Transient Modeling of Viscosity.
"The mountain flowed before the lord" (prophetess Deborah, 1200 B.C.). As Reiner stated, everything flows; it just needs enough time , The viscosity of polymer melts is strongly dependent on their temperature and deformation history , Many different constitutive models have been developed to describe the shear rate dependence of the viscosity for generalized Newtonian fluid (GNF) fluids including Newtonian , power-law as defined by de Waele-Ostwald [4-7], Cross [7, 8], Carreau , Ellis , Dekke , and others. The temperature dependence of the viscosity can then be separately described with a model such as Arrhenius, Williams, Landel, and Ferry (WLF), and others [14, 15].
The Cross-WLF model is named after a combination of two constitutive models: a "Cross" model that describes the shear rate dependence and a "WLF" model that describes the temperature dependence. This model has become widely implemented in various process simulations due to its intuitive form and excellent predictive capability across a relatively wide range of processing conditions. The reference viscosity, [eta] 0, provides the value of the viscosity at a specified temperature, T, and zero shear rate. Many fluids demonstrate a transition from this Newtonian viscosity to a shear thinning or powerlaw behavior at higher shear rates. The transition occurs at a critical shear stress, [[sigma].sup.*], above which the slope of the viscosity is described by a power law index, n, such that the viscosity is modeled as:
[eta](T, [??]) = [[eta].sub.0](T)/1 + ([[eta].sub.0](T)[??]/[[sigma].sup.*]) (1)
The temperature dependence of the viscosity is now most often estimated with the WLF model  using the concept of the time-temperature superposition [16, 17]. This theory states that a master relation exists across a broad range of temperatures and shear rates in which the shift factor, [a.sub.T], for WLF temperature dependence is described as:
[a.sub.T] (T, [??]) = exp ([A.sub.1](T - [T.sup.*]/[A.sub.2]+(T - [T.sup.*])) (2)
where the reference temperatures [T.sup.*] and [A.sub.2] correspond to the glass transition and critical temperature offset for the fluid. Given the temperature dependence of the viscosity, the zero shear rate viscosity, [[eta].sub.0], at a specified temperature, T, is then described relative to the fitted reference viscosity, [D.sub.1], as [[eta].sub.0](T)=[D.sub.1][a.sub.T](T).
The viscoelastic behavior of fluids has been long recognized; Isayev  aptly states that "[a] basic characteristic of polymer systems is their relaxation spectrum, which determines all manifestations of their viscoelastic properties." From first principles, Maxwell showed that imposed stresses in a viscoelastic material should decay at an exponential rate . According to Maxwell, a solid body free from viscosity will exhibit a stress, [sigma], proportional to strain, [epsilon], wherein [sigma]=E[epsilon] for a coefficient of elasticity, E. Any changes in the strain will cause a proportional change in the stress, such that d[sigma]/dt = E d[epsilon]/dt. If the body is viscous, the stress will not remain constant but will tend to disappear at a rate depending on the value of stress and the nature of the body. Maxwell proposed that this rate of relaxation is proportional to the imposed stress, such that:
d[sigma]/dt = E d[epsilon]/dt - [sigma]/[lambda] (3)
where [lambda] is a characteristic relaxation time. For a constant applied strain, the stress relaxation is:
[sigma] = E[epsilon][e.sup.-t/[lambda]] (4)
with the viscosity defined as [eta] = E[lambda] in the presence of a steady flow, that is, constant d[epsilon]/dt.
Since Maxwell's early work, there have been many constitutive models developed to describe viscoelastic behavior, including both differential and integral formulations , Of the integral type models, the Kaye-Bernstein, Kearsley and Zapas (K-BKZ) model [21, 22] with has been widely implement with various memory functions such as the PSM (Papanastasiou, Scriven and Macosko ). The K-BKZ provides broad control the shear and elongational behavior, but with relatively high complexity and number of coefficients. The Wagner model [24-26] is another widely used model providing an analogy between elasticity of rubber and flow of viscous polymers. Researchers have investigated fundamental mechanisms of nonlinear viscoelasticity in different regimes. For example, Yao and Zatloukal models have developed models for different deformation/flow modes [27, 28]. Yao  proposed a non-Newtonian fluid model including separate relaxation processes to treat material stain and rotation with good fitting to observed rheological properties of polymeric flow. Zatloukal  showed that chain branching of LDPE causes the strain hardening in both uniaxial and planar extensional viscosities with the maximum viscosity shifting higher strain rates with increasing temperature.
To summarize the literature and capabilities of the prevailing differential viscoelastic models, the following general form is adapted from Osswald and Rudolph :
[mathematical expression not reproducible] (5)
where [sigma] and [??] are the stress and rate of deformation tensors in which the subscripts J and C, respectively, refer to the Jaumann and convected derivatives defined as:
[[sigma].sub.J] = D[sigma]/Dt = D[sigma]/Dt - 1/2 ([omega] x [sigma] - [sigma] x [omega]) [[sigma].sub.C] = D[sigma]/Dt + ([([nabla]u).sup.T] x [sigma] - [sigma] x ([nabla]u)) (6)
for a velocity gradient, [nabla]u, and a vorticity tensor, [omega], defined as [nabla]u - [([nabla]u).sup.T]. The coefficients Y and [[lambda].sub.0,i,...,4] may be selected to model various viscoelastic behaviors; Table 1 provides some definitions corresponding to different well-known constitutive models. Peters et al.  suggest that of these differential constitutive models, the Phan-Thien-Tanner [31, 32] (PTT) and Giesekus [33-35] models have been applied most frequently. The Giesekus model is based on a kinetic theory analysis of closely packed polymer chains, with a key characteristic being the inclusion of only one adjustable parameter to control nonlinearity of the stress. The PTT model has two adjustable parameters with the use of the mixed upper and lower convected derivatives, and thus provides some ability to describe shear and elongational properties independently. The White-Metzner model  is also notable for its usefulness given its simplicity with a nonlinear relaxation time, [[lambda].sub.1], that is modeled as the shear rate dependent viscosity, [eta]([??]), divided by the shear modulus, G.
Generally, there is a tradeoff between the model complexity (with increasing number of coefficients) and the types of viscoelastic behaviors that can be explained. There have been many studies comparing the performance of the various models. Markovitz  provided an early comparison of the behavior of viscoelastic models. Spriggs et al.  evaluated eleven differential and nine integral rheological models for viscoelastic fluids against representative experimental data. Dhanasekharan et al.  compared the PTT, Giesekus, Leonov, and WhiteMetzner models finding that all models had spurious predictions in certain regimes, with the White-Metzner model best replicating the observed behavior. A team of 22 researchers led by Mackley  considered the usefulness of viscoelastic constitutive models with respect to the simulation of exemplary polymer processing applications concluding "the choice of the appropriate constitutive equation that captures the correct form of the nonlinear response is still an open debate. In addition, the way in which nonlinear rheological data are obtained over a wide range of strains and strain rates is also still a very open experimental question."
EXPERIMENTAL CHARACTERIZATION AND ANALYSIS OF TRANSIENT FLOW
Determination of the type of test, with a suitable range of temperatures and shear rates, is critical in the characterization of polymer melts. The fidelity of constitutive models will subsequently depend on the validity of the test assumptions as well as the correlation of the model topology to that of the observed behavior. However, rheologists have observed that there can be a significant transient response in the settling of the viscosity when transitioning between shear rates. For example, Fig. 1 characterizes the typical lack of correlation between the Cross-WLF model, fit to the apparent shear rates, and the transient rheological data observed in 0.2 s increments; this rheology data were generated for a neat, low density polyethylene (LDPE XSH 7071, Dow Chemical) having a melt flow index (MFI) of 0.3 g/10 min at 200[degrees]C for a capillary die with a length of 30 mm and a diameter of 1 mm. The coefficient of determination, [R.sup.2], is 0.871. The vertical excursions from the Cross-WLF model correspond to a transient change in the apparent viscosity with a shear rate change. A "better" fit can be obtained by fitting the rheological model to only the last data point at a given shear rate is used for rheological characterization as an assumption of the steady state response, yet this standard practice ignores the true transient behavior and does not ensure that a steady state has been achieved.
In previous research , auxiliary instrumentation and numerical simulations were implemented in order to investigate sources of variation in capillary rheometry. One significant source is the polymer melt's compressibility, which is typically not considered in the calculation of the apparent shear rate in standard rheological analysis. Figure 2 provides the programmed apparent shear rates as a function of time during the rheological characterization of Fig. 1 along with the estimated apparent shear rate with compressibility effects. To analyze the effect of compressibility, the specific volume of the polymer melt in the rheometer barrel was modeled using the Tait equation , wherein changes in the specific volume as a function of pressure can cause the shear rate in the capillary to vary significantly from the apparent values. As shown in Fig. 2, the melt compressibility not only induces a delay, but also reduces the range of shear rates across the test. The differences between the incompressible steady shear rate profile (filled dots) and the compressible shear rate profile (open circles) are large and indicate that to better model the apparent viscosity as a function of apparent shear rate, one must take into consideration the compressibility. The variation in the shear rates at low values is due to the quantization of the plunger displacement coupled with small electrical noise on the pressure instrumentation, and cannot be filtered without inducing false transient modeling behavior.
If compressibility is not considered, then the viscosity and shear stress are modeled as a function of stepped apparent shear rates. Figure 3 provides the observed and predicted shear stresses for the apparent shear rate data plotted in Figs. 1 and 2; the coefficient of determination, [R.sup.2], between the predicted and observed shear stresses is 0.773 when incompressibility is assumed. The model predicts a step change in the shear stress given the step change in the apparent shear rate. The observed shear stress, however, has a much slower response and does not, in fact, reach steady state at any of the conditions between 50 and 250 s. As a result, Fig. 3 implies that the steady state and incompressible assumptions are invalid at shear rates below 100 [s.sup.-1], even with extended hold times.
The prior research  also found viscous heating and pressure dependence to be significant contributors to the variations in capillary viscometry. As the polymer melt flows through the capillary, viscous dissipation results in melt temperature increases. This viscous heating could reduce the melt viscosity and contribute to systematic, transient behavior. Accordingly, the viscous heating analysis implemented a transient, lumped capacitance model with four heat transfer sources including: (1) input convective heat flow, (2) internal heat generation due to viscous heating, (3) heat conduction to the di.e., and (4) output convective heat flow. The results indicated that viscous heating was shown to cause a significant temperature increase of up to 7[degrees]C, yet was not found to explain the observed transient behavior in capillary viscometry.
In addition to compressibility and viscous heating, the pressure dependence of the viscosity was modeled using Macedo's approach  where the reference temperature [T.sup.*] in the WLF model was shifted with pressure according to a pressure sensitivity coefficient, [beta]:
[T.sup.*] = [T.sup.*.sub.P.0] + [beta]P (7)
One benefit of the use of Macedo's approach with Eq. 7 is that the pressure sensitivity coefficient, [beta], is actually the same coefficient used in the double-domain Tait equation that shift the transition temperature as a function of pressure. Thus, it is available from the characterization of the pressure-volume-temperature behavior. When the pressure dependence of the polymer melt is modeled along with compressible flow and viscous heating, the resulting shear stresses are modeled as shown in Fig. 4. It is observed that the model now mimics the transient behavior of the observed shear stress, albeit with significant scatter. The fit may appear to be quite good, and the implemented model explains ~93% of the observed variation in the rheological data. Yet part of the behavior (7%) remains unexplained. Consequently, the current work seeks to demonstrate that the observed transient behavior in the viscosity data is due to polymer melt stress relaxation.
PROPOSED CONSTITUTIVE MODEL AND TRANSIENT ANALYSIS
The simplest possible viscoelastic analysis is sought that captures the transient shear stress behavior observed in Fig. 3. The approach here is to extend Maxwell's model of linear viscoelasticity to a nonlinear analysis through the use of numerical simulation that incorporates a constitutive model describing the relaxation time as a function of applied stress. Maxwell  proposed a theory of linear viscoelasticity by suggesting that the rate of relaxation is proportional to the applied stress according to Eq. 3. For transient analysis of the dynamic rheological data, an incremental solution of the stress is sought that allows for ongoing analysis of varying strain. Accordingly, multiplying Eq. 3 by the time constant, /, provides:
[lambda] d[sigma]/dt = [lambda]E d[epsilon]/dt - [sigma] (8)
For a simple shear flow, the rate of change of strain with respect to time is the shear rate. The viscosity, [eta], can also be substituted according to Maxwell's definition of viscosity as the product of the modulus and the time constant, E[lambda]. The resulting model is:
[lambda] d[sigma]/dt = [eta][??] - [sigma] (9)
Simple rearrangement provides an incremental formulation of the stress:
[mathematical expression not reproducible] (10)
The described transient analysis thus implements linear viscoelasticity when [lambda] is a constant, or nonlinear viscoelasticity with [lambda] as a function of the stress or other states. A salient feature of this analysis is that it should be capable of modeling the shear stresses observed during a dynamic rheological experiment wherein the material will tend to relax even as new stresses are applied. The model is fully consistent inasmuch as it supports pure stress relaxation without flow, shear stress development from a relaxed state, and steady shear flow. The governing model of Eq. 10 can also implement a variety of constitutive models for the viscosity and the relaxation time. For the current research, the Cross-WLF model has been adopted to describe the viscosity while a power law model has been adopted to model the relaxation time as a function of the applied shear stress.
For a shear driven flow, the numerical solution is obtained by integrating the differential equation at a given time step, [t.sub.k]:
[mathematical expression not reproducible] (11)
applying Euler's forward method  provides a numerical approximation of the modeled stress, [??], as:
[mathematical expression not reproducible] (12)
where [[eta].sub.k] and [[??].sub.k] are the viscosity and shear rate at a given time, [t.sub.k].
Many researchers have hypothesized that stress relaxation is fundamentally due to molecular and, thus, strain relaxation [45-48]. The described incremental formulation and numerical solution advantageously provides for the use of different rheological models for the viscosity, arbitrarily imposed shear rate profiles, and (most importantly) relaxation times, [lambda] (t). The relaxation time function takes many forms, with the most widely used including the Generalized Maxwell Model (GMM) in which sets of relaxation pairs ([G.sub.i], [[tau].sub.i]) are defined across the relaxation spectrum [49, 50]:
[[lambda].sub.i] = [[eta].sub.i]/[G.sub.i] (13)
Models of the relaxation time have also been defined and validated as a power-law function of the modeled storage modulus, [??] [51, 52]:
[mathematical expression not reproducible] (14)
as well as an exponential function of the modeled reversible extension, [[??].sub.R] :
[lambda]([[??].sub.R])=[C.sub.1]exp(-[C.sub.2] x [[??].sub.R]) (15)
In both Eqs. 14 and 15, the hat accent ([??]) denotes that the modulus and reversible extension are modeled and not directly observable states. In this research, the relaxation time is defined as a function of the observed stress, [sigma]:
[mathematical expression not reproducible] (16)
This form is similar to that of Eq. 14 based on the storage modulus, but the stress is directly observable and so this implementation reduces the number of simplifying assumptions. Furthermore, each of the above constitutive models for the relaxation time has been implemented, with relaxation time model of Eq. 16 providing the most stable and accurate results.
The rheological model coefficients including the apparent viscosity, [eta], and the relaxation time, [tau], may be fit by minimizing an error norm such as the sum of the squared error of the predictions, SSE, between the modeled and observed stresses across m time steps:
SSE=[m.summation over (k=1)][([??]([t.sub.k])-[sigma]([t.sub.k])).sup.2] (17)
A log transformation of the stress' prediction error may be used in Eq. 17 to provide a more uniform weighting across shear stress observations having very different orders of magnitude. Constrained optimization algorithms such as the interior point method  or sequential quadratic programming  may then be employed to minimize the SSE by determining the best fitting model coefficients. Regression statistics such as the coefficient of determination or standard prediction error quantify the fidelity of different rheological models fit to the rheological dataset.
RESULTS OF CONSTITUTIVE MODEL FITTING TO TRANSIENT RHEOLOGICAL DATA
The results for the predicted shear stress using the viscoelastic model with incompressible and isothermal flow assumptions are provided in Fig. 5. It is observed that the predicted shear stress closely models the observed behavior, explaining 99% of the observed variation in the rheological data. There are some discrepancies, with the curvature of the model not quite matching that of the observed shear stress. Notably, there is a significant discrepancy between the observed and predicted shear stress at 225 s, when the shear rate begins to increase. It is unclear from this graph if the discrepancy is due to errors in modeling the relaxation response, or if the other factors (compressibility, viscous heating, and pressure dependence) are responsible.
The next logical step was composing a model that includes the described transient analysis as well as all the corrections: compressibility, pressure dependence, and viscous heating. Figure 6 depicts this model's behavior. While the coefficient of determination, [R.sup.2], has increased only slightly to 0.993, there are some significant differences. The transient behavior of each shear rate step is correctly modeled along with the final value. Furthermore, the highly transient shear stress at 20 and 270 s (corresponding to shear rates of 1,000 [s.sup.-1]) are also now modeled.
Figure 7 plots the modeled viscosity as a function of the shear rate for the viscoelastic model with the corrections for compressibility, pressure dependence, and viscous heating (model #4 in Table 2). Figure 7 uses the same underlying rheology data as Fig. 1. However, the shear rate distribution differs from that of Fig. 1 due to the modeling of the polymer's compressibility and its effect on the flow rates through the capillary. As a result, the viscosity predictions for each observation now closely track the observed transient viscosity, with the most significant variation at the start of the test (starting at bottom right of Fig. 7). The dashed trace represents the Cross-WLF model for purely viscous flow with the fit coefficients. Clearly, the fidelity of the implemented transient model is excellent.
RESULTS AND DISCUSSION
Comparison to Linear Viscoelasticity
Parallel-plate viscometers provide for dynamic rheological testing through the application of an oscillatory strain to a sample, a capability that does not exist with the capillary rheometry viscometer. During the dynamic test, a resulting sinusoidal stress is measured as a function of the input strain, and the viscous and elastic properties of the sample are measured simultaneously. For viscoelastic materials, the stress can be separated into two components: an elastic stress that is in phase with strain, and a viscous stress that is 90[degrees] out of phase with strain. In the same manner, the viscosity, which is calculated as the ratio between the stress and the strain rate, can be separated into two components: an in-phase viscosity or dynamic viscosity, [eta]':
[eta]' = [[eta].sup.*] * sin([delta]) (18)
and an out-of-phase viscosity, [eta]" :
[eta]" = [[eta].sup.*] * cos([delta]) (19)
where [[eta].sup.*] is the complex viscosity and ([delta]) is the phase shift. The complex viscosity is defined as:
[[eta].sup.*] = [G.sup.*]/[omega] (20)
where [G.sup.*] is the complex shear modulus and [omega] is the frequency.
In order to validate the transient analysis, further characterization was performed on parallel plate rheometer, ARES G2 by TA Instruments (New Castle, DE). Dynamic oscillatory shear tests are common in rheology and have been used to investigate a wide range of materials including polymer melts. Generally, Small Amplitude Oscillatory Shear (SAOS) tests have become the acknowledged method for probing the linear viscoelastic properties. In terms of the experimental input, SAOS requires the user to select appropriate ranges of strain amplitude ([[gamma].sub.0]) and frequency ([omega]).
In order to analyze time dependent behavior of the polymer melt and validate the viscosity model, a frequency sweep test method was implemented. During the frequency sweep test, successive measurements across a range of frequencies were recorded while the magnitude of the temperature and oscillatory strain were held constant. Ideally, the selected strain should be within the linear viscoelastic region of the sample, an important fact that will be discussed in due course. The analysis of the parallel plate data was analyzed with both the Trios software provided by the instrument manufacturer. Figure 8 plots the storage and loss moduli for the LDPE tested by the parallel plate rheometer for a sample specimen having a diameter of 25 mm and a thickness of 3 mm with a strain of 1% at 200[degrees]C. The tangent of the phase angle, [delta], shows the decrease in the loss modulus to the elastic modulus at higher oscillatory frequencies. It can be seen that as the oscillatory frequency declines, the storage and loss modulus decrease due to polymer's time-dependent behavior.
The transient data from the parallel plate rheometer was also analyzed by the described transient analysis. For the purpose of relating the transient analysis to the conventional linear viscoelasticity model, each set of cycles at a specified frequency was fit to provide a pair of (viscosity, relaxation time) values. This analysis corresponds to a linear viscoelastic model with the shear modulus, G, defined as the viscosity divided by the relaxation time.
Figure 9 depicts the observed shear stresses, during a frequency sweep test, which have been plotted as a function of shear rate for 13 different frequencies. The frequency sweep proceeds from lower to higher frequencies that correspond to the progression from the inner to outer ellipsoids in Fig. 9; the last few ellipsoids are labeled. In addition, the predicted stress-strain that was modeled with the transient analysis has been superimposed on the acquired transient data. The transient model captures the linear viscoelastic behavior with high fidelity; the coefficient of determination, [R.sup.2], exceeds 0.999 for every one of the oscillatory frequencies.
The model coefficients output from the conventional linear viscoelastic analysis and the described transient analysis are provided in Table 2. It is observed that the two sets of results have similar values and trends. The results are not identical, but are actually not expected to be. The reason is that the storage modulus in the linear viscoelastic analysis is computed as the cosine of the phase angle times the stress divided by the strain:
G'=[sigma]/[epsilon] cos [delta] (21)
while the relaxation time [lambda] is computed as the complex viscosity, [[eta].sup.*], divided by the complex modulus, [G.sup.*], which equates to simply the reciprocal of the parallel plate's angular velocity, [omega]:
[eta] = [[eta].sup.*]/[G.sup.*] = 1/[omega] (22)
By comparison, the transient analysis outputs a direct estimate of the viscosity and relaxation time, so (for comparison) the shear modulus, G, is computed as the viscosity divided by the relaxation times.
G = [eta]/[lambda] (23)
Figure 10 plots the modulus as a function of the relaxation time for the two sets provided in Table 2. Figure 10 also provides the shear modulus defined per Eq. 23 for the transient capillary data. It is observed that the relaxation times and modulus from the transient analysis of the capillary data have a magnitude and trend like those computed for the parallel plate data albeit with some observable variances. Most significantly, the calculated shear moduli for the capillary and parallel plate data diverge at lower relaxation times, which correspond to higher shear rate regimes. This divergence is likely due to strain hardening effects [e.g., 28] in capillary modeling at high shear rates and accumulated strains than occur in parallel plate geometries with finite imposed strain. Still, it is believed that the transient analysis is truly modeling the viscoelastic behavior occurring in the capillary rheometer.
Analysis of Variance
Table 3 provides the Cross-WLF model coefficients and coefficients of determination for four different rheological models including (1) GNF fluid assuming isothermal, incompressible, and steady state conditions; (2) GNF fluid with non-isothermal, compressible, and transient conditions; (3) viscoelastic fluid with isothermal and incompressible but transient conditions; and (4) viscoelastic fluid with non-isothermal, compressible, and transient conditions. In Table 3, SE represents the standard error between the observed shear stresses and the predicted shear stresses for each model while the coefficient of determination, [R.sup.2], defines the fraction of observed rheological variation that is explained by the rheological model:
[R.sup.2] = 1 - [SS.sub.res]/[SS.sub.obs] (15)
where for each observation of the shear stress, [[tau].sub.i], and model prediction, [f.sub.i], the sum of squares of the residuals and observed behavior are, respectively:
[SS.sub.res] = [summation] [([[tau].sub.i] - [f.sub.i].sup.2] [SS.sub.obs] = [summation][([[tau].sub.i] - [bar.[tau]]).sup.2] (16)
Inspecting the results of Table 3, it is observed that the power law index (n), critical shear stress ([[tau].sup.*]), and reference viscosity ([D.sub.1]) vary somewhat with the type of model and assumptions. The temperature sensitivity coefficient ([A.sub.1]) is within the exponential function of Eq. 2 so the modest changes listed in Table 3 are quite significant with respect to modeling behavior; the modeling of viscous heating and viscoelasticity both indicate a potential increase in the temperature sensitivity of the polymer than conventional model fitting with the Cross-WLF model. While the magnitude of D1 (the reference viscosity index) changes by an order of magnitude, the temperature sensitivity coefficient Al also changes. As such, the product of DI and the shift factor, which is a function of Al, remain consistent across the various models. Perhaps most interestingly, the inclusion of the compressibility and transient behavior increases the temperature sensitivity coefficient and critical shear stress. The fact that the viscoelastic model yields coefficients that are self-consistent with those of the purely viscous models and does not shift the coefficients dramatically increase the confidence that the viscoelastic model is capturing the true stress relaxation in the polymer's flow. The viscoelastic model's power law coefficients ([C.sub.1] and [C.sub.2]) were similarly valued regardless of assumptions related to viscous heating, compressibility, and pressure dependence.
Increasing the modeling complexity as in model (2) that relaxes the assumptions, or model (3) incorporating viscoelastic terms, or model (4) their combination consistently resulted in improving the model fidelity as measured by either the standard error (SE) or the coefficient of determination ([R.sup.2]). The modeling of the corrections for compressibility, pressure dependence, and viscous heating roughly halves the standard error, to 28,500 Pa. The viscoelastic model, without the corrections, has a standard error of 10,600 Pa while the viscoelastic model with the corrections has a standard error of 8,930 Pa.
Twenty four sets of models were fit according to a blocked, full-fractional design of experiments (3 X .[2.sup.3]) to ensure that the factors governing the various model types and assumptions were balanced , meaning that the effect of each contributing factor may be assessed without confounding of other factors. Here, the coefficient of determination, [R.sup.2], was used as the response for analysis of variance (ANOVA, ) since the [R.sup.2] is itself a measure proportional to the modeled rheological variation of interest in this application. Accordingly, a multiway analysis  of variance regression was implemented with the results provided in Table 4 for each investigated source of variance including their first order interactions. For presentation purposes, "Relax" corresponds to the proposed relaxation model of Eq. 11 to 17, "Comp" corresponds to modeling of compressibility, "Pres" corresponds to modeling of pressure dependence, and "Heat" corresponds to modeling of viscous heating effects.
The F values define the ratio of the amount of variation explained by each modeling factor relative to the unexplained variation; a higher F value indicates that the factor is more likely to be statistically significant. The right-most column of Table 4 provides the probability that the effect is not significant as computed from the cumulative density function for the F-distribution. The results indicate that all the modeled factors are significant at the 95% confidence level. Compressibility is known to effect the flow rate in capillary rheometers as shown by Hatzikiriakos and Dealy [59, 60], a result that is verified here with an F value above 8,000. However, modeling of viscoelastic relaxation is statistically more significant, with an F value exceeding 16,000. These F values are extremely high, suggesting that the underlying modeled phenomena are occurring.
The existence of relaxation in melt flow has long been understood [61-63]. However, many practitioners have failed to appreciate its magnitude and potential significance. If not taken into consideration, the relaxation phenomenon in capillary rheological characterization may lead to incorrect decisions for materials, products, or processes. The fact that the observed stress was recorded by two different rheometers and generated by different test methods, yet provide similar coefficients when analyzed by the same viscoelastic model, validates the hypothesis that relaxation behavior is an important factor in transient rheological testing and should be taken into consideration during experimental work and subsequent analysis to achieve high fidelity models. Errors in rheological characterization are most likely to cause problems in applications, such as thin wall molding and blown film, that are pushing the limits of the material and constrained by the capabilities of the machinery. In such applications, transient modeling of the viscosity can provide improved model coefficients for improved guidance regarding the feasibility and optimality of the material, product, and process.
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Amir Moshe, (1) David O. Kazmer, (1) Margaret J Sobkowicz-Kline, (1) Stephen P. Johnston, (1) Shmuel Kenig (2)
(1) Plastics Engineering Department, University of Massachusetts Lowell, Lowell, Massachusetts 01854
(2) Plastics Engineering Program, Shenkar College of Engineering and Design, Ramat-Gan, Israel
Correspondence to: A. Moshe; e-mail: email@example.com
Published online in Wiley Online Library (wileyonlinelibrary.com).
Caption: FIG. 1. Apparent viscosity as a function of apparent shear rate. Note the significant variance related to the transient shear rate. The Cross-WLF model fits the observed data (zero average error) but does not capture the transient behavior.
Caption: FIG. 2. Shear rate hysteresis loop used in capillary testing; dots represent apparent shear rate set-points while circles represent the estimated shear rates by modeling the melt compressibility.
Caption: FIG. 3. Observed shear stress as a function of time throughout the hysteresis test. The polymer exhibits a significant transient response given the step change in shear rates plotted in Fig. 1. The purely viscous Cross-WLF model predicts shear stresses having a step response in accordance with the applied apparent shear rates. The coefficient of determination, [R.sup.2], between the observed stress and the modeled stress is 0.773.
Caption: FIG. 5. Observed shear stress as a function of time including the modeled stress predicted with viscoelasticity but without compressibility, pressure dependence, and viscous heating. This viscoelastic model, without corrections, suggests that the transient response is due to viscoelasticity. The coefficient of determination, [R.sup.2], between the observed stress and the modeled stress is 0.990.
Caption: FIG. 6. Observed shear stress as a function of time including the modeled stress predicted with viscoelasticity, compressibility, pressure dependence, and viscous heating. This viscoelastic model, without corrections, suggests that the transient response is due to viscoelasticity and other physics. The coefficient of determination, [R.sup.2], between the observed stress and the modeled stress is 0.993.
Caption: FIG. 7. Shear rate hysteresis loop used in capillary testing; dots represent apparent viscosity assuming shear rate compressibility while stars represent the modeled viscosity predicted with viscoelasticity, compressibility, pressure dependence, and viscous heating.
Caption: FIG. 8. Storage and loss moduli for the LDPE tested by parallel plate rheometer for a sample specimen having a diameter of 25 mm and a thickness of 3 mm. The tangent of the phase angle, 5, shows the decrease in the loss modulus to the elastic modulus at higher. [Color figure can be viewed at wileyonlinelibrary.com]
Caption: FIG. 9. Observed shear stress as a function of rate from parallel plate testing (dots). The dashed lines correspond to the predicted shear stress values using the described transient analysis for a first order linear viscoelastics model with (viscosity, relaxation time) fit to each oscillatory frequency.
Caption: FIG. 10. Modulus as a function of the relaxation time for the conventional parallel plate analysis (open circles) providing the storage modulus, transient analysis of the same parallel plate data (solid circles), and the transient analysis of the capillary data (triangles).
TABLE 1. Constitutive model structures. Constitutive model Y Generalized Maxwell 1 Upper Convected Maxell 1 Corotational Maxwell 1 Convected Jeffrey 1 Corotational Jeffrey 1 White-Metzner 1 Phan-Thien-Tanner 1+[epsilon][lambda]tr (Linear) ([sigma])/[eta] Phan-Thien-Tanner exp (-[epsilon][lambda]tr (Exponential) ([sigma])/[eta]) Giesekus 1 Leonov 1 Proposed 1 Constitutive model [[lambda].sub.0] [[lambda].sub.1] Generalized Maxwell 0 0 Upper Convected Maxell 0 [[lambda].sub.1] Corotational Maxwell [[lambda].sub.0] 0 Convected Jeffrey 0 [[lambda].sub.1] Corotational Jeffrey [[lambda].sub.0] 0 White-Metzner 0 [eta]([??])/G Phan-Thien-Tanner 0 [lambda] (Linear) Phan-Thien-Tanner 0 [lambda] (Exponential) Giesekus 0 [[lambda].sub.1] Leonov [[lambda].sub.0] [[lambda].sub.1] Proposed [(C/tr([sigma])).sup.B] 0 Constitutive model [[lambda].sub.2] [[lambda].sub.3] Generalized Maxwell 0 0 Upper Convected Maxell 0 0 Corotational Maxwell 0 0 Convected Jeffrey 0 0 Corotational Jeffrey 0 [[lambda].sub.3] White-Metzner 0 0 Phan-Thien-Tanner [lambda][xi]/2 0 (Linear) Phan-Thien-Tanner [lambda][xi]/2 0 (Exponential) Giesekus 0 [alpha][[lambda].sub.1]/ [[eta].sub.0] Leonov 0 1/2 Proposed 0 0 Constitutive model [[lambda].sub.4] Generalized Maxwell 0 Upper Convected Maxell 0 Corotational Maxwell 0 Convected Jeffrey [[lambda].sub.4] Corotational Jeffrey 0 White-Metzner 0 Phan-Thien-Tanner 0 (Linear) Phan-Thien-Tanner 0 (Exponential) Giesekus 0 Leonov 0 Proposed 0 TABLE 2. Relaxation spectrum (modulus, relaxation time pairs) from parallel plate testing with comparison of the described transient conventional linear viscoelastic analyses. Transient analysis Frequency (Hz) [eta] [lambda] G 0.133 2.49 E + 04 1.04 2.39 E + 04 0.215 2.08 E + 04 0.677 3.07 E + 04 0.350 1.69 E + 04 0.430 3.93 E + 04 0.560 1.41 E + 04 0.282 5.00 E + 04 0.509 2.05 E + 04 0.324 6.33 E + 04 0.829 1.65 E + 04 0.207 7.93 E + 04 1.09 1.44 E + 04 0.161 8.96 E + 04 2.25 1.02 E + 04 0.0841 1.22 E + 05 2.48 1.20 E + 04 0.0804 1.49 E + 05 3.17 1.18 E + 04 0.0665 1.77 E + 05 5.07 0.965 E + 04 0.0437 2.21 E + 05 5.57 1.148 E + 04 0.0417 2.73 E + 05 6.05 1.48 E + 04 0.0427 3.45 E + 05 Linear viscoelasticity Frequency (Hz) G' [lambda] 0.133 9.97 E + 03 1.19 0.215 1.38 E + 04 0.742 0.350 1.79 E + 04 0.455 0.560 2.44 E + 04 0.2844 0.509 3.12 E + 04 0.313 0.829 4.20 E + 04 0.192 1.09 7.03 E + 04 0.145 2.25 8.98 E + 04 0.0705 2.48 1.12 E + 05 0.0641 3.17 1.12 E + 05 0.0506 5.07 1.43 E + 05 0.0315 5.57 1.85 E + 05 0.0286 6.05 2.39 E + 05 0.0263 TABLE 3. Different model coefficients and statistics for capillary data of Fig. 1. Model 1 2 Transient? No Yes Viscoelastic? No No Compressible? No Yes Pressure dependent? No Yes Viscous heating? No Yes N 0.257 0.271 [[tau].sup.*] (Pa) 4.35 E + 04 4.75 E + 04 [D.sub.1] (Pa s) 1.37 E + 17 1.48 E+ 18 [A.sub.l] 34.7 36.7 [beta] (Pa) 0 2.39E-07 [C.sub.1] 0 0 [C.sub.2] 0 0 [SE.sub.stress] (Pa) 5.02 E+ 04 2.85 E + 04 [R.sup.2] 0.773 0.927 Slope 1.000 1.000 Model 3 4 Transient? Yes Yes Viscoelastic? Yes Yes Compressible? No Yes Pressure dependent? No Yes Viscous heating? No Yes N 0.257 0.259 [[tau].sup.*] (Pa) 4.35 E + 04 6.82 E + 04 [D.sub.1] (Pa s) 6.04 E + 17 1.37 E + 18 [A.sub.l] 36.0 37.3 [beta] (Pa) 0 2.39E-07 [C.sub.1] 3.08 E + 05 1.76 E + 05 [C.sub.2] 2.80 2.04 [SE.sub.stress] (Pa) 1.06 E+ 04 8.93 E + 03 [R.sup.2] 0.990 0.993 Slope 1.000 1.000 TABLE 4. Analysis of Variance (ANOVA) for the models' coefficients of determination. Factor Sum_Sq. d.f. Relaxation 0.12305 2 Compressibility 0.03045 1 Pressure Dependence 0.00007 1 Viscous Heating 0.00002 1 Relax*Comp 0.01210 2 Relax* Pres 0.00013 2 Relax*Heat 0.00010 2 Comp*Pres 8.71 x [10.sup.-10] 1 Comp*Heat 0.00006 1 Pres*Heat 0.00401 1 Error 0.00003 9 Total 0.1660 23 Factor F Prob>F Relaxation 16,361.51 9.48 x [10.sup.-17] Compressibility 8,096.44 1.31 x [10.sup.-14] Pressure Dependence 18.49 0.0020 Viscous Heating 6.25 0.0338 Relax*Comp 1,608.34 3.20 x [10.sup.-12] Relax* Pres 17.33 0.0008 Relax*Heat 13.31 0.0020 Comp*Pres <0.01 0.9882 Comp*Heat 14.77 0.0039 Pres*Heat 1.78 0.215 Error Total
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|Author:||Moshe, Amir; Kazmer, David O.; Sobkowicz-Kline, Margaret J.; Johnston, Stephen P.; Kenig, Shmuel|
|Publication:||Polymer Engineering and Science|
|Date:||Oct 1, 2017|
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