Effect of sparse long-chain branching on the step-strain behavior of a series of well-defined polyethylenes.
Step-strain tests are used to determine the strain dependence of the relaxation modulus. These tests are defined by a sudden shearing displacement of a fluid at rest to some strain, [[gamma].sub.0]. Beyond some characteristic time, [[tau].sub.k], after the imposition of a step shear strain the relaxation modulus, G(t,[gamma]), for a wide range of polymers [1-3] can be factored into separate time and strain functions:
G(t, [gamma]) = [G.sup.0](t)*h([gamma]) (1)
where [G.sup.0](t) is the linear relaxation modulus, and h([gamma]) is the strain-dependant damping function. This phenomenon is known as time-strain separability [4-6] and is illustrated in Fig. 1 for a sparsely branched HDPE resin. The behavior of the relaxation moduli for polymeric melts, both before and after [[tau].sub.k], is not fully understood, thereby leading to much debate in the literature as to why differences in the shape of the relaxation and damping function curves are seen for different materials.
[FIGURE 1 OMITTED]
Much of the disagreement in the literature regarding a material's behavior at times shorter than [[tau].sub.k] stems from the mechanical issues inherent in a rheometer's imposition of an instantaneous strain. Three main issues are the focus of much of the discussion. First is the amount of time needed to impose this "instantaneous" strain, called the finite rise time. The second issue is that of transducer overloads and material rupture. Third is wall-slip at the polymer-plate surface interface.
Finite rise times are created by the fact that it is physically impossible with current rheometer technology to instantaneously shear a fluid to a desired strain. Instead, there exists some finite rise time for the imposition of strain. This finite rise time is a point of contention as it is unknown if the data preceding [[tau].sub.k] is truly accurate or an artifact of the experimental apparatus. Vega and Milner  proposed that in the framework of the BKZ constitutive model, the finite rise time was the primary reason time-strain separability was not observed until after some time [[tau].sub.k]. Sui and McKenna  attributed deviations at short times to different loading times of the normal force and torque response (i.e. reaching maximum value) in linear and dendritic polyisobutylene (PIB). Venerus  simulated the effect of finite rise time on the response of the relaxation modulus, finding that differences at times less than [[tau].sub.k] were still seen as a function of the imposed strain. This suggested that the short time response prior to [[tau].sub.k] was realistic and not an artifact of the experimental apparatus. Stadler et al.  stated that the finite rise (step) time had no relevant influence on the relaxation modulus after times longer than 10 times the step time. Venerus and Kahvand  found for a PS solution with an average of seven entanglements per molecule that the measured shear stress was independent of the step time for times roughly three times larger than the step time.
Transducer overloads and material ruptures can occur at higher strains for materials with a high plateau modulus, such as some entangled polyethlyenes . Transducer overloads have been shown to be avoidable by utilizing plate fixtures with smaller diameters [11-13]. This technique has been shown to suffer from a reduced accuracy and increased error at long times due to the smaller sample size . Stadler et al.  argued that sample rupture was not present in melts that had a measureable damping function, h([gamma]). The authors argued that the sample rupture would have changed the shape of the relaxation modulus thereby making the measurement of a realistic h([[gamma].sub.0]) impossible. This finding made the detection of rupture simpler in that a complicated and expensive rheo-optical measurement systems was not necessary .
Wall-slip at the polymer-plate interface was found to be a major source of error in step-strain [6, 14]. Wall slip occurs when the wall stress exceeds some threshold value greater than the surface bond between the polymeric melt and the wall. If slip occurs, the applied strain is lower than the desired strain. Wall slip depends on the surface characteristics and the dynamic modulus at timescales on the order of the applied strain. This causes the measured damping function, h([[gamma]), to be too small , Slip was also found to cause a "kink" in the damping function curve at higher strains . Vega and Milner  observed the deformation of a vertical line initially drawn on the undeformed sample to confirm the presence/absence of wall slip. If the line deformed affinely with the plates, slip was not present. Likewise if the line was underdeformed, slip was present. Vega and Milner  also observed that slip occurred in well-entangled polyethylenes (PEs) at strains as low as 2.0. Sanchez-Reyes and Archer  found that slip could be mitigated in well-entangled polystyrenes (PSs; [greater than or equal to]37 entanglements) by attaching a single layer of micron-sized glass beads to the plate surface. Vega and Milner  reduced wall-slip and transducer overload by blending low molecular weight ([M.sub.W]) PE with entangled PE.
It has also been proposed that the relaxation modulus' behavior prior to [[tau].sub.k] is not a mechanical artifact. Vega and Milner  found disagreement in the literature as to the reason for a material's behavior at times less than [[tau].sub.k]. In some works, it has been claimed that the value of [[tau].sub.k] was related to the time required for a complete relaxation of the contour length,[[tau].sub.s], [15, 16] via [[tau].sub.k] ~ [[tau].sub.s] ~ 4[[tau].sub.R], where [[tau].sub.R] is the longest Rouse time of the chain. Other, more contemporary, results [6, 17] suggested that [[tau].sub.k] was not dictated by [[tau.sub.R]. Measurements of the dynamic moduli for well-entangled linear PS solutions displayed time-strain separation at times comparable with the terminal relaxation of the chain [[tau].sub.d]. For weakly entangled systems, values of [[tau].sub.s] approach [[tau].sub.k].
Regardless of the origin of such short-time behavior, the imperfect strain history must be accounted for in any constitutive modeling. Sui and McKenna  cited a correction necessary for any constitutive modeling of G(t,[gamma]) originally put forward by Zapas and Craft  to correct for the imperfect strain history that arose from the finite rise time
[t.sub.corrected] = t - [[tau]/2] (2)
where t was the raw measured experimental time,[t.sub.corrected] was the corrected time, and [tau] was the finite time required to impose the desired strain, [tau] was determined to be the time at which the normal force and torque response reached their maximum value . [tau] was found to occur on timescales less than the onset of time-strain separability, [[tau].sub.k], for linear and dendritic PIB .
Although the short times of the relaxation modulus are of primary interest to this work, the behavior beyond [[tau].sub.k] is useful in that it has been linked to the molecular structure of the melt. Stadler et al.  summarized contemporary works that highlighted the effect of molecular structure on the behavior of time-strain separation in the literature. The damping function h([[gamma].sub.0]) was found to be independent of molar mass [19, 20] and temperature [19, 21, 22] in linear polymer melts. Molecular weight distribution, MWD, has been shown to affect the value of h([[gamma].sub.0]) [23-25], The chemical structure has been shown to affect h([[gamma].sub.0]), for example, in commercial linear PE and polypropylene (PP) . The damping function of long chain branching (LCB) polymers with multiple branch points (both commercial resins and model systems) has been shown [5, 11, 22, 26] to have a weaker dependence on strain than their linear counterparts. Hepperle and Munstedt  observed that increasing arm length in nearly monodisperse combs reduced the strain dependence of h([[gamma].sub.0]). Polymeric melts of three- and four-armed star (only one LCB branch point) have shown only weak deviation from the nonlinear behavior of linear polymer melts [27-29]. The exact reason for such a wide variety of strain dependences of h([[gamma].sub.0]) is unknown. It has been theorized that the nonlinear strain dependence originated from the induced chain anisotropy and the decrease in entanglement density induced by the large shear amplitude . For branched polymers, the retarded relaxation of the backbone chain due to the pinned branch points was theorized to be the reason for the weaken strain dependence of h([[gamma].sub.0]) .
The strain dependence of h([[gamma].sub.0]) has been modeled to describe the nonlinear response of polymeric melts. An expression based on the Doi-Edwards (DE) tube model  with independent alignment was found to accurately model the damping function seen in linear and some star polymer melts with narrow MWD [22, 28, 29]. Stadler et al.  observed that the shape of log h([[gamma].sub.0]) versus log [[gamma].sub.0] resembled that of a simple shear viscosity curve, and proposed a numerical damping function based on the Carreau-Yasuda model originally developed for shear viscosity profiles [31, 32].
The objective of this paper is to investigate the specific relationship between LCB content and the step-strain behavior using a series of well-characterized, sparsely branched HDPE resins. Specifically, it is the goal of this work to illustrate that rheological data measured prior to the onset of time-strain separability is indeed a property of the material and not a mechanical artifact. The materials selected are well understood in terms of their branching content and rheological behavior [33-37], and make excellent candidates for such an analysis. The shape and smoothness of the damping function is investigated to rule out the presence of slip and material rupture during testing. The finite rise time to impose the desired strain is carefully monitored and compared to the Rouse relaxation time of the linear HDPE resin studied. Transducer overloads are mitigated by using smaller cone-and-plate fixtures. All materials are checked to see if time-strain separation is observed beyond some characteristic time, [[tau].sub.k]. The effect of LCB is then correlated to the magnitude of [[tau].sub.k]. The behavior of the relaxation modulus at times shorter than [[tau].sub.k] is investigated by an analysis of the enhancement seen to the linear relaxation modulus, [G.sup.0](t), as a function of strain and LCB content. Correlation between this enhancement and LCB content is made. Discussion is also given regarding the differences seen in the short-time behavior of materials with similar shear viscosity curves, but different LCB content. Information reported in this study can possibly be extrapolated in future works using constitutive relationships to determine information regarding specific molecular structure.
EXPERIMENTAL MATERIALS AND METHODS
A series of sparsely long-chain branched and linear metallocene-catalyzed high-density PE (HDPE) homopolymers PEs with varying degrees of LCB made available by the Dow Chemical Company were used. [M.sub.w], MWD, and LCB content for these resins are given in Table 1. The sparsely long-chain branched metallocene-catalyzed PE resins include HDB1, HDB2, HDB3, HDB4, HDB5, HDB6, and HDB7 and were synthesized using a constrained geometry catalyst of the type described by Stevens [38, 39] in a continuous, stirred-tank reactor as described by Lai et al. . No information was provided as to the method of synthesis for the linear resin HDPE resin HDBL. Both the sparsely branched resins and the linear resin are homopolymers with narrow molecular weight distributions, MWD, characteristic to metallocene type catalysts. These materials are henceforth referred to collectively as the HDB series.
TABLE 1. Material properties. Material [M.sub.w] [M.sub.w]/[M.sub.n] LCB/10,000 C Avg. # of (g/mol) branches per chain HDBL 113,000 2.5 Linear Linear HDB1 77,100 2.0 0.26 0.14 HDB2 82,000 1.9 0.37 0.22 HDB3 85,700 2.0 0.42 0.26 HDB4 96,300 2.1 0.80 0.55 HDB5 79,000 -- 0.90 0.51 HDB6 68,000 -- 1.88 0.91 HDB7 70,000 -- 3.33 1.66
The HDB resins are ideal for this study for three primary reasons. First, the HDB resins were well characterized in previous studies [33-37, 41], most notably the linear rheology of HDBL, HDB1, HDB3, and HDB4 by Wood-Adams and Dealy [35, 36]. Second, they were readily available in sufficient quantities to accurately measure the viscoelastic properties. Finally, their branching content is well known based on (13) C NMR, providing an excellent opportunity to measure the effect of slight variations in the long-chain branching content when subjected to step-strain deformation.
Shear step-strain experiments were conducted using a Rheometrics RMS-800 fitted with 25 mm cone and plate fixtures with a cone angle of 0.1 radians and 10 mm cone and plate fixtures with a cone angle of 0.035 radians. Tests were carried out at a temperature of 170[degrees]C. Strains of 0.01, 0.1, and 1 strain units were evaluated using the 25 mm diameter plates and strains of 1, 5, 7.5, 10, and 12.5 strain units were evaluated using the 10 mm diameter plates , Finite rise time to impose the desired strain was found to range from 0.055 to 0.065 sec and increased with strain. The presence of wall-slip and rupture were investigated via inspection of the experimentally determined damping function as per the recommendations of Stadler et al.  and Vega and Milner . Linear viscoelastic shear measurements were conducted using a Rheometrics RMS-800. Cone-and-plate (25 and 10 mm) fixtures with cone angles of 0.1 and 0.05 radians, respectively, were used. An angular frequency range of 0.01-100[s.sup.-1] was probed. All tests were performed under an inert [N.sub.2] atmosphere to prevent oxidative degrade The test samples were prepared by compression molding preforms at 170[degrees]C under nominal pressure and allowing them to cool slowly under no pressure. This method provided homogenous samples with minimal residual stress.
The results of both the step-strain and shear viscosity curves are presented in this section for all eight resins of interest. Several materials are found to have relative identical shear viscosity curves, allowing for comparisons of the resins' relaxation modulus behavior on the basis of not only LCB content but relative to simple shear flow behavior as well. General trends and observations are given here, while a more detailed analysis is reserved for the discussion section presented later.
Shown in Fig. 1a are the relaxation moduli, G(t,[gamma]), for HDB1. Increasing the strain decreases the overall magnitude of G(t,[gamma]), especially at longer times. However, the general shape of the curve appears similar at each strain as time is increased. G(t,[gamma]) is then divided through by an experimentally determined damping function to gauge time-strain separability and is shown in Fig. 1b. At long times, the shifted moduli curves overlap, obeying Eq. 1, while at short times, G(t,[gamma])/h([gamma]) increased with the value of strain. [[tau].sub.k] was found to be equal to 0.135 sec. for HDBL The remainder of the relaxation moduli results presented in this work are given as G(t,[gamma])/h([gamma]) versus time to emphasize both the onset of time-strain separability and the behavior of the resins at short times.
In Figs. 2-7 are given the dampened step-strain behavior for the remainder of the sparsely-branched resins, HDB2, HDB3, HDB4, HDB5, HDB6, and HDB7. Each resin was observed to obey Eq. 1 beyond some value of [[tau].sub.k] that was found to range from 0.135 to 0.490 s. Furthermore, for each branched resin, the magnitude of G(t,[gamma])/h([gamma]) and the value for of [[tau].sub.k] as a function of strain was observed at short times to increase with the value of strain. Finally, in Fig. 8 are given the curves showing the shifted relaxation moduli for the linear ITDPE resin HDBL. Similar to the branched resins, HDBL obeys Eq. 1 beyond some characteristic time [[tau].sub.k]. The value of [[tau].sub.k] was found to be equal to 0.11 s. Also, the magnitude of the enhancement of G(t,[gamma])/h([gamma]) as a function of strain was much lower than that for the sparsely branched resins. Further discussion regarding the ramifications and implications of the linear resin's behavior relative to the sparsely branched system is reserved for discussion later in this work.
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
The magnitude of complex viscosity versus angular frequency curves at 170[degrees]C for each material is given in Fig. 9. The Cox-Merz relationship  was validated for the HDPE materials in previous studies [33, 34], suggesting that the magnitude of complex viscosity determined from small-angle oscillatory shear is a good approximation of the steady-shear viscosity. All branched-resins were found to have an enhanced zero-shear viscosity relative to the linear HDPE. The correlation of LCB content to the level of zero-shear viscosity enhancement was found to pass through a maximum, which is in agreement with the findings of Janzen and Colby , among other works [45-47]. Furthermore, HDB3, HDB5, and HDB6 were found to have similar shear viscosity flow curves despite differences in LCB content (0.37 versus 0.90 versus 1.66 LCB/[10.sup.4] C, respectively). This allows for direct comparison of the effect of LCB content on the step-strain for materials with relatively identical shear viscosity curves.
[FIGURE 9 OMITTED]
Considerations for Finite Rise Time
Finite rise times arise from the inability of a rheometer to instantly apply the desired strain to a material. Instead, a relatively small amount of time passes prior to the implementation of the desired strain. This can affect the stress relaxation modulus output by artificially damping the results due relaxation occurring prior to complete deformation . The maximum time required for implementation of the highest strain used in this work ([gamma] = 12.5) was 0.065 seconds. No experimental data was used until at least 95% of the desired strain was reached. The importance of the finite rise time for linear resins lies in its comparison with the Rouse relaxation time, [[tau].sub.R] For the relaxation modulus curves to be unaffected by relaxation that takes place prior to reaching the applied strain, [[tau].sub.R] should be sufficiently greater than the finite rise time. [[tau].sub.R] for HDBL was calculated following the method of Elbirli and Shaw , given by Eq. 3:
[[tau].sub.R] = [[6[[eta].sub.0][M.sub.w]]/[[[pi].sup.2][rho]RT]] (3)
where [[eta].sub.0] is the zero-shear viscosity at the test temperature, [M.sub.w] is the molecular weight, and [rho] is the material density at the test temperature, T. Because Rouse dynamics were derived under the assumption of a linear architecture, it is inappropriate to calculate values of [[tau].sub.R] for LCB resins. However, as it has been shown  that the relaxation time for LCB resins was longer than linear resins of similar molecular weight, it is expected that the value of [[tau].sub.R] for HDBL to be the minimum value of the resins investigated in this work. Using the shear viscosity data given in Fig. 9, resin data from Table 1, and the density of HDPE at 170[degrees]C equal to 0.7719 g/[cm.sup.3] (see Baird and Collias , p. 105), [[tau].sub.R] for HDBL was calculated to be 0.170 s, a value that is nearly three times that of the longest finite rise time, suggesting that the materials were not sufficiently relaxed prior to the inception of the full strain. Furthermore, from a qualitative perspective, if the finite rise time was allowing for significant relaxation of the material, one would expect to see a trend of decreasing values of G(t,[gamma])/h([gamma]) at very low times (say t = 0.05 s) as a function of strain. This trend was not observed in any of the resins (see Figs. 1-8), further suggesting that the finite rise time is not significantly affecting the material data at times shorter than [[tau].sub.R].
Behavior After Time--Strain Separation (t > [[tau].sub.k])
In Fig. 10 is given the damping function, h([gamma]), for each of the eight resins investigated. Also plotted are a h([gamma]) curve predicted by DE theory  and the h([gamma]) curve for a sparsely branched HDPE resin from a previous study . According to Stadler et al.  and Vega and Milner , the presence of an irregularly or illogically shaped h([gamma]) curve (i.e., a "kink") would correspond to material rupture or slip at the plate-polymer interface. From a physical point of view, if the material underwent slip during a test, the resulting applied strain would be less than the desired strain. Therefore, the value of h([gamma]) at the applied strain would be higher than the value of h([gamma]) at the desired strain, resulting in a "kink." One can imagine the same scenario for a material rupture, which is simply slip at some polymer-polymer interface within the homogeneous polymer matrix. An investigation of the data given in Fig. 10 shows that the data set is devoid of any major "kinks" for any of the resins investigated. This suggested that the results presented in Figs. 1-8 were devoid of any major slip or rupture. An observed trend that was consistent with previous works [4, 9] was the reduction of the strain dependence in sparsely-branched LCB resins. This is evidenced in Fig. 10 by h([gamma]) having a smaller slope than that of the linear HDBL at higher strains. Vega and Milner  attributed this behavior to the quick relaxation times of the branched arms, suggesting that a substantial portion of the branched material has relaxed prior to [[tau].sub.k]. Regardless of the mechanism, the data in Fig. 10 suggest that wall-slip and sample rupture were not responsible for the relaxation behavior of the HDPE resins.
[FIGURE 10 OMITTED]
At times longer than [[tau].sub.k], values of G(t,[gamma]) as a function of LCB were reminiscent of the behavior of [[eta].sub.0] as a function of LCB content. In Fig. 11, G(t,[gamma] = 1.0) is shown as function of time for the eight HDPE resins. It was observed that HDBL had the lowest relaxation modulus value at t equal to 1 s, suggesting that the material had relaxed its stress faster than the sparsely branched resins. At the same time, the sparsely branched resins exhibited an enhanced relaxation modulus. This enhancement increased up through LCB contents of 0.80 LCB per [10.sup.4] carbons (corresponding to HDB4). At higher LCB concentrations (HDB5, HDB6, and HDB7), this enhancement was reduced as a function of LCB content but still remained higher than the linear HDBL resin. This dependence of G(t,[gamma]) on LCB content is similar to that observed in Fig. 9 for [[eta].sub.0] as a function of LCB content. The data in Fig. 11 suggest that LCB has an enhancing effect on the magnitude of the relaxation modulus at times longer than [[tau].sub.k], up until some critical value before this enhancement begins to decrease approaching values similar to the linear resin.
[FIGURE 11 OMITTED]
Behavior Before Time-Strain Separation (t [less than or equal to] [[tau].sub.k])
Now that wall-slip, sample rupture, and transducer overload were eliminated as potential explanations for the behavior of G(t,[gamma]) at times less than [[tau].sub.k], the effect of sparse LCB content at these short times was investigated. This was done by investigating two characteristics of the G(t,[gamma])/h([gamma]) curve at short times. First, the magnitude of [G.sup.0](t = 0.07 s, [gamma])/[G.sup.0](t = 0.07 s, [gamma] = 0.01) enhancement as a function strain, LCB content, average number of branches per chain was studied. Second, the effect of LCB content of the value of [[tau].sub.k] was investigated. Both characteristics were found to correlate positively with LCB content and average number of LCB per chain.
In Fig. 12, [G.sup.0](t = 0.07 s, [gamma])/[G.sup.0](t = 0.07 s, [gamma] = 0.01) is plotted against LCB per 10,000 carbon atoms. In an effort to eliminate the slight differences in the MW of the resins, the same enhancement values are plotted as function of the average number of LCB per backbone chain in Fig. 13. The two strains investigated were [gamma] = 5.0 (x's) and [gamma] = 12.5 (squares). These values correspond to the relative magnitude of the inflection at short times given in Figs. 1-8. Increasing LCB content served to enhance the initial magnitude and inflection of the shifted relaxation modulus at short times. Furthermore, materials with similar simple shear viscosity curves (HDB3, HDB5, and HDB6) were observed to exhibit different short-time behavior, suggesting that step-strain was more sensitive to LCB content, akin to the response seen in uniaxial extension .
[FIGURE 12 OMITTED]
[FIGURE 13 OMITTED]
Finally, in Fig. 14 is shown the effect of LCB content per [10.sup.4] carbon atoms (squares) and average number of LCB per backbone (x's) on the time for the occurrence of time-strain separation (t = [[tau].sub.k]). Solid and dashed lines are given only to aid the eye and show proof of a strong positive correlation. Both increasing LCB content per [10.sup.4] carbons and average number of LCB per backbone were shown to increase the value of [[tau].sub.k]. This suggested that an increased LCB content served to increase the time required for sparsely-branched LCB systems to enter the linear relaxation region given by [G.sup.0](t, [gamma]) in Eq. 1.
[FIGURE 14 OMITTED]
The effect of sparse LCB on the shear step-strain relaxation modulus was analyzed using a series of eight well-characterized HDPE resins. Strains of 1 to 1250% were imposed on materials with LCB content ranging from 0 to 3.33 LCB per 10,000 carbons. The shape and smoothness of the damping function was investigated to rule out the presence of wall-slip and material rupture during testing. The finite rise time to impose the desired strain was carefully monitored and compared with the Rouse relaxation time of the linear HDPE resins studied. Transducer overloads were mitigated by using smaller cone-and-plate fixtures. All materials were observed to obey time-strain separation beyond some characteristic time, [[tau].sub.k]. LCB content in sparsely branched resins was observed to increase the value of [[tau].sub.k] relative to the linear resin. Furthermore, the amount of LCB content was seen to correlate positively with increasing [[tau].sub.k]. The values of G(t,[gamma]) at long times were seen to behave as a function of LCB content similar to that of the zero-shear viscosity. Enhancement of the magnitude of G(t,[gamma]) relative to that of the linear HDBL as a function of LCB passed through a maximum at a LCB content of 0.80 LCB/[10.sup.4] carbon atoms before decreasing in value, while remaining above the linear resin. The behavior of the relaxation modulus at times shorter than [[tau].sub.k] was investigated by an analysis of the enhancement seen to the linear relaxation modulus, [G.sup.0](t), as a function of strain and LCB content. All eight HDPE resins were seen to exhibit enhancement. It was observed that that this enhancement was significantly larger in the sparsely branched HDPE resins relative to the linear HDPE resin. Furthermore, the enhancement was seen to increase in magnitude with increasing LCB content. The results presented in this work suggest that increasing sparse LCB served to enhance both the amount of time needed for and magnitude of stress relaxation relative to the linear resin.
This research is a collaborative effort for the World Network of Materials-UK Leeds. Gratitude is given to the Dow Chemical Company for making their respective PE materials available.
(1.) M.H. Wagner, J. Non-Newtonian Fluid Mech., 4, 39 (1978).
(2.) A.C. Papanastasiou, L.E. Scriven, and C.W. Macosko, J. Rheol., 27, 387 (1983).
(3.) P.R. Soskey and H.H. Winter, J. Rheol., 28, 625 (1984).
(4.) D.A. Vega and S.T. Milner, J. Polym. Sci. Part B: Polym. Phys. 45, 3117 (2007).
(5.) K. Osaki and M. Kurata, Macromolecules, 13, 671 (1980).
(6.) J. Sanchez-Reyes and L.A. Archer, Macromolecules, 35, 5194 (2002).
(7.) C. Sui, G.B. McKenna, and J.E. Puskas, J. Rheol., 51, 1143 (2007).
(8.) D.C. Venerus, J. Rheol., 49, 277 (2005).
(9.) F.J. Stadler, D. Auhl, and H.M. Munstedt, Macromolecules, 41, 3720 (2008).
(10.) D.C. Venerus and H. Kahvand, J. Polym. Sci. Part B: Polym. Phys., 32, 1531 (1994).
(11.) L.A. Archer and S.K. Varshney, Macromolecules, 31, 6348 (1998).
(12.) L.J. Kasehagen and C.W. Macosko, J. Rheol., 42, 1303 (1998).
(13.) T.C.B. McLeish, J. Allgaier, D.K. Bick, G. Bishko, P. Biswas, R. Blackwell, B. Blottiere, N. Clarke, B. Gibbs, D.J. Groves, A. Hakiki, R.K. Heenan, J.M. Johnson, R. Kant, D.J. Read, and R.N. Young, Macromolecules, 32, 6734 (1999).
(14.) H. Gevgilili and D.M. Kalyon, J. Rheol., 45, 467 (2001).
(15.) M. Doi and S.F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, New York (1988).
(16.) K. Osaki, K. Nishizawa, and M. Kurata, Macromolecules, 15, 1068 (1982).
(17.) T. Inoue, T. Uematsu, Y. Yamashita, and K. Osaki, Macromolecules, 35, 4718 (2002).
(18.) L.J. Zapas and T. Craft, J. Res. Natl. Bur. Stand., 69A, 541 (1965).
(19.) L.A. Archer, J. Rheol., 43, 1555 (1999).
(20.) R. Fulchiron and V. Verney, J. Non-Newtonian Fluid Mech., 48, 49 (1993).
(21.) M.H. Wagner and H.M. Laun, Rheol. Acta, 17, 138 (1978).
(22.) J. Hepperle and H. Munstedt, Rheol. Acta, 45, 717 (2006).
(23.) O. Urakawa, M. Takahashi, T. Masuda, and N.G. Ebrahimi, Macromolecules, 28, 7196 (1995).
(24.) K. Osaki, E. Takatori, and M. Kurata, Macromolecules, 20, 1681 (1987).
(25.) D.C. Venerus, E.F. Brown, and W.R. Burghardt, Macromolecules, 31, 9206 (1998).
(26.) P.J.R. Leblans, J. Sampers, and H.C. Booij, Rheol. Acta, 24, 152 (1985).
(27.) CM. Vrentas and W.W. Graessley. J. Rheol., 26, 359 (1982).
(28.) K. Osaki, E. Takatori, M. Kurata, H. Watanabe, H. Yoshida, and T. Kotaka, Macromolecules, 23. 4392 (1990).
(29.) L.J. Fetters, A.D. Kiss, D.S. Pearson, G.F. Quack, and F.J. Vitus, Macromolecules. 26, 647 (1993).
(30.) R.G. Larson, J. Rheol., 29, 823 (1985).
(31.) P.J. Carreau, Trans. Soc. Rheol., 16, 99 (1972).
(32.) K. Yasuda, R.C. Armstrong, and R.E. Cohen, Rheol. Acta, 20, 163 (1981).
(33.) C. Das, N.J. Inkson, D.J. Read, M.A. Kelmanson, and T.C.B. McLeish, J. Rheol., 50, 207 (2006).
(34.) P.M. Wood-Adams. J. Rheol., 45, 203 (2001).
(35.) P.M. Wood-Adams and J.M. Dealy, Macromolecules, 33, 7481 (2000).
(36.) P.M. Wood-Adams, J.M. Dealy, A.W. deGroot, and O.D. Redwine, Macromolecules, 33, 7489 (2000).
(37.) C.D. McGrady and D.G. Baird, J. Rheol., 53, 539 (2009).
(38.) J. Stevens., J. Studies Surf. Sci. Catalysts, 89, 277 (1994).
(39.) J. Stevens, J. Studies Surf. Sci. Catalysts, 101, 11 (1996).
(40.) S.Y. Lai, J.R. Wilson, J.R. Knight, and G.W. Stevens, PCT Int. Appl. (1993).
(41.) P. Wood-Adams and J.M. Dealy, 57th Annu. Tech. Conf. Soc. Plast. Eng., 1, 1205 (1999).
(42.) C.D. McGrady, PhD thesis, Virginia Polytechnic Institute and State University, Blacksburg (2009).
(43.) R.B. Bird, R.C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, 2nd ed., Vol. 1, Wiley, New Jersey, 633 (1987).
(44.) J. Janzen and R.H. Colby, Polym. Mater. Sci. Eng., 82, 128 (2000).
(45.) P.J. Doerpinghaus and D.G. Baird, Macromolecules, 35, 10087 (2002).
(46.) P.J. Doerpinghaus and D.G. Baird,.J. Rheol., 47, 717 (2003).
(47.) S.E. Bin Wadud and D.G. Baird, J. Rheol., 44, 1151 (2000).
(48.) H.M. Laun, Rheol. Acta, 17, 1 (1978).
(49.) B. Elbirli and M.T. Shaw, J. Rheol, 22, 561 (1978).
(50.) D.G. Baird and D.I. Collias, Polymer Processing, Wiley, New York (1998).
Christopher D. McGrady, Christopher W. Seay, Syed M. Mazahir, Donald G. Baird
Department of Chemical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0211
Correspondence to: Donald G. Baird; e-mail: email@example.com
Contract grant sponsor: National Science Foundation; contract grant number: DMR-052198.
Published online in Wiley InterScience (www.interscience.wiley.com).
[C] 2010 Society of Plastics Engineers
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|Author:||McGrady, Christopher D.; Seay, Christopher W.; Mazahir, Syed M.; Baird, Donald G.|
|Publication:||Polymer Engineering and Science|
|Date:||Jul 1, 2010|
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