Effect of long branches on the rheology of polypropylene

Effect of long branches on the rheology of polypropylene* A. D. Gotsisa) and B. L. F. Zeevenhoven Department of Polymer Materials and Engineering, Delft University of Technology, P.O. Box 5045, 2600 GA Delft, The Netherlands C. Tsenoglou Department of Chemical Engineering, National Technical University of Athens, 15780 Zografos, Greece (Received 8 October 2003; final revision received 23 April 2004) Synopsis In order to study the rheology of long chain branched polymers, branches have been added on linear polypropylene precursors in varying amounts using reactive modification with peroxydicarbonates. The branched polypropylene samples show distinct strain hardening, something absent from the linear melt; this considerably improves the melt strength of the modified polymer. The zero shear viscosity and the elasticity measured by the relaxation spectrum also increase with the number of branches per molecule. Two models are applied to describe strain hardening of the viscosity during the course of elongation. The model parameters were found to vary systematically with the degree of branching and, therefore, their values can be used as a measure of this. Consequently, fluidity, elasticity, strain hardening, and melt strength are all related to the degree of long chain branching. © 2004 The Society of Rheology. 关DOI: 10.1122/1.1764823兴 I. INTRODUCTION Polypropylene 共PP兲 is the polymer that has captured the largest share in the plastics market. This is due to its many desirable properties compared to other thermoplastics such as polyethylene and PVC: It has a higher melting point and lower density, it shows excellent chemical resistance, has a higher tensile modulus, and is lower in cost. Commercial PP is produced by Ziegler-Natta or metallocene catalysts, resulting in highly linear chains with a relatively narrow molecular weight distribution. A consequence of this, however, is low melt strength and poor processing characteristics in extensional flow-dominated processes. In order for PP to be used in foaming, thermoforming, extrusion coating, blow molding, and similar processes, modifications are needed to enhance its strain hardening behavior in such flows. Even though a very broad molecular weight distribution 共e.g., bimodal兲 can achieve this, it is most efficiently done by the addition of long chain branches 共LCBs兲. It is expected that if the behavior of PP in elongational flows is improved, its position in the plastics market will become even more prominent. Efforts in this direction have already been made and a representative example is high melt strength polypropylene 共HMS-PP兲, a name used for the commercial long chain *This paper is dedicated to W. W. Graessley on occasion of his 70th birthday. a兲 Author to whom correspondence should be addressed, electronic mail: a.d.gotsis@tnw.tudelft.nl © 2004 by The Society of Rheology, Inc. J. Rheol. 48共4兲, 895-914 July/August 共2004兲 0148-6055/2004/48共4兲/895/20/$25.00 895 896 GOTSIS, ZEEVENHOVEN, AND TSENOGLOU branched PP. Several grades of it are already available; they are produced by modifying linear PP either by using electron beam 共EB兲 irradiation 关Rätzsch 共1999兲兴 or low decomposition temperature peroxides 关Lagendijk et al. 共2001兲兴. These modifications result in the grafting of long chain branches on the PP backbone. A. Melt strength and molecular structure The melt strength 共MS兲 of a polymer is defined as the maximum force at which a molten thread can be drawn under standard conditions before it breaks 关Ghijsels et al. 共1990兲兴. High values of MS are desired in processes where the material is stretched in its molten state, such as in film blowing, thermoforming, or foaming. The melt strength is enhanced by the presence of strain hardening in elongational viscosity 关Lagendijk et al. 共2001兲兴. Strain hardening is the acceleration of elongational viscosity growth with strain that takes place beyond some characteristic strain. It has been observed in several polymer melts. Increasing the average molecular mass of a polymer results in higher shear viscosity, as well as higher MS. The melt strength also increases when the molecular mass distribution 共MWD兲 becomes broader. Nevertheless, MS increases much more dramatically than shear viscosity upon the addition of long chain branches on the polymer backbone 关Ghijsels et al. 共1990兲; Ghijsels and De Clippeleir 共1994兲; Ghijsels et al. 共1997兲兴. The MS of LDPE, e.g., was found to be twice 关Gotsis and Ke 共1999兲兴 or five times 关De Maio and Dong 共1997兲兴 higher than the melt strength of LLDPE and HDPE for the same melt flow index 共MFI兲. This effect is attributed to the strain hardening behavior of the elongational viscosity of the melt and is manifested more strongly in ‘‘tree-type’’ rather than ‘‘combtype’’ LCB. LLDPE of comonomers with different lengths 共1-butene, 1-hexene, and 1-octene兲 do not show such difference in strain hardening behavior of the melt, indicating that these short chain branches 共SCBs兲 cannot improve the melt strength. Branches are referred to as ‘‘long chain’’ when they can have at least two to three entanglements or a length at least 2.5 times the molecular weight at the onset of entanglements, M C 关Gell et al. 共1997兲兴. Otherwise they are characterized as ‘‘short.’’ Comparison among several linear and branched polypropylenes obtained by electron beam irradiation has shown that the melt strength of branched PP can be up to 10 times higher than that of linear PP with the same MFI 关De Maio and Dong 共1997兲兴. In that report the melt strength was related to the loss tangent 共tan ␦兲, suggesting that higher elasticity leads to higher melt strength. Similar findings were reported by Gotsis et al. 共2004兲; they found the enhancement of the melt strength is related to the increase of the weight average number of long chain branches per molecule, B n . B. Molecular structure and rheology The molecular architecture of the polymer 共short and long chain branching and MWD兲 affects the rheology of the melt. A broad or a bimodal MWD, e.g., can cause strain hardening in elongational viscosity 关Cogswell 共1981兲兴: While a PMMA with narrow MWD shows only slight strain hardening, a homogeneous blend of this polymer with 1.5 wt % ultrahigh molecular weight 共UHMW兲 PMMA results in very pronounced strain hardening 关Takahashi et al. 共1999兲兴, apparently due to an increase in the number of entanglements and better network connectivity. In another report, polybutadienes with narrow MWD and different M w were blended to obtain bimodal distributions 关Berger and Meissner 共1992兲兴. While a narrow distribution came close to the linear viscoelastic limit for elongational viscosity (3 ␩ 0 ), the bimodal distributions were strain hardening. Simi- EFFECT OF LONG BRANCHES 897 larly, PS samples with a bimodal MWD exhibited clear strain hardening, whereas samples of PS with a simple broad MWD showed almost none 关Berger and Meissner 共1992兲兴. In a thorough review on the effect of LCB on the linear viscoelasticity of polyolefins recently published by Vega et al. 共2002兲, it was shown that the introduction of LCB induces higher levels of elasticity than broadening of the MWD of a melt of linear chains of similar molecular weight. On the other hand, it is current understanding that SCB cannot cause large increases in elasticity. Yet, Vega et al. 共1998兲 reported that SCB resulted in higher zero shear rate viscosities, ␩ 0 , higher relaxation times, higher values of the elastic 共storage兲 modulus, and higher activation energies of flow compared to linear polymers. Analysis, by size exclusion chromatography 共SEC兲 coupled with intrinsic viscosity measurements, revealed, however, that the SCB polymers utilized in that study also possessed some LCB. Therefore, the behavior reported by Vega et al. 共1998兲 should primarily be attributed to these long branches. Yan et al. 共1999兲 synthesized polyethylene samples of comparable molecular weight and molecular weight distribution but with varying LCB content from 0 to 0.044 branches per 1000 carbons 关determined by nuclear magnetic resonance 共NMR兲兴. These branched samples showed more pronounced shear thinning, higher zero shear viscosity, lower viscosity at high frequencies, and much longer relaxation times. Further, the loss tangent 共tan ␦兲 of the samples decreased with an increase in branching content, whereas the extrudate swell increased. Tsenoglou and Gotsis 共2001兲 showed that the increase in zero shear viscosity of samples made by modifying the same linear PP by progressive addition of increasing amounts of long chain branching could be related to B n . The activation energy of flow, E a , also seems to increase with an increase in branching content, at least at branching levels that are not very low 关Vega et al. 共1998兲兴. Strain hardening in extensional flows has been shown to be caused by long chain branches in several other polymers and in particular in LDPE 关see, e.g., Cogswell 共1981兲; Dealy and Wissbrun 共1990兲兴. The differences in elongational viscosity, ␩ E , between long chain branched LDPE, which shows strong strain hardening, and linear HDPE, which shows much less strain hardening, have been studied extensively. Bin Wadud and Baird 共2000兲 performed shear and elongational measurements on conventional LDPE and metallocene-catalyzed LLDPE with approximately the same shear viscosity. The long chain branched LDPE showed distinct strain hardening in uniaxial elongation, while the short chain branched LLDPE did not. Branched and linear metallocene-catalyzed polyethylenes were also compared in that study. The linear polymer showed no strain hardening and its extensional behavior was comparable to that of LLDPE. The branched samples did show strain hardening and possessed higher zero shear viscosity, earlier and more pronounced shear thinning, and a higher storage modulus at low frequencies than their linear analogs. Gotsis and Ke 共1999兲 indicated that short chain branches cannot enhance the melt strength because they do not cause strain hardening in elongational viscosity. The HDPE, LDPE, and LLDPE that were tested in that work had approximately the same shear viscosity over a broad range of shear rates. While LDPE showed maximum strain hardening and the highest melt strength, LLDPE showed the lowest, even lower than HDPE. However, since no molecular weight data were available for the samples, this may have also been due to differences in the molecular weight distribution. The effect of the extent of branching on the extensional viscosity has been described by Kasehagen and Macosko 共1998兲. Polybutadienes were synthesized with 10, 26, and 39 wt % branching contents from the same precursor material using different coupling agents. The extent of strain hardening, as well as of zero-shear viscosity and the degree 898 GOTSIS, ZEEVENHOVEN, AND TSENOGLOU TABLE I. Grades of polypropylene used as precursors for modifications. The first three grades have linear chains, while the fourth is a commercial branched PP. Code Commercial Name Manufacturer Mw M w /M n B F93 F96 PF Borealis HC 1000 Fortilene F9300 Fortilene F9600 Profax-814 HMS-PP Borealis BP/Amoco 共Solvay兲 BP/Amoco 共Solvay兲 Basell 422 000 333 000 326 000 629 000 6.2 5.7 7.3 9.3 of shear thinning, increased with increases in branching content. Kasehagen and Macosko 共1998兲 concluded that long chain branched molecules had broader relaxation spectra and longer relaxation times in comparison to a linear molecule. This causes more resistance to elongation. However, increases in the high end of the molecular weight distribution should have the same effect. Therefore, the presence of strain hardening alone cannot always be used for the detection of branching. Gabriel and Münstedt 共2003兲 showed that even very small amounts of long chain branching increase both the zero-shear viscosity and the strain hardening of elongational viscosity. High values of B n that are typical in LDPE reduce ␩ 0 , while they still enhance strain hardening of ␩ E . Those authors surmised that the number of entanglements per branch is a decisive parameter that influences shear and elongational behavior. Much less research has been done on the extensional properties of PP melts because appropriate samples have been difficult to obtain. Hingmann and Marczinke 共1994兲 found that ‘‘branched’’ PP samples showed distinct strain hardening and concluded that a few branches on the chain had an enormous effect on the extensional behavior of the melt. However, the presence of branches in these samples was never confirmed by SEC. Kurzbeck et al. 共1999兲 reported that a linear ethylene–propylene copolymer had six times higher zero-shear viscosity than a PP with LCB introduced by electron beam irradiation and somewhat higher weight average molecular weight, M w . The branched PP showed greater die swell and pronounced strain hardening in elongational viscosity at all strain rates, stronger even than that one shown by a typical LDPE. Strain hardening was not observed in the linear polymer. The strain hardening in the branched PP started for most strain rates at around 1.5 strain units 共s.u.兲. A high molecular weight shoulder was detected in the MWD curve of the branched PP and the authors attributed the pronounced strain hardening to a combination of branches and the fraction of this high molecular weight material. Strain hardening of branched PP samples has also been reported by Gabriel and Münstedt 共2003兲. II. EXPERIMENT The branched samples for this study were obtained by two methods. First, several linear polypropylene grades were used as precursors and they were modified using peroxydicarbonates 共PODICs兲. Long chain branched samples were the products of this modification. The precursors are listed in Table I. The commercial names, the chemical formulas, and the characteristics of the peroxydicarbonates were presented by Lagendijk et al. 共2001兲. In the present article we also report results from samples obtained by modifications using a myristyl-peroxydicarbonate 共P-26兲 and ethyl-hexylperoxydicarbonate 共EHP兲. These peroxydicarbonates are produced by Akzo Nobel. Second, blends of a commercial branched polypropylene ‘‘PF’’ 共see Table I兲 and linear B were prepared. EFFECT OF LONG BRANCHES 899 TABLE II. Molecular weight averages 共g/mol兲 and weight average branching numbers (B n , branches per molecule兲 of polymer B modified using increasing amounts of EHP PODIC 关Lagendijk et al. 共2001兲兴; ␤: strain sensitivity exponent for the damping function 关Eq. 共8兲兴; ␤⬘: parameter of the molecular stress function 共Sec. IV B兲. Mn Mw M w /M n Bn ␤ ␤⬘ 74 000 69 000 62 000 68 000 71 000 410 000 400 000 410 000 460 000 485 000 5.54 5.74 6.56 6.81 6.81 0 0.05 0.15 0.23 0.36 1.1 0.95 0.75 0.6 0.45 1.5 1 0.8 0.6 0.5 Sample PP virgin With 0.5 mmol EHP 1.0 mmol EHP 2.0 mmol EHP 3.0 mmol EHP The molecular weights and the degrees of branching of the modified samples and the blends were measured using a triple sensor, high temperature SEC at Akzo Nobel Research, Arnhem, The Netherlands. The theory of Zimm and Stockmayer 共1949兲 was used to extract the branching number from the intrinsic viscosity and its difference from that of the linear polymer. Details on the modification procedure, the SEC measurements, and the method for characterization are given in theses of Lagendijk 共1999兲 and Zeevenhoven 共2003兲. Tables II–IV show molecular structure data of the samples. The melt strength of the samples was measured using a Göttfert Rheotens device in combination with a Göttfert Rheograph 2001 capillary rheometer. The standard MFI for polypropylene is measured at 230 °C and uses a total load of 2.16 kg. For comparisons between melts with different shear viscosities we present the ‘‘relative melt strength’’ in Tables III and IV. This is the value of the melt strength, measured by the Rheotens test under standard conditions, multiplied by the MFI, the fluidity index used in engineering TABLE III. Properties of polypropylenes with initially different molecular weights modified using varying amounts 共mmol/100 g PP兲 of P-26 共dimyristyl peroxydicarbonate兲. The properties of PF are also included for comparison. The column labels are similar to those in Table II. In addition, ␩ 0 is zero-shear viscosity at 190 °C; relative melt strength ⫽ melt strength (cN)⫻melt flow index 共g/10 min at 2.16 kg兲, and T ⫽ 230 °C. Polymer code B P-26 共mmol兲 ␩0 共kPa s兲 Relative MS Bn ␤ ␤⬘ Mw ⫻1000 M w /M n Ea 共kJ/mol兲 3.5 4.1 2.5 8.2 5.6 0 0.4 0.6 0.7 0.8 1 0.3 0.2 0.1 0 1.2 0.6 0.5 0.5 0.5 422 575 574 581 571 6.2 7.8 7.6 8.6 8.9 38.4 36.8 33.3 34.1 33.0 0 1 2 3 5 9.1 20 48 56 F93 0 1 2 3 5 2.5 5.7 12.3 16.3 22 3.6 11.9 10.8 10.4 9.7 0 0.2 0.4 0.5 0.6 1.8 0.4 0.25 0.15 0.2 2 0.6 0.5 0.5 0.5 333 381 413 456 458 5.8 7.1 7.8 7.9 7.7 40.5 39.3 41.1 40.0 38.5 F96 0 1 2 3 5 4.5 6.3 7.2 8 8.5 9.8 11.2 16.8 15.5 11.5 0 0.1 0.3 0.5 0.7 0.7 0.5 0.3 0.15 0.1 1 0.6 0.5 0.5 0.5 314 363 389 407 414 7 8.1 9.3 9.2 8.6 41.3 40.4 43.9 39.8 35.7 PF ¯ 25.8 5 0.2 1.2 629 9.5 43.9 a a 21 The zero-shear rate viscosity of this sample could not be estimated with enough confidence using the Cross expression. 900 GOTSIS, ZEEVENHOVEN, AND TSENOGLOU TABLE IV. Molecular data and elastic properties of melts of B/PF blends. The column labels are similar to those in Tables II and III. ␩0 PF 共%兲 共kPa s兲 Relative MS Bn ␤ ␤⬘ Mw ⫻1000 M w /M n Ea 共kJ/mol兲 0 12.5 25 50 75 100 9.1 11 12 16 20 21 3.5 6.1 8.4 11.6 15.7 25.8 0 0.2 0.4 1.7 3.3 5 1 0.8 0.4 0.3 0.2 0.2 0.5 0.5 0.5 0.55 0.6 1.2 422 388 400 485 569 629 6 6.3 6.4 8 7.9 9.5 38.4 43.5 43.7 42.2 41.5 43.9 comparisons. Complete results of these measurements are given in work of Lagendijk 共1999兲 and Zeevenhoven 共2003兲. The samples for the rheology measurements were cut from a plate of 1.5 mm thickness that had been pressed at 230 °C for 10 min at 200 kN. The material was stored overnight in a vacuum oven at 120 °C to remove possible reaction by-products left from the modification process. The dynamic shear rheology measurements were performed in a controlled rate plate/ plate rotary rheometer 共RMS 800, Rheometric Scientific兲 at 180, 190, 210, and 230 °C. Frequencies of 0.01–100 rad/s were used at a strain amplitude of 10% in order to be within the linear viscoelastic region of the materials. The transient extensional viscosity was measured at 190 °C in a RME® rheometer from Rheometric Scientific. This controlled strain rate extensional rheometer was designed and described by Meisner and Hostletter 共1994兲. Rectangular strips with dimensions of 60⫻7⫻1.5 mm3 were used as samples. These samples were stress free at the beginning of measurement, something that is necessary to obtain accurate results at high strain. The nominal strain rate used was 0.11 s⫺1. Due to the possibility of slippage at clamping fixtures of the instrument, the actual strain rate was determined by monitoring the motion of glass beads added on top of the strips and the change in sample width by means of a video camera. The true strain rate was found to be less than the nominal (␧˙ true ⬃ 0.1 s⫺1 ) but constant throughout measurement. The strain and stress data given by the instrument were corrected for the true strain rate. III. SHEAR RHEOLOGY A. Relaxation spectrum The addition of long chain branches affects the elasticity of the melt. Its effects may be assessed by monitoring variations in dynamic moduli; this is examined here in Sec. III A. Time–temperature superpositioning was used to expand the frequency domain and the data were shifted to a temperature of 190 °C. The shift was reasonable for all samples, indicating that the modified melts remained rheologically simple. The activation energy of the flow was calculated from shift factors and is listed in Tables 3 and 4. The dynamic behavior of the melts of linear polymer B and a sample resulting from this PP in which long chain branches were added are shown in Fig. 1. Figure 1 shows that the values of G ⬘ at low frequencies increase, while the cross-over frequency shifts toward lower values, from 20 to around 1 rad/s, when the branching number of the samples increases. The loss modulus, G ⬙ , is affected less by the addition of branches. The changes in storage modulus that result from increased levels of branching are shown in Fig. 2 for F93 and its modifications. 关A complete set of curves for these and the EFFECT OF LONG BRANCHES 901 FIG. 1. Dynamic moduli of the linear and the modified B melt. Modification was done using 3 mmol P-26/100 g PP. measurements that follow for all samples listed in Tables I–IV are included in an Appendix. See the reference section for EPAPS material.兴 It can be seen there that the values of G ⬘ at low frequencies increase with the number of branches per molecule, B n , while, predictably, they are not affected at higher frequencies, where they approach a B n -independent rubber plateau. This is because the high frequency dynamic response reflects small segment molecular dynamics. Similar graphs were obtained for all other sample sets. The plot of the loss angle, ␦ (tan ␦ ⫽ G⬙/G⬘), as a function of the frequency has been used as a rheological probe for the presence of LCB 关Wood-Adams et al. 共2000兲兴. This plot is shown in Fig. 3 for the melts of F93 and its modifications. The linear polymer melt demonstrates monotonic decrease in loss angle. The branched samples, however, show inflection in the curve of the loss modulus, which hints to the development of a plateau at frequencies around 1 rad/s. According to Wood-Adams et al. 共2000兲 the magnitude and breadth of this plateau should depend on B n . This is also implied in Fig. 3, where the inflection is stronger at higher levels of modification, i.e., at higher LCB content. This FIG. 2. Storage modulus at 190 °C of F93 modified with different amounts of P-26. 902 GOTSIS, ZEEVENHOVEN, AND TSENOGLOU FIG. 3. Loss angle vs the frequency at 190 °C for various degrees of LCB. Variation of parameters according to the number of branches per molecule, B n , for a series of P-26 modified melts of F93. plot, however, only indicates that there is an increase of the elasticity of the melt and no information can be extracted as to the type of branching or even of the origin of the enhanced elasticity. Quantitative conclusions from these graphs, therefore, cannot be drawn. The relaxation spectra of the melts were evaluated from results of the dynamic moduli using the software supplied with the rheometer. The relaxation spectra of the linear polymers and the samples with increasing values of B n produced by modifying the linear polymer are shown in Fig. 4. Following the trends of the dynamic moduli, H( ␶ ) increases with B n at higher relaxation times, whereas there is little change at short relaxation times. That is, the spectra become broader as B n increases. This is another indication that the elasticity of the melt increases with the degree of branching. The activation energy of flow has been used to estimate the degree of branching of polyethylene 关see, e.g., Vega et al. 共1998兲兴. The activation energies of the melts that were used in the present work were evaluated from the shift factors of G ⬘ and G ⬙ and are listed in Tables III and IV. The values of E a for all linear and branched PP samples are FIG. 4. Relaxation spectra of linear and branched PP-F93 melts and their dependence on branching number, Bn . EFFECT OF LONG BRANCHES 903 FIG. 5. Dynamic shear viscosity vs the frequency at 190 °C for the melts of B and its modification with P-26. around 40 kJ/mol and there is no definitive trend of reduction or increase of this value with variation of B n . It appears that the degree of branching in the present samples is too small to have a perceptible influence on the activation energy of flow. The value of E a of the highly branched commercial PF is somewhat higher than that of the three linear polymers. The blends of PF and B all have an activation energy for flow with a value around that of pure PF. B. Shear viscosity The molecular weight and MWD of the polymers did not change dramatically with the modification. Tsenoglou and Gotsis 共2001兲 reported that the zero-shear viscosity of samples produced in a similar way was higher at higher values of the degree of branching, while the viscosity in the shear thinning region changed very little at these low values of branching. The same was found also in the present set of experiments 共e.g., in Fig. 5兲. The shear viscosity of the melts was estimated here from the complex viscosity using the Cox–Merx rule. The data were fitted by a Cross viscosity function in order to estimate the zero-shear viscosity, ␩ 0 , by extrapolation, ␩⫽ ␩0 1⫹共␭␥˙ 兲n . 共1兲 Since the molecular weight of the modified polymers did not change much, timemolecular weight superposition, as suggested by Doerpinghaus and Baird 共2003兲, was not necessary for the present comparisons. Janzen and Colby 共1999兲 showed that the zero shear rate viscosity depends on the degree of branching and found a maximum in that dependence. For sparsely branched polymers with high molecular weight, as in the present case of the modified samples, ␩ 0 increased with B n . Such an increase has also been reported in several previous investigations, especially for sparsely branched PE made by metallocene catalysis 关Gabriel and Münstedt 共2002兲; Graessley and Roovers 共1979兲; Wood-Adams et al. 共2000兲; Doerpinghaus and Baird 共2003兲兴. On the other hand, at high values of B n , the zero-shear viscosity decreased with B n 关Janzen and Colby 共1999兲兴. The highly branched PP sample PF in the present work may be such an example: Even though its average molecular mass is higher than the other branched and linear samples, it has a much lower value for ␩ 0 共Table III兲. 904 GOTSIS, ZEEVENHOVEN, AND TSENOGLOU Similar decreases in the value of ␩ 0 have also been reported for LDPE 关Doerpinghaus and Baird 共2003兲; Gabriel and Münstedt 共2003兲兴. The value of the zero-shear viscosity has been used in the literature to infer the level of long chain branching from rheological measurements. Development of methods that characterize the degree of branching in polymers rheologically is justified by the fact that rheological measurements are much easier to conduct and more accurate than the traditional methods 关gel permeation chromatography 共GPC兲 or NMR兴 used to measure B n . Several indices have been proposed based on ␩ 0 , its deviation from the ␩ 0 value of the linear polymer, and its dependence on the molecular weight. Two such examples are the Dow rheology index 共DRI兲 关Lai et al. 共1994兲兴 and the long chain branching index 共LCBI兲 关Shroff and Mavridis 共1999兲兴. Since they were both developed specifically for polyethylene, adopting these indices for PP necessitates the reevaluation of all associated parameters by fitting them to appropriate PP data. The dependence of the zero-shear viscosity on the degree of branching for low branching numbers of PP has been studied by Tsenoglou and Gotsis 共2001兲. Gradual modification of a linear polymer generated more and more three-arm branched molecules, where each arm had roughly equal length. The viscosity of these samples was related to the fraction of stars in the polymer. An expression was then derived that gives the degree of branching, B n , in terms of the weight average molecular weight of the linear precursor, M L , its zero-shear viscosity ␩ L , and the zero-shear viscosity, ␩ B , of the branched polymer that resulted from modification of the linear precursor, Bn ⫽ ln ␩r 冋冉 冊 册 ␣ ML MC ⫺1 ⫺3 ln ML . 共2兲 MC Here ␩ r ⫽ ␩ B / ␩ L is the reduced zero-shear viscosity of the branched sample and ␣ is an adjustable parameter, which is also included in the original molecular theory of tethered arm relaxation 关de Gennes 共1979兲; Doi and Edwards 共1988兲兴. The molecular weight at the onset of entanglements, M C , has a value of 11.2 kg/mol for PP 关Pearson and Helfland 共1984兲兴. The predictions of Eq. 共2兲 are shown in Fig. 6 for branched polymers that resulted from modifying the linear PP precursors. The exact value of coefficient ␣ varies somewhat from precursor to precursor, implying that ␣ is not exactly a universal constant but depends on the average molecular weight and MWD of the parent linear polymer. It lies, however, close to the 0.42 value found to apply for similar PP melts by Tsenoglou and Gotsis 共2001兲 and is consistent with values extracted from earlier experiments by Roovers 共1985兲, Ball and McLeish 共1989兲, Milner and McLeish 共1997兲, and Struglinski et al. 共1988兲. Nevertheless, according to Janzen and Colby 共1999兲 a simple relation between ␩ r and B n should not be expected to hold in general because of the large confounding effect of the length of entangling edges. Apparently the theory of Tsenoglou and Gotsis 共2001兲 is applicable to melts that have been produced using the same precursor and a specific method of modification that alters the molecular architecture without causing dramatic shifts in the average molecular weight. 关In that respect, M L in Eq. 共2兲 for evaluation of B n may in general be replaced by the weight average molecular weight of all chains in the melt of mixed architecture, M w , and not just that of the linear precursor. Similarly, ␩ L may signify the zero-shear viscosity of a linear chain of molecular mass M w .] EFFECT OF LONG BRANCHES 905 FIG. 6. Relation of the degree of branching, B n , to the reduced zero-shear rate viscosity, ␩ r , for the melts of the three modified precursors. The lines show the predictions of the theory of Tsenoglou and Gotsis 共2001兲 for specified values of parameter ␣. By far, the most spectacular changes in the rheology of the melt upon addition of long chain branches are found in their behavior in elongational flow. This is described in Sec. IV. IV. EXTENSIONAL RHEOLOGY The extensional viscosity of the tested PP melts initially shows simple monotonic growth as a function of strain with a decrease in first derivative that compares well with the theory of linear viscoelasticity. Beyond deformation of around 1 s.u., however, some of the curves show an increase in their slope as the growth of viscosity accelerates; this is the region of strain hardening. All the modified samples broke more or less by elastic fracture, whereas the linear samples did not break up to 7 s.u., where the measurements ended. However, the force measured at those high strains was very low, that is, below the sensitivity of the rheometer, and elongational viscosities could not be calculated at strain higher than 6. Points with unreliable force values are not included in any of the figures. The elongational viscosity of the linear PP grades were essentially nonstrain hardening. Long chain branches added on these polymers during peroxide modification, however, increased the temporary network connectivity in the melt and reduced the rate of disentanglement. Thus, the new polymers showed enhanced strain hardening of their viscosities in uniaxial elongational flow. In general, the more branches added on the chains and the longer these branches, the steeper the stress growth at strain above 1 s.u. The viscosity growth curve of the highly branched PF melt was also strongly strain hardening. This melt starts with a relatively lower viscosity at low strain, something which is also consistent with its lower shear viscosity. The elongational viscosity growth curves are usually given as log–log plots. However, when a semilog plot is used, these curves may become straight lines at high strain. They can be approximated, then, by exponential functions of the strain: ␩ E⫹ ⬇ C 1 exp兵k1␧其. The exponential form of the function ␩ E⫹ (␧) suggests the use of a specific formalism to describe the nonlinear viscoelastic response during elongational viscosity growth; this is outlined in Sec. IV A. 906 GOTSIS, ZEEVENHOVEN, AND TSENOGLOU A. Uniaxial elongation and nonlinear viscoelasticity The growth of stress during uniaxial elongation of polymeric melts can be described by several existing viscoelastic models. One of the most successful for this appears to be the rubber-like liquid model 关Lodge 共1964兲兴; it is here modified to account for nonlinear, strain thinning effects by incorporation of a damping function into the viscoelastic memory, which assumes separability of time and strain, ␴⫽ 冕 t ␮共t⫺t⬘兲h共␭兲F共 t⫺t ⬘ 兲 dt ⬘ . 共3兲 ⫺⬁ In this equation F is the Finger deformation tensor, h(␭) is the damping function, ␭ is the stretch as a function of time present and past 关 ␭(t,t ⬘ ) 兴 , and ␮ is the memory function. The Finger tensor the case of uniaxial elongation 共constant strain rate, ␧˙ ) is given by 关Bird et al. 共1977兲兴 F共 t⫺t ⬘ 兲 ⫽ 冋 exp关2␧˙ 共t⫺t⬘兲兴 0 0 0 exp关⫺␧˙ 共t⫺t⬘兲兴 0 册 . 共4兲 0 0 exp关⫺␧˙ 共t⫺t⬘兲兴 The memory function, ␮, is a material function of time and depends on the relaxation spectrum of the melt. A form of ␮ that is widely used comes from the multimode Maxwell model, ⬁ ␮共t⫺t⬘兲 ⫽ gi 兺 exp i ⫽ 1 ␶i 再 ⫺共t⫺t⬘兲 ␶i 冎 , 共5兲 where ␶ i and g i are the sets of relaxation times and moduli evaluated by fitting the dynamic shear moduli of the melts at the same temperature. The damping function, h(␭), physically signifies the extent of stress loss due to reduction of the entanglement density and segment orientation following deformation of a given magnitude, ␭. Several forms for h have been proposed and tried with variable degrees of success 关Wagner 共1976兲; Larson 共1988兲兴. We introduce here the simplest possible form that accommodates the essential phenomenology, i.e., controlled exponential growth of ␩ E⫹ and LCB effects of variable intensity, h共␭兲 ⫽ ␭⫺␤. 共6兲 Here, ␤ is an adjustable parameter that should depend on the branching number: we intuitively expect that increasing B n contributes to better network connectivity, improved resistance to strain-induced network destruction and, therefore, to less stress damping and smaller ␤ values. The viscosity in uniaxial elongational flow is defined as follows: ␩E ⫽ ␴11⫺ ␴ 22 ␧˙ . 共7兲 Stretch ␭ is related to Hencky strain, ␧, and the constant strain rate, ␧˙ , by an exponential function, ␭ ⫽ exp兵␧其 ⫽ exp兵␧˙ t其. The damping function, then, in terms of ␧˙ is h共␭兲 ⫽ exp关⫺␤␧˙ 共t⫺t⬘兲兴. 共8兲 With this form for the damping function, Eq. 共3兲 for viscosity growth in uniaxial elongation becomes EFFECT OF LONG BRANCHES 907 FIG. 7. Elongational viscosity growth of PP at 190 °C for the samples resulting from modification of F93 with P-26 and for B/PF blends. The fit of Eq. 共9兲 is also shown. The extension rate is around 0.1 s⫺1. ␩E⫹ ⫽ 冕 ⑀ 1 t ˙ ⫺⬁ ␮共t⫺t⬘兲exp关⫺␤␧˙ 共t⫺t⬘兲兴兵exp关2␧˙ 共t⫺t⬘兲兴⫺exp关⫺␧˙ 共t⫺t⬘兲兴其dt⬘. 共9兲 When ␤ ⫽ 0 there is no damping and one recovers the original rubber-like liquid model of Lodge. Strain hardening is then predicted for strain rates higher than the inverse of the longest relaxation time used in ␮. The slope of the viscosity growth curve versus strain at high strains is 2 in a semilog plot, similar to that characterizing stretching of a neo-Hookean spring. There is no steady state for elongational viscosity, except for ␤ ⫽ 2, in which case there is complete damping and no strain hardening. Then, the elongational viscosity does not accelerate; it merely increases monotonically and reaches a steady state value asymptotically. The fit of this model for the branched polypropylene melts is shown in Fig. 7. The relaxation data used in ␮共␶兲 were evaluated from the dynamic moduli measurements of each melt. It can be seen in Fig. 7 that the model of Lodge with our proposed damping function, Eqs. 共8兲 and 共9兲, does a relatively good job in describing growth of the elongational viscosity of the linear and branched samples, thus proving it is adequate to model strain hardening of these polymer melts. The same is true for all samples tested. When one compares the samples produced from a single precursor, e.g., B, F93, or F96, then it can be seen that the strain sensitivity exponent ␤ of the damping function de- 908 GOTSIS, ZEEVENHOVEN, AND TSENOGLOU creases when B n increases. There is, therefore, a relation between the branching number and this parameter. When B n is zero, ␤ takes a value of around 1 for melts B and F96 and a value of 2 for F93. At the highest values of B n , ␤ comes close to zero. Figure 7 also shows the elongational viscosity growth and corresponding fits of Eq. 共9兲 for melts of the blends of B and PF. Even though the molecular weight, its distribution, and the degree of branching change among samples in this case, there is still a monotonic increase in the degree of strain hardening according to the number of branches present; this translates into a corresponding decrease of ␤ from a value of 1 for purely linear B to a value of 0.2 for blends containing 75% or higher PF. B. Molecular stress function theory Based on modification of the reptation model 关Doi and Edwards 共1988兲兴, Wagner et al. 共2000兲 have proposed the following model for elongational stress growth of entangled polymer melts: ␴ ⫽ ⫺pI⫹ 冕␮ 共 t⫺t ⬘ 兲 f 2 S共 t⫺t ⬘ 兲 dt ⬘ , 共10兲 where S is the second-order orientation tensor that describes the average orientation of primitive paths in the reptation model. The molecular stress function, f, is essentially a damping function that gives the contribution of relative stretching of the macromolecular chain. It accounts for the change in tube diameter, a, when the material is stretched: f ⫽ a 0 /a, where a 0 is the initial tube diameter. The molecular stress function has been derived using generalization of the strain energy function of the Doi–Edwards model 关Wagner and Schaeffer 共1994兲; Wagner et al. 共2000兲兴. Two cases were distinguished for the form of f. Assuming affine deformation and constant volume for the average tube it was shown that a reasonable form of f for linear chains is the linear molecular stress function 共LMSF兲 f 2 ⫽ exp具ln u⬘典0 , 共11兲 where 具 ln u⬘典0 is the average logarithmic stretch 共expressed as the length of deformed unit vector of the primitive chain, u ⬘ , averaged over an isotropic distribution function兲. For branched chains, f appears to be of higher order quadratic molecular stress function 共QMSF兲: f 2 ⫽ 21 exp 2具ln u⬘典0⫹ 21. 共12兲 In a more recent article 关Wagner et al. 共2001兲兴 f was generalized to the following expression: f 2 ⫽ ␤⬘ exp 1 ␤⬘ 具ln u⬘典0⫹共1⫺␤⬘兲, 共13兲 where the parameter ␤⬘ represents the effect of the branching topology on the orientational free energy and should take a value in the range of 0 ⭐ ␤⬘ ⭐ 1. For ␤⬘ ⫽ 1, one recovers the LMSF model, while for ␤⬘ ⫽ 1/2 the QMSF model is obtained. The model in this form predicts an unbounded increase of elongational viscosity with the strain. By specifying a nonlinear material parameter, f max , which gives the limit for allowable stretching of the tube, the viscosity growth stops at high strain and the elongational viscosity approaches a steady state 关Wagner et al. 共2000兲兴, something that has been observed experimentally in some strain hardening melts. EFFECT OF LONG BRANCHES 909 FIG. 8. Fit of the elongational viscosity growth of PP for the samples resulting from modification of F93 with P-26 and for B/PF blends with the molecular stress function model of Wagner et al. 共2001兲 关Eqs. 共10兲 and 共13兲兴 and for variation of parameter ␤⬘. The extension rate is around 0.1 s⫺1 and the temperature is 190 °C. It has been found that linear polymer melts in the highly entangled regime follow the LMSF model quite accurately, at least in stress growth experiments in uniaxial, biaxial, and equibiaxial elongational flows 关Wagner et al. 共2000兲兴. As soon as the chains have a few long branches, they follow the QMSF model. Both models predict strain hardening in uniaxial elongation but the viscosity in the QMSF model increases much faster. In our samples the number of branches is rather low: B n ⬍ 1 means that only a fraction of the total number of chains has long chain branches. The melt in this case is a mixture of linear 共the majority兲 and branched chains. By expanding on the above arguments, then, it seems reasonable to relax the requirement that ␤⬘ take values of either 1 or 1/2; this parameter may also be let to assume intermediate values that depend on the fraction of branched chains in the mixture. Furthermore, one might extrapolate this model by applying it in melts of lesser network connectivity, such as linear polypropylenes with narrow molecular weight distribution, which show less or no strain hardening. When ␤⬘ assumes values higher than 1, Eq. 共13兲 predicts no strain hardening up to a high degree of deformation. Implementation of this approach is illustrated in Fig. 8. Since steady state could not be achieved in our measurements for any of the strain hardening melts, a finite value for 910 GOTSIS, ZEEVENHOVEN, AND TSENOGLOU f max was not used. The predictions for viscosity growth are shown in Fig. 8 only in the range of strain where we could obtain accurate measurements: 0– 6 Hencky strain units. It can be clearly seen in Fig. 8 that one needs a continuous spectrum of ␤⬘ values to fit the elongational viscosity growth of melts with different branching numbers. The linear PP melt does not show any strain hardening in the range of strain that could be achieved, and the value of ␤⬘ is higher than 1. As B n increases, however, the value of ␤⬘ decreases abruptly. For all samples with branching numbers larger than 0.5 the appropriate value of ␤⬘ is 1/2. The picture is similar for the blends of B with PF. Even here, where one is certain that these samples are mixtures of linear and branched chains, the change in the value of ␤⬘ is very abrupt, from a little over 1 for purely linear B to ␤⬘ ⫽ 1/2 for the blends with just over 25% PF. However, since these blends have branching numbers higher than 0.4, they already correspond to samples with the highest degree of branching among the series made by modification of the linear PP. Furthermore, with B n ⬇ 5, the PF sample has too many branches per molecule for the above arguments to be entirely valid. V. DISCUSSION Both models tried in the present work fitted the experimental data of the linear and branched polypropylenes rather well. Only the highest branched samples produced from the PODIC-induced modification seem to deviate from these models. A possible explanation for this is the presence of some crosslinked 共gelled兲 material in these samples 关Gotsis et al. 共2004兲兴. In a recent attempt to relate the parameters of a such model to the degree of branching, Doerpinghaus and Baird 共2002兲 have demonstrated the capability of the multimode pompom model 关McLeish and Larson 共1998兲兴 to describe strain hardening of the elongational viscosity of sparsely branched polyethylenes (B n ⬍ 2). Besides changes in the relaxation spectrum, the most important parameter of the model that needs adjustment in order to properly fit elongational viscosity growth is the number of pompom arms for the mode of the longest relaxation time. This value increases when B n increases, while it retains a value of 1 for linear polymers that show no strain hardening. A similar approach is made here to relate the value of B n with the values of model parameters ␤ and ␤⬘. Figure 9 shows the monotonic decrease of ␤ with an increase of B n for the model in Eq. 共9兲. For low values of B n ( ⭐ 0.8) one is therefore tempted to use the value of ␤ as a measure of the degree of branching. Most data in Fig. 9共a兲 seem to fall around the same curve; one, though, has to keep in mind that all the corresponding samples were produced identically and, therefore, have similar molecular architecture. The data for the B/PF blends in Fig. 9共b兲 fall on a different curve. It appears, therefore, that the relationship between B n and ␤ 关similar to parameter ␣ in Eq. 共2兲兴 is not universal but depends also on M w , MWD, and details of branching. Furthermore, if one could extrapolate the data in Fig. 9, then all melts that have more than 0.8 long chain branches per molecule would be able to fit the original Lodge model without the need of a damping function 共␤ ⫽ 0兲 and would behave as neo-Hookean solids even at large deformation. Parameter ␤⬘ of the molecular strain function model, on the other hand, decreases very rapidly with an increase of B n and obtains the limiting value of 0.5 at around 1/2 branch per molecule on average. This corroborates the claim of Wagner et al. 共2000兲 that, essentially, all entangled melts behave according to either the LMSF model 共for no or very little branching兲 or to the QMSF. For linear PP melts that show no strain hardening, however, the value of ␤⬘ of this model should be greater than 1. EFFECT OF LONG BRANCHES 911 FIG. 9. Branching number, B n , as a function of strain sensitivity parameter ␤ of the damping function 关Eq. 共8兲兴. Data in 共a兲 for 共modified兲 B, F93, and F96 samples and in 共b兲 for B/PF blends. When the values of the relative melt strength in Table III are compared with the values of strain sensitivity parameter ␤ of each sample, then it can be seen that, indeed, strain hardening of the elongational viscosity enhances the melt strength. In order to be able to use PP efficiently in thermoforming or foaming processes, where high melt strength is required, strain hardening melts are needed. The relation between ␤ and B n could, in principle, give an indication of how many branches would be needed for a certain degree of strain hardening and a corresponding enhancement of the melt strength. The fast drop of ␤ with the degree of branching means that one does not always need a very high degree of modification in order to achieve optimum melt strength behavior. Table III shows that the optimum value for the relative melt strength may be achieved at rather low values of branching, where ␤ has already reached a value close to 0. Even higher branching levels, on the other hand, could decrease the maximum extensibility of the melt during stretching. The result could be that the thermoforming processing properties of the melt and its capability to produce high quality foam worsen at very high degrees of branching, as has also been observed experimentally 关Gotsis et al. 共2004兲兴. 912 GOTSIS, ZEEVENHOVEN, AND TSENOGLOU VI. CONCLUSIONS The modification of polypropylene with specific peroxydicarbonates leads to long chain branching and improved melt strength. While most linear precursors show a monotonic increase in elongational viscosity until a steady state is reached, all branched samples show strain hardening and relatively higher melt strength. The shear viscosity at high shear rates is not affected much by the modifications. The zero-shear viscosity, however, increases considerably with the addition of a few branches, and the increase may be related to the degree of branching. The relaxation spectrum of the melts shifts to higher relaxation times, indicating that the elasticity increases. The activation energy for flow, on the other hand, does not change perceptibly in the presence of low levels of branching. The strain hardening in elongational viscosity can be described by a Lodge rubber-like liquid model modified by incorporation of a damping function of the form h(␭) ⫽ ␭ ⫺ ␤ . The exponent, ␤, of the strain dependence of the damping function decreases with the average number of branches per molecule. The strain sensitivity parameter ␤ can, thus, be used as a relative indicator of the degree of branching at low branching levels. Elongational viscosity growth of the samples tested can also be described well by the molecular stress function model of Wagner et al. 共2000兲. It appears that the two forms of this model, LMSF and QMSF, are adequate to describe the highly entangled linear chain melts and the branched chain melts, respectively, while the transition between the two takes place at very low degrees of branching and is very abrupt. ACKNOWLEDGMENTS The authors wish to thank A. Buijtenhuijs for performing most of the SEC measurements at Arnhem; B. Norder and G. de Vos for many of the rheological measurements; Professor M. Wagner for introducing the authors to the molecular stress function model; and André Hogt of Akzo-Nobel Polymer Chemicals Research for his ideas on the PODICs and for financing part of this work. References See EPAPS Document No. E-JORHD2-48-010404 for figures. 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