On the Origins of the Salt-Concentration-Dependent Instability and

Oct 9, 2009 - Interactions between Grafted Cationic Dendrimers and Anionic Bilayer Membranes. Thomas Lewis , Venkat Ganesan. The Journal of Physical ...
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On the Origins of the Salt-Concentration-Dependent Instability and Lateral Nanoscale Heterogeneities of Weak Polyelectrolyte Brushes: Gradient Brush Experiment and Flory-Type Theoretical Analysis Jaehyun Hur, Kevin N. Witte, Wei Sun, and You-Yeon Won* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907 Received July 13, 2009. Revised Manuscript Received September 15, 2009 We investigated, by experiment and theory, the lateral structure of a weak polyelectrolyte brush at various added salt concentrations and chain grafting densities. Model poly(2-(dimethylamino)ethyl methacrylate) (PDMAEMA) brushes with grafting density gradients were developed for this study by using a novel “Langmuir-Blodgett-depositionunder-compression” (LB\C) method. Fluid AFM images of these brushes indicate that the lateral structure of the brush system is sensitive to both added salt concentration and grafting density. Under low salt conditions, 0-20 mM NaCl, the brush structure shows strong microscopic lateral heterogeneities at high grafting densities; both the width and height of the heterogeneities increase with increasing grafting density but are independent of the salt concentration. As the bulk salt concentration is increased to an intermediate regime, 60-100 mM NaCl, these heterogeneities become smaller in size and number, coexisting with smooth homogeneous regions. At high enough concentrations, 300-500 mM NaCl, the entire surface becomes homogeneous. A simple free energy-based Flory-type argument is presented which explains the essential features of the thermodynamic behavior of the brush system. In the zero-salt limit, relatively few monomers are charged, and the hydrophobicity of the backbone chain drives the collapse/aggregation of the chains. At high salt concentrations, the brush chains become sufficiently charged to overcome the hydrophobic nature of the monomers and stabilize the homogeneous state. However, at intermediate salt concentrations, it is found that the osmotic pressure of the counterions surrounding the charged polymer moieties can be decreased by collapsing the chain structure while simultaneously decreasing the number of charges along the backbone and releasing small ions into the bulk solution. This effect, which we term “osmotic instability”, serves to destabilize the homogeneous brush configuration.

1. Introduction Attachment of charged moieties to polymers, in general, enhances the solubility of the polymers in water. Charged polymer segments repel each other primarily by Coulomb forces. In water, this Coulomb repulsion is relatively short-ranged; the Bjerrum length of water is about 7 A˚. In most practical situations where the presence of small ions (e.g., H3O+, OH-, or ions from added salt or contaminants) is unavoidable, the like-charge repulsion is further mediated by the osmotic effect of an increased counterion concentration surrounding the charged segments. When this screened electrostatic repulsion dominates over other opposing forces (such as hydrophobicity of the polymer), the charged polymers (“polyelectrolytes”) become molecularly dispersed in the medium. This dissolved state of polyelectrolytes can be perturbed rather easily. One reason for this lies in the fact that most of the common polyelectrolytes are inherently hydrophobic in nature. For instance, amine-based polycations (e.g., poly(2-(dimethylamino)ethyl methacrylate) (PDMAEMA)1 and polyethylenimine (PEI)2) normally become water-insoluble at high pH where the fraction of charged monomers is too low to produce sufficient monomer-monomer repulsion necessary to overcome the tendency of the chains to collapse or aggregate in water. The same happens for polyanions (e.g., poly(acrylic acid) (PAA)3 and *To whom correspondence should be addressed. E-mail: yywon@ecn. purdue.edu. (1) Plamper, F. A.; Schmalz, A.; Ballauff, M.; Muller, A. H. E. J. Am. Chem. Soc. 2007, 129, (47), 14538-þ. (2) Weyts, K. F.; Goethals, E. J. Makromol. Chem., Rapid Commun. 1989, 10, (6), 299-302. (3) Hu, K.; Dickson, J. M. J. Membr. Sci. 2007, 301, (1-2), 19-28.

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poly(methacrylic acid) (PMAA)4) in the opposite, low-pH, limit. It is known that the delicate interplay between the two opposing forces (i.e., the electrostatic repulsion vs the hydrophobicity) can give rise to rich structural and phase behaviors of polyelectrolyte molecules in water.5-8 This balance is easily affected by added salt. For instance, in so-called strongly coupled systems where the spatiotemporal arrangements of counterions near a charged polymer are highly correlated (i.e., where the Bjerrum length rescaled by the valency of the ions far exceeds the Gouy-Chapman length),9 a known mechanism by which charged polymers can become destabilized in water is through the attractive interactions between likecharged polyelectrolyte segments caused by the presence of multivalent counterions; condensation of multivalent counterions onto highly charged polymers can lead to overcompensation of the charges of the polymers9,10 and can thus create a net attraction between the polyelectrolyte segments. A prominent example of this phenomenon is the condensation of DNA into compact structures induced by multivalent cations (e.g., histone proteins, (4) Burkhardt, M.; Martinez-Castro, N.; Tea, S.; Drechsler, M.; Babin, I.; Grishagin, I.; Schweins, R.; Pergushov, D. V.; Gradzielski, M.; Zezin, A. B.; Muller, A. H. E. Langmuir 2007, 23, 12864-12874. (5) Dobrynin, A. V.; Rubinstein, M.; Obukhov, S. P. Macromolecules 1996, 29, 2974-2979. (6) Mahdi, K. K.; de la Cruz, M. O. Macromolecules 2000, 33, 7649-7654. (7) Ross, R. S.; Pincus, P. Macromolecules 1992, 25, 2177-2183. (8) Volk, N.; Vollmer, D.; Schmidt, M.; Oppermann, W.; Huber, K. Conformation and phase diagrams of flexible polyelectrolytes. In Polyelectrolytes with Defined Molecular Architecture II, Springer-Verlag Berlin: Berlin, 2004; Vol. 166, pp 29-65. (9) Naji, A.; Jungblut, S.; Moreira, A. G.; Netz, R. R. Physica A 2005, 352, (1), 131-170. (10) Manning, G. S. J. Chem. Phys. 1969, 51, 924.

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cationic lipid assemblies, polycations, and high-valency metal ions).11,12 Under weakly coupling conditions which are commonly encountered, for instance, in water solutions of moderately charged polyelectrolytes with added monovalent salt (i.e., when the rescaled Bjerrum length is less than the Gouy-Chapman length),9 addition of salt also significantly impacts the solubility of polyelectrolytes. However, the trends in how the stability of dissolved polyelectrolytes is affected by the addition of monovalent salt (such as NaCl) can be qualitatively different depending on the chemical nature of the charge groups on the polyelectrolyte (hereafter, we will use the term “stability” to refer, in short, to the thermodynamic stability of the dissolved state of the polyelectrolyte). In situations involving strong (or quenched) polyelectrolytes (i.e., polyelectrolytes that contain permanently charged ionic groups along the chain such as poly(diallyldimethylammonium chloride)13 or sodium polystyrenesulfonate14), increasing salt concentration simply monotonically lowers the water solubility of the polyelectrolyte because of the increased screening of the electrostatic repulsion between the charged groups.5,15 It has been reported that in the brush conformation, strong polyelectrolyte chains of hydrophobic character undergo microphase separation into bundle-like structures under high salt conditions where electrostatic repulsion between monomers is sufficiently screened.16,17 In weak (or annealed) polyelectrolytes whose ionization is reversible and thus dictated by local pH, addition of monovalent salt normally causes an increase in the degree of ionization of the polyelectrolyte.18,19 However, this increased ionization can lead to both a decrease or increase in the polyelectrolyte stability. The former behavior is believed to occur when the ionized groups of polyelectrolytes form ion pairs with specific counterions;20 an example of a system which exhibits such behavior is a solution of PAA (or PMAA) in alcohol with added sodium ions.21-24 However, it is more common in aqueous solutions of weak polyelectrolytes that the increased degree of ionization of the polyelectrolytes by addition of salt makes the polymer molecules more soluble, contrary to the behavior of strong polyelectrolytes.25 Recently, it has been predicted by a single-chain mean-field (SCMF) theory that brushes of hydrophobic weak polyelectrolytes can become thermodynamically destabilized in water when no (or little) salt is present in the system, while the systems are stable at high concentrations of added salt.26 Our research group (11) Bloomfield, V. A. Biopolymers 1997, 44, 269-282. (12) Gelbart, W. M.; Bruinsma, R. F.; Pincus, P. A.; Parsegian, V. A. Phys. Today 2000, 53, 38-44. (13) Wandrey, C.; Hernandez-Barajas, J.; Hunkeler, D., Diallyldimethylammonium chloride and its polymers. In Radical Polymerisation Polyeletrolytes: Springer-Verlag Berlin: Berlin, 1999; Vol. 145, pp 123-182. (14) Wang, L. X.; Yu, H. Macromolecules 1988, 21, 3498-3501. (15) Israels, R.; Leermakers, F. A. M.; Fleer, G. J. Macromolecules 1994, 27, 3087-3093. (16) Gunther, J. U.; Ahrens, H.; Forster, S.; Helm, C. A. Phys. Rev. Lett. 2008, 101, 258303. (17) Sandberg, D. J.; Carrillo, J. M. Y.; Dobrynin, A. V. Langmuir 2007, 23, 12716-12728. (18) Lyatskaya, Y. V.; Leermakers, F. A. M.; Fleer, G. J.; Zhulina, E. B.; Birshtein, T. M. Macromolecules 1995, 28, 3562-3569. (19) Netz, R. R. J. Phys.-Condes. Matter 2003, 15, (1), S239-S244. (20) Khokhlov, A. R.; Kramarenko, E. Y. Macromolecules 1996, 29, 681-685. (21) Azzaroni, O.; Moya, S.; Farhan, T.; Brown, A. A.; Huck, W. T. S. Macromolecules 2005, 38, 10192-10199. (22) Klooster, N. T. M.; Vandertouw, F.; Mandel, M. Macromolecules 1984, 17, 2070-2078. (23) Makhaeva, E. E.; Tenhu, H.; Khokhlov, A. R. Macromolecules 2002, 35, 1870-1876. (24) Morawetz, H.; Wang, Y. C. Macromolecules 1987, 20, 194-195. (25) Wang, S. Q.; Granick, S.; Zhao, J. J. Chem. Phys. 2008, 129, 241102. (26) Gong, P.; Genzer, J.; Szleifer, I. Phys. Rev. Lett. 2007, 98, (1), 018302. (27) Witte, K. N.; Hur, J.; Sun, W.; Kim, S.; Won, Y. Y. Macromolecules 2008, 41, 8960-8963.

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has reported experimental results which support these theoretical predictions.27 In this previous study, we prepared a model polyelectrolyte brush system by dispersing a polyelectrolytecontaining diblock copolymer at the air-water interface. As shown in Figure S1 of the Supporting Information, this diblock copolymer comprises a positively chargeable poly(2-(dimethylamino)ethyl methacrylate) (PDMAEMA) block in which each repeat unit contains a tertiary amine group attached to the hydrophobic backbone (monomer pKa = 8.4),28 and a water-insoluble, nonglassy poly(n-butyl acrylate) (PnBA, Tg = -55 °C29) block which anchors the accompanying PDMAEMA segment to the air-water interface. Our previous work has demonstrated that unlike other hydrophobic polymers (e.g., polystyrene30,31) that normally dewet from the water surface, the PnBA chains completely wet the air-water interface and form a molecularly flat anchoring layer thereon. Therefore, the Langmuir monolayer of PDMAEMA-PnBA offers a convenient model system for studies of the structure and behavior of the PDMAEMA polyelectrolyte brushes; see ref 27 for details. Also as will be discussed further in this paper, the surface pressure behavior of the PDMAEMA-PnBA monolayer indicates that the PDMAEMA chains do not form so-called carpet-like structures (under any of the grafting density and salt conditions tested) that have been reported for other polyelectrolyte brush systems prepared at the air-water interface.32,33 All of these properties make the PDMAEMA brush system amenable to a simple theoretical analysis (as will be demonstrated in the present work). In our previous publication,27 we have also demonstrated that the PDMAEMA-PnBA monolayer at the air-water interface can be reliably transferred onto a graphite substrate for AFM imaging under aqueous conditions. Using these procedures, we performed an investigation of how the thermodynamic stability of a weak polyelectrolyte brush is influenced by variations of the grafting density and the ionic strength of the medium. Most notably, it was observed that the PDMAEMA brush system becomes laterally heterogeneous (i.e., the homogeneous state becomes unstable) at zero or low concentrations of NaCl, and this heterogeneity disappears with increasing NaCl concentration; these results are in agreement with the SCMF theoretical prediction. Extending this initial investigation, the present work aims to address in greater detail the following two fundamental questions. What are the exact mechanisms by which added NaCl influences the thermodynamic stability of the PDMAEMA brush at various grafting densities? Why do the heterogeneities occur at microscopic length scales (rather than resulting in macroscopically phase-separated structures)? To this end, we conducted more detailed AFM measurements on the PDMAEMA brushes over wider ranges of NaCl concentration and polymer grafting density; for this study we developed a new method of effectively producing PDMAEMA brushes with grafting density gradients. For comparison and analysis of experimental results, we developed a simple free-energy model. This model captures all the essential features of the experimental results. Theoretical analysis based on (28) van de Wetering, P.; Zuidam, N. J.; van Steenbergen, M. J.; van der Houwen, O.; Underberg, W. J. M.; Hennink, W. E. Macromolecules 1998, 31, 8063-8068. (29) Roff, W. J., Handbook of Common Polymers: Fibres, Films, Plastics, and Rubbers; CRC Press: Cleveland, OH, 1971. (30) Kumaki, J. Macromolecules 1988, 21, 749-755. (31) Zhu, J. Y.; Eisenberg, A.; Lennox, R. B. J. Am. Chem. Soc. 1991, 113, 5583-5588. (32) Kaewsaiha, P.; Matsumoto, K.; Matsuoka, H. Langmuir 2004, 20, 6754-6761. (33) Mouri, E.; Kaewsaiha, P.; Matsumoto, K.; Matsuoka, H.; Torikai, N. Langmuir 2004, 20, 10604-10611.

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this model suggests a very interesting possibility that at moderately low concentrations of added NaCl (e.g., 20 mM), the PDMAEMA brush chains become laterally aggregated primarily, not because of an insufficient degree of protonation and the hydrophobicity of the monomers, but instead because of the collapse-induced charge neutralization of the ionizable groups which reduces the osmotic stress caused by the accumulation of counterions in the vicinity of the charged polyelectrolyte segments. It should be noted that this mechanism, termed “osmotic instability” in this article, does not require the polymer to be hydrophobic in character, or any multivalent counterions to be present, in order to induce a tendency for the brush chains to collapse or aggregate.

2. Experimental Procedures Synthesis and Characterization of the Diblock Copolymer Used in This Study. The PDMAEMA-PnBA and PnBA polymers used in this study were synthesized by atom transfer radical polymerization (ATRP) as described in our earlier publication.27 The number-average degrees of polymerization of the PDMAEMA and PnBA blocks in the copolymer were measured (by 1H NMR) to be 80 and 94, respectively, and the polydispersity index of the copolymer (determined by GPC) was 1.31. The separately prepared PnBA homopolymer was found to have a number-average degree of polymerization of 90 and a polydispersity of 1.07.

Preparation of the Polyelectrolyte Brushes with Grafting Density Gradients. The PDMAEMA polyelectrolyte brush samples were prepared by LB deposition of a Langmuir monolayer of the PDMAEMA-PnBA diblock copolymer onto a graphite substrate using a Kibron Langmuir trough apparatus (Microtrough X-LB, 23.0 cm  5.9 cm  6.0 cm). Using a microsyringe, an appropriate amount of a 0.25 mg/mL PDMAEMA-PnBA solution in chloroform was spread on the ultraclean surface of subphase water solution containing a specified amount of NaCl (prepared using Milli-Q water of 18 MΩ 3 cm resistivity). After this, the spreading solvent was permitted to evaporate completely (typically for 3-4 h). For the preparation of the gradient brush samples (shown in Figures S3, S4, S6, 2-4, S7 and S9), the Langmuir film prepared as described above was first compressed using two Teflon barriers to an area of 1/σ = 1942 A˚2/chain prior to the LB deposition. This copolymer monolayer was LB-transferred onto a freshly cleaved graphite substrate placed on an AFM specimen disk by lowering the substrate vertically through the monolayer at a speed of 1.0 mm/min until a 5.0 mm length of the graphite substrate is immersed into the subphase; during this process, the monolayer was compressed at a speed of 5.0 mm/min from an initial monolayer area of 1942 A˚2/chain to a final area of 719 A˚2/chain at a constant temperature of 25 °C, in order to create a PDMAEMA brush with a varying grafting density from one end of the graphite surface to the other. Subsequently, the deposited sample was transferred to and stored (typically for ∼ 24 h prior to the AFM study) in the subphase solution collected from the PDMAEMA-PnBA Langmuir monolayer system used for the preparation of the AFM specimen. The pH and conductivity values of the subphase solutions prepared at various concentrations of added NaCl are presented in Table S1 of the Supporting Information; as shown in the table, the measured pH values were within about (0.2 pH units of the neutral value (pH 7), and were found to be invariant over time within measurement error (for details, see the explanation provided in the table caption). Fluid AFM Imaging of the Polyelectrolyte Brushes. The gradient PDMAEMA brushes, prepared as described above under various salt concentrations, were AFM-imaged under aqueous conditions using a NanoScope IV from Digital Instruments (with software version 5.12r3). The sample was transferred to the AFM stage as follows. The wet specimen was taken out of Langmuir 2010, 26(3), 2021–2034

the subphase medium, and a silicon O-ring (Veeco, FCO-10) was quickly placed on top of the specimen. Immediately after this, a small amount (∼30 μL) of the subphase solution (containing a designated concentration of NaCl) was added to the sample surface to ensure the maintenance of the same aqueous environment during the rest of the AFM operation. Next the liquid cell was assembled on top of the O-ring above the sample, and the cell volume was filled with an appropriate amount (∼5 mL) of the subphase solution using a syringe. AFM imaging was conducted after the temperature inside the liquid cell was fully equilibrated. All imaging measurements were performed under tapping-mode conditions using an appropriate cantilever (Veeco, NP-20, spring constant = 0.32 N/m, tip diameter = 10 nm). Isotherm Measurements. Surface pressure-area isotherms were collected using a KSV 5000 Langmuir trough (15.0 cm  51.0 cm  1.0 cm) within a plexiglass chamber. The hydrophobic trough and hydrophilic symmetric moving barriers are cleaned by repeated rinsing with ethanol followed by deionized water. The surface pressure is measured by the Wilhelmy method using a platinum plate, located equidistant from each barrier, which is cleaned by rinsing with ethanol and water and then flamed. Subphase solutions were prepared by dissolving 0, 4.9090, 6.5453, 8.1816 or 40.9080 g of NaCl into 1.4 L of deionized Millipore-purified water (18 MΩ 3 cm resistivity). The surface is cleaned via aspiration prior to spreading. The cleanliness of the surface and probe are checked by compressing the water surface at 3 mm/min and monitoring the surface pressure. If the total change in the pressure is greater than 0.2 mN/m, the surface and probe are recleaned, and the measurement is retaken until the surface pressure change is below this threshold. A 1 mg/mL solution of PDMAEMA-PnBA or PnBA polymer in chloroform is prepared fresh for each trial and spread dropwise on the water surface using a Hamilton microsyringe. The surface pressure is monitored after deposition until it reaches a plateau value, typically 3-4 h. The isotherm is then recorded while compressing the barriers at 3 mm/min.

3. Results and Discussion 3.1. Preparation of PDMAEMA Brushes with Grafting Density Gradients (for Efficient Investigation of the Effect of Added NaCl on the Lateral Structure of the Brush at Various Grafting Densities). For the present investigation, we use, as model systems, PDMAEMA brushes prepared by LB deposition of an amphiphilic PDMAEMA-PnBA diblock copolymer monolayer initially constructed at the air-water interface onto a hydrophobic graphite substrate; the validity of this model system has been discussed in detail in our previous publication.27 Under this approach, systematic investigations of the lateral nanostructures of the PDMAEMA brushes at various grafting densities and various concentrations of added NaCl would require preparation of a large number of samples, each at a time by LB deposition under a specific set of parameter conditions, and therefore, would be a very time-consuming task. For more efficient investigation of the effect of added NaCl on the structure of the PDMAEMA brush at various grafting densities, we developed, in the present study, a method of producing PDMAEMA brushes with grafting density gradients. This method uses, in essence, the same LB deposition technique to transfer a PDMAEMA-PnBA monolayer from the air-water interface to the surface of graphite immersed in water as reported previously in ref 27. However, in the present technique, in order to create a gradient in polymer grafting density within the LB-deposited polymer layer, we continuously vary the surface area of the precursor monolayer at the air-water interface (through lateral compression of the monolayer) during the LB transfer process (Figure 1); therefore this method will be termed hereafter as the “LB-deposition-under-compression” (or “LB\C”) technique for convenience. To our knowledge, this DOI: 10.1021/la902549b

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Figure 1. PDMAEMA brushes with grafting density gradients can be prepared by Langmuir-Blodgett (LB) deposition of a Langmuir monolayer of PDMAEMA-PnBA onto graphite under continuous compression of the precursor monolayer at the air-water interface; this procedure will be referred to as the “LB-deposition-under-compression” (or “LB\C”) method.

method has not previously been presented in the literature. Therefore, we will devote this subsection to a demonstration of the effectiveness of the LB\C procedure for application to the present study. In the earlier publication, we have presented preliminary results demonstrating that the structure of the PDMAEMA polyelectrolyte brush layer is laterally heterogeneous under neutral pH and low-salt conditions, and the length scale of the heterogeneities is dependent on the grafting density of the brush chains, as also illustrated in Figure S2 of the Supporting Information. We use these properties of the weak polyelectrolyte PDMAEMA brush to validate the existence and stability of the grafting density gradients in the PDMAEMA brushes created using the LB\C technique. A gradient PDMAEMA brush sample was prepared (at neutral pH with no added salt) by LB deposition of a Langmuir monolayer of PDMAEMA-PnBA (DPn,PDMAEMA = 80, DPn,PnBA = 94, PDI = 1.31) onto graphite while the area of the monolayer at the air-water interface was continuously decreased by lateral compression from an initial monolayer area of 1/σ = 1943 A˚2 per chain to a final area of 1/σ = 842 A˚2 per chain. Figure S3 displays representative fluid AFM height images of the resultant PDMAEMA brush taken at four different locations on the sample surface, which confirms the gradual change in the size of the heterogeneities and thus the gradient of PDMAEMA grafting density within the specimen. For obtaining the images shown in Figure S3, we conducted the AFM experiments within a day after the preparation of the specimen. An important question that arises regarding the applicability of the LB\C technique in a wide range of experimental circumstances is whether the grafting density gradient produced by the LB\C procedure would remain unaltered (i.e., “unrelaxed”) over a sufficiently long period of time so that property measurements can be reliably performed at reasonably long times after the preparation of the specimen. To illustrate this aspect, a gradient PDMAEMA brush specimen (prepared by the LB\C method) was stored in the subphase solution collected from the original PDMAEMA-PnBA Langmuir monolayer system for a much longer period of time (i.e., 2 weeks), and afterward, liquid AFM measurements were performed to determine whether any change had occurred in the gradient structure of the PDMAEMA brush 2024 DOI: 10.1021/la902549b

during the time period. As shown in Figure S4, the resulting images exhibit no considerable difference in the characteristics of the lateral heterogeneities from those presented in Figures S2 and S3. For further quantitative comparison of the results, we calculated the average width (i.e., distance between two adjacent peaks) and height (i.e., peak-to-base distance) of the lateral heterogeneities from the AFM height profiles of the two-week-old gradient PDMAEMA brush (including the height profiles shown in Figure S4) for each of the five grafting density conditions examined in Figure S4. The results of these analyses are compared in Figure S5 with those obtained from the analyses of PDMAEMA brush specimens prepared by using the normal LB procedure at the various grafting densities (Figure S2). As shown in the figure, the size characteristics of the heterogeneous structures in the gradient brush sample stored for 2 weeks after it had been prepared are identical with the size characteristics determined individually for the various grafting density points, confirming the appropriateness of using the LB\C procedure in the present study. Therefore, all experimental results presented in section 3.2 and Figures S6-S9 of the Supporting Information have been obtained using PDMAEMA brush samples prepared by the LB\C technique. 3.2. Effects of Added NaCl and Chain Grafting Density on the Lateral Structures of PDMAEMA Brushes: Experimental Results. Using the PDMAEMA brush samples prepared by the LB\C method, we performed extensive AFM imaging measurements of the lateral microstructure of the PDMAEMA brush layer at neutral pH under various salt concentrations (CNaCl = 0, 20, 60, 80, 100, 300 and 500 mM) and inverse grafting density (i.e., area per chain) values (1/σ = 1942, 1650, 1357, 1064 and 770 A˚2/chain). Representative results are presented in Figures S6, 2, 3 and 4, and also in Figures S7-S9 of the Supporting Information. As shown in Figures S6 and 2, when no salt (Figure S6) or a relatively small amount (20 mM) of NaCl (Figure 2) is added, the PDMAEMA brush is thermodynamically unstable, resulting in the formation of lateral heterogeneities in the structure. These heterogeneities are microscopic with estimated length scales (i.e., average peak-to-peak distances in the AFM height profiles) ranging from 35 nm (at 1/σ = 1942 A˚2/ chain) to 240 nm (at 1/σ = 770 A˚2/chain); possible explanations Langmuir 2010, 26(3), 2021–2034

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Figure 2. Representative fluid AFM tapping-mode height images (2 μm  2 μm) of PDMAEMA brushes with added 20 mM NaCl under pH

7 at five different inverse grafting densities of 1/σ = (A) 1942 ( 24, (B) 1650 ( 24, (C) 1357 ( 24, (D) 1064 ( 24 and (E) 770 ( 24 A˚2/chain. The PDMAEMA brush sample was prepared by using the LB\C method; a Langmuir monolayer of PDMAEMA-PnBA was LB-transferred at a rate of 1.0 mm/min onto a graphite surface in water (20 mM NaCl, pH 7) over a 5 min period, and during this process, the PDMAEMA-PnBA monolayer at the air-water interface was compressed at a rate of 5.0 mm/min from an initial monolayer area of 1942 A˚2/chain to a final area of 719 A˚2/chain. The AFM measurements were carried out about 24 h after the preparation of the sample. AFM images were taken from 5 different positions in the gradient brush specimen: i.e., at (A) 0.0 ( 0.1, (B) 1.2 ( 0.1, (C) 2.4 ( 0.1, (D) 3.6 ( 0.1 and (E) 4.8 ( 0.1 mm distance from the leading front of the brush layer, which correspond to the above respective area per chain (1/σ) values.

for the microscopic nature of the heterogeneities and also for the dependence of the size of the heterogeneities on the grafting density are discussed in section 3.4. With further addition of the salt, the homogeneous state of the brush becomes more stable. For instance, under 80 mM NaCl Langmuir 2010, 26(3), 2021–2034

conditions (Figure 3), both the spatial and size distributions of the lateral heterogeneities become significantly irregular. Typically, a relatively small number of isolated regions of higher brush densities coexist in the background of a homogeneous brush state, and the size of the heterogeneous domains increases with the DOI: 10.1021/la902549b

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Figure 3. Representative fluid AFM tapping-mode height images (2 μm  2 μm) of the PDMAEMA brushes with added 80 mM NaCl under pH 7 at five different inverse grafting densities of 1/σ = (A) 1942 ( 24, (B) 1650 ( 24, (C) 1357 ( 24, (D) 1064 ( 24 and (E) 770 ( 24 A˚2/chain. The PDMAEMA brush sample was prepared by using the LB\C method.

grafting density. These observations appear to indicate the possible existence of metastable states; i.e., one plausible interpretation is that at low overall grafting densities, the homogeneous (i.e., uniformly dispersed) state is more stable than the heterogeneous (i.e., locally aggregated) state, while the homogeneous state becomes metastable relative to the heterogeneous state at high grafting densities (as will be further discussed in section 3.3). At 60 mM (Figure S7) and 100 mM (Figure S8) NaCl concentrations, the behaviors of the PDMAEMA brush are qualitatively similar to the behavior at the 80 mM NaCl 2026 DOI: 10.1021/la902549b

condition. However, it is obvious from the data shown in Figures S7, 3 and S8 that the heterogeneous state becomes less and less stable as the salt concentration is increased. As shown in Figures S9 and 4, at sufficiently high salt concentrations (e.g., at 300 mM or higher NaCl concentrations), the homogeneous state of the PDMAEMA brush becomes stable, and the brush structure becomes completely uniform in the lateral directions at all grafting densities. It should be noted that the observed trend of increased polyelectrolyte brush stability (i.e., solubility) with increasing salt concentration may seem contrary Langmuir 2010, 26(3), 2021–2034

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Figure 4. Representative fluid AFM tapping-mode height images (2 μm  2 μm) of the PDMAEMA brushes with added 500 mM NaCl under pH 7 at five different inverse grafting densities of 1/σ = (A) 1942 ( 24, (B) 1650 ( 24, (C) 1357 ( 24, (D) 1064 ( 24 and (E) 770 ( 24 A˚2/ chain. The PDMAEMA brush sample was prepared by using the LB\C method.

to what one would normally expect for hydrophobic polyelectrolytes (such as the PDMAEMA polymer used in this study); one would ordinarily expect that with an increase in salt concentration, the repulsive interactions between charged segments become more screened, and the hydrophobic polyelectrolyte molecules should thus be rendered less soluble in water at high salt concentrations.34 In the next subsection, using a Flory-type theoretical model, we will show that while the latter type of behavior is expectable for strong (or quenched) polyelectrolytes, (34) Dautzenberg, H. Macromolecules 1997, 30, 7810-7815.

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systems of weak (or annealed) polyelectrolytes (in which the ionization of the monomers is reversible) can indeed become more stable (counterintuitively) when the concentration of added salt is high; as will be discussed in the next section, in the low-salt limit, collapse (or aggregation) of hydrophobic weak polyelectrolytes is accommodated by a shift of the polyelectrolyte ionization equilibrium toward a less charged state (which reduces the repulsive forces between monomers) upon the collapse of the chains, but at high concentrations of added salt, the magnitude of this charge re-equilibration becomes too small to have any effect toward the collapse of the chains. DOI: 10.1021/la902549b

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Before we theoretically discuss this issue (section 3.3), it is worthy of remark that the PDMAEMA brush system is an excellent experimental model system for comparison with simple theories. There are two aspects to this argument. The first is related to the fact that many amphiphilic block copolymers have been reported to form surface micelles when dispersed at the air-water interface.31,35,36 This property makes the Langmuir monolayers of such polymers inappropriate model systems for studies of polymer/polyelectrolyte brushes. In contrast, as discussed in our previous publication,27 the PnBA block of a PDAMEAM-PnBA diblock copolymer forms a molecularly flat layer on the water surface that provides a smooth anchoring surface for the submerged PDMAEMA chains. Another complicating situation observed in many Langmuir-monolayer-type brush systems is when the brush chains adsorb strongly to the underlying interface, leading to formation of a dense layer of collapsed brush chains (termed in the literature as the “carpetlike” structure).37,38 A rigorous examination of whether the Langmuir monolayers of PDMAEMA-PnBA diblock copolymers exhibit such behavior requires the use of X-ray or neutron reflectivity measurement techniques. We plan to conduct these experiments in the future. In the present study, we used measurements of surface pressure vs area isotherms in order to (indirectly) probe the formation of a dense adsorbed layer in the PDMAEMA brush system. It has been reported that a carpet-forming system typically exhibits, in the pressure-area isotherm, a broad tail which starts at large monolayer areas, prior to the upslope of the isotherm curve, during lateral compression of the monolayer.33,37-39 As shown in Figure 5, no evidence of such behavior has been observed in any of the PDMAEMA-PnBA isotherm data measured under various salt concentration conditions. Rather, the surface pressure behavior at monolayer areas greater than the critical area for complete coverage of the air-water interface by PnBA27 (denoted in the figure as 1/σo) appears to be dictated by the presence of the PnBA domains, supporting the claim that the PDMAEMA chains do not have a strong tendency to adsorb to the grafting surface. To explain this point in detail, we first note the following two discrepancies between the PDMAEMA-PnBA isotherms and the PnBA isotherm displayed in Figure 5. First, the PnBA isotherm is shifted to the left along the x-axis; e.g., for the PnBA homopolymer, the 1/σo value is estimated to be 1881 ( 27 A˚2/ chain, while it is 1988 ( 18 A˚2/chain for the diblock copolymer. This is due to the difference in molecular weight of the PnBA homopolymer (degree of polymerization equal to 90) and the PnBA block of the copolymer (degree of polymerization equal to 94). The validity of this explanation can be seen by calculating the area per monomer at the critical area (1/σo). The area per PnBA monomer calculated from the critical area value for the PDMAEMA-PnBA diblock is 21.1 ( 0.2 A˚2, while that calculated for the PnBA homopolymer is 20.9 ( 0.3 A˚2. Second, the surface pressure at the critical area, 1/σo, is higher for the PDMAEMA-PnBA isotherms. We believe that this surface pressure difference is mainly due to the presence of PDMAEMA solute in the aqueous subphase. The PDMAEMA segments (35) Chung, B.; Choi, M.; Ree, M.; Jung, J. C.; Zin, W. C.; Chang, T. H. Macromolecules 2006, 39, 684-689. (36) Cox, J. K.; Yu, K.; Constantine, B.; Eisenberg, A.; Lennox, R. B. Langmuir 1999, 15, 7714-7718. (37) Matsuoka, H.; Furuya, Y.; Kaewsaiha, P.; Mouri, E.; Matsumoto, K. Langmuir 2005, 21, 6842-6845. (38) Matsuoka, H.; Furuya, Y.; Kaewsaiha, P.; Mouri, E.; Matsumoto, K. Macromolecules 2007, 40, 766-769. (39) Mouri, E.; Furuya, Y.; Matsumoto, K.; Matsuoka, H. Langmuir 2004, 20, 8062-8067.

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Figure 5. Surface pressure (π) vs area (1/σ) isotherms for PDMAEMA-PnBA (DPn,PDMAEMA = 80, DPn,PnBA = 94, PDI = 1.31) and PnBA (DPn,PnBA = 90, PDI = 1.07) at various added NaCl concentrations ranging from 0 to 500 mM. The point marked with 1/σo indicates the critical monolayer area at which the air-water interface becomes completely covered by a film of PnBA.

dissolved in the subphase will lower the PnBA-water interfacial tension (γPnBA-water), which will lead to an increase in the value of π at the critical area (πo = γair-water - γPnBA-air - γPnBA-water).27 The electrostatic/osmotic repulsion between the PDMAEMA brush chains will also contribute to the increase in the surface pressure. In fact, this effect seems to have resulted in a higher pressure for intermediate salt concentrations (60 and 80 mM NaCl) where electrostatic type interactions dominate and a slightly lower pressure for high salt concentrations (500 mM NaCl) where these effects are highly screened. However, the magnitude of this effect appears to be small relative to the overall difference in the πo values between the homopolymer and the diblock copolymer. Also of note, addition of NaCl has a significant influence on the detailed shape of the diblock isotherm at high grafting densities (i.e., at 1/σ < 1/σo), which we plan to investigate in future work. Taking the above differences into account, we constructed a fictitious PnBA isotherm from the measured data, first by addition of a constant to the area/chain value to make the critical area equal to that observed for the PDMAEMA-PnBA (this procedure is to correct for the molecular weight difference, and is justifiable because the shape and features of the PnBA isotherm are molecular weight independent),40 and then by rescaling the surface pressure of the homopolymer isotherm by a constant multiplicative factor to make it comparable to that from the diblock isotherm at the critical area (this is to account for the difference in γPnBA-water). As presented in Figure 5, the resultant isotherm (dotted line) reasonably reproduces the low-grafting-density portion of the diblock isotherm, which suggests that the PDMAEMA chains do not form an adsorbed layer structure beneath the PnBA domain. On the basis of these observations, for the theoretical analysis of the PDMAEMA brush system (next section) we will assume that the grafting surface is planar and nonadsorbing. 3.3. Discussion of the Origin of the Lateral Heterogeneities in PDMAEMA Brushes and the Effect of Added NaCl on the Lateral Structure of the PDMAEMA Brush: FloryType Theoretical Analysis. In order to explain why, under low (40) Witte, K. N.; Kewalramani, S.; Kuzmenko, I.; Sun, W.; Fukuto, M.; Won, Y. Y. Macromolecules 2009, submitted for publication.

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Figure 6. (A) Schematic illustration of positively charged polyelectrolyte brushes. The polyelectrolyte chains are end-grafted to a surface at a density of σ chains per unit area. Each chain has N monomers. In each chain, only a fraction (f) of the monomers are charged (0 e f e 1). Within the brush layer (i.e., 0 < z < H), because of the positive charges on the polyelectrolyte molecules, the concentrations of the small negative ions (i.e., the counterions generated by the ionization of the polyelectrolyte segments plus the negative ions produced from water molecules or added salt) are higher than their respective bulk values; that is, ci- > Ci-. Similarly, the opposite happens for the small positive ions (i.e., ci+ < Ci+). Shown in P(B) is a graphical explanation of how an increase of H at constant σ and f affects the small ion concentrations within the brush layer (c( = ici(). The height increase reduces the concentration of the charged monomers (fco = fNσ/H), resulting in a decrease in the difference c- - c+, due to the charge neutrality constraint. Note these changes in the ci+ and cj- values occur such that the Donnan condition Ci+ = Cj- = (ci+cj-)1/2 is satisfied (where i+ and j- are conjugate ions with a valency of (1, respectively, and the salt is assumed to be symmetric). Now, when f is allowed to vary (i.e., in the “weak polyelectrolyte” cases), the situation becomes far more complicated; in this case, f is dependent upon cH3O+ and co according to the relations given in eqs 2 and 3. See the main text for a discussion of how this variability of the degree of monomer ionization impacts the overall behavior of the system in response to an increase of H.

salt conditions, the homogeneous state of the weak polyelectrolyte PDMAEMA brush system becomes thermodynamically unstable, it will be useful to discuss first why, in the case of strong (or quenched) polyelectrolytes, electrostatic charges help stabilize (i.e., solubilize) the polymers in water. For instance, let us consider a system of polyelectrolyte chains (of size N) end-grafted to a surface (at a grafting density of σ chains per unit area) (Figure 6). For now, we will assume that in this polyelectrolyte brush system, each chain contains a certain number of ionic groups (fN where 0 e f e 1) which are permanently positively charged. The Coulomb repulsion between these charged groups will certainly contribute to the solubilization of the polymer molecules in water. However, due to water’s high dielectric permittivity and also due to the presence of small ions, often added or, generated by the ionization of the water molecules, such effect is typically only secondary. Instead, it is the diffuse layer of small ions (especially those possessing the opposite charge) surrounding the charged segments of the polyelectrolyte molecules that causes the repulsion between the charged segments; an overlap of these small ion clouds causes a buildup of osmotic pressure, which is thermodynamically unfavorable. In the polyelectrolyte brush system described above, these oppositely charged small ions localized within the brush layer will always drive the polymer chains to stretch out from the grafting surface (Figure 6). Now, to consider this problem theoretically, we will make the following two additional assumptions. The first is that within the Langmuir 2010, 26(3), 2021–2034

brush layer (i.e., in the region defined as 0 < z < H in Figure 6), the positive and negative charges are exactly balanced, that is, fco + c+ = c- where co denotes the concentration of the polymer segments (monomers) in the brush layer, and c+ and c- are, respectively, the total concentrations of P the positive and negative mobile ions within the brush layer (c( = ici() (all concentrations in units of number per volume); the reasonableness of this local electroneutrality assumption has been confirmed by rigorous self-consistent field theoretical calculations.41 Second, we assume, for simplicity, that all concentrations are uniform within the brush layer. Within a mean-field approximation, it is not difficult to show that the individual small ion concentrations follow the Boltzmann statistics;41 i.e., for monovalent ions ci (

  eψ ¼ Ci ( exp kB T

where Ci(, e and ψ denote the bulk concentration of a specific type of ion, the elementary charge and the electrostatic potential within the brush layer, respectively. From this, it follows, naturally, that ci(/Ci( = cj(/Cj( , and ci(cj- = Ci(Cj-; these relations are known as the Donnan equilibrium equations.42 Under these (41) Witte, K. N.; Won, Y. Y. Macromolecules 2006, 39, 7757-7768. (42) Donnan, F. G. Chem. Rev. 1924, 1, 73-90.

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Figure 7. Plots of the osmotic free energy (Fosm) vs scaled brush height (H/b, where b denotes the monomer size), estimated using eq 1, at four different added NaCl concentrations of CNa+ (= CCl-) = (A) 0, (B) 20, (C) 80 and (D) 500 mM, representing the three different regimes of weak polyelectrolyte brush behavior discussed in the text. The parameter values used are N = 100 (chain length), b = 7 A˚ (monomer size), v = -50 A˚3 (excluded volume), w = 104 A˚6 (three-body interaction coefficient), CH3O+ = 10-7 M (bulk hydronium ion concentration), KA = 108.4 M-1 (protonation equilibrium coefficient), and σ = 3/Nb2 (=6.1  10-4 A˚-2) (grafting density).

simplifications, it can be visualized clearly why the localization of small ions causes the swelling of the brush layer when the fraction of charged monomers (f) is fixed. Let us consider a situation in which the brush height (H) has been increased at a fixed grafting density (σ) (Figure 6). This height increase will cause a dilution of the concentration of the charged polyelectrolyte segments in the brush volume (fco = fNσ/H), and due to the charge neutrality condition, this variation in fco will result in a decrease in the difference, c- - c+. As depicted schematically in Figure 6, under the constraint of the Donnan rule, the said variation in c- - c+ can only be effected by a simultaneous decrease and increase respectively in the c- and c+ values, and therefore reduces both the ci- - Ci- and Ci+ - ci+ quantities, which will allow the release of the osmotic stress in the brush layer. The change in the small-ion osmotic free energy associated with a variation of the brush height (H) can be quantitatively modeled by using the established formalism43

Fosm

2 ( )   H 4X ci þ þ þ þ þ ðCi -ci Þ ¼ ci ln σ Ci þ i 8 93 ! = X< cj þ ðCj - -cj - Þ 5 þ cj - ln : ; Cj j

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K

aMH þ aH2 O cMH þ cH2 O fcH2 O ≈ ¼ aM aH3 O þ cM cH3 O þ ð1 -f ÞcH3 O þ

ð1Þ

ð2Þ

where K denotes the equilibrium constant for the reaction of a polyelectrolyte monomer with hydronium ion (i.e., M + H3O+ h MH+ + H2O) and is assumed (to an approximation) to be pH and f-independent, and ai denotes the activity of species i. For a brush system immersed in water with added NaCl, for instance, f can be separately estimated from the combined electroneutrality and Donnan equilibrium conditions (as has been first shown by Zhulina et al.44): fco 1 -f 1 f ¼ KA CH3 O þ f KA CH3 O þ 1 -f CH3 O þ þ CNa þ

where Fosm denotes the osmotic free energy per polyelectrolyte chain in units of kBT. Though not shown in this article, it is now trivial to show that at fixed σ values, Fosm is a monotonically decreasing function of H; i.e., the mobile ions always force the polyelectrolyte chains to adopt stretched conformations. (43) Seifert, U. Adv. Phys. 1997, 46, (1), 13-137.

Let us now consider a brush system consisting of weak polyelectrolyte (e.g., PDMAEMA) chains in which the fraction of positively charged monomers (f) is dependent upon the pH of the local environment within the brush layer; this dependence follows the equation

ð3Þ

where KA  K/cH2O. By incorporating these conditions into the osmotic free energy expression of eq 1, one can evaluate how Fosm varies as a function of H at fixed σ for weak polyelectrolyte brushes. As demonstrated in Figures 7 and S10, the equilibrium nature of the monomer ionization process causes the behavior to become qualitatively different from the previously discussed case involving permanently charged monomers. In the example demonstrated in Figures 7 and S10, it is found that depending on the amount of salt (e.g., NaCl) added, the weak polyelectrolyte brush system exhibits (44) Zhulina, E. B.; Birshtein, T. M.; Borisov, O. V. Macromolecules 1995, 28, 1491-1499.

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three different patterns of behavior in response to a variation of H. As shown in Figure 7(A), in the zero-salt limit, the brush layer is osmotically unstable; the osmotic free energy Fosm decreases, as H is decreased, and thus favors a collapsed state. This counterintuitive effect can be explained in the following way. As discussed previously (and also with Figure 6), an increase in the height of a permanently charged polymer brush results in a decrease in the CH3O+ - cH3O+ value and therefore a decrease in the local pH within the brush layer. In weak polyelectrolyte brushes, as can be deduced from eq 3, the degree of protonation (f) tends to be very small (i.e., f , 1) in the zero-to-low salt concentration regime (CNa+ = CCl- f 0), and as a consequence, the protonation level varies sensitively in response to a variation of the local pH (i.e., f ∼ cH3O+ from eq 2). Therefore, the lowered pH due to the height increase will cause a tendency for an increase in f, which in fact overcompensates the effect that the decrease in co would have under fixed f when the brush height is increased, thereby giving rise to a net increase in the osmotic free energy (Fosm); therefore, as H is increased, the system becomes more osmotically stressed. This phenomenon is termed in this article as “osmotic instability”. It should be noted that this phenomenon is (due to the mechanism of collapse-induced deionization of the chains) only possible in weak polyelectrolyte-based systems, and therefore facilitates destabilization of (hydrophobic) weak polyelectrolytes in water. In highly salted water, the brush recovers the behavior of permanently charged systems (Figures 7(D) and S10(C)). In this regime (CNa+ = CCl+ . fco), the protonation degree of the polyelectrolyte molecules (f) is only slightly less than its bulk value (see eq 3), and thus becomes relatively less sensitive to the local pH change than in the low-salt case (as can be seen from eq 2). At intermediate salt concentrations, Fosm increases with H initially, but the trend becomes reversed upon further increase of H (Figures 7B, 7C, S10A and S10B). We will now assess theoretically whether under practical situations, this osmotic effect can play a significant role in determining the structure and phase behavior of a weak polyelectrolyte brush system. In thermodynamic terms, the overall behavior of the system is governed by the total system free energy (F). There are other contributions to the free energy that need to be considered: the free energies associated with (i) polyelectrolyte chain conformations, (ii) electrostatic and (iii) nonelectrostatic monomer interactions, (iv) the chemical transformation of monomers from neutral to charged states, and (v) the multiplicity of the distribution of charged monomers along the chain; these terms will be denoted hereafter by Fcon, Fele, Fint, Fche and Fmul, respectively. Therefore, it can be written F = Fosm + Fcon + Fele + Fint + Fche + Fmul. In this work, we will assume, to a reasonable approximation, that the electrostatic-energy contribution to F is negligible (i.e., Fele ≈ 0). This is consistent with the electroneutrality and uniform charge distribution assumptions. Due to Doi and Edwards,45 the chain conformational free energy (in units of kBT per chain) can be written approximately as Fcon ¼

3 H2 π2 Nb2 þ 2 2 Nb 6 H2

ð4Þ

conditions (Fint = 0), should adopt Gaussian conformations (i.e., H ≈ N1/2b) in the uncharged limit (Fosm = 0). The monomer-monomer interaction free energy (in units of kBT per chain) can be expressed as a virial expansion in powers of the monomer concentration (co = Nσ/H),46 and by taking only the leading-order terms, one can write Fint ¼ v

N2 N3 þw H=σ ðH=σÞ2

ð5Þ

where v and w are the two-body interaction (i.e., the excluded volume) coefficient and the three-body interaction coefficient, respectively. Taking into account the fact that typical aminebased polycations are hydrophobic in the uncharged state, we will assume a value of v = -50 A˚3 for the excluded volume parameter (which is equivalent to setting T/Θ = 0.87); other values which correspond to the θ and good-solvent conditions (i.e., v = 0 and 50 A˚3, respectively) will also be considered later in this section. The three-body parameter (w) is set to be equal to 10 000 A˚6 so that in the noncharged state (Fosm = 0), the system free energy F (= Fcon + Fint) has a minimum at a finite H value which reasonably represents a collapsed state of the brush (i.e., b j H , N1/2b). We note that all these parameter values are, though somewhat arbitrary, quite representative of the experimental system used in this study. The free energy change (per chain) for the monomer protonation reaction can be calculated, under constant temperature and pressure, from the fundamental thermodynamic relation: Fche = P fN iωiμi, where ωi and μi are, respectively, the reaction stoichiometric coefficient and P chemical potential (in units of kBT) of species i, and therefore, i(ωiμi) = μMH+ + μH2O - μM - μH3O+. By substituting the definition of the activity μi = μoi + ln ai (where μoi is the standard-state chemical potential of species i) into the above equation for the free energy of reaction, one obtains X Fche ¼ fN½ ðωi μoi Þ þ ln K ð6Þ i

Consider that for polyelectrolyte monomers in its unpolymerized form in the bulk water solution (i.e., F ≈ Fche), at ionization equilibrium (Fche = 0) the equilibrium constant is specified as X ðωi μoi Þ K  ¼ exp½ i

Because we assume, for simplicity, that the monomer protonation equilibrium constant (K) is independent of the local pH condition and the protonation degree (f), the value of this constant for the monomers in the brush should equal the bulk monomer value (K*), and due to this “pre-minimization” approximation, the chemical reaction free energy, in our model, makes no contribution to the minimization of the total free energy of the system (i.e., Fche = 0). For the entropic contribution to the free energy due to the degeneracy of the charged states of the polyelectrolyte molecule, we will use the Hildebrand-type expression proposed by Raphael and Joanny:47 Fmul ¼ N½f ln f þ ð1 -f Þ lnð1 -f Þ

ð7Þ

where b is the monomer size and will be set equal to 7 A˚ throughout this work. This expression guarantees that the brush chains, when grafted sparsely to a surface under θ-solvent

Now, using eqs 1, 4, 5 and 7, we can evaluate the total system free energy per chain F (=Fosm + Fcon + Fint + Fmul) as a function of σ and H, for analysis of the thermodynamic stability

(45) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford University Press: New York, 1995.

(46) Rubinstein, M.; Colby, R. H., Polymer Physics; Oxford University Press: New York, 2003. (47) Raphael, E.; Joanny, J. F. Europhys. Lett. 1990, 13, 623-628.

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Figure 8. Plots of the polyelectrolyte chemical potential (μ) as a function of σNb2, calculated according to eq 8, at four different added NaCl concentrations of CNa+ (= CCl-) = (A) 0, (B) 20, (C) 80 and (D) 500 mM, representing the three different regimes of weak polyelectrolyte brush behavior discussed in the text. For each salt condition, the effects of the three different (i.e., poor, θ and good) solvent-quality conditions were evaluated, respectively, at v = -50 (solid curve), 0 (dotted-broken curve) and +50 (broken curve) A˚3. The other parameter values used are N = 100 (chain length), b = 7 A˚ (monomer size), w = 104 A˚6 (three-body interaction coefficient), CH3O+ = 10-7 M (bulk hydronium ion concentration), and KA = 108.4 M-1 (protonation equilibrium coefficient).

of the system with respect to changes in σ. Specifically, for each fixed value of σ, we calculate the minimum value of F (Fmin) with respect to H, and repeat this calculation for different values of σ to determine Fmin as a function of σ. From the Fmin values, the chemical potential of the polyelectrolyte molecule (μ) is estimated as a function of σ by the equation: μ ¼

DFmin σ þ Fmin Dσ

ð8Þ

Shown in solid lines in Figures 8(A)-8(D) and S11(A)-S11(C) are μ vs σ plots estimated under the same sets of parameter conditions as in Figures 7(A)-7(D) and S10(A)-S10(C), respectively. As shown in Figures 8(A) (0 mM NaCl) and 8(B) (20 mM NaCl), the homogeneous state of the brush system is thermodynamically unstable (i.e., ∂μ/∂σ < 0) in the zero-to-low-salt regime (Figures 7(A) and 7(B)); if the end-grafted chains are laterally mobile, the chains may spontaneously condense to form denser brush domains, thereby also yielding domains devoid of the polymer chains; if the grafting points are fixed in space, this densification will likely involve local (microscopic) rearrangements of chain conformations in both lateral and/or vertical directions. At high salt concentrations where the extent of the shift of the protonation equilibrium is not sufficient to cause the brush instability (Figures 7(D) and S10(C)), the homogeneous state of the brush system is thermodynamically stable (∂μ/∂σ > 0) (Figures 8(D) (500 mM NaCl) and S11(C) (300 mM NaCl)), and the chains will form a laterally uniform brush structure. At intermediate salt concentrations (Figures 8(C) (80 mM NaCl), S11(A) (60 mM NaCl) and S11(B) (100 mM NaCl)), the system is stable at low σ, while it becomes unstable at high σ, which is consistent with the behavior seen in the corresponding 2032 DOI: 10.1021/la902549b

Fosm vs H profile (Figures 7(C), S10(A) and S10(B), respectively). Of note, in this regime, especially at grafting densities lower than the σ value corresponding to the peak in the μ vs σ curve, higher -σ domains may coexist as a metastable (or even as a more stable) structure within a homogeneous matrix of the low-σ phase. At higher σ, the whole system will become unstable. All these predictions are fully in agreement with the experimental results discussed in section 3.2, which supports the correctness of our theoretical explanation for the observed salt-concentration dependence of the lateral heterogeneities of the PDMAEMA brushes. We believe that it is the combined effects of the osmotic instability (due to the reversible nature of the protonation reaction) and the hydrophobicity of the polymer that cause the instability of the weak polyelectrolyte brush in the low NaCl concentration region. To illustrate this point, we present in Table S2 of the Supporting Information a summary of the values of the degree of monomer protonation (f) and the local pH of the solution within the brush layer estimated using eqs 2 and 3 for the various concentrations of added NaCl. It should be pointed out that at pH 7 with no added NaCl, the monomers of PDMAEMA in their unpolymerized form (pKa = 8.428) are expected to be mostly protonated in the bulk water solution; i.e., using eq 2, the value of f is estimated to be 0.96. To the contrary, as shown in Table S2, the PDMAEMA monomers in the brush (N = 100, b = 7 A˚, σ = 3/Nb2) will exist mostly in the uncharged state (f = 0.0019) under the same condition because of the depletion of the hydronium ions in the brush region (pH = 14). Therefore, in the zero-salt limit the brush instability is driven predominantly by the hydrophobic nature of the polymer. When a small amount of salt is added, the situation becomes qualitatively different. For instance, at 20 mM NaCl, the local pH is lower than the pKa value of the PDMAEMA monomer (see Table S2), and as a result, more Langmuir 2010, 26(3), 2021–2034

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than half of the monomers become protonated (f = 0.55). At this level of protonation, although the electro-osmotic repulsion between PDMAEMA monomers is sufficient to overcome the effect of the hydrophobicity of the polymer,48 the PDMAEMA brush system is predicted to be unstable (Figure 8(B)); in this case, the brush instability is enabled by spontaneous shrinkage of the polyelectrolyte chains through the mechanism of osmotic instability. As the NaCl concentration is further increased, the PDMAEMA brush chains become more charged (Table S2), making both the effects of the hydrophobicity and the osmotic instability more and more insignificant. To further elucidate the role of polyelectrolyte hydrophobicity in the instability of the brush structure, we performed the same set of μ vs σ analyses for better solvent-quality conditions (e.g., v = 0 and 50 A˚3) than used earlier, and the results of these predictions are displayed as broken (v = 50 A˚3) and dotted-broken (v = 0 A˚3) plots for comparison with the previous case of v = -50 A˚3 in Figures 8(A)-8(D) and S11(A)-S11(C). When no salt is added, as the solvent quality is changed from poor (v = -50 A˚3) to θ (v = 0 A˚3), or to good (v = 50 A˚3), the region of thermodynamic instability (where ∂μ/∂σ < 0) completely disappears (Figure 8A), which supports that in the zero-salt case, the hydrophobicity plays the key role in causing the brush instability. It is interesting to note, however, that the model predicts that even if the polyelectrolyte is inherently water-soluble (i.e., v = 50 A˚3), the brush layer can become destabilized at low salt concentrations (e.g., see Figures 8(B) and S11(A)) solely because of the effect of the osmotic instability. Presently, it remains unanswered whether this prediction is due to the various simplifying assumptions made in the derivation of the model (e.g., the preminimization of the free energy of the protonation reaction, the uniformity of the species concentrations within the brush layer, etc.), or that the osmotic instability alone can indeed cause the polyelectrolyte chains to become insoluble in water, for instance, even when water is a good solvent for the polymer. A more rigorous study of this problem is currently underway using a full-scale self-consistent field (SCF) model,49 and the results of this study will be discussed in a future publication. It should also be noted that the observed trend of diminishing brush heterogeneities with increasing NaCl concentration is caused by the reversibility of the protonation of the polyelectrolyte monomers. As shown in Figure S12 of the Supporting Information, and also as has been implied from the Fosm analysis presented earlier in this section, in the absence of this charge reversibility (i.e., when the value of f is fixed), hydrophobic polyelectrolyte brushes can become unstable only at high concentrations of added NaCl, which is opposite to what has been observed with the weak polyelectrolyte brush system. Now, in this case of strong (or quenched) polyelectrolytes, the homogeneous state of the brush system is expected to become thermodynamically stable at all salt conditions, when the solvent quality is better than the θ solvent (Figure S13). 3.4. Discussion of the Nanoscale Nature of the Lateral Heterogeneities in the PDMAEMA Brushes. There are two crucial questions that still need to be discussed: Why do the lateral heterogeneities occur at microscopic (i.e., submicrometer) length scales? Why does the length scale of the heterogeneities increase with the polymer grafting density? We believe that the answer to the first question is that, in the experimental system (48) Sharma, R. An ABC Triblock Copolymer-Based Approach for Non-Viral Gene Delivery. Ph.D. Thesis, Purdue University, West Lafayette, IN, 2008. (49) Witte, K. N.; Kim, S.; Won, Y. Y. J. Phys. Chem. B 2009, 113, 11076-11084.

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Figure 9. Log-log plots of the width of the brush heterogeneities

(ξ) vs the excess area per chain (1/σo - 1/σ) for two different concentrations of added NaCl; i.e., 0 and 20 mM. The ξ values were measured from liquid AFM images of the respective samples, and represent the average peak-to-peak distances in the AFM crosssectional height profiles; the sampling numbers (i.e., the numbers of objects used for calculating the average sizes) were N = 826, 448, 284, 274 and 166, respectively, for the cases of 1/σ = 1942, 1650, 1357, 1064 and 770 A˚2/chain. See Figures S6 and 2 for respective representative AFM images used in the analyses. Here, σo denotes the grafting density at which the water surface becomes fully covered by the PnBA monolayer (for the PDMAEMA-PnBA material used in this study, σo = 0.00050 A˚-2). The solid line represents the linear regression fit to the data, and the slope of the line is measured to be 0.50.

used in this study, the covalent linkage between the PDMAEMA and PnBA chains restricts macroscopic lateral aggregation of the PDMAEMA chains. Such macroscopic phase separation should produce areas uncoated with the copolymer at the air-water interface which would involve a huge energetic cost. This explanation is supported by the fact that the size of the heterogeneities (ξ) increases linearly with the square root of the excess area per chain, i.e., ξ ∼ (1/σ - 1/σo)1/2 (see Figure 9) where σo denotes the grafting density at which the water surface becomes completely covered by the PnBA monolayer (for the PDMAEMA-PnBA material used in this study, the 1/σo value was measured to be 1942 A˚2/chain. See Figure 5 and also ref 27); as the monolayer is compressed, the PnBA domain becomes thicker and gains more conformational degrees of freedom to accommodate the deformation associated with the local lateral clustering of the PDMAEMA chains, and therefore, the lateral dimension of the heterogeneous structures that result at a given 1/σ(ξ) is the maximum size of the PDMAEMA clusters that can be tolerated by the PnBA film without creating an empty water surface, which should be comparable, within a numerical factor of the order of unity, to (1/σo - 1/σ)1/2 (i.e., the maximum change in interface coverage area that can be locally produced by lateral flattening of a collapsed PnBA chain is (1/σo - 1/σ), and thus the corresponding length scale is the square root of it). This explanation also answers the second question. We also found that in the unstable regime (i.e., in the low-salt limit), the average length scale of the heterogeneities depends only on the grafting density (or the thickness of the anchoring layer) but not on the salt concentration (see Figure 9), which is also consistent with the above explanations. Within the above model, the number of PDMAEMA brush chains contained in each lateral aggregate (n) can be calculated by n = σξ2 ∼ (σ/σo - 1). The aggregation number is estimated to be, for instance, n ≈ 60 at a monolayer area of 1/σ = 1942 A˚2/chain (ξ ≈ 34 nm), and the aggregation number becomes increased to n ≈ 6.9  103, when the area is reduced to a value of 1/σ = 770 A˚2/ chain (ξ ≈ 230 nm). DOI: 10.1021/la902549b

2033

Article

Hur et al.

4. Summary and Conclusion PDMAEMA brushes with stable grafting density gradients were fabricated by LB deposition of Langmuir monolayers of PDMAEMA-PnBA diblock copolymers (DPn,PDMAEMA = 80, DPn,PnBA = 94, PDI = 1.31) onto graphite substrate surfaces under continuous lateral compression of the precursor monolayers at the air-water interface; this procedure (termed in this paper as the “Langmuir-Blodgett-deposition-under-compression” or “LB\C” method) allowed us to significantly reduce the number of samples necessary for extensive investigations of the structures of the PDMAEMA brushes under various combinations of parameter conditions. Specifically, using this sample preparation approach, the lateral nanoscale structural characteristics of the PDMAEMA brushes were studied by AFM at various chain grafting densities and various concentrations of added NaCl under neutral pH conditions. These studies revealed the following intriguing findings. Finding no. 1: As a function of NaCl concentration, the PDMAEMA brush system exhibits three qualitatively distinct classes of structural behavior; at low NaCl concentrations (e.g., at 0 and 20 mM NaCl), the PDMAEMA brush chains are locally clustered, giving rise to lateral nanoscale heterogeneities in the spatial distribution of the height of the PDMAEMA brush (as demonstrated, for example, in Figures S6 and 2); when a sufficient amount of NaCl is added (e.g., at 300 mM or higher), the PDMAEMA brush structure becomes completely laterally homogeneous (Figures S9 and 4); at intermediate NaCl concentrations (e.g., at 60-100 mM NaCl), the aggregated structure and the homogeneous region coexist in single PDMAEMA brush systems (Figures S7, 3 and S8). Finding no. 2: In the nanoscale aggregated regime (i.e., at low NaCl concentrations), both the width and height of the heterogeneities increase with increasing grafting density but are independent of the salt concentration. Quantitative analysis of the relationship between the size of the heterogeneities and the chain grafting density suggests that the microscopic nature of the lateral heterogeneities is due to the covalent connectivity between the PDMAEMA and PnBA chains. Finding no. 3: In the coexisting cases (e.g., CNaCl = 60-100 mM), the size and abundance of the lateral clusters of the brush chains increase with grafting density, but both quantities decrease as more NaCl is added, until the heterogeneous domains disappear completely at higher NaCl concentrations. In an attempt to explain these observations, we present a simple theoretical analysis. This analysis suggests that PDMAEMA brush chains can indeed become thermodynamically destabilized in the low-salt limit under neutral pH conditions. This instability is due to charge neutralization of the ionizable moieties of the polyelectrolytes in the brush layer (relative to the degree of monomer protonation for the same polyelectrolyte in the bulk monomer limit) which is driven by the depletion of the hydronium ion (i.e., the increase in the local pH) in the brush region. The influence of this monomer overneutralization behavior at low salt conditions is two-fold. First, in the case that the degree of monomer protonation is extremely low (i.e., in the zero-NaCl limit), the brush chains become insoluble in water simply because of the hydrophobic nature of the polymer. The second aspect is that when the polyelectrolytes are mildly overneutralized (i.e., at low salt concentrations), the tendency of the polyelectrolyte brush toward a less charged and thus less osmotically stressed state

2034 DOI: 10.1021/la902549b

(via chain shrinkage) becomes a dominant factor that drives the brush chains to collapse or laterally aggregated; we call this effect “osmotic instability”. Now whether this effect alone can make a hydrophilic polyelectrolyte brush thermodynamically unstable is a question that remains to be further explored. With increasing NaCl concentration, the local pH in the interior of the brush becomes less and less different from the bulk pH; the PDMAEMA chains become significantly more protonated, and the extent of an increase in the local pH that can be induced by a collapse (or aggregation) of the brush chains becomes insufficient to cause the osmotic instability effect. As a result, at sufficiently high concentrations of added NaCl, the homogeneous state of the brush becomes thermodynamically stable. The detailed μ vs σ analysis suggests the possible coexistence of metastable states at intermediate NaCl concentrations; in this regime, the stability of the brush system depends on polymer grafting density, with the homogeneous state of the brush being more stable than the collapsed (or aggregated) state at low grafting densities and with the reverse situation at high grafting densities. All these predictions are, though qualitative, in close agreement with the experiments (findings nos. 1 and 3), suggesting that the proposed mechanism for the formation of heterogeneous structures in weak polyelectrolyte brushes is plausible and worthy of further investigation. Acknowledgment. The authors are grateful for financial support of this research from the National Science Foundation (DMR-0906567), the donors of the Petroleum Research Fund, administered by the American Chemical Society (Grant No. 46593-G7), the 3M Company (Nontenured Faculty Award), and the Purdue Research Foundation Shreve Fund. Supporting Information Available: Figure S1 (the chemical structure of PDMAEMA-PnBA, and a schematic depiction of a Langmuir monolayer of PDMAEMA-PnBA), Figure S2 (representative fluid AFM images of PDMAEMA brushes prepared by normal LB techniques), Figures S3 and S4 (representative fluid AFM images of the gradient PDMAEMA brush sample prepared using the LB\C procedure, taken at 24 h and 2 weeks after the preparation of the sample, respectively), Figure S5 (comparison of the results between the normal LB vs LB\C techniques), Figures S6-S9 (additional representative fluid AFM images of the PDMAEMA brushes at various grafting densities under four different NaCl concentrations, i.e., 0, 60, 100 and 300 mM), Figures S10 and S11 (plots of Fosm vs H/b and of μ vs σNb2, estimated for the weak polyelectrolyte brushes at added NaCl concentrations of 60, 100 and 300 mM), Figures S12 and S13 (plots of μ vs σNb2 for strong polyelectrolyte brushes under various salt concentrations and fixed charge group densities at two different solvent quality conditions: v = -50 A˚3 (Figure S12), and v = 0 A˚3 (Figure S13)), Table S1 (measured pH values and NaCl concentrations in the subphase solutions used for the preparation of the PDMAEMA brush samples), and Table S2 (values of f and the local pH in the interior of the PDMAEMA brush, calculated for various NaCl concentrations). This material is available free of charge via the Internet at http://pubs. acs.org.

Langmuir 2010, 26(3), 2021–2034