mo/mog) = -8.5, a horizontal distance of about 8 oC exists between these 2 curves, which is quite significant rheologically speaking. The rms deviation between the curves in Fig. 10 is 12%, with systematically curved residuals, and the r2 for the superposition is equal to 0.976, which indicates a poor correlation. In conclusion, although it is hardly visible when the whole curve is shown in Fig. 9, the curves actually do not superpose. This means that the “constants” K and K’ of Eqs (1) and (2) are actually dependent on Mw, and are not simply related to the monomeric friction coefficient, as classically admitted [1]. Fig.11 Fig.12 In summary, the universality of the WLF constants is, at best, a gross approximation. The Vogel-Fulcher equation, which translates the hyperbolic nature of the temperature dependence of Log(mo), is verified for all molecular weights, but, as shown in Part II of this review article, other equations, equally simple, linearize the data as well, or perhaps even better. In fact, it is quite possible that the apparent success of free volume expressions [16] to explain the curvature of Arrhenius plots, i.e. of Log(viscosity) -1/T , has inhibited the search for other curve fitting expressions susceptible to bring a different perspective to the flow behavior. In other words, temperature and molecular weight do not seem to accurately separate according to the classical formula of viscosity, Eqs 1, 2, 3. If so, the exponent 3.4 in equation (2) should itself be critically re-examined, as done in the next section. Validity of the 3.4 power exponent for M > Mc. Fig. 13a displays, for 4 temperatures, the log ETAo – Mw curves for all monodispersed polystyrene melts across Mc ~33,000. The data are cross-plotted from the original data which were provided against temperature, for each Mw. The original data are first curvefitted with Eq. (3bis), which is in fact the Vogel-Fulcher formula, and retabuulate every 10 degrees from 120 to 210 oC. Viscosity values, for different Mw, which correspond to the same T are then collected in a new file. Plots of Log(mo) vs Log(Mw) are analyzed with linear regression, for each T. Let’s concentrate first on the M > Mc grades in Fig. 13a. In this region , the plot of Log(mo) versus Log (Mw), at each temperature, should be linear and the slope should equal 3.4. Fig.13a The first observation is that a straight line can, indeed, be drawn through the M > Mc data points, at each temperature, and, furthermore, that the slope of that line seems constant for all temperatures, from T=130 oC to T=210 oC, an apparent excellent validation of the classical views regarding the melt viscosity description. However, let’s look at the results more closely. Fig. 13b Fig. 13b clearly reveals a trend for curved residuals above M > Mc. Of course, one could also draw a short horizontal line to cover the points to the left. However, a fair assessment, based on residuals, would conclude that a better fit, above Mc (perhaps even starting at Me), is a curve, not a straight line, showing a systematic deviation from the behavior corresponding to a 3.4 exponent. Fig. 14 is a plot of the slope of Log ho vs Log M, for M > Mc, obtained by regression analysis, against temperature. One sees 1st that the best slope is not 3.4 but closer to 3.327, on average, and 2nd that the value of the slope is not temperature independent, the data indicating a systematic increase of the slope as temperature is decreased. The deviation is not large, but systematic, which is the point we want to make. In conclusion, for M > Mc, PS monodispersed fractions, not only is the theoretical value of the exponent, 3.4, not validated, but the value of the best curvefit for the exponent is temperature dependent, not fixed. Fig.14 This demonstrates a result well known to regression analyst experts: starting from the wrong formula can lead to crude mathematical inconsistencies. One should not only test the coefficient of determination r2 and the chi square k2 [28] of the regression, but analyze the randomness of the residuals before any claim to the validity of the fit is concluded. For the data of Fig. 13a above Mc, linearity seems admitted, not only based on visual inspection, but by regression analysis, with an acceptable r2 of 0.996 and a k2 of 2.5%, but the residuals are badly curved, which is the reason for the pronounced curvature when the data are plotted a different way, such as in Fig. 13b. Also, Fig. 14 clearly demonstrates that the exponent is temperature dependent. As already pointed out before, the reason for the small but systematic deviation is to be sought in the fact that T and Mw do not truly separate in the expression of viscosity, contrarily to what the admitted models assume, Eqs (1) and (2). For Polycarbonate, the Log(mo) -T data for the two linear PC grades can be calculated with an accuracy better than 0.2 % over a temperature range between 165 and 280 oC. Knowing the average molecular weight Mw for these two grades at better than 5% [32], one can calculate, at each temperature, the ratio: ( Log mo2 -Log mo1 ) /(Log Mw2 -Log Mw1 ) which should equal 3.4 and be independent of temperature. Fig. 15 reports the result using two molecular weights for Polycarbonate, grade 2 and grade 1. It is clear that the power exponent is not constant and systematically increases as T decreases; it becomes 5 at a temperature 30 oC above Tg in Fig 15. One might argue that using only two molecular weights to test a formula is not justified, even if the points are generated with great accuracy. Yet, these results for Polycarbonate are showing the same trend than those found for a series of 23 monodispersed Polystyrene (Fig. 14).. Fig.15 In summary, for both PC and PS, the criteria of constancy of the 3.4 exponent is shown to be a curvefitting approximation, which theoretical models should not try (nor succeed) to explain. For M < Mc Viscosity is not proportional to M, contrary to Rouse’s model. Fig. 16 is a Log mo -Log Mw plot for Mw < Mc at T = 210 oC, showing that it is not linear. In other words, Newtonian viscosity is not proportional to Mw, at T constant, below Mc. This seems to contradict the Rouse’s model known to apply well to unentangled melts[1]. Fig.16 It has been argued [1] that viscosity -Mw plots, below Mc, should be considered at constant friction factor, instead of constant temperature. Only under those conditions is viscosity proportional to Mw [1]. The requirement of a constant friction coefficient is met when the systems are compared at constant (T-Tg), which is assumed to correspond to constant free volume, hence constant friction factor conditions. This happens if one studies the molecular weight dependence, below Mc, at a “moving” temperature (Tg+X), where X is a constant, and Tg is function of molecular weight Mn (Mw ~ Mn in monodispersed systems). We know the relationship between Tg and Mw in Fig. 11: Tg (oC) = -253.0 + 353.65 . Mw /(Mw + 330.1) Fig. 17 is a plot of Log(mo/Mw) against Log(Mw) for a series of Mw selected across Mc, at different temperatures, all equal to (Tg+85) oC. A very good agreement with theory is seen in this graph: not only the M>Mc behavior is well described by the expected 2.4 power exponent (the y-axis is Log(mo/Mw)), but the MMc the behavior is more curved than straight. Also, for X=30, below Mc, the slope is definitely negative, a deviation which, seemingly, cannot be understood in terms of coefficient of friction. In summary, Eqs (1) and (2), traditionally accepted to represent viscosity behavior, should be used with great caution. Molecular weight and temperature dependence on viscosity do not separate out. They do not separate out below Mc, because the data must be compared at constant (T-Tg), where Tg is molecular weight dependent. They do not separate out above Mc, because not only the temperature constant in Eqs (1) and (2) is shown to vary with molecular weight, as proven in Figs 9,10,12, but also because the 3.4 exponent varies with temperature, as clearly demonstrated in Figs 14 and 15. Fig.18 In conclusion, the well admitted view that the viscosity of polymer melts can be described by well known Eqs (1), (2), and (3) is far from satisfactory. At best, the well known equations should only be used if no other experimental data is available, as also admitted by Ferry [16]. For each material, one must specify the actual value of Log( mog), C1g and C2g in Eq. (3), and of the power exponent in Eq. (2). Furthermore, to be accurate, the temperature dependence of the power exponent should be known (Fig. 14). While this approach can arguably present some merit for prediction of the flow behavior at 20 to 50% accuracy [10], theoretical models should not attempt to physically understand these formulas, simply because they are wrong and do not describe the data correctly. Other formulations must be determined in order to find answers to the fundamental question of polymer melt rheology: what are entanglements and how do they influence viscosity and shear-thinning? Accuracy in the Determination of the Newtonian Viscosity. A possible explanation for the failure of traditional models and equations to describe well our data is a lack of accuracy in the determination of Newtonian viscosity. The quality of the experimental data and of the sample preparation has already been discussed in a previous section. But Newtonian viscosity values are essentially derived by extrapolation from the non-Newtonian range, and the method of extrapolation might be questioned. In particular, Newtonian viscosity calculated from different curvefitting formulas, either at vanishing strain rate (or w) or vanishing stress, might be different. The common plot to present viscosity results, at constant temperature and variable frequency (or strain rate) uses a double logarithmic scale, Log(m) vs Log(w), which is shown in Fig. 19 for polycarbonate Grade 1 at T = 223oC. Fig.19 The generalized Cross-Carreau equation, Eq. (5) curvefits such a plot almost perfectly, with r2 better than 0.99995 and k2 =5.7 10-4. 4 curvefitting constants are determined from the fit, including the Newtonian viscosity corresponding to w =0. The value of mo is 26,790 Poises. Fig. 20a is another representation of the same data, using a linear scale: m versus G*. The Newtonian viscosity corresponds to G*=0, which is easily found if the curve is linearized. Fig. 20b shows a linear plot of m against sN, where sN is a function of G*, with the dimension of a stress, such as the curve of Fig. 23a has become a linear function: (8) sN = (a3 + 105/G*)-1 m = mo + a1 . sN a3 is a curvefitting constant, easily found by regression analysis. For Fig. 20b: a3 = 0.01456 mo = 26,776 r2 = 0.99994 and k2 = 4.3 10-4. Only 3 curvefitting constants are required. Fig. 20a Fig. 20b Finally, Newtonian viscosity is sometimes calculated from the extrapolation at G* = 0 of a Log(m) vs Log(G*), shown in Fig. 21 for the same data at the same temperature. Such a plot is extremely well fitted by the following expression: (9) Log m = a1 +a2*tanh(a3*LogG* +a4) with a1, a2, a3 and a4 obtained by non-linear regression. Log mo corresponds to G*=0: Log(mo) = (a1-a2). For Fig. 21, Log(mo) = 4.419853, so mo = 26,294, and r2 = 0.99999, k2 = 3.5 10-4. Another excellent correlation with perfectly random residuals. The fit is shown as the line passing through the data points. Fig. 21 One sees that the same value is found for the Newtonian viscosity, at 0.16% determination error, whether it is extrapolated from a Log-Log scale, from linear axes, from G*=0 or w=0. But this was done at a single temperature: 223 oC. The same analysis can be carried out at other temperatures, which allows the determination of Log mo (T) from several extrapolation techniques. The fits obtained at all temperatures are excellent, comparable to the results quoted for T=223oC. Fig. 22 gives the result for polycarbonate Grade 2. Fig.22 In this figure, the two curves, obtained from extrapolating either Eq. (5) or Eq. (9), are identical from 280 oC to 210 oC, but separate out at lower temperatures, even showing a difference of more than a decade at 180 oC. A Vogel-Fulcher or the WLF equation, Eq. (3), can be used to curvefit either the top or the bottom curve of Fig. 22, albeit with different C1g , C2g and Log(mog) constants. However, neither of these constants confirm the universal value provided by Ferry [16], and in any case, even if it seems impossible to conclude, at low temperature, which curve corresponds to the true Newtonian viscosity (in Fig. 9, it is assumed that it is the lower one), the behavior between 210 oC and 280 oC is unambiguous. Even in this temperature range, where the extrapolated Newtonian values are not questioned, Fig. 9 clearly demonstrates a molecular weight dependence of the friction factor term in Eq.(2) which should only vary with temperature, should temperature and molecular weight truly separate. The next section confirms the distinct behavior one should expect below 190 oC for PC, and the reasons theoretical models should explain it. Time-Temperature Superposition. As time temperature superposition principle is applied in this section, for both dynamic and capillary data, many important issues of viscoelastic melts become apparent. As already pointed out before, for a Vinogradov’s plot of Log(m/mo) vs Log(mo.w) all data points collect onto a single mastercurve. Eq. (5) predicts such a mastercurve, as long as the fitting parameters are truly independent of temperature. If Log(m*) is plotted against Log(w) at various T (Fig.2), the superposition principle stipulates that all curves superpose by horizontal and vertical shifting, both shift factors LogaT and LogbT being a function of temperature only. The superposition can be performed by computer, as shown by Ibar [26]. The regression technique is described in Ref. [27]. Eq. (5) can first be applied to a reference curve, at temperature T1 , to find the various fitting parameters: m = b1 /( 1+ b2. wb3)b4 and, at temperature T, the equation of the curve is the same, but shifted on both scales: m. bT = b1 /( 1+ b2. (waT) b3)b4 with the same constants b2 b3 b4. This can be rewritten: m = b’1 /( 1+ b’2. wb3)b4 with aT = mo/mo1 and bT = 1/ aT if Eq. (5) is correct, (the subscript 1 refers to the reference curve). The r2 and k2 of the regression fit, and analysis of the residuals, at each T, should tell us the quality of the superposition according to Eq. (5). Fig. 23 clearly shows that, indeed, bT = 1/ aT , except in the lower T region (at the bottom right end side of the curve) for which a systematic deviation is visible. Fig. 24 also clearly demonstrates that aT = mo/mo1, with perhaps a small deviation in the lower temperature region. T ref is equal to 225 oC. Fig.23 Fig.24 Notice that we do not see any apparent upper melt temperature discontinuity on neither Fig. 23 or 24: no Tl,l transition in these two graphs. The classical views on melt flow are apparently forcefully validated here: the superposition principle is validated, Eq. (5) is validated, and there is absolutely no transition in the Tl,l temperature region. This behavior is typically reported for polymer melts. Figs 25 and 26 plot the statistical results of the double-shifting procedure. Fig. 25 provides (r2 -1) at each T and Fig. 26 displays k2. Fig. 25 Fig. 26 The superposition correlation is excellent for T values between 220 and 245 oC only. Below 220 oC, the superposition becomes worse and worse as T decreases, and, additionally, there seems to be a systematic deviation of the curve in Fig. 25 above 245oC. This trend is unambiguously confirmed in Fig. 26 which clearly identifies a transitional behavior at 245 oC = (Tg + 100 oC). Furthermore, the regression of the superposition by double-shifting is totally inadequate below 215 oC as shown by the rapid rise of k2. The lower portion of the curve is purposely cut off below T = 200 oC in Fig. 26, in order to focus on the high temperature range, but k2 actually reaches 160% for T=185 oC, a temperature which is still 40 oC above Tg, right in the middle of the rubbery region [16]. Nothing in the classical theories predicts such a departure (remember, the superposition principle is supposed to work, and the WLF equation supposed to describe it, between Tg and Tg+100oC).. In other words, the same results which, plotted as Figs 23 and 24, apparently illustrate so powerfully the claims of classical views about the melt flow behavior, are, under close examination (Figs 25 and 26), revealing quite a different picture: superposition is not validated from Tg to Tg+100, Eq. (5) is wrong (the Carreau’s “constants” are actually temperature dependent), and there is a transition temperature in the upper melt, located precisely where Boyer predicted it, at 245 oC [see later]. These same conclusions are found regardless of the equation used to curvefit the reference curve (onto which the other curves are superposed by computer shifting). For instance, in Fig. 27, a3 of Eq. (6) and of Fig. 20b, is plotted at different temperatures: Fig. 27 a3 remains constant on a small temperature range only, between 210 and 260 oC, and strongly deviates from a constant outside that range. The same behavior is observed for the other two polycarbonate grades and for polystyrene. The results for polystyrene will be discussed later. The Upper Melt Temperature Departure from Superposition. Let’s now study capillary viscosity data. The existence of the transitional behavior at around 245 oC for Polycarbonate is affirmed and well demonstrated when Stress is plotted against strain rate, instead of Viscosity. This is done in Fig. 28 for Grade 1 Polycarbonate. Temperature spans across 245 oC from 235 oC to 310 oC. Fig. 29 displays the mastercurve obtained by computer double-shifting, with Tref = 235 oC, demonstrating, once again, an excellent superposition of the curves, in the traditional sense (bT in Fig. 29 has a different meaning than in Fig. 23, since the variable used to superpose is different). However, the r2 for the superposition fit, shown in Fig. 30, splits the temperature range into two clear regions, 1 and 2. In region 1, covering 235 oC to 260 oC, r2 remains excellent (0.998) for all isotherms. R2 sharply drops to 0.987 for the last 3 isotherms, 270 oC to 310 oC, which we designated region 2. The curves of region 2 do not superpose well with the curves of region 1, although they certainly do superpose between themselves. This confirms for Grade 1 what was found for Grade 2, e.g. the existence of a subtle transitional behavior in the upper melt. Fig.28 Fig. 29 Fig.30 The use of an analytical superposition method, instead of a graphical one, is necessary to determine the existence of such a change of viscous mechanism, which, perhaps, explains its lack of recognition in the literature. In 1981, Ibar [26] introduced such analytical shifting methods based on regression minimization algorithms [27], performing the shifting on both the vertical and horizontal axes. The temperature variation of the shift factors revealed the existence of the upper melt transition [26]. It was also shown that the vertical shift factor bT , as determined from the optimized shifting of the curves, is not equal to (T1 r1 /T r ), as it should be according to classical views [1,8]. The distinct behavior across the two regions can further be analyzed by calculating what would be the stress in region 1 if we were to extrapolate results from region 2. This is done by choosing T = 290 oC as new reference and finding the doubleshifftin factors to superpose the other two isotherms of region 2. Hyperbolic functions fit perfectly the temperature dependence of both log (shift factors), providing a way to calculate, by extrapolation from the high temperature region, what the stress would be, for a given temperature of region 1, and compare that to what the real stress is in this region. Fig. 31 compares the calculated curve from high T region extrapolation, to real data for T = 245 oC. It is clear that the real stress data are all located below the calculated stress, perhaps an indication that the change of flow mechanism across this (Tg+100) “transition” is similar to the effect of strain rate when the melt starts to shear-thin (see later). The behavior described in Fig. 31 seems to be general. It applies to all three Polycarbonate grades studied in this paper, to Polystyrene, and to many other melts analyzed in the same way. In the case of Polystyrene, for instance, the authors of Ref. [17] fit the variation of the horizontal shift factor over the broad temperature range with two WLF hyperbolas, calling the transition in between “a network transition”. Again, the stress (and therefore the viscosity) calculated for the lower temperature region, from extrapolation using the upper temperature WLF equation, gives too large stresses: something happens in the system of interactions between the bonds which makes it easier to flow at lower temperature. Notice that, contrary to common sense, flow ability increases when temperature goes down across that transition. This observation is quite important for theoretical reasons: the reptation model for entanglement [4-6, 9] does not predict the existence of such behavior, and is possibly not adequate to account for the characteristic melt behavior described here, i.e. a temperature-thinning effect. Another important observation is the location of that “melt transition”, it occurs precisely where R.F. Boyer defined his Tll transition [ see “Tll and Related Liquid State Transitions-Relaxations: A Review,” Polymer Yearbook 2, Harwood Academic Publishers, Edited by Richard A. Pethrick, pp234-343 (1985)]. The existence of Tll has been denied by the polymer scientific community for decades. Flory denied it first with vehemence [ ]. Plazek followed, de Genne ignored it, probably did not know what to say. Boyer must be given the full credit, even if, as it turns out, his explanation for it was wrong, perhaps actually responsible for the controversy (non-believers associate Tll with local order, Boyer’s explanation). Fig. 31 The temperature dependence of viscosity beyond (Tg+100oC), where WLF equation is known to fail [16], is described by an activated process, of the Eyring or Arrhenius type. Hence, it is well known and admitted that the WLF equation is valid between Tg and (Tg +100 oC) only, and resumes to an Arrhenius form, corresponding to an activated mechanism of flow, beyond that temperature range. If one can find a formulation for viscosity which does not seem to require a change from a free volume to an activated process mechanism in the middle of the temperature range, does it not make more sense to use such an expression? Viscosity is a parameter representative of the internal resistance to flow under stress, which has to do with the reorganization of the interactions between the bonds which form the macromolecules. Which thermodynamic quantities can be correlated with molecular parameters? The reason for the failure of the free volume based WLF equation at high temperature is left largely unexplained, ironically by the same scientists [29] who refute the existence of an upper transition in the melt, the Boyer’s Tl,l transition [30]. If there is such a transition in the upper melt, what causes it, and why are the accepted models of flow [4-6] not addressing such an important issue? What happens to the reptation tube below and above Tl,l ?. Obviously, if one needs to change the form of the viscosity equation across a certain temperature in the melt, it seems very important to know why, not to mention the influence that such a complexity brings to the prediction of the flow viscosity when performing computer simulation of molding processes [31]. The issue of the existence of a transition in the upper melt must be re-addressed by rheologists and polymer scientists. The Lower Melt Temperature Departure from Superposition. Let’s now turn to the departure from the superposition principle at the lower temperature end, as Tg is approached. The nature of the transitional behavior at around 190 oC -210 oC in Figs 25-28 can be better characterized by plotting the relative elasticity (G’/G*) against the strain rate (or frequency) at the various temperatures. Fig. 32 is actually a plot of the square of (G’/G*), for reasons which will be apparent later. Fig. 32 There is no doubt that superposition by horizontal shift is possible in Fig. 32, which covers the temperature range between 200 and 280 oC. However, the lower melt temperature range, shown in Fig. 33 from 163 to 200 oC, cannot be superposed with any reference curve of the previous graph, confirming the departure from superposition noticed earlier. This behavior is also true for the data on Polystyrene[17] re-plotted in Fig. 34. One distinctively observes, for both Polycarbonate in Fig. 33 and Polystyrene in Fig. 34, the presence of a maximum for the lower temperature isotherms. The frequency of the maximum increases with decreasing temperature. The traditional way for handling the superposition principle is to shift over the whole range of temperature above Tg, accepting somehow arbitrarily less precision for the lower temperature isotherms, for which the principle seems to deviate the most. The result is an “acceptable” mastercurve, as shown for Polystyrene in Fig. 35. However, to obtain superposition of the tails located after the maxima in Fig. 34, a sort of two-way shift is necessary, one shift for the lower tail, another shift for the upper tail, casting doubt on to the validity of the superposition principle for these data points. In fact, if we eliminate the points located beyond the maximum, the superposition seems to be much more accurate, even for the low temperature isotherms. Furthermore, the points located on the high frequency tail, beyond the maximum, at different temperatures, superpose between themselves, with a different temperature dependence for the shift factor than the data localized on the lower frequency side of the maximum. This means that the mastercurve in Fig. 35, plotted against a single variable, Log(aT w), is not representative of the real situation beyond the maximum. There is a change of the shift factor for the data below the maximum and for the data above the maximum: there is no single mastercurve for the whole temperature-frequency range. This result is also true for the 3 Polycarbonate grades studied. Fig. 33 Fig. 34 Fig. 35 In terms of traditional superposition of the data plotted as Log(m) vs Log (w), it appears, from the previous analysis, that those points which correspond to a frequency above the maximum in Fig. 34 should not superpose with the same shift factor, and should be eliminated from the regression analysis when superposing analytically, or from the pool of data points when curvefitting with the generalized Cross-Carreau equation, Eq. (5). Fig. 36 shows that those points to eliminate, the high frequency data on the lower isotherms, are traditionally assigned to the existence of a second Newtonian regime, which Hieber and Chiang [10] try to incorporate into a new equation, Eq. 4. This approach has possibly little merit, from a physics point of view. By way of analogy, it amounts to put in the same bag, for analysis, data from the elastic state and from the plastic state of a stretched material, say a metal, or mixing data above and below Tg when fitting PVT results. In Fig. 35, the data from the left end side of the maximum correspond to one mechanism of deformation, an elastic deformation in our analogy; this mechanism is interrupted and triggers another deformation mechanism when frequency exceeds a certain value, corresponding to yielding of the melt, in our analogy. The data contributing to the second Newtonian regime, which is often perceived as a limitation to shearthinnning should not be mixed with the data describing shear-thinning, because they “pollute” the regression coefficients obtained. Practically speaking, it seems difficult to determine which data should be eliminated, from a viscous -strain rate plot (Fig. 36). For instance, the last three high frequency points which look perfectly OK on isotherm 150 oC (the 4th curve from the top in Fig. 36) are actually located beyond the maximum in Fig. 34 and should be eliminated from the regression giving the parameters of Eq. 5. This might explain why the regression parameters for superposition always seem to diverge sharply at lower temperatures (Figs 25-27). Fig. 36 Is the Superposition Principle ever Valid? The idea behind superposition is scaling, or the renormalization of the variables to describe temperature and strain rate effects. Over a 100 oC span above Tg+30, and within a 25%-50% error tolerance, the answer is yes, suffice it to quote all the papers which have tested its validity under those conditions, and routinely present their data after superposition has been performed. However, if the tolerated error is of the same order of magnitude than the experimental error, then the answer is no, even on a short temperature span of 30 oC , if the scaling variables to superpose have not been truly defined. Consider Figs. 37 and 38 for a PS of Mw=250,000 (polydispersity 2.5). The temperature span is 30 oC, from 140 to 170 oC. The longest relaxation time (the terminal time) is determined from the maximum of h” = G’/w in Fig. 37. to = 1/wo .. In theory, the scaling factor for the horizontal axis should be w/wo and h / ho for the vertical axis. These are the reducing variables, according to the traditional approach. It is, indeed, almost true. Fig. 38 is a close-up view of the masterplot and superposition is almost OK. Yet, if one is concerned whether the curves truly superpose, a fair answer is that there is a systematic deviation visible for all curves, and that these curves, in fact, do not superpose. It should be noted the large number of points per decade to define the frequency sweeps at each temperature. Perhaps very significantly, if we filter out, starting from the same data set, the number of points down to 3 points per decade, a procedure followed in most routine sweeps, the superposition now appears quite satisfactory. Fig. 37 Fig. 38 The only way to make the superposition really work is to force it to work, by double-shifting, done analytically by regression. But, as explained by Ibar [26], when one analytically forces the data to fit an equation which includes fixed parameters, in order to accomplish a double-shift procedure, one invariably observes that the regression statistical parameters decline systematically on both sides of the reference temperature. This remains true even after the high frequency data corresponding to the second Newtonian regime have been eliminated. This observation is the proof that there is no validity of the time temperature superposition over the range of temperature suggested by Ferry [8], i.e Tg and Tg+100oC. As shown earlier, it is possible to establish two ranges of strain rate and temperature, within which the time superposition does work, but if this is the case, one needs to admit the existence of a transition-relaxation above Tg, and theories should explain it: this is not the case at present, and, furthermore, the existence of such a transition is vehemently opposed by the current school of polymer physics. The ultimate test of validity for the time temperature superposition is to let all constants loose in the curvefitting function and determine if there is any trend with temperature for those parameters which are supposed to remain constant. Does any break in the trend show up? This will be done in part II of this critical review. The Question of Understanding Rheology with a Spectrum of Relaxation. Like with any concept that has shown extremely useful over the years, it is very tempting to explain new phenomena with the same concepts, by the same type of equations. The concept of relaxation time goes back to the explanation of resonance phenomena, in mechanics, in electrical circuits, in magnets orientation, in just about every single field of physics. When the time scale to observe is tuned to the time scale of what is changing, resonance occurs: wt=1. Mechanical and electrical analogs for resonant systems are taught in elementary physics classes. The simplicity of their equations is the elegant driver that makes us look for them. A resonant system can be defined by its relaxation time t. When the experimental results are more complex than what can be described by a single resonant system, then a network of dashpots and springs or capacitors and resistors is defined, and “a spectrum of elementary relaxation times” relates to the complexity of the response. For dashpot and spring in series, the relaxation time is simply the ratio of the viscosity of the dashpot by the modulus of the spring (the spring constant). In the case of understanding the visco-elastic properties of polymer melts, nothing seemed more appropriate than a network of dashpots and springs, a Maxwell network for stress relaxation experiments, a Voigt network for creep, and a set of fancy equations to coordinate one network to the other, in order to make it general. The next step was to find what was moving and define it in molecular terms. Macromolecules are long chains of mers and their melt properties must be described by the constant motion of the bonds to re-organize within the restrictions imposed by other bonds, those on the same chain, and those located near-by on adjacent chains. The simplest well known theory to characterize statistically a set of atoms or molecules submitted to thermal agitation is the statistical gas kinetic theory due to Maxwell. Pressure of the gas can be quantified from the average velocity of the molecules, so can diffusion in the gas. The link between macroscopic variables and molecularly defined entities is what makes Maxwell’s description of the properties of a perfect gas a breakthrough. Of course, macromolecules are not as simple as simple molecules, because their mers are covalently linked to each other along a chain, which drastically reduces the degree of freedom of their motion. But the temptation seems natural to redefine the statistical units present in the theory of gases in terms of mers dimensions, and define a relaxation time (or a spectrum of those) that correlate to the local “spring and dashpot” properties of the mers. This is the successful Rouse’s theory of diffusion in polymers [32bis]. Simply stated the relaxation time is the ratio of a spring constant, the same as for a statistical gas, 3kT/b2, and a viscosity, N xo, where N is the number of mers per chain, and xo is the local friction coefficient per mer; b is the quadratic distance of two beads defining a chain segment ( b= Rg, the radius of gyration). (10) xR = r2 1 kT po N < Rg2> The diffusion constant is defined as the ratio of thermal energy to the total friction (from Stoke-Einstein): (11) D = Npo kT and the shear modulus G(t) is: (12) G(t) = M tRT e-p2 xR f p t p ! where p=1,2,3…defining a family of discrete relaxation times: (13) xp = p 2 xR The steady state viscosity is: (14) ho = 6 r2 M tRT c mxR This formula gives an explanation to what was stated in Eq. (1). The Rouse time, in Eq. (10) varies with M2 since Rg varies with M0.5 , so viscosity scales with M according to Eq. (14). We have shown that this was not true, at constant T (Fig.16 ) for M < Mc monodispersed PS, also confirmed by more recent results by Majeste et al [35], but could be be true (Fig. 17) at constant (T-Tg) where Tg is made a function of M ( this could be defined as “a constant free volume” friction coefficient approach. See part II of this article). In the frequency domain, Eq. (12) above can be transposed to give the Rouse complex modulus as a function of w (15) GRouse * (~) = M tRT 1 + ~2xp2 ~2xp2 + j ~xp 1N! If one calls Go,N = r RT/M , and if, for the sake of illustration, one limits G* to the first term ( p=1, terminal relaxation) of the series, then the above equations can be rewritten as: (15bis) G' = 1 + ~2xo2 ^ h ~2xo2 GNo G" = 1+ ~2xo2 ^ h ~xo GNo tan d = G' G" = ~xo 1 (16) cos 2 d = 1 + tan 2 d 1 = G* G' c m2 = GNo G' (17) ~G' = GNo ~ G * G' b l2 The reasons to write Eqs (16) and (17) will become apparent in the section below. The Rouse model is admitted to describe well polymers with M < Mc (we saw earlier that it does only if data are re-tabulated at (T-Tg) constant), but is unable to explain the effect of entanglements on viscosity, or the existence of a molecular weight independent plateau modulus Go,N = = r RT/Me for M > Mc melts (where Me is the weight of the strands between entanglement points). Rouse predicts that Go,N decreases with M. Another problem, the contribution of the high frequency terms (the influence of the transitional relaxation terms), in addition to the Rouse term needs to be addressed. The following expression is due to A. Allal [36]: (18) GHF * (~) = G3 1 -1 + j~x'o ^ h1/2 1 > H x'o = r2 1 kT po b' 2 and now: (19) G*(~) = GRouse * + GHF * For entangled polymers, the de Gennes [4] and Doi-Edwards’s [5, 6] reformuulatio of the motion of a chain embedded in a sea of topological obstacles created by the presence of other chains, resulted in the definition of a tube into which macromolecules could only move by reptation. The confined motion explained the viscosity increase. The tube could fluctuate in length, in proportion to (M/Me)0.5 , and could locally be renewed resulting in local constraint release from topological restrictions (Montfort, Des Cloizeaux, Klein, Graessley). The details are very sophisticated. For our purpose, the new model is merely a justification to refine Rouse’s family of discrete relaxation times to make it fit to the increase of viscosity with M due to entanglements. Thus, de Gennes’ school redefines the terminal relaxation time to, corresponding to p=1 in Eqs. 12, 15, with respect to the molecular weight, introducing the value of Me, proposes a different “structuring” of the relaxation time family, for instance with respect to the number of terms (p=1,3,5… only the odd terms relax), modifies the weight of each relaxation process (Gp = Go/p2 in the original Doi-Edwards’s model), but keeps the same family description of the relaxation times tp as a function of p (Eq. 13, with tR replaced by td). The whole exercise is driven by the need to explain the molecular weight dependence of viscosity, which is assumed to vary like M3.4 , with all the reservations made about this relationship in this article. Many authors gravitated around the reptation platform, modifying some aspects of it, debating about the improvements, creating a real school of thoughts about polymer flow based on reptation. As an example of the new reptation ideas, the following formulas, given by Matsuoka [33], are claimed to describe well dynamic rheological data of monodispersed PS melts by Marin and Graesley[34]: (20) G' = Gp 1 + ~2x p2 _ i ~2x p2 p odd ! G" = Gp 1 + ~2x p2 _ i ~xp p odd ! (21) with: xp = p3.4 xo Gp = p Go p = 1, 3, 5, ..., pmax pmax = Me M Matsuoka makes a variation in the definition of the Doi Edwards parameters in Eq. (20-21), but it should be pointed out that the Rouse’s model and the reptation models have a common framework, only the definition of Gp and tp change. In the following graphs, we apply formula (20) and (21) to the case of Marin’s data on monodispersed Polystyrene, M= 110,000 (M/Me=6.47). The temperature is 140.9oC. The exercise is to learn how well a modified spectrum of relaxation, according to the reptation school, describes results obtained in dynamic shear. The terminal time to is determined from the maximum of G’/w vs logw and is equal to 13.15 sec at 140.9 oC. The elastic plateau modulus of Polystyrene is well known and equal to 0.2 MPa. Me is taken at 17,000 g/mole (consistent with Mc=34,000). The transition end of the entanglement plateau is determined by tT = to *(Me/M)3.4 = 0.023 sec, corresponding to a frequency wT of 43.48 sec-1 The simulation is very straightforward: all parameters entering the terms of the series in Eq. (18) and (19) are defined as a function of the terminal time to and, therefore, G’p and G”p can be tabulated for a range of w values, as well as the sum G*, h*(w), h*o, and the other functions included in Eq. (16) and (17). Figs 39-45 display the simulation results. The points colored in red are for w > wT, i.e. beyond the transition frequency. In Fig. 39 and 40, we learn how the elementary relaxation components combine, respectively for G” (Fig.39) and for G’ (Fig.40), to add up to the blue curve, the simulated total loss and elastic moduli. The low w behavior is totally dominated by the terminal time, and one sees the broadening of the G” peak by the addition of the secondary terms p=3, 5, 7. The influence of the secondary relaxations on G’ is only visible above wo, which also defines the sharp departure from the Newtonian viscosity value in Fig.41 (the Newtonian value is 2.65 Mpa-s.) This wo does correspond to the maximum of G’/w in Fig. 42, which is the inverse of the terminal time to. Fig. 39 Fig.40 Fig. 41 Fig.42 In Fig. 43, a plot of normalized G” vs G’ on a double-log axis, we determine that the slope is 0.5 for the low w points, almost all the way up to the cross-over (G’=G”), but that the slope decreases as we get closer to the cross-over, to become zero at the crossovver Additionally, at the cross-over (G’/GoN) = 0.5. From Eqs. (16) and (17), it should become apparent that the cross-over should be equivalent to the point of maximum of G’/w to define wo Fig. 43 Fig. 44 is a plot of (G’/G*)2 vs (G*/GoN). One sees that the upward curvature sharply reverses for G* = GoN, and that (G’/G*)2 is very close to 1 at its maximum, 0.9684 to be precise. Also, another important feature of this plot is the good match between the values found on each axis corresponding to the cross-over point (G’=G”). It is equal to 0.5 on the y-axis and 0.707 on the x-axis (the square root of 0.5). At this stage of the simulation, it must be admitted, many correlations seem to confirm and validate the feeling that classical concepts look very satisfactory. Additionally, their application seems quite simple and elegant: once the plateau modulus and the terminal time are known, one can easily determine the family of discrete relaxation times. Furthermore, the time-temperature superposition is inherently included from the definition of the relaxation times that all scale with xo, the local friction, that varies with temperature according to a molecular weight independent WLF equation. However, let’s have a closer comparison with the data of Marin (M=110,000 T=140.9oC). These data correspond to to=13.15 sec, as confirmed by a plot of G’/w vs logw. Fig.45 is plot of (G’/G*)2 vs G*/GoN for real data, to be compared with Fig.44. The differences are simply staggering. Unlike for the simulation in Fig. 44, the real data curvature changes in the middle of the data range, showing an inflection point; the maximum is 0.839 compared to 0.968; the value of G*/GoN corresponding to the crossovverpoint on the y-axis, i.e. corresponding to (G’/G*)2=0.5, is 0.35 for the real curve and 0.707 for the simulation, and the y-value corresponding to the x-axis cross-over (G*/GoN = 0.707) is 0.8289 for the real data, almost the value of the maximum, and 0.5 for the simulation. The definition of the cross-over, from a spectrum of relaxation point of view, no longer works. The discrepancy is confirmed when comparing Figs 46 and 47, which are plots according to Eq. (17). The simulation validates this equation extremely satisfactorily (Fig. 46), but the real data fail it badly (Fig. 47). One could pass a parallel line to the simulation line (with slope -1) through the first few points at the left of the graph, corresponding to w < wo. One could also pass another line, but with slope -0.82, not -1, through several points for w > wo. However, there is an upturn departure from that line at high w that does not follow the -0.82 straight line behavior. All these features are significant and will be discussed in part II of this review article. They cannot simply be explained by the equations derived from the Rouse or the Doi Edwards models. In appearance, one might find “good” fit between data and simulation: Newtonian viscosity is found to be “slightly” different, 1 MPa-s instead of 2.63 MPa-s, (+163% difference), and the slope of G”/GoN vs G’/GoN on a log-log plot (as in Fig. 44 for the simulation) is 0.514 instead of 0.5. One could also argue that the value of to and GoN in the simulation could be fine-tuned to obtain a better fit. Indeed, regression analysis on G’(w) and G”(w), using an even or uneven weighted modulus series matching Eq.(20), with all unknown parameters to be determined by regression will be successful. The purpose here is to suggest from these selected observations our strong reservations regarding the interpretations of rheological data with the use of a spectrum of discrete relaxation. In other words, the classical approach itself, that claims to understand the physics of the visco-elastic phenomena in polymer melts, through the search of the dependence of the relaxation times with physical variables, such as molecular weight or temperature, is, in our views, is approximate, limited, and theoretically wrong. One might argue that so many other good scientists have checked the relationships carefully before, and that it must only be coincidental that claims of failure would be advanced. Fig.44 Fig. 45 Fig. 46 Fig.47 In summary, the Rouse model is an application of Maxwell’s statistics to beads and springs connected together, each bead having a friction force exerted on it created by the presence of the neighbors. The main idea behind the Rouse model is, in fact, the definition of the local friction coefficient xo. Otherwise, this model is nothing more than a cartoon representation of a set of springs and dashpots, and the Rouse’s relaxation time nothing less than the ratio of the viscosity of the dashpot by the modulus of the spring. This model assumes that chains are Gaussian, deformation is affine, and there are no excluded volume, in the Flory’s sense, i.e. no perturbation to the local conformation of the bead-springs due to the presence of adjacent chains. It is rather intriguing, to say the least, to see such an enthusiasm for such a simplistic and crude model. The irony is that the popularity comes from the hidden foundation that the model is nothing more than a Maxwell spring and dashpot network, so popular by its own merits in the 60s and 70s. However, one realized immediately, then, the limitations of a network of dashpots and springs: it was clear to everyone that it had nothing to do with a real polymer systems of intra and inter molecular interactions. One might argue that Rouse’s model introduced the motion of a family of relaxation times, which restricted the number of constants needed to define the network to the definition of the terminal time. True. But this simplification also amplified the error found when fitting equations to real data. Checking the success claimed by others [35] to the applicability of the Rouse’s model to the dynamics of M < Mc data, it is surprising how bad the fits actually work out to be, with residuals severely curved, and fitting r2 criteria hardly passing the 0.96 mark. The Doi-Edwards-de Gennes model also assumes Gaussian chain, affine deformation and the existence of a family of discrete relaxation times generated by the terminal time. All these assumptions are severe limitations to the domain of applicability of the model, but, in the end, the apparent success comes from the fact that the melt behaves, within the restricted range, like a network of spring and dashpots, whose relaxation times have been forced to comply with the molecular weight 3.4 variation. Yet again, upon checking the validity of the results, we are facing mediocre performance, and unsatisfactory projections and extrapolations. The Myths of rheology start with the belief that linear visco-elasticity is well understood and under full control. We suggest otherwise. Conclusions Let’s summarize the great Myths of rheology. as reviewed in this article: -in the expression of viscosity, the effect of temperature and molecular weight do not separate, whether M < Mc or M> Mc. -The 3.4 exponent is not strictly temperature dependent, it increases as T decreases. -For M < Mc, viscosity is not proportional to M, disproving the Rouse’s model even for these low molecular weight fractions. Working at (T-Tg) constant seems to work within a restricted zone of temperature ( T> Tg +30oC). -The WLF equation is just an hyperbolic curvefitting function. Expressing viscosity as a function of (T-Tg) does not make it molecular weight independent. There is not universality of the WLF constants for monodispersed PS fractions or well characterized linear grades of PC. -The time-temperature superposition principle is not valid on the temperature range claimed by its present users. One requires at least two WLF equations to cover the range Tg+100oC, and another Arrhenius fit beyong ~ 1.2 Tg. This diversity of response does not seem to receive an easy explanation from existing molecular dynamic models. -There is a transition above Tg, as claimed by Boyer, called Tll by Boyer. A careful analysis of data, when those same data do not show any transition according to the classical tools of analysis, reveals the existence of this transition-relaxation. “Thermal-thinning” occurs when crossing the Tll transition, going downward: Newtonian viscosity is less compared to extrapolations made from using data above Tll. -There is another time-temperature failure transition at higher frequency or lower temperature (around Tg+30oC or its frequency equivalent), corresponding to a different mode of melt deformation, where enthalpic forces interplay with entropic ones (see part II). Beyond this transition (corresponding to a maximum of cos2q, i.e. a minimum of tanq), the time-temperature superposition applies with different shift factors than for the temperature on the other side (the melt side) of this transition. -The time-temperature is only an approximation, a convenient tool for engineers. It should not be the foundation of existing theoretical interpretations of melt deformation. In fact, theoretical models should explain why it is only an approximation and correctly describe the true behavior. -Rheology data in the literature should be presented in their “raw” state, without shifting, because the shifted data are probably wrong, providing the wrong experimental facts to theorists. -The “molecular dynamic” description of visco-elastic data in terms of a family of discrete relaxations generated by a terminal time that varies with a local friction coefficient (thus providing the temperature dependence), and with topology (to explain the M or M3.4 dependence) is an elegant and simple mathematical tool to compare the effect of structural and chemical parameters on melt properties, but, as we suggest, too simplistic to have any value in terms of the physics of deformation of a set of long chains. -The Rouse’s or reptation models (de Gennes, Doi-Edwards), based on a such spectrum of relaxation, are probably not describing at all the basic deformation process giving rise to visco-elastic effects, shear-thinning, normal stresses, extensional flow and the numerous other phenomena observed in non-linear deformation, at very high shear rate, at high amplitude of strain, causing melt yielding, melt fracture and astonishing memory effects. The models’ shortcoming is probably deeply rooted in the misunderstanding of the concept of chain entanglement , and of the entropy of the melt deformation process. Part II of this review intends to bring some light towards a new understanding of these concepts. July 25, 2008 REFERENCES [1] W.W. Graessley, Advances in Polymer Science, Vol. 16, “The Entanglement Concept in Polymer Rheology”, Springer (1974). [1a] J.M. McKelvey, “Polymer Processing”, John Wiley, New York (1962), Ch. 2, p. 32, Table 2-4. [2] G.C. Berry and T.G. Fox, Adv. Polymer. Sci., 5, 261 (1968). [3] F. Bueche et Al, J. Chem. Phys., 20, 1956 (1952). [4] P.G. de Gennes, J.Chem. Phys., 55, 572 (1971). [5] S.F. Edwards, J.W.V. Grant, J. Phys. A: Math.Nuclear.Gen, 6, 1169 (1973). [6] M. Doi and S.F. Edwards, “The Theory of Polymer Dynamics”, Oxford Univ. Press, Oxford, UK (1986) [7] S. Onogi et Al, Kolloid-Z, Z Polymere, 222, 110 (1968). [8] J.D. Ferry et Al, J. Phys. Chem., 67, 2297 (1963). [9] P. 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Negami, Polymer 8, 161 (1967). [24] J.F. Pierson, Ph-D Thesis, CRM Strasbourg (1968). Numero d’ordre at Centre Documentation CNRS, Rue Boyer, Paris: AO 2106. [25] R. Susuki, Ph-D Thesis, CRM Strasbourg (1970). Numero d’ordre at Centre Documentation CNRS, Rue Boyer, Paris: T 32307. [26] J. P. Ibar, J. Macromol. Sci. Phys., B19 (2), 269 (1981): Non-Newtonian Flow Behavior of Amorphous Polymers in the T > Tg Temperature Range: A New Analysis of the Data According to the "Double-Shift" Procedure. [27] Non linear regression routine is based on the Lenvenberg-Merquardt algorithm described in “Numerical Recipes in C”, Press, Flennery, Teukolsdy and Vetterling, Cambridge University Press (1988) [28] The Chi Square expression which is minimized during the routine is: c2 = S (Yi -f(Xi))2 /s2 where s is the standard deviation. [29] L.E. Nielsen, Polym. Eng. Sci,17, 713(1977). The general consensus in the polymer scientist community is that Tll only existed in the imagination of R.F Boyer. 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Table 1 Fitting Parameters in WLF Eq. (3) for the 3 PC grades of Fig. 9 TgoC Log(mog) C1g C2g GRADE 1 136.0 9.842 10.22 76.70 GRADE 2 145.4 11.58 10.46 51.88 GRADE 3 151.0 12.06 11.87 84.16