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Molecular Weight Dependence of the Shear Rheology of Poly (methyl methacrylate) Langmuir Films: A Comparison between Two Different Rheometry Techniques :: A. Maestro,*,† F. Ortega,† F. Monroy,† J. Kragel,‡ and R. Miller‡ †
Departamento de Quımica Fısica I, Facultad de Quımica, Universidad Complutense, 28040-Madrid, Spain, and :: :: ‡ MPI fur Kolloid- and Grenzflachenforschung, D14476 Potsdam/Golm, Germany Received January 23, 2009. Revised Manuscript Received February 26, 2009
The surface shear rheology of Langmuir monolayers of poly(methyl methacrylate) (PMMA) has been studied as a function of polymer concentration (Γ) and molecular weight (N). Two different rheology techniques were used, one based on free damped oscillations of a ring with a sharp edge and the other based on a forced oscillation of a biconical disk. Both instruments were used in the oscillatory mode at comparable oscillation frequency and amplitude, which gave access to the viscoelastic shear modulus (S). The two instruments, working in different viscosity ranges, provide complementary and mutually compatible data. The results obtained for four PMMA samples of molecular weight between 8 103 and 2.7 105 g 3 mol-1 show powerlike behavior as S ∼ Γ10 and S∼ N 4. These strong dependences suggest a structural scenario based on the 2D percolation of the polymer pancakes.
Introduction Polymer films at fluid interfaces have been the subject of extensive research for many years.1 The reason for these studies is that the possibility to effectively confine the polymers in twodimensional systems appears important from both the theoretical and technological points of view.2,3 Polymer monolayers adsorbed at fluid interfaces are considered indeed as a paradigmatic system for exploring soft matter physics at two-dimensional topologies.4 In the side of applications, polymer films on solid substrates are productively exploited in nonlinear optical devices, in electronics, and as biosensors.3 In most applications, correct function implies high performances in terms of thickness and chemical homogeneity. The Langmuir-Blodgett5 and the layer-by-layer2 techniques provide suitable methods for transferring well homogeneous fluid layers to solid substrates. However, optimal transfers require mechanical processability in terms of adequate film viscosity and rigidity. While the equilibrium features of insoluble polymers monolayers seem to be well understood, there are not many studies on their mechanical properties. Although the mechanical response of polymer monolayers is crucial for understanding not only the transfer process but also their own long-term stability, very little is indeed known about the rheological behavior related to the physicochemical state of the polymer film (defined in terms of surface concentration, molecular weight, and temperature). Other important processes such as foam drainage or emulsion stabilization also involve adsorbed polymer layers. These dynamical effects underlie mechanisms mainly controlled by surface tension but also by the
mechanical response of the layer against stresses induced by the flow.6 Structurally, Langmuir films represent a near flat two-dimensional topology where polymer chains are confined between two fluid phases, and consequently, they lose one degree of freedom with regard to the bulk conformation. As a result, an entropic penalty must be paid upon spontaneous adsorption, resulting anyway in a net decrease of the surface free energy which is conveniently accounted for by the macroscopic surface tension. For a Langmuir film compared to a bulk solution, the interface plays the same role as the solvent and the adsorption energy as the solvation interactions. Consequently, as for bulk solutions, it is possible to characterize the Langmuir polymer film by a “solvent quality” parameter which accounts for the balance between the polymer interface adsorption interaction and polymer cohesion. For Langmuir polymer films, it is the quality of the air-water interface as a surface solvent for the polymer that defines the distinct conformational scenarios, from a swollen random-coil at good-solvent conditions to a near-collapsed chain at poor-solvent conditions. The results of capillary-wave experiments reported by Esker and co-workers suggest that there is a strong correlation between the rheological behavior of the film and the solvent quality of the air-water interface.7,8 In the present paper, we report an experimental study on the shear mechanical properties of Langmuir monolayers of poly (methyl methacrylate) (PMMA). The equilibrium behavior of these PMMA films has been studied extensively by surface pressurearea isotherms,7,9,10 surface light scattering,8,10 ellipsometry,10,11 atomic force microscopy,11,12 and dielectric spectroscopy.13,14
*Corresponding author. E-mail:
[email protected]. (1) Scheutjens, J. M. H. M.; Fleer, G.; Cohen-Stuart, M.; Cosgrove, T.; Vincent, B. Polymer at Interfaces; Chapman and Hall: London, 1993. (2) Decher, G.; Schlenoff, J. B. Multilayer Thin Films; Wiley-VCH: Weinheim, 2003. (3) Jones, R. A. L.; Richards, R. W. Polymers at Surfaces and Interfaces; Cambridge University Press: Cambridge, U.K., 1999. (4) Forrest, J. A.; Jones, R. A. L. The glass transition and relaxation dynamics in thin polymer films. In Thin polymer surfaces and thin films; Karim, A., Kumar, S., Eds.; World Scientific Publishing: Singapore, 2000. (5) Roberts, G. Langmuir-Blodgett films; Plenum: New York, 1990. (6) Dickinson, E.; Walstra, P. Food Colloids and Polymers: Stability and Mechanical Properties; Royal Society of Chemistry: Cambridge, 1993.
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(7) Esker, A. R.; Zhang, L. H.; Sauer, B. B.; Lee, W.; Yu, H. Colloids Surf., A 2000, 171, 131. (8) Esker, A. R.; Kim, C.; Yu, H. Adv. Polym. Sci. 2007, 209, 59. (9) Vilanove, R.; Rondelez, F. Phys. Rev. Lett. 1980, 45(18), 1502. (10) Kawaguchi, M.; Sauer, B. B.; Yu, H. Macromolecules 1989, 22, 1735. (11) Kumaki, J.; Kawauchi, T.; Okoshi, K.; Kusanagi, H.; Yashima, E. Angew. Chem., Int. Ed. 2007, 46, 5348. (12) Kumaki, J.; Kawauchi, T.; Yashima, E. Macromolecules 2006, 39, 1209. (13) Forrest, J. A.; Sharp, J. S. Phys. Rev. E 2003, 67, 031805. (14) Hartmann, L.; Gorbatschow, W.; Hauwede, J.; Kremer, F. Eur. Phys. J. E 2002, 8, 145.
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From these studies, PMMA is known to adsorb at the air-water interface at poor-solvent conditions,9 and thus, PMMA Langmuir films might provide an adequate system for probing the shear properties of a two-dimensional arrangement of self-avoiding chains conformed as polymer pancakes. The dilational properties of PMMA monolayers have been previously studied from surface light scattering experiments.7,10 Cardenas-Valera and Bailey15 used the stress-relaxation method to study the dilational behavior of Langmuir films of poly(ethylene oxide) (PEO)/PMMA graft polymers. By using the torsion pendulum technique, Cardenas-Valera and Bailey15 and Peng et al.16 have also studied the interfacial shear rheology of PEO-g-PMMA spread at an oil-water interface. A strong correlation between the shear properties of the adsorbed film and emulsion stability was deduced in these works, particularly, emulsions with a higher interfacial shear viscosity were found with a higher stability. This is due to the enhancement of the mechanical resistance of the interfacial films and their ability to respond to local variations in film thickness. To our knowledge, only very few data have been published about the interfacial shear rheology of PMMA monolayers adsorbed at the air-water interface. Using a torsion surface rheometer, Peng et al.16 have studied Langmuir films of atactic PMMA with different molecular mass under different states of compression. While at low densities the torsion pendulum technique works well in oscillatory mode, at a certain stage of compression the interfacial shear viscosity became so large that the measuring body got stuck in the interfacial layer and the rotation mode had been employed. However, published data on the interfacial shear viscosity versus surface pressure dependencies show different slopes for different measuring modes, and therefore, the experimental results have been discussed for the different states of compression separately. It was also shown that the viscosity depends on the molecular weight, where very small differences were observed when comparing data at the same surface pressure. The aim of the present study is to determine the precise dependence of the interfacial shear rheological properties of PMMA Langmuir monolayers with the molecular weight (Mw) and discuss it in structural terms. Because substantial changes in viscosity are expected at increasing Mw’s, two interfacial shear rheometers combined covering a large experimental interval have been used. This study demonstrates that it is possible to obtain complementary results with two interfacial shear rheometers based on different measuring principles and different sensitivity. It will be demonstrated that the shear parameters follow scale with the molecular weight of the polymer chains.
Langmuir Polymer Monolayers Equilibrium Properties. The surface pressure of the film (Π = γ0 - γ) is measured as the decrease in surface tension (γ) with respect to the bare interface (γ0). Consequently, the equilibrium Π-Γ isotherm represents the surface energy gained upon adsorption at surface states characterized by different polymer density, Γ. At diluted states, the isotherm follows gaslike behavior, Π ∼ Γ, as corresponds to surface areas per molecule much larger than the cross section of an isolated chain, A0 ≈ πRF.2 This elemental size is determined by RF, (15) Cardenas-Valera, A. E.; Bailey, A. I. Colloids Surf., A 1993, 79, 115. (16) Peng, J. B.; Barnes, G. T.; Abraham, B. M. Langmuir 1993, 9, 3574.
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the Flory radius which, for a given polymer chain N monomers sized, scales as RF ∼N ν
ð1Þ
where ν is the Flory exponent accounting for the “thermodynamic quality” of the solvent. At good-solvent conditions, ν = 3/4, the polymer chains conform as swollen random-coils; contrarily, at poor-solvent conditions, ν = 1/2, they collapse down as flat pancakes. In compressing further, at surface areas close to A0, the chains come into lateral contact and the gel state is entered. At this overlapping regime, good-solvent conditions means extended flexible conformation and thus the chains mutually overlap, resulting in a swollen gel. However, at poor-solvent conditions, each molecule excludes out its own space for the others, rather than resulting in a percolated gel. Surface Rheology. The modes of deformation possible for a polymer monolayer adsorbed at a fluid interface can be classified into two main classes:17 (a) out-of-plane modes and (b) inplane modes, including shear and compression contributions. The transverse out-of-plane deformations result in curvature motions which are ultimately restored by surface-tension-driven Laplace forces. On the other hand, if a surfactant monolayer is present, longitudinal in-plane modes are restored by surface tension gradients. For pure compression motions, the change in surface energy upon lateral compression is accounted for by the compression modulus, K. Its equilibrium value K0 can be obtained from the relative slope of the equilibrium isotherm measured at different surface concentration, and this is18,19 K0 ¼ Γ
DΠ DΓ T
ð2Þ
The longitudinal mechanical response of Langmuir polymer monolayer is indeed characterized by two dynamic elastic moduli: K, the compression modulus, and S, the shear modulus.20 Because the dynamical response can be different depending on the time scale, each modulus is actually characterized by a storage term, which considers the elastic energy released, and an imaginary loss term, which considers the energy dissipated.21 Similarly to bulk rheology, the common notation to define the complex shear modulus S* reads: S ðωÞ ¼ G0 ðωÞ þ iG00 ðωÞ G0 þ iωη
ð3Þ
where the real element G0 is the shear storage component and the imaginary part G00 is the loss component. For a small amplitude oscillatory motion of frequency ω, the loss modulus is related to the viscous friction G00 = ωη, where η is the shear viscosity. The viscoelastic behavior of interfacial layers actually contains compression and shear components. For fluid monolayers made of small surfactants, compression usually dominates22 while shear vanishes.23 However, solid or gel-like systems (17) Kawaguchi, M. Prog. Polym. Sci. 1993, 18, 341. (18) Loglio, G.; Teseo, U.; Cini, R. J. Colloid Interface Sci. 1979, 71, 316. (19) Hilles, H.; Maestro, A.; Monroy, F.; Ortega, F.; Rubio, R. G.; Velarde, M. G. J. Chem. Phys. 2007, 126(12), 124904. (20) Miller, R.; Wfistneck, R.; Kruegel, J.; Kretzschmar, G. Colloids Surf., A 1996, 111, 75. (21) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; Wiley: New York, 1980. (22) Monroy, F.; Giermanska-Kahn, J.; Langevin, D. Phys. Rev. E 1999, 60(6), 7164. (23) Cicuta, P.; Stancik, E. J.; Fuller, G. G. Phys. Rev. Lett. 2003, 90, 236101.
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support shear elasticity, with S in these cases being a relevant contribution.24 For dense polymer monolayers, as those studied here, shear effects could indeed become comparable to the compression component. Most techniques of interfacial rheology deal with longitudinal deformations applied directly onto the interface, for example, relaxations after a sudden compression of the monolayer19,25,26 or oscillatory barrier.19,27,28 Other classes of techniques are based in the coupling between capillary waves and the longitudinal ones, for example, electrocapillary waves (ECW)29 and light scattering of thermally excited capillary waves (SQELS).30 The first two techniques are used in the low frequency range (below 1 Hz), while those based on capillary waves probe high frequencies (10 Hz -1 kHz for ECW and above 1 kHz for SQELS). Determination of Surface Shear Rheology. It is very useful to use a oscillatory test to determine the interfacial shear properties of the polymer monolayers31 and to present the experimental results in terms of the components of the complex modulus: the interfacial shear storage modulus G0 and the interfacial shear loss modulus G00 . According to the oscillatory methodology used, the dynamic surface modulus is defined as the proportionality factor related to the stress and the strain:21 σ ¼ S u
ð4Þ
where u = u0 eiωt is the applied sinusoidal strain of frequency, ω, and amplitude, u0. σ = σ0 ei(ωt + δ) is the resultant stress at the same frequency with the amplitude σ0, and δ the phase shift relative to the strain. Rearranging eq 3 and using the assumpiδ(ω) = tions of the oscillatory test, we can obtain S* = σ0u-1 0 e 0 00 G (ω) + iG (ω). Finally, each dynamic measurement at a given frequency provides simultaneously two independent parameters G0 and G00 related by tan δ = G00 /G0 . σ0 cosðδÞ ð5Þ G0 ¼ u0 σ0 G ¼ sinðδÞ u0 00
ð6Þ
Two different interfacial shear rheometers have been used in this work, both of them operating in an oscillatory mode. We have verified that the influence of the bulk motion on the dissipation of the interfacial stress can be neglected for the two techniques.32 The degree of hydrodynamic interaction between the monolayer and the subphase is conveniently characterized by the Boussinesq number Bo.32,33 Bo ¼
interfacial viscosity η ¼ bulk viscosity length scale ðη1 þ η2 ÞR
ð7Þ
(24) Maru, H. C.; Mohan, V.; Wasan, D. T. Chem. Eng. Sci. 1979, 34, 1283. (25) Monroy, F.; Hilles, H. M.; Ortega, F.; Rubio, R. G. Phys. Rev. Lett. 2003, 91(26), 268302. (26) Hilles, H. M.; Ortega, F.; Rubio, R. G.; Monroy, F. Phys. Rev. Lett. 2004, 92(25), 255503. (27) Hilles, H.; Monroy, F.; Bonales, L. J.; Ortega, F.; Rubio, R. G. Adv. Colloid Interface Sci. 2006, 122, 67. (28) Hilles, H. M.; Sferrazza, M.; Monroy, F.; Ortega, F.; Rubio, R. G. J. Chem. Phys. 2006, 125(7), 074706. (29) Monroy, F.; Munoz, M. G.; Rubio, J. E. F.; Ortega, F.; Rubio, R. G. J. Phys. Chem. B 2002, 106(22), 5636. (30) Munoz, M. G.; Encinar, M.; Bonales, L. J.; Ortega, F.; Monroy, F.; Rubio, R. G. J. Phys. Chem. B 2005, 109(10), 4694. (31) Brooks, C. F.; Fuller, G. G.; Frank, C. W.; Robertson, C. R. Langmuir 1999, 15, 2450. (32) Edwards, D. A.; Brenner, H.; Wasan, D. T. Interfacial Transport Processes and Rheology; Butterworth-Heinemann: Boston, 1991. (33) Boussinesq, M. J. Ann. Chim. Phys. 1913, 29, 349.
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where η1 and η2 are the lower and upper fluid viscosities, respectively, η is the interfacial shear viscosity, and R is the characteristic distance of the flow geometry. In our devices, R is the radius of the probe located at the interface. For small Boussinesq numbers (Bo , 1) the interfacial flow is controlled by the bulk phase stresses. For intermediate values of Bo, subphase contributions to the interfacial viscosity are significant. Finally, for high values of Boussinesq number (Bo . 1), the bulk viscous effects are negligible and, as a result, the interfacial flow dominates over the flow in the subphase. For the two techniques used here, we have confirmed that Bo . 1. This fact means that the interfacial flow is predominant to the flow of the adjacent phases and it is possible to measure the viscoelastic properties of the interface, considered as an isolated twodimensional fluid.
Materials and Methods Chemicals. The four different samples of poly(methyl methacrylate)-atactic (PMMA) used in this study were obtained from Polymer Source, Canada, with molecular weights ranging from 80 103 to 270.8 103 g 3 mol-1. The microstructure of these atactic samples has the following percentages: syndio about 56%, hetro 38%, and iso 6%. Scheme 1 shows the constitutive unit of this polymer, and the properties of these samples have been summarized in Table 1. Chloroform (Sigma Aldrich, 99% purity) was used as spreading solvent. The concentration of the spreading solution was 0.01 mg 3 mL-1. Double distilled and deionized water from a Milli-Q-RG system was used to prepare the subphase, having a resistivity higher than 18 mΩ 3 cm-1 and a surface tension of γ = 72.6 mN 3 m-1 at 20 °C. For the equilibrium measurements, the spreading solution was slowly applied by a micro syringe at different places onto the surface. The surface concentration was changed by subsequent additions of the polymer solution, waiting for adequate time to reach the equilibration of surface pressure. Surface Pressure-Surface Concentration Isotherms. The surface pressure versus surface concentration isotherms Π(Γ) of polymer monolayers were performed on a commercial Teflon Langmuir trough, Nima 720 model (U.K.). A Pt Wilhelmy balance placed at the air-water interface was used as the surface force sensor. The temperature of the monolayers was controlled by a thermostat and a water jacket of the trough. Near the interface, the temperature was measured with a precision of 0.01 °C, and the temperature control was better than (0.05 °C. Care was taken to avoid any changes in the height of the monolayer during the experiments due to evaporation. For each experiment, the surface concentration Γ was increased by subsequent additions of the polymer solutions, using a Hamilton microsyringe. After solvent-evaporation, equilibration times of at least 30 min were required before the surface pressure was measured (more than 45 min was necessary for highly concentrated monolayers). Each Π-value was determined with a precision of ( 0.05 mN 3 m-1. Interfacial Shear Rheology. Two different experimental devices have been used to measure the interfacial shear storage modulus, G0 , and loss modulus, G00 , of PMMA monolayers at the air-water interface. The first is the Interfacial Shear rheometer ISR-1, from Sinterface, consisting of a ring with a sharp edge hanging on a tungsten torsion wire. After applying an impulsive torque by an instantaneous movement of the torsion head, the pendulum generates damped oscillations characterized by a damping factor R and a radian frequency ω.34 The angular position of the measuring body is registered by means of a mini laser and a position-sensitive photodiode with an accuracy (34) Kragel, J.; Siegel, S.; Miller, R.; Born, M.; Schano, K. H. Colloids Surf., A 1994, 91, 169.
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Scheme 1. Chemical Structure of the Repetitive Units of Poly(methyl methacrylate)
follows: trough inner radius, R1 = 40 mm; whole trough height, h = 45 mm; disk radius, R2 = 34.14 mm; and cone angle, R = 5°. For measurements at the air-liquid interface, the cell is filled to a height of about 25 mm. Also, a Peltier element is used to control the temperature of each experiment. The direct DSO mode allows us to work with a low angular resolution (0.1 μrad) and low torques (0.01 μN 3 m with DSO)36,37 and to perform measurements of the rheological parameters with the lowest detection limits for the interfacial shear viscosity of about 10-2 mN 3 s 3 m-1 and interfacial shear elasticity modulus of about 10-2 mN 3 m-1. The maximum measurable values for both parameters are higher than those for the ISR-1.
Table 1. Properties of Different Samples of Atactic Poly(methyl methacrylate)
Results Equilibrium Properties. Figure 1 shows the dependence of the Π-Γ isotherms for the PMMA monolayers of different molecular weights spread at the air-water interface at 25 °C. The isotherms display similar qualitative behavior, however shifting at lower polymer concentration as molecular weight increases. This behavior points out the strong dependence of the different concentration regimes on the chain size. Three welldefined concentration regimes are clearly visible in the log-log plots (see inset): (a) A diluted regime at low pressures (Π < 1mN 3 m-1), which displays an N-dependent shape as a consequence of the influence of the interchain pair potential. In this regime, Π follows ideal-gas behavior at high dilution, Π ∼ Γ1. (b) At Γ = Γ*, defined as the overlapping concentration, the adsorbed chains come into lateral contact and a semidilute regime is entered. From a structural point of view, the chains become strongly packed in this concentration regime (Γ* > Γ > Γ**), and consequently, Π strongly increases. (c) At Γ > Γ**, the concentrated regime is entered. Here, the surface solvent is almost excluded and the film becomes purely polymeric. At this state (Γ > Γ**), the isotherm reaches a plateau value corresponding to the saturation of the film. For the present system, no clear dependence of the plateau pressures on the molecular weight is observed (Πplateau ≈ 16 ( 1 mN 3 m-1), suggesting the structural similarity of the concentrated regime with the bulky molten state, where chains mutually overlap and each one forgets its own entity in a continuous entangled state. The overlapping concentration Γ* has fundamental importance because it provides direct experimental values for the Flory radius of the coils:
commercial Name
Mw (103 g 3 mol-1)
Mn (103 g 3 mol-1)
Mw/Mn
PMMA10 P4938MMA P3243MMA P5177MMA
270.8 159 104.3 80
146.4 105 64.6 40
1.85 1.5 1.61 2.0
of (0.01°. The mathematical relationships for an oscillating torsion pendulum in surface films was derived by Tschoegl.35 The measuring cell consists of a circular trough. The geometric dimensions of the device are as follows: trough inner radius, R1 = 35 mm; measuring body radius, R2 = 25 mm. The measuring cell is surrounded by a temperature controlled water jacket. The accessible range of data for the interfacial shear viscosity is from 10-4 to 1 mN 3 s 3 m-1 and for the interfacial shear elasticity from 10-3 to 1 mN 3 m1. The registered oscillation curve is fitted with a parameter model, and consequently, R and ω of the torsion oscillation are determined.34 From the difference in the values of the two variables in relation to those for the oscillation in a pure air-water interface, the rheological parameters are calculated via the following expressions: η ¼ 2HS Ir ðR -R0 Þ
ð8Þ
G0 ¼ HS Ir R -R0 2 þ ω2 -ω0 2
ð9Þ
where η is the shear viscosity modulus, G0 is the shear storage modulus, HS is a device constant, which depends on the shear field geometry, Ir is the moment of inertia of the measurement system, and R0 and ω0 are determined from calibration measurements at the air-water interface. The second rheometer is an Interfacial Shear rheometer (model MCR301-IRS) from Anton Paar. The set up consists of a biconical disk rigidly coupled with a driven motor and also a torque and normal force transducer unit. The edge of the disk is placed in the interface between the two different fluids, airliquid or liquid-liquid. Both rotational and oscillatory mode can be performed using this instrument. For the best sensitivity in the measurements, the direct strain oscillation (DSO) mode was selected. The device is able to measure in stress- and strain-controlled oscillation experiments, while we used the strain-controlled option. In order to perform reproducible measurements, the biconical disk must be placed correctly in the interface. In the positioning procedure, the biconical disk is lowered slowly to the interface while the normal force is measured. When the edge of the biconical disk touches the surface a jump in normal force is observed. The positioning height can be calculated from the point of contact and the geometry of the disk.36 The measuring cell consists of a circular trough made from glass. The geometric dimensions of the device are as (35) Tschoegl, N. W. Kolloid-Z 1961, 19, 181. :: (36) Erni, P.; Fisher, P.; Windhab, E. J.; Kusneszov, V.; Stettin, H.; Lauger, J. Rev. Sci. Instrum. 2003, 74(11), 4916.
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Γ ¼
1 πRF 2
ð10Þ
This is defined as the hydrodynamic radius of an isolated polymer coil, which gives an idea of the molecular dimensions in the monolayer. Figure 2 shows that the experimental values of RF scale with the chain length as eq 1. Experimentally, a scaling regime is found as RF ∼ N 0.55 ( 0.02. The Flory exponent ν ∼ 0.55 ( 0.02 is found to be compatible with poor-solvent conditions, which is structurally compatible with a flat coiled conformation as near-collapsed pancakes for which intrachain interactions dominates over polymer-surface forces. In the semidilute regime, it is usual to describe the Π-Γ isotherms as a power law:38 Π∼Γy K0 ∼yΠ∼Γy
ð11Þ
:: (37) Lauger, J.; Wollny, K.; Huck, S. Rheol. Acta 2002, 41, 356. (38) de Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979.
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Figure 1. Π-Γ isotherms of the Langmuir monolayers of PMMA at 25 °C obtained by the subsequent aliquot-addition method. Molecular weight (in g 3 mol-1): (0) 8 104, (O) 1.043 105, (4) 1.59 105, and (r) 2.708 105. (Inset) The log-log plot points out three well-defined concentration regimes: diluted (D), semidilute (SD), and concentrated (C). The overlapping concentration Γ* separates the dilute from the semidilute regimes. Γ** defines the onset of the concentrated regime.
Figure 2. Molecular weight dependence of the Flory radii as calculated from Γ* (see eq 10). Data are well described by a scaling law RF ∼ N ν with a critical exponent ν ∼ 0.55 ( 0.02, characteristic for Langmuir films at poor-solvent conditions.
where y = 2ν/(2ν - 1), with ν being the Flory exponent for the 2D radius of gyration previously defined in eq 1. From the linear slopes of the log-log plots, the power-law exponent is obtained ca. y ∼ 10 ( 2 for all molecular weights. This is compatible with a value of the Flory exponent ν ∼ 0.55 ( 0.02, corresponding to poor-solvent conditions, in agreement with previous experiments by Vilanove and Rondelez.9 Shear Viscoelasticity. The interfacial shear storage modulus G0 and the interfacial shear loss modulus G00 have been studied as a function of the surface concentration Γ for the PMMA monolayers with different molecular weights. Two different rheometers using different principles of measurement have been used: the ISR-1 working in the free oscillatory regime of the ring and the MCR301-IRS exerting a forced oscillatory stress on a biconical disk. At Γ < Γ*, the shear parameters vanish at values smaller than the instrumental sensibility, as expected for a diluted solution. The experiments have been restricted to the semidilute regime, between Γ* and Γ**. Langmuir 2009, 25(13), 7393–7400
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Experiments have been carried out under identical experimental conditions, at a frequency and strain rate compatible with the linear response. In both experiments, the frequency was fixed at 0.7 rad 3 s-1 and the strain amplitude at 4%. Figure 3 shows the interfacial shear storage modulus G0 as function of the surface concentration for the four different molecular weights studied. Experimental values of G0 are found between 10-4 and 15 mN 3 m-1, at high concentration and molecular weight. Values of the shear modulus lower than 10-4 mN 3 m-1, below the sensitivity limit of the ISR-1 rheometer, are not adequately resolved. Values higher than 1 mN 3 m-1 are measured with the MCR301-ISR rheometer. Data measured with the two rheometers are in quantitative agreement and show identical trends; thus both techniques can be considered to produce complementary and reproducible measurements. For each Mw, the shear modulus is found strongly 0 dependent on Γ, exhibiting a power-law behavior as G0 ∼ Γy . The power-law exponent is found to be y0 ≈ 10, irrespectively of the molecular weight of the polymer sample. This value is very reproducible for the different PMMA monolayers, thus suggesting a strong correlation between shear elasticity and molecular packing. For the sake of comparison, according to eq 3, the compression modulus, K0, has been calculated from the slope of the equilibrium Π-Γ isotherms. The calculated values of K0 in the semidilute range are plotted in Figure 3 as a function of the polymer concentration. Similarly to the shear modulus, a power-law behavior is found as K0 ∼ Γ11(3, which is in agreement with the power law described in eq 10, K0 = Γ(∂Π/∂Γ) = y Π ∼ Γy, and with the Γ-dependence of the shear modulus, thus suggesting a similar energetic mechanism for hydrostatic and shear elasticity. However, while the values of K0 are found to be essentially independent of the molecular weight, those of G0 display a strong increase for larger Mw’s. Consequently, for molecular weights, high enough shear and compression elasticity are found to be progressively similar (see Figure 3). Furthermore, while K0 is found to be essentially chainsize independent, K0 ∼ N 0, G0 displays scaling behavior. As discussed below, if data at equal packing (constant Γ) are compared, the shear modulus shows a strong increase with the molecular weight. Our results confirm that the compression modulus dominates the viscoelastic response but the system also supports shear elasticity which eventually becomes similar to hydrostatic compression at molecular weights high enough. The dominance of compression over shear is a well-known fact already described for different surface systems in the classical textbook by Edwards, Brenner, and Wasan.32 Recently, this dominance also has been discussed for protein monolayers by Cicuta and Terentjev.39 Figure 4 shows the shear loss modulus G00 as a function of the surface concentration. For comparison, data corresponding to two different rheometers are plotted. The lower G00 values were measured with the more sensible ISR-1 rheometer , while all values higher than 1 mN 3 m-1 were measured with the MCR301IRS rheometer. Data obtained by the two different devices are found in quantitative agreement. Similarly to the storage modulus,0 0 the shear viscous losses display power-law behavior G00 ∼ Γy , with an experimental exponent ∼10 for all molecular weights. It is noteworthy that, for a given packing state, values of the loss modulus are found similar to the shear elasticity, that is, G0 ≈ G00 . As for G0 , the absolute values of G00 are found strongly dependent on the molecular weight. (39) Cicuta, P.; Terentjev, E. M. Eur. Phys. J. E 2005, 16(2), 147.
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Figure 3. Shear (G0 ; symbols) and compression (K0; dashed lines) elasticity as a function of the polymer concentration in the semidilute regime. Each panel corresponds to a sample with different molecular weight. From left to right: 2.708 105, 1.59 105, and 8 104 g 3 mol-1. Measurements were made with two different rheometers: (O) ISR from Sinterface0and (9) MCR301 from Anton Paar. Straight lines represent power-law fits of the shear modulus against the polymer concentration, G0 ∼ Γy , with exponent ∼10 ( 1 in all cases.
Figure 4. Shear loss modulus (G’’) as a function of the polymer concentration in the semidilute regime. Each panel corresponds to a sample with different molecular weight (as in Figure 3). Measurements were made with two different rheometers: (O) ISR from Sinterface and (9) 00 MCR301 from Anton Paar. Straight lines represent power-law fits of the shear modulus against the polymer concentration, G00 ∼ Γy , with exponent ∼10 ( 1 in all cases.
Discussion Equilibrium Behavior. From a conformational point of view, an adsorbed polymer chain is considered as the result of the interplay between different forces, monomer-monomer and adsorption forces between the polymer and the interface.40 From the Π-Γ isotherms, the poor-solvent character of the airwater interface for PMMA has been evidenced, as deduced from values of the Flory exponent, calculated from the G-dependence of the surface pressure (Π ∼ Γ10, thus ν = 0.55 ( 0.02) and from the chain dimensions deduced from Γ* (RF ∼ N 0.55). From these data, the flattened pancake conformation is expected.38 This structural picture agrees with the conclusions drawn by Yu and co-workers7,10 about the influence of
solvent quality on the structure and dynamics of insoluble polymer monolayers. Because each chain excludes out its own volume, PMMA monolayers above Γ* can be structurally conceived as a percolation network, viewed as a disordered gel phase of connected chains thus characterized by a finite shear modulus, S. Thus, when the monolayer is compressed from the diluted state, a sol-gel transition is expected at a percolation threshold in the polymer fractional area defined as φ = A0/A.41 Figure 5 shows the dependency of the Π isotherms on the fractional area, φ. This plot shows the fact that when each isotherm is rescaled by the size of its structural unit, A0 = πR2F, all of them merge in a near-universal isotherm dependent on the rescaled density φ. Therefore, the existence of a crossover from a
(40) Monroy, F.; Ortega, F.; Rubio, R. G.; Velarde, M. G. Adv. Colloid Interface Sci. 2007, 134, 175.
(41) Stauffer, D.; Aharony, A. Introduction to Percolation Theory; Taylor & Francis: London, 1992.
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diluted regime to a percolated one at Γ* can be plausibly hypothesized. At the percolation threshold, φc ∼ 0.78, close to the random-lattice close packing, the surface pressure is characterized by a sudden increase typical of a percolation network of compressed circles.42 Beyond the percolated threshold, φ > φc, Π can be plausibly described by the Princen’s equation of state of a disordered 2D array of soft disks:43-45 1=2 φ ΠðφÞ=ðσ=RÞ∼ ðφc -φÞ φc
ð12Þ
where the elemental interaction energy σ/R is governed by the lateral contact between neighboring discs (σ is a line tension and R the circle radius). Taking plausible values, σ ≈ 10-9N, R ≈ 10 nm, the ratio σ/R ≈ 100 mN 3 m-1. We have included in Figure 5 a prediction from the Princen’s equation in eq 12, which reproduces rather well the experimental Π-φ isotherms, making plausible the sketched percolation scenario. Shear Viscoelasticity: Power-Law Behavior. Surface shear viscoelasticity provides relevant information on the dynamics of structural changes occurred in the monolayer film. As the area is kept constant, only the shape of the interfacial area is distorted under shear.31 The absolute values of the shear parameters found here are similar to those previously reported by Saccheti et al.46 for Langmuir films of poly(tert-butyl methacrylate). These results measured by canal viscosimetry actually correspond to a polymer at good-solvent conditions. Several groups have reported shear parameters for Langmuir protein films.39,47,48 Those data are quantitatively similar to the shear viscoelasticity reported here. In the present work, we show the existence of significant shear viscoelasticity in percolated polymer gels at poor-solvent conditions. Above the overlapping threshold, shear viscoelasticity is found to increase strongly with polymer concentration. Furthermore, a strong power-law behavior was found as G0 ∼ Γ10(1and G00 ∼ Γ10(2 (see Figures 3 and 4). Similar power-law behavior is found for the compression modulus K0 ∼ Γ11(3 (see dashed lines in Figure 3). Because the powerlaw exponents are found close for every mechanical property (G0 , G00 , and K0), a similar compositional mechanism can be hypothesized under shear and compression, not only for energy storage but also for dissipation. On the other hand, in the semidilute regime, shear parameters correlate with chain size but hydrostatic compression is found however independent of the molecular weight, K0 ∼ N 0. In other words, the shear elastic energy is strongly influenced by changes in chain shape or distortions of the two-dimensional percolated gel. However, lateral compression is neither sensible to chain size nor to the relative distance between chains. Contrarily to entangled gels, for which hydrostatic and shear elasticity are governed by an Nindependent distance between entanglement points, a percolation network supports shear elasticity which is ultimately governed by the size of the structural unit. The observed elasticity suggests the plausibility of the percolation scenario instead of the entangled gel-like scenario typical of polymers at good-solvent conditions. A plausible explanation for the strong power-law dependencies found above the percolation threshold (42) Monroy, F.; Ortega, F.; Rubio, R. G.; Rittaco, H.; Langevin, D. Phys. Rev. Lett. 2005, 95, 056103. (43) Princen, H. M. Langmuir 1987, 3, 36. (44) Mason, T. G.; Weitz, D. A. Phys. Rev. Lett. 1995, 75, 2770. (45) Hutzler, S.; Weaire, D. J. Phys.: Condens. Matter 1995, 7, L657. (46) Sacchetti, M.; Yu, H.; Zografi, G. Langmuir 1993, 9, 2168. :: (47) Kragel, J.; Derkatch, S. R.; Miller, R. Adv. Colloid Interface Sci. 2008, 144, 38. (48) Bantchev, G. B.; Schwartz, D. k. Langmuir 2003, 19, 2673.
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Figure 5. Experimental isotherms of Figure 1 replotted in terms of the fractional area occupied by the polymer φ (φ = A0/A = A0Γ; symbols as in Figure 1). Below a percolation threshold, φ < φc, the isotherms obey gaslike behavior. Above φc, the Π data follow the prediction from the Princen’s equation (eq 12) for φc = 0.78.
Figure 6. Chain size dependence of the surface shear parameters: (a) shear elasticity modulus and (b) loss modulus. Scaling description in the semidilute regime G0 ∼ N 4.1(0.1 and G00 ∼ N 4.2(0.1.
is the high cooperativity between the interacting pancakes in the percolation network. Molecular Weight Dependence: Scaling Behavior. The well-known scaling of the shear viscosity of bulky polymer solutions, η ∼ N3.4, has been traditionally invoked as a strong evidence for the existence of entanglements in polymer gels21,49 and of a flow mechanism dominated by terminal reptation motions as described by de Gennes.38 Our experiments point out a stronger dependence of the shear viscoelasticity on the molecular weight of the PMMA. From the results in Figures 3 and 4, we have interpolated data at a constant surface coverage compatible with the onset of the concentrated regime; that is, at Γ = Γ**, one gets φ = φ** ≈ 1. As shown by Figure 6, the viscoelastic parameters (G0 and G00 ) at the packed state φ** follows a well-defined scaling as G0 ∼ N 4.1(0.1 and G00 ∼ η ∼ N4.2(0.1. While the experimental values of the loss modulus are characterized by a scaling exponent higher, but similar, to the prediction Grep00 ∼ ηrep ∼ N3 from the reptation model,50,51 the storage modulus shows an amazing scaling G0 ∼ N 4 which is at (49) Watanabe, H. Prog. Polym. Sci. 1999, 24, 1253. (50) Gennes, P. G. d. J. Chem. Phys. 1971, 55, 572. (51) Doi, M.; Edwards, S. F. The theory of polymer dynamics; Clarendon: Oxford, 1986.
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odds not only with theoretical predictions Grep0 ∼ N 0 but also with experiments on polymer solutions, where the storage modulus is almost found nearly independent of the molecular weight.21 The above sketched percolation mechanism compatible with a structural scenario based on collapsed pancakes could underlie the amazing viscoelastic behavior observed for PMMA Langmuir monolayers.
Conclusion The equilibrium and rheological properties of PMMA monolayers have been measured for four samples with different molecular weights. From the overlapping concentrations and the power-law description of the equilibrium Π-Γ isotherms in the semidilute regime, a conformational scenario as collapsed pancakes at poor-solvent conditions (ν ∼ 0.55 ( 0.02) has been plausibly hypothesized. The surface shear mechanical properties
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have been measured by two different rheology techniques which get access to complementary experimental ranges. The results obtained by both techniques are mutually equivalent and have allowed for obtaining the shear storage modulus G0 and the viscous losses G00 for the different molecular weights at different concentrations in the semidilute regime. The viscoelastic properties measured upon linear shear show strong power-law dependencies on both polymer concentration ∼ Γ10 and chain size ∼ N4. From this behavior, a percolation structural scenario has been suggested. Acknowledgment. We gratefully thank R. G. Rubio for fruitful discussions. The work was financially supported by a project of the European Space Agency (FASES MAP AO-99052), the DFG SPP 1273 (Mi418/16-1), and the DAAD bilateral project Acciones Integradas Hispano Alemanas and MICINN (CTQ2006-06208). A.M. benefits from a FPU-grant by the Spanish MICINN.
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