Wetting of a Polymer Brush by a Chemically Identical Polymer Melt

For much longer chains in the melt, the calculations predict that the contact ..... Finally, when the grafting density increases further, we see finit...
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Langmuir 2002, 18, 8871-8880

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Wetting of a Polymer Brush by a Chemically Identical Polymer Melt: Phase Diagram and Film Stability J. H. Maas, G. J. Fleer, F. A. M. Leermakers, and M. A. Cohen Stuart* Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, 6703 HB Wageningen, The Netherlands Received May 7, 2002. In Final Form: July 29, 2002 We report experimental results for the wetting states of a melt of polystyrene in contact with a brush of polystyrene chains end-attached to a substrate. Wettability was assessed by monitoring the stability of an ultrathin film of the molten polymer; if the film remained stable and uniform for several days, it was considered to be a case of complete wetting, whereas spontaneous breakup (initiated by hole formation) was interpreted as partial wetting. The bare (oxidized silicon) substrate without the brush is partially wet. When the grafting density is varied, two wetting/dewetting transitions are found, depending on the length P of the chains in the melt. At low grafting densities, a transition from partial to complete wetting is observed, which is driven by the swelling of the attached chains and their mixing with the melt. At a higher grafting density, there is a second wetting transition back to partial wetting which we ascribe to the poor mixing between stretched chains in the brush and free chains. The experimental results are compared with scaling relations and numerical self-consistent-field (SCF) calculations. For melts of chains that are short compared to the grafted chains, these calculations confirm the occurrence of two transitions. For much longer chains in the melt, the calculations predict that the contact angle remains finite, but its value remains very low over a certain window of grafting densities. Moreover, the calculations indicate that the zero contact angle can become metastable in this regime. This is consistent with the experimental finding that there is, also at high P, a window of grafting densities where no instabilities are found. Around the onset of instability, the films disproportionate into complicated patterns of dry surface, mesoscopic films, and droplets. These results suggest that the disjoining pressure has a double minimum structure, which is consistent with SCF calculations.

I. Introduction Thin polymer films play an important role in technology. For example, successful application of coatings, fabrication of core-shell latices, and dispersion of pigment particles in polymer melts all require that a given polymer forms a stable film on a solid substrate, that is, the polymer must completely wet the substrate. When the film is not stable because the polymer wets partially, one may attempt to remedy this by appropriate additives or chemical treatments. An obvious approach is to introduce polymer chains end-attached to the substrate. For example, it has been reported that a partially wet surface can be made completely wet by means of tethered polymers compatible with the melt.1-3 However, this does not always work; it sometimes happens that the tethered chains have the opposite effect, that is, they induce a transition from complete to partial wetting. For the special case where the grafted chains are chemically identical to those in the melt, this situation is referred to as “autophobicity”.4-7 Autophobic behavior of polymer melts on brushes has been observed in some experimental cases.7,8 The behavior of melt + brush systems poses interesting problems concerning the role of entropy near interfaces and therefore (1) Liu, Y; Rafailovich, 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. (2) Yerushalmi-Rozen, R.; Klein, J.; Fetters, L. J. Science 1994, 263, 793-795. (3) Maas, J. H.; Cohen Stuart, M. A.; Leermakers, F. A. M.; Besseling, N. A. M. Langmuir 2000, 16, 3478-3481. (4) Shull, K. R. Faraday Discuss. 1994, 98, 22. (5) Matsen, M. W.; Gardiner, J. M. J. Chem. Phys. 2001, 115, 27942804. (6) Mueller, M.; McDowell, L. G. Europhys. Lett. 2001, 55, 221-227. (7) Reiter, G. Phys. Rev. Lett. 2000, 85, 5599-5602. (8) Reiter, G.; Khanna, R. Langmuir 2000, 16, 6351-6357.

has attracted attention from polymer physicists. Various theoretical calculations are available that analyze the interface between brushes and free chains in a melt, underpinning the autophobic effect.4-7 So far, however, a systematic experimental investigation of the wetting behavior of melt/brush systems, paying attention to key variables such as grafting density and chain length, has not been carried out. There is one study on the effect of the brush length N and the free chain length P on wetting2 which arrives at the conclusion that the wetting-dewetting transition is only a function of the ratio N/P. The effect of the grafting density was studied much less systematically. The work of Reiter mainly addresses kinetic issues and the effect of long-range (dispersive) forces.7-10 A critical test of theories therefore has not been possible. It is the aim of the present paper to fill this gap. We report a complete set of experiments on the wetting behavior of substrates with grafted chains by a polymer melt, and we compare the results with theoretical calculations. We chose polystyrene as the polymer. The surfaces with end-attached chains used in the experiments were prepared in two ways, either by adsorbing block copolymers or by chemical grafting of endfunctionalized polystyrene. We focus on the effects of the chain length P of the free polymer melt and the grafting density σ of the attached chains. The length N of the grafted chains was kept constant. The idea is that by localizing the transitions between stable (completely wetting) and unstable (partially wetting) films we should be able to obtain a phase diagram in the (σ, P) plane. The paper is organized as follows. First, we discuss in section II.A the behavior of polymers at interfaces from (9) Reiter, G.; Sharma, A.; Casoli, A.; David, M.-O.; Khanna, R.; Auroy, P. Europhys. Lett. 1999, 46, 512-518. (10) Reiter, G.; Khanna, R. Phys. Rev. Lett. 2000, 85, 2753-2756.

10.1021/la020430y CCC: $22.00 © 2002 American Chemical Society Published on Web 10/17/2002

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a theoretical perspective and we develop some simple scaling relations for (two) wetting transitions that one might expect to occur upon changing either σ or P. At low σ, a transition from partial to complete wetting is possible upon increase of σ. At higher σ, a return from complete to partial wetting may occur upon increase of either P or σ. We subsequently report, in section II.B, numerical calculations obtained on the basis of a self-consistentfield (SCF) approach and compare the results with the scaling result obtained in section II.A. The experimental methods are given in section III, after which we present the experimental results (section IV), with their interpretation in terms of the theory, and a short discussion (section V) and conclusion section (section VI). II. Theory A. Scaling Approach. When a polymer chain is in the vicinity of a hard wall, its number of accessible conformations is restricted, and the average shape of the molecule is different from that of an ideal unperturbed coil. Due to the entropy reduction associated with these restrictions, the interfacial Gibbs energy is increased. Polymer chains will therefore tend to avoid a hard surface, unless their segments experience a compensating attractive interaction from the wall. One might call surfaces which neither attract nor repel the segments indifferent. In dilute solution, polymer molecules are depleted from indifferent surfaces. In polymer melts, similar restrictions are operative, but because of the strong cohesion between the segments the density must remain high and there are a number of chains in the interfacial region that cannot escape from the wall. The interface between a polymer melt and an indifferent surface is thus characterized by zero excess energy but a negative excess entropy. The interfacial tension of such an interface is entirely of entropic origin, and it is likely that a polymer melt on an indifferent surface tends to display partial wetting. Calculations confirm this.11 Let us consider the situation of an initially bare substrate that is partially wet. By attachment of chains that can mix with the melt, the Gibbs energy of the melt/ substrate interface can be lowered, and when the grafting density σ reaches a certain level σ*, a compensation of the initially unfavorable Gibbs energy may be obtained so that complete wetting is achieved. We denote the region bounded by σ ) 0 and σ ) σ* as the allophobic regime: the melt “does not like” the substrate enough to wet it completely. To develop an expression for σ*, we first consider the bare surface (σ ) 0). The spreading parameter S characterizes the wettability of a given surface. It is defined by the following combination of interfacial tensions γ:

S ) γsv - (γsl + γlv)

(1)

where the subscripts s, l, and v refer to solid, liquid, and vapor (vacuum), respectively. Partial wetting corresponds to S < 0, whereas complete wetting is found for S > 0. In our case, the bare surface is characterized by a negative spreading parameter S0: S0 < 0. Using Young’s law, cos θ ) (γsv - γsl)/γlv, the spreading parameter can also be expressed in the contact angle and the liquid/vapor interfacial tension γlv. Hence, for the bare substrate

S0 ) γlv(cos θ0 - 1)

(2)

where θ0 is the contact angle of the melt on the bare surface. (11) Leermakers, F. A. M.; Schlangen, L. J. M.; Koopal, L. K. Langmuir 1997, 13, 5751-5755.

Attached chains will modify the properties of the substrate and hence change the value of γsl - γsv. We denote this change due to the attached polymer as S0 - S; it is equal to minus the Gibbs energy change per unit area. If we denote the change in Gibbs energy due to grafting and per grafted chain by ∆G, we have as a criterion for the wetting transition

S ) S0 - σ*∆G ) 0

(3)

As long as the density of the grafted chains is sufficiently low, we can neglect any interactions between them and ∆G will be independent of σ. In the absence of the melt (i.e., when the surface is dry), grafted chains of length N will assume the form of collapsed globules of size R ∝ N1/3. Upon being immersed in a melt of chains of length P, the grafted chains will (1) swell to adopt the ideal coil size, thus regaining the full conformational entropy, and (2) mix with the free chains. The Gibbs energy contributed by a grafted chain thus has two terms, one for the mixing and one for the swelling. The (ideal) mixing contribution is easily shown to be given by -kT(N/P). The entropy loss of a collapsed chain can be expressed by means of the Flory expansion coefficient R defined by R2 ) R2/R02, where R0 is the radius of the unperturbed coil, characterized by R02 ∝ N. The Gibbs energy associated with a collapsed coil (R , 1) is given by kTR-2.12 Upon swelling, the Gibbs energy is obviously decreased by this amount, so that we finally arrive at

N ∆G ) - + N1/3 kT P

(

)

(4)

Hence, using eqs 2, 3, and 4 we find for the grafting density at the transition

σ* )

γlv(1 - cos θ0) N kT + N1/3 P

(

)

(5)

The surface tension γlv of the melt generally increases somewhat with molar mass, that is, with P. However, this variation rarely amounts to more than a few percent and occurs mainly at low P. Because we will primarily deal with large values of P, we ignore it in the present scaling approach. Also, θ0 may depend somewhat on P. For large values of P (as occur in our experiments), N/P is small with respect to N1/3; in this limit σ* increases weakly with P. In the limit of small P, σ* becomes very small. As the density of the brush is further increased, the grafted chains dissolved in the melt will eventually begin to interact and stretch. This situation is now commonly called a brush. A seminal discussion of the properties of brushes is that by Alexander and de Gennes.13,14 These authors used a scaling approach in which a constant density is assumed throughout the brush: all the brush chains are assumed to be equally stretched and to end at a distance from the substrate equal to the thickness of the brush. Such a treatment clearly neglects the entropy associated with the spatial distribution of chain ends. Some 10 years later, a numerical self-consistent-field calculation was reported in which the density profile is no longer assumed to be a block profile and where the end points (12) Grosberg, A. Y.; Khokhlov, A. R. Giant Molecules; Academic Press: San Diego, 1997. (13) Alexander, S. J. Phys. Fr. 1977, 38, 983. (14) de Gennes, P. G. Macromolecules 1980, 13, 1069.

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of the chains are distributed throughout the brush.15 Analytical equations based on a similar model were developed by Milner et al.16,17 and by Zhulina et al.18-22 These refinements did not affect the main conclusions of the original scaling approach, however. The Gibbs energy G of a chain in a brush (with respect to a free chain at infinite dilution) has essentially two contributions, one from the stretching entropy and one due to chain-chain interaction:23

H2 a3N2σ G ≈ 2 + kT a N PH

(6)

where H is the thickness of the brush and a is a monomer size. Minimalization of the free energy with respect to H gives for the brush height

H ≈ NP-1/3(σa2)1/3 a

(7)

Increasing the grafting density (creating a denser brush) thus leads to an increase in brush height. Furthermore, in a “melt” of monomers (P ) 1) the brush will be strongly swollen, but upon increasing P the brush contracts gradually, and the monomer density due to grafted chains increases as well. Since there is no solvent, the average brush density 〈φ〉 of chains in the brush (swollen with some free chains) can be written as

〈φ〉 )

Nσa3 ≈ (σa2)2/3P1/3 H

(8)

where Na3 is the collapsed volume of one chain. Note that 〈φ〉 does not depend on N. At sufficiently high [σ, P], the brush density approaches that of the melt (〈φ〉 f 1), that is, the melt becomes a poor solvent for the brush and the brush expels the free chains and collapses. From eq 8, it follows that this “dry” brush appears when7

σ**a2 ∝ P-1/2

(9)

We denote this limiting behavior as the “drying limit”. The demixing and collapse of the brush are likely to affect the wetting behavior because, for wetting to occur, some interpenetration is necessary: a fully collapsed brush with 〈φ〉 ) 1 and no interpenetration would be equivalent to an indifferent surface for which one expects partial wetting.11 We thus expect that the transition from complete to partial wetting more or less coincides with the collapse of the brush. To find a more precise criterion, we would have to calculate ∆S as a function of σ, P in order to find out where S equals zero. This requires a more careful analysis of the free energy at partial (15) Cosgrove, T.; Heath, T.; van Lent, B.; Leermakers, F. A.M; Scheutjens, J. M. H. M. Macromolecules 1987, 20, 1692. (16) Milner, S. T.; Witten, T. A.; Cates, M. E. Europhys. Lett. 1988, 5, 413. (17) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, 2610. (18) Zhulina, E. B.; Borisov, O. V.; Priamitsyn, V. A. J. Colloid Interface Sci. 1990, 137, 495. (19) Birshtein, T. M.; Liatskaya, Y. V.; Zhulina, E. B. Polymer 1990, 31, 2185. (20) Borisov, O. V.; Birshtein, T. M.; Zhulina, E. B. J. Phys. II Fr. 1991, 1, 5. (21) Borisov, O. V.; Birshtein, T. M.; Zhulina, E. B. Macromolecules 1994, 27, 4795. (22) Zhulina, E. B.; Birshtein, T. M.; Borisov, O. V. Macromolecules 1995, 28, 1491. (23) Aubouy, M.; Frederickson, G. H.; Pincus, P.; Raphae¨l, E. Macromolecules 1995, 28, 2979.

interpenetration. Leibler24 made an estimate of the extent of interpenetration between a melt and a brush and concluded that the chain length P** at which the spreading parameter changes sign scales as P** ∝ σ-2/3. Conversely, one may expect a wetting transition at σ** ∝ P-3/2. We refer to this result of Leibler as the “interpenetration limit”. However, the expulsion of mobile chains from the brush occurs very gradually, and this may render a scaling analysis of the wettability very difficult. We thus simply suppose that the grafting density of the brush at the autophobic wetting transition more or less follows the collapse (hence, decays continuously with P), and we deal with its location more quantitatively by means of numerical calculations. Finally, for large P, our eqs 5 and 9 would imply that σ* and σ** should eventually cross. If the autophobic transition σ** scales as P-1/2 this should be at P′′ ≈ (σ*a2)-2, and for σ** ∝ P-3/2 this is even at lower P, that is, P′′ ≈ (σ*a2)-2/3. Hence, for sufficiently large P (at fixed N), the complete wetting regime must eventually disappear, provided the grafted chains interact strongly enough to justify the application of brush theory. This is a surprising result, which we investigate more closely in the next section. B. SCF Calculations. So far, we considered simple approximate scaling relations for the wetting transitions. For more detailed calculations, we use numerical SCF calculations. SCF theories have been applied for a variety of systems, among which are polymer melts4 and brushes;15,25-27 a detailed account of the method has been given elsewhere.28 Each polymer chain i is considered as a series of Ni linked segments numbered t ) 1...Ni. For discretization, we choose M parallel layers numbered from the surface (s) at z ) 0 to z ) M. Since we deal with a solvent-free polymer system, each unit of volume is occupied either by a segment (p) or by a vacancy (v); the vacancies represent the free volume in the melt and ensure that the system has a finite coefficient of thermal expansion. End-attached chains start from the layer adjacent to the surface, at z ) 1. Each segment experiences an average potential energy field u(z); this field represents the interactions of a segment of the chain with its environment. It includes the hard-core interactions (through the requirement that each layer has to be completely filled) as well as short-range segment-segment and segment-surface interactions which are expressed by Flory-Huggins χ parameters. We use subscripts p, s, and v to refer to polymer segments, surface sites, and vacancies, respectively. In the present computations, we used χps ) -1 (a slight effective attraction between polymer and surface), χsv ) 0 (no interaction between surface and vacancies), and χpv ) 1 (for a melt far from the critical temperature). After proper normalization, the equilibrium volume fraction profiles Fi(z) and thermodynamic quantities (energy and entropy) are calculated. Integration of densities over these profiles gives the total amount Γi of a component in the layer. Plotting the amount of free (24) Leibler, L.; Ajdari, A.; Mourran, A.; Coulon, G.; Chatenay, D. Ordering in Macromolecular Systems. In Proceedings of the OUMS Conference, Osaka, Japan, 1993; Springer Verlag: Berlin, 1994; p 301. See also: Semenov, A. N. Macromolecules 1992, 25, 4967-4977. (25) Wijmans, C. M.; Scheutjens, J. M. H. M.; Zhulina, E. B. Macromolecules 1992, 25, 2657. (26) Klushin, L. I.; Birshtein, T. M.; Mercurieva, A. M. Macromol. Theory Simul. 1998, 7, 483-495. (27) Currie, E. P. K.; Wagemaker, M.; Cohen Stuart, M. A.; Fleer, G. J. Macromolecules 1999, 32, 9041. (28) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman & Hall: London, 1993.

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Figure 1. Adsorption isotherms for (a) complete wetting and (b) partial wetting, in the form Γ(µ), where µ is the chemical potential (the chemical potential at bulk coexistence is denoted as µ#). In the partial wetting case, part of the van der Waals loop at high values of µ - µ# is cut off. The parameters are P ) 100, N ) 200, and σ ) 0.01 (a) and 0.005 (b).

polymer as a function of its chemical potential µ gives the adsorption isotherm. An adsorption isotherm for a complete wetting situation is shown in Figure 1, curve a. The isotherm rises monotonically and diverges when the chemical potential approaches its coexistence value µ#, implying that a stable macroscopic film is formed. Partial wetting refers to an equilibrium situation in which droplets with a finite contact angle coexist with (nearly) dry areas with only a very low coverage of (free) polymer. The adsorption isotherm for such a situation intersects with the saturation axis (µ#) at the dry coverage Γ*. A typical example is shown in Figure 1, curve b; the (very low) value of Γ* is indicated. At each point on the isotherms, the interfacial tension is known. The contact angle is then found from the interfacial tensions at coexistence, as follows. Rewriting Young’s law, we have

cos θ ) 1 +

γsv - (γsl + γlv) γlv

(10)

The excess free energy associated with the point (Γ*, µ#) equals γsv. A point in the vertical part of Figure 1, curve b, corresponding to a macroscopic film, is associated with the sum γsl + γlv. To compute γlv, we carried out a separate calculation of a melt/vapor interface, omitting the substrate. Substituting these values into eq 10 directly gives the contact angle. At a transition from partial wetting to complete wetting, the adsorbed amount at µ# goes from a finite value to an infinite one, that is, to a macroscopic value: the parameter Γ* in Figure 1 then diverges. The way in which this occurs characterizes the order of the transition; at a first-order transition this occurs discontinuously (jumplike), whereas for a second-order transition there is a gradual divergence of Γ*. The wetting behavior of a polymer melt on a brush surface was investigated numerically for different grafting densities σ and chain lengths P of the polymer melt. The length N of the brush chains was fixed at N ) 200. For the present combination of parameters (χps ) -1, χpv ) 1, χsv ) 0), there is a (weak) net attraction between polymer segments and the surface, leading to some adsorption, but this attraction is not enough for wetting; in the absence of a brush (σ ) 0) we have partial wetting. By way of example, adsorption isotherms for polymer melts of different chain length P at two values of σ are given in Figure 2. The case of a dilute brush (σ ) 0.005) is shown in Figure 2a. At high

Figure 2. Adsorption isotherms for polymer melt chains (with length P, as indicated) on a polymer brush with length N ) 200. The grafting density is σ ) 0.005 (a) and 0.025 (b). In the left diagram, the isotherm is plotted up to Γ ) 10; the curves continue as in Figure 1 and approach µ ) µ# from below saturation (first-order wetting transition). In the right diagram, the isotherms approach the binodal from the supersaturated side (second-order wetting transition).

P, we have a pronounced loop into the supersaturated region, corresponding to partial wetting; Γ* is very small. When P decreases, the loop rapidly becomes smaller until, around P* ≈ 90, a Maxwell construction (not shown) proves that the wetting transition is found. The associated contact angle then becomes zero (complete wetting). The scenario that the curves of Figure 2a represent is typically that of a first-order wetting transition: microscopic values for Γ* exist for P > P*, and at P* the equilibrium film thickness (and thus Γ) jumps to infinity. Because of this jump, the derivative of the contact angle θ with respect to σ is undefined at P ) P*. In Figure 2b, we present results for a somewhat higher grafting density, σ ) 0.025. Again, we find partial wetting (i.e., an excursion into the supersaturated region) for long chains (P ) 600, 400) and complete wetting for lower P (e200). Hence, a transition must have been crossed at some P value between 400 and 200. However, in contrast to the situation presented in Figure 2a, this transition appears to be of second order: coming from high P the adsorbed amount at µ ) µ# increases gradually with decreasing P, diverging at the transition. For this transition, dθ/dσ is continuous. The calculated adsorption isotherms in Figure 2 thus show that the wetting can be controlled by changing either the grafting density σ or the length P of the free chains in the melt. Similar adsorption isotherms, now at a constant P (100 units) and N (200 units) but with varying grafting densities σ, are given in Figure 3. At low grafting densities (σN ) 0.1, 0.5), the adsorption isotherm again intersects the saturation axis µ ) µ# at a very (invisibly) low value of Γ*,

Wetting of a Polymer Brush by a Polymer Melt

Figure 3. Adsorption isotherms of a polymer melt (P ) 100) on a brush (N ) 200) at varying grafting densities. The curves with low overall grafted amount σN ) 0.1, 0.5, 3, and 10 are given as solid curves; those for a higher grafted amount (σN ) 20, 30, and 60) are dashed.

Figure 4. Calculated contact angle θ (in degrees) for different values of the length of the melt chains P as a function of the grafting density, N ) 200.

implying that the polymer melt film will be unstable. A first-order transition to a complete wetting situation is obtained when the grafting density is increased up to σN ≈ 3. A further increase of the grafting density up to σN ) 30 again results in a second-order transition to a partial wetting situation. Figure 4 shows the corresponding contact angles obtained with eq 10, for a polymer droplet on a brush with N ) 200 and P in the range 50-500. At a very low grafting density, the contact angle is high and essentially independent of the molar mass of the melt. When the grafting density is increased, the contact angle falls steeply; at approximately σ ) 0.01 the decreasing curve hits the abscissa, apparently implying a zero contact angle (complete wetting). What happens at a higher grafting density depends on P. For relatively short chains, complete wetting is maintained until a nonzero contact angle reappears at some grafting density σ**. Due to the progressive but gradual expulsion of the free melt out of the brush, the contact angle increases very gradually with σ; this makes it rather difficult to precisely localize the transition. As can be seen, the contact angle for P ) 100 is very close to zero over a significant range of grafting densities and then increases again. It is hard to decide where σ** is exactly located. Nevertheless, it is clear that there is a window of grafting densities [σ*, σ**] where the contact angle is nearly zero; the width of this window decreases with increasing P. Experimentally, this implies that with decreasing chain length of the free polymer, the interval (Γ*, Γ**) where there is complete wetting widens; within the parameter range covered by Figure 4 a polymer melt with P ) 50 does not reach the autophobic regime.

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Figure 5. Theoretical phase diagram in terms of the grafting density σ and the length of the chains of the polymer melt P in double-logarithmic coordinates, for N ) 200. The parameter σ* (open symbols) represents the first-order wetting transition at low grafting densities, whereas σ** (filled symbols) corresponds to the second-order transition at high grafting densities. The allophobic, complete wetting, and autophobic regimes are indicated. The two crossover σ* and σ** meet at P ) P′′, σ ) σ′′. Two short dashed lines at the top of the graph indicate the -3/2 slope and the -1/2 slope, predicted by the Leibler (interpenetration) scaling and the drying limit, respectively.

Extrapolating the trend, one expects the autophobic wetting transition for P ) 50 to occur at extremely high grafting densities (σ** > 0.3). For P > 200, the situation is different. Now, the contact angle goes through a minimum but does not become zero anymore. Hence, in a strict sense, there is no longer a wetting transition in which the contact angle vanishes. Also, it seems that although the function θ(σ) has no zeros, its derivative has a discontinuity around σ ≈ 0.01, suggesting a special kind of (wetting) transition. Clearly, this is a sign that the two transitions, namely, the firstorder allophobic/complete and the second-order complete/ autophobic, have merged. We return to this special point below. From plots such as Figure 4, taking care to properly localize the zeros of θ(σ), values of σ* and σ** can be found for various values of P. In this way, we can obtain a theoretical phase diagram as shown in Figure 5. The (σ, P) plane (shown in double-logarithmic coordinates) has three regions. At low grafting density, there is partial wetting for any value of P because of the lack of attraction between polymer and substrate; this is the allophobic regime. Upon increasing the grafting density for not too high P, we cross a first-order wetting transition, entering the complete wetting regime. Finally, when the grafting density increases further, we see finite contact angles reappear. This is the autophobic region. Beyond σ**, the contact angle increases with increasing grafting density up to a plateau value of about 79° for all molecular weights. As alluded to before and conjectured in section II.A, for the parameters chosen the two wetting transitions meet, that is, there is a value P′′ for which σ* ) σ** ) σ′′. For N ) 200, this occurs at P′′ ≈ 200, σ′′ ≈ 0.0075. Hence, the calculations imply that for P < P′′ the contact angle is finite at σ < σ*, zero for σ* < σ < σ**, and nonzero again for σ > σ**, whereas for P > P′′ there is no completewetting window anymore. In this region, θ(σ) shows a minimum but this minimum remains finite: the substrate dewets for all grafting densities of the brush. This confirms our earlier conclusion based on the scaling approach (see the discussion following eq 9). An important feature of the transition between complete wetting and autophobic dewetting (the σ** boundary in Figure 5) is that it is very gradual. When grafting density

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or P is increased, the melt is very gently expelled from the brush. The contact angle also changes very gradually from zero to a finite value, and the thickness of the film in equilibrium with macroscopic liquid decreases slowly as the grafting density and/or P increases. This characterizes the transition as second order (there is no jump in thickness). When we try to interpret the power-law behavior in Figure 5, we see that the transition σ* (allophobic/ complete) follows a P1/2 power law. This gradual increase is not inconsistent with the slow increase predicted by eq 5. In fact, forcing a power law (with the present set of parameters) through the prediction of eq 5 is close to the SCF result. The upper transition σ** (complete/ autophobic) does not follow a well-defined power law. This may be due to the fact that the transition is second order and thus very gradual. For such a transition, scaling arguments may not work. However, forcing a power law σ ∝ PR through the points (near P′′) gives a value of R ≈ -4. This is much stronger than predicted by the Leibler result, -3/2 (interpenetration limit), and the drying limit, -1/2. However, when P , N the P dependence tends to a power law not very different from P-3/2 or P-1/2. These two slopes are indicated as short dashed lines in Figure 5. Summarizing, there are three predictions for σ**(P), and it is of interest to analyze what is seen in experiments. III. Experimental Section A. Materials. As substrates, we used either silicon wafers with an oxide layer of approximately 80 nm (prepared by thermal oxidation) or hydrogen-terminated silicon wafers, depending on the technique that was used to prepare brushes. Polystyrene (PS) does not wet oxidized silicon wafers, whereas hydrogenated (passivated) Si seems to be completely wetted by PS; the latter substrate was used only at high grafting densities where the role of the substrate in the wettability is of no consequence. B. Preparation of Brushes. Brushes with relatively low grafting density were prepared by adsorbing poly(4-vinylpyridine)-polystyrene (PVP/PS) block copolymers on oxidized silicon. The adsorption was carried out using either dip-coating in a dilute solution (using a nonselective solvent) or adsorption from the melt, depending on the desired density of the brush. A detailed description is given elsewhere.3 This method gives grafting densities up to about 0.2 nm-2 (i.e., 5 chains per nm2). To obtain denser brushes (up to about 1 nm-2, one chain per nm2), we applied chemical grafting of vinyl-terminated PS on hydrogen-terminated silicon to form an alkyl (Si-CH2-) bond. The reaction can be done either from solution or directly from the melt. An important advantage of using polymer melts for this reaction is that one can get much denser brushes than with grafting from solution. We used two PS samples with degrees of polymerization of 20 and 200, respectively; the polymers are referred to by a subscript indicating the number of monomeric units in the chain. The highest grafting density (σ) of the PS20 layer is 0.95 nm-2 (i.e., every chain has 1.05 nm2 available on the surface), and the dry thickness is 3.0 nm. For PS200, the highest σ was 0.55 nm-2, with a corresponding dry layer thickness of 19.2 nm. A detailed description of the chemical grafting method is given elsewhere.29 C. Overlayers. PS overlayers on the brushes were prepared by spin-coating the polymer from CHCl3 at 3000 rpm, using appropriate concentrations (1-5 mg/mL) to obtain films thin enough to have a measurable rate of destabilization. The initial film thickness was varied to ensure that stability/instability transitions were not missed because of nucleation barriers. For very thick films, spontaneous nucleation is too slow, and one has to rely on induced nucleation, for example, edge-induced dewetting by breaking the sample.7,10 We did not use such stimuli in our study. (29) Maas, J. H.; Cohen Stuart, M. A.; Sieval, A. B.; Zuilhof, H.; Sudho¨lter, E. J. R. Thin Solid Films, accepted.

Maas et al. The thickness of the dry polymer layer was determined by computer-controlled null ellipsometry (Sentech SE-400). The refractive index of PS for λ ) 632.8 nm was taken as 1.58.29 The atomic force microscopy (AFM) measurements were performed with a Digital Instruments Nanoscope III AFM equipped with a scanner, capable of scanning an area of 100 µm2. All AFM measurements were performed in air at room temperature and a relative humidity of 50-60%. Commercially available tappingmode tips were used on cantilevers with a resonance frequency in the range of 350-400 kHz. The sample surfaces were stored in vacuo (because atmospheric oxygen attacks the samples) at a temperature (145° C) well beyond the Tg of PS (≈100° C). After 12 days (needed for the highest M polystyrene to develop dewetting patterns), the samples were quenched to room temperature and the surface topography of the top films was investigated by means of AFM.

IV. Results and Discussion A. PS Layer on a Chemically Identical Brush. Figure 6 shows seven representative AFM images (a-g) obtained for PS films with Mn ) 183 kg/mol on brushes with an increasing grafting density of the PS brush. The bright areas are elevated, and the darker patches indicate depressed regions. A very dilute brush is seen in Figure 6a: σ is low (0.09 nm-2), and the PS film is unstable and dewets. We observe a surface covered with small polymeric patches separated by dry surface, a pattern very similar to that discussed by Sharma and Khanna.31,32 The contact angle can be estimated to be about 20°. Complete wetting (no pattern at all) is found for a brush with σ ) 0.35 nm-2 (Figure 6d). At very high amounts of grafted PS (σ ) 0.95 nm-2, Figure 6g), the free PS film is again unstable and dewets into droplets. Hence, when the grafting density is increased, the system traverses through three main regimes, namely, a complete wetting regime (at intermediate σ) bounded by two partial wetting regimes at low and high σ. The crossover between these three major regimes is characterized by two narrow regimes, one around each transition (Figure 6b,c and Figure 6e,f). In cases b (σ ) 0.19 nm-2) and f (σ ) 0.52 nm-2), the film has developed a large number of droplets protruding above the film, as well as depressions (often surrounding the droplets). We refer to this situation as “holes and hills”. In cases c (σ ) 0.21 nm-2) and e (σ ) 0.50 nm-2), droplets (hills) have appeared on top of a smooth film of free PS but there are no depressions. The development of these intermediate surface patterns in thin films on a flat surface will be discussed in section IV.B. We repeated the experiments with PS of different Mn, spanning a range of 1-104 kg/mol. The results, all for an initial film thickness of 5 nm, are summarized in the (σ, P) plane as shown in Figure 7. We are aware that our samples dewetted very slowly and that they may sometimes not have reached full equilibrium, even though we waited for 12 days. Assuming nevertheless that we reached equilibrium, Figure 7 may be regarded as a phase diagram. Regions of complete wetting and partial wetting and the intermediate regimes are indicated. The two intermediate regimes occur in narrow bands, both at the first transition (σ ≈ 0.19 nm-2) and at the second transition (σ ≈ 0.55 nm-2 for high Mn). Let us now compare the experimental data (Figure 7) with the theoretical picture discussed in section II (Figure 5). At low molar mass of the free polymer, there is good agreement. We find three regions: one of complete wetting, (30) Koneripalli, N.; Levicky, R.; Bates, F.; Matsen, M. W.; Satija, S. K.; Ankner, J.; Kaiser, H. Macromolecules 1998, 31, 3498. (31) Sharma, A.; Khanna, R. Phys. Rev. Lett. 1998, 81, 3463. (32) Sharma, A.; Khanna, R. J. Chem. Phys. 1999, 110, 4929.

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Figure 6. AFM images of a PS melt (Mn ) 183 kg/mol) on a PS brush. Image, 10 × 10 µm2; maximum height difference between the dark depressed regions and the elevated bright patches, 30 nm. (a) Partial wetting, (b) holes + hills, (c) hills, (d) complete wetting, (e) hills, (f) holes + hills, (g) dewetting. From (a) to (g), σ increases as indicated: σ ) 0.09 (a), 0.19 (b), 0.21 (c), 0.35 (d), 0.50 (e), 0.52 (f), and 0.95 (g) nm-2.

Figure 7. Wetting phase diagram of a PS melt of varying Mn on a PS brush of varying grafting density σ after annealing for 12 days at 145 °C under reduced pressure. The initial film thickness in all experiments is 5 ( 0.1 nm.

bordered by two partial wetting regimes at low and high grafting density. Hence, we conclude that the latter two correspond to the allophobic and autophobic regimes defined in section II. The transition from allophobic to complete takes place at a grafting density (σ*) which is virtually independent of the length P of the free polymer. This is not very different from the theoretical result; the P dependence found in the calculations is so weak that it is hardly detectable experimentally. The transition from complete wetting to the autophobic regime (at σ**) decreases with increasing P, just as found in the calculations. However, at large P there is a clear difference. In the theoretical phase diagram, the upper transition continues to decrease and eventually hits the lower transition at P′′, and for P > P′′ complete wetting no longer exists. In the experimental phase diagram, we still see a regime of stable films up to very long chains (molar mass of 104 kg/mol). Figure 8 gives the stability boundaries of Figure 7 over a limited range of P values (5-40), plotted on a doublelogarithmic scale. Within experimental error, we can consider the lower curve as horizontal (we find a slope of

Figure 8. Stability boundaries in the (σ, P) plane (doublelogarithmic) showing the scaling behavior. Data were taken from Figure 7.

-0.02), but the upper transition σ** clearly goes down with increasing P. We find a slope of -0.5 ( 0.05 which is consistent with the drying limit and definitely less steep than the -1.5 predicted by Leibler’s interpenetration limit. In Figure 7 (linear σ-axis), we found a window of complete wetting up to very high P, whereas in the theoretical prediction of Figure 5 (logarithmic scales) we found an “upper wetting limit” at P ) P′′. In the representation of Figure 8, we may extrapolate the experimental data for σ* and σ** and find P′′ to be of the order of 350. Hence, for not too high P there seems to be, to some extent, agreement between experiment and theory. The reason that for very high P this agreement is absent may be kinetic factors. As explained in the theoretical section, one expects at high P a range of grafting densities where the contact angle is not zero but very small. Dewetting of the (very viscous) films may be very slow. Yet, it seems unlikely that no instabilities would be detectable after times as long as 12 days. However, the shape of the isotherms in this range might provide an explanation. As an example, we consider a disjoining pressure Π(d) isotherm, see Figure 9. In such an isotherm, the chemical

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Figure 9. Calculated disjoining pressure as a function of film thickness for a grafting density close to the minimum contact angle. Note the maximum at d ) 3. Parameters: P ) 500, N ) 200, σ ) 0.0113.

potential µ, interpreted as the disjoining pressure Π, is plotted as a function of the adsorbed amount Γ which is translated into a film thickness d. We chose a case close to the minimum contact angle in Figure 9. As can be seen, this curve has two minima, one at very small d (essentially a dry surface) and one at somewhat larger d (a mesoscopic film). Between these two minima, there is a maximum. This maximum is very effective in suppressing the formation of holes in the film, thus making it metastable. We conjecture that, experimentally, it is very likely that we find, at high P and intermediate σ, very long-lived, metastable films (corresponding to the shallow minimum in Figure 9) rather than the nearly dry (dewetting) surface, that is, the film near the deep minimum in Figure 9. Let us now consider the scaling descriptions for σ* and σ** as given in eqs 5 and 9. The wetting transition σ* occurs at a grafting density around 0.15 chains/nm2. Equation 5 predicts a value of σ ≈ 0.08 chains/nm2 (for large P). Taking into account that this is a scaling result derived from very simple arguments for very diluted brushes, it agrees reasonably well with the experimental data. The upper transition σ** shows (for not too large values of P) a decrease with P compatible with P-1/2. This is in line with the drying limit predicted by eq 9. This is in conflict with the numerical SCF calculations (Figure 5) which show a much stronger dependence even in the limit P , N where the Leibler interpenetration scaling P-3/2 is approached. This is remarkable, because of the three predictions for σ**, the drying argument is the most hand waving. In this argument, it was just conjectured that the wetting transition coincides with the condition that there is not enough space for the free chains in the brush. We argued above that there might be kinetic reasons why the experimental system gives a “phase diagram” with a wider window of complete wetting than expected from equilibrium considerations. The kinetics seems to be relatively fast when N is not much different from P and in particular when the brush becomes dry. For P . N, the kinetics progressively slows. More evidence of this is discussed below. B. Pattern Formation around Transitions. The double-well structure of the disjoining pressure as predicted by the numerical SCF theory has been used above to explain the deviations found between theory and experiments for large values of P: for intermediate grafting density the films remained perfectly stable. However, when the grafting density was changed to lower or higher values, that is, toward the allophobic and the autophobic dewetting transitions, interesting dewetting

Figure 10. An AFM image of a 6 × 6 µm area of the dewetting patterns found in the intermediate regime between stable (complete wetting) and unstable (partial wetting) regimes. Data were taken from Figure 6b and plotted as a 3D diagram. Droplets, surrounded by holes, and (meta)stable film are clearly seen. The highest point is 40 nm above the substrate.

patterns were observed (cf. Figures 6 and 7). We believe that these patterns can be explained again by the doublewell structure of the disjoining pressure. We were able to distinguish two intermediate regimes (Figure 7) both near σ* and near σ**. The first intermediate regime (named “hills” in Figure 7) shows two thicknesses in one sample: a thin polymer film on top of the brush and droplets of bulk liquid. In the second intermediate regime (named “holes and hills” in Figure 7), three thicknesses occur simultaneously: macroscopic droplets, nearly dry surface, and a mesoscopic polymeric film having almost the thickness of the applied overlayer. Indeed, we find side by side all possible combinations: a droplet surrounded by a very thin film, droplets surrounded by mesoscopically thin films, and the mesoscopic film with holes in it. Figure 10 gives an example; the data were taken from Figure 6b and plotted in 3D to show the morphology. These patterns are a consequence of the way in which slowly evolving instabilities develop. Simulations of instabilities in dewetting films have recently become available.31-33 The evolution of a thin film with time is described by a kind of Navier-Stokes equation, with the surface tension and the disjoining pressure of the film as driving forces and a Newtonian viscosity term counteracting. For the special case where there was a double minimum in the free energy curve as a function of the thickness, the simulations on the corresponding film properties gave the following results.33 (i) Thick films disproportionate in the usual way by creating cylindrical holes. With increasing annealing time, the holes grow in size until they touch, leading to a polygonal network of liquid ribbons. Eventually these ribbons decay into droplets due to the wellknown instabilities of a ridge. (ii) Films with thicknesses below the spinodal minimum decay into droplets via a spinodal dewetting mechanism with some well-defined wavelength. (iii) Films of intermediate thickness are the more interesting ones: in the course of time they develop a variety of structures on the substrate. These structures depend not only on the initial film thickness but also on the precise shape of the free energy curve. Our experiments show structures very similar to those obtained in calcula(33) Singh, J.; Sharma, A. J. Adhes. Sci. Technol. 2000, 14, 145166.

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Figure 11. Wetting behavior as a function of the initial film thickness of the free PS, for Mn ) 183 kg/mol.

tions of Sharma et al.,31-33 indicating that the pattern formation may well be due to a double minimum in the free energy. Also, such a double minimum in the free energy was not only found in earlier self-consistent-field calculations5 but also confirmed by the calculations in the present work (see section II.B and Figure 9), thus supporting this explanation. V. Discussion The experimental results presented so far give reasons to believe that the dewetting may be a very slow process and that kinetic effects should be taken into account. If this is true, one would expect that the results also depend on (nonthermodynamic) experimental parameters of the system. In Figure 11, the influence of the initial film thickness d of the free PS films on the stability/instability transition is shown, as a function of σ. Indeed, the grafting densities where transitions occur depend slightly on d, the lower transition shifting down with decreasing d and the upper one shifting up. For example, at σ ) 0.2 nm-2 and d ) 2 nm there is a stable film; increasing d to 9 nm leads to instability (arrow a f b in Figure 11). Likewise, one crosses the upper stable/unstable transition (arrow) upon increasing d at about 0.6 nm-2. This may suggest that for d f ∞ there is no transition left in this particular example. Around the upper transition, we have a brush and a very thin overlayer (point c in Figure 11). A large fraction of the overlayer can be used to swell the attached chains on the surface. The result is that the intrinsically nonwettable surface seems to display wetting behavior. Basically, there is (on the time scale of the experiment) not enough free polymer available to form droplets. Increasing the thickness of the overlayer to point d in Figure 11 then results in a dewetting behavior. The effect can be qualitatively understood in terms of adsorption isotherms such as, for example, Figure 3. The slopes of such isotherms represent the thermodynamic stability: a positive slope corresponds to a stable film and a negative slope to an unstable film. The point of reversal between negative and positive shifts to lower Γ (or d) at higher σ (Figure 3, dashed curves), so that a thin film is indeed less prone to break up upon increase of σ. A similar reasoning applies also to the very sparsely grafted layer: thermodynamic arguments would lead to the conclusion that with increasing thickness the transition can only remain at the same grafting density or shift downward. We see just the opposite. It seems more plausible that very thin films on a sparsely grafted layer are kinetically stabilized against the forces that would otherwise destabilize them. Indeed, to a first approximation, the rate of film breakup would scale as d, so that the destabilization time diverges at small d unless there is a brush to facilitate mobility and transport of the free polymer.

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The experiments show that it is possible to have very thin films on top of, or submersed into, a brush. These thin films may or may not be at bulk coexistence. We believe that the 12 days of relaxation of these films as allowed by our experimental methods is often not enough to guarantee equilibrium. Thus, the brushes can trap a wetting film. Thin stable films of a wetting component inside brushes have been studied by analytical34 and numerical techniques for the case that the wetting component is low molecular weight.35 In a recent study, even polyelectrolyte brushes were considered.36 In these papers, the chain conformations have been studied in detail. These systems have many interesting features in common with the current system. It is worth investigating whether it is feasible to probe by AFM experiments brushed surfaces that have a film of low molecular weight solvent submerged into them. Numerical SCF calculations were used to mimic the experimental system. To translate the dimensionless grafting density as used in the SCF calculations to the real dimensions (chains per unit area), it is necessary to know the cross-sectional area of a segment. However, a quantitative comparison between the theory and experiments was not attempted. For a more detailed comparison, it is essential to tune the SCF parameters more exactly to the system under investigation. Moreover, in the numerical SCF calculations the van der Waals contributions are only partially included. The short-range part is accounted for in the Flory-Huggins interaction parameters. The long-range aspect has been ignored. We expect, however, that especially when the brush is thick (i.e., for large values of N and σ) the disjoining pressure as found by the SCF calculations dominates over the van der Waals contribution. However, a more detailed study on the interplay of the double-well contributions due to the brushed surface and the van der Waals part may still be useful, because the combination can give an even more rich disjoining pressure isotherm than that predicted by the numerical SCF model. VI. Conclusions We have studied the wetting behavior of a polystyrene film on a surface carrying grafted polystyrene, as a function of the grafting density and the chain length of the free polymer. From a simple scaling analysis, we find that if the bare surface is partially wet (i.e., has a finite contact angle), grafted chains may induce two wetting transitions: one from partial to complete wetting at low grafting density and one back from complete to partial wetting at high grafting density. We call the region of partial wetting at low grafting density the allophobic regime and the second region of partial wetting at high grafting density the autophobic regime. The lower transition (allophobic to complete) occurs at a grafting density which increases weakly with the molar mass P of the free chains. In contrast, the grafting density at the upper transition (complete to autophobic) decreases strongly with P, roughly as P-1/2. The calculations predict an intersection point of the two transitions at high P, where the complete wetting regime ends. These scaling predictions are qualitatively confirmed by numerical self-consistent-field (34) Johner, A.; Marques, C. M. Phys. Rev. Lett. 1992, 69, 18271830. (35) Leermakers, F. A. M.; Mercurieva, A. A.; van Male, J.; Zhulina, E. B.; Besseling, N. A. M.; Birshtein, T. M. Langmuir 2000, 16, 70827087. (36) Mercurieva, A. A.; Birshtein, T. M.; Zhulina, E. B.; Iakovlev, P.; van Male, J.; Leermakers, F. A. M. Macromolecules 2002, 35, 47394752.

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calculations. Moreover, the calculations show that at high P there is a window of grafting densities where films may be metastable rather than stable. Experimentally, for short chains in the melt the existence of the two wetting transitions separating an allophobic, a complete wetting, and an autophobic regime is fully confirmed. The dependences of the transitions on the grafting density and on the molar mass P are qualitatively understood. However, in the experiments the upper limit of the complete wetting regime at high P, as predicted by the theory, is not found. This is ascribed to the fact that at high chain length of the melt, the equilibrium contact angle is finite, but there is a secondary minimum of (nearly) zero contact angle. The films then do not dewet because a pronounced maximum in the disjoining pressure prevents them from dewetting, making thick films metastable. Our experiments cannot distinguish such metastable films from true wetting films.

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Around the wetting transitions, peculiar patterns are formed which were concluded to be signatures of the destabilization process driven by the disjoining pressure in the film. In line with recent calculations reported in the literature, the occurrence of droplets together with depressed regions was attributed to the presence of two minima in the free energy as a function of the film thickness. Such a double minimum was indeed found in self-consistent-field calculations, thus supporting this conclusion. Acknowledgment. This work was supported financially by the Dutch National Research School PolymerenPTN. Discussions with Professor A. Skvortsov (St. Petersburg) are gratefully acknowledged. We are also grateful to the referees for several constructive suggestions. LA020430Y