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Relaxation Dynamics of Entangled Polymer Liquids near Solid Substrates Tien T. Dao† and Lynden A. Archer*,‡ Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843, and School of Chemical Engineering, Cornell University, Ithaca, New York 14853-5201 Received January 5, 2001. In Final Form: April 16, 2001 We investigate relaxation dynamics of entangled 1,4-polybutadiene (PBD) melts and concentrated polystyrene/diethyl phthalate (PS/DEP) solutions near an attractive glass substrate. Narrow molecular weight distribution PBD melts and PS/DEP solutions covering a broad range of molecular weights (6.7 × 103 e M h w,PBD e 5.15 × 105; 1.3 × 105 < φM h w,PD < 1.1 × 106 g/(mol)) were used in the study. A new experimental method, evanescent wave laser polarimetry (EWLP), was developed to investigate nearsurface relaxation dynamics in these materials. The method relies on total internal reflection of phasemodulated laser light at an interface between a high refractive index transparent hemisphere and the polymer liquid to probe shear-induced changes in molecular orientation in a fluid layer within approximately 80 nm of the polymer/substrate interface. A characteristic near-surface relaxation time τsurface determined from step shear EWLP relaxation experiments using PBD melts was found to be a much stronger function of bulk polymer molecular weight, τsurface ∼ Mw4.2(0.03, than the corresponding terminal properties in bulk, h w3.4(0.02 and τd0 ∼ M h w3.4(0.04. In the case of PS/DEP solutions, τsurface was consistently larger than η0 ∼ M bulk terminal relaxation times but displayed the same dependence on polystyrene molecular weight as τd0. A new gel-like surface relaxation process was also identified in the PS/DEP solutions that is not present in bulk. These experimental results are discussed in terms of the equilibrium structure of physically adsorbed polymer chains and classical relaxation mechanisms available to entangled polymer liquids near rigid substrates.
Introduction The static structure and relaxation dynamics of polymers near rigid surfaces are believed to be fundamentally different from those in bulk materials. Substrates are for instance argued to decrease molecular configurational freedom,1-4 induce collective glasslike dynamics,5-9 promote phase segregation,9-13 and alter stress relaxation mechanisms available to adsorbed polymer chains.13-18 Materials engineering applications impacted by one or more of these surface effects are both numerous and † ‡
Texas A&M University. Cornell University.
(1) deGennes, P. G. Macromolecules 1980, 13, 1069; Adv. Colloid Interface Sci. 1987, 27, 18. (2) Scheuttjens, J. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619; 1980, 84, 178. (3) Eisenriegler, E. Polymers Near Surfaces; World Scientific Press: Singapore, 1993. (4) Bitsanis, I. A.; ten Brinke, G. J. Chem. Phys. 1993, 99, 3100. (5) Montford, J. P.; Hadziioannou, G. J. Chem. Phys. 1988, 88, 7187. (6) Johnson, H. E.; Granick, S. MRS Bull. 1996, 33. (7) Kajiyama, T.; Tanaka, K.; Tanaka, A. Macromolecules 1995, 28, 3482; 1997, 30, 280. (8) van Zanten, J. H.; Wallace, W. E.; Wu, W.-I. Phys. Rev. E 1996, 53, R2053. (9) Keddie, J. L.; Jones, R. A. L. Europhys. Lett. 1994, 27, 59. (10) O’Malley, J. J.; Thomas, H. R.; Lee, G. M. Macromolecules 1979, 12, 996. (11) Jones, R. A. L. Phys. Rev. Lett. 1991, 66, 1326. (12) Saout-Elhak, A.; Benhamou, M.; Daoud, M. J. Phys. II 1997, 7, 503. (13) Mhetar, V. R.; Archer, L. A. Macromolecules 1998, 31, 6639. (14) Silberzan, P.; Leger, L. Macromolecules 1992, 25, 1267. (15) Ajdari, A.; Brochard-Wyart, F.; deGennes, P. G.; Leibler, L.; Viovy, J.-L.; Rubinstein, M. Physica A 1994, 204, 17. Brochard-Wyart F.; Ajdari, A.; Leibler, L.; Rubinstein, M.; Viovy, J. L. Macromolecules 1994, 27, 803. (16) Zheng, X.; Sauer, B. B.; Van Alsten, J. G.; Schwarz, S. A.; Rafailovich, M. H.; Sokolov, J.; Rubinstein, M. Phys. Rev. Lett. 1995, 74, 407. (17) Klushin, L. I.; Skvortsov, A. M. Macromolecules 1991, 24, 1549. (18) Semenov, A. N. Phys. Rev. Lett. 1998, 80, 1908.
multidisciplinary.13,19-23 Despite the vast literature on the subject, however, direct evidence of surface-induced changes in liquid state polymer properties is surprisingly scant. In this article, we report the first direct experimental results that conclusively show that proximity to solid substrates changes relaxation dynamics of entangled polymer liquids in multiple ways. To investigate relaxation dynamics of polymer liquids near solid substrates, we have developed a new experimental method that relies on total internal reflection of laser light at an interface between a high refractive index transparent hemisphere and a liquid polymer to probe shear-induced changes in molecular orientation near the interface. The technique has been labeled evanescent wave laser polarimetry (EWLP) because it relies on a phasemodulated evanescent laser field to probe time-dependent changes in optical birefringence of oriented polymer chain segments near the substrate (Figure 1). In its present configuration, the method uses visible laser light (λ ) 542.3 nm) at grazing incidence to measure time-dependent changes in birefringence within about 80 nm of a polymer/ solid interface. This article focuses on results from EWLP measurements using narrow molecular weight distribution entangled polystyrene/diethyl phthalate solutions and 1,4-polybutadiene melts covering a broad range of entanglement densities, N/Ne. In both situations, spontaneously adsorbed (physisorbed) polymer chains are used to (19) Norde, I. In Surface and Interfacial Aspects of Biomedical Applications; Andrade, J. D., Ed.; Plenum Press: New York, 1995. (20) Brochard-Wyart, F.; deGennes, P.-G. Langmuir 1992, 8, 3033. Brochard-Wyart, F.; Gay, C.; deGennes, P. G. Macromolecules 1996, 29, 377. (21) Mhetar, V. R.; Archer, L. A. Macromolecules 1998, 31, 8607; 1998, 31, 8617. (22) Wu, S. Polymer Interface and Adhesion; Marcel Dekker: New York, 1982. Comyn, J. Adhesion Science; Royal Society of Chemistry Pubs: Cambridge, U.K., 1997. (23) Granick, S. Science 1992, 255, 966.
10.1021/la0100183 CCC: $20.00 © 2001 American Chemical Society Published on Web 05/30/2001
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Langmuir, Vol. 17, No. 13, 2001 4043 Table 1. Structural and Linear Viscoelastic Data for All Polymers Used in the Study sample
Figure 1. Schematic of the EWLP apparatus used in the study (P, linear polarizer; PEM, photoelastic modulator; Q, quarter wave plate; CP, analyzing circular polarizer; D, detector).
report relaxation dynamics near the interface when bulk materials are subjected to small-amplitude homogeneous step shear strains in a planar-Couette shear flow configuration (see Figure 1). Despite the apparent simplicity of both the chosen polymer systems (weak dispersion forces and topological constraints compose the only important intermolecular interactions) and of the flow-field used to characterize their dynamics, several classical mechanisms can be proposed to explain how proximity to a solid substrate could alter dynamics in these liquids. One of our objectives in the present study is to determine which, if any, of the proposed mechanisms explain observed near-surface properties of entangled polymer liquids. It is possible, for example, that denser (or, alternatively, more sparse) packing of molecular segments near a solid substrate24 simply alters the monomeric friction coefficient and/or equilibrium tube diameter of molecules located within the first few molecular layers from the substrate. By analogy to entangled bulk polymer liquids, simple reptation diffusion would be anticipated to be the dominant relaxation process near such a substrate. A rescaled version of the bulk reptation relaxation time could therefore be defined for fluid near the surface τd ) τd,Rep ≈ τ1sN3.5(0.1Nes-1.16,18 Here, τ1s is a new monomeric surface relaxation time that reflects differences in monomermonomer and monomer-substrate interactions and Nes is the number of monomeric units between entanglement points of surface chains. Alternatively, it can be argued that physical attachment of surface chains (N-mers) to a substrate prohibits reptational diffusion altogether.13,15,21 The rigorous requirement for this situation to occur is that the timescale for surface reconstruction should be about 4-5 times larger than the characteristic terminal relaxation time of molecules physically adsorbed to the substrate. Once longrange translational diffusion of surface adsorbed molecules is prohibited, these molecules might instead relax in a sequence of Rouse-like jumps after constraints imposed by surrounding polymer chains (P-mers) diffuse away (relax). If P-mers are assumed to visit the interface but are not confined by it and their concentration near the surface is much higher than that of the N-mers, the terminal “constraint release” (CR) relaxation time of N-mers is τd ) τd,CR ≈ τ1N2.4(0.1P2(0.2Ne-2(0.3.13,15 Arguments developed by Brochard-Wyart et al. suggest that this formula for τd,CR should hold only if N > P; these authors have proposed an alternative estimate, τd ) τd,CR ≈ τ1N3P1.5Ne-2.5, for the case N < P.15 Here, Ne is the average number of segments between entanglement points and τ1 (24) See, for example: Jones, R. A. L.; Richards, R. W. Polymers at Surfaces and Interfaces; Cambridge University Press: New York, 1999; Chapter 5.
PS-P67M PS-P93M PS-1P8M PS-2P9M PS-3P8M PS-5P5M PBD67 PBD86 PBD129 PBD176 PBD336 PBD650
M hn× 10-6 0.67 0.935 1.810 2.890 3.840 5.480 0.067 0.087 0.129 0.176 0.336 0.515
PI
η0 [Pa s]
GN [Pa]
ωc-1 [s]
τd0 [s]
1.01 1.01 1.03 1.09 1.04 1.15 1.04 1.04 1.03 1.03 1.07 1.05
1.3 × 7.1 × 102 7.5 × 103 5.4 × 104 1.4 × 105 4.4 × 105 7.9 × 103 2.0 × 104 8.7 × 104 2.1 × 105 1.6 × 106 7.5 × 106
4.8 × 4.8 × 103 4.6 × 103 4.6 × 103 4.7 × 103 4.8 × 103 9.3 × 105 9.2 × 105 9.7 × 105 9.5 × 105 1.0 × 106 1.2 × 106
7.2 × 3.2 × 10-1 4.4 4.1 × 101 7.6 × 101 3.2 × 102 1.7 × 10-2 3.9 × 10-2 1.9 × 10-1 4.1 × 10-1 2.1 8.7
6.5 × 10-2 3.6 × 10-1 5.7 4.3 × 101 7.2 × 101 3.4 × 102 2.0 × 10-2 4.7 × 10-2 2.2 × 10-2 5.3 × 10-1 3.8 1.5 × 101
102
103
10-2
is a monomeric jump relaxation time. The effect of the substrate on monomer jump time and entanglement spacing may be taken into account in any of the above expressions by making the substitutions τ1 f τ1s and Ne f Nes. It is also possible that entangled molecules attached to a surface could relax stress by simply withdrawing themselves toward the attachment point (down the tube formed by surrounding molecules) and then diffusing away from that point in a random direction. This process, known as arm retraction (AR), is an activated process and the terminal relaxation time for N-mer chains executing it is to a first approximation λd ) λd,AR ≈ τ1N2 exp(15N/8Ne).25 More refined estimates of λd,AR are available that account for dilution of the mean tube diameter enmeshing the N-mer by relaxed outer arm segments.26 In situations where the concentration of N-mer chains is high enough that entanglement between N-mers is important, a combination of CR and AR dynamics should describe polymer relaxation near the surface. Finally, it has been argued in the literature that proximity to surfaces increases cooperativity of molecular motion.5-9 When collective glasslike dynamics dominate, long-range translational diffusion of surface copolymer chains is arrested and relaxation progresses by shortrange motions of small molecular subunits.27 In this situation, the time-dependent evolution of stress following step-strain could be described by stretched exponential functions of the Kohlrausch-Williams-Watts (KWW) type,27,28 σ(t) ) σ|t)0 exp[(-t/λ)β]. An average polymer relaxation time λ much larger than the terminal relaxation time of the bulk polymer liquid could be recovered by fitting the experimental data to this equation. At low surface polymer molecular weights, this relaxation time would be expected to depend at most linearly on molecular weight and to become independent of molecular weight above about 105 Da.29 Experiment Materials. Polystyrene (PS) samples used in the study were purchased from Tosoh Corp., Japan, and from Aldrich Chemicals. Weight-averaged molecular weights and polydispersity indices of the polymers are provided in Table 1. Solutions containing 18 wt % polystyrene in diethyl phthalate, DEP (Aldrich), were formulated in a dichloromethane (“methylene chloride”, Aldrich) cosolvent. Following dissolution of polystyrene and diethyl (25) Doi M.; Kuzuu, N. Y. J. Polym. Sci., Polym. Lett. Ed. 1980, 18, 775. (26) Milner, S. T.; Mcleish, T. C. B. Macromolecules 1997, 30, 2159. (27) Matsuoka, S. Relaxation Phenomena in Polymers; Hanser Verlag Pubs: New York, 1992. (28) Kohlrausch, F. Poggendorff’s Ann. Phys. 1847, 12, 393. Williams, G.; Watts, D. C. Trans. Faraday Soc. 1970, 66, 80. (29) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; Wiley: New York, 1996; Chapter 10.
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Figure 2. (a and b) Frequency-dependent normalized imaginary component of complex viscosity η′′(ω)/η0 at 25.5 °C for 1,4polybutadiene melts (a) and polystyrene/diethyl phthalate solutions (b) used in the study. (c and d) Frequency-dependent real component of complex viscosity η′ at 25.5 °C for 1,4-polybutadiene melts (a) and polystyrene/diethyl phthalate solutions (b) used in the study. phthalate, the cosolvent was driven off at room temperature and the weight of the remaining polystyrene/diethyl phthalate solutions measured until constant. Various narrow molecular weight distribution (MWD) polybutadienes (PBD) (1,4 addition > 90%) were purchased from Polymer Source, Inc. Weightaveraged molecular weights of the polymers used in the study h w ) 5.15 × 105 g/mol. Molecular ranged from M h w ) 6.73 × 104 to M weight and polydispersity indices of these materials are also provided in Table 1. Rheological Characterization. Bulk rheological properties of polymer samples used in the study were characterized by smallamplitude oscillatory shear measurements. These measurements were performed at a nominal temperature of 25.5 °C using a Paar Physica universal dynamic spectrometer (UDS), equipped with stainless steel cone-and-plate fixtures (8 mm diameter, 2° gap angle). Frequency-dependent imaginary and real components of the complex shear viscosity (η′′ ≡ G′/ω and η′ ≡ G′′/ω, respectively) are provided in Figure 2a-d for several PS/DEP solutions and PBD melts. The η′′ data in Figure 2a,b are
normalized by the limiting or zero shear viscosity η0 ≡ limωf0 η′(ω). For a single Maxwell viscoelastic element (linear elastic spring and viscous Newtonian fluid dashpot arranged in series), η′′(ω)/η0 ) ωτ(/1 + ω2τ2), which exhibits a local maximum value of 0.5 at ω ) ωc ) τ-1. Here, τ ≡ η0/E is an effective viscoelastic relaxation time, E is the elastic constant of the spring, and η0 is the viscosity of fluid in the dashpot. The close correspondence between linear viscoelastic properties expected for a single Maxwell element and the measured η′′(ω)/η0 results (Figure 2a,b) confirms the narrowness of the molecular weight distribution of the materials used in the study. It also allows us to estimate the characteristic terminal relaxation times ωc-1 for the materials (see Table 1). Zero shear viscosities η0 estimated from Figure 2c,d are provided in Table 1 for each material. Plateau viscoelastic moduli GN, estimated from the linear viscoelastic data as the storage modulus G′ value at which the loss modulus G′′ manifests a local minimum, are also provided in Table 1. The near constancy
Relaxation Dynamics of Entangled Polymer Liquids of GN, with varying M h w, for a given polymer type underscores the fact that the entanglement molecular weight is essentially constant for all 1,4-polybutadiene and polystyrene/diethyl phthalate samples used in the study. Knowledge of η0 and GN also allows us to obtain a second estimate of the terminal relaxation time τd0 ≈ 2.4η0/GN (see Table 1). Previous work by one of us indicates that this procedure for estimating terminal times yields τd0 values about one-half those obtained from stress relaxation measurements following small-amplitude step strain.30 Evanescent Wave Polarimetry. When light of wavelength λ travels from a medium of refractive index n1 to one of lower refractive index n2, it is totally reflected at the interface for incident angles beyond a critical value φ1c ) asin(n2/n1).31 The amplitude of the evanescent field created at the interface decays exponentially in medium 2, allowing a characteristic penetration depth dp ) λ/(2πxn12sin2φ1-n22) to be defined. Therefore, in addition to providing information about optical properties at the 1-2 interface, the reflected light also carries information about material in medium 2 located within dp, actually about 3dp,31 of the interface. If visible laser radiation of well-defined polarization is used for the light source, time-dependent changes in polarization following total internal reflection at a glass/polymer interface could be used to study configurational relaxation dynamics of polymer molecules within the first few molecular diameters of the glass surface. The apparatus depicted in Figure 1 is used in the present study to investigate near-surface relaxation behavior of entangled polymer liquids. The instrument utilizes a helium-neon laser with λ ) 543.5 nm incident at a grazing angle (φ ≈ 4°) to the neutral direction, z, to generate an evanescent field in a polymer liquid subjected to shear in the x-y plane. Flow-induced changes in polymer orientation in the shear plane (x - y) can therefore be probed using this instrument. The glass hemisphere employed in the present work is a specialty optical quality high refractive index (n ≈ 1.9) glass blend formulated to produce a material with a very low stress optical coefficient at the optical wavelength used in the work (C < 10-15 Pa-1 at λ ) 542 nm). For this arrangement, dp ≈ 77 nm, which is about 3 times the root-mean-squared end-to-end distance and about 1/10 the fully stretched out length of polymer chains in PBD67, the lowest molecular weight 1,4-polybutadiene melt used in the study. The hemispherical shape of the glass substrate is advantageous for at least two reasons. First, because light travelling radially always achieves normal incidence, measurements at multiple incident angles (penetration depths) are possible with the same optical arrangement. Second, polarization and intensity changes at the glass/air interface can be safely ignored, which greatly simplifies analysis of the experimental results. The low stress optical coefficient of the hemisphere material also guarantees that shearinduced changes in optical anisotropy are localized in the soft polymer layer. Dynamic contact angle (DCA) measurements performed at 25.5 °C using thin rectangular plates of the hemisphere material and a Cahn Instruments DCA 315 device indicate that both polymer systems used in the study (PBD melts and concentrated PS/DEP solutions) spontaneously wet the material. Under the conditions of our experiments, the glass hemispheres may therefore be considered attractive substrates for both polymer liquids. To generate step shear flow, polymer samples are sandwiched to a specified thickness, h ≈ 0.5 mm, between the hemisphere and a fused silica glass plane supported on an x-y-z micropositioning platform. Step shear flow is generated in the instrument by rapidly translating the glass plane parallel to x -y while the hemisphere is held rigidly in place. By use of this procedure, small-amplitude step strains could be imposed in the studied materials in around 100 ms. Both substrates are thoroughly cleaned by sonication in methylene chloride to reduce surface contamination and to promote wetting. Rest times ranging from 2 days (PS-P67M) to 5 weeks (PBD650) are also allowed between sample loading and EWLP measurements to equilibrate polymer surface configurations adopted during loading. These times were determined empirically by evaluating variability (30) Archer, L. A. J. Rheol. 1999, 43, 1555. (31) Harrick, N. J. Internal Reflection Spectroscopy; Harrick Scientific Corp.: Ossining, NY, 1987; Chapter 2.
Langmuir, Vol. 17, No. 13, 2001 4045 between repeat transient step shear EWLP experiments performed 2-10 h apart. To facilitate time-dependent orientation measurements, the incident laser light is first polarized parallel to the direction of shear x using a linear polarizer (P). A photoelastic modulator (PEM) and quarter wave plate oriented in the configurations shown are subsequently used to modulate the phase of the incident light. Following total internal reflection at the hemisphere/polymer interface, the laser light is demodulated and the information it carries recovered using a circular polarizer (CP), photodetector (D), and lock-in amplifiers. The time-dependent intensity I(t) measured by the detector D can be related to the incident optical intensity I0, the polarization modulation frequency ωm, the sample’s birefringence retardation (δ′ ) 2πdeff∆n′/ λ) and apparent orientation angle χ′,
I(t) ≈
I0 [1 - 2J1(A) sin(δ′) cos(2χ′) sin(ωmt) + 2 2J2(A) sin(δ′) sin(2χ′) cos(2ωmt)] (1)
Here, deff ≈ 2dp/(cos φ1) (i.e., ignoring the Goos-Hanchen shift32) is about 30 times the penetration depth dp for the optical configuration used in the present study. J1(A) and J2(A) are Bessel function coefficients that arise from the modulation.33 Both coefficients can be determined from separate calibration experiments using materials of known birefringence retardation and with defined orientation angles relative to the laser polarization direction (e.g., a second quarter wave plate oriented at some known angle θ).33 We have also implicitly assumed that the photoelastic modulator is tuned to achieve the condition J0(A) ) 0. An alternative form of eq 1, I(t) ≈ Rdc - Rωm sin(ωmt) + R2ωm cos(2ωmt), is particularly revealing because it indicates that two lock-in amplifiers tuned to ωm and 2ωm with the appropriate phases and a low-pass (dc) filter are sufficient to measure all Fourier amplitudes, Rk, of the optical signal. Provided J1(A) and J2(A) are known, the measured Fourier amplitudes can be used to compute the two unknowns, δ′ and χ′, in eq 1. δ′ and χ′ are in turn related to the three principal values of the refractive index tensor (n1, n2, and n3) in a molecule-fixed coordinate frame (12-3) and the molecular orientation angle χ in the plane of shear. ∆n′ ) (n1 - n2)(c2χ + sχ2 sφ2) + (n2 - n3)sφ2, and tan 2χ′ ) (n1 n2)s2χ/∆n′, where c2χ ) cos 2χ, and so forth. If the angle of incidence is chosen to be large (i.e., φ1 ≈ 90°, φ ≈ 0°), it is apparent that ∆n′ ≈ (n1 - n2)c2χ ) nxx - nyy and χ′ ≈ χ. Under these circumstances, optical equivalents of shear stress, nxy ) (1/2)(n1 - n2)s2χ, and first normal stress difference nxx - nyy ) (n1 - n2)c2χ can be computed from the experimental measurables. Large φ1 values are also desirable because they increase deff and thereby enhance the experimental signal-to-noise ratio, while preserving the surface selectivity of the EWLP measurements.
Results and Discussion Time-dependent EWLP step shear birefringence results at a nominal temperature of 25.5 °C are provided in Figure 3a for PBD176. The strain amplitude γ used was deliberately maintained below 5% so that only the linear viscoelastic response of the polymer is probed. This choice of strain is motivated both by oscillatory shear measurements that support bulk linear response at shear strains as high as 10% and by previous step shear rheological studies using similar well-entangled polymer systems, which show deviations from linear response at shear strains as low as 20%.30,34 At these strains, χ ≈ 45 and δ′ is small, so that only the lock-in amplifier tuned to sin(ωmt) registers a measurable signal. A limited number of experiments performed at large shear strains γ ≈ 2 revealed measurable responses in both lock-in amplifiers, lending support to this point. Furthermore, in these (32) Goos, F.; Hanchen, H. Ann. Phys. 1947, 1, 33. (33) Fuller, G. G. Optical Rheometry of Complex Fluids; Oxford University Press: New York, 1995; Chapter 3. (34) Juliani; Archer, L. A. J. Rheol., in press.
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Figure 3. (a) EWLP small-amplitude step shear relaxation data for PBD176. At the small strain amplitudes used in these experiments, Rωm(t) ∝ δ′(t); the experimental results were therefore normalized by Rωm(t ) ˆt) to capture the time dependence of the near-surface optical birefringence retardation δ′. (b) Molecular weight dependence of τsurface and terminal bulk material properties (τd0 and η0) of 1,4-polybutadiene melts at 25.5 °C. Best-fit lines through the data support the following approximate scaling relationships between material properties and bulk PBD molecular weight: τsurface ∼ Mw4.2 and η0(Mw) ∼ τd0(Mw) ∼ Mw3.4.
experiments it was observed that when the direction of the applied step strain was reversed the measured Fourier amplitudes, Rωm and R2ωm, displayed exactly opposite symmetries (specifically, whereas Rωm changed sign on flow reversal, R2ωm did not). This observation is precisely what one would anticipate from eq 1. The Rωm data in Figure 3a were obtained by normalizing experimental Rωm results by the instantaneous sin(ωmt) response immediately following imposition of the step (i.e., Rωm(t) ) Rωm(t)/Rωm(t ) ˆt). For the small shear strains used in these measurements, Rωm ∝ δ′, and the results can be correctly viewed as time-dependent changes in birefringence of a film of polymer within dp of the glass
Dao and Archer
substrate. The solid line through the data points represents a single exponential function fit to the experimental results. The same result was observed for all PBD melts studied, except the decay time constant of the exponential function increased rapidly with bulk polymer molecular weight. That a single exponential function correctly captures polymer orientation relaxation following smallamplitude step shear is expected for narrow molecular weight distribution bulk polymer melts. This simple response could be revealing for a polymer melt adsorbed to a rigid substrate. Specifically, it suggests that little, if any, change in collective dynamics of molecular segments is induced by adsorption. This statement is, however, only tentative because the number of molecular layers probed by the experiment (dp g 3R, where R ≈ 27 nm is the rootmean-squared end-to-end distance of PBD67 in bulk) could well be too large to report surface-induced collective behavior in polymer liquids. Characteristic near-surface relaxation times τsurface deduced from the exponential fits are provided in Figure 3b for a wide range of polymer molecular weights. The error bars on the data reflect confidence intervals based on five repeat experiments per sample. These results are compared with the bulk limiting shear viscosities η0 and terminal relaxation times τd0 for the same materials. It is apparent from the data that even after accounting for the approximate factor of 2 difference between τd0 and the terminal time estimated from stress relaxation experiments, the near-surface terminal relaxation times are significantly larger than those for the bulk polymer. It is further apparent that τsurface is more strongly dependent on polymer molecular weight than are the corresponding bulk linear viscoelastic properties. Specifically, whereas the zero-shear viscosity and terminal time in bulk obey h w3.4(0.02 classical entangled polymer scaling laws η0 ∼ M and τd0 ∼ M h w3.4(0.04, the near-surface terminal time manifests a rather different scaling form τsurface ∼ Mw4.2(0.03. The effect of polymer molecular weight on τsurface is actually in remarkably good accord with expectations for constraint release dynamics of noninteracting, endtethered entangled polymer chains dispersed in a melt of chemically identical linear polymers with the same molecular weight.13,15 Indeed, even the ratio τsurface/τd0 observed experimentally compares favorably with the theoretical expectation for surface dynamics dominated by a tube renewal mechanism, τCR/τd0 ≈ N/(3Ne). Here, Ne is the average number of monomers between entanglements and can be estimated from the plateau modulus of 1,4-polybutadiene.29 Although these observations could be easily rationalized for EWLP measurements using endtethered polymer chains entangled with molecules in a nonadsorbing melt, they are surprising (at least naively) for experiments using unfunctionalized molecules physically adsorbed to an attractive substrate. The equilibrium structure of polymer molecules adsorbed at flat, neutral, impenetrable substrates from dense monodisperse melts has been the subject of several studies.35-39 The main conclusion is that the configuration distribution of adsorbed chain segments satisfies random walk statistics at a reflecting boundary. Thus, if all the surrounding unadsorbed molecules are stripped away, the thickness of the resulting surface layer is of the order of 2Rg ∼ N1/2, where N is the degree of polymerization of (35) deGennes, P. G. C. R. Acad. Sci. Paris 1980, 290, 509. (36) Silberg, A. J. J. Colloid Interface Sci. 1988, 125, 14. (37) Cohen-Addad, J. P. Polymer 1899, 30, 1820. (38) Guiselin, O. Europhys. Lett. 1992, 17, 225. (39) Durning, C. J.; O’Shaughnessy, B.; Sawhney, U.; Nguen, D.; Majewski, J.; Smith, G. S. Macromolecules 1999, 32, 6772.
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molecules in the melt. The number of molecular segments per chain in contact with the substrate is on the order of Rg3/Nb3 ∼ N1/2, where b is the statistical segment length. Adsorption studies using uncharged polymer melts and attractive substrates also show that surface chemical effects are extremely short ranged (i.e., they extend only to a distance on the order of b from the substrate), and provided sufficient time is allowed for surface chains to equilibrate have little if any effect on the equilibrium structure of adsorbed molecules.2,3 If each surface contact is assumed to yield an effectively permanent physical bond between segment and substrate (e.g., PDMS adsorbed at clean silica glass substrates), extra mobility at the tails is unimportant and loops should dominate the surface landscape. In this situation, each tail can be modeled as a half loop and the number of monomers per loop is therefore roughly N/xN.37 The tail contour length is then 1/2xN/Neb, which is on the order of xN times smaller than the molecular contour length of the bulk P-mers. The constraint release time predicted by Brochard-Wyart et al.15 is therefore on the order of N3.75, which is slightly weaker than the result seen experimentally. However, the large density of molecular units near a loopy melt/substrate interface should lead to significant molecular overlap and entanglements between loops at the substrate. The simple constraint release picture assumed in ref 15 is not applicable to this situation. Furthermore, if the configuration distribution of segments within loops is assumed to follow Gaussian statistics, the thickness of the dry surface layer hdry for PBD67 would be on the order of N1/4b ≈ 5 nm, which is less than 10% of the surface layer thickness probed by the EWLP experiment. The scaling relation hdry ∼ N1/4 is also inconsistent with experimental results for adsorption of polymer melts, which support hdry ∼ xN.39 Alternatively, if the initial segment-substrate contacts result in strong (i.e., > kT) but nonpermanent bonds, the extra mobility at the chain ends is important and a structure dominated by tails (i.e., small loops and long trains compose the near-surface landscape) should result. As before, details of the segment surface interactions are screened over distances on the order of the monomer length, and a Gaussian coil structure at the reflecting interface is expected. The number of segments in contact with the substrate is again of order xN and the desorption energy per polymer chain is consequently very large (at least on the order of xNkT). The probability of spontaneous desorption of an entire molecule is likewise quite small (approximately exp(-xN)), and the degree of polymerization per tail is approximately 0.5(N - xN). At high N, the degree of polymerization per tail is, therefore, to leading order proportional to the degree of polymerization of molecules in the bulk melt, and the surface density of tails is low. In this case, we may estimate the terminal constraint release time of surface chains to be on the order of N4.4,13,15 which is in good agreement with the experimental result. Furthermore, it is readily seen that in the tail-dominated case hdry ≈ xN/2b, which is consistent with experimental observations.39 The surface layer thickness is however on the order of 20 nm for PBD67, indicating that the dynamics of a significant amount of near-surface, untethered molecules is probed by the experiment, perhaps explaining the observed deviation from the theoretical molecular weight scaling. The surface energy of high molecular weight polystyrene and 1,4-polybutadiene were determined to be γPS ) 38.9 mJ/m2 and γPBD ) 40.4 mJ/m2 at 25.5 °C using a Wilhelmy
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plate DCA technique employing polymer-coated fused silica disks. The polystyrene result compares favorably with literature values and is close to the reported surface energy of diethyl phthalate (γDEP ) 37.5 mJ/m2).24 These surface energies are all substantially lower than that found for the EWLP hemisphere material, γs ) 78.6 mJ/m2, using DCA. It is therefore reasonable to anticipate that the structure of PS chains physically adsorbed from concentrated PS/DEP solutions is similar to that of PBD melts under the conditions of the EWLP experiments. Near-surface step strain relaxation dynamics in entangled polystyrene/diethyl phthalate solutions covering a broad range of entanglement densities 4 e N/Ne e 35 were investigated using EWLP. Time-dependent EWLP step shear birefringence (Rωm) results for a typical polystyrene/diethyl phthalate solution (PS-1P8M) are provided in Figure 4a. Again, the strain amplitude γ used to induce orientation was deliberately kept low to ensure that only the linear viscoelastic response of the material is probed in step strain experiments. Unlike the PBD melts, however, the time-dependent Rωm observed for most of the PS/DEP solutions is best described by a linear combination of two exponential functions (see Figure 4a). For PS molecular weights above 1.8 × 106, the time constant τfast for the faster of the two relaxation processes was found to be around 6 s, irrespective of the molecular weight of polystyrene used. The characteristic time τsurface for the slower process increased rapidly with molecular weight, however. A conservative estimate of the entanglement molecular weight Me for a 18 wt % PS solution can be obtained using the formula Me ≈ (Me0/φ4/3) ) (1.8 × 104 g/(mol)/0.194.3) ) 1.7 × 105 g/(mol), where Me0 is the entanglement molecular weight of the polystyrene melt29 and φ is the volume fraction of polymer in solution. The entanglement jump time τe for the polymer can therefore be estimated from the terminal time as τe ≈ τd0(Me/M)3.6 ≈ (1.1 ( 0.4) ms, which is clearly much smaller than τfast. Archer recently reported results from a detailed rheological study of step strain relaxation dynamics in entangled PS/DEP solutions.30 The materials used in that study were in most cases identical to the ones considered here. At large shear strains, the author reported a fast and slow relaxation mode in these materials. However, the fast relaxation time was found to increase with M2. Thus, we conclude that τfast is not related to entanglement reorganization or nonlinear relaxation dynamics of the bulk PS/DEP solutions used in the present work. Several studies of semidilute polystyrene solutions in good solvents have identified a fast “gel-time” that has been associated with relaxation of concentration fluctuations in a frozen gel network.40 In concentrated solutions, this so-called gel mode is typically masked by slower reptation relaxation modes because the elastic modulus of the entanglement network is substantially larger than the osmotic modulus. Nonetheless, both the molecularweight dependence of τfast and the relaxation strength of the fast mode seen from the evanescent wave birefringence relaxation data (see Figure 4a) are consistent with expectations for a gel mode. However, the ratio of τfast to τd0 is about 10-50 times larger than expected from previous research.40 Figure 4b summarizes the effect of polystyrene molecular weight on τsurface. The plot also includes bulk zeroshear viscosity η0 data and terminal step shear relaxation times λd,bulk ≈ 2τd0 estimated from bulk τd0 data for the same materials. Again, the effect of polymer molecular (40) Stepanek, P.; Brown, W. Macromolecules 1998, 31, 1889.
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Figure 4. (a) EWLP small-amplitude step shear relaxation data for PS-1P8M. At the small strain amplitudes used in these experiments, Rωm(t) ∝ δ′(t); the experimental results were therefore normalized by Rωm(t ) ˆt) to capture the time dependence of the near-surface optical birefringence retardation δ′. (b) Molecular weight dependence of τsurface and terminal bulk material properties (λd,bulk and η0) of various 18 wt % polystyrene/ diethyl phthalate solutions at 25.5 °C. Best-fit lines through the data support the following approximate scaling relationship between material properties and PS molecular weight: τsurface(Mw) ∼ η0(Mw) ∼ λd,bulk(Mw) ∼ Mw3.6.
weight on bulk material properties follows power-law scalings, η0 ∝ λd,bulk ∼ Mw3.6, similar to those reported in the literature for entangled polystyrene melts29,37 and concentrated polystyrene/diethyl phthalate solutions.30 It is also apparent from Figure 4b that except for the fact that the experimental near-surface relaxation times are consistently about 2-3 times larger than the bulk step shear terminal times, the molecular weight dependence seen in τsurface is essentially the same as that observed in bulk. Thus, near-surface step shear relaxation dynamics in PS/DEP solutions is better described by models based on reptational diffusion of physically adsorbed polymer chains than those based on constraint release. As pointed out in the Introduction, it is possible that the quantitative differences between λd,bulk and τsurface could
Dao and Archer
arise from an enhancement of the monomeric friction coefficient near the solid substrate. Such an enhancement could result from either direct chemical interactions between molecular segments and the substrate and/or denser packing of polymer segments near the substrate. The latter explanation is perhaps preferred over the former because simulation results suggest that surface chemical effects disappear rapidly (i.e., over a distance on the order of the segment length, which is about 100 times smaller than the penetration depth of the evanescent optical field in our EWLP experiments) in polymer melts and concentrated solutions. Indeed, even the latter explanation appears inadequate when considered in the context of the apparently shorter lifetime of surface bonds between PS trains and the substrate. The three major findings for the PS/DEP solutions ((i) existence of a fast gel-like relaxation mode near the surface that is absent in bulk, (ii) apparent dominance of reptation diffusion near the substrate, and (iii) longer than expected terminal times) appear to suggest that diethyl phthalate preferentially wets the substrate. The interaction between adsorbed polystyrene molecules and the substrate therefore appears to be mediated by a highly mobile layer of solvent or semidilute polymer solution adsorbed to the substrate. Thus, unlike 1,4-polybutadiene melts, the surface renewal time for PS/DEP solutions should be quite fast, which implies that on timescales on the order of τsurface PS chains near the substrate are just “visitors”. In this scenario, the observed enhancement of nearsurface reptation time could simply result from the increased concentration of polymer in the solvent-depleted layer adjacent to the substrate but visible to the EWLP experiment. Assuming that bulk scaling relations between reptation time and concentration29 hold near the surface, an increase in the concentration of polymer within dp from 19.1% in bulk to about 29.5 vol % could account for a 2-fold difference between the measured near-surface and bulk reptation times. Such an enhancement would require an approximately 0.1dp ≈ 7 nm layer of DEP to accumulate at the substrate. This layer thickness is well within the depletion zone size Λ ≈ Rg anticipated from polymer solution thermodynamics arguments,24 lending some support to the mechanism proposed. This mechanism is also supported by earlier work by Mhetar and Archer,13 showing enhanced composition fluctuations in concentrated polystyrene/diethyl phthalate solutions near fused silica glass walls. The results are also consistent with wall slip experiments using entangled polystyrene13 and polybutadiene41 solutions, which fail to show stick-slip dynamics as in entangled melts near the critical stress for gross slip. Conclusions We have investigated step shear relaxation dynamics of 1,4-polybutadiene melts and concentrated polystyrene/ diethyl phthalate solutions near a rigid attractive substrate. Both polymer systems were formulated to span a wide range of polymer entanglement densities M/Me, at fixed entanglement molecular weight Me. To investigate near-surface relaxation dynamics in these materials, a new experimental method termed evanescent wave laser polarimetry was developed. The method relies on total internal reflection of laser light at an interface between a high refractive index transparent hemisphere and a liquid polymer to probe time-dependent, shear-induced changes in molecular orientation in a liquid layer within about 80 nm of the polymer/substrate interface. (41) Dao, T. T.; Archer, L. A. Langmuir, in preparation.
Relaxation Dynamics of Entangled Polymer Liquids
Time-dependent step strain EWLP relaxation experiments using 1,4-polybutadiene melts indicate that a single exponential function completely describes orientation relaxation dynamics of fluid near the substrate. The characteristic decay time τsurface determined from these experiments was found to be a much stronger function of bulk polymer molecular weight, τsurface ∼ Mw4.2(0.03, than the corresponding terminal properties of 1,4-polybutadiene h w3.4(0.02 and τd0 ∼ M h w3.4(0.0. Comparison melts in bulk, η0 ∼ M of the Mw dependence of τsurface with theoretical expectations for several alternative molecular mechanisms indicates that a constraint release tube renewal mechanism best explains the experimental observations. Step shear EWLP experiments using entangled PS/DEP solutions showed no conclusive difference between τsurface and τd0. Specifically, apart from a 2- to 3-fold decrease in polymer relaxation rate near the substrate, τsurface manifested a nearly identical dependence on polystyrene molecular weight as τd0 and η0. The results are consistent with expectations for a stress relaxation mechanism based
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on retarded reptation diffusion of chains near a nonadsorbing substrate. On the basis of the known favorable interaction of both polystyrene and diethyl phthalate with the substrate, we have concluded that polystyrene chains near the substrate preserve their bulk translational freedom by interacting with the substrate via a mobile layer of diethyl phthalate molecules or a semidilute PS/ DEP solution adsorbed to the substrate. Acknowledgment. We are grateful to the National Science Foundation Career Program, the Texas Higher Education Coordinating Board Advance Research Program, and the 3M Corporation Nontenured Faculty Award Program for supporting this study. We are also grateful to Mr. Randy Marek for valuable suggestions regarding design and construction of the optical polarimeter used in the study. LA0100183
