Collapse of Polyelectrolyte Brushes Probed by Noise Analysis of a

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Collapse of Polyelectrolyte Brushes Probed by Noise Analysis of a Scanning Force Microscope Cantilever Martin Gelbert, Markus Biesalski, Ju¨rgen Ru¨he, and Diethelm Johannsmann* Max-Planck-Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany Received December 21, 1999. In Final Form: March 24, 2000 We report on the dynamics of surface-attached polyelectrolyte (PEL) brushes under compression and varying conditions of added salt and pH. Fluctuations of the brush thickness are probed by bringing the sample in contact with a small sphere attached to the cantilever of a scanning force microscope (SFM). The noise power spectral density (PSD) of the cantilever motion reflects the interaction between the surface-attached polyelectrolyte layer and the sphere. The brush-sphere system mimics the interaction between a brush-coated solid wall and a colloidal particle. A shrinkage of the brush height was induced by addition of salt and variation of the pH. Close to the collapse the brush is highly compressible with an increased dissipation between the brush and the cantilever. This suggests that polyelectrolyte brushes under near-collapse conditions could be efficient rheology modifiers for colloidal dispersions.

Introduction Tethered polymer chains can provide a convenient and well-defined model system for a large class “soft interfaces”. These soft surfaces are characterized by structural integrity, on one hand, and swellability, compressibility, permeability, and configurational variability, on the other. A common feature of these layers is their covalent attachment to the substrate, which prevents desorption, delamination, or dewetting processes frequently encountered with physisorbed layers. Stability being guaranteed by the chemical bond, the layers are in a liquidlike state in most other respects. Entropic forces usually govern their behavior. Tethered polymers have attracted a lot of interest with regard to their structure, their dynamics, and their response to vertical compression or shear.1 Polymer “brushes”, in particular, have been intensely investigated.2-5 In these densely grafted systems the osmotic pressure inside the polymer layers is so high that the chains actually stretch away from the surface. The attachment of the chains to the substrate leads to polymer conformations and dynamics much different from the bulk. Additional features arise when the polymer chains carry charge. The thickness of polyelectrolyte brushes then depends not only on chain length and grafting density but also on ion concentration6,7 and pH.8-10 The presence of added salt reduces the osmotic pressure of the counterions and thereby induces the shrinkage of the layer. By * Author for correspondence. Phone 49-6131-379 163; Fax 496131-379 360; E-mail [email protected]. (1) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman and Hall: London, 1993. (2) Milner, S. T. Science 1991, 251, 905. (3) Halperin, A. Halperin, A.; Tirrell, M.; Lodge, T. P. Adv. Polym. Sci. 1991, 100, 31. (4) Taunton, H. J.; Toprackciogly, C.; Fetters, L. J.; Klein, J. Macromolecules 1990, 23, 571. (5) Doyle, P. S.; Shaqfeh E. S. G.; Gast, A. P. Phys. Rev. Lett. 1997, 78, 1182. (6) Ahrens, H.; Fo¨rster, S.; Helm, C. A. Phys. Rev. Lett. 1998, 81, 4172. (7) Mir, Y.; Auroy, P.; Auvray, L. Phys. Rev. Lett. 1995, 75, 2863. (8) Lyatskaya, Yu. V.; Leermakers, F. A. M.; Fleer, G. J.; Zhulina, E. B.; Birshtein, T. M. Macromolecules 1995, 28, 3562. (9) Biesalski, M.; Ru¨he, J.; Johannsmann, D. J. Chem. Phys. 1999, 111, 7029. (10) An, S. W.; Thirtle, P. N.; Thomas, R. K.; Baines, F. L.; Billingham, N. C.; Armes, S. P.; Penfold, J. Macromolecules 1999, 32, 2731.

lowering the pH, one can partially neutralize the charges on “weak” polyelectrolyte brushes and thereby again decrease the brush thickness. Being tunable with regard to thickness and chain conformation by external parameters, polyelectrolyte brushes have a very intriguing potential as functional surfaces. Polyelectrolyte tethered layers are believed to play a prominent role in the mechanical and hydrodynamic interaction of biological cells with their environment.11,12 Polymer brushes rather dramatically reduce sliding friction and presumably play a major role in lubricating the joints in the human body.13,14 Brushes are also used for “steric” stabilization of colloidal dispersions15 and to adjust the rheology16,17 and the tribology13 of heterogeneous particle mixtures. When performing such functions, polyelectrolyte brushes interact with other bodies. These can be either macroscopic objects (like the balls of a ball bearing) or smaller particles (like colloidal spheres). In general, interacting brushes will experience compression. The static and dynamic behavior of neutral brushes under compression has in some depth been investigated with the surface forces apparatus.13 Here we describe an alternative approach that is experimentally rather easy and very flexible with regard to sample preparation and technical modifications. It is based on the thermally activated motion of a scanning force microscope (SFM) cantilever in contact with a brush (Figure 1). A sphere with a diameter of 10 µm is glued to the tip of the cantilever to achieve a well-defined geometry in the interaction zone. The fundamental vibration of the cantilever is a single degree of freedom and therefore is activated with an average energy of kT. For typical cantilevers this motion corresponds to a root-mean-square displacement 〈δz2〉1/2 of about 3 Å. A motion of this amplitude is easily detected with the standard optical lever technique. The fluctuation dissipation theorem states that there is a one-to-one correspondence between the noise power spectral density (PSD) of the cantilever (11) Alberts, B.; Bray, D.; Lewis, J.; Raff, M.; Roberts, K.; Watson, J. D. Molecular Biology of the Cell; Garland: New York, 1994. (12) de Kruif, C. G.; Zhulina, E. B. Colloids Surf. A 1996, 117, 151. (13) Klein, J.; Kumacheva, E.; Mahalu, D.; Perahia, D.; Fetters, L. J. Nature 1994, 370, 634. (14) Klein, J. Annu. Rev. Mater. Sci. 1996, 26, 581. (15) Pincus, P. Macromolecules 1991, 24, 2912. (16) Parnas, R. S.; Cohen, Y. Rheol. Acta 1994, 33, 485. (17) Potanin, A. A.; Russel, W. B. Phys. Rev. E 1995, 52, 730.

10.1021/la991664t CCC: $19.00 © 2000 American Chemical Society Published on Web 05/23/2000

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Figure 2. Brush-sphere system mimics the interaction between a colloidal particle and a brush-covered solid wall. Presumably, the interaction between two brush-covered spheres shows similar features.

Figure 1. Experimental setup. The dynamic information is contained in the random motion of a sphere which is in contact with the brush and suspended on the cantilever of a scanning force microscope (SFM).

movement and the cantilever’s response to a hypothetical external force.18 The power spectrum d|δz2(ω)|/dω is proportional to the imaginary part of the susceptibility χ, the latter being defined as the ratio of cantilever deflection and external force. The outcome of SFM noise analysis is equivalent to a force modulation spectrum.19-24 Importantly, the amplitude of thermally activated motion is so small that linear response holds, which is not the case in some other dynamic modes of the SFM like the tapping mode.25 A similar approach has been taken recently with colloidal spheres, the motions of which were observed with optical microscopy.26-28 In a way, SFM noise analysis is a variant of tracer microrheology, which has been reviewed in ref 29. Advantages of SFM noise analysis are that the SFM tip can be actively positioned with a piezodrive and that rather large static forces can be applied. Also, being based on an optical lever technique, the technique is very sensitive. We view the brush-sphere system as a model system for the interaction between a colloidal particle and a brush-coated wall (Figure 2). Presumably, the interaction between two brush-coated particles will display similar features as the interaction between the colloidal particle and the brush. The osmotic pressure in concentrated polymer dispersions has been measured to be well in the range of 1 atm,30 which matches the pressures exerted by the SFM cantilever. The collision frequencies in dense dispersions are in the kilohertz to megahertz range, making the individual particle collision a rather fast event. Again, SFM noise analysis covers this frequency range well. (18) See, for example: Kubo, R.; Toda, M.; Hashitsume, H. Statistical Physics; Springer: Heidelberg, 1985; Vol. 2. (19) Radmacher, M.; Tillmann, R. W.; Gaub, H. E. Science 1990, 268, 257. (20) Ratcliff, G. C.; Erie, D. A.; Superfine, R. Appl. Phys. Lett. 1998, 72, 1911. (21) Oulevey, F.; Gremaud, G.; Se´menoz, A.; Kulik, A. J.; Burnham, N. A.; Dupas, E.; Gourdon, D. Rev. Sci. Instrum. 1998, 69, 2085. (22) Roters, A.; Johannsmann, D. J. Phys.: Condens. Matter 1996, 8, 7561. (23) Roters, A.; Gelbert, M.; Schimmel, M.; Ru¨he, J.; Johannsmann, D. Phys. Rev. E 1997, 56, 3256. (24) Roters, A.; Schimmel, M.; Ru¨he, J.; Johannsmann, D. Langmuir 1998, 14, 3999. (25) Martin, Y.; Williams, C. C.; Wickramasinghe, H. K. J. Appl. Phys. 1987, 61, 4723. (26) Gittes, F.; Schnurr, B.; Olmsted, P. D.; Mackintosh, F. C.; Schmidt, C. F. Phys. Rev. Lett. 1997, 79, 3286. (27) Florin, E. L.; Pralle, A.; Horber, J. K. H.; Stelzer, E. H. K. J. Struct. Biol. 1997, 119, 202. (28) Gisler, T.; Weitz, D. A. Phys. Rev. Lett. 1999, 82, 1606. (29) Gisler, T.; Weitz, D. A. Curr. Opin. Colloid Interface Sci. 1998, 3, 586.

This paper specifically focuses on the collapse of charged brushes upon addition of salt or variation of the pH of the solution. A collapse in bad solvent has been studied in detail for neutral brushes.31 Here, the degree of swelling can be understood by minimization of a Flory-Hugginstype free energy with a repulsive three-body interaction and chain elasticity introduced as additional contributions. For charged chains the electrostatic repulsion in combination with counterion screening results in a rich phenomenology.32 Mean-field calculations show that the free energy as a function of layer thickness may have two separate minima, resulting in bistability and a first-order stretched-collapsed transition as the external parameters are changed.33-35 Even for conditions where no true firstorder transition is found, the brush thickness as a function of ion concentration or pH has a sigmoidal shape with a threshold behavior. It may be assumed that close to the threshold the brush responds very sensitively not only to variations of ion content and pH but also to vertical pressure. Rabin and co-workers have investigated the relation between the salt-induced collapse and the compressibility in more detail.36 They find a remarkable difference between the static compressibility which becomes large close to the transition and the dynamic compressibility which increases much less. They trace this difference back to a difference between the electrostatic and the hydrodynamic screening length. This prediction emphasizes how important it is to specify the time scale on which the interactions between the brush and its counterpart occur. Being dense polymeric systems, brushes will display a wide spectrum of internal relaxations. Fast compression will be resisted much stronger than quasi-static compression. SFM noise analysis is wellsuited to investigate these questions. The dynamic compressibility is essentially the inverse of the friction coefficient ξ of the cantilever and is directly accessible with the experiments described below. Materials The preparation of tethered polymer layers by the grafting from technique has been described in detail elsewhere.37,38 Figure 3 outlines the essential steps. A selfassembled monolayer of an azo-initiator is immobilized (30) Bonnet-Gonnet, C.; Belloni, L.; Cabane, B. Langmuir 1994, 10, 4012. (31) Zhulina, E. B.; Borisov, O. V.; Pryamitsyn, V. A.; Birshtein, T. M. Macromolecules 1991, 24, 140. (32) von Goeler, F.; Muthukumar, M. Macromolecules 1995, 28, 6608. (33) Ross, R. S.; Pincus, P. Macromolecules 1982, 25, 2177. (34) Borisov, O. V.; Birshtein, T. M.; Zhulina, E. B. J. Phys. II 1991, 1, 521. (35) Von Goeler, F.; Muthukumar, M. J. Chem. Phys. 1996, 105, 11335. (36) Rabin, Y.; Fredrickson, G. H.; Pincus, P. Langmuir 1991, 7, 2428. (37) Prucker, O.; Ru¨he, J. Macromolecules 1998, 31, 592. (38) Prucker, O.; Ru¨he, J. Macromolecules 1998, 31, 602.

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Figure 3. Synthesis of surface-attached polyelectrolyte brushes by the grafting from approach. As a strong polyelectrolyte a MePVP (poly(4-vinylpyridine) quarternized with methyl iodide) brushes was used. In addition, a weak PMAA (poly(methacrylic acid)) brush was generated by the same technique.

on the substrate (LaSFN9, Helma, Jena, n ) 1.844) via a monochlorosilyl moiety. Glass prisms were used as substrates because all samples were optically investigated with multiple-angle null ellipsometry9,39 prior to the SFM measurements. With this procedure the optical thickness as a function of salt concentration and pH is known. After addition of monomer and activation of the initiator layer with heat, a covalently attached polymer monolayer forms by free radical chain polymerization. During the initiation process the AIBN unit splits into two parts which both start the growth of a chain. Because one of the two parts is detached from the surface, free polymer is generated as well. It is washed away in the cleaning process and analyzed with regard to its molecular mass distribution. Experiments on similar systems have shown that the molecular mass distribution of the brush is about the same as the one of the free polymer grown from the second free radical.40 We report on brushes of strong and weak polyelectrolytes. As an example of a strong polyelectrolyte we prepared poly(4-vinylpyridine) brushes which were subsequently quarternized with methyl iodide (MePVP) to form a positively charged brush. The synthesis is described in detail in ref 41. The thickness of the MePVP brush in the dry state was Ldry ) 14 nm. The molecular mass as measured with gel permeation chromatography on the nonattached fraction of the polymer was Mn ≈ (2.6 ( 0.5) × 106 g/mol with a polydispersity index of Mw/Mn ∼ 2. From the molecular mass Mn and the dry thickness Ldry the distance between two grafting sites dg is derived as dg ) (Mn/FLdryNA)1/2 ≈ 15 nm (F ) 1.4 g/cm3 the density and NA is Avogadro’s number). As a weak polyelectrolyte we used poly(methacrylic acid) (PMAA). Details of the synthesis are given in refs 9 and 42. The polymerization of methacrylic acid from the surface-bound initiator was carried out in pure monomer (39) Habicht, J.; Schmidt, M.; Ru¨he, J.; Johannsmann, D. Langmuir 1999, 15, 2460. (40) Schimmel, M.; Ru¨he, J., submitted for publication. (41) Biesalski, M.; Ru¨he J. Langmuir 2000, 16, 1943.

at a temperature of T ) 60 °C.9 After the polymerization the prisms were extracted with methanol and water (pH ) 6) for 15 h. Methanol and water are both good solvents for the polyacid. After extraction the physisorbed polymer chains have been removed. The thickness of the PMAA brush in the dry state was Ldry ) 23 nm. The molecular mass was about Mn ≈ (2.0 ( 0.5) × 106 g/mol. With a density of PMAA of F ) 1.12 g/cm3 this results in a distance between two grafting sites of dg ≈ 11 nm. In the swollen state, the ratio between thickness and grafting distance can be up to 100. Neighboring chains strongly overlap. The systems therefore are well in the brush regime and where chain stretching is significant. Experimental Section The technique of SFM noise analysis has been described in detail in previous publications.22,23 Briefly, we used a TMX 2010 instrument (TopoMetrix) with standard silicon nitride tips. The thermal rms noise 〈δz2〉1/2 was about 3 Å, which corresponds to a spring constant of κ ) 0.05 N/m according to the relation 〈δz2〉 ) kT/κ. Only the fundamental resonance was in the accessible range of frequencies. In water the resonance frequency was about 10 kHz with a bandwidth of ∆fHBFW ∼ 10 kHz. We glued polystyrene spheres with a diameter of 10 µm (purchased from Bangs Laboratories Inc.) to the tips in order to achieve chemically well-defined surfaces and a well-defined geometry in the interaction zone. The brushes did not adhere to the polystyrene spheres as evidenced by the absence of attractive forces on separation. The viscoelastic measurements were correlated with optical measurements of the segment density profiles on identical samples. All samples were synthesized on the base of high-index prisms and investigated with multiple-angle null ellipsometry.9,39 The prisms were then sawed into the appropriate pieces and mounted in the sample compartment of the SFM. A drop of water with the pH and the salt content properly adjusted was placed between the sample and the window of the liquid cell. Since data acquisition took about 10 min, evaporation of water was not a problem. Diffusion of CO2 from the atmosphere to the sample (42) Biesalski, M.; Johannsman, D.; Ru¨he J. Langmuir, submitted for publication.


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Figure 4. Derivation of the distance parameter z. Because no absolute measure of the substrate-sphere distance is available, the distance has to be inferred from the line of constant compliance. Constant compliance, however, may be reached at different levels of compression. surface (lowering the local pH) is not a problem on these time scales, either. The vertical tip-sample distance was controlled by the z-piezo of the scanner, while the x- and the y-piezo were disconnected. A low-pass filter was inserted into the drive electronics to suppress electronic noise from the scanner. All data were taken during approach. The laser diode was replaced by a He-Ne laser fed into the sample compartment with a single mode optical fiber in order to suppress noise caused by mode hopping of the laser diode.43 While continuously approaching the sample with the tip, we monitored the noise with a Fourier analyzer (HP 35670A) connected to the analogue output of the quadrant detector. Averages of four power spectra were stored in the analyzer’s internal memory and retrieved for analysis after completion of the experiment. Data acquisition for a single data point takes about 4 s. Distance calibration is not trivial. For “hard” surfaces there is a well-defined kink in the force-distance curve which is a natural choice for z ) 0. When spheres are glued to the tip the spheres usually prevent the tip from diving into the brush. In this case such a kink should be interpreted as the tip touching the top of the brush. Note that the situation is reversed if there is no sphere glued to the tip.24 In these cases the local pressure at the tip apex is so high that the tip will usually dive into the brush unless the brush is completely collapsed. A well-defined kink in these cases occurs if the tip can dive to the bottom the brush. For brushes with a finite compressibility the choice of z ) 0 entails ambiguities. We calculated z from the difference between the force-distance curve and the “line of constant compliance”. We manually select a point on the force distance curve where we judge that the line of constant compliance has been reached and calculate a corrected tipsurface distance from the slope of that line as indicated in Figure 4.

Data Analysis For sufficiently low damping the noise power spectrum can be well fitted with resonance curves,23,44,45 where the extracted parameters are an effective spring constant κ, a friction coefficient ξ, and an effective mass m:

d|δz(ω)2| ξ ) 2kT ) dω (κ - mω )2 + ξ2ω2 2

1 (1) 2 2 2 2 2 0 - ω ) + γ ω

γ 2kT m (ω

where δz is the displacement, ω the angular frequency, (43) Cleveland, J. P.; Scha¨ffer, T. E.; Hansma, P. K. Phys. Rev. B 1995, 52, R8692.

kT the thermal energy, ω0 ) (κ/m)1/2 the resonance frequency, and γ ) ξ/m the damping constant. A fourth quantity automatically obtained is the time-averaged displacement of the cantilever 〈δz〉 which is proportional to the static force FDC. The friction coefficient (sometimes called “drag coefficient” as well) here is to be understood in the Stokes sense, where the viscous drag is proportional to the velocity of the particle. It should not be confused with the use of the term friction coefficient in tribology, where it signifies the ratio between the friction force and the normal load. For high damping the simple harmonic oscillator approximation fails. This failure could be caused either by hydrodynamic effects or by viscoelastic relaxation. In the Appendix we argue that the hydrodynamics in the gap between substrate and the sphere actually is rather simple. It is governed by “lubrication forces” first discussed by Reynolds in 1886.46 In particular, neither the mass nor the friction coefficient becomes a function of frequency for reasons of hydrodynamics alone. Therefore, we attribute the failure of the simple harmonic oscillator model to viscoelastic relaxation. The stress only partly decays on the time scale of the cantilever motion, and the friction coefficient becomes a function of frequency. For a quantitative treatment of viscoelasticity one uses the generalized Langevin equation

m

d2δz(t) 2

dt

+

∫-∞t ξ(t - t′)

dδz(t′) dt′ + κδz(t) ) R(t) (2) dt′

It is a force balance equation where t is the time and R(t) the random force. The generalized Langevin equation differs from the conventional Langevin equation by the memory integral in the friction term. The conventional Langevin equation without memory leads to the simpleharmonic oscillator model. The fact that “friction with memory” is related to viscoelastic dispersion becomes evident by considering the two limiting cases of ξ(t-t′) ) ξ0δ(t-t′) and ξ(t-t′) ≡ ξ1. In the first case the normal Langevin equation with the friction coefficient ξ0 is recovered. In the second case ξ(t-t′) ≡ ξ1 is a constant, and the integral ∫ξ1(dδz(t′)/dt′) dt′ can be trivially integrated to ξ1δz(t). The friction term turns into an elastic contribution. One could formally omit the elastic term κδz(t) completely in eq 2 and cover all elastic contributions by the nondecaying part of ξ(t-t′). However, from the experimentalist’s point of view this obscures the fact that long time scales are not accessible in the experiment described here. We therefore separately keep the elastic term. This spring constant κ includes all forces which relax slower than the experimental time scale. It can be shown that the noise power spectrum is explicitly related to the Fourier transform ξ(ω) of the memory kernel of friction ξ(t-t′). The quantity ξ(ω) ) ξ′(ω) + iξ′′(ω) is complex because ξ(t-t′) ) 0 for t < t′. One has18,47

(

i mω -

)

[

]

d|δz2(ω)| κ + ξ(ω) ) 2kT ω2 ω dω

-1

(3)

The quantity in square brackets is complex, the imaginary part being derived from its real part by the Kramers(44) Walters, D. A.; Cleveland, J. P.; Thompson, N. H.; Hansma, P. K.; Wendman, M. A.; Gurtley, G.; Ellings, V. Rev. Sci. Instrum. 1996, 67, 3583. (45) Sader, J. E. J. Appl. Phys. 1998, 84, 64. (46) Reynolds, O. Philos. Trans. R. Soc. London 1886, 177:157. (47) Gelbert, M.; Roters, A.; Schimmel, M.; Ru¨he, J.; Johannsmann, D. Surf. Interface Anal. 1999, 27, 572.

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Kronig relations. The real part of the friction coefficient ξ′(ω) is directly given by the real part of the right-hand side of eq 3. The imaginary part ξ′′(ω) is related to ξ′(ω) by the Kramers-Kronig relation

ξ′′(ω) ) -

ξ′(ω′)

∫0∞ω′2 - ω2 dω′

2ω P π

(4)

where P is the principal value. Clearly, the spectrum of the friction coefficient contains the information on the local relaxation times. For example, a Debye-type single relaxation would appear as a shoulder in a plot of ξ′(ω) vs log(ω). We searched for such characteristic individual relaxation times and could not unambiguously identify them up to now. Generally speaking, the spectrum of relaxations should be broad, and a single relaxation time is not expected. On the other hand, the spectrum of relaxations should not be featureless, either. For example, the slowest relaxation should be given by the “breathing mode”.48 The search for such modes will require longer integration times to improve the signal-to-noise ratio. At this point the integration time is limited by the drift of the piezoactuator. Given that characteristic relaxation times could not be identified, we have confined ourselves to fitting the spectra ξ′(ω) with power laws of the form

ξ′(ω) ≈ ξ0(ωref)

( ) ω ωref

γ

(5)

where ξ0 is the friction coefficient at the reference frequency ωref and γ is a characteristic exponent. As the reference frequency we chose ωref ) 2π × 1 kHz. The exponent γ typically is in the range between 0 and -0.5. After ξ′′(ω) has been calculated from ξ′(ω) with the Kramers-Kronig relations, the imaginary part of the right-hand side of eq 3 can be used to obtain the effective mass m and the spring constant κ according to

mω -

[(

)]

d|δz2(ω)| κ ) A(ω) ≡ 2kΤ Im ω2 ω dω

-1

- ξ′′(ω) (6)

While the function A(ω) is derived from the experimental data, the left-hand side provides a fit function for A(ω). The friction coefficient ξ(ω) can be viewed as an integrated effective viscosity. For example, for a sphere with radius R immersed in a homogeneous Newtonian liquid with viscosity η, one has ξ ) 6πηR. For a sphere at a distance D , R from a flat wall one has ξ ) 6πηR2/z.46,49 The latter relation is sometimes termed the “lubrication approximation” and is discussed in detail with regard to its hydrodynamic implications in the Appendix. For homogeneous liquids the lubrication approximation could be inverted to derive an “effective viscosity”. For instance, at D ) 1 µm, a friction coefficient of 1 µN m-1 s-1 translates to ηeff ∼ 2 × 10-3 Pa s. Similarly, the quantity κ + ωξ′′(ω) can be converted to an effective modulus of compression. However, because the brush clearly is an inhomogeneous system, we do not usually make this conversion and show the friction coefficient ξ and the spring constant κ rather than the viscosity and the modulus of compression. These quantities are convolutions of the intrinsic material properties with the geometry. The same convolutions with (48) Fytas, G.; Anastasiadis, S. H.; Seghrouchni, R.; Vlassopoulos, D.; Li, J.; Factor, J. C.; Theobald, W.; Toprakcioglu, C. Science 1996, 274, 2041. (49) Happel, J. R.; Brenner, H. Low Reynolds Number Hydrodynamics; Kluwer: Dordrecht, 1983.

geometry will govern the interaction between brushes and particles in realistic applications as well. From the viscoelastic spectra, we usually extract the spring constant κ, the low-frequency friction coefficient ξ0, and the exponent γ. These parameters are displayed as a function of cantilever position z. In addition, we always show the static force FDC. These “viscoelastic profiles” give insight into the dynamic aspects of the brush-sphere interaction. Results and Discussion Dependence of the Viscoelastic Spectra on the Cantilever Position. Figure 5 shows an example of the noise spectra d|δz(ω)|2/dω, the frequency-dependent friction coefficient ξ(ω), and the function A(ω) as derived from eq 5. The data were taken on a MePVP brush (Ldry ) 15 nm, Mn ) 2.6 × 106 g/mol) in aqueous solution of 0.01 mol/L potassium iodide (KI). At large distances (z ) 0.93 µm), the noise power is well described by resonance curves (solid lines in Figure 5a). The corresponding spectra of ξ′(ω) are rather flat (Figure 5b). The friction coefficient in this case is largely independent of frequency. A single parameter ξ could be used as well, and the simpleharmonic-oscillator model (eq 1) is recovered. When the sphere and the brush touch, one observes an excess noise at small frequencies which cannot be accounted for with the simple harmonic oscillator model. The friction coefficient now decreases with frequency. Evidently, there are internal modes of stress relaxation in the experimental frequency window. At high frequencies the slow modes behave elastically, and viscous dissipation becomes less efficient than at low frequencies. At the same time the elastic part ξ′′(ω) increases, as required by the Kramers-Kronig relations. In Figure 5c we show the function A(ω) and fits to κ/ω - mω. The good agreement between the experimental data calculated with eq 6 and the fit to κ/ω - mω proves the internal consistency of the formalism. Figure 6 shows the derived fit parameters from the data set of Figure 5 as a function of the cantilever position z. Note again that z is the distance from the “line of constant compliance” which is not necessarily the distance to the substrate or the brush surface. The general features are similar to what was found with neutral brushes:23,24 (i) The dissipative interaction as given by the friction coefficient ξ0 is of a longer range than the elastic interaction as given by the static force FDC or the spring constant κ. It may be mediated by the flow of solvent and does not require direct physical contact between the brush and the sphere. (ii) The range of the elastic interaction as indicated by the spring constant κ is longer than the range of the repulsive static force FDC. This is a consequence of the different time scales of measurement. The dilute outer regions of the brush more efficiently transmit highfrequency stress than low-frequency stress. On time scales less than the longest relaxation time, part of the energy is stored rather than dissipated. Viscoelastic dispersion is also evidenced by a nonzero exponent γ. (iii) The effective mass also increases upon approach because the substrate takes part in the motion to some extent. The effective mass is of minor interest in this context and is not displayed in the following. To support the hypothesis that the frequency-dependent friction coefficient ξ(ω) goes back to viscoelastic dispersion in the polymer, we performed bulk measurements on polymer solutions. We reason that chain relaxation should be observed in the bulk as well, whereas surface effects


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Figure 5. Typical data set: (a) noise power spectral density d|δz(ω)2|/dω; (b) real part of the frequency-dependent friction coefficient ξ′(ω); (c) the function A(ω) and fits with κ/ω - mω. Here and in the following graphs the data have been vertically displaced for clarity. The data were taken on a 15 nm thick MePVP brush in aqueous solution of 0.01 mol/L KI. The cantilever positions z are indicated in part c. The noise spectra are well fitted by resonance curves at large cantilever-substrate separation (z ) 0.93 µm). The friction coefficient is largely independent of frequency in this range. As the sphere comes closer to the substrate, the resonance fits describe the spectra less well. There is an excess noise at low frequencies which is caused by viscoelastic dispersion. The friction coefficient decreases with frequency. The function A(ω) is explicitly derived from the spectra with eq 6. It allows for a determination of the mass m and the spring constant κ which takes dispersion into account.

Figure 6. Set of viscoelastic parameters extracted from the spectral data in Figure 5 as a function of cantilever position z. The labels “1” to “5” indicate the corresponding spectra in Figure 5.

should be absent. The polymer used was poly(styrenesulfonate) (PSS). We immersed the cantilever into solutions of various PSS concentrations and analyzed the noise power spectra in the usual way. Note, however, that the

lubrication approximation discussed in the Appendix does not apply to measurements in the bulk. Hydrodynamic effects come into play more prominently since the entire cantilever contributes to the signal. Still, the phenomenology of the noise spectra (Figure 7) and the derived parameters (Figure 8) shows a remarkable resemblance to the spectra obtained when approaching the sphere to a brush. In particular, viscoelastic dispersion is found in concentrated solution as well. Salt-Induced Collapse of a Strong Polyelectrolyte Brush. The collapse behavior of strong MePVP brushes upon the addition of potassium iodide has been in detail investigated with ellipsometry. The dependence on graft density, chain length, and the type of salt will be given in a later publication.50 It is known from studies on the bulk material that the addition of salt not only affects the Debye screening length but also the amount of charge on the chain.51 (Even charge overcompensation is found at ion concentrations beyond the ones shown here. This effect is specific to the addition of KI. It does not occur in the same way with other salts.) This neutralization effect leads to a steep collapse when adding potassium iodide to the aqueous solution. In Figure 9 we show the ellipsometric thickness as a function of salt concentration. In the presence of salt the brush height shrinks to almost the dry thickness. The collapse concentration of KI is about 0.05 mol/L. Given that the collapse is so steep, we looked into the question of whether the collapse was a true first-order stretched-collapsed transition.33-35 Such a discontinuous transition may arise for systems with hydrophobic backbones which are poorly soluble in water. Note that the backbones of polyelectrolytes based on vinyl polymers are rather hydrophobic. Water usually is a bad solvent for the (50) Biesalski, M.; Johannsmann, D.; Ru¨he, J. Manuscript in preparation. (51) Beer, M.; Schmidt, M.; Muthukumar, M. Macromolecules 1997, 30, 8375.

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Figure 7. Data set taken on a bulk polymer solution at various concentrations: (a) noise power spectral density d|δz(ω)2|/dω; (b) real part of the frequency-dependent friction coefficient ξ′(ω); (c) the function A(ω) and fits to κ/ω - mω. The data were taken on a bulk solution of sodium poly(styrenesulfonate) salt (NaPSS). They look qualitatively similar to the spectra obtained when approaching the brush. In particular, viscoelastic dispersion is also observed. This proves that viscoelastic dispersion is caused by relaxation processes in the polymer.

Figure 9. Ellipsometric thickness of a MePVP brush as a function of salt concentration. The brush collapses at about cs ) 0.05 mol/L KI. Figure 8. Set of viscoelastic parameters extracted from the spectral data in Figure 7. The NaPSS concentrations are indicated at the bottom. Several values are given for each concentration to indicate the range of scatter.

backbones, the solubility of the polymer being brought about by the side groups and the electric charge. While the charges on the chain feel a repulsion, the uncharged portions of the chain attract each other. The functional form of the two terms in the free energy is such thatsfor certain parameterssthe free energy as a function of brush thickness has a double-well shape with two separate minima (Figure 10a). We are not aware of direct experimental proof for bistable swelling behavior, although a hysteresis was found in experiments with the surface forces apparatus, which may be indicative of an underlying first-order transition.52 Clearly, the question of compressibility is linked to the collapse behavior. The compressibility essentially is the inverse second derivative of the free energy with respect

to thickness. When the minimum of the free energy is shallow (Figure 10a), the compressibility is large. If no true double-well potential develops, the minimum of the free energy may still be broad in the transition range. In this case the brush thickness as a function of the driving parameter (salt content or pH) varies in a sigmoidal form with a threshold dividing the collapsed and the expanded regime (Figure 10b). Close to the threshold one expects a large compressibility as well. We could not find evidence of a true first-order stretched-collapsed transition for the MePVP brushes investigated here. Still, we observe a sigmoidal dependence of the thickness on the ion concentration, which should be associated with a large compressibility. We suspect that polydispersity in our case tends to smoothen out the first-order transition. (52) Watanabe, H.; Patel, S. S.; Argillier, J. F.; Parsonnage, E. E.; Mays, J.; Dan-Brandon, N.; Tirrell, M. Mater. Res. Soc. Symp. Proc. 1992, 249, 255.


Langmuir, Vol. 16, No. 13, 2000 5781

Figure 10. (a) Free energy as a function of thickness for polyelectrolytes brushes with hydrophobic backbones. Because of the hydrophobicity there is a tendency toward the formation of a double-well potential (dotted line). Even if no true bistability exists, the minimum of the free energy will still be broad close to the collapse condition, resulting in a high compressibility. The width of the central minimum is related to the steepness of the stretched-collapsed transition (b).

In Figure 11 we show the viscoelastic profiles at four selected ion concentrations of “Milli-Q” (that is Milli-Q water with no added salt), cs ) 0.01 mol/L, cs ) 0.03 mol/L, and cs ) 0.1 mol/L KI. The results are summarized as follows: (i) At no point does the viscoelastic profile extend to a length corresponding to the optical thickness of the brush of 1 µm. The viscoelastic effects appear in a range of up to 200 nm above the line of constant compliance. This can only be understood if the line of “constant” compliance is reached somewhere close to the top of the brush. The modulus of compression of the brush times the contact area must exceed the spring constant. This argument can be made more quantitative by assuming that the mechanical strength in the fully expanded state is given by the osmotic pressure of the counterions. This corresponds to the osmotic brush regime.15 The counterion concentration is given by F ∼ Ldry/Lswollen (3Vmon)-1 ∼ 0.05 mol/L (Vmon is the monomer volume, and the factor of 3 accounts for the fact that there is about one effective charge per three segments). The osmotic pressure of the counterions is given by p ) FkT ) 105 Pa. The isothermal modulus of compression of an ideal gas KT ) V(dp/dV)T is equal to the pressure p. The contact area can be estimated from the chord theorem53 as Ac ∼ 2πR∆z with ∆z the amount by which the sphere has penetrated into the brush and R the radius of the sphere. Using ∆z ∼ 100 nm (see Figure 11), one arrives at Ac ∼ 3 µm2. At an impression of ∆z ∼ 100 nm the net vertical force exerted by the brush is of the order of KTAc∆z swollen ∼ 40 nN. This exceeds the static force of about 5 nN and the brush appears as rigid. (ii) At a moderately low salt concentration of 0.01 mol/L, the profiles most closely resemble a soft brush. In this range the compressibility is comparable to the inverse spring constant of the cantilever. One also finds a clear increase of the spring constant. By far the most striking (53) Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1992; p 143.

Figure 11. Viscoelastic parameters measured during approach to a MePVP brush under various conditions of added salt. Data have been vertically displaced for clarity. The friction coefficient is increased at intermediate salt concentrations.

feature is the large increase of the friction coefficient. This confirms the predictions by Rabin et al.,36 who concluded that the friction coefficient (or the inverse dynamic compressibility) does not follow the static elastic behavior. At a concentration of 0.01 mol/L the brush has softened to the extent that the sphere can compress the brush. The brush does, however, still exert a large amount of friction. (iii) At a moderate salt concentration of cs ) 0.03 mol/L the profile of the static force also resembles that of a hard surface. However, there is now some upward bending before the line of constant compliance is reached. Also, the profile of the friction coefficient ξ1kHz at this point still is significantly different from the profile at cs ) 0.1 mol/L, where the collapse is complete. Interestingly, the ellipsometric data indicate that the brush is still expanded at this state. Apparently, the brush at cs ) 0.03 mol/L is so compressible that the cantilever cannot pick up mechanical resistance. Whether this is a consequence of increased electrostatic screening or partial neutralization cannot be inferred from the data. (iv) At a high salt concentration of cs ) 0.1 mol/L one finds what one expects for a fully compressed layer. The slope in the static force curve is caused by a system drift. There is a small increase in the friction coefficient at z < 100 nm caused by the conventional lubrication force always found on solid surfaces (see Appendix). pH-Induced Collapse of a Weak Polyelectrolyte Brush. On weak polyelectrolyte brushes the charge on the chain is not constant but subject to recombination with ambient counterions. The most prominent example

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Figure 12. Ellipsometric thickness of a PMAA brush as a function of pH. The brush collapses at about pH ) 4, which corresponds to the pKa of PMAA.

Gelbert et al.

(ii) The friction coefficient ξ shows a very pronounced increase at intermediate pH. It is a maximum at pH ) 6. This is another manifestation of the prediction by Rabin and co-workers. An increased static compressibility does not necessarily imply a large dynamic compressibility as well. On the contrary, the viscoelastic profiles display an extended range of a substantially enlarged friction coefficient. The increased friction is the result of the interplay between a large static compressibility allowing for a significant impression of the sphere into the brush, on one hand, and a small dynamic compressibility resisting fast movements of this kind, on the other. (iii) From the force-distance profile one would judge that the brush collapses between pH ) 6 and pH ) 5. This again is a variance with the ellipsometric data, which show a threshold around pH ) 4. As in the case of the salt-induced collapse, we argue that the brush is highly compressible before it actually collapses, such that forcedistance curves give a misleading picture of the equilibrium brush thickness. (iv) For the collapsed brushes (pH ) 3) we find adhesion on separation. This aspect will be described in detail in a separate publication.56 Apparently, the hydrophobic chains adhere to the polystyrene sphere provided that they are only weakly charged. This interplay between static and dynamic compressibility should have applications in the rheology of dispersions of spheres coated with such brushes. Under conditions far away from the collapse, the interaction potential will be rather steep. Particle collisions will not be accompanied by large amounts of dissipation. When the brushes are softened by bringing them closer to collapse two things will happen: first, the overall density of the dispersion will increase, because the particles can approach each other more closely. Second, the viscosity of the dispersion will be much enlarged because of the high dissipation encountered during particle collisions. Also, viscoelastic dispersion will manifest itself not only on the local but also on the global scale. At frequencies in the kilohertz range a substantial shear elasticity is expected. Conclusions

Figure 13. Viscoelastic parameters measured during approach to a PMAA brush under various conditions of pH. The friction coefficient is increased at intermediate pH.

of weak polyelectrolytes are polyacids, which are only charged at a pH above the pKa. If the abundance of protons is too high, they will recombine and the chain becomes neutral. Figure 12 shows the ellipsometric thickness of the weak polyacid PMAA as a function of pH. As expected, the thickness decreases below the pKa of PMAA, which is about 4.3.54 The increase is more gradual than in the saltinduced collapse.9 In Figure 13 we show the corresponding viscoelastic profiles. The results are summarized as follows: (i) The static force profiles indicate that in this case the brush is softest at high degrees of swelling. This correlates with the finding that weak polyelectrolyte brushes appear to have extended outer tails.55 Such a tail region was found with ellipsometry on this sample as well. (54) Arnold, R. J. Colloid Sci. 1957, 12, 549.

We have shown that SFM noise analysis is a versatile tool to probe the dynamic interactions between soft surfaces. In close contact we find viscoelastic dispersion which is caused by relaxation processes in the interaction zone. When a sphere is glued to the end of the cantilever, the lubrication approximation holds which much simplifies the hydrodynamic aspects of the interaction. Polyelectrolyte brushes show an increased compressibility when they are in a near-collapse condition. We have shown two examples of a salt-induced collapse of a strong polyelectrolyte brush and a pH-induced collapse of a weak polyelectrolyte brush. While the static compressibility close to collapse is high, the dynamic compressibility as given by the inverse friction coefficient is not. This finding should translate into the large-scale properties of colloidal dispersions stabilized by such brushes. A large amount of energy will be dissipated during particle collisions, resulting in a much increased viscosity. Acknowledgment. We thank Markus Preuss and Hans-Ju¨rgen Butt for help with the preparation of the cantilevers. This work was funded by the Deutsche (55) Lyatskaya, Yu. V.; Leermakers, F. A. M.; Fleer, G. J.; Zhulina, E. B.; Birshtein, T. M. Macromolecules 1995, 28, 3562. (56) Gelbert, M.; Biesalski, M.; Ru¨he, J.; Johannsmann, D. Manuscript in preparation.


Langmuir, Vol. 16, No. 13, 2000 5783

Figure 14. Geometry of the gap region. If the radius of the sphere R is much larger than the gap width D, the “lubrication approximation” holds and the hydrodynamic situation is much simplified.

Forschungsgemeinschaft (Schwerpunkt: “Polyelektrolyte mit definierter Moleku¨larchitektur”). Appendix. Hydrodynamic Behavior of the Brush-Sphere System The hydrodynamics of an oscillating rigid body in a viscous environment is, generally speaking, quite a complicated issue. The special case of an isolated sphere is treated in the textbook by Landau and Lifshitz.57 Sader has treated rectangular cantilevers clamped on one side.45 We are not aware of a realistic treatment of V-shaped SFM cantilevers. However, when a sphere glued to the tip of the cantilever approaches a flat surface, the situation simplifies because the “lubrication approximation” holds.46 The hydrodynamic forces are dominated by the pressure in the gap, and it suffices to consider the gap region. If the radius of the sphere R is much larger than the distance across the gap D, a separation of length scales occurs. The geometry and the definition of some variables are shown in Figure 14. Oron and co-workers have reviewed the application of the lubrication approximation to the physics of thin films.58 We start out from the Navier-Stokes equation

[∂v∂t + (v‚∇)v] ) -∇p + η∆v

F

(A1)

with F the density, v the velocity, p the pressure, and η the viscosity. If the nonlinear term (v‚∇)v can be neglected, one is in the Stokes regime where all quantities are linear in velocity v. The magnitude of the nonlinear term is estimated by the Reynolds number Re, which for oscillatory motion is usually given by Re ) R2ωF/η, with R the typical dimension of the oscillating object.45,59 R here would have to be the size of the cantilever (10-100 µm), not the radius of the sphere. Inserting the numbers R ) 10-100 µm, ω ) 2π × 20 kHz, F ) 103 kg/m3, and η ) 10-3 Pa s, one finds a Reynolds number of Re ∼ 10-1000. The Reynolds number therefore generally is larger than 1. We will show below that the Reynolds number in the lubrication approximation is given by Re ) DaωF/η, with D the gap width and a the oscillation amplitude. Since both the oscillation amplitude a and the gap width D are small, the Reynolds number is low in the gap. Given a low Reynolds number, the second issue is the magnitude of the inertial term F ∂v/∂t. Below a critical frequency ωc the shear waves emanating from the oscillating object will have a penetration depth δ much larger than the width of the gap, and the flow field is the same as the stationary flow field. The space and the time dependence then factorize, and the time dependence is (57) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics; Butterworth: London, 1987. (58) Oron, A.; Davis, S. H.; Bankoff, S. G. Rev. Mod. Phys. 1997, 69, 931. (59) Batchelor, G. K. Fluid Dynamics; Cambridge University Press: Cambridge, UK, 1974.

just given by a prefactor exp(iωt). The penetration depth δ is given by δ ) (2η/(ωF))1/2, which (with a gap width of D ∼ 1 µm) results in a critical frequency ωc ∼ 2π × 300 kHz. The cantilever motion is much slower than this, and the flow field instantaneously adjusts to the boundary conditions. In the following we make the above arguments more quantitative. We use cylindrical coordinates r and z (Figure 14). The flow field has a radial component vr(r,z) and a vertical component vz(r,z). The Navier-Stokes equation then reads

(

∂2vr

η

(

η

∂z

2

∂2vz 2

∂z

+

+

)

∂2vr 2

∂r

)

∂2vz 2

∂r

-

-

(

)

∂vr ∂vr ∂vr ∂p )F + vr + vz ∂r ∂t ∂r ∂z

(

)

∂vz ∂vz ∂vz ∂p )F + vr + vz ∂z ∂t ∂r ∂z

(A2)

The vertical component of the velocity vz is on the order of the velocity of the sphere ∂D/∂t ) iωa with a the amplitude of oscillation. The radial component vr is larger than the vertical component because the liquid has to escape to the side. From volume considerations, one arrives at vr(r,z) ∼ vz(r,z) r/h. h itself is a function of r given by h(r) ) D + r2/(2R). For an order of magnitude estimation one may approximate the Navier-Stokes equation by

iωaη

( (

iωaη

) )

( (

(

))

∂p r r r 1 r ∼ iωaF iω + iωa 2 + 2 + ∂r h h3 hr h h

))

1 ∂p 1 1 1 ) iωaF iω + iωa + + ∂z h h h2 r2

(

(A3)

The Reynolds number Re is the ratio of the term nonlinear in velocity v ) iωa and the viscous term. One finds

Re ∼

haωF DaωF ∼ η η

(A4)

where we have set h ∼ D. The Reynolds number Re is much less than 1 for all realistic choices of parameters. All terms in the lower line apart from the pressure dependence are smaller than the corresponding terms in the upper line by a factor of h/r, and one can write

∂p ≈0 ∂z

(A5)

The inertial term is smaller than the viscous term by a factor ωρh2/µ ∼ 10-2, which justifies its neglect as well. The flow field can be approximated by stationary flow. With no brush present the viscosity η is a parameter independent of z. The radial component of flow results from the condition that

η

∂2vr(r,z) ∂z2

≈

∂p ∂r

(A6)

η∆v is independent of z, and the well-known parabolic profile (“Poiseulle flow”) results:

vr(r,z) ) D˙

3r 2 (z - hz) h3

(A7)

The prefactor 3r/h3 is a consequence of volume conservation. Since the vertical component of flow is much smaller

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Gelbert et al.

than the lateral component, we do not discuss it in the following. The pressure p(r) is given by

p(r) ) D˙

(

∫0rη

∂vr(r,z) ∂z2

)

dr′ - C ) D˙

12ηR3 (A8) (2DR + r2)2

where the constant of integration C was chosen such that p(r) vanishes at r ) ∞. The force onto the sphere F(D) is obtained by integration of the pressure over the entire area. One finds the time-honored Reynolds lubrication law46

ξ(D) )

F(D) ) D˙

In the presence of the brush, the viscosity η is not a constant parameter but instead a complex function of space and frequency η(z,ω) ) η′(z,ω) + iη′′(z,ω). If η(z,ω) is known, the friction coefficient ξ(D,ω) can be obtained by numerically solving eqs A6, A8, and A9. Clearly, the inverse problem of calculating η(z,ω) from ξ(D,ω) is much more complicated. However, the relations A6-A9 do not affect the frequency dependence of η and ξ in any way. If a frequency-dependent friction coefficient ξ(ω) is found, this frequency dependence necessarily goes back to viscoelastic dispersion given by a frequency-dependent viscosity η(ω). A frequency-dependent friction coefficient ξ(ω) therefore has to be attributed to stress relaxations in the brush. It is not caused by hydrodynamic effects.

3

12ηR R dr ) 6πηR ∫0∞(2DR 2 2 D +r )

(A9) LA991664T

Collapse of Polyelectrolyte Brushes Probed by Noise Analysis of a

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