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Experimental Study and Modeling of Wall Slip of Polymethylmethacrylate Considering Different Die Surfaces.

INTRODUCTION

Polymer melt flow calculation models for extruder screws and dies commonly assume wall adhering conditions, which means that the polymer melt sticks to the wall and the flow velocity at the boundary to the surface is equal to the surface velocity of the wall. This assumption is not always valid in polymer processing. Under certain process conditions, some polymers exhibit slip at the wall.

Different methods are known to determine wall slip velocity. According to Mooney [1], wall slip can be evaluated from data obtained in pressure driven flows with high pressure capillary rheometers or slit dies. It is assumed that the slip velocity is solely a function of the wall shear stress. The reduced volume flow rate or the apparent wall shear rate is drawn against the reciprocal die diameter or die height in a Mooney plot. The slope of the line is the wall slip velocity at a certain wall shear stress. The Mooney plot is assumed to be linear. Geiger [2] proposed an improved evaluation method, which can be applied to non-linear Mooney plots.

Migler et al. [3] used Fluorescence Recovery after Photobleaching (FRAP) to study slip phenomena for polydimethylsiloxane (PDMS) in a sliding plate geometry. Miinsted et al. [4] employed laser-Doppler velocimetry to investigate the slip behavior of linear low density polyethylene (LLDPE) and high density polyethylene (HDPE) in a slit die flow. Apostol et al. [5] studied the effect of moisture and chain extenders on wall slip of polylactidacid (PLA) using pulsed wave sonography with a Y-rheometer in the extrusion process.

Wall slip can be induced in different ways. When processing aids such as slip additives migrate towards the wall they form a thin film of low viscosity between the flow channel surface and the polymer melt. According to J.-M. Piau et al. [6] the polymer may then slip on this thin lubricating layer. The exudation of low molecular weight chains can induce wall slip in a similar way to a slip additive.

Wall slip can also take place under other conditions, where different cases can be distinguished. Hatzikiriakos [7-9] reported that weak slip is induced by desorption of a few chains from the flow channel surface or by a partial disentanglement of the bulk from the molecular layer strongly absorbed to the wall.

Strong slip occurs when the polymer chains are completely disentangled from the molecular layer strongly absorbed to the wall [10-12].

Typical polymers which are known to slip at the wall when a critical wall shear stress is exceeded are polyvinyl chloride (PVC), high molecular weight polyethylene (PE), high density polyethylene (HDPE), linear low density polyethylene (LDPE), branched polypropylene (PP), polybutadiene, polyisobutylene (PIB), polylactides (PLA), fluoropolymers, highly filled polymers and elastomers [8, 13-22].

The die surface can influence wall slip. Ramamurthy [23] reported that beryllium copper as a die material leads to a lower critical wall shear stress for wall slip compared to steel. Seidel et al. [24] studied the influence of steel, brass, PTFE and polyetheretherketone (PEEK) as die material on wall slip and the extrudate surface of a linear low density polyethylene (LDPE). Zitzenbacher et al. [25] investigated the influence of ground steel, polished steel, titanium nitride (TiN), titanium aluminum nitride (TiAIN), diamond like carbon (DLC) and polytetrafluoroethylene (PTFE) on wall slip of a polypropylene blockcopolymer and a high density polyethylene blow moulding grade.

Hatzikiriakos [7] reported the influence of two different fluoropolymer coatings on wall slip and extrudate appearance of a high density polyethylene (HDPE), where one fluoropolymer was an adhesion promoter and the other a slip promoter. Although only the slip promoter caused more wall slip, both coatings improved the extrudate appearance compared to clean walls.

The effect of different kinds of slip additives on wall slip of polymer melts was reported by White et al. [26] and Ahn et al. [27, 28]. Octadecanoic acid and zinc stearate induced the greatest slippage for polyethylene (PE) and polypropylene (PP). Octadecanamide and octadecanol also induced noticeable slippage in PE and PP. Polyamide 12 (PA 12) exhibited no slippage. Slippage occurred for octadecanoic acid and zinc stearate in polymethylmethacrylate (PMMA) and polystyrene (PS). They reported that the level of slippage depends on the ability of the end-groups of the slip additive to interact with the steel walls. It also depends on the polarity of the polymer. Polar polymers such as polyamide have greater adhesion to steel than PE.

When processing PMMA in extrusion technology and injection moulding, an unstable output or the deposition and degradation of the polymer on the surfaces of screws and dies can occur. The effect of processing aids such as slip additives and the proper surface treatment or coating of tools and screws has to be further investigated.

Wall slip of a polymethylmethacrylate extrusion grade on different tool surfaces was studied in this paper to gain a deeper knowledge about the processes taking place between the polymer melt and the surface of the flow channel. This work comprises an experimental and a modelling part. In the experimental part, rheological tests were conducted using a slit die with exchangeable flow channel inserts mounted to a high pressure capillary rheometer. The influence of different flow channel surfaces on wall slip was studied in the tests. The concept of reduced volume flow rate was employed to determine wall slip qualitatively dependent on wall shear stress. Wall slip velocity was evaluated quantitatively as a function of wall shear stress using the Mooney Method for flow channels with a rectangular cross section. Wall slip was obtained on polished steel, but on ground steel and on the DLC-coating no slip was observed.

In the modelling part, a mathematical wall slip model was derived assuming a lubricating film between the polymer and the die surface. The calculated slip velocity has a power law dependency on wall shear stress. Furthermore, wall slip increases with rising temperature. It is shown that the predicted slip behavior correlates well with the experimental results.

EXPERIMENTAL

Methods Used

The rheological experiments were carried out using a high pressure capillary rheometer Rheograph 6000, Goettfert, Buchen, Germany and a rheological slit die. The slit die concept allows a direct measurement of pressure without doing a Bagley-correction. Figure 1 shows a detailed view of the rheological slit die. The main components are two halves of the die body (1, 6), a carrier (2) for the upper flow channel insert (3) and the lower flow channel insert (4). The flow channel inserts (3, 4) allow experiments with different die materials and coatings. Carriers (2) with different depths were used to perform experiments with flow channel heights of 1 and 2 mm. The length of the flow channel is 100 mm and the width is 10 mm.

The melt pressure was measured before the entrance of the die and also with three pressure transducers located 30.5 mm (pi, (9)), 55 mm (p2, (14)) and 81 mm (p3, (10)) away from the die entrance. The measurement of the pressure at three positions allows the evaluation of the pressure distribution along the flow channel. Pressure holes were used because a flush mounting solution of the pressure transducers would entail larger areas of deviating surface compared to the flow channel. A different surface could significantly influence the flow at the wall of the die.

The thermocouples were situated in the inserts 3 mm below the flow channel surface at three positions along the flow channel at the counterface to the pressure transducers.

The rheological experiments were conducted at the temperatures 220[degrees]C, 230[degrees]C and 250[degrees]C. First the polymeric material was put into the reservoir channel, melted and heated up to the defined temperature. The slit die was heated to the same temperature. When extruding the melt, the piston velocity was increased gradually in order to obtain rheological data at different apparent wall shear rate values.

As a first test to detect wall slip the concept of reduced volume flow rate was employed. Rheological experiments at constant temperature with at least two different channel heights H are needed to determine wall slip of a polymer melt qualitatively. The reduced volume flow rate [Q.sub.red] (Eq. 1) has to be plotted dependent on the wall shear stress [[tau].sub.w] for the different flow channel heights H

[Q.sub.red] = Q/W[H.sup.2], (1)

where Q is the volume flow rate and W is the width of the flow channel. The effect of the side walls of the flow channel on the evaluated wall shear stress was taken into account according to Walters [29] (Eq. 2)

[[tau].sub.w] = [DELTA]p x H/2L x [(1 + H/W).sup.-1] (2)

where [DELTA]p is the measured pressure loss and L is the length between the pressure probes. The reduced volume flow rate curves for different flow channel heights are in the case of wall adherence the same. Deviations between these curves indicate wall slip [1, 30].

The wall slip velocity [v.sub.s] was evaluated quantitatively in this paper using a modified Mooney method for flow channels with a rectangular cross section

[Q.sub.red] = [v.sub.s]/H + f([[tau].sub.w]), (3)

where f corresponds to the contribution of shear flow to the total reduced volume flow rate [25]. The reduced volume flow rate has to be evaluated at a certain wall shear stress for different channel heights. Later, these reduced volume flow rate values are plotted dependent on the reciprocal flow channel height at constant wall shear stress and temperature. Then, the wall slip velocity [v.sub.s] at a certain wall shear stress can be calculated to

[v.sub.s] = [Q.sub.red,2] - [Q.sub.red,1]/1/[H.sub.2] - 1/[H.sub.1]. (4)

The surface energy of the flow channel surfaces was determined at room temperature using a contact angle measurement system Kruess DSA30S, Hamburg, Germany. The contact angles of the measuring liquids deionised water, diiodomethane and 1,5-pentanediol were detected employing the sessile drop method. The tests were carried out five times for each pairing of measuring liquid and flow channel insert. Then the dispersive crSidiS and the polar fraction [[sigma].sub.s,pol] of surface energy were evaluated according to Owens et al. [31]

1 + cos [??]/2 x [[sigma].sub.L]/[square root of [[sigma].sub.L,dis]] = [square root of [[sigma].sub.s,pol]] x [square root of [[sigma].sub.L,pol]/ [[sigma].sub.L,dis]] + [square root of [[sigma].sub.s,dis]], (5)

Where [??] is the contact angle, [[sigma].sub.L,dis] is the dispersive fraction of the surface energy of the liquid, [[sigma].sub.L,pol] is the polar fraction of surface energy of the liquid and [[sigma].sub.L] is the total surface energy of the liquid.

The surface roughness of the flow channel inserts was determined by means of a confocal microscope DCM3D, Leica Microsystems, Wetzlar, Germany. Scans were carried out to measure the surface topology at the positions 17.5 mm, 42.5 mm, 68 mm and 90 mm along the flow channel (measured from the inlet of the die) for both inserts. The rectangular measuring window around each position had the dimensions 484 [micro]m x 535 [micro]m. An area weighted surface roughness [S.sub.a] was evaluated from these data using the software Leica Map.

Materials

The experiments were conducted with a polymethylmethacrylate (PMMA) PLEXIGLAS 7M from EVONIK, Darmstadt, Germany, which is suitable for extruded profiles and panels in lighting engineering. PMMA PLEXIGLAS 7M has a melt flow rate (MFR) of 3.45 g [(10 min).sup.-1] at (230[degrees]C/3.80 kg).

Polished and ground X153CrMoV12 steel inserts as well as silicon doped amorphous Diamond Like Carbon (DLC) coated steel inserts were used for the rheological experiments. First, the steel inserts, but not the ground steel, were polished to a surface roughness of app. 14 nm. Then, a heat treatment of the samples was performed to achieve a sufficient high surface hardness to deposit the coating. The DLC coating was produced using a PA-CVD process (plasma assisted chemical vapor deposition). The mean values and standard deviations of the surface roughness and the polar and dispersive fraction of surface energy of the investigated flow channel inserts are shown in Table 1.

RESULTS

Experimental Results

The reduced volume flow rate was determined dependent on wall shear stress for a channel height of 1 mm and 2 mm as a first test to detect wall slip. The reduced volume flow rate for PMMA obtained with polished steel, ground steel, and DLC as die surfaces is shown dependent on wall shear stress in Figs. 2, 4 and 6 in a wall shear stress range between 0 and 0.15 MPa. Figs. 3, 5 and 7 are enlargements of these diagrams for a wall shear stress below 0.05 MPa. The magnitude of the error bars indicates the good reproducibility of the measured values.

With polished steel as the die surface material, the reduced volume flow rate curves for different flow channel heights are not the same. The determined values at a certain wall shear stress are higher at a channel height of 1 mm than at a channel height of 2 mm (Fig. 2). This result is valid for all investigated temperatures, which can clearly be seen in the enlarged diagram (Fig. 3). Moreover, this observation indicates wall slip of the PMMA melt on polished steel. Only slight deviations were obtained between the curves determined at a lower temperature (220[degrees]C), which means that wall slip is low at this temperature. With rising temperature, the differences between the reduced volume flow rate data obtained with different channel heights increase. It is obvious that wall slip of PMMA on polished steel is temperature dependent and shows an increase with rising temperature.

The DLC-coating exhibits similar physical surface properties in terms of roughness (polished steel: 14.3 nm, DLC: 22.3 nm) and surface energy (polished steel: 34.8 mN [m.sup.-1], DLC: 36.3 mN [m.sup.-1]) like polished steel. Thus, it is astonishing that the reduced volume flow rate curves obtained with different flow channel heights are the same (Fig. 4, and the enlarged diagram Fig. 5). The DLC-coated flow channel causes wall adhering of the PMMA melt, whereas polished steel enhances wall slip. Only at a temperature of 250[degrees]C slight deviations between the reduced volume flow rate values obtained with different flow channel heights arise. Due to these low differences compared to the size of the error bars, these deviations are not significant and were attributed to measurement errors.

The ground steel flow channel surface exhibits a significantly higher surface roughness (164.3 nm) compared to polished steel (14.3 nm) and DLC (22.3 nm). When considering the reduced volume flow rate values determined with different flow channel heights it turns out that using the rougher surface a similar effect is obtained as with the DLC-coating (Fig. 6 and the enlarged diagram Fig. 7). The curves are the same which proves the wall adhering conditions of PMMA on ground steel. The diagrams show slight deviations between the reduced volume flow rate curves in the other direction than observed with polished steel and DLC, which means that the reduced volume flow rate data increase slightly with rising channel height. Considering the rheological evaluation method point of view such a result should not occur and is attributed to the influence of measurement errors.

In the next step the wall slip velocity dependent on wall shear stress was evaluated to get more information about wall slip. The evaluation was solely carried out with polished steel because the previously performed qualitative wall slip test indicated that the other die materials show no slip. An example for the Mooney plot used for the evaluation of the wall slip velocity is shown in Fig. 8 at a temperature of 250[degrees]C. The reduced volume flow rate was plotted dependent on the reciprocal flow channel height at constant wall shear stress. The slope of the line was taken as the slip velocity at a certain wall shear stress. The Mooney plot shows an increase in the slope of the lines with rising wall shear stress.

The wall slip velocity data obtained using the Mooney method are depicted dependent on wall shear stress in Fig. 9 with logarithmic scaling for different temperatures. The measurement error was estimated from the errors of the input data using Gaussian law of error propagation. The magnitude of the error bars exhibits an increase with rising wall shear stress.

The wall slip velocity of PMMA reveals an increase with rising wall shear stress and temperature. The maximum wall slip velocity is 0.59 mm [s.sup.-1], 1.89 mm [s.sup.-1] and 6.81 mm [s.sup.-1] at a temperature of 220[degrees]C, 230[degrees]C and 250[degrees]C.

In polymer rheology, a power law expression is often used to approximate slip velocity

[v.sub.s] = A x [[tau].sub.w.sup.B]. (6)

The wall slip velocity data in Fig. 9 were approximated with Eq. 6 using the least squares method. The parameters A, B and the coefficient of determination [R.sup.2] are given in Table 2. The coefficient of determination values of 0.96, 0.98 and 0.99 confirm the good approximation of the wall slip data using Eq. 6.

The wall slip curves exhibit a similar slope at different temperatures, which can be seen in the parallelism of the approximation lines in the diagrams. This is also proved with the power law parameter B, which is 0.44, 0.52 and 0.52 at a temperature of 220[degrees]C, 230[degrees]C and 250[degrees]C. The power law parameter A increases with rising temperature. The wall slip curves are shifted parallel to higher values with increasing temperature, because the distance on the wall slip axis is equal to the logarithm of A.

Wall Slip Model

For this wall slip model the flow of a wall slipping polymer melt in a flow channel with rectangular cross section was considered. The slip additive needs some time to migrate towards the die wall and to establish a lubricating film there which allows the polymer melt to slip. In this model, only the flow region in the die with a lubricating film was considered. Then, the following assumptions and prerequisites for the derivation of the wall slip model were taken into account:

* The calculations were carried out under isothermal and steady state conditions.

* Gravity and inertia forces were neglected.

* A fully developed flow was assumed.

* The flow channel height H is much smaller than the flow channel width W. Thus the flow can be described as the flow between two parallel plates with infinite extension in the lateral direction.

* An incompressible melt was assumed.

* The polymer melt and the lubricant reveal shear thinning behavior.

The simplified equation of motion is

0 = - [partial derivative]p/[partial derivative]z + [partial derivative][[tau].sub.yz]/ [partial derivative]y,(7)

where p is pressure, [[tau].sub.yz] is shear stress, z is the flow direction, and y is perpendicular to the flow direction. Integration of Eq. 7 yields

[[tau].sub.yz] = [partial derivative]p/[partial derivative]z x y+[C.sub.1]. (8)

The shear thinning behavior of the polymer melt (Index P, Eq. 9) and the lubricant (Index L, Eq. 10) is described with a power law model

[mathematical expression not reproducible] (9)

[mathematical expression not reproducible] (10)

where [v.sub.z,P] is the flow velocity of the polymer melt, [v.sub.z,L] is the flow velocity of the lubricant, [[PHI].sub.P] is the fluidity of the polymer melt, [[PHI].sub.L] is the fluidity of the lubricant, [m.sub.P] is the power law exponent of the polymer melt and [m.sub.L] is the power law exponent of the lubricant.

The origin of the coordinate axis is placed in the symmetry plane of the flow channel. Symmetry of the flow yields

[partial derivative][v.sub.z,P]/ [partial derivative]y|[sub.y=0] = 0. (11)

Equation 11 results in

[C.sub.1] =0. (12)

With Eqs. 8, 10, and 12 the differential equation for the flow velocity [v.sub.z,L] of the lubricant is obtained

[mathematical expression not reproducible] (13)

Integration of Eq. 13 results in

[mathematical expression not reproducible] (14)

Wall adherence of the lubricant is described by the boundary condition

[v.sub.z,L] (y = H/2)=0. (15)

The flow velocity of the lubricant can be rewritten considering Eq. 15

[mathematical expression not reproducible] (16)

It is assumed that the slip velocity v, is the velocity in the interface between the lubricating film and the polymer melt

[mathematical expression not reproducible] (17)

where h is half the height of the polymer layer. With Eqs. 8 and 12 the shear stress at the wall [[tau].sub.w] follows and the pressure gradient can be expressed as

[[tau].sub.w] = [partial derivative]p/[partial derivative]z x H/2 [??] [partial derivative]p/[partial derivative]z = 2/H x [[tau].sub.w]. (18)

After introducing Eq. 18 in Eq. 17 and considering

H/2 = h+[delta] (19)

the exact solution for the slip velocity [v.sub.s] is obtained

[mathematical expression not reproducible] (20)

where [delta] is the thickness of the lubricating film. Eq. 20 can be further modified by the approximation

[mathematical expression not reproducible] (21)

which is valid for [delta] much smaller than h. Then Eq. 20 for the slip velocity can be rewritten to

[mathematical expression not reproducible] (22)

The slip velocity [v.sub.s] {Eq. 22) is dependent on the power law parameters [[PHI].sub.L] and [m.sub.L] of the lubricant, the film thickness of the lubricant [delta] and the wall shear stress [[tau].sub.w].

The exact solution of slip velocity (Eq. 20) and the solution with the simplification (Eq. 22) are compared in Fig. 10. The film thickness of the lubricant was [delta] = [10.sup.-4] m, [delta] = [10.sup.-5] m and [delta] = [10.sup.-6] m. Fluidity was set to [[PHI].sub.L] = [10.sup.7] [Pa.sup.-m] [s.sup.-1] and the power law exponent was L = 0.5. For Eq. 22 half the height of the polymer melt layer h = 0.5 mm was assumed.

Both equations show a linear increase in the slip velocity with a higher wall shear stress in the double logarithmic diagram. With increasing thickness of the lubricating film an increase in the wall slip velocity occurs. Only at a film thickness of [delta] = [10.sup.-4] m a slight difference between the evaluation of the exact solution and the solution with simplification can be seen. This result justifies that for the further work Eq. 22 can be used.

Temperature dependency is incorporated by using the exponential law for the temperature shift factor

[mathematical expression not reproducible] (23)

where [T.sub.0] is the reference temperature of the viscosity master curve and T is the actual temperature, [[PHI].sub.L,0] is the fluidity at [T.sub.0] and b is a material specific parameter for the temperature shift of viscosity. Considering temperature dependency of viscosity, the slip velocity [v.sub.s] is

[mathematical expression not reproducible] (24)

It can be seen that with rising temperature an increase in the slip velocity is predicted.

RESULTS AND DISCUSSION

Comparison of the Experimental Results with the Wall Slip Model

As reported in literature [6-12], different mechanisms come into consideration to explain wall slip of this PMMA material on polished steel. Wall slip can be induced by processing aids, the exudation of low molecular weight chains, the loss of adhesion on the die and a disentanglement of polymer chains from a polymer monolayer on the flow channel surface.

PMMA would not be expected to slip because it is a polar polymer, but the slip velocities are still lower than data reported for HDPE [14]. When comparing the experimental results with the obtained wall slip model and taking literature into account, it becomes clear that the studied PMMA slips due to the presence of a lubricant, for which there are several reasons:

1. PMMA is a polar polymeric material which exhibits a greater adhesion to the steel die wall than polymers with low polarity such as PE [26, 32], This stronger bonding to the steel surface usually results in wall adherence.

2. The measured rheological data revealed wall slip even at low wall shear stress values. Other authors [7-10, 13, 24], who observed different slip mechanisms reported the first appearance of wall slip after exceeding a critical wall shear stress value.

3. Generally, the experimentally determined slip velocity data correlate well with the dependencies predicted with the presented slip model in this paper, which considers a lubricating film between the die wall and the polymer melt.

4. The experimentally obtained slip velocity increases with rising wall shear stress. The presented slip model predicts a power law dependency on wall shear stress. The same dependency on wall shear stress was also found for the experimentally determined slip velocity data.

5. The measured slip velocity data exhibit an increase with rising temperature. In the double logarithmic plot of the slip velocity versus wall shear stress the wall slip curves are shifted parallel to higher values. The slope of the curves, which is equal to the power law parameter of wall shear stress, remains constant. This observation is in good agreement with the presented wall slip model, which yields the same influence of temperature on wall slip.

6. With rising temperature a decrease in the viscosity of the lubricating film occurs, which causes better conditions for wall slip. The lower viscosity of the lubricant induces higher wall slip velocities.

Influence of Die Surface on Wall Slip

Generally, it would be assumed that the addition of a lubricant to a polymer would cause wall slip independent of the die material or coating used. The experiments revealed the opposite: that wall slip of PMMA occurred only on polished steel, but ground steel and the DLC coating exhibited no slip.

In order to achieve wall slip, the slip additive must migrate toward the die wall and form a thin lubricating layer between the die surface and the polymer melt. In this process, the slip additive develops bonds to the die wall with the polar end of the molecules. A high polarity and the chemical properties of the die wall are of great importance for this bonding [6, 26].

The amount of slip additive in the polymer used was between 400 ppm and 1500 ppm [33, 34]. This means that the thickness of the lubricating film is rather low.

The ground steel die surface inserts have a rougher surface ([S.sub.a] = 164 nm) than polished steel ([S.sub.a] = 14.3 nm). Due to the small concentration of the slip additive the rough surface cannot completely be filled and no stable and complete lubricating film is developed. The polymer interacts with the peaks of the rough steel surface and no slip conditions can be gained.

The DLC coating exhibits a similar roughness ([S.sub.a] = 22.3 nm) like polished steel ([S.sub.a] = 14.3 nm). The polar fraction of surface energy is the same for polished steel and DLC ([[sigma].sub.s,pol] = 1.9 mN [m.sup.-1]), the dispersive fraction shows only a small difference (DLC: [[sigma].sub.s,dis] = 34.4 mN [m.sup.-1], polished steel: [[sigma].sub.s,dis] = 32.9 mN [m.sup.-1]). The physical surface properties exhibit no significant difference and give no explanation for wall adherence on the DLC coating, but the surface properties were determined at room temperature. At processing temperature, larger differences in the surface energy between DLC and polished steel could arise.

Beside the physical properties, the chemical properties of the die surface are of great importance. It has to be considered that DLC is a metastable form of amorphous carbon containing a significant fraction of [sp.sup.2] and sp3 bonds [35]. These bonds are also mainly found in polymeric materials or in the low polar molecule section of slip additives [27]. As the slip additives usually develop a lubricating layer on the polar steel die surface, they are not able to form a stable film on the low polar DLCcoating. Now, the polymer melt interacts with the die surface and no slip is obtained.

Sanchez-Lopez et al. [36] reported that the doping of DLC with Si causes an increase in the contact angle of water to the surface (more hydrophobic), which means that the polar fraction of surface energy is reduced. They supposed that these elements force carbon into a sp3 bonding state. This observation supports the proposed explanation that slip additives are not able to bond to the DLC surface.

CONCLUSIONS

A new model to describe wall slip of polymer melts including a slip additive is presented in this paper. This model predicts an increase in slip velocity with rising wall shear stress which is described by a power law. The slip velocity increases with rising temperature, but the exponent of wall shear stress is not influenced by temperature. This dependency leads in a double logarithmic plot to a parallel shift of the slip velocity lines to higher values with increasing temperature.

The predicted dependences of the slip model in terms of wall shear stress and temperature are in good agreement with the experimentally determined slip velocity values on polished steel. It can be concluded that wall slip of this PMMA material is induced by a slip additive and slip velocity can be described with the presented wall slip model.

The rheological tests conducted with other die surfaces revealed that processing aids did not induce wall slip independent of the die material used. With Si doped DLC no slip was observed, although the physical surface properties such as roughness and surface energy are similar to polished steel. DLC is a metastable form of amorphous carbon and contains sp2 and sp3 bonds. Si in this DLC coating makes the surface even more hydrophobic, which explains why slip additives are not able to develop a stable lubricating film on this surface.

Whether wall slip can be obtained using a lubricant also dependents on the roughness of the surface. With increasing roughness the surface structure cannot completely be filled with the slip additive and no entire lubricating film is developed. The polymer bonds to the steel peaks of the surface and as a consequence no wall slip is obtained.

As a prospect for future work, it has to be studied if with an increasing concentration of lubricant in the polymer melt wall slip on rough steel surfaces can be obtained.

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Gernot Zitzenbacher (ID), (1) Zefeng Huang, (1) Clemens Holzer (2)

(1) Department of Materials Technology, School of Engineering, University of Applied Sciences Upper Austria, Wels, Austria

(2) Department Polymer Engineering and Science, Chair of Polymer Processing, Montanuniversitaet Leoben, Leoben, Austria

Additional Supporting Information may be found in the online version of this article.

Correspondence to: G. Zitzenbacher; e-mail: g.zitzenbacher@fh-wels.at

DOI 10.1002/pen.24727

Published online in Wiley Online Library (wileyonlinelibrary.com).

Caption: FIG. 1. Cross sectional view of the slit die. Die body (1, 6), carrier (2), flow channel inserts (3, 5), flow channel (4), heater band (7), adjust pins (8, 11, 12, 16), pressure transducer [p.sub.1] (9), pressure transducer [p.sub.2] (14), pressure transducer [p.sub.3] (10), thermocouples (13, 15).

Caption: FIG. 2. Reduced volume flow rate dependent on wall shear stress for two different channel heights at a temperature of 220, 230 and 250[degrees]C with polished steel.

Caption: FIG. 3. Reduced volume flow rate dependent on wall shear stress for two different channel heights at a temperature of 220, 230 and 250[degrees]C with polished steel (enlargement for a wall shear stress below 0.05 MPa).

Caption: FIG. 4. Reduced volume flow rate dependent on wall shear stress for two different channel heights at a temperature of 220, 230 and 250[degrees]C with DLC.

Caption: FIG. 5. Reduced volume flow rate dependent on wall shear stress for two different channel heights at a temperature of 220, 230 and 250[degrees]C with DLC (enlargement for a wall shear stress below 0.05 MPa).

Caption: FIG. 6. Reduced volume flow rate dependent on wall shear stress for two different channel heights at a temperature of 220, 230 and 250[degrees]C with ground steel.

Caption: FIG. 7. Reduced volume flow rate dependent on wall shear stress for two different channel heights at a temperature of 220, 230 and 250[degrees]C with ground steel (enlargement for a wall shear stress below 0.05 MPa).

Caption: FIG. 8. Mooney plot for the evaluation of wall slip velocity on polished steel at a temperature of 250[degrees]C for different wall shear stresses.

Caption: FIG. 9. Wall slip velocity dependent on wall shear stress at a temperature of 220, 230 and 250[degrees]C on polished steel. The lines show the approximation functions of the measured data.

Caption: FIG. 10. Slip velocity dependent on wall shear stress for different lubricating film thicknesses, evaluated with the exact slip velocity equation (Eq. 20, continuous lines) and with the slip velocity equation with approximation (Eq. 22, dashed lines).
TABLE 1. Surface roughness [S.sub.a], surface energy
[[sigma].sub.s], dispersive fraction of surface energy
[[sigma].sub.s,dis] and polar fraction of surface energy
[[sigma].sub.s,polar] of the flow channel inserts including
their standard deviations.

Surface            [S.sub.a] (nm)      [[sigma].sub.s]
                                       (mN [m.sup.-1])

Polished steel   14.3 [+ or -] 2.1    34.8 [+ or -] 0.2
Ground steel     164.3 [+ or -] 9.6   36.3 [+ or -] 0.3
DLC coated       22.3 [+ or -] 2.1    36.3 [+ or -] 0.3

Surface          [[sigma].sub.s,pol]   [[sigma].sub.s,dis]
                   (mN [m.sup.-1])       (mN [m.sup.-1])

Polished steel    1.9 [+ or -] 0.1      32.9 [+ or -] 0.1
Ground steel      6.5 [+ or -] 0.1      29.8 [+ or -] 0.2
DLC coated        1.9 [+ or -] 0.1      34.4 [+ or -] 0.2

TABLE 2. Parameters A and B of the power law expression for
wall slip including the coefficient of determination [R.sup.2].

Temperature                  A                  B     [R.sup.2]
([degrees]C)   (mm [s.sup.-1] [MPa.sup.-B])    (1)       (1)

220                        1.89                0.44     0.96
230                        6.27                0.52     0.98
250                        24.52               0.52     0.99
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Author:Zitzenbacher, Gernot; Huang, Zefeng; Holzer, Clemens
Publication:Polymer Engineering and Science
Article Type:Report
Date:Aug 1, 2018
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