Stick−Slip Dynamics of Entangled Polymer Liquids - American

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Stick-Slip Dynamics of Entangled Polymer Liquids Tien T. Dao† and Lynden A. Archer*,‡ Department of Chemical Engineering, Texas A&M University, College Station, Texas 77840, and School of Chemical Engineering, Cornell University, Ithaca, New York 14853-5201 Received August 9, 2001. In Final Form: January 10, 2002 Stick-slip dynamics of polymer liquids sheared between aluminum, stainless steel, and R-brass substrates are investigated using model 1,4-polybutadiene (PBD) melts with a broad range of molecular weights 6.7 × 104 e Mn e 5.15 × 105. A variable gap, 50 µm e H e 750 µm, planar-Couette shear flow apparatus equipped with a centrally mounted shear force transducer is used to produce uniform steady shear flow in the polymers and to measure shear stresses over a constant area near the shear platen’s center. The same instrument is used to quantify extrapolation lengths b from steady shear flow measurements at multiple gaps. Experiments performed using aluminum (Al) and stainless steel substrates reveal dramatic transitions from simple shear flow to stick-slip dominated flows at shear stresses near a critical value σ* ≈ 0.26 ( 0.02 MPa. The stick-slip regime is evidenced by asymmetric transient stress oscillations that at first observation possess periods close to the longest relaxation times of polymers studied. At higher shear rates, stress oscillations become more pronounced and their frequency increases nearly in proportion to the applied shear rate. At even higher rates, the oscillations disappear altogether and are replaced by constant, steady-state shear stresses that are substantially lower than those observed prior to the onset of the stick-slip flow regime. Several features of the stick-slip transition, including complete reversibility of the effect and a nearly quadratic dependence of extrapolation length on polymer molecular weight, appear consistent with expectations for apparent slip by flow-induced disentanglement of surface adsorbed and bulk polymer molecules. Planar-Couette flow experiments performed using polished R-brass substrates reveal very different physics, however. These experiments, for example, show no evidence of the stick-slip transition observed with Al and stainless steel; they in fact yield steady-state shear stresses that are consistently about 2 orders of magnitude lower, and with no noticeable oscillatory character. Oxidation of the R-brass substrates at elevated temperatures dramatically increases steady-state shear stresses and yields transient stresses that bear a weak signature of the stick-slip process. Both sets of results are discussed in terms of interfacial slip mechanisms for polymer liquids at fluid/solid boundaries.

Introduction After nearly 2 decades of active investigation by several groups, the consensus is that unlike simple liquids, polymeric fluids can and do violate the no-slip boundary condition.1-14 In some cases the slip violations appear at already developed polymer/solid interfaces and therefore require sufficiently large shear stresses to overcome interactions between polymer and substrate.1-3,12-16 This type of slip has been termed true slip because relative motion between an adsorbed liquid polymer and solid substrate occurs, in direct violation of the no-slip condition. In other situations slip appears to occur at a polymer/ † ‡

Texas A&M University. Cornell University

(1) Vinogradov, G. V.; Gotasov, V. P.; Dreval V. E. Rheol. Acta 1984, 23, 46. (2) Schowalter, W. R. J. Non-Newtonian Fluid Mech. 1988, 29, 25. (3) Denn, M. M. Annu. Rev. Fluid. Mech. 1990, 22, 13. (4) Brochard-Wyart, F.; deGennes, P. G. Langmuir 1992, 8, 3033. (5) Hatzikiriakos, S. G.; Dealy, J. M. J. Rheol. 1991, 35, 497; 1992, 36, 703. (6) Migler, K. B.; Hervet, H.; Leger, L. Phys. Rev. Lett. 1993, 70, 287. Leger, L.; Hervet, H.; Massey, G.; Durliat, E. J. Phys.: Condens. Matter 1997, 9, 7719. (7) Ajdari, A.; Brochard-Wyart, F.; deGennes, P. G.; Leibler, L.; Viovy, J. L.; Rubinstein, M. Physica A 1994, 104, 17. (8) Henson, D. J.; Mackay, M. E. J. Rheol. 1995, 39, 359. (9) Wang, S.-Q.; Drda, P. Macromolecules 1996, 29, 4115. (10) Mhetar, V. R.; Archer, L. A. Macromolecules 1998, 31, 6639. (11) Black, W. B.; Graham, M. D. Phys. Rev. Lett. 1996, 77, 956. (12) Hill, D. A. J. Rheol. 1998, 42, 581. (13) Joshi, Y. M.; Lele, A. K.; Mashelkar, R. A. J. Non-Newtonian Fluid Mech. 2000, 89, 303. (14) Denn, M. M. Annu. Rev. Fluid Mech. 2001, 33, 265. (15) Ghanta, V. G.; Riise, B. L.; Denn, M. M. J. Rheol. 1999, 43, 435. (16) Mhetar, V. R.; Archer, L. A. Macromolecules 1998, 31, 8607, 8617.

polymer interface and has been termed apparent slip because the tangential velocity of fluid in molecular contact with the substrate is that of the substrate, as required by the no-slip condition.4,7,9-10,17 In yet other situations apparent slip violations may occur as a result of less effective momentum transport between a low viscosity, polymerdepleted layer of fluid that remains attached to the substrate and a bulk highly viscous polymer solution.10,18 The characteristics of slip and location of the plane of slip in polymer liquids are undoubtedly complicated functions of polymer structure (segmental scale chemistry, architecture, molecular weight, molecular weight distribution, solution concentration, etc.) and physicochemical properties (e.g., surface energy, roughness, and composition) of the substrate at which the slip violations occur. To control where and how slip violations occur in polymers, it is essential that each of these effects be isolated and their fundamental role understood. This article focuses on the role of polymer molecular weight and substrate/ polymer interactions on stick-slip dynamics of entangled polymer liquids. Our work is motivated by an earlier study by Ramamurthy19 in which the effect of various die materials (aluminum, R-brass, bronze, carbon steel, and stainless steel) on spontaneous surface roughening of polymers (sharkskin and melt fracture) was investigated in capillary rheometers and in blown film extrusion processes. In the capillary rheometer study, Ramamurthy observed mild extrudate surface roughening for all die material types (17) Awati, K. M.; Park, Y.; Weisser, E.; Mackay, M. E. J. NonNewtonian Fluid Mech. 2000, 89, 117. (18) Paul, D. R.; Southern, J. H. J. Appl. Polym. Sci. 1975, 19, 3375. (19) Ramamurthy, A. V. J. Rheol. 1986, 30, 337.

10.1021/la0112662 CCC: $22.00 © 2002 American Chemical Society Published on Web 03/01/2002

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investigated. However, in the case of blown film extrusion experiments, sharkskin could not be detected in extrudates processed over R-brass surfaces following an “induction time” of 30-45 min. Ramamurthy contended that the induction period was required to oxidize the die, which he thought promoted stick-slip flow because oxidation of R-brass increases adhesion between polymer and substrate. Similar behavior was later observed by Ghanta et al. using R-brass dies in a capillary rheometer.15 In that study, only extruder dies precleaned with Unipurge (a mixture of polymer and abrasive particles) under a blanket of nitrogen showed the sharkskin suppression capability reported by Ramamurthy. Ghanta et al. explained their observation by suggesting that the cleaned brass extruder die enhanced adhesion between polymer and substrate, reducing slip at the die walls. Recently, Mhetar and Archer reported “direct” evidence of a transition from weak to strong slip in entangled 1,4polybutadiene melts sheared between silica glass substrates in a planar-Couette drag flow device.16 Using a combination of microtracer particle velocimetry and total shear force measurements, the authors observed a remarkable sequence of flow dynamics ranging from stick, to time-dependent stick-slip, and finally to slip-dominated flows, as shear rate was increased. Mhetar and Archer also observed that when the glass substrates were grafted with long entangled 1,4-polybutadiene molecules, asymmetric stress oscillations associated with stick-slip dynamics became quite pronounced and contributed about 10% of the measured steady-state shear stress. On the other hand, these authors reported that when the same polymers were sheared over a glass substrate grafted with a monolayer of a low molecular weight perflorinated silane, slip velocities a factor of about five times higher were observed at all rates, and no evidence of stick-slip flow detected. Extrapolation lengths over the fluorinated substrates were also found to be at least an order of magnitude lower than expected for slip of an entangled polymer over nonadsorbing substrates.16 Mhetar and Archer explained their observations using a model for stick-slip based on disentanglement/re-entanglement of surface adsorbed and bulk polymer chains sheared above a critical stress σ* ≈ Ge/Ne1/2, where Ge is the plateau modulus of the bulk polymer and Ne is the number of monomer units spanning a single entanglement.4,7,10 The objective of the present study is to fundamentally understand stick-slip dynamics of entangled polymer melts near metallic substrates. We therefore study slip and stick-slip dynamics in a planar-Couette flow device (Figure 1) comprised of metallic shearing platens with known roughness and surface energy. The chosen instrument is advantageous because it allows stick-slip dynamics to be studied in a setting free from effects of pressure on polymer compressibility17 and in a setting where errors due to common-line (three-phase line of contact) motion,20 secondary flow,21 and viscous heating can be minimized. A centrally mounted, fixed-area shear force transducer developed by Dealy et al.22 and a rigid low aspect ratio (gap/width) planar-Couette shear flow device is used in the study to investigate stick-slip dynamics in 1,4-polybutadiene melts sheared between aluminum, stainless steel, and R-brass substrates. The instrument facilitates shear stress measurements under controlled shear rate conditions at gaps ranging from 0.75 (20) Dhori, P. K.; Giacomin, J. A.; Slattery, J. C. J. Non-Newtonian Fluid Mech. 1997, 71, 215. (21) Mhetar, V. R.; Archer, L. A. J. Rheol. 1996, 40, 549. (22) Dealy, J. M.; Doshi, S. R.; Bubic, F. R., U.S. Patent 1991.

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Figure 1. Schematic of the planar-Couette shear flow apparatus used in the study. Table 1. Molecular Characteristics and Rheological Properties of 1,4-Polybutadiene Melts Used in the Study sample

Mn

Mw/Mn N/Ne η0 (Pa‚s)

PBD67 PBD86 PBD129 PBD173 PBD253 PBD315 PBD650

6.73 × 8.65 × 104 1.29 × 105 1.73 × 105 2.53 × 105 3.15 × 105 5.15 × 105 104

1.04 1.04 1.03 1.03 1.04 1.04 1.05

31 40 60 81 117 145 239

7.53 × 1.88 × 104 7.45 × 104 2.03 × 105 3.51 × 105 1.07 × 106 7.30 × 106 103

GN (Pa)

τd (s)

7.87 × 0.022 7.82 × 105 0.058 5 8.25 × 10 0.22 8.51 × 105 0.57 8.69 × 105 0.97 8.36 × 105 3.07 8.35 × 105 22.38 105

mm to 50 µm, allowing slip extrapolation lengths to be quantified under a variety of flow conditions. Experimental Section Materials. Various narrow molecular weight distribution (MWD) polybutadienes (PBD) (1,4 addition >90%) were purchased from Polymer Source, Inc. Weight-averaged molecular weights of the polymers used in the study ranged from 6.73 × 104 to 5.15 × 105. The entanglement molecular weight Me ) 1900 ( 100 g/(mol) for dominantly 1,4-polybutadiene, so the chosen materials provide a correspondingly wide selection of entanglement densities 31 e M/Me e 238 for studying slip violations in polymers. Rheological properties of each material were characterized using a Paar Physica universal dynamic spectrometer (UDS200) equipped with cone-and-plate fixtures (15-25 mm diameter, 1° gap angle), and are summarized in Table 1. Three different metallic substrates were selected for the study: aluminum, R-brass, and stainless steel. Prior to performing planar-Couette shear flow measurements using these substrates, each material was subjected to a polishing protocol designed to equalize their surface roughness. A Stylus profilometer (Federal Surface Analyzer 5000) was used to quantify root mean squared roughness of the three materials. Root mean square roughness values obtained using this procedure (aluminum, 0.3 ( 0.1 µm; R-brass, 0.3 ( 0.1 µm; and stainless steel, 0.4 ( 0.1 µm), verify the near-constancy of surface roughness achieved. These values bracket mean surface roughness of the optical quality amorphous silica glass planes (root mean square roughness ∼10 nm) and roughened silica glass substrates (root mean square roughness ∼5 µm) used to study apparent slip violations in 1,4-polybutadiene melts by Mhetar and Archer.16 In that study the authors reported a 2-fold reduction in the critical stress for the onset of strong (rheologically significant) slip violations for an almost 1000-fold increase in surface roughness, indicating that slip violations in the flow regime of interest are only weakly affected by substrate roughness, at least on the scales considered here. Minimum wetting surface energies of the substrate materials were estimated from dynamic contact angle measurements (CAHN Instruments DCA 120) using three different liquids (water, glycerol, and ethylene glycol) with known surface energies.

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A procedure proposed by Owens and Wendt,23 was used to estimate the surface tension for each substrate from the measured interfacial energies and the known liquid surface energies. Substrates used for the contact angle measurements were subjected to repeated cycles of cleaning with dichloromethane (methylene chloride) and toluene prior to the measurements. This procedure yielded a minimum wetting surface energy for aluminum of 46.7 dyn/cm and 46.8 dyn/cm for stainless steel. Both values are substantially lower than expected for freshly prepared metals, indicating that significant levels of oxidation and/or carbonaceous contaminants are present on the “clean” surfaces. Minimum wetting surface energy values for freshly polished R-brass specimens cleaned in the same manner were found to be 34.59 dyn/cm. Methods. Shear stresses during start-up of steady shear flow were measured using a planar-Couette shear flow apparatus with a centrally mounted shear force transducer (Figure 1). The planar-Couette shear device consisted of two platens supported by metal frames. The upper platen was held stationary and the lower one translated on a crossed/roller platform connected to a microstepper motor, which imposes the desired shear flow. The upper platen was designed so that the surface of a shear force transducer (SST), developed by Dealy and co-workers,22 could be mounted centrally and flush with its surface. The upper shear cell platen and active (sensing) region of the transducer were designed to facilitate easy removal of polymer debris from the spacing between transducer and shear platen following sample loading and gap changes, and to allow straightforward interchange of transducer heads. This device can be used to investigate slip violations in polymers sheared between any surface, transparent or opaque. The surface area of the transducer was maintained fixed at 127 mm2 in all the experiments reported here. Separation between the two plates was set by six precision stainless steel shims mounted at discrete locations around the perimeter of the shear cell. The sample aspect ratio (gap/width) was maintained below 0.03 and the shear stress transducer mounted at the center of the stationary upper shear platen to minimize stress measurement errors due to secondary flow.21 Time-dependent changes in shear force during start-up of steady shear flow were recorded using a Labview (National Instruments) software program.

Results and Discussion Steady shear flow dynamics of 1,4-polybutadiene melts were investigated using the planar-Couette flow device depicted in Figure 1. Time-dependent shear stress measurements were performed at 25.5 °C on aluminum and stainless steel substrates. Typical results obtained using a highly entangled polymer sample, PBD315 with Mn ) 3.15 × 103 g/mol, are summarized in Figures 2 and 3. At low shear rates, γ˘ e 0.52 s-1, the measured shear stresses (shear force/sensor area) increase as approximately exponential functions of time before reaching stable steadystate values at long times. Following cessation of shearing, the shear stresses are observed to decay nearly exponentially with time, eventually returning to their preshear, zero values at long times. At shear rates γ˘ e 0.52 s-1, the steady-state shear stress is also found to increase linearly with shear rate, irrespective of the substrate material. All of these features are well-known for polymeric liquids. At shear rates slightly above 0.52 s-1 the shear stress manifests a transient overshoot and appears to become locked to a critical steady-state value σ* ≈ 0.22 MPa. At even higher shear rates the shear stress is clearly observed to manifest time-dependent asymmetric oscillations about a steady-state stress value slightly lower than σ* for the aluminum substrate (Figure 2a). Time-dependent stress oscillations are also observed in experiments using stainless steel substrates (Figure 3), but in this case stress oscillations are preceded by a sequence of large transient overshoots followed by sudden decreases in shear stress. (23) Owens, D. K.; Wendt, R. C. J. Appl. Polym. Sci. 1969, 13, 1741.

Figure 2. Time dependent shear stress at 25.5 °C for PBD315 sheared between aluminum substrates at various shear rates γ˘ (s-1). The results clearly show the onset of stick-slip flow at shear rates γ˘ > 0.42 (Wi ) γ˘ τd0 > 1.3), higher stick-slip frequencies as shear rate increases, and ultimate disappearance of stress oscillations at very high shear rates.

Several features of the shear stress oscillations observed using Al and stainless steel are, nonetheless, quite similar. For example, for both materials stress oscillations become more noticeable and their asymmetry more pronounced as the nominal shear rate increases (see Figures 2b, 2c, and 3b). The average shear stress and stress amplitude during oscillations are also observed to decrease with

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Figure 4. Stick-slip frequency of PBD315 at 25.5 °C versus shear rate increment ∆γ˘ following onset of stick-slip flow between Al and stainless steel substrates. Straight line through the data is consistent with a linear increase of stick-slip frequency with shear rate.

Figure 3. Time dependent shear stress at 25.5 °C for PBD315 sheared between stainless steel substrates at various shear rates γ˘ (s-1).

increasing shear rate. Stress oscillations also disappear at nominal shear rates slightly above 4 s-1 for both materials. Finally, the oscillations in shear stress, however large, disappear immediately following cessation of shearing between Al and steel substrates, indicating that the oscillations are a characteristic of the flowing polymer liquid. Shear stress oscillations in planar-Couette shear flow experiments have previously been reported by Mhetar and Archer from simultaneous stress and microtracer particle velocimetry measurements over bare and polymergrafted silica glass substrates.16 These authors suggested that stress oscillations in sheared polymer melts are characteristic of the metastability of the stick and slip dynamic states of an entangled liquid at an adsorbing surface. The asymmetry of shear stress oscillations was contended to reflect gradual build up of stresses up to σ*, followed by a sudden decrease when momentum transport between bulk and surface polymer chains is interrupted by disentanglement.4,9,10 Thus, σ* was thought to be associated with the critical stress required for flow-induced disentanglement of surface adsorbed and bulk polymer molecules.4,9,10 Mhetar and Archer further contend that at sufficiently high shear rates (of order the longest Rouse relaxation time of disentangled surface chains) stickslip dynamics should be suppressed during steady shear

flow and steady-state shear stresses should be reduced by a factor of order (N/Ne)2 due to complete slippage (gross slip) between surface adsorbed and bulk polymer molecules.10 Apparent slip by a disentanglement/re-entanglement process is not the only mechanism by which our observations can be explained. An alternative possibility is that σ* is associated with the work needed to desorb physisorbed polymer molecules.12-14 In that case, stress oscillations would be argued to reflect the dynamic processes of attachment and detachment of polymer chains to and from the shear substrates during flow. The ultimate disappearance of stress oscillations at high rates would then imply total debonding of polymer from the shear cell walls. This last true slip state is also anticipated to be accompanied by a reduction in steady-state shear stress by a factor of order (N/Ne)2. The two mechanisms are, therefore, in principle difficult to separate and may in fact coexist under certain conditions.16 Additional details of the stick-slip process can be uncovered from more careful scrutiny of the experimental results. It is, for example, evident from Figures 2 and 3 that in the stick-slip flow regime nearly perfect stress recovery occurs after each successive slip event. Another remarkable feature of the stress oscillations is that the transition from monotonic stress growth at low shear rates to stick-slip flow at higher rates can be recovered qualitatively if experiments are repeated at these rates immediately following measurements at high rate, in the gross slip regime. Both findings point to a rather rapid restoration of surface structure both during flow and following flow cessation. More careful examination of the stress oscillations in the stick-slip regime also indicate that both the onset shear rate and initial stick-slip frequency ω* (see Figure 4 and Table 2) are close to the reciprocal terminal relaxation times τd0-1 of the materials studied. This last observation is in excellent accord with results reported by Mhetar and Archer16 from studies of PBD melts sheared between silica glass substrates. Taken together, the two observations also appear to simultaneously demonstrate that the stick-slip behavior is a feature of the nonNewtonian liquid state and that stress recovery following slip occurs by the same molecular processes responsible for stress growth during start-up of steady shear flow. This last lends support to the idea that sequential re-

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Table 2. Slip and Stick-Slip Properties of 1,4-Polybutadiene Melts sample

σ* (MPa)

PBD67 PBD86 PBD129 PBD173 PBD205 PBD253 PBD315 PBD335 PBD650

0.32 0.25 0.25 0.26 0.28 0.26 0.28 0.24 0.28

ω* (s-1)

1.92 1.3 1.21 0.36 0.196 0.046

b (µm) 4.5 ( 2.2 13.8 ( 2.1 19.1 ( 2.9 47.8 ( 7.5 56.1 ( 11.2 58.2 ( 34.7 85.38 ( 12.7 80.0 ( 9.6

entanglement/disentanglement of polymer chains near the surface are the cause of stick-slip flow in the materials studied here. Thus, any of the disentanglement theories for slip4,7,10 would predict stick-slip dynamics that are qualitatively consistent with these observations. Stickslip dynamics predicted by an extension of Schallamach’s theory for sliding friction of elastomers to the case of sliding friction between an adsorbing substrate and a transient network17 would also seem capable of describing the stress oscillation patterns observed. Our observations therefore appear to be consistent with a stick-slip process that begins with slow molecular orientation, followed by sudden disentanglement by fast Rouse-like retraction processes when the shear stress exceeds σ*. The re-entangled but disoriented molecules are subsequently reoriented by the shear flow, causing the stress to again slowly rise and the entire sequence repeated.4,7,9,10 Additional support for this mechanism is evident in Figure 4, where the shear stress oscillation frequency is plotted against shear rate increment ∆γ˘ ≡ γ˘ - 0.52, i.e., relative to the first observation of stress oscillations. It is apparent from this figure that, but for the highest shear rates considered, the measured stickslip frequency increases approximately linearly with ∆γ˘ for both Al and stainless steel substrates. This finding is in remarkably good accord with a well-known result from bulk steady shear flow rheological experiments using polymer liquids. In these systems, the time constant for stress growth is initially a constant (of order the terminal time), but at shear rates γ˘ > τd0-1, the rate of stress growth increases nearly in proportion to the imposed shear rate.24 If the stick-slip process does indeed occur by a disentanglement/re-entanglement mechanism, shear stress decreases during the slip section of the cycle must be rapidly arrested by a fast Rouse-like re-entanglement process. For entangled polymer liquids, the time constant for re-entanglement would be anticipated to be of order τe ≈ Ne2τm ) (N3/Ne)-1τd0, where τm is a monomeric hopping time. Thus, the longest characteristic Rouse relaxation time of surface-adsorbed chains, τRouse ≈ N2τm, would be naı¨vely anticipated to determine the characteristic timeconstant for slip. This expectation does not appear to be borne out by the experiments. Specifically, at nominal shear rates substantially below τRouse-1 of the bulk molecules, both the shear stress oscillation amplitude and average stress value during oscillations are observed to decrease with increasing nominal shear rate. Such a decrease would be expected only if the rate of stress growth is competitive with the re-entanglement rate (i.e., if entanglements between surface and bulk molecules are incompletely restored, before stress growth resumes). Indeed if steady shearing occurs at a rate faster than the time scale for re-entanglement, the state of slip should persist until shearing stops, lowering the average steady(24) Larson, R. G. Constitutive Equations for Polymer Melts and Solutions; Butterworths: Boston, 1988.

Figure 5. Molecular weight dependence of apparent shear rate range (γ˘ 2*/γ˘ 1*) spanned by the stick-slip flow regime. The line through the data satisfies the relation γ˘ 2*/γ˘ 1* ∼ Mn2.9.

state shear stresses observed.10 Our results therefore appear to point to a slower surface entanglement restoration process than that recognized by any of the slip theories based on the disentanglement hypothesis. It is nonetheless recognized that a much longer time (of order τCR ≈ (N/ Ne)3Nτm) is needed for reorganization of entanglement structure in bulk polymer systems, perhaps explaining the gradual destruction of the near-surface entanglement structure apparent from the results. The nominal or apparent shear rate (i.e., shear rate not corrected for slip) at the first observable appearance of stress oscillations γ˘ 1* and at complete disappearance of oscillations γ˘ 2* were determined for the three highest molecular weight polymer samples studied. The results are supplemented by data from a fourth material, PBD205 (Mn ) 2.05 × 105 g/mol, PI ) 1.03, and τd0 ) 0.78 s). Though shear stress oscillations were observed in two of the lower molecular weight polymers PBD173 and PBD129, the oscillations in these systems were neither as large nor as regular as in the other high molecular weight materials to allow γ˘ 1* and γ˘ 2* to be determined reliably. The ratio of the two shear rates γ˘ 2*/γ˘ 1* is plotted in Figure 5 for various bulk polymer molecular weights, Mn. The line through the data satisfies the relation γ˘ 2*/γ˘ 1* ∼ Mn2.9, with a small numerical prefactor. These results can be further refined by correcting for differences in slip velocity before and after the transition from stick-slip to strong slip. However, to perform such correction rigorously, independent measurements of slip velocity at the transition are required. For simplicity, we use a theoretical result from analyses of slip by disentanglement of long surface bound and bulk entangled molecules. Namely, that the jump in slip velocity at the transition is of order N/Ne.7,10 Thus, for large slip lengths the real shear rate range spanned by the two slip transitions scales roughly as Mw1.9. An upper bound estimate of this range is τd0(N)/τRouse(P) (i.e., the ratio of the shear rate where the orientation mismatch between bulk and surface chains first becomes important to the shear rate where retraction of surface chains is prevented by shear). Here N is the degree of polymerization of polymer chains in bulk and P is the polymerization index of surface chains. An additional approximation, P ≈ N, yields γ˘ 2*/γ˘ 1* ≈ τd0(N)/τRouse(N) ∼ N1.4, which is somewhat narrower than the range observed. Next we study variations in the critical stress σ* required for the onset of gross slip for variable polymer molecular weight (Table 2). In each case, the critical stress

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was estimated as the maximum shear stress value from plots of mean steady-state shear stress versus nominal shear rate (see Figure 6a). It is clear from the results in the figure that though the shear rate at which the stress maxima are observed changes substantially with gap and polymer molecular weight, the maximum stresses themselves manifest remarkably little sensitivity to these variables. In fact a critical shear stress σ* ) 0.26 ( 0.02 can be identified for all polymers studied (Table 2). These σ* values are within a factor of 2 of critical stresses for spurt flow of narrow distribution polybutadienes with higher vinyl content in capillary rheometers25,26 and are slightly higher than critical stress values reported from microtracer particle slip measurements in polybutadienes with similar microstructures to those studied here.16 The critical stresses also compare favorably with the theoretical result, σ* ≈ GN/Ne1/2 ) (0.16 ( 0.02) MPa, for the stress required to disentangle nonoverlapping tethered (graft density ν ≈ Rg-2) and bulk polymer molecules,10 and are about four times lower than the critical stress computed using Schallamach’s theory for slip of a polymer network over a strongly adsorbing substrate.17 The changing slope of the stress-shear rate diagram with gap (Figure 6a) suggests that substantial levels of slip are present at steady state even at shear stresses below the critical value. The magnitude of slip at a particular shear stress can be quantified from steadystate stress data obtained at any pair of shear cell gaps using the relation b ) (R - 1)H1H2/2(H1 - RH2). Here b is the steady-state extrapolation or slip length, R ) γ˘ a(H2)/γ˘ a(H1), and γ˘ a(Hi) is the nominal or apparent shear rate (uncorrected for slip) at a shear cell gap Hi. In cases where data are also available at large gaps, i.e., where slip has no measurable effect on the stress velocity diagram, an equivalent expression γ˘ rel ) 1 + 2b/H can be used to determine the extrapolation length b from plots of γ˘ rel ≡ γ˘ a(H)/γ˘ (H f ∞) versus 1/H, at fixed shear stress. Both procedures were used in the present study to compute b. Figure 6b summarizes the effect of shear stress on b for three polymers. The error bars on the data are confidence intervals obtained from experiments using multiple gap pairs and 1/H plots to determine b. Two features of the results in Figure 6b stand out. First, in every case the extrapolation length is essentially independent of stress up to stresses of order σ*. Close to σ* the shear stress/apparent shear rate diagram becomes multivalued and the extrapolation length diverges. Second, the extrapolation length increases with polymer molecular weight. The second observation is explored in greater detail in Figure 6c where the line through the data suggests an approximate scaling relationship b ∼ Mw1.8. Except for the fact that the extrapolation lengths reported here are consistently about a factor of 2 higher than those reported by Mhetar and Archer from microtracer particle velocimetry measurements near clean silica glass substrates,16 both observations are in surprisingly good accord with the earlier results. In fact, both sets of extrapolation lengths are consistent with expectations for slip in the pretransitional linear slip regime labeled regime c by Mhetar and Archer.10 The slip length estimated for PBD315 in regime c, for example, is (1/4)(Mw,315/M0)(Mw,315/Me)a ≈ 137 µm, where the statistical segment length a of PBD is taken to be 6 Å and its entanglement molecular weight Me ≈ 2 × 103 g/(mol). The corresponding result for PBD253 is b ≈ 89 µm; PBD173 b ≈ 42 µm; PBD86 b ≈ 10 µm; and PBD67 b ≈ 6 µm. (25) Lim, F. J.; Schowalter, W. R. J. Rheol. 1989, 33, 1359. (26) Yang, X.; Wang, S.-Q.; Halasa, A.; Ishida, H. Rheol. Acta 1998, 37, 415.

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Figure 6. (a) Steady-state stress σss versus apparent shear rate γ˘ for PBD315 sheared between aluminum substrates at various substrate separations H. σss becomes multivalued above a critical stress σ* ≈ 0.27 MPa, irrespective of the substrate separation. Even at stresses σss < σ*, the shear rate required to produce the same level of stress is observed to increase significantly as H decreases, suggesting that measurable levels of slip are present at low stresses. (b) Extrapolation lengths, b, at various shear stresses for three of the PBD melts studied. The results show that b is essentially independent of stress over a broad range of stresses and that at fixed stress the extrapolation length increases with polymer molecular weight. (c) Molecular weight dependence of extrapolation length at 25.5 °C for σss < σ*. Best fit solid line through the data supports an approximate relation b ∼ Mw1.8 over a broad range of polymer molecular weights. Dashed line supports a weaker scaling relationship b ∼ Mw0.85 over a narrower range of PBD molecular weights.

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More careful scrutiny of Figure 6c suggests a possible reason for uncertainty about the molecular weight dependence of the extrapolation length, however. It is apparent from the figure that extrapolation lengths measured using the four highest molecular weight polymers manifest a weaker dependence on molecular weight b ∼ Mw0.85 than seen overall. Indeed a much weaker dependence, if any, of extrapolation length on polymer molecular weight was recently reported by Wise et al. for PBD melts extruded between metallic and ZnSe substrates.27 The authors used a combination of attenuated total reflectance (ATR) infrared spectroscopy and transport theory with slip boundary conditions to quantify extrapolation lengths. It is nonetheless unclear whether the transition in slip behavior implicit in separating the molecular weight dependences in this way is real or merely a manifestation of uncertainty in the slip lengths measured over too narrow a molecular weight range. Even modest levels of uncertainty in the small extrapolation lengths (b < 1 µm) obtained using the ATR/transport analysis procedure could in fact mask a b ∼ Mw0.85 dependence over a sufficiently narrow range of polymer molecular weights. Direct measurements of extrapolation lengths in polybutadiene layered systems (h-PBD/d-PBD) similar to those studied by Wise et al. are planned using the planar-Couette shear flow device to help establish the origin of the much smaller extrapolation lengths observed in this system. An additional effect of apparent slip on polymer dynamics is evident from stress relaxation experiments at variable gap. Specifically, it is observed that at shear rates well below critical values for observation of dynamic stick-slip, relaxation of shear stress following cessation of steady flow becomes gap-dependent. At large gaps, normalized shear stresses σ-(t) ) σ(t)/σss show little dependence on H, while at smaller gaps there is a clear slowing down of stress relaxation with decreasing gap (Figure 7a,b). An obvious, but troublesome source of this behavior is more efficient entrapment of polymer in the space between the active transducer surface and shear platen as the gap is lowered. The trapped polymer damps motion of the transducer element, which would increase the time to steady-state and stress relaxation times following cessation of shear. A second potential source of artifact is a small squeezing flow component that can arise from lack of parallelism between the shearing substrates. The first source of error was minimized by transducer placement and design: (i) The shear transducer was mounted into a low-angle cone-shaped opening in the stationary shear platen. This arrangement causes polymer debris to be squeezed into the transducer housing rather than accumulate in the space between the transducer head and shear platen. (ii) The shear transducer was mounted on the upper stationary shear platen. In this configuration flow of liquid polymer into the space between the transducer head and platen is resisted by gravitational forces. (iii) The transducer was designed to facilitate easy removal of polymer debris from the space between transducer and shear platen prior to performing experiments. Checks performed after numerous experiments at fixed gap revealed no noticeable accumulation of polymer, indicating that normal forces in shear are insufficient to drive flow of the highly viscous materials into the small space between the force transducer and shear platen. The second source of artifact was mitigated through careful mechanical design of the instrument. Time-dependent (27) Wise, G. M.; Denn, M. M.; Bell, A. T. J. Rheol. 2000, 44, 549567.

Dao and Archer

Figure 7. Normalized shear stress versus time for (a) PBD315 and (b) PBD253, sheared at low shear rates (Wi < 1) between aluminum substrates at various gaps H. The results show a small, but systematic, slowing down of stress relaxation with decreasing gap that mirrors the effect of shear cell gap on apparent shear rate. (c) Apparent terminal relaxation times λd,app for various high molecular weight polybutadiene melts as a function of reciprocal gap, H-1. All results were obtained by fitting stress relaxation results at low shear rates Wi < 1 to single-exponential functions. Results provided for PBD315 are a composite from fits to relaxation data at multiple shear rates.

shear stress growth and relaxation measurements in both the forward and reverse directions (x and -x, Figure 1) yielded virtually identical stress relaxation times at each gap, confirming that local loss of parallelism cannot be the source of the retarded relaxation observed at lower gaps. At each gap studied, shear stress decay following cessation of flow is very well described by a single-

Entangled Polymer Liquids

exponential relaxation function. This indicates that the gap-dependent stress relaxation dynamics observed do not result from a new dynamic process superposed on normal relaxation of polymer chain segments oriented by flow. It is also apparent from Figures 6a and 7a that the trend toward gap-dependent stress relaxation somewhat mirrors the trend toward gap-dependent flow curves, already attributed to apparent slip near the shear cell walls. Characteristic relaxation times extracted from singleexponential fits to stress relaxation plots at variable gap are provided in Figure 7c for three of the polymers studied. Results presented for PBD315 are a compilation from measurements at four different shear rates (γ˘ ) 0.02, 0.05, 0.1, and 0.2 s-1), while those for PBD253 and PBD173 are for γ˘ ) 0.1 s-1 and γ˘ ) 0.5 s-1, respectively. Comparison of these times with terminal times measured in oscillatory shear (Table 1) indicate that in every case the large gap stress relaxation time is the terminal time of the material under study. The results also indicate that the actual retardation of relaxation required to produced the effects seen in parts a and b of Figure 7 are quite modest. λd,app results for PBD315 and PBD173, for instance, are enhanced by a factor less than 3 for a nearly 10-fold reduction in H. Likewise, λd,app for PBD253 increases by a factor less than 4 for a nearly 20-fold reduction in the shear cell gap. Figure 7c also reveals an approximately linear relationship between apparent terminal time and H-1 up to gaps approaching the computed extrapolation length of the polymer studied. On the basis of the large H values at which slowing down of stress relaxation is observed, an explanation based on confinement-induced changes in dynamics can be safely ruled out. Likewise, the fact that a single-exponential function captures the entire stress relaxation process at large H (i.e., negligible rheological influence of slip) and that the stress relaxation time at large gaps is the terminal time estimated from linear viscoelasticity indicate that the effect observed cannot originate from a nonlinear rheological source. Thus, the source of the slowing down of stress relaxation is presently unknown. Recently, Dao and Archer28 used an evanescent wave optical polarimetry technique to study relaxation dynamics of entangled polymer chains within three molecular diameters of high refractive index glass substrates. These authors found that a slow constraint release-like process governed dynamics near the substrate. A similar process may be responsible for the observed slowing down of stress relaxation and may also help explain the slower than expected re-entanglement dynamics apparent in the stick-slip flow regime. We conclude with a brief look at apparent slip violations in 1,4-polybutadiene melts sheared between R-brass substrates. As pointed out in the Introduction, at least two studies have reported delay in the onset of polymer surface instabilities in polymer melt extrusion processes when this material is used to construct the die. If, as suggested in the literature, sharkskin melt fracture originates from stick-slip flow in the capillary and/or near its exit, a suppression of stress oscillations would be expected in our planar-Couette flow experiments if R-brass is used to fabricate the substrates. Figure 8a summarizes our observations for transient shear stress measurements of PBD335 (Mn ) 3.35 × 105 g/mol, PI ) 1.07, and τd0 ) 3.9 s) sheared over polished R-brass. It is immediately evident from the data that at any given shear rate, shear (28) Dao, T. T.; Archer, L. A. Langmuir 2001, 17, 4042.

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Figure 8. Time-dependent shear stress at 25.5 °C for PBD335 sheared between various substrates: (a) freshly polished R-brass; (b) R-brass oxidized in air for 3 h; and (c) R-brass oxidized in air for 24 h substrates at various shear rates.

stresses measured at the brass substrate are substantially lower than those seen with Al and stainless steel fixtures. Further evaluation of the data also reveals a definitive lack of the shear stress oscillations observed with the Al and stainless steel. Taken together, the two sets of observations indicate that the ability of R-brass to suppress

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stick-slip likely comes from weaker binding of the polymer to the polished R-brass substrates. Stick-slip dynamics of 1,4-polybutadiene melts sheared between oxidized R-brass substrates were also investigated using the planar-Couette flow device. Substrates were oxidized in humid air at elevated temperatures for periods ranging from 3 to 24 h. Rutherford backscattering spectroscopy measurements using a helium ion stream were used to determine the surface chemical makeup of R-brass specimen subjected to air exposure times of 3, 12, and 24 h. These experiments reveal a surprisingly gradual but steady masking of the Zn scattering edge by an oxygenrich overcoat as the air exposure time is increased. Parts b and c of Figure 8 summarize results from transient planar-Couette shear flow measurements using the 3- and 24-h materials. It is evident from both figures that much higher, better quality, shear stresses develop during shearing over the oxidized brass substrates. It is also apparent that at shear stresses close to the critical values for gross slip on aluminum and stainless steel substrates, the steady-state shear stress measured using R-brass begins to decrease with increasing shear rate, an observation credited to apparent slip at the walls in the case of aluminum and stainless steel substrates. Both findings appear to confirm our previous assertion that PBD binds better to oxidized R-brass than to polished brass. The ability of polished R-brass dies to delay or prevent surface instabilities in extrusion would therefore seem to arise from enhancement of slip at the poorly adsorbing metal substrate. Unlike aluminum and stainless steel, however, the shear stress reductions observed on oxidized R-brass are not preceded by stress oscillations, or stick-slip dynamics, indicating that binding of PBD to these substrates may be more complicated. A slight hint of oscillatory, stick-slip, dynamics is nonetheless evident for the R-brass substrate oxidized for 24 h (Figure 8c), but only at high shear rates. The unexpectedly slow oxide formation kinetics provides a clue that this difference in apparent slip dynamics between oxidized R-brass and aluminum may arise from either a patchy covering of oxide or a thin