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Why Surfaces Modified by Flexible Polymers Often Have a Finite Contact Angle for Good Solvents M. A. Cohen Stuart, W. M. de Vos, and F. A. M. Leermakers* Laboratory of Physical Chemistry and Colloid Science, Wageningen UniVersity, Dreijenplein 6, 6703 HB Wageningen, The Netherlands ReceiVed October 7, 2005. In Final Form: December 20, 2005 Polymers adsorbing from a dilute solution onto the solvent-vapor interface generate a nonzero surface pressure. When the same polymers are end-grafted onto a surface such that a so-called polymer brush is formed, one will find that the solvent wets this compound interface partially. The partial wetting and the finite surface pressure are intimately linked properties of the polymer-solvent-vapor combination. It is shown that the spreading parameter in the wetting problem is proportional to the surface pressure in the adsorption case. Complete wetting is only possible when this surface pressure is nonpositive. The wetting characteristics are hardly influenced by the grafting density and chain length characterizing the brush. We argue that the grafted polymer chains can bridge to the solvent-vapor interface, thereby preventing the wetting film to become macroscopically thick. We present experimental data underpinning our self-consistent field analysis. Indeed, finite contact angles should be expected in various systems in which bridging attraction contributes to the disjoining pressure in wetting films.
Introduction The properties of surfaces can be profoundly modified by means of thin films of attached polymer molecules.1 In this paper, we are interested in the effect of layers of end-attached polymers (commonly called “brushes”) on the wettability of such surfaces. Brushes of hydrophilic polymers such as poly(ethylene oxide) (PEO) or poly(hydroxyethyl methacrylate) (PHEMA) have been found to be capable of protecting surfaces against the deposition of proteins and microorganisms; the effect is believed to be due to the hydrophilic nature of the brush. One might argue that a good solvent on a polymer brush will form a stable wetting layer, provided the grafting density is sufficiently high: for a large number of long chains, the polymer’s solvation free energy is large and will easily outweigh any unfavorable solvent-substrate interactions. Hence, on this basis, one expects zero contact angles for a good solvent on a dense brush. Yet, when one measures contact angles on, e.g., PEO brushes, one usually finds values well above zero (quite commonly about 40°); increasing the grafting density does not seem to lower this value.2 Evidently, the simple argument based on solvation seems to be missing the point. The viewpoint pursued here is that many water-soluble polymers also interact with the water-air interface and that this may well lead to the effects observed. In the context of wetting, polymer brushes are interesting systems. For example, quite a lot of experimental and theoretical work has been done on polymer brushes in contact with liquid polymer made up of the same monomers as the brush.3-7 A detailed analysis of such systems revealed a complex wetting * To whom correspondence should be addressed. (1) Currie, E. P. K.; Norde, W.; Cohen Stuart, M. A. AdV. Colloid Interface Sci. 2003, 100-102, 205-266. (2) Roosjen, A.; Kaper, H. J.; van der Mei, H. C.; Norde, W.; Busscher, H. J. Microbiology 2003, 149, 3239-3246. (3) Maas, J. H.; Fleer, G. J.; Leermakers, F. A. M.; Cohen Stuart, M. A. Langmuir 2002, 18, 8871-8880. (4) Liu, Y.; Rafialovich, M. H.; Sokolov, J.; Schwarz, S. A.; Zhong, X.; Eisenberg, A.; Kramer, E. J.; Sauer, B. B.; Satija, S. Phys. ReV. Lett. 1994, 73, 440-443. (5) Gay, C. Macromolecules 1997, 30, 5939. (6) Kerle, T.; Yerushalmi-Rozen, R.; Klein, J. Macromolecules 1998, 31, 422429. (7) Reiter, G.; Auroy, P.; Auvray, L. Macromolecules 1996, 29, 2150-2157.
phase diagram, which is particularly rich when the free polymers do not wet the substrate onto which the brush is grafted3. As long as the length of the free chains (of length P) does not exceed the length of the grafted ones (of length N), one can expect two wetting transitions upon increasing the grafting density, one from partial to complete wetting, and another transition back to partial wetting. The window of complete wetting becomes narrower the longer the chain length of the free chains and closes at sufficient asymmetry P > N (depending on the wettability of the substrate). In contrast to simple solid/liquid interfaces, surfaces carrying polymer brushes typically interact with the wetting component on two different length scales. The interactions with the solid substrate are short-ranged (except van der Waals interactions which may be important in some cases), whereas the interactions with the brush occur on the much longer length scale of the brush height H. These competing length scales may interfere: the wetting component may want to wet the substrate when the shortrange interactions favor this but may be prevented to wet the brush, e.g., when the solvent quality is only marginally good. As a macroscopic thick wetting layer is needed on the surface before one can declare the surface to be wet, it is clear that the interactions on the longest length scale are decisive whether the surface is wet or not. As a result, one can only expect complete wetting when the solvent is (very) good for the polymer segments. The fact that a polymer is in a good solvent does not exclude that it anchors to the solvent/vapor interface. In particular for water, there is ample evidence of the surface activity of fully water soluble polymers.8-11 This means that one has to consider not only the interaction between polymer and solvent inside the brush but also what happens at the edge of the brush where the solvent meets the vapor phase. As soon as adsorption occurs at the (narrow) L/V interface, polymer bridges between the substrate and the solvent-vapor interface are formed. These bridges can (8) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman & Hall: London, 1993. (9) Lu, J. R.; Su, T.J.; Thomas, R. K.; Penfold, J.; Richards, R. W. Polymer 1996, 37, 109. (10) Kopperud, H. B. M.; Hansen, F. K. Macromolecules 2001, 34, 56355643. (11) Bosker, W. T. E.; Agoston, K.; Cohen Stuart, M. A.; Norde, W.; Timmermans, J. W.; Slaghek, T. M. Macromolecules 2003, 36, 1982-1987.
10.1021/la052720v CCC: $33.50 © 2006 American Chemical Society Published on Web 01/20/2006
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only be broken when the polymer is forced to desorb from the solvent-vapor interface, at the expense of some anchoring (free) energy, and thus we should anticipate that the solvent does not spontaneously detach from the solvent swollen brush. This implies partial wetting. Below we will theoretically explore the consequences of such bridging effects on the thermodynamics of wetting and investigate how the measured contact angle on the (compound) interface correlates with the strength of adsorption as quantified by means of the surface pressure exerted by an adsorbed polymer. As our tool, we employ the self-consistent field theory for inhomogeneous polymer systems. We finally confront our theoretical results with experimental data on contact angles for water on PEO brushes; most of these data were determined in the context of this work but we also found a value in the literature.2
SCF Theory, Parameter Choice, and Contact Angles Most, if not all, analytical theories for polymer brushes are based on the Edwards equation in which the excluded-volume effects are dealt with on a mean-field level by introducing selfconsistent potentials.12 In the limit of strongly stretched chains, i.e., at sufficiently high grafting density, it is possible to make use of the most likely trajectories. In such an approach, one accurately describes the universal properties of polymer brushes analytically.13 It appears however much more difficult to also account for nonuniversal effects that occur near the surface, i.e., adsorption phenomena, and to describe the fluctuations of chain conformations near the edge of the brush. As we will see below, the wetting of the polymer brush by a low molecular weight solvent is intricately coupled to what exactly happens at these extremities. As a result, one needs to solve the self-consistent field equations for polymers at interfaces rigorously. This is possible only numerically. Here we will use the discretisation scheme of Scheutjens and Fleer to account for all relevant interactions of a polymer brush in a liquid solvent coexisting with solvent vapor. The Scheutjens Fleer self-consistent field (SF-SCF) theory has been presented in large detail in several publications before,8 and we refrain from a discussion here. Instead, we briefly describe the system and discuss the input parameter set. We consider homopolymer chains composed of N isotropic polymer segments with length l of type A, with segment s ) 1 localized next to a flat solid boundary. The grafting density (number of chains per unit area) equals σ. The solid phase, composed of units of type S, has a fixed Fresnel-like volume fraction profile φS(z) ) 1 ∀ z e0 and φS(z) ) 0 for all coordinates z ) 1, 2, ..., M, where the spatial coordinate is in units l. In all cases M . N which ensures that the system is always large compared to the relevant length scales in the problem. Besides polymer segments there are monomeric components V (“vacancies”) and W (“water”) which stand for free volume and solvent, respectively. Strong repulsion between V and W allow us to consider a liquid/vapor interface. Evidently, at each coordinate the total volume fractions add up to unity (incompressible limit), i.e., φA(z) + φV(z) + φW(z) ) 1 ∀ z > 0. The polymer chains are made of freely jointed segments, and the set of all possible conformations is generated using first-order Markov statistics. Only short-range nearest-neighbor contact interactions are accounted for. For incompressible three-component systems (A, V, W) there are three relevant Flory-Huggins interaction parameters. The solvent/ (12) Edwards, S. F. Proc. Phys. Soc. 1965, 85, 613; 1966, 88, 265. (13) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, 2610.
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vacancy demixing is controlled by χVW. Its value is fixed at 3.5, which gives a strong segregation corresponding to a solvent far from the critical point. As a consequence, the V-W interface is sharp (width of order l). The SCF theory immediately gives the interfacial tension for the V-W interface; for the parameter chosen, we obtain γ ) 0.638 kBT/l2. The binodal is symmetric, and the volume fractions for the dilute (vapor) and dense (solvent) phase are φ ) 0.03787 and 0.96213, respectively. To further mimic W as a good solvent and V as a free volume component, we choose the polymer units to have a strong bias in their interactions (favoring W over V) as follows: χAW ) 0 (good solvent) and χAV g1 (very) poor solvent. Finally, there are three interaction parameters involving the substrate (S). Because the substrate’s density is fixed, we can arbitrarily choose one of the interactions to be zero without loosing generality; that is, we choose χSV ) 0. If we choose the polymer to have some extra adsorption energy over water such that the conformational entropy loss caused by the impenetrable surface is approximately compensated, i.e., when χAS ) χWS - 1, we will have effectively no extra affinity of the polymer segments for the substrate if the chain is submerged in the W environment.8 Unless specified otherwise, we will generally choose the symmetric parameter setting χAV ) -χWS ensuring that we have the same wetting behavior both in the limit of a bare surface (σ ) 0) as well as in the limit of maximum grafting density, i.e., σ ) 1. With the above choices, the remaining key control parameters are the grafting density σ and χAV; the smaller χAV, the stronger the adsorption of segments at the liquid interface. As explained, we will correlate our findings to results for free homopolymers adsorbing onto the W-V interface. This case is completely determined by the parameters involving the components A, W, and V. In a SCF solution, the volume fraction profiles are obtained as a functional of the corresponding self-consistent field potentials. Moreover, the method specifies how to compute the self-consistent field potentials from the volume fraction profiles. A high-precision fixed point of these equations, obeying the incompressibility constraint and the proper boundary conditions, is found routinely by a numerical approach.14 For each SCF solution, the partition function is available in terms of volume fraction and self-consistent potentials. This partition function gives access to the Helmholtz energy per unit area F and related thermodynamic potentials. Of special interest is the grand potential
Ω ) F - µWnW - µVnV
(1)
Here, µi is the chemical potential of component i and nW ) M φW(z) is the number of W molecules per unit area; a similar ∑z)1 equation holds for nV. For each SCF solution, it is easy to compute the excess amount of W units at the interface, ΓW ) ∑zφW(z) - φbW where φbW is the volume fraction of W units far from the surface, i.e., in the bulk. In systems with a steep density gradient (in the present system this is the V-W interface), one will encounter so-called lattice artifacts. These artifacts present themselves, e.g., in adsorption isotherms where the excess adsorbed amount of water ΓW is plotted as a function of the chemical potential of water (or more practically log φbW), as very small “van der Waals loops” upon the increase of the adsorbed amount by one unit. Taking sufficient points, e.g., order 10, per loop, one can obtain an (14) Evers, O. A.; Scheutjens, J. M. H. M.; Fleer, G. J. Macromolecules 1990, 23, 5221.
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averaged adsorbed amount per loop. Such smoothed isotherms are presented below. The wetting behavior is characterized in terms of the W-adsorption isotherm ΓW(µW). Since we are dealing with a system with a solubility gap, we have a binodal specifying compositions of the coexisting phases W and V. When the isotherm does not cross the binodal, we have a case of complete wetting. In the case of partial wetting one can find at least two coexisting film compositions (ΓW values), i.e., the so-called thin and the thick film at the binodal chemical potential. It is possible to evaluate the surface excess grand potentials Ωthin and Ωthick corresponding to these two situations. The contact angle of the macroscopic drop on top of the swollen brush is commonly expressed in terms of Young’s law:15
γSV ) γSP + γWV cos θ
(2)
Making use of the identities Ωthin ) γSV and Ωthick ) γSP + γWV, we can evaluate the contact angle as follows:
cos θ )
Ωthin - Ωthick +1 ΩWV
(3)
As the |cos θ| cannot exceed unity, one often introduces a spreading parameter S ≡ Ωthin - Ωthick.16 When S > 0, the surface is completely wet, and partially wet otherwise. To facilitate the conversion from S to contact angles, we will normalize the spreading parameter with the interfacial tension of the water/ vapor interface, i.e., S˜ ≡ (Ωthin - Ωthick)/ΩWV. In the partial wetting regime, we thus have S˜ ) cos θ - 1. Experimental Section PEO brushes of varying grafting density were prepared by means of a Langmuir-Blodgett method described by Currie et al.17 As substrates, pieces of silicon wafers were used, which were covered with a film of polystyrene. All solvents used were of PA grade (Sigma-Aldrich). The coating of substrates with polystyrene was carried out in the following way. First, the silica wafer was cut into strips. The strips were rinsed with alcohol and water and further cleaned using a plasma-cleaner (10 min). The strips were then covered with a solution of 1 g/L vinyl-terminated polystyrene (vinyl-PS200; Mw ) 2100, Mw/Mn ) 1.11, Polymer Source inc.) in chloroform and, after evaporation of the chloroform, heated overnight at 150 °C under vacuum. In this way, the vinyl-PS200 becomes covalently bound to the Si/SiO2 surface.18 Nonbound vinyl-PS200 was washed off with chloroform. The strips were spin-coated with polystyrene using a solution of 13 g/L PS (Mw ) 870 kDa, Mw/Mn ) 1.05, Polymer Source inc.) in toluene at 2000 rpm for 30 s. This yields a stable PS layer with a thickness of approximately 90 nm. For the brush layer fabrication, monolayers of PS-PEO block copolymers (PS36-PEO148, Mw/Mn ) 1.05; PS36PEO377, Mw/Mn ) 1.03; PS38PEO773, Mw/Mn ) 1.05) were prepared at the air-water interface, by dissolving the copolymers in chloroform and spreading these solutions very carefully on water in a Langmuir-trough using a microsyringe. After evaporation of the chloroform and a quick compression-decompression cycle, the films were compressed to the appropriate surface density, and the films were transferred to the substrates in a single-passage Langmuir-Blodgett transfer. The surfaces so prepared were carefully stored in clean water until use. (15) Young, Th. Philos. Trans. R. Soc. 1804, 65-87. (16) de Gennes, P. G. ReV. Mod. Phys. 1985, 57, 827. (17) Currie, E. P. K.; Sieval, A. B.; Avena, M.; Zuilhof, H.; Sudholter, E. J. R.; Cohen Stuart, M. A. Langmuir 1999, 15, 7116-7118. (18) Sieval, A. B.; Demirel, A. L.; Nissink, J. W. M.; Linford, M. R.; van der Maas, J. H.; de Jeu, W. H.; Zuilhof, H.; Sudholter, E. J. R. Langmuir 1998, 14, 1759-1768.
Figure 1. Selection of (smoothed) water adsorption isotherm γW versus log φbW into a brush with grafting densities σ ) 0.0040, 0.0051, 0.0054, and 0.01 as indicated. N ) 100, χAV ) 1.5, χAS ) -2.5, and χWS ) -1.5. Note that the concentration axis is on a logarithmic scale. For other parameters see parameters section. The vertical line gives the bulk binodal. Contact angles corresponding to thermodynamic equilibrium as closely as possible require that the chemical potential of the liquid is constant throughout the system. This is most easily achieved by using air bubbles as the vapor phase, because the vapor pressure inside the bubbles is virtually at saturation. Therefore, we measured contact angles by means of captive bubbles introduced under the brush sample. An optical system was used to produce an image of the bubble’s contour, and the contact angle was calculated from a fit of the Laplace equation to the contour. For every different brush layer (different grafting density and polymer length), at least two prepared surfaces were measured and each surface was measured two times. The measurements were reproducible within approximately 3°. It was found that when the contact angle was measured using a sessile water droplet put on top of a brush sample, much higher values were found than for the case of a captive air bubble under a brush sample. This may well indicate that the contact angles measured with the sessile water droplet method do not correspond to a proper thermodynamic equilibrium, presumably because the vapor phase is not saturated with water.
Results and Discussion This section splits naturally into a theoretical and an experimental part. We will first present the theoretical analysis. Wetting problems are inherently complex because there are three phases involved and the contact angle is determined by three interfacial tensions. It is therefore constructive to first present some adsorption isotherms for a particular parameter setting and discuss the structure of the swollen brush, before embarking on a more detailed thermodynamic analysis of wetting. In an adsorption isotherm one presents the excess number of relevant molecules (W, which is the solvent) accumulated at the surface as a function of the (logarithm of the) bulk concentration of W (chemical potential). The bulk binodal corresponding to the demixing of W and V is a very important point in this isotherm because in thermodynamic equilibrium one cannot cross this value. However, in SCF calculations, one can oversaturate the system with W, and it is instructive to investigate and interpret such supersaturated regions of the adsorption isotherm. As an example, we choose χAV ) 1.5, which according to the default settings implies the values χAS ) -2.5 and χWS ) -1.5, leaving the polymer grafting density as the only free parameter. In Figure 1, we show four isotherms displaying two distinct kinds of behavior. For grafting densities well below σ ) 5 × 10-3, the isotherms cross the binodal already when there is only a microscopically thin film of water on the surface. For an example, see Figure 1 for σ ) 0.004. The isotherm does not return to the binodal until the amount of water on the surface is large (a macroscopically thick film). For grafting densities well above σ ) 5 × 10-3, isotherms also cross the bulk binodal, but now at some mesoscopic value for the adsorbed amount (cf. Figure 1 σ ) 0.01 with Γ#W ≈ 10, dashed curve). As we will see, for
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Figure 2. Selection of segment volume fraction profiles for the grafted chains (A), the vapor phase (V), and the water phase (W) at the chemical potential of the solvent and vapor corresponding to the bulk binodal. Other parameters as in Figure 1. (a) For fixed grafting density σ ) 0.0051 the microscopically thin film (dotted lines) and the coexisting mesoscopically sized film (solid lines). (b) For two values of the polymer grafting density σ ) 0.01 and 0.05, respectively, as indicated.
Figure 3. (a) Absorbed amount of water in the brush at bulk coexistence Γ#W as a function of the logarithm of the grafting density σ. (b) The normalized spreading coefficient S˜ as a function of the (logarithm) of the grafting density. The affinity of the polymer for the V-W interface decreases with increasing χAV. For each value of χAV the other parameter have the corresponding values χAS ) -χAV - 1, χAW ) -χWS.
these systems, there is a well-solvated polymer brush on the surface. Again, the isotherm returns to the binodal from the supersaturated side only when there is a macroscopic amount of solvent on the surface. Both types of isotherms are characteristic for systems with a finite (nonzero) contact angle; that is, the surface is partially wet. The isotherms have an additional van der Waals loop around the crossover value σ ) 5 × 10-3 (see Figure 1). Such a loop is often found in wetting studies and is characteristic for a firstorder wetting transition. When the loop occurs at sub-saturated concentrations (Figure 1, σ > 0.0051), there is a first-order transition from a microscopic to a mesoscopic adsorbed amount, known as a pre-wetting transition. It is significant that in the present system the step in the isotherm does not diverge when it occurs at coexistence (this is seen in Figure 1 σ ≈ 0.0051). In Figure 2, we present a selection of volume fraction profiles of solvated polymer brushes at the bulk coexistence of the monomeric solvent. We give the full profiles of all of the species in the system: V, W, and the polymer A. In Figure 2a the interfacial compositions are shown for the two coexisting layers and the special grafting density σ ) 0.0051 corresponding to the case that the van der Waals loop exactly occurs at the binodal condition. At this special condition, there is a microscopically thin film (dotted curves) that coexists with a mesoscopic one (solid curves). From the profiles, it is clear that there is a significant sudden jump in uptake of solvent W along with the increase in thickness. For the microscopically thin film, the amount of solvent on the surface is not sufficient to have a well-defined isolated V-W interface away from the surface; instead the V phase extends up to the surface. As a result, the polymer chains are more or less collapsed. In the films with a mesoscopic size however, there is a distinct solvent-vapor interface and the grafted chains are swollen by the solvent. The chains tend to avoid the substrate (the density next to the surface is slightly reduced) due to the conformational entropy loss that is not compensated for by adsorption energy. The chains are homogeneously stretched throughout the swollen brush, which leads to an approximately homogeneous polymer density within the solvent film. Finally,
there exists a significant adsorption peak of polymers at the free interface. Note that the polymer remains on the solvent-rich side of the interface as it does not like to dissolve in the V phase (χAV > 0.5). Although the swelling transition at low grafting density is interesting, the focus of the present paper is more on the mesoscopic films at grafting densities relevant for polymer brushes (i.e., σ > 1/N). Therefore, we will not analyze the metastable parts and the spinodal points that accompany this first-order surface transition any further. In Figure 2b, we present two sets of profiles, for grafting densities σ ) 0.01 and 0.05, respectively, again for the case that the solvents are at the bulk bimodal. We see that the uptake of solvent increases with increasing grafting density. The amount of polymer units at the V-W interface is not a strong function of the grafting density and as the film thickness does not grow proportionally to σ, the polymer concentration (φA) in the solvated brush increases with increasing grafting density. When the polymer brush remains in the partial wetting regime, we should anticipate that in the limit of σ f 1 the solvent uptake vanishes. A logical consequence of this is that the uptake of solvent in the brush should go through a maximum. In Figure 3a, we present the saturated uptake of W in the brush as a function of the grafting density for three values of the effective affinity (χAV) of the polymer for the solvent-vapor interface. The more the polymer is repelled from the vapor phase (the higher χAV), the lower the affinity. As expected, the curves have a jump in water uptake at very small values of the grafting density and go through a maximum at elevated values of σ. Note that the relevant grafting densities that can be reached in experiments typically will be well below the maximum. Over a large range of the relevant grafting densities, we find that the water uptake grows logarithmically with increasing number of polymer chains attached to the surface. This means that the overall layer thickness of the swollen brush is only weakly dependent on the grafting density. To measure experimentally the amount of polymers grafted at an interface, it is therefore much more sensitive to record the grafted amount in the dry state (e.g., with ellipsometry) than to determine the total layer thickness in the wet state. However, the large amount of water that is absorbed in the swollen
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Figure 4. Normalized spreading coefficient S˜ as a function of the interaction parameter of the polymer units for the V phase, i.e., χAV for the grafting density σ ) 0.01.
brush may be measured more accurately than the amount of polymer in the dry state. In this case one may make use of this logarithmic dependence of the swollen brush thickness on σ to estimate the grafting density. From Figure 3a, we also conclude that the lower the affinity, i.e., the more exclusively it is solvated by W, the more W is taken up by the brush. Eventually, when the uptake Γ#W increases more and more, the thin film gives way to a macroscopic solvent layer and we go into the direction of a wetting transition. As soon as Γ#W does no longer remain finite, we say that the brush is wet by W. The wettability of the polymer brush can be characterized by the normalized spreading coefficient S˜ , e.g., as a function of the grafting density. Examples of this relationship S˜ (σ) are shown in Figure 3b for the same parameters as in Figure 3a. As long as the brush chains can anchor on the solvent-vapor interface, the spreading parameter is negative and thus the contact angle remains finite over the whole range of grafting densities. Recall that the parameters are chosen such that S˜ is the same for the two limiting cases σ ) 0 and 1. For intermediate values of σ, we find S˜ to be less negative than in these limiting cases. This means that the polymer chains facilitate the wetting of the surface. There is a sharp kink in the S˜ (σ) curve. This kink occurs at the first order adsorption transition (jump) mentioned above. Again, we choose not to present the metastable branches near this transition region and give only the stable parts of the curve. This transition point splits the S˜ (σ) curve naturally into two parts: one part where the grafting density increase leads to a less negative S˜ and the other part where S˜ decreases smoothly with grafting density. In the latter range of grafting densities (which is quite relevant for many experiments), we find that S˜ is a very weak function of the grafting density. We therefore conclude that for brushes with σ gN-1 contact angle measurements (in the partial wetting regime cos θ ) S˜ + 1) are not suitable to determine the brush’s grafting density. Only in the pre-transition region of σ’s do we find a strong dependence of the contact angle on the grafting density. Increasing the value of χAV pushes the spreading coefficient S˜ toward the complete wetting value S˜ ) 0. For the present set of parameters, the value χAV ) 2 is such that the brush is almost completely wet. The S˜ (σ) curve just does not touch the S˜ ) 0 value. Comparison between panels a and b in Figure 3 shows that the amount of solvent inside the brush can grow significantly (with increasing σ) while the contact angle remains virtually constant. This means that, if one aims at characterizing the polymer brush by challenging it with a low molecular weight solvent, contact angle measurements are less informative than solvent uptake data. From Figure 3, it is clear that upon increasing χAV the system approaches complete wetting. This trend is more accurately quantified in Figure 4 where we plot the normalized spreading
Cohen Stuart et al.
Figure 5. Density profiles across the liquid-vapor interface W-V onto which a nongrafted homopolymer A100 is adsorbed for which the concentration of the free polymer in the W-rich phase equals φAW ) 0.001. The z coordinate is chosen such that z ) 0 is approximately at the position where φV ) φW.
coefficient S˜ as a function of χAV for the case σ ) 1/N ) 0.01 which is at the crossover from isolated (“mushrooms”) to interacting chains (brush). It can be seen that complete wetting is reached just above χAV ) 2 and that the curve smoothly tends to S˜ ) 0. This means that the wetting transition would classify as second order. We note that in the present analysis long-range van der Waals interactions are not included. If, on the length scale of the brush, the van der Waals forces are still important, i.e., for not too large values of N, we might expect that the wetting transition is altered, depending on whether the van der Waals forces are favoring or counteracting wetting. Any further discussion of these matters is outside the scope of this paper. Correlation to Homopolymer Adsorption from Solution. In this subsection, we will discuss the adsorption of homopolymers from solution and its effect on the interfacial tension. The discussion will be restricted to parameter sets used in the previous section. This means that we restrict ourselves to the adsorption of a homopolymer, i.e., A100 onto the (narrow) V-W interface. In Figure 5, we present an example of density profiles. Note that not only the solvent but also the polymer can penetrate a little bit into the vapor phase. The parameters are chosen such that the polymer strongly prefers the W-rich phase (which is positioned at positive values of the z coordinate). As before, we observe that the polymer adsorbs onto the W-V interface but that it prefers the solvent side of the interface. The (experimental) quantity of interest is the surface pressure Π exerted by the adsorbed polymer. Figure 6a presents the surface pressure as a function of the area per molecule a. At very large area, i.e., a > 105, the ideal gas law Π ) kBT/a is recovered. At smaller areas per molecule, the surface pressure increases more strongly. In this region, the adsorbed molecules develop loops and tails. The corresponding adsorption isotherm (Figure 6b) is plotted up to the point where the bulk concentration of polymer enters the overlap region, i.e., upto φ(W) A ) 0.01. At this concentration, the surface pressure is close to its maximum value. The fact that the polymer can anchor at the liquid/vapor interface not only leads to a finite surface pressure Π but also (as we have seen) affects Ωthick and, hence, S. We therefore now correlate these two quantities (Π and S) for various values of the adsorption energy. Because Π depends on polymer concentration, we need a reference concentration at which we evaluate the surface pressure, for which we choose the overlap concentration. This value will be labeled Π*. In Figure 7a, we present the surface pressure Π* generated by the adsorbing homopolymers at the interface for φ(W) A ) 0.01, as a function of the affinity χAV of the polymer units for the V-W interface. Positive values for the surface pressure persist up to χAV ≈ 2.1.
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Figure 8. Normalized spreading parameter as a function of the amount of polymers grafted at the surface, i.e., σN for three values of the chain length as indicated. Parameters are similar as in Figure 3 with χAV ) 1.
Figure 6. (a) Logarithm of the surface pressure (made dimensionless by kBT/l2, with l the segment size) versus the logarithm of the area per molecule a. This slope -1 (ideal surface pressure) at large areas is indicated. (b) The corresponding adsorption isotherms where the logarithm of the excess adsorbed amount of polymer per unit area is plotted as a function of the logarithm of the polymer concentration in the water rich phase. The slope 1 in the Henry region is indicated. Results are plotted up to the overlap concentration for polymers in the bulk which is φ(W) A ) 0.01.
Figure 7. (a) Surface pressure generated by the adsorbed polymer molecules at the overlap bulk concentration Π* as a function of the parameter χAV. (b) The normalized spreading parameter S˜ ) S/γWV plotted versus the normalized surface pressure Π ˜ ) Π*/γWV of the liquid-vapor system.
As pointed out above, we expect a correlation between Π*(χAV) for the V-W interface and the function S˜ (χAV) shown in Figure 4. To quantify this we simply plot, in Figure 7b, S˜ versus the corresponding values of the normalized surface pressure Π ˜ ) Π*/γWV, where γWV ) 0.638kBT/l2, as discussed above. It is gratifying to note that the correlation is a simple proportionality. The curve crosses the origin (S˜ ,Π ˜ ) ) (0,0) within numerical accuracy, proving that the wetting behavior is completely controlled by the affinity for the polymers to adsorb on the L/V interface. It thus follows that the contact angle for a brush is given by
S˜ ) cos θ - 1 ) - 1.391Π ˜ (σN > 1)
(4)
a very simple result indeed. Hence, we find that, once the
Figure 9. Cosine of the contact angles θ (in degrees) of a captive air bubble under a PEO brush measured as a function of grafted amount σN (with σ in units of chains per nm2), for 〈N〉 ) 148 (O), 〈N〉 ) 370 (0), and 〈N〉 ) 770 (4). The horizontal line indicates the plateau value of the cos(40°) as expected from surface pressure data. The ascending line at low σN has been chosen as a linear fit to the data in this range.
(saturated) surface pressure of a given polymer at the solvent/ vapor interface is known, the contact angle for a brush made up of that same polymer directly follows from eq 4; properties of the substrate do not seem to play a role. This need not surprise us, as for grafted chains of reasonable length and density the contribution of solvation to the free energy easily dominates over any solvent-substrate interactions. Note that we studied here strictly monodisperse chains; however, it is quite unlikely that some polydispersity will have a significant effect because all that is needed is that substrate-anchored chains can populate the solvent-vapor interface up to a certain local monomer density. Whether these monomers come from long or short chains is irrelevant for the free energy they contribute. However, at low grafting density, when the contact angle has not yet reached its plateau value, solvation effects are the dominating contribution and we expect a stronger dependence on N. To illustrate this, we plot in Figure 8 the normalized spreading parameter as a function of the amount of monomers grafted to the surface, i.e., S˜ (σN) for three values of the chain length N. For low grafting densities, we see that the spreading parameter S˜ is only a function of the product of the grafting density and the chain length, i.e., a function of the grafted amount. Also the value of the spreading parameter at higher grafting densities (grafted amount in excess of unity) is not extremely sensitive to the chain length. It appears that one can slightly improve the wettability of the surface by increasing the chain length. The first-order transition, which occurs at the kink in the curves of Figure 8 (just below a grafted amount of unity) occurs almost at the same value of the adsorbed amount for various N. Close inspection shows that the transition moves slightly to lower grafted amounts with increasing chain length. Comparison with Experimental Results. In Figure 9, we present our data for contact angles on PEO brushes as a function of grafting density, for three different chain lengths N ) 148, 370, and 770 monomer units, respectively. The contact angle of
1728 Langmuir, Vol. 22, No. 4, 2006
the bare and very hydrophobic polystyrene substrate is 92°. With increasing grafting density, the contact angles drop, longer chains being more effective than short ones. At about σΝ ) 35 nm-2 a more or less constant value of 40° is attained. Qualitatively, this behavior is similar to what we find theoretically (Figure 8): a steep ascending part of the cosine of the contact angle with grafted amount abruptly followed by a plateau. Only for the longest chains, N ) 770, does the crossover to the plateau value appear more gradual. It should be noted that the contact angles are not entirely free of hysteresis, particularly at lower grafting densities. This is to be expected, as surface inhomogeneities would be most important in this range. Hysteresis appears to decrease in a systematic fashion with increasing grafting density, which is manifested by the observation that air bubbles move more easily over dense brushes than over dilute brushes. This may explain the somewhat deviating data for the longest chain length. For a more quantitative check, we use the correlation between the plateau value of S˜ and Π ˜ , as found theoretically (eq 4). The surface pressure of a saturated PEO adsorption layer on water has been reported in a number of publications; one generally finds a value of 12 mN/m. Inserting this, we arrive at θ ≈ 39.8° remarkably close to the experimental value of 40 ( 5°. We note that in Figure 8 the grafting density is made dimensionless by the area per lattice site. Ideally one would also like to use a dimensionless grafting density in Figure 9 as well. There are several ways to do this. A reasonable Ansatz is to use a lattice site length l ≈ ∼0.5 nm, which implies that there are about 4 lattice sites per nm2. Implementing this will set the kink slightly above σN ≈ 5 (in dimensionless units). Comparison with Figure 8 then shows that the kink is experimentally somewhat higher than theoretically predicted. We may suggest two reasons for this discrepancy. One of the aspects is that the contact angle in the bare surface is higher experimentally than assumed in the idealized calculations. This may shift the kink in the curve to higher grafting densities. Second it is known that PEO adsorbs onto the PS surface (as used in the experiments). This adsorption has the effect that grafting density is effectively lower than its nominal value. Although contact angles on brushes seem to be routinely measured, systematic studies as a function of grafting density are hard to find. However, some support comes from data of Roosjen et al.,2 who prepared PEO brushes by chemical grafting of vinyl terminated PEO on glass. The grafting density of these brushes was about 0.6 nm-2. The authors report contact angles of about 40°, in good agreement with our data.
Discussion Many water soluble polymers adsorb significantly at the solvent/vapor interface. This adsorption correlates to the observation of partial wetting of the same (good) solvent for surfaces onto which these polymers are attached; a simple and quantitive relation is found between the surface pressure Π* generated by
Cohen Stuart et al.
the polymer (when it adsorbs) on one hand and the contact angle (θ) (when it bridges the solvent film) on the other hand. It is tempting to consider the question whether these ideas can be carried over to other cases; considering the arguments given above it is to be expected that for all systems in which the bridging mechanism provides an attractive contribution to the disjoining pressure the contact angle will bear the mark of this. A case in point is a polymer gel. The surface of a gel will also expose loose chain ends and strands that can adsorb to the solvent-vapor interface. Hence, not only should we expect that the solvent contact angle on a swollen gel is nonzero but also that its value is related to the surface pressure of the polymer in question. This is expected to hold for dilute gels; very dense gels impose entropic constraints on the formation of an adsorbed layer that may affect the value of the contact angle as well. Presently, we have no evidence to substantiate these conjectures, although finite contact angles have been reported for waterswollen gels.19 Another interesting case is that of polyelectrolyte multilayers. It has been found that the contact angle on such films is controlled to a large extent by the outermost layer and can be rather different, depending on whether the polycation or the polyanion was applied last.20 It would follow from our findings that the surface pressures of the two polyelectrolytes differ in the same way, too, but this has not been verified so far. We also want to point out here that most water-soluble polymers tend to adsorb at water-air interfaces, because of the high interfacial tension of water and its rather unusual solvent properties. In organic solvents, adsorption at the solvent surface occurs, too,21 but it is less common.
Conclusions The thermodynamic wettability (equilibrium contact angle) of polymer brushes (formed by terminally attached chains of sufficient length and grafting density) by a good solvent is often nonzero, despite the substantial and negative free energy contributed by chain solvation. The cause of such nonzero contact angles is that the brush chains adsorb at the solvent-vapor interface thereby forming bridges between the solvent-substrate and the solvent-vapor interface. These bridges lead to an attraction which accounts for the partial wettability. The contact angle is in this case uniquely determined by the surface pressure that the polymer exerts when adsorbed from solution onto the liquid-vapor interface. We provide an explicit and quantitative relation between these two quantities. Experimental surface pressure and contact angle data for poly(ethylene oxide) agree with this relation. LA052720V (19) Grundke, K.; Werner, C.; Poeschel, K.; Jacobasch, H. J. Colloids Surf. A 1999, 156, 19-31. (20) Kolasioska, M.; Warszynski, P. Bioelectrochemistry 2005, 66, 65-70. (21) Ober, R.; Paz, L.; Taupin, C.; Pincus, P.; Boileau, S. Macromolecules 1983, 16, 1981.
