Rheological properties of branched polystyrenes: nonlinear shear and extensional behavior
Abstract
Nonlinear shear and uniaxial extensional measurements on a series of graft-polystyrenes with varying average numbers and molar masses of grafted side chains are presented. Step-strain measurements are performed to evaluate the damping functions of the melts in shear. The damping functions show a decreasing dependence on strain with an increase in mass fraction of grafted side chains. Extensional viscosities of the melts of graft-polystyrenes exhibit a growing strain hardening with increasing average number of grafted side chains as long as the side branches have a sufficient molar mass to be entangled. Graft-polystyrenes with side arms below the critical molar mass M_{c} for entanglements of linear polystyrene but above the entanglement molar mass M_{e} of linear polystyrene (M_{e} < M_{w,br} < M_{c}) still show a distinct strain hardening. With decreasing molar mass of the grafted side chains (M_{w,br} < M_{e}) the nonlinear-viscoelastic properties of the graft-polystyrene melts approach the behavior for a linear polystyrene with comparable polydispersity.
Keywords
Graft-polystyrenes Long-chain branching Shear and elongational viscosity Damping function in shear and elongation Elongational viscosity Strain hardening EntanglementsIntroduction
Many nonlinear rheological properties of polymer melts are influenced by long-chain branches. For example, an increased shear thinning behavior compared to the properties of the corresponding linear polymers is found for long-chain branched polyethylenes, polystyrenes and polycarbonates (e.g., Heino et al. 1992; Gabriel and Münstedt 1999; Ferri and Lomellini 1999; Grigo et al. 1996).
Another important nonlinear feature is the strain-hardening behavior in uniaxial extensional flow. Some chemically branched or electron beam irradiated polypropylenes show an increased strain hardening when compared to their linear counterparts (Hingmann and Marczinke 1994; Kurzbeck et al. 1999; Gotsis et al. 2004). For polyethylene, it is well known that long-chain branched LDPE exhibits a pronounced strain hardening in experiments with constant strain rate. Experiments at constant stresses have shown that the maximum of the steady-state elongational viscosity increases with the degree of branching (Münstedt and Laun 1981). However, the authors pointed out that there is a lack of reliable characterization methods for these long-chain branched polymers. Moreover, the molar mass distribution of the higher branched LDPE studied by Münstedt and Laun (1981) is broader compared to the one with a lower branching content, which makes an unambigous interpretation of the influence of branching on strain hardening more difficult.
The strain-hardening due to long-chain branching has important implications for processing operations, and particularly for those in which free surface flow occurs, e.g., in film blowing, blow molding, thermoforming or foaming. As an example, a better homogeneity of the thickness of blown films from low-density polyethylene (LDPE) compared to that one for a nonstrain hardening linear-low density polyethylene (LLDPE) has been found (Kurzbeck 1999).
To understand the influence of long-chain branching on rheological properties, melts or solutions of model polymers like stars and H-type structures have been studied mainly in linear viscoelastic flow. Only a very few experimental studies of nonlinear-viscoelastic rheological properties on well-defined star- or H-type polymers are reported (Pearson et al. 1983; McLeish et al. 1999). Detailed studies on nonlinear-viscoelastic properties of well-defined graft-polymers in shear and in elongational flow are not available from literature. Therefore, from the rheological experiments on grafted polystyrenes in the nonlinear regime reported in this paper, new insights into the role of long-chain branches can be expected.
The linear-viscoelastic properties of the materials studied here are discussed in another paper (Hepperle et al. 2005). To study the influence of molecular topology of graft-polystyrenes on nonlinear-viscoelastic properties, two experimental methods were chosen:
- (1)
Stress relaxation after shear steps to obtain the time and strain-dependent modulus G(t,γ) and the damping function h(γ).
- (2)
Deformation at a constant Hencky-strain rate \(\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon}_{0} \) in uniaxial elongation to determine the tensile stress \(\sigma ^{+}_{\rm E} (t,\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon})\) and the tensile growth coefficient \(\eta _{\rm E}^{+} (t,\;\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon}) =\sigma ^{+}_{\rm E} (t,\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon})/\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon}.\)
For these two types of flow, the influence of the molecular topology on the material functions is studied. From the tensile stress growth coefficients in uniaxial extension, the damping functions h_{u}(ε) are calculated according to Wagner (1979). The dependence of h(γ) and h_{u}(ε) on molecular structure is compared.
Experimental
Samples
Shear rheometry
Data on sample composition and experimental conditions chosen. Sample constitution is described as \({\text {PS}}{\text {-}}x{\text {-}}\ifmmode\expandafter\bar\else\expandafter\=\fi{p}G{\text{-}}y\) with x number average molecular mass M_{n,bb} (kg/mol) of the backbone, \(\ifmmode\expandafter\bar\else\expandafter\=\fi{p}:\) average number of grafted side chains per molecule, y number average molar mass M_{n,br} (kg/mol) of the side chains. w_{MM} denotes the mass fraction of low-molecular weight residue, Φ _{n,br} denotes the number average mass fraction of grafted side chains
Sample | w_{MM} | Φ _{n,br} | Step- Strain | Damping function parameter | Uniaxial elongation | ||||
---|---|---|---|---|---|---|---|---|---|
T (^{°}C) | η_{0} (T)^{a} (Pa s) | T (^{°}C) | η_{0} (T)^{b} (Pa s) | ||||||
PSM | Soskey–Winter | ||||||||
α | a | b | |||||||
PS-60-1.4G-42 | 0.17 | 0.49 | 190 | 1.94×10^{3} | 16.6 | 12.3 | 1.75 | 160.0 | 3.63×10^{4} |
PS-60-0.5G-55 | 0.07 | 0.30 | 190 | 3.61×10^{3} | 10.7 | 9.9 | 1.88 | 158.0 | 1.02×10^{5} |
PS-60-0.5G-55* | 0.23 | 0.30 | 180 | 5.14×10^{3} | 10.9 | 9.1 | 1.84 | 154.0 | 1.02×10^{5} |
PS-70-3.2G-22 | 0.24 | 0.50 | 180 | 2.13×10^{3} | 17.4 | 10.5 | 1.55 | 148.5 | 8.46×10^{4} |
PS-60-2.1G-27 | 0.01 | 0.49 | 190 | 1.33×10^{3} | 19.7 | 14.0 | 1.78 | 151.0 | 7.69×10^{4} |
PS-55-2.1G-27* | 0.12 | 0.51 | 180 | 1.77×10^{3} | 18.8 | 14.4 | 1.74 | 147.5 | 8.07×10^{4} |
PS-90-1.2G-27 | 0.05 | 0.27 | 220 | 0.72×10^{3} | 10.1 | 6.0 | 1.54 | 168.0 | 3.94×10^{4} |
PS-90-1.2G-27* | 0.15 | 0.27 | 200 | 1.47×10^{3} | 9.4 | 6.2 | 1.63 | ||
180 | 7.08×10^{3} | ||||||||
PS-80-0.6G-22 | 0.06 | 0.14 | 190 | 2.64×10^{3} | 8.7 | 5.9 | 1.64 | 159.0 | 6.06×10^{4} |
PS-55-4.0G-13 | 0.04 | 0.48 | 170 | 4.55×10^{3} | 17.0 | 13.0 | 1.78 | 150.0 | 5.27×10^{4} |
PS-105-6.7G-6 | 0.01 | 0.28 | 190 | 8.39×10^{3} | 7.5 | 5.9 | 1.77 | 170.0 | 5.22×10^{4} |
200 | 4.03×10^{3} | ||||||||
210 | 2.12×10^{3} | ||||||||
PS-r-95 | – | – | 200 | 2.8×10^{3} | 7.4 | 6.1 | 1.82 | 169.0 | 4.01×10^{4} |
180 | 13.4×10^{3} |
Extensional rheometry
The samples were prepared by pressing the powder under vacuum in a cylinder-shaped mold followed by an extrusion at T=170^{°}C through a capillary using a melt flow indexer. The extruded rod was annealed in an oil bath at a temperature adjusted to the measuring temperature of the extensional tests (see Table 1). The rod was then cut into cylinder-shaped samples with lengths of 20–25 mm and diameters of 4–5 mm. The samples were fixed with a glue to aluminum clamps.
The elongational experiments were carried out using the commercial elongational rheometer RER 9000 (Rheometrics). Its principal construction follows the design published by Münstedt (1979). The sample is stretched vertically in a silicone oil bath. The density of the oil bath matches the density of the sample at the measuring temperature as closely as possible. Measuring temperatures from 147.5 to 170^{°}C were used. The tensile force is measured by a force transducer which is submerged within the oil bath. With a maximum displacement of the pull rod of the elongational rheometer of 500 mm and a typical initial sample length L_{0} between 20 and 25 mm, stretching ratios of λ=20–25 can be achieved which correspond to Hencky-strains of ε=3–3.2 according to \(\varepsilon = \ln \lambda = \ln \frac{L}{{L_{0}}}.\)L is the sample length after elongation.
Results and discussion
Step-strain in shear
Step-strain experiments are widely used for the investigation of nonlinear-viscoelastic properties of polymer melts. From measurements of such kind new insights into effects of different branching topologies of the graft-polystyrenes on the nonlinear shear behavior are expected.
The time-dependent function G(t) is the linear relaxation modulus, h(γ) is called the damping function.
Time-strain separability
Influence of temperature and low-molar mass residues
As can be seen from Fig. 3a and b on two different samples, the damping function is independent of temperature for the range of temperatures shown. It was also proven that the low-molecular weight residues do not alter the strain dependence (Fig. 3c): The damping functions of PS-60-0.5G-55 and PS-60-0.5G-55* are indistinguishable, as well as the ones of PS-60-2.1G-27 and PS-55-2.1G-27*. Values for the mass fraction of the low-molar mass residues w_{MM} of the samples are given in Table 1.
Influence of branching on the damping function in shear
Figure 4 shows the experimentally determined damping functions for all graft-polystyrene melts investigated, ordered in groups with longer side chains (Fig. 4a), shorter side chains (Fig. 4b) and shortest side chains (Fig. 4c).
As can be seen from the dotted lines in Fig. 4 representing Eq. 3, the experimental data are slightly overpredicted at low strains and underpredicted at high strains. The full lines in Fig. 4 show the numerical descriptions obtained with Eq. 4. From the comparison it can be concluded that Eq. 4 is more appropriate for a numerical description than Eq. 3. Data for α, a and b are given in Table 1.
For the damping function according to the theory of Doi and Edwards (1986), several simple approximations have been proposed (Larson 1985; Takahashi et al. 1993), of which the one by Larson is applied in this work. It corresponds to Eq. 3 with α=5. This approximation for the Doi–Edwards damping function is marked by DE in the figures.
The fits for PS-r-95 are given in Fig. 4a. The damping function of the linear polystyrene PS-r-95 is always included in the diagrams (open squares). The value of α=7.4 for PS-r-95 is close to the one from previous studies on melts of linear PS, for which values of 5.5 (M_{w}=398,000 g/mol, M_{w}/M_{n}=2.9) and 7.9 (M_{w}=220,000 g/mol, M_{w}/M_{n}=2.44) are given by Khan et al. (1987).
Figure 4a shows two graft-polystyrenes with the same number average molar masses M_{n,bb} of the backbone (60 kg/mol) and with grafted side chains of the molar masses 42 and 55 kg/mol, which lie above the critical molar mass M_{c}≅35 kg/mol for entanglements. The two graft-polystyrenes in Fig. 4a have average numbers of 1.4 and 0.5 grafted chains per molecule. PS-60-1.4G-42 shows a damping function which is less dependent on strain than that one of PS-60-0.5G-55 with a lower average number of grafted side chains. These two branched polystyrenes exhibit a weaker damping than the linear polystyrene, PS-r-95, which has a similar polydispersity as the graft-polystyrenes. It can therefore be supposed that at nearly constant backbone and side chain molar masses, an increase in the average number of grafted side chains significantly reduces the dependence of the damping function h(γ) on strain γ.
Graft-polystyrenes with side chains of similar molar masses M_{w,br} around 25 kg/mol, but varying average numbers of side chains per molecule are shown in Fig. 4b. A weaker damping with increasing average number of grafted side chains is found again. But comparing PS-80-0.6G-22 and PS-90-1.2G-27 with the damping function of PS-r-95, the influence of grafting is found to be not as strong as in the case of PS-60-1.4G-42 and PS-60-0.5G-55.
Two graft-polystyrenes with mass average molar masses of side chains well below the critical molar mass for entanglements M_{c} of linear chains and even well below the entanglement molar mass M_{e} were also studied. Their shear damping functions are shown in Fig. 4c. The damping function of PS-105-6.7G-6 is very similar to that one for the linear PS-r-95. This result can be interpreted as such that the side branches which are very short compared to the molar mass for entanglements M_{e} do not contribute to a changed strain dependence due to the lack of entanglements.
In contrast to this result, the graft-polystyrene PS-55-4.0G-13 with the longer side branches shows clearly a weaker dependence of h(γ) on strain than PS-105-6.7G-6. It can be compared to that one of the graft-polystyrenes PS-70-3.2G-22 or PS-60-1.4G-42. This result means that in addition to the length of the side chains another molecular property may influence the damping function.
Figure 5 shows the damping functions of all graft-polystyrenes together with the various number average mass fractions Φ _{n,br} of grafted side chains. Three groups of graft-polystyrenes can be distinguished: the graft-polystyrenes with the highest mass fraction Φ _{n,br} of about 0.50 which give the weakest damping and the graft-polystyrenes with a mass fraction Φ _{n,br} between 0.14 and 0.30 which show a stronger damping. The third group is PS-105-6.7G-6 and PS-r-95 with the strongest damping. The similarity of h(γ) for PS-105-6.7G-6 with that of the linear PS-r-95 can be understood, as the short side chains of M_{w,br}=6,800 g/mol are well below the entanglement molecular weight of 18,000 g/mol and therefore do not affect the entanglement network. Somewhat surprising is the finding that h(γ) for PS-55-4.0G-13 and PS-70-3.2G-22 coincide with the other two samples of Φ _{n,br} ≅ 0.5 although this have molar masses significantly below M_{c}. As a consequence it can be concluded that also short side branches with M_{e}<M_{w,br}<M_{c} do alter the strain dependence of h(γ) and that the mass fraction Φ _{n,br} of grafted side chains is the decisive parameter for the strain dependence of h(γ) for the grafted polystyrenes studied.
Uniaxial elongational flow
Influence of the number of grafted side chains
As it is well known from literature that besides branching high-molar mass components and the polydispersity can have an influence on strain hardening (Hepperle 2003; Münstedt and Laun 1981; Münstedt 1980) the graft-polystyrenes in this study were synthesized in a way that similar molar masses and molar mass distributions are obtained (Haug 2000).
The influence of the average number of grafted side chains on strain hardening is investigated on a series of graft-polystyrenes with similar molar masses of the grafted side chains (Fig. 8a–c). The melt of the graft-polystyrene PS-70-3.2G-22 with an average of about three grafts per chain shows the highest strain hardening (Fig. 8a), followed by PS-60-2.1G-27 with an average of about two grafts per chain (Fig. 8b). The slightly branched PS-80-0.6G-22 (Fig. 8c) for which on average only every second molecule is branched, exhibits a lower degree of strain hardening than the other two graft-polystyrenes. From these results it can be concluded that a growing average number of grafted side chains leads to an increased amount of strain hardening, if melts of a similar topology are compared.
Influence of the molar mass of grafted side chains
Figure 9b displays the tensile stress growth coefficients for the melts of the graft-polystyrenes PS-60-1.4G-42 and PS-90-1.2G-27. The latter has a backbone mass average molar weight that is a factor of 1.5 higher (13.7×10^{4} g/mol compared to 9.3×10^{4} g/mol) and side branches that have mass average molar masses significantly lower, resulting in a mass fraction of grafted side chains of only 0.27 compared to a fraction of 0.49 for PS-60-1.4G-42. Despite the different topology, both melts show a similar degree of strain hardening. This result confirms that one obtained from the graft-polystyrenes with the lower amount of grafted side chains: not the length of the grafted side chains increases the amount of strain hardening, but the average number of grafts per molecule.
For all melts studied, the accuracy of the measurements is confirmed by the fact that the Trouton ratio η_{E}^{+}(t) = 3 η(t) is fulfilled in the linear-viscoelastic range for all the strain rates applied. This is indicated in Figs. 7, 8, 9, and 10 by the solid lines representing 3η(t) derived from the dynamic data (Hepperle et al. (2005)) and shifted according to the WLF-equation with respect to the measuring temperature.
Quantification of the strain hardening properties
In comparison to the determination of S=f(ε) for one strain rate, the determination of \(\hbox{S}=\hbox{f}(\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon})\) at a fixed elongation allows one to account for the varying zero-shear viscosities of the different materials, as the strain hardening factors S can be compared at reduced strain rates \(\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon} \cdot \eta _{0} \). A similar suggestion was made by comparing strain hardening factors as a function of strain at a constant reduced strain rate \(\tau _{0} \cdot \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon}\) for different materials with τ_{0} being the terminal relaxation time (Takahashi et al. 1993). Values of S are determined from Figs. 7, 8, 9 and 10, for a strain of ε=3.0.
Figure 11a shows the strain hardening factors of the graft-polystyrenes with a similar mass average molar mass of the grafted side chains, but different average numbers \(\ifmmode\expandafter\bar\else\expandafter\=\fi{p}\) of grafted side chains per molecule. The strain hardening factor S increases with increasing average numbers \(\ifmmode\expandafter\bar\else\expandafter\=\fi{p}\) of grafted side chains per molecule. The lowest strain hardening is obtained for the melt of the linear polystyrene PS-r-95 which has a polydispersity comparable to the graft-polystyrenes (Hepperle et al. 2005). The two graft-polystyrenes PS-90-1.2G-27 and PS-80-0.6G-22 with a low average number of grafts per molecule show a higher strain hardening compared to the melt of the linear PS-r-95. PS-80-0.6G-22 and PS-90-1.2G-27 have comparable mass average molar masses of the backbone (M_{w,bb}=122,100 g/mol and M_{w,bb}=136,600 g/mol, respectively), but PS-90-1.2G-27 has twice the mass fraction Φ _{n,br} of grafted side chains. The higher strain hardening of PS-90-1.2G-27 compared to PS-80-0.6G-22 can therefore be attributed to the higher average number \(\ifmmode\expandafter\bar\else\expandafter\=\fi{p}\) of grafted side chains per molecule. It is also evident from the stress growth coefficients shown in Fig. 11a that the slope of the dependence of the strain hardening factor S on the reduced strain rate \(\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon} \cdot \eta _{0} \) increases with the amount of grafted side chains.
A comparison of the higher branched graft-polystyrenes PS-70-3.2G-22 and PS-60-2.1G-27 reveals an obvious information about the role of the number of grafted chains per molecule on the strain hardening effect, as the number average mass fraction Φ _{n,br} of grafted chains is constant. PS-70-3.2G-22 shows a more pronounced strain hardening than the graft-polystyrene with the lower number of grafted side chains (PS-60-2.1G-27) indicating once more the growing strain hardening with increasing number of grafted chains.
Figure 11b shows the comparison of the strain hardening factors for pairs of graft-polystyrenes with side chains of different molar masses. The graft-polystyrenes with a higher average number of grafted chains (PS-60-1.4G-42 and PS-90-1.2G-27) show slightly higher values for the strain hardening factor S than the graft-polystyrenes with lower average number of chains per molecule (PS-60-0.5G-55 and PS-80-0.6G-22).
Figure 11c presents the graft-polystyrenes with grafted side chains of molar masses below M_{c} and M_{e}. The graft-polystyrene PS-105-6.7G-6 with short side chains (M_{w,br}=6,800 g/mol) does only show a slightly increased strain hardening factor compared to the linear PS-r-95, which can also be caused by the somewhat higher polydispersity of PS-105-6.7G-6 with M_{w}/M_{n}=1.9 in comparison to PS-r-95 with M_{w}/M_{n}=1.65. PS-55-4.0G-13 and PS-70-3.2G-22, however, possess a pronounced strain hardening. From these results follows that the side chains with M_{w,br} = 6,800 g/mol are too short to be entangled, whereas side chains with M_{w,br}=14,170 g/mol are already long enough to contribute to a higher stress level in uniaxial elongation.
Damping function in uniaxial elongational flow
The calculated damping functions h_{u}(ε) of the linear PS-r-95 and four branched graft-polymers with Φ _{n,br} ≅ 0.5 are shown in Fig. 12. The damping functions h_{u}(ε) are calculated for strain rates \(1.0 \geqslant \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon} \geqslant 0.1\,s^{-1}\) and deformations up to ε_{H}=3. Within experimental accuracy, the damping functions are the same for different strain rates for one material. A time-strain separability was already experimentally found for many branched and linear polymer melts (Wagner et al. 2000).
Conclusions
The graft-polystyrenes studied here possess a certain polydispersity of the backbone. Therefore, all graft-polystyrenes also include topologies with a higher amount of branches than the average value, especially for molecules of higher molar masses resulting in a variety of structures like asymmetric stars, H-shaped structures and brushes. Their rheological properties show distinct differences in nonlinear-viscoelastic flows, depending on their molecular structure and on the type of flow applied to their melts. In uniaxial elongational flow, the tensile stress growth coefficient is mainly dependent on the average number of grafted chains per molecule, whereas the molar mass of grafted side chains does not significantly change the amount of strain hardening.
Surprisingly, rather short branches do show an influence on the damping function and on strain hardening. For example, the graft-polystyrene with M_{w,br}/M_{e} = 0.8, i.e., PS-55-4.0G-13, still possesses an increased damping function in shear and in elongation and a higher degree of strain hardening than the linear PS with comparable polydispersity. However, if the side chains of the graft-polystyrene are very short (M_{w,br}/M_{e} = 0.4 for PS-105-6.7G-6) the melt shows a damping function h(γ) similar to the one of a linear polystyrene, whereas in uniaxial deformation the strain hardening for this branched graft-polystyrene is only slightly higher than the one of a linear PS.
The damping function in elongation is less dependent on strain the higher the amount of strain hardening. The damping function in shear flow shows a dependence on the number average mass fraction of grafted side chains.
The comparison of the damping functions determined from step-strain experiments in shear flow and calculated with the relaxation time spectra and the extensional viscosities in uniaxial elongational flow shows that the influence of molecular topology on nonlinear-viscoelastic strain functions is quite different for various types of flow. Elongational flow is much more sensitive to differences in molecular structure than shear flow.
Notes
Acknowledgements
Financial support from the German Research Foundation (DFG) (grant numbers Mu 1336/2-1 and Mu 1336/2-3) is gratefully acknowledged. J.H. wants to thank Prof. Dr. M. H. Wagner for the source code of the program for the damping function calculations.
Supplementary material
References
- Doi M, Edwards SF (1986) The theory of polymer dynamics. Clarendon Press, OxfordGoogle Scholar
- Ferri D, Lomellini PJ (1999) Melt rheology of randomly branched polystyrenes. J Rheol 43:1355–1372CrossRefGoogle Scholar
- Ferry JD (1980) Viscoelastic properties of polymers. Wiley, New YorkGoogle Scholar
- Gabriel C, Münstedt H (1999) Creep recovery behavior of metallocene linear low-density polyethylenes. Rheol Acta 38:393–403CrossRefGoogle Scholar
- Gotsis AD, Zeevenhoven BLF, Tsenoglou C (2004) Effect of long branches on the rheology of polyproylene. J Rheol 48:895–914CrossRefGoogle Scholar
- Grigo U, Kircher K, Müller PR (1996) Polycarbonates. In: Bottenbruch L (ed) Engineering thermoplastics: polycarbonates, polyacetals, polyesters, cellulose esters. Carl Hanser Verlag, MunichGoogle Scholar
- Haug PK (2000) Polystyrol-Pfropfpolymere definierter Struktur als Modellsystem für Langkettenverzweigungen in Polyolefinen. Doctoral Thesis, University of StuttgartGoogle Scholar
- Heino EL, Lehtinen A, Tanner J, Seppälä J (1992) Theoretical and Applied Rheology. In: Proceedings of the XIth international congress on rheology, pp 360–362Google Scholar
- Hepperle J (2003) Einfluss der molekularen Struktur auf rheologische Eigenschaften von Polystyrol- und Polycarbonatschmelzen. Doctoral Thesis, University Erlangen-NürnbergGoogle Scholar
- Hepperle J, Münstedt H, Haug PK, Eisenbach CD (2005) Rheological properties of branched polystyrenes—linear viscoelastic behavior. Rheol Acta (in print)Google Scholar
- Hingmann R, Marczinke BL (1994) Shear and elongational flow properties of polypropylene melts. J Rheol 38:573–587CrossRefGoogle Scholar
- Khan SA, Prud’homme RK, Larson RG (1987) Comparison of the rheology of polymer melts in shear, and biaxial and uniaxial extensions. Rheol Acta 26:144–151CrossRefGoogle Scholar
- Kurzbeck, S (1999) Dehnrheologische Eigenschaften von Polyolefinschmelzen und Korrelationen mit ihrem Verarbeitungsverhalten beim Folienblasen und Thermoformen, Doctoral thesis, University Erlangen-NürnbergGoogle Scholar
- Kurzbeck S, Oster F, Münstedt H, Nguyen TQ, Gensler RJ (1999) Rheological properties of two polypropylenes with different molecular structure. J Rheol 43:359–374CrossRefGoogle Scholar
- Larson RG (1985) Nonlinear shear relaxation modulus for a linear low-density polyethylene. J Rheol 29:823–831CrossRefGoogle Scholar
- McLeish TCB, Allgaier J, Bick DK, Bishko G, Biswas P, Blackwell R, Blottière B, Clarke N, Gibbs B, Groves DJ, Hakiki A, Heenan RK, Johnson JM, Kant R, Read DJ, Young RN (1999) Dynamics of entangled H-polymers: theory, rheology, and neutron-scattering. Macromolecules 32:6734–6758CrossRefGoogle Scholar
- Münstedt H (1979) New universal extensional rheometer for polymer melts. Measurements on a polystyrene sample. J Rheol 23:421–436CrossRefGoogle Scholar
- Münstedt H (1980) Dependence of the elongational behavior of polystyrene melts on molecular weight and molecular weight distribution. J Rheol 24:847–867CrossRefGoogle Scholar
- Münstedt H, Laun HM (1981) Elongational properties and molecular structure of polyethylene melts. Rheol Acta 20:211–221CrossRefGoogle Scholar
- Papanastasiou AC, Scriven LE, Macosko CW (1983) An integral constitutive equation for mixed flows: viscoelastic characterization. J Rheol 27:387–410CrossRefGoogle Scholar
- Pearson DS, Mueller SJ, Fetters LJ, Hadjichristidis N (1983) Comparison of the rheological properties of linear and star-branched polyisoprenes in shear and elongational flows. J Polym Sci Polym Phys Ed 21:2287–2298CrossRefGoogle Scholar
- Soskey PR, Winter HH (1984) Large step shear strain experiments with parallel-disk rotational rheometers. J Rheol 28:625–645CrossRefGoogle Scholar
- Takahashi M, Isaki T, Takigawa T, Masuda T (1993) Measurement of biaxial and uniaxial extensional flow behavior of polymer melts at constant strain rates. J Rheol 37:827–846CrossRefGoogle Scholar
- Wagner MH (1976) Analysis of time-dependent non-linear stress-growth data for shear and elongational flow of a low-density branched polyethylene melt. Rheol Acta 17:136–142CrossRefGoogle Scholar
- Wagner MH (1979) Zur Netzwerktheorie von Polymer-Schmelzen. Rheol Acta 18:33–50CrossRefGoogle Scholar
- Wagner MH, Laun HM (1978) Nonlinear shear creep and constrained elastic recovery of a LDPE melt. Rheol Acta 17:138–148CrossRefGoogle Scholar
- Wagner MH, Bastian H, Hachmann P, Meissner J, Kurzbeck S, Münstedt H, Langouche F (2000) The strain-hardening behaviour of linear and long-chain-branched polyolefin melts in extensional flows. Rheol Acta 39:97–109CrossRefGoogle Scholar