Surface Forces between Telechelic Brushes Revisited: The Origin of a Weak Attraction

Transcription

1 2712 Langmuir 2006, 22, Surface Forces between Telechelic Brushes Revisited: The Origin of a Weak Attraction Dapeng Cao and Jianzhong Wu* Department of Chemical and EnVironmental Engineering, UniVersity of California, RiVerside, California ReceiVed October 13, In Final Form: January 12, 2006 Telechelic polymers are useful for surface protection and stabilization of colloidal dispersions by the formation of polymer brushes. A number of theoretical investigations have been reported on a weak attraction between two telechelic brushes when they are at the classical contact, i.e., when the surface separation is approximately equal to the summation of the brush thicknesses. While recent experiments have confirmed the weak attraction between telechelic brushes, its origin remains elusive because of conflicting approximations used in the previous theoretical calculations. In this paper, we have investigated the telechelic polymer-mediated surface forces by using a polymer density functional theory (PDFT) that accounts for both the surface-adhesive energy and segment-level interactions specifically. Within a single theoretical framework, the PDFT is able to capture both the depletion-induced attraction in the presence of weakly adhesive polymers and the steric repulsion between compressed polymer brushes. In comparison of the solvation forces between telechelic brushes with those between brushes formed by surfactant-like polymers and with those between two asymmetric surfaces mediated by telechelic polymers, we conclude that the weak attraction between telechelic brushes is primarily caused by the bridging effect. Although both the surfactant-like and telechelic polymers exhibit a similar scaling behavior for the brush thickness, a significant difference has been observed in terms of the brush microstructures, in particular, the segment densities near the edges of the polymer brushes. 1. Introduction Polymers tethered onto a solid surface are useful for colloidal stabilization and for prevention of often undesirable, nonspecific adsorptions. 1,2 For example, hydrophilic polymers such as poly- (ethylene glycol) (PEG) or poly(ethylene oxide) (PEO) are widely used for improving the biocompatibility of implanted medical devices and for stabilizing liposomes devised as drug carriers. 3 A similar concept has been proposed recently for controlling adsorption of biomacromolecules secreted by microorganisms. 4 The conformations of one-end-grafted, uncharged linear polymers in a good solvent and interactions between polymer-grafted surfaces have been well-documented. 5-9 A widely held theoretical approach was developed by Milner, Witten, and Cates (MWC) 10 on the basis of earlier concepts proposed by Alexander 11 and more comprehensively by de Gennes. 12 This analytical selfconsistent mean-field theory predicts that, in a good solvent and at moderately high surface coverage, the end-grafted polymer * To whom correspondence should be addressed. jwu@engr.ucr.edu. Current address: P.O. Box 100, College of Chemical Engineering, Beijing University of Chemical Technology, Beijing , People s Republic of China. (1) Halperin, A.; Tirrell, M.; Lodge, T. P. AdV. Polym. Sci. 1992, 100, (2) Currie, E. P. K.; Norde, W.; Stuart, M. A. C. AdV. Colloid Interface Sci. 2003, 100, (3) Castner, D. G.; Ratner, B. D. Surf. Sci. 2002, 500, (4) Hoipkemeier-Wilson, L.; Schumacher, J.; Carman, M.; Gibson, A.; Feinberg, A.; Callow, M.; Finlay, J.; Callow, J.; Brennan, A. Biofouling 2004, 20, (5) Netz, R. R.; Andelman, D. Phys. Rep. 2003, 380, (6) Kreer, T.; Metzger, S.; Muller, M.; Binder, K.; Baschnagel, J. J. Chem. Phys. 2004, 120, (7) Singh, C.; Pickett, G. T.; Balazs, A. C. Macromolecules 1996, 29, (8) Zhulina, E. B.; Borisov, O. V.; Priamitsyn, V. A. J. Colloid Interface Sci. 1990, 137, (9) Szleifer, I. Biophys. J. 1997, 72, (10) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, (11) Alexander, S. J. Phys. IV 1977, 38, (12) de Gennes, P. G. Macromolecules 1980, 13, brushes exhibit a parabolic local density profile and the surface potential between two brush layers varies as the cube of the compression distance. These theoretical predictions have been validated with results from molecular simulations for various coarse-grained models of grafted polymers. 6 Telechelic polymers are linear chains with functional groups at both ends. If the polymer backbone is highly soluble and the ends can be attached to a solid surface, telechelic polymers also provide a steric repulsive layer useful for surface protection. Two types of telechelic polymers have been extensively studied in the literature. One is represented by ABA triblock copolymers, such as PEO-polystyrene (PS)-PEO, where block A can be absorbed onto the surface but block B cannot. 13 The other consists of end-modified homopolymers such as long hydrophilic polymers with a small hydrophobic or ionic group at each end. 14,15 Previous experiments and theoretical calculations both indicate that the interaction between two highly stretched telechelic brushes is primarily repulsive, resembling that between two singly tethered polymer brushes Nevertheless, a weakly attractive force has also been observed when the two telechelic brushes are near contact; i.e., the separation is about twice the thickness of an isolated brush. 14,20,21 The origin of such a weak attraction has been subjected to a number of theoretical analyses Witten (13) Dai, L. M.; Toprakcioglu, C. Macromolecules 1992, 25, (14) Eiser, E.; Klein, J.; Witten, T. A.; Fetters, L. J. Phys. ReV. Lett. 1999, 82, (15) Berret, J. F.; Serero, Y. Phys. ReV. Lett. 2001, (16) Bhatia, S. R.; Russel, W. B. Macromolecules 2000, 33, (17) Klein, J. J. Phys: Condens. Matter 2000, 12, A19-A27. (18) Pham, Q. T.; Russel, W. B.; Thibeault, J. C.; Lau, W. Macromolecules 1999, 32, (19) Pham, Q. T.; Russel, W. B.; Thibeault, J. C.; Lau, W. Macromolecules 1999, 32, (20) Wijmans, C. M.; Leermakers, F. A. M.; Fleer, G. J. J. Colloid Interface Sci. 1994, 167, (21) Zhulina, E.; Singh, C.; Balazs, A. C. Langmuir 1999, 15, (22) Milner, S. T.; Witten, T. A. Macromolecules 1992, 25, (23) Johner, A.; Joanny, J. F. J. Chem. Phys. 1992, 96, la CCC: $ American Chemical Society Published on Web

2 The Origin of a Weak Attraction Langmuir, Vol. 22, No. 6, and co-workers suggested that the attraction is due to thermal fluctuations of the middle segments near the edge of the polymer brush, which allow the polymer ends to penetrate into the opposite brush to form the bridges. 22,25 They predicted that, near the classical contact, the number of bridges is proportional to the mean-square end-to-end distance of the polymer and rises linearly with the compression distance as the telechelic brushes are compressed. Bjorling and Stilbs, on the other hand, argued that the bridge formation suppresses the monomer number density near the surface that results in an entropic attraction. 26,27 Another explanation was offered by Zilman and Safran who attributed the attraction to the association of the functionalized chain ends to form clusters. 28 More recently, Meng and Russel 29 extended the MWC theory to telechelic polymers and predicted that the attraction is due to the combinatorial entropy of the polymer end distribution between two surfaces. The attraction predicted by Meng and Russel is much stronger than that suggested by Witten and co-workers. In all of these theoretical investigations, it was assumed that the anchoring energy is sufficiently strong such that all free ends are permanently attached to the surface. Nevertheless, numerical results from lattice-model self-consistentfield theory 20 and Monte Carlo simulations indicate that both the fraction of bridging and the telechelic-mediated surface forces are sensitive to the chain end-surface interactions explicitly. In this paper, we examine the surface forces mediated by telechelic polymers by using a polymer density functional theory (PDFT). 35 A similar calculation has been recently applied to solvation forces because of multivalent polymers. 36 The objective here is to conciliate various interpretations of the telechelic mediated forces by inspecting the solvation forces between the symmetric and asymmetric surfaces and those between surfaces mediated by telechelic polymers and by single-end attachable polymers. In comparison with alternative theoretical methods, PDFT takes the advantages of specifically accounting for the chain end-surface interactions along with the segment-level excluded-volume effects and van der Waals attractions. Furthermore, PDFT allows us to consider, at a certain degree, the correlation effects that are neglected in a typical mean-field theory. We expect that our calculations will also reveal the structural difference between a singly tethered brush and a telechelic brush when both are isolated. 2. Molecular Models We consider a coarse-grained model of telechelic polymers dissolved in a good solvent. The backbone of the telechelic polymers is represented by a freely jointed tangent-sphere chain, and the sticky ends are adhesive to a solid surface. While this simplified model is not intended to represent any specific telechelic polymer used in experiments, it retains the generic features including explicit endsurface interactions, excluded-volume effect, and chain connectivity. For simplicity, the solvent is treated as a continuous medium and all polymer segments are assumed to have the same size. Figure 1a (24) Avalos, J. B.; Johner, A.; Joanny, J. F. J. Chem. Phys. 1994, 101, (25) Tang, W. H.; Witten, T. A. Macromolecules 1996, 29, (26) Bjorling, M.; Stilbs, P. Macromolecules 1998, 31, (27) Bjorling, M. Macromolecules 1998, 31, (28) Zilman, A. G.; Safran, S. A. Eur. Phys. J. E 2001, 4, (29) Meng, X. X.; Russel, W. B. Macromolecules 2003, 36, (30) Misra, S.; Nguyen-Misra, M.; Mattice, W. L. Macromolecules 1994, 27, (31) Misra, S.; Mattice, W. L. Macromolecules 1994, 27, (32) Nguyen-Misra, M.; Misra, S.; Mattice, W. L. Macromolecules 1996, 29, (33) Peng, C. J.; Li, J. K.; Liu, H. L.; Hu, Y. Eur. Polym. J. 2005, 41, (34) Li, J. K.; Peng, C. J.; Liu, H. L.; Hu, Y. Eur. Polym. J. 2005, 41, (35) Cao, D. P.; Wu, J. Z. Macromolecules 2005, 38, 971. (36) Cao, D. P.; Wu, J. Z. Langmuir 2005, 21, Figure 1. (a) Schematic of a telechelic polymer. (b) Definitions of the surface separation H and the range of adhesion potential w. (c) Solvation forces because of associating polymers. Case I, telechelic polymers between adhesive surfaces; case II, surfactant-like polymers between adhesive surfaces; case III, telechelic polymers between a neutral and an adhesive surface. shows a schematic of a telechelic chain according to our coarsegrained model. The nonsticky backbone segments, denoted as B, are represented by neutral hard spheres, and the sticky ends, denoted as A segments, are represented by attractive spheres. The interaction between A and B segments is assumed to be the same as that between two B segments, and the interaction between two A segments is described by a square-well potential 35 { r < σ φ AA (r) ) -ɛ AA σ e r e γσ 0 r > γσ where σ is the segmental diameter and γσ is the square-well width. Throughout this paper, we assume γ ) 1.2 and the attractive energy between two A segments is ɛ AA ) 1kT, where k stands for the Boltzmann constant and T stands for the absolute temperature. To examine the adsorption behavior and solvation forces, we consider telechelic polymers confined between two infinitely large parallel plates that are neutral to B segments but can be either attractive or neutral to A segments. In other words, we consider the solvation forces between symmetric surfaces and that between asymmetric surfaces. For an attractive plate, the surface energy for each A segment is represented by a square-well potential att ψ AW (z) ) { -ɛ AW 0 < z < w 0 otherwise where z denotes the perpendicular distance from the surface, w is the width of the surface potential, and ɛ AW stands for the strength of the surface attraction. The range of the surface attraction is fixed at w ) 0.5σ. Figure 1b shows the definitions of the surface width parameter (w) and the perpendicular distance from the surface (H). For a comparison, we consider also the solvation forces as a result of polymers with only single sticky ends (case II in Figure 1c). In these cases, polymer brushes are formed but no bridge conformations are expected between two surfaces. 3. Polymer Density Functional Theory (PDFT) A. Helmholtz Energy Functional. The Helmholtz energy functional of a polymeric fluid is conventionally expressed in (1) (2)

3 2714 Langmuir, Vol. 22, No. 6, 2006 Cao and Wu terms of that for a system of ideal chains and an excess part taking into account the nonbonded interactions 37 where R ) (r 1, r 2,..., r M ) represents a composite vector defined by the positions of individual segments and F M (R) stands for a multidimensional density profile, which is related to the segmental densities by F AorB (r) ) F[F M (R)] ) F id [F M (R)] + F ex [F M (R)] (3) j A orb F sj (r) ) j AorB The Helmholtz energy functional for ideal chains is exactly known βf id [F M (R)] ) drf M (R)[lnF M (R) - 1] + drδ(r - r j )F M (R) (4) β drf M (R)V bond (R) (5) where β ) (kt) -1 and V bond (R) stands for the bond potentials. For a tangent-sphere chain consisting of M identical segments, V bond (R) is related to the Dirac-delta function by M-1δ( r i+1 - r i - σ) exp[- βv bond (R)] ) (6) i)1 4πσ 2 The excess Helmholtz energy functional, on the other hand, takes into account the contributions as a result of the hard-sphere repulsions, the van der Waals attractions, and the intramolecular chain correlations F ex [F i (r)] ) F hs [F i (r)] + F vdw [F i (r)] + F chain [F i (r)] (7) In writing eq 7, it is assumed a priori that the excess Helmholtz energy functional can be completely specified by the segmental density profiles F i (r). As in our previous papers, 35,37-39 the excess Helmholtz energy functionals for the hard-sphere repulsion and chain correlation are represented by a modified fundamental measure theory 38,40 and a generalized first-order perturbation theory, 41 respectively βf hs ) dr{ -n 0 ln(1 - n 3 ) + n 1 n 2 - n V1 n V n 3 (n n 2 n V2 n V2 )[ ln(1 - n 3 ) 1 + (8) 2 12πn 3 12πn 3 (1 - n 3 ) 2]} βf chain ) dr 1 - M M n 0 ξ ln yhs (σ,n R ) (9) where n R (r), R)0, 1, 2, 3, V1, V2 are the scalar and vector weighted densities, 42 ξ ) 1 - n V2 n V2 n 2 2, and y hs (σ,n R )isthe contact value of the cavity correlation function between segments y hs (σ,n R ) ) 1 + n 2 ξσ 1 - n 3 4(1 - n 3 ) + n 2 2 ξσ (10) 2 72(1 - n 3 ) 3 The excess Helmholtz energy as a result of chain correlations (37) Cao, D. P.; Wu, J. Z. J. Chem. Phys. 2004, 121, (38) Yu, Y. X.; Wu, J. Z. J. Chem. Phys. 2002, 117, (39) Li, Z.; Cao, D. P.; Wu, J. Z. J. Chem. Phys. 2005, 122, (40) Roth, R.; Evans, R.; Lang, A.; Kahl, G. J. Phys: Condens. Matter 2002, 14, (41) Yu, Y. X.; Wu, J. Z. J. Chem. Phys. 2002, 117, (42) Rosenfeld, Y. J. Phys: Condens. Matter 2002, 14, is not directly related to the bonding potentials of polymeric molecules that have been already included in eq 5. Instead, it takes into account the effect of chain connectivity on the nonbonded correlations between polymeric segments. Finally, the excess Helmholtz energy functional as a result of van der Waals attractions is represented by the mean-field approximation 35 βf vdw ) 1 drdr F i (r)f j (r)βφ att ij ( r - r ) (11) 2 i,j)a,b where φ att ij (r) is given by eq 1. In a previous work, 35 we have shown that the predictions of the PDFT agree with simulation results quantitatively. At equilibrium, the molecular density profile F M (R) can be solved by minimization of the grand potential Ω[F M (R)] ) F[F M (R)] + [ψ M (R) - µ M ]F M (R)dR (12) where ψ M stands for the external potential for the entire molecule and µ M denotes the chemical potential. For a given bulk density, µ M is calculated from the corresponding equation of state for the polymeric fluids. 35 Following the variational principle, the stationary condition of the grand potential satisfies δω[f M (R)] ) 0 (13) δf M (R) A combination of eqs 4, 12, and 13 gives a set of coupled integral equations for the segmental density profiles F si (r) ) drδ(r - r i ) exp[βµ M - βv bond (R) - where the self-consistent field λ i)a,b (r) is defined in terms of the excess Helmholtz energy functional F ex and the external potentials for individual segments ψ A (r) and ψ B (r) λ A (r j ) ) δf ex β j A λ A (r j ) - β The external potentials for A and B segments are given by eq 2. For polymers confined between two planar surfaces, the density profiles vary only in the direction perpendicular to the surface (z), i.e., F si (r) )F i (z). In this case, the average segmental densities can be simplified to where G L i (z) and G R i (z) are, respectively, the left and right recurrence functions with G L 1 (z) ) 1, G R M (z) ) 1, and i ) 1, 2,..., M. Equation 16 can be solved by using the standard Picard iteration. Unlike those k B λ B (r k )] (14) δf A (r j ) + ψ A (r j ) λ B (r k ) ) δf ex δf B (r k ) + ψ B (r k ) (15) F AorB (z) ) exp(βµ M ) G i L (z) exp[-βλ i (z)]g i R (z) (16) i AorB G i L (z) ) 1 z+σ exp[-βλi-1 (z)]g i-1 2σ z-σ L (z) dz (17) G i R (z) ) 1 z+σ exp[-βλi+1 (z)]g i+1 2σ z-σ R (z) dz (18)

4 The Origin of a Weak Attraction Langmuir, Vol. 22, No. 6, Figure 2. Solvation forces between two surfaces mediated by telechelic polymers: (a) chain length M ) 11, and (b) M ) 32. The inset plots are magnified views near the classical contact. for homopolymers, the left and right recurrence functions for the telechelic polymers bear no symmetry. 35,37,43 B. Solvation Forces. For two planar surfaces mediated by a polymer solution, the solvation force per unit area is calculated from 44 F H (2AkT) )- 1 A βω H ) j ψ j dz F j (z) z (19) where (F H A) is the force per unit area on a single surface, a factor of 2 means that the force is exerted on two surfaces, and ψ j is the external potential applied to segment j. At a given surface separation H, the reduced solvation force per unit area is given by 36 Fσ 3 (2AkT) ) (F H - F )σ 3 (2AkT) (20) where F is the force per unit area when the two surfaces are infinitely apart. 4. Results and Discussion Unless the polymer ends are covalently bonded to the surface, the interaction between two telechelic brushes is sensitive to both the surface-adhesive energy and the polymer chain length. To inspect such effects, Figure 2 shows the predicted solvation forces between two identical surfaces induced by telechelic polymers of various surface-adhesive energies and mid-block chain lengths. In all cases, the polymer volume fraction in the bulk is fixed at φ b ) 0.05, assuming an ideal scenario where only (43) Cao, D. P.; Wu, J. Z. J. Chem. Phys. 2005, 122, (44) Maciolek, A.; Drzewinski, A.; Bryk, P. J. Chem. Phys. 2004, 120, a small amount of polymer is used for surface protection or stabilization. For relatively short telechelic chains (M ) 11, Figure 2a), the solvation force varies from that dominated by shortrange attraction to that with long-range repulsion as the surface energy for the end segments is increased from ɛ AW ɛ AW kt ) 1, 8, 12, to 15. The former resembles that induced by nonadsorbing polymers, wherein the short-range attraction is due to the depletion of polymers near the surfaces. Conversely, the latter corresponds to that between two permanently tethered telechelic brushes. The inset plot in Figure 2a provides a magnified view of the solvation forces when the two polymer layers are near classical contact, i.e., when the separation between the two surfaces is about twice the brush thickness. For weakly adhesiveor nonadsorbing chains, the PDFT captures both entropic depletion and a weak, secondary repulsion between two nonadsorbing surfaces that is missed by the Asakura-Oosawa (AO) theory. 45 The weak repulsion as a result of nonadsorbing polymers was first claimed by Napper and co-workers 46 and has been extensively documented in the literature (e.g., see ref 47). However, direct experimental verification of this weak repulsion is often below the sensitivity limit of typical force measurement techniques. 48,49 For strongly adsorbed telechelic polymers, on the other hand, the PDFT predicts a weak attraction at the classical contact. Because the chain length of telechelic polymers is M ) 11, the maximum attraction that occurred at about H* Hσ ) 8 corroborates the highly extended polymer configurations conjectured in previous self-consistent field investigations. As discussed in the Introduction, the weak attraction between two identical telechelic brushes has been observed in experiments and also predicted by a number of theoretical calculations. However, different from previous investigations, no permanent adsorption is assumed in the PDFT calculations. Clearly, both the range and strength of the attraction vary with the tethering energy of the polymer ends. Figure 2a also indicates that the strength of repulsion is directly correlated with the tethering energy. As shown in Figure 2b, no qualitative difference is observed as the chain length is increased from M ) 11 to 32, suggesting that the solvation forces as discussed above for relatively short chains are equally applicable to long telechelic polymers. Qualitatively, however, we notice that the strength of the solvation force is closely related to the tethering energy per segment, and a stronger tethering energy (ɛ AW ) 30) is therefore required for the formation of longer polymer brushes. Furthermore, the brush thickness (or the range of repulsionattraction) is directly correlated with the polymer chain length. Interesting enough, the strength of attraction at the classical contact of two telechelic brushes varies little with the chain length. As for shorter chains, the telechelic polymers for M ) 32 are also highly extended in the brush limit. Figure 3 presents the reduced local density profiles for the polymeric segments at the low and high surface energies ɛ AW )1 and 15 as discussed in Figure 2a. For clarity, the density profiles for H* ) 6 and 4 have been consecutively moved upward by 0.5 unit for the case at the low surface energy and by 5 units for the case at the high surface energy. Because of the symmetry of the telechelic polymers, the density profiles are also symmetric. As expected, a low polymer density is observed near weakly (45) Asakura, S.; Oosawa, F. J. Chem. Phys. 1954, 22, (46) Bolhuis, P. G.; Louis, A. A.; Hansen, J. P.; and Meijer, E. J. J. Chem. Phys. 2001, 114, (47) Napper, D. H. Polymeric Stabilization of Colloidal Dispersions; Academic Press: London, U.K., (48) Verma, R., Crocker, J. C.; Lubensky, T. C.; and Yodh, A. G. Macromolecules 2000, 33, (49) Martin, J. I.; Wang, Z. G.; Schick, M. Langmuir 1996, 12, 495.

5 2716 Langmuir, Vol. 22, No. 6, 2006 Cao and Wu Figure 4. Solvation forces because of the surfactant-like and telechelic polymers of similar brush thicknesses. Figure 3. Reduced density profiles of the telechelic polymers near a weakly adhesive (ɛ AW ) 1) and near a strongly adhesive (ɛ AW ) 15) surface. adhesive surfaces and a surface accumulation appears near the high energy surfaces. The oscillatory density profiles shown in Figure 2b arise from the close packing of polymer segments. However, no oscillation is observed in the solvation forces because of the strong attachment of telechelic polymers. Unlike the solvation forces introduced by nonadsorbing polymers or solvent molecules, here, the solvation force is primarily due to the compression telechelic brushes instead of the depletion effect. To explore the origin of attraction near the classical contact of two telechelic brushes (H* ) 7-8 in Figure 2a, and H* ) in Figure 2b), we study in parallel two different scenarios (see cases II and III in Figure 1c), where bridging between two surfaces is prohibited. In the first scenario, we consider the solvation forces between two identical surfaces mediated by surfactant-like polymers, each having only one surface-adhesive segment at the end. In the second scenario, we consider the telechelic polymer-mediated solvation forces between two asymmetric surfaces, where one surface is attractive to the sticky ends of a telechelic polymer and the other is neutral. In both cases, we compare the solvation forces with those induced by telechelic polymers, where the bridging between two surfaces is allowed. We show in Figure 4 the solvation forces between two symmetric surfaces induced by surfactant-like polymers that each has only one end adhesive to the surface. Two types of surfactantlike chains are considered, with the number of segments per chain equal to M ) 11 and 16 and the adhesive energies for the end segments 15kT and 30kT, respectively. For a direct comparison, also presented in Figure 4 are the solvation forces introduced by telechelic polymers with the end segments identical to the adhesive ends of the surfactant-like polymers. These telechelic polymers are able to form bridge configurations across the surfaces. Their chain lengths, M ) 22 and 32, are selected such that the thicknesses of the telechelic brushes are identical to those of the brushes formed by surfactant-like polymers with M ) 11 and 16, respectively. According to our calculations, the thickness of a telechelic brush is not appreciably different from that of a brush formed by surfactant-like polymers of half chain length. Whereas there is a close match of the brush thicknesses, the surface forces as a result of surfactant-like polymers are distinctively different from those as a result of telechelic polymers, near the classical contact in particular: One is purely repulsive, and the other shows a weak attraction as discussed earlier. At the same surface energy per segment and brush thickness, the short-range repulsion as a result of the telechelic polymers is significantly stronger than that as a result of the surfactant-like polymers, which suggests a cooperative behavior of the two sticky ends of a telechelic polymer. Because the attraction emerges only between telechelic brushes but not between those formed by surfactant-like polymers, we conclude that the correlation of the end-segment distributions at the brush edges plays only a minor role in the solvation forces, and therefore, the bridging effect is the main cause of the weak attraction. The microscopic structure of a polymer brush varies with the degree of compression or equivalently, the surface separation. Such a variation is expected to be most significant for those segments near the edge of the polymer brush, i.e., the nonsticky ends of surfactant-like polymers or the middle segments of telechelic polymers. For uncompressed brushes of surfactantlike polymers, the end segments are broadly distributed and decay slowly near the brush surface (Figure 5a). When two identical brushes are pushed together, only the periphery end segments are compressed in the beginning and beyond the classical contact and the density profile for the end segments exhibits a bell-shape curve. In other words, an inhomogeneous variation of the microscopic structure is observed as the polymer brush is under compression. Figure 5b shows a similar variation of the microscopic structure for the compression of telechelic brushes. Because of the chain connectivity, the density profiles for the middle segments of telechelic polymers near the brush surface are noticeably different from those for the nonadhesive ends of surfactant-like polymers. In a telechelic brush, the middle-segment density near the brush surface declines sharply beyond its maximum, and upon a mild compression, no noticeable change is observed for the density profiles of the middle segments near

6 The Origin of a Weak Attraction Langmuir, Vol. 22, No. 6, Figure 7. Solvation forces between symmetric and asymmetric surfaces induced by telechelic polymers. Figure 5. Reduced density profiles (a) of the end segments of the surfactant-like polymers (M ) 16) and (b) of the middle segments of the telechelic polymers (M ) 32). Figure 6. Normalized brush thicknesses of the surfactant-like and telechelic polymers versus the surface separation. the adhesive surface, suggesting that the bridging formation has little to do with the variation of polymer configurations inside the brushes. The thicknesses of the single-end and telechelic brushes can be determined by the maximum values of the end- and middlesegment distributions, respectively. Figure 6 shows the normalized layer thickness LL infinite versus the surface separation H(2L infinite ), where L infinite represents the thickness of an isolated polymer brush. When the surface separation is less than 2L infinite, the brush thickness is essentially a linear function of the separation. When the surface separation is larger than 2L infinite, however, the brush thickness becomes the same as L infinite and is independent of the surface separation. Except some small deviations at the separation around 2L infinite, the normalized layer thicknesses for both singleend and telechelic brushes agree well with the predictions of Figure 8. Microstructures of telechelic polymers between two asymmetric surfaces. Milner and Witten 22 and of Meng and Russel 29 based on the self-consistent mean-field calculations. Figure 7 presents the telechelic polymer-mediated forces between two asymmetric surfaces; i.e., one surface is adhesive to the polymer ends, but the other surface is neutral to all segments. For a comparison, Figure 7 also shows the solvation forces between two symmetric surfaces mediated by surface-adhesive telechelic polymers. When the two surfaces are mediated by telechelic brushes of the same thickness, the solvation force between asymmetric surfaces is significantly stronger than that between the symmetric surfaces. In other words, at the same degree of compression, a telechelic brush repels a neutral surface more than a similar compressed telechelic brush. No attraction is observed between two asymmetric surfaces near the classical contact, suggesting that the weak attraction between two telechelic brushes is solely due to the bridging effect. For telechelic polymers of the same chain length, the solvation force between two asymmetric surfaces is in a shorter range than that between two symmetric surfaces. However, at small separations, the solvation force between two asymmetric surfaces is still noticeably stronger than that between two symmetric surfaces. The reduced repulsion between two symmetric faces is probably due to the increased probability of bridging at small separations. Because only one surface is adhesive to the polymer ends, the density profile of the telechelic polymers shows no symmetry (Figure 8). At a small separation (H* ) 2), the local densities near the attractive and neutral surfaces are similar because, in this case, the density profile is essentially determined by the confinement effect. At larger separations (H* ) 4 and 6), however,

7 2718 Langmuir, Vol. 22, No. 6, 2006 Cao and Wu the local density near the attractive surface is significantly greater than that near the neutral surface. 5. Conclusions We have investigated the solvation forces induced by telechelic polymers via a PDFT. The focus was on the origin of weak attraction between two telechelic brushes. Three cases were considered. Case I was concerned with the telechelic polymermediated forces between two symmetric surfaces that were adhesive to the polymer ends; case II was the solvation force as a result of surfactant-like polymers with only one sticky end; and case III dealt with the telechelic polymer-mediated interactions between an adhesive surface and a neutral surface. In case I, we expected loop- and bridge-like microscopic conformations for the telechelic polymers. In case II, the surfactant-like polymers formed polymer brushes similar to those by telechelic polymers but without loop and bridge conformations. In case III, the telechelic brushes interacted with a bare surface such that the bridge conformations were prohibited. With the segment-level interactions and polymer end-surface adhesion specifically taken into account, the PDFT was able to capture both the depletion attraction as a result of weakly adhesive polymers and strong repulsion between polymer brushes within the same theoretical framework. It also reproduced the weak attraction between two telechelic brushes when their edges were in contact. Because such attraction was absent in the solvation force between two single-tethered polymer brushes and that between a telechelic brush and a neutral surface, we concluded that the fluctuation of the end-segment distributions or the association of polymer ends played only a minor role in the telechelic polymer-mediated interactions. Instead, the attraction must be directly related to the bridging effect. While a stronger attraction between two telechelic brushes could also be reproduced by introducing a combinatory free energy for the end-segment distributions, 29 we found that, in general, the microscopic structure of a telechelic brush was quite different from that of a singly tethered brush, for the distributions of the endmiddle segments near the brush boundaries in particular. The nonsticky ends of a single-tethered polymer brush exhibited a broad density distribution near the brush surface. Conversely, the middlesegment density in a telechelic brush was narrowly distributed and declined sharply at the brush surface. When two polymer brushes were pushed together, an inhomogeneous variation of the segmental density profiles was observed for both singly tethered and telechelic brushes. In agreement with the earlier predictions, 22,29 the thickness of telechelic brushes scaled linearly with the surface separations when they were compressed. With the same end-surface adhesion and similar brush thickness, the interaction between two telechelic brushes was significantly more repulsive (except near classical contact) than that between two single-end tethered brushes. This cooperative behavior suggested that the telechelic polymers were probably more effective than surfactant-like polymers for surface protection or colloidal stabilization. In addition, we found that under a similar level of compression, the repulsion between a telechelic brush and a neutral surface was much stronger than that between two telechelic brushes. Although relatively short chains were used in this paper, we believe that similar conclusions should be valid for longer telechelic polymers. Figure 2a shows that the AO-type depletion potential has a length scale of about H* 2, which is clearly decoupled from that of the repulsion between two strongly adsorbed telechelic brushes that has a length scale of about H* 8. Whereas we have not performed the calculations by a systematic variation of the polymer chain length, no qualitative difference was observed for two telechelic polymers with M ) 11 and 32. Moreover, Figure 4 indicates that the strength of attraction appears independent of the chain length. Necessarily, the longer the polymer chain length, the stronger adhesion energy is required for the formation of the telechelic brushes. Acknowledgment. The authors are benefited from insightful comments by William Russel at Princeton University. This work is supported by the National Science Foundation (CTS and CTS ). LA