Dynamic Slip and Nonlinear Viscoelasticity.
Several significant problems arise when film is fabricated on a large scale. One of these is the appearance of irregularities on the extrudate surfaces when the polymer melt is extruded at high rates. These irregularities vary in intensity and form and are generally known as sharkskin melt fracture. This phenomenon, which occurs when the wall shear stress exceeds a critical value, is a limiting factor for production rates in many industrial extrusion operations such as film blowing of polyethylene. We used a sliding plate rheometer incorporating a shear stress transducer to study slip in both steady and unsteady flows. By combining a dynamic slip model with a nonlinear viscoelastic constitutive model, we determined the slip model parameters for LLDPE film resin with and without a fluoropolymer sharkskin suppressant. The models give good prediction of our slip data in steady shear but show insufficient gap dependence in exponential shear. Our own film blowing studies demonstrated the efficiency of the sharksk in suppressant; it has more than doubled the throughput in our laboratory setup. The fluoropolymer additive was found to profoundly affect both the steady and dynamic slip parameters. Hence, the sharkskin suppressant alters how the LLDPE remembers its past slipping motions.
In the early 1950s, work with film grade resins such as linear low-density polyethylene was done on small-scale equipment and at low extrusion rates. On this scale, the film properties looked attractive. As greater quantities of such resin became available so that the fabrication of film on a large scale could be undertaken, several significant problems quickly developed. One major problem was the appearance of irregularities on the extrudate surfaces when the polymer melt was extruded at high rates. These irregularities vary in intensity and form and are generally known as sharkskin melt fracture . This phenomenon, which occurs when the wall shear stress exceeds a critical value, is a limiting factor for production rates in many industrial operations such as blown film extrusion of polyethylene.
Considerable research over the past forty years to determine the origin of sharkskin melt fracture has reached no unified conclusion. There are different explanations for the causes of sharkskin melt fracture. According to Howells and Benbow , Cogswell , Kurtz  and Moynihan , the polymer extrudate fractures at the die exit due to an abrupt change in boundary conditions that lead to high stretching rates exceeding the melt strength. The onset of these extrudate distortions is also accompanied by wall slip and failure of adhesion at the polymer-metal interface in the die land .
In this investigation, we study the relationship of sharkskin to wall slip. With the use of dynamic slip models and different constitutive equations such as Wagner's equation and the Liu model, we will determine the slip model parameters for [Dowlex.sup.TM] 2045 with and without processing aids. We hope to get a better understanding of the role of slip in extrusion sharkskin melt fracture by studying how the processing aids affect the dynamic slip behavior. The reviewer added, "while the wall-slip/sharkskin connection is perhaps a viable hypothesis, it is definitely one that needs much more study."
Both linear and nonlinear differential and integral constitutive equations will be studied in this section. The Lodge rubber-like liquid equation is a linear integral constitutive equation 
[[tau].sub.ij] = [[[integral of].sup.t].sub.-[infinity]] m(t - t')[B.sub.ij](t, t') dt' (1)
where [B.sub.ij] is the Finger tensor that has the following form in simple shear
[B.sub.ij](t,t') = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
and m (t - t') is the memory function. The non-zero dynamic normal stresses in the reference state can be eliminated with the strain tensor [[gamma].sub. of Bird et al  which subtracts off the unit tensor. Equation 1 is our defining relation for linear viscoelasticity. It is good for describing behavior for small or slow deformations. Based on Lodge's network theory, the memory function, a linear viscoelastic property is:
m(t - t') = [[sigma].sup.N].sub.i=1] [G.sub.t]/[[lambda].sub.t] exp [- (t - t')/[[lambda].sub.t] (3)
where [G.sub.t] is the relaxation modulus at relaxation time [[lambda].sub.t]. The rubber-like liquid model does not provide a quantitative description of the molten plastic undergoing large and rapid deformations. However, it is a simple equation with all the parameters determined from the linear viscoelastic behavior. The model gives the correct prediction of actual material response at low shear rate or strain amplitude, and any material response deviating from the Lodge rubber-like liquid equation is nonlinear viscoelasticity.
The rubber-like liquid model can be modified so that it fits experimental data more accurately. The most common generalization is to preserve the single integral form and modify the memory function to describe the deformation processes. One useful approach is to let the memory function depend on strain and time. Wagner then proposed the following
[[tau].sub.ij] = [[[integral of].sup.t].sub.-[infinity]] m(t - t') h([[gamma].sub.tt']) [B.sub.ij](t,t') dt' (4)
where h([[gamma].sub.tt]') is the damping function. This nonlinear integral constitutive equation with a factorable memory function is called Wagner's equation (8). When the damping function is unity, we recover the Lodge rubber-like liquid model, which is a good representation of small deviations from linear viscoelasticity.
The upper convected Maxwell model is a linear differential constitutive equation that has the following form in simple shear
[sigma](t) = [[[sigma].sup.n].sub.i=1] [[sigma].sub.t] (5)
[[sigma].sub.t] + [[lambda].sub.t] [delta][[sigma].sub.t]/[[delta]t = [G.sub.t] [[lambda].sub.t] [gamma] (6)
The upper convected derivative, [delta]/[delta]t, in the model makes the material behavior properly independent of translation, rotation, and deformation of the coordinate frame. This model is a differential form of the Lodge rubber-like liquid in Eq 1.
Kinetic network theory models nonlinearize the upper convected Maxwell model by making both the relaxation time and the relaxation moduli functions of a set of structural parameters, [x.sub.i], the ratio of the equilibrium entanglement molecular weight to the molecular weight between entanglement contributing to the ith relaxation time.
[G.sub.t] = [G.sub.oi][X.sub.i] (7)
[[lambda].sub.t] = [[lambda].sub.oi][[X.sub.i].sup.1.4] (8)
where [G.sub.oi] and [[lambda].sub.oi] are the values at no deformation ([x.sub.t] = 1). The structural parameters vary between 0 and 1 and Liu et al  introduced the following kinetic rate equation:
[dx.sub.t]/dt = [k.sub.1](1 - [x.sub.i])/[[lambda].sub.i] - [k.sub.2][x.sub.i] [[[[pi].sub.D].sup.1/2] (9)
where [[pi].sub.D] is the second invariant of the rate of deformation tensor. [k.sub.1] is the kinetic rate constant for thermal regeneration of the entanglements, and [k.sub.2] is the kinetic rate constant for flow induced disentanglement. In simple shear, the Liu model reduces to:
[sigma](t) = [[[sigma].sup.N].sub.i=1][[sigma].sub.t] (10)
[N.sub.1](t) = [[[sigma].sup.N].sub.i=1][N.sub.i] (11)
[[sigma].sub.t] + [[lambda].sub.i] d[[sigma].sub.i]/dt - [[lambda].sub.i][[sigma].sub.i]/[x.sub.i] [dx.sub.i]/dt = [G.sub.i][[lambda].sub.t] [gamma] (12)
[N.sub.i] + [[lambda].sub.i] [d[N.sub.i]/dt - [[lambda].sub.i][N.sub.i]/[x.sub.i] [dx.sub.i]/dt = 2[[sigma].sub.i][[lambda].sub.i] [gamma] (13)
where [sigma] is the shear stress, [N.sub.1] is the first normal stress difference, and [[sigma].sub.i] and [N.sub.i] are their ith spectral components.
When polymeric liquids are subjected to high shear stress, the classical no-slip boundary condition at the solid-liquid interface may not be valid. Instead of having a zero velocity at the wall, the fluid begins to move with a velocity [u.sub.s] (difference between the fluid velocity and the wall velocity) as shown in Fig. 1. The nominal shear rate is related to the actual shear rate by:
[gamma] = [[gamma].sub.n] - [2[u.sub.s]/h] (14)
where [[gamma].sub.n] is the nominal shear rate, [gamma] is the actual shear rate, [u.sub.s] is the slip velocity, and h is the gap. Instabilities and complex time dependence in the flow of polymeric melt are often associated with wall slip. It has been suspected to be the cause of sharkskin melt fracture during extrusion when polymer melts are subjected to high wall shear stress causing slip. Other evidence of slip includes the dramatic increase in flow rate through a capillary die at a critical value of shear stress (6) and the deviation of the stress response from sinusoidal behavior once the critical stress for slip is exceeded in a large amplitude oscillatory shear flow .
The slip phenomenon is often empirically modeled as an algebraic relationship between the slip velocity [u.sub.s] and the shear stress [sigma]:
[u.sub.s] = [phi]([sigma]) 
Specifically, Hatzikiriakos and Dealy  have shown that the following model, called the steady slip model, gives a good fit to their data from steady shear experiments:
[u.sub.s] = O ([sigma] [less than] [[sigma].sub.c]) 
[u.sub.s] = a[[sigma].sup.m] ([sigma] [greater than] [[sigma].sub.c] 
where [[sigma].sub.c] is the critical shear stress, which is defined as the threshold for the onset of melt slip. The slip coefficient a has units of m [s.sup.-1][(MPA).sup.-m] and seems to depend on temperature while the exponent m does not .
The above model assumes that the slip velocity adjusts instantaneously to the shear stress at the wall and this assumption may not be valid. For a smooth transition from the no slip to the slip condition, Pearson and Petrie  introduced the "memory" slip mode or the dynamic slip model:
[u.sub.s] + [[lambda].sub.s][u.sub.s] = [phi]([sigma]) 
This model assumes that the slip velocity is a function of the shear stress history with a relaxation time of [[lambda].sub.s]. It is a dynamic model for the slip and reattachment process. To incorporate the steady state observation into the dynamic slip model, Hatzikiriakos and Dealy  proposed
[u.sub.s] + [[lambda].sub.s][u.sub.s] = a[[sigma].sup.m] 
for steady slip flow, [u.sub.s] is zero and the steady slip model is recovered from the dynamic slip model.
The dynamic slip model in Eq 18 is evaluated at the boundary under the assumption that the fluid velocity is always tangential to the boundary. Renardy  showed that this boundary applied to inertialess shear flow of an upper convected Maxwell fluid leads to an ill-posed problem. Black and Graham  thus proposed the following slip model:
[u.sub.s] = [epsilon] 1 - X/X [sigma] 
where [epsilon] is a constant and [sigma] is the wall shear stress. X is the fraction of available polymer segments that are strongly interacting with the solid surface and has the following evolution equation:
DX/D[theta] = 1/[We.sub.s] [(1 - X) - sX tr [sigma]] 
where [We.sub.s] and s are positive constants. Here the dimensionless time has been scaled with the inverse nominal shear rate, [theta] = t/[[gamma].sub.n] and stress with the largest shear modulus, tr [sigma] = [N.sub.1]/[G.sub.1]. s is the ratio between the dimensionless detachment and attachment rate constants and [We.sub.s] is the dimensionless attachment time constant (slip Weissenberg number):
[We.sub.s] = [[lambda].sub.s][gamma] 
with [[lambda].sub.s] as the characteristic time for the slip process.
Slip, which occurs at high stresses, is always related to nonlinear viscoelastic behavior. So to infer dynamic slip behavior accurately, we must begin with an accurate nonlinear viscoelastic model.
All rheological tests were done on the sliding plate rheometer that was developed at McGill University and was commercialized by the Interlaken Technology Corporation (ITC) . The rheometer is designed for use with molten polymers, concentrated solutions, raw elastomers, cheese and other viscoelastic materials. In this device, the molten sample is subjected to simple shear. Capable of generating high shear rates ranging from [10.sup.-3] to [10.sup.-3] [s.sup.-1] and total shear strains up to 500, the rheometer can measure both linear and nonlinear viscoelastic behaviors. The heart of the system is the patented  shear stress transducer (SST), which uses a noncontact capacitance probe (by Capacitec, Boston). The active face of the transducer is flushed with the wall of the fixed plate to measure the shear stress directly, eliminating the need to indirectly calculate the stress from the applied force and sample contact area. Since the SST is at the center of the sample, far from its edges, edge effects a re avoided. Also, sample life is longer because only the edges degrade.
The rheometer is mounted vertically in an ITC Series 3300 test frame. The moving plate is coupled to a servo hydraulic actuator with the hydraulic pump in an isolated room to keep the work area quiet. The plates are fixed to the rheometer frame. Both the sliding plate and fixed plates are made from type 420 stainless steel and the gap between the plates is set precisely with stainless steel shims. Three gaps were used in our experiments: 0.5, 0.36 and 0.23 mm. Sliding plate flow can be used to study the melt slip phenomenon because it generates a homogeneous flow field with no pressure gradient. The rheometer is mounted in a forced convention oven with an ITC Series 3260 Temperature Controller. The maximum achievable temperature is 300[degrees]C. A deep-well thermocouple is mounted in the fixed plate nearly touching the sample. Another deep-well thermocouple is just beneath the surface of the moving plate and is used as the sensing element for the temperature controller. We used a software called QuikTest to determine the linear and nonlinear viscoelastic properties.
Before we run the test, the plates were cleaned thoroughly by using "Easy Off" heavy-duty oven cleaner to keep them free of any degraded polymer from previous experiments. The plates were then rinsed several times with Ace hardware all-purpose household spray and then with toluene. To make sure no residues were left behind from the solutions, the plates were then wiped clean with a wet cloth. Also, the shear stress transducer was removed from the stationary plate to be cleaned thoroughly before each test. The oven was then heated for 90 min to reach the test temperature of 215[degrees]C.
To study the effect of processing aids on the polymer, we used [Dynamer.sup.TM] 9613, a popular 3M product for suppressing sharkskin in commercial film resins. This fluorocarbon elastomer is in the form of a white powder and was applied to the surface as follows. First the [Dynamar.sup.TM] 9613 was dissolved in acetone to yield a 1% solution (by weight). This solution was let stand overnight and insoluble particles were precipitated. The clarified solution was applied to the horizontal stationary plate after it was cleaned thoroughly as described above. After the solvent had evaporated, the plate was heated to 215[degrees]C and stayed at that temperature for approximately 30 min for the coating to be stabilized before we ran the test.
At the desired temperature, the stress transducer was calibrated. Without any sample in the rheometer, the hydraulic pump was turned on and the sliding plate was then centered at the zero position. Finally, the sample was placed properly onto the stationary plate and the oven was allowed to heat up again for another 45 min for the sample to reach the steady test temperature.
Exponential shear experiments were performed on the sliding plate rheometer for three gaps, 0.23, 0.36 and 0.5 mm at 215[degrees]C. The exponential shear strain is programmed using the QuikTest designer development package:
[gamma](t) = A([e.sup.[alpha]t] - 1) (23)
This flow is of special interest because it tends to generate a high degree of molecular stretching. The strain rate is given by:
[gamma](t) = [alpha][Ae.sup.[alpha]t] (24)
The response is always nonlinear at some value of t because the shear rate increases exponentially. However, both linear and nonlinear viscoelastic information can be obtained from this exponential shear flow. At the beginning of the deformation, when [gamma] is still small and the rate of deformation is still low, we expect the material to display linear viscoelastic properties. As time increases, the response of the polymeric melt begins to deviate from linear viscoelasticity and enter the nonlinear viscoelastic region where it is expected to agree with the Wagner or Liu prediction. At longer time, the shear rate becomes so high that the behavior becomes highly nonlinear. Eventually, the stress reaches a critical value where slip occurs.
Film blowing experiments were performed on the Wayne "Yellow Jacket" Blown Film Tower 6536 with an annular die, manufactured by the Wayne Machine and Die Company in Minnesota. Table 1 shows the die and screw dimensions.
IV. RESULTS AND DISCUSSION
The material used in our study is [Dowlex.sup.TM] 2045, a linear low-density polyethylene film resin for heavy-duty applications, provided by Dow Plastics. This material had also been studied by several other research groups such as Hatzikiriakos et al. , Dealy and Hatzikiriakos , Jeyaseelan and Giacomin , and Tzoganakis and Price . The physical properties of [Dowlex.sup.TM] 2045 are listed in Table 2.
We observed sharkskin melt fracture when the film blowing tower is operated under the conditions listed in Table 3. In general, sharkskin melt fracture is observed when we increased the extrusion rate or decreased the die zone temperature, both of which will cause greater wall shear stress. However, an increase in the extrusion rate on our equipment in turn caused an increase in the die zone temperature, owing to higher shear rate. The parameters on the setup needed to be adjusted carefully before we could observe sharkskin melt fracture.
We repeated the film blowing experiments with 0.1 percent (by weight) of [Dynamar.sup.TM] 9613 added to the [Dowlex.sup.TM] 2045 pellets in the hopper. The extruder is allowed to run for 20 min at 10 rpm to make sure that the equipment is fully conditioned with the additives. The test is carried out under the same conditions listed in Table 3 and we did not observe any sharkskin melt fracture. The extrusion rate is increased to the maximum capacity of our extruder, 50 rpm, and still there is no sign of sharkskin melt fracture.
Dow Plastics provided the small amplitude oscillatory shear data G'([omega]) G"([omega]) of the [Dowlex.sup.TM] 2045 at 215[degrees]. We employed linear regression with regularization to get the discrete relaxation spectra ([G.sub.i],[[lambda].sub.i]) from the equations
G'([omega] = [[[sigma].sup.N].sub.i=1] [G.sub.t][([omega][[lambda].sub.i]).sup.2]/1 + [([omega][[lambda].sub.i])].sup.2] (25)
G"([omega] = [[[sigma].sup.N].sub.i=1][G.sub.t][([omega][[lambda].sub.i])/1 + [([omega][[lambda].sub.i])].sup.2] (26]
The best fit is in Fig. 2 for five relaxation times ranging from 0.01 to 100 s. The discrete relaxation spectra of [Dowlex.sup.TM] 2045 at 215[degrees]C is shown in Table 4.
The steady shear viscosity curve is generated from small amplitude oscillatory shear data using the Cox-Merz rule
[eta]([gamma]) = \[eta]*\ = [square root of][(G'/[omega]).sup.2] + (G"/[omega]).sup.2], [omega] = [gamma] (27)
Figure 3 shows the derived steady shear viscosity curve of [Dowlex.sup.TM] 2045 at 215[degrees]C. Tzoganakis and Price  reported a power law index, n of 0.618 and a consistency index. K of 0.0085 MPa [s.sup.n] for the material. Their values are compared with our Cox-Merz calculation in the power-law region in Fig. 4. A good agreement is achieved between the two steady shear viscosities.
The steady slip model in Eq 17 contains three parameters, a, m and [[sigma].sub.c] that must be determined experimentally. Dealy and Hatzikiriakos  performed extrusion experiments on an Instron piston-driven, constant-speed capillary rheometer and plotted the pressure-corrected slip velocity of [Dowlex.sup.TM] 2045 at 215[degrees]C as a function of the wall shear stress. Using Eq 17, they determined m = 4.7 and a = 2.94*[10.sup.-10] mm [s.sup.-1] [kPa.sup.-m] for the uncoated steel capillary. We independently carried out steady shear viscosity experiments on our sliding plate rheometer to check the two parameters and found that our measurements agreed with theirs.
We used the Mooney analysis outlined by Dealy and Hatzikiriakos  to get the steady slip parameters from our steady shear data for the case when the polymer-wall interface is coated with [Dynamar.sup.TM] 9613. Figure 5 compares the steady shear responses for the three gaps. The data nearly coincide at shear stress below 20 kPa and separate at higher shear stress when slip occurs. The existence of a critical shear stress for the onset of slip implies that the coating ([Dynamar.sup.TM] 9613) does not behave as a liquid lubricant and this agreed with Dealy and Hatzikiriakos . Figure 6 is a plot of shear rate versus reciprocal gap for several shear stresses. Straight lines are obtained, implying that the slip velocity depends only on the shear stress. Figure 7 shows the dependence of the slip velocity on shear stress. Equation 17 is fitted to the data using linear regression giving a = 3.8*[10.sup.-5] mm [s.sup.-1] [(MPA).sup.-m] and m = 2.53 (within 95% confidence limit). Compared with the results of Dea ly and Hatzikiriakos , a increases considerably and m decreases slightly when the polymer-wall interface is treated with [Dynamar.sup.TM]. This agrees with the findings of Tzoganakis and Price . Figure 8 compares the slip velocity for both cases and we concluded that [Dynamar.sup.TM] 9613 promotes slip during extrusion.
Figure 9 shows the exponential shear response of the three gaps for the case when the strain scale factor, A is 0.076 and the exponential rate constant, [alpha] is 5.0. The Lodge rubber-like liquid equation is solved and the result is compared with the three responses. For exponential shear, the critical shear stress for the onset of slip is around 60 kPa. By onset, we mean there is a difference of approximately 2.5 kPa detected between the shear responses of the three gaps. The Lodge rubber-like liquid accurately predicts the response at the low shear strain region where linear viscoelasticity still holds.
To study the effect of processing aid on slip, we repeated the above exponential shear experiments with the polymer-wall interface coated with [Dynamar.sup.TM] 9613. Figure 10 shows the shear responses for the three gaps. With [Dynamar.sup.TM] added, the critical shear stress for the onset of slip in exponential shear occurs at 72 kPa. Although the slip analysis could in principle be carried down to lower stresses, the proximity of the curve below 70 kPa introduces too much noise.
The damping function in the Wagner equation is obtained from single step shear strain experiments, a method that has been most widely used to study nonlinear viscoelastic behavior. Step shear strain experiments with magnitude ranging from 2.0 to 10.0 were performed on our rheometer to determine the damping function of [Dowlex.sup.TM] 2045 at 215[degrees]C. For a step shear strain of magnitude [gamma], the shear stress with a separable function is given by:
[sigma](t, [gamma]) = [gamma]h ([gamma]) G(t) (28)
By definition, the nonlinear relaxation modulus is:
G(t, [gamma]) = h([gamma]) G(t) (29)
where G(t) is the linear relaxation modulus and h is the damping function. The stress responses. [sigma](t, [gamma]) were divided by [gamma] to obtain the nonlinear stress relaxation moduli. Each test is carried out on a new 0.36 mm thick specimen and repeated once (with a fresh sample) to show reproducibility. In Fig. 11, we superimposed the nonlinear relaxation modulus, G(t, [gamma]) of each step strain onto one plot together with the linear relaxation modulus, G(t) from the generalized Maxwell model. The linear modulus is larger than all the nonlinear moduli so that h [less than or equal to] 1.0 and as the step strain magnitude increases, the nonlinear modulus shifted down and move further away from the linear modulus indicating that the damping function decreases too. To calculate the damping function, we picked the time to be 0.32 s and divided all the nonlinear modulus by a corresponding damping factor ([less than]1.0) so that they shifted up to coincide with the linear modulus curve. Finally the dampin g factor is plotted against its step strain magnitude to obtain the damping function shown in Fig. 12. The damping function is unity and has a zero slope at zero shear strain. It also decreases as shear strain increases, inflects at a finite strain, vanishes at large strain, and easily fits the equation proposed by Soskey and Winter :
h([gamma])=1/1 + [[0.153.sub.[gamma]].sup.1.665] (30)
To obtain the kinetic rate constants, we solve the Liu model in steady shear and viscosity equation simultaneously for various ratio of [k.sub.2]/[k.sub.1]
[x.sub.i] = [[1 + [[k.sub.2]/[k.sub.1] [[lambda].sub.oi][gamma] [[x.sup.1,4].sub.i]].sup.-1] (31)
[eta][gamma] = [G.sub.ot][[lambda].sub.oi][[X.sup.2.4].sub.i] (32)
The results are compared to the steady shear viscosity curve in Fig. 13. Both kinetic rate constant ratios of 0.35 and 0.4 give good fits to the steady shear viscosity curve. To obtain individual values of the [k.sub.1] and [k.sub.2], we solve the Liu model with exponential shear simultaneously and compare the results to the exponential shear response in Fig. 14. Equally good fits are obtained for [k.sub.1] = 0.3 with [k.sub.2]/[k.sub.1] = 0.35 and for [k.sub.1] = 0.25 with [k.sub.2]/[k.sub.1] = 0.4 respectively in the no-slip region. The first case over-predicts the shear response while the latter case under-predicts the shear response in the slip region. The Liu model is supposed to over-predict the exponential shear response at high shear region because it has not yet taken slip into account. Hence the correct kinetic rate constants ratio, [k.sub.2]/[k.sub.1] is 0.35 and [k.sub.1] is 0.3. The above example illustrates that both steady shear viscosity and exponential shear are essential in determining the k inetic rate constants.
Before we determine the dynamic slip coefficient, we would like to see how the constitutive models compare to our exponential shear data without taking slip into account. The solution to the Lodge rubber-like liquid equation (upper convected Maxwell model), Wagner's equation, and the Liu model are compared with the measured values in Fig. 15. Both Lodge and Maxwell model gives the exact prediction and they describe the low shear region of the data accurately. Both the Wagner and Liu models are able to predict the response accurately even when the response departs slightly from the linear viscoelastic region. However, at the high shear region, the Wagner equation does not give the correct shape and it underpredicts the response as the shear stress increases. The Liu model gives the correct shape and the over prediction worsens with the shear stress. This is due to the increasing amount of slip in the high shear region. We concluded that the Liu model works best for the exponential shear of [Dowlex.sup.TM] 204 5. This model will be used to determine the dynamic slip coefficient.
To obtain the dynamic slip coefficient using a nonlinear constitutive model, we solved the Liu model with dynamic slip (Eq 19) for exponential shear with and without [Dynamar.sup.TM]. We made calculations with different initial guess values of dynamic slip coefficient, [[lambda].sub.s], for each gap. The best curve fit for the three gaps in each case is presented in Figs. 16 and 17. The waviness in the stress is caused by slip-stick. The Liu model predicts the responses accurately for 0.5 and 0.36 mm gaps but overshoot the data for 0.23 mm gap with and without [Dynamar.sup.TM], The model also gives a higher critical shear stress and strain for the onset of slip than we observed. The dynamic slip coefficient reported is 0.1 s and it increases to 0.5 s when the [Dynamar.sup.TM] is added. The Liu model shows that the addition of [Dynamar.sup.TM] increases the [[lambda].sub.s].
Searching for a closer fit, the Black-Graham slip model is tested on our steady shear data.
[u.sub.s] = [epsilon] S(tr [[tau].sub.w]) [[tau].sub.w] (33)
(tr [[tau].sub.w]) = 2[[gamma].sup.2] [sigma] [G.sub.i] [[[lambda].sup.2].sub.t] [[x.sup.3,4].sub.i] (34)
[[tau].sub.w] = [gamma] [sigma] [G.sub.t] [[lambda].sub.t] [[x.sup.2,4].sub.i] (35)
The above equations are solved for various values of [epsilon]s and compared to our measured values. Figure 18 is a comparison of the Black-Graham (Eq 33) and Dynamic slip model (Eq 19) to the steady shear data with [Dynamar.sup.TM] for [epsilon]s 2.55*[10.sup.2] mm [kPa.sup.-1][s.sup.-1]. Both models give equally good curve fits at the low shear region and predict the critical shear stress for the onset of slip correctly. Individual value of [epsilon] or s and [[lambda].sub.s] must be known before we can test the Black-Graham slip model on our exponential shear measurements. We solved the Liu's equation using the Black-Graham slip model for exponential shear with several guess values of s and [[lambda].sub.s]. The best fit is given by [epsilon] = 0.5 and [[lambda].sub.s] = 0.5 s in Fig. 19. The Black-Graham and dynamic slip model are compared in Fig. 20 for exponential shear. Both models give insufficient gap dependence results.
The Liu model with dynamic slip gives accurate steady shear response but does not predict enough gap dependence in dynamic shear. To improve the dynamic slip model, we tried adding a parameter by raising the time-derivative in the model (Eq 19) to a power p:
[u.sub.s] + [[lambda].sub.s][([du.sub.s]/dt).sup.p]= a[[sigma].sup.m] (36)
This modified model preserves the steady shear properties and alters the dynamic response. The Liu model is solved with Eq 36 for p = 0.5 and several guess values of [[lambda].sub.s] Figure 21 compares both original and modified dynamic slip model and we do not get better gap dependence in the exponential shear response with the modified model.
Both the steady shear viscosity and exponential shear are essential in determining the kinetic rate constants for the Liu model. Without slip in the calculation, the Wagner equation does not give the correct shape as our exponential data at the high shear region and it under-predicts the response as shear stress increases. The Liu model gives the correct shape and the amount of over prediction increases with the shear stress due to the increasing amount of slip. Thus, the Liu model works best for the exponential shear of [Dowlex.sup.TM] 2045. The critical shear stress for the onset of slip for exponential shear is around 60 kPa and it increases to 72 kPa with the addition of [Dynamar.sup.TM].
Using the Liu model, both Dynamic and Black-Graham slip models give equally good fits to the steady shear data and predict the critical shear stress for the onset of slip correctly. The Liu model also predicts the exponential responses accurately for the gaps of 0.5 and 0.36 mm. However, the models give a higher critical shear stress and strain for the onset of slip than is observed, and they give insufficient gap dependence. [Dynamar.sup.TM] increases the slip relaxation time, [[lambda].sup.s], from 0.1 s to 0.5 s. The critical extrusion rate for sharkskin melt fracture is more than doubled by the [Dynamar.sup.TM] slip suppressant.
We thank the 3M Company and the office of University-Industry Relations of the University of Wisconsin Madison for the financial support of this work. We acknowledge Professor Mike D. Graham for his helpful suggestion.
(*.) To whom correspondence should be addressed.
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Die and Extruder Screw Dimensions. Screw Diameter (in.) 1 Barrel Length to Diameter ratio 24:1 Outer Die Diameter (in.) 0.513 Inner Die Diameter (in.) 0.501 Die Land Length (in.) 0.751 Physical Properties of [Dowlex.sup.TM] 2045 at 215[degrees]C. Melt index 1.0 g/10 min. Density 0.92 g/cc Weight average molecular weight, [M.sub.w] 118 000 g/mol Polydispersity, [M.sub.w]/[M.sub.n] 3.93 Power law constant, n 0.618 Consistency index, K 0.0080 [MPa.s.sup.n] Operating Conditions of the Film Blowing Tower. Die Extruder Extruder Speed Pressure Temp. ([degrees]C) Temp. ([degrees]C) (rpm) (psi) 127 150 30 3654 Discrete Relaxation Spectrum of [Dowlex.sup.TM] 2045 at 215[degrees]C. [[lambda].sub.i](S) [G.sub.i](kPa) 0.01 176.927 0.1 25.141 1 2.698 10 0.175 100 0.006767
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|Author:||LAN, SEON KEAT; GIACOMIN, A. JEFFREY; DING, FAN|
|Publication:||Polymer Engineering and Science|
|Date:||Feb 1, 2000|
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