Morphologies of Planar Polyelectrolyte Brushes in a Poor Solvent: Molecular Dynamics Simulations and Scaling Analysis

Transcription

1 pubs.acs.org/langmuir 2009 American Chemical Society Morphologies of Planar Polyelectrolyte Brushes in a Poor Solvent: Molecular Dynamics Simulations and Scaling Analysis Jan-Michael Y. Carrillo and Andrey V. Dobrynin* Polymer Program, Institute of Materials Science and Department of Physics, University of Connecticut, 2152 Hillside Road, U-3046, Storrs, Connecticut Received May 22, Revised Manuscript Received July 14, 2009 Using molecular dynamics simulations and scaling analysis, we study the effect of the solvent quality for the polymer backbone, the strength of the electrostatic interactions, the chain degree of polymerization, and the brush grafting density on conformations of the planar polyelectrolyte brushes in salt-free solutions. Polyelectrolyte brush forms: (1) vertically oriented cylindrical aggregates (bundles of chains), (2) maze-like aggregate structures, or (3) thin polymeric layer covering a substrate. These different brush morphologies appear as a result of the fine interplay between electrostatic and short-range monomer-monomer interactions. The brush thickness shows nonmonotonic dependence on the value of the Bjerrum length. It first increases with the increasing value of the Bjerrum length, and then it begins to decrease. This behavior is a result of counterion condensation within a brush volume. *Corresponding author. dobr@math.nsc.ru. (1) Ballauff, M.; Borisov, O. Polyelectrolyte brushes. Curr. Opin. Colloid Interface Sci. 2006, 11, (6), (2) Minko, S., Responsive polymer brushes. Polym. Rev. 2006, 46, (4), (3) Naji, A.; Seidel, C.; Netz, R. R. Theoretical approaches to neutral and charged polymer brushes. Adv. Polym. Sci. 2006; Vol. 198, (1), (4) Netz, R. R.; Andelman, D. Neutral and charged polymers at interfaces. Phys. Reports 2003, 380, (1-2), (5) Ballauff, M. Spherical polyelectrolyte brushes. Progr. Polym. Sci. 2007, 32, (6) Ruhe, J.; Ballauff, M.; Biesalski, M.; Dziezok, P.; Grohn, F.; Johannsmann, D.; Houbenov, N.; Hugenberg, N.; Konradi, R.; Minko, S.; Motornov, M.; Netz, R. R.; Schmidt, M.; Seidel, C.; Stamm, M.; Stephan, T.; Usov, D.; Zhang, H. N. Polyelectrolyte brushes. Adv. Polym. Sci. 2004, 165, (7) Huck, W. T. S. Responsive polymers for nanoscale actuation. Mater. Today 2008, 11, (7-8), (8) Lu, W. H.; Wang, R. M.; He, Y. F.; Zhang, H. F. Preparation and application of smart coatings. Progr. Chem. 2008, 20, (2-3), (9) Raviv, U.; Giasson, S.; Kampf, N.; Gohy, J. F.; Jerome, R.; Klein, J. Lubrication by charged polymers. Nature 2003, 425, (10) Wong, J. S.; Granick, S. Open questions about polymer friction. J. Polym. Sci., Part B: Polym. Phys. 2007, 45, (24), (11) Bae, S. C.; Granick, S. Molecular motion at soft and hard interfaces: From phospholipid bilayers to polymers and lubricants. Annu. Rev. Phys. Chem. 2007, 58, (12) Klein, J. Chemistry Repair or Replacement-A Joint Perspective. Science 2009, 323, (5910), Introduction Polyelectrolyte brushes consist of charged polymers end grafted to substrates of different geometries. 1-6 In polar solvents, the ionizable groups on the polymer backbone dissociate by releasing the counterions into solution and leaving uncompensated charges on the polymer chains. The morphology of the grafted polyelectrolyte layers depends on the solvent quality for the polymer backbone, the fraction of the charged groups, the chain s degree of polymerization, the polymer grafting density, and the salt concentration. By varying these parameters one can control both brush thickness and structure (see for review refs 1-6). The tremendous interest in these polymeric systems was dictated by their applications for colloidal stabilization, drug delivery, biocompatible coatings, ph-controlled gate devices (filters), smart surfaces, and biosensor technology. 1,2,4,5,7-12 While the properties of the polyelectrolyte brushes in good and θ-solvent conditions for the polymer backbone were extensively studied over the years, the analysis of the phase diagram of the polyelectrolyte brushes in poor solvents is still incomplete. 1-5 There were only a few attempts to analyze this type of polymeric system. 6,13-17 Recently, we used molecular dynamics simulations in combination with scaling analysis to study the effects of the solvent quality for the polymer backbone and the strength of the electrostatic interactions on the morphology of the spherical polyelectrolyte brushes in salt-free solutions. 14 We have shown that the morphology of the spherical polyelectrolyte brush is controlled by a fine interplay between the long-range electrostatic interactions between charged groups and the short-range monomer-monomer interactions. It was demonstrated that the spherical polyelectrolyte brush could be in a star-like spherical conformation, a star of bundles conformation in which polyelectrolyte chains self-assemble into clusters of pinned cylindrical micelles, a micellelike conformation with a dense core and charged corona, or could form a thin polymeric layer uniformly covering the particle surface. Counterions play an important role in controlling brush properties. Counterion condensation inside the brush results in nonmonotonic dependence of the layer thickness on the strength of the electrostatic interactions which is controlled by the value of the Bjerrum length. We have found that the thickness of the brush layer first increases with the increasing value of the Bjerrum length then it begins to decrease. The decrease of the brush thickness was explained by a combination of two effects associated with the counterion condensation. The first effect is due to compensation of the brush charge by condensed counterions, which weakens the electrostatic repulsion between polyelectrolyte chains forming a brush. The second effect is due to a correlation-induced attraction (13) Pryamitsyn, V. A.; Leermakers, F. A. M.; Fleer, G. J.; Zhulina, E. B. Theory of the collapse of the polyelectrolyte brush. Macromolecules 1996, 29, (25), (14) Sandberg, D. J.; Carrillo, J. M. Y.; Dobrynin, A. V. Molecular dynamics simulations of polyelectrolyte brushes: From single chains to bundles of chains. Langmuir 2007, 23, (25), (15) Zhulina, E.; Singh, C.; Balazs, A. C. Behavior of tethered polyelectrolytes in poor solvents. J. Chem. Phys. 1998, 108, (3), (16) Gong, P.; Genzer, J.; Szleifer, I. Phase behavior and charge regulation of weak polyelectrolyte grafted layers. Phys. Rev. Lett. 2007, 98, (1), (17) Gunther, J. U.; Ahrens, H.; Forster, S.; Helm, C. A. Bundle Formation in Polyelectrolyte Brushes. Phys. Rev. Lett. 2008, 101, (25), DOI: /la901839j Published on Web 08/10/2009

2 Article the simulation box to prevent counterions from escaping. The system was periodic in the lateral x and y directions. All particles in the system interacted through the truncatedshifted LJ potential: 18,19 ¼ 8 >< >: 4ε LJ 2 4 U LJ ðr ij Þ 12 3 σ - σ 6 - σ 12 þ σ 6 5 r e r cut r ij r ij r cut r cut 0 r > r cut ð1þ Figure 1. Snapshot of the simulation box. The system is periodic in x and y directions and the grafting surface is located at z = 0. Grafted polyelectrolyte chains are shown in blue, counterions are shown in green, and particles belonging to a substrate are colored in red. between condensed counterions and charged monomers resulting in an effective attraction between polyelectrolyte chains. In this paper we extend our analysis to morphological transformations of planar polyelectrolyte brushes in poor solvent conditions for the polymer backbone. In particular, we study the effect of solvent quality and strength of the electrostatic interactions on the brush conformations. We have established the range of parameters for which a planar polyelectrolyte brush can form a forest of cylindrical micelles (bundle of chains), a maze-like aggregate structure (vertically standing lamellar domains), and a polymeric film covering a substrate. All these different brush morphologies appear as a result of competition between the counterion configurational entropy, the solvent quality for the polymer backbone, and the strength of the electrostatic interactions. The rest of the manuscript is organized as follows. In Section 2, we present the results of the molecular dynamics simulations of the planar polyelectrolyte brush in saltfree solution. In Section 3, we develop a scaling model of a polyelectrolyte brush in a poor solvent and compare model predictions with the simulation results. Finally, Section 4 summarizes our results. 2. Molecular Dynamics Simulations of Planar Polyelectrolyte Brushes 2.1. Simulation Details. We performed molecular dynamics simulations 18,19 of the planar polyelectrolyte brushes. The polyelectrolytes were modeled by chains of charged Lennard-Jones (LJ) particles (beads) with diameter σ, degree of polymerizations (number of monomers) N=60 and N=180, and fraction of charged monomers f = 1/3, corresponding to every third monomer carrying a negative electrical charge, -e. Polyelectrolyte chains, N ch, were attached at one end to particles forming a substrate at several grafting densities. To neutralize the negatively charged polyelectrolyte chains, positively charged counterions were added to a simulation box. The system configuration is shown in Figure 1. The simulation box had dimensions L x L y L z. The grafting surface was modeled by a periodic hexagonally packed lattice of beads with a diameter σ and located at z=0. A similar nonselective surface was located at the opposite side of where r ij is the distance between i-th and j-th particles, and σ is the particle diameter chosen to be the same regardless of the particle type. The cutoff distance, r cut =2.5σ, was selected for polymerpolymer interactions, and r cut ¼ 6p ffiffiffi 2 σ was chosen for all other pairwise interactions. The interaction parameter ε LJ was equal to k B T for polymer-substrate, polymer-counterion, counterioncounterion, and counterion-substrate interactions, where k B is the Boltzmann constant, and T is the absolute temperature. The value of the LJ interaction parameter for the polymer-polymer pair was varied between 0.5 k B T and 1.5 k B T, which allowed to change a solvent quality for the polymer backbone. Our choice of parameters for the polymer-polymer interactions corresponds to poor solvent conditions for the polymer backbone. The chain s connectivity was maintained by the finite extension nonlinear elastic (FENE) potential: 18,19 U FENE ðrþ ¼- 1 2 k springr 2 max r2 ln 1 - R 2 max with the spring constant k spring =30k B T/σ 2,andthemaximum bond length R max =1.5σ. Electrostatic interactions between any two charged particles with the charge valences q i and q j and separated by the distance r ij were given by the Coulomb potential: U Coul ðr ij Þ¼k B T l Bq i q j ð3þ r ij where l B =e 2 /εk B T is the Bjerrum length, defined as the length scale at which the Coulomb interaction between two elementary charges e in a medium with the dielectric constant ε is equal to the thermal energy k B T. In our simulations, the value of the Bjerrum length l B was varied between 1/128 and 16 σ. The particle-particle particle-mesh (PPPM) method for the slab geometry implemented in LAMMPS 19 with a sixth-order charge interpolation scheme and an estimated accuracy of 10-5 was used for calculations of the electrostatic interactions between all charges in the system. The 2D periodic images of the system were periodically replicated along the z-direction with distance L=3L z between their boundaries. This reduced the problem of calculation of the electrostatic interactions in a 2D periodic system to those in a 3D system. Simulations were carried out in a constant number of particles, volume V=L x L y L z and temperature-(nvt) ensemble. The constant temperature was maintained by coupling the system to alangevinthermostat. 18,19 In this case, the equation of motion of the i th particle is ð2þ (18) Frenkel, D.; Smit, B. Understanding Molecular Simulations. Academic Press: New York, (19) Plimpton, S. LAMMPS Molecular Dynamics Simulator, SANDIA: m d ν iðtþ dt ¼ FB i ðtþ - ξvb i ðtþþfb R i ðtþ ð4þ DOI: /la901839j 13159

3 Article Table 1. System Sizes and Interaction Parameters grafting density, F g = σ -2 degree of polymerization, N = 60 degree of polymerization, N = 180 solvent quality, ε LJ = 0.5 k B T, 1.0 k B T, 1.5 k B T solvent quality, ε LJ = 1.5 k B T N ch L x [σ] L y [σ] L z [σ] l B [σ] N total N ch L x [σ] L y [σ] L z [σ] l B [σ] N total Table 2. System Sizes and Interaction Parameters solvent quality, ε LJ = 1.5 k B T degree of polymerization, N = 60 degree of polymerization, N = 180 N ch L x [σ] L y [σ] L z [σ] F g [σ -2 ] N total N ch L x [σ] L y [σ] L z [σ] F g [σ -2 ] N total l B = σ l B = σ l B = 1.0 σ l B = 1.0 σ where vb i (t) is the particle velocity, and FB i (t) is the net deterministic force acting on i th particle of mass m. FB R i (t) is the stochastic force with the zero average value ÆFB R i (t)=0æ and δ-functional correlations ÆFB R i (t)fb R i (t 0 )æ=6ξk B Tδ(t-t 0 ). The friction coefficientpξ was set to ξ=m/τ LJ,whereτ LJ is the standard LJ time τ LJ ¼ σ ffiffiffiffiffiffiffiffiffiffiffiffi m=ε LJ. The velocity-verlet algorithm with a time step Δt =0.01τ LJ was used for integration of the equations of motion (eq 4). Simulations were performed using the following procedure: Polyelectrolyte chains were grafted to the substrate in the fully extended conformation pointing normal to the substrate surface. Neutralizing monovalent counterions were uniformly distributed over the volume of the simulation box. The system was preequilibrated for MD steps. This was followed by the production run lasting MD steps. During the production run, we averaged the brush thickness and the fraction of the condensed counterions and determined the counterion distribution function and the distribution function of the number of chains forming an aggregate. We have performed two sets of simulations. In the first set, we have fixed the chain grafting density F g and varied the degree of polymerization N, the solvent quality for the polymer backbone, and the strength of the electrostatic interactions, which was controlled by the value of the Bjerrum length l B. The system parameters are summarized in Table 1. In the second set of simulations, we varied the brush grafting density F g, the value of the Bjerrum length l B, and the chain s degree of polymerization N, keeping the solvent quality for the polymer backbone unchanged (see for details Table 2). The results of our simulations are discussed in Section Diagram of States and Brush Thickness. Figure 2 shows the diagram of state of the planar polyelectrolyte brush in the salt-free solutions as a function of the value of the LJ interaction parameter and the Bjerrum length. Note, that the value of the LJ interaction parameter determines the solvent quality for the polymer backbone in molecular simulations without the explicit solvent, while the value of the Bjerrum length DOI: /la901839j

4 Article Figure 4. Dependence of the average aggregation number <M > Figure 2. Diagram of states of the polyelectrolyte brushes with the degree of polymerization N = 60, grafting density Fg=0.058 σ-2, and the fraction of charged monomers f=1/3: cylindrical micelles (squares), maze-like aggregates (rhombs), and uniform polymeric layers (triangles). The dash lines separating different regimes are not actual transition lines. on the value of the Bjerrum length for weakly charged, f = 1/3, planar brushes with the degree of polymerization N = 60, and grafting density Fg = σ-2 at different values of the LJ interaction parameter: εlj = 1.5 kbt (circles) and εlj = 1.0 kbt (squares). Inset shows distribution function of the micelle aggregation number M for: εlj = 1.0 kbt and lb = 0.5 σ (a), and εlj = 1.5 kbt and lb= σ (b). the first moment of the polymer density distribution F(z), RLz RLz ÆHæ ¼ 2 zfðzþdz= FðzÞdz.) The thickness (radius) of the 0 Figure 3. Dependence of the average brush height on the value of the Bjerrum length for weakly charged, f = 1/3, planar brushes with the degree of polymerization N=60, and grafting density Fg= σ-2 at different values of the LJ interaction parameter: εlj= 1.5 kbt (circles), εlj = 1.0 kbt (squares), and εlj = 0.5 kbt (rhombs). Dependence of the Gouy-Chapman length, λgc = (2πlBFgfN)-1, on the value of the Bjerrum length is shown by a dashed line. determines the strength of the electrostatic interactions. In real experiments the variations in the value of the Bjerrum length can be achieved by varying a solvent dielectric constant, which can be done, for example, by using mixed solvents. There are three different conformation regimes on the diagram of state (Figure 2): forest of cylindrical aggregates (bundle of chains), vertically standing lamellar domains (maze-like aggregates), and thin polymeric layers. Most of the diagram is occupied by a region with the cylindrical aggregates. For small values of the Bjerrum lengths, lb < 1/64 σ, these cylindrical aggregates are relatively short with the height being only slightly larger than their thickness. However, the aggregates become taller as the strength of the electrostatic interactions increases (see Figure 3). (The brush thickness or average aggregate height was calculated by using 0 cylindrical aggregates and their aggregation number (see Figure 4) both decrease with an increasing value of the Bjerrum length, while the height of the micelles increases. Note that, for small values of the Bjerrum length, we observed a bimodal distribution of the bundle aggregation number. This form of the distribution function is a result of the random distribution of the grafting points over the substrate surface. The effect of the grafting point distribution becomes less important for larger values of the Bjerrum length when the aggregates become smaller. The average aggregation number decreases with the value of the Bjerrum length as ÆMæ lb-0.4. This is close to the scaling model prediction which predicts the aggregation number to scale with the Bjerrum length as ÆMæ lb-0.44 (see discussion in Section 3). The number of chains in a cylindrical aggregate is also related to the average distance between aggregates d by the following simple relation d2 ÆMæ/Fg or d ÆMæ1/2. To obtain the distance between aggregates, we have calculated a 2D Fourier transform of the monomer-monomer correlation function S(q) of the brush slice with thickness 2σ located at the half height of the brush, z=h/2. SðqÞ ¼ ÆΓðqÞΓð-qÞæ ð6þ where brackets correspond to an ensemble average over all distributions of the polymer density within the slice. We have used the fast Fourier transform (FFT) method for calculation of the function S(q). In order to utilize the FFT procedure, the distribution of the monomer density in a slice Γ(x,y) was meshed over grid points. The meshing was implemented by using the linear interpolation scheme with 2D periodic boundary conditions for the boundary grid points of the array. This procedure provides a discrete representation of the function Γ(x,y) on the grid points. The calculation of the 2D FFT of the function Γ(x,y) was performed for each realization of the function Γ(x,y) collected during the simulation run. The resultant function S(q) was obtained by averaging the product of function Γ(q) and its complex conjugated Γ(q)* over all realizations of the monomer density distribution obtained during the production run. The distance between aggregates d was DOI: /la901839j 13161

5 Article Figure 5. Dependence of the average distance between aggregates d on the value of the Bjerrum length for weakly charged, f=1/3, planar brushes with the degree of polymerization N = 60 and grafting density F g =0.058 σ -2 at different values of the LJ interaction parameter: ε LJ =1.5 k B T (circles), ε LJ =1.0 k B T (squares), and ε LJ =0.5 k B T (rhombs). Insets show snapshots of the brush structure and corresponding monomer-monomer correlation functions S(q). estimated from the radius q* of the high-intensity ring of the function S(q) asd=2π/q*. Figure 5 shows the dependence of the spacing between aggregates d on the value of the Bjerrum length. As expected the aggregate spacing is proportional to <M > and scales with the Bjerrum length as d µ l B -2. The insets show the evolution of the monomer-monomer correlation function S(q) with the Bjerrum length. One can easily identify the high-intensity ring with characteristic six peaks corresponding to a hexagonal arrangement of the aggregates. A different scaling dependence of the spacing between aggregates d µ l B -0.4 and a different form of the monomer-monomer correlation function is observed for systems with the LJ interaction parameter ε LJ =0.5 k B T and the value of the Bjerrum length l B <0.25 σ. In this range of parameters, a polyelectrolyte brush forms maze-like aggregates. At the large values of the Bjerrum lengths, l B >2σ, andforall three values of the LJ interaction parameters, a polyelectrolyte brush collapses forming a dense layer covering a substrate. Collapse of a brush is a result of a counterion condensation. 14,20-23 At large value of the Bjerrum length, almost all counterions are localized within a brush volume. In this case, the structure of the brush layer is similar to a structure of the strongly correlated Wigner liquid. 22,24 In this regime, the collapse of a brush is due to a correlation-induced attraction between charged monomers on the brush backbones and counterions. Thus, the equilibrium monomer density inside a brush and the brush thickness are a result of optimization of the correlation-induced attraction and short-range repulsion between monomers and counterions Counterion Distribution. As we have already mentioned, a counterion condensation plays an important role in (20) Dobrynin, A. V.; Rubinstein, M. Theory of polyelectrolytes in solutions and at surfaces. Progr. Polym. Sci. 2005, 30, (21) Gonzalez-Mozuelos, P.; de la Cruz, M. O. Ion condensation in salt-free dilute polyelectrolyte solutions. J. Chem. Phys. 1995, 103, (8), (22) Levin, Y. Electrostatic correlations: from plasma to biology. Rep. Prog. Phys. 2002, 65, (11), (23) Solis, F. J.; de la Cruz, M. O. Collapse of flexible polyelectrolytes in multivalent salt solutions. J. Chem. Phys. 2000, 112, (4), (24) Grosberg, A. Y.; Nguyen, T. T.; Shklovskii, B. I. Colloquium: The physics of charge inversion in chemical and biological systems. Rev. Mod. Phys. 2002, 74, (2), determining the brush height and structure. For example, the brush height demonstrates a nonmonotonic dependence on the value of the Bjerrum length, l B (see Figure 3). It first increases with the increasing value of the Bjerrum length, and then it begins to decrease. This behavior is due to counterion condensation within the brush volume and counterion correlation-induced attraction at large values of the Bjerrum length. There are different length scales which are involved in a counterion condensation process. First, the counterion condensation occurs within a brush layer. With increasing the strength of the electrostatic interactions, the Gouy-Chapman length, λ GC =(2πl B F g fn) -1, determining a length scale within which a half of the brush counterions are localized, becomes on the order of the brush thickness. This corresponds to a crossover to the so-called osmotic brush regime when the stretching of the brush is caused by a translational entropy of localized within brush counterions. 1,6 In addition to counterion condensation within a brush layer, we can have a counterion condensation on cylindrical aggregates resulting in reduction of the effective linear charge density along cylindrical micelles to the critical Manning-Oosawa value. 20,25-27 Note, that in the case of the maze-like aggregates, the distribution of counterions between the aggregates is similar to that observed for a planar substrate, with about half of the counterions localized within the corresponding Gouy-Chapman length determined by the surface charge density of the maze-like aggregates. 28 The counterions can also penetrate inside the cylindrical and maze-like aggregates neutralizing their bulk charge and producing correlation-induced attraction between charged monomers and counterions. Below we will illustrate this multiscale picture of counterion distribution within a brush. We will first consider the distribution of the counterions outside the brush layer. Localized within the brush layer, counterions reduce an effective surface charge density of the brush layer to Σ=F g f(1 - x)n, where x is the fraction of localized within brush counterions. For the remaining outside the brush counterions, their density distribution is given by 28 F c ðzþ ¼ 1 s 2 2πl B h 2 cos 2 ðsð1 - ðz - HÞ=hÞÞ where we defined h=l z - H, and the parameter s is a solution of the equation s tan(s)=2πl B Σh. Equation 7 is a solution of the nonlinear Poisson-Boltzmann equation which couples the distribution of counterions with electrostatic potential. In Figure 6, we plotted distribution of the counterion density as a function of the distance from the substrate surface. The solid line corresponds to eq 7 with the effective surface charge density of the brush layer evaluated at the height of the brush layer H=21 σ. The agreement between theoretical expression for the counterion density distribution and simulation results is very good. Note, that qualitatively similar counterion density profile was also observed in simulations of polyelectrolyte brush in good solvent conditions for the polymer backbone by Hehmeyer et al. 29 This supports an argument that in the regime of the weak (25) Manning, G. S. Counterion condensation theory constructed from different models. Physica A 1996, 231, (1-3), (26) Manning, G. S. The critical onset of counterion condensation: A survey of its experimental and theoretical basis. Berichte Der Bunsen-Gesellschaft - Phys Chem Chem Phys 1996, 100, (6), (27) Barrat, J. L.; Joanny, J. F. Theory of polyelectrolyte solutions. Adv. Chem. Phys. 1996, 94, (28) Evans, D. F.; Wennerstrom, H. The Colloidal Domain. Wiley-VCH: New York, (29) Hehmeyer, O. J.; Arya, G.; Panagiotopoulos, A. Z. Monte Carlo simulation and molecular theory of tethered polyelectrolytes. J. Chem. Phys. 2007, 126, /1-11. ð7þ DOI: /la901839j

6 Article Figure 7. Wigner-Seitz cell of a cylindrical aggregate of polyelectrolyte chains with εlj = 1.5 kbt, lb = σ, Fg = σ-2, and N=180. Figure 6. Density distribution of charged monomers (open circles) and counterions (filled circles) along z direction for polyelectrolyte brush with εlj=1.5 kbt, lb=0.125 σ, N=60, and grafting density Fg = σ-2. The solid line corresponds to counterion distribution given by eq 7. electrostatic interactions the counterion distribution outside a brush layer is determined by the effective surface charge density of the polyelectrolyte brush. Inside a brush layer the distribution of counterions is determined by the structure of the brush layer. In the case of the cylindrical aggregates (see Figure 7), we can apply a cylindrical cell model to approximate distribution of counterions around cylindrical domains. To describe distribution of the counterion density we will approximate a Wigner-Seitz cell by a circle and evaluate distribution of the counterion density by assuming a cylindrical symmetry of a cell surrounding each cylindrical aggregate (see Figure 7). It is important to point out that the net charge of the cylindrical cell is not equal to zero since some of the counterions escape a brush (see discussion above). The solution of the nonlinear Poisson-Boltzmann equation for the cylindrical cell with nonzero net charge was obtained by Deshkovskii et al.30 This solution gives the following counterion density profile around a cylindrical aggregate as a function of the distance r from the aggregate axes 2 R2 ξ2r r2r - 2 Fc ðrþ ¼ πlb ðr2r - ξ2r Þ2 ð8þ where parameters R and ξ are solutions of the following equations r2r 0 γ0-1 - R γ - 1- R ¼ ξ2r ¼ R2R R γ0-1 þ R γr - 1 þ R ð9þ and we defined r0 as a size of the cylindrical aggregate, R as a cell size (in our case it is a half distance between centers of the cylindrical aggregates (see Figure 7)), and introduced reduced linear charge density of the cylindrical aggregate with the aggregation number M γ0 = fnmlb/h, on the aggregate surface at r=r0, and γr=(1-x) fnmlb/h, on the outer cell surface at r=r. Note, that eqs 9 are the boundary conditions for the electric filed at the surface of the cylindrical aggregate and at the outer cell boundary. In Figure 8 we fitted a counterion density distribution to the analytical function given by the eq 8. Our analysis shows that parameter R is a pure complex, which corresponds to a (30) Deshkovski, A.; Obukhov, S.; Rubinstein, M. Counterion phase transitions in dilute polyelectrolyte solutions. Phys. Rev. Lett. 2001, 86, (11), Figure 8. Radial density distribution of charged monomers (open circles) and counterions (filled circles) for a polyelectrolyte brush with εlj=1.5 kbt, lb=0.125 σ, N=180, and grafting density Fg = σ-2. The solid line is the counterion distribution described by the two-zone model with model parameters γ0 = , γr =0.0177, r0 =5.25 σ, R=9.25 σ, R=1.024 i, and ζ= σ. counterion condensation regime with a universal scaling for the counterion distribution profile.30 Thus, our analysis of the counterion density distribution in the regime of the diagram of states with cylindrical aggregates shows that one can use analytical solutions of the nonlinear 1D and 2D Poisson-Boltzmann equations to describe a counterion distribution outside and inside the brush layer. It is important to point out that the agreement between the analytical results and the computer simulations worsens with increasing the value of the Bjerrum length, lb > σ. In this range of parameters, the correlation effects begin to play an important role and control counterion distribution.22, Dependence of the Brush Thickness on Grafting Density. Grafting density is one of the most important parameters that could be controlled during the brush polymerization process. At low brush grafting densities, the distance between molecules is large such that a brush layer consists of individually collapsed chains (see insets in Figure 10). This is the so-called mushroom regime where individual chain conformations are controlled by intrachain electrostatic interactions. In this regime there is only a weak dependence of the chain size on the brush grafting density. It is interesting to point out that in a mushroom regime polyelectrolyte chains in poor solvent conditions for the polymer backbone could form a necklace-like globules of dense DOI: /la901839j 13163

7 Article Figure 9. Dependence of the average brush height on the brush grafting density F g for weakly charged, f=1/3, planar brushes with the value of the LJ interaction parameter ε LJ =1.5 k B T, Bjerrum length l B =1.0 σ, and degree of polymerization, N=60 (circles) and N=180 (squares). Insets show typical brush conformations. beads connected by strings of monomers. 15,20,31-36 For our longest chains with the number of monomers N=180, we have observed the formation of dumbbells. At brush grafting densities, F g >10-3 σ -2, the brush structure is determined by a fine interplay between the electrostatic interactions due to counterions and charged monomers and short-range monomer-monomer interactions. The combined effect of the chain aggregation and the electrostatic interactions leads to a stronger dependence of the brush thickness on the brush grafting density, H µ F g 1/3. The chains in this regime are strongly stretched which is supported by the fact that the brush thickness scales linearly with the degree of polymerization, N. We have repeated the same set of simulations for the case of weak electrostatic interactions with the value of the Bjerrum length l B =0.125 σ. In this case for the shorter chains with N=60, we have observed three different scaling regimes. At low brush grafting densities, the layer thickness shows a weak dependence on the brush grafting density; this follows by a regime with H µ F g 1/3 at the intermediate values of the brush grafting densities. Finally, at high brush grafting densities, the layer thickness shows a slightly stronger increase with the brush grafting density. For the systems with longer chains, N = 180, the simulation data follows H µ F g 1/2 scaling dependence. 3. Scaling Model In this section we will develop a scaling model of a polyelectrolyte brush in a poor solvent condition for the polymer backbone to explain our simulation results. We will analyze in detail the case of the cylindrical aggregates, since this type of aggregate (31) Dobrynin, A. V.; Rubinstein, M.; Obukhov, S. P. Cascade of transitions of polyelectrolytes in poor solvents. Macromolecules 1996, 29, (8), (32) Dobrynin, A. V.; Rubinstein, M. Hydrophobic polyelectrolytes. Macromolecules 1999, 32, (3), (33) Dobrynin, A. V. Theory and simulations of charged polymers: From solution properties to polymeric nanomaterials. Curr. Opin. Colloid Interface Sci. 2008, 13, (6), (34) Liao, Q.; Dobrynin, A. V.; Rubinstein, M. Counterion-correlation-induced attraction and necklace formation in polyelectrolyte solutions: Theory and simulations. Macromolecules 2006, 39, (5), (35) Jeon, J.; Dobrynin, A. V. Necklace globule and counterion condensation. Macromolecules 2007, 40, (36) Solis, F. J.; de la Cruz, M. O. Variational approach to necklace formation in polyelectrolytes. Macromolecules 1998, 31, (16), Figure 10. Dependence of the brush height on the brush grafting density F g for weakly charged, f=1/3, planar brushes with the value of the LJ interaction parameter ε LJ =1.5 k B T, Bjerrum length l B =0.125 σ, and degree of polymerization, N=60 (filled circles) and N=180 (filled squares). Insets show typical brush conformations. Figure 11. A cylindrical aggregate. occupies a majority of the space on the diagram of states. Consider a brush of polyelectrolyte chains with the degree of polymerization N attached to the planar surface with the grafting density F g and the total brush surface area S.Eachpolyelectrolyte chain is partially charged with the fraction of charged monomers f. Let us assume that a polyelectrolyte brush forms cylindrical aggregates consisting of M polyelectrolyte chains with height H and radius D (see Figure 11). The surface energy of the cylindrical aggregate is equal to F surf 2πγðDH þ D 2 Þ ð10þ where γ is the surface energy of the polymer solvent interface. For our choice of the LJ interaction parameters, the average monomer density inside an aggregate is on the order of F µ σ -3.Thus,we can estimate the surface energy as γ µ k B Tε c /σ 2, where ε c is a monomer cohesive energy inside an aggregate in terms of the thermal energy k B T. Note, that the strength of the cohesive energy ε c k B T is proportional to the value of the LJ interaction parameter ε LJ. In writing the eq 10, we have also assumed that the value of the surface energy of the polymer-substrate interface is on the order of that for the polymer-solvent interface. The grafting points of the M chains assembled into an aggregate are distributed over the surface area πr 2 M/F g.thus, each chain with a grafting point located at the distance r from the center of the aggregate should extend a distance on the order of r - D before reaching the core of the cylindrical aggregate. This will impose an additional energetic penalty on the order of k B Tε c per each monomer in a string of l(r) monomers of the length r - D DOI: /la901839j

8 (By pooling monomers into a string from a core of the cylindrical aggregate, one eliminates a cohesive energy between monomers.) The free energy of a string connecting a cylindrical aggregate to the chain s grafting point is equal to FðrÞ k B T 3 ðr - DÞ 2 2 σ 2 þ lε c ð11þ l Optimizing eq 11 with respect to the number of monomers in a pstring, l, we obtain that the optimal string has lðrþ ffiffiffi pffiffiffiffiffiffi 3 ðr -DÞ=ðσ 2ε cþ monomers. The total contribution to the aggregate free energy from all the strings connecting the chain s grafting points located at the distances r larger than the aggregate radius D is obtained by adding contributions from all such chains. This results in the following expression for the strings free energy: F string k B T 2πF g Z R D FðrÞ k B T rdr 241=2 πε 1=2 c F g σ -1 R DR2 þ D3 2 6 ð12þ In evaluating the electrostatic energy of a brush of cylindrical aggregates we will assume that the charged monomers are uniformly distributed over the volume of the cylindrical aggregate and counterions occupy a space between aggregates (see Figure 7). Each aggregate has fmn charged monomers distributed within a cylinder with a radius D and a height H.Thisresults in an average charge density inside the aggregate to be equal to F ch fmn/(πhd 2 ). The counterions are distributed outside the core of the aggregate with the average density F c fmn/(πhr 2 - πhd 2 ), where R (M/F g π) 1/2. In this approximation, the electrostatic energy per aggregate can be approximated as follows: 2 U elect k B T π2 l B HD 4 4F 2 ch 4 þ 4F2 t ln R R - 2 4F D t F c D þ F 2 c # R 4 D 4-1 ð13þ where we introduced F t = F ch þ F c. The localized within brush layer counterions provide an additional osmotic contribution to the brush free energy: þ F osm k B T βfnm ln βfnmσ 3 πhðr 2 -D 2 Þ πhðr2 - D 2 Þ σ 3 - βfnm ln 1 - βfnmσ 3 πhðr 2 - D 2 Þ ð14þ where β is the fraction of the osmotically active counterions, and the second term in the right hand side of the eq 14 describes the excluded volume interactions between counterions. Polyelectrolyte chains in each aggregate are strongly stretched, and the elastic energy contribution to the aggregate free energy is equal to Article F elast k B T 3 MH 2 2 σ 2 ð15þ N The shape dependent part of the free energy of the brush layer is obtained by combining the expressions for the aggregate electrostatic energy, the osmotic free energy, the aggregate surface energy, the string s free energy, and the chain s elastic energy together and by multiplying it by the number of the cylindrical aggregates, S/πR 2 SF g /M, where S is the area of the brush layer. Below we will use a scaling approach neglecting all numerical prefactors to obtain the power law dependences of the brush height, the aggregate radius and the aggregation numbers on the Bjerrum length, the brush grafting density, and solvent quality for the polymer backbone. In this approximation, the expression for the brush free energy is equal to F cyl k B TSF g l BðfNÞ 2 M H þ ε chd Mσ 2 þ ε cd 2 Mσ 2 þ H2 σ 2 N þ ε1=2 c M 1=2 F 1=2 g σ βf F g Nσ3 ð16þ H þ βfn ln Simplifying eq 16, we have assumed that the distance between the aggregates, R, is larger than their radius, D, which is true for the following range of the aggregation numbers M>F g D 2,and neglected a logarithm in the expression of the brush electrostatic energy. In order to find the equilibrium parameters of the aggregate, eq 16 has to be minimized with respect to the height of the cylindrical aggregate H and its width D at an additional constraint: D 2 H σ 3 MN ð17þ which fixes the total volume of an aggregate to a value controlled by the short-range monomer-monomer interactions. This equation provides a relation between the height of the aggregate H,its radius D, and the aggregation number M, M HD 2 /Nσ 3. Using this relation, we can rewrite eq 16 as follows: 0 F cyl N@ l Bf 2 D 2 k B TSF g σ 3 þ ε cσ D þ H2 σ 2 N þ βf ln βfnf g σ3 2 H þ ε cσn H þ ε1=2 c D H 1=2 ð18þ F 1=2 g σ 2 σn The radius of the aggregate D is obtained by minimizing the first and the second terms in the right hand side of eq 18 with respect to aggregate size D. Thisleadsto D cyl σðε c =uf 2 Þ 1=3 1 A ð19þ where u is the ratio of the Bjerrum length l B to the bond length σ. Thus, the radius (thickness) of the aggregate is a result of the optimization of the electrostatic interactions between the charged monomers forming an aggregate core and its surface energy. Note that the expression for the aggregate width is similar to the one obtained for the bead size of a polyelectrolyte chain in a poor solvent for the polymer backbone. A new feature in eq 19 is an explicit dependence on the strength of the cohesive energy ε c.the brush height is obtained by balancing the last two terms in the eq 18: H cyl σðf g σ 2 Þ 1=3 ε 1=9 c ðuf 2 Þ 2=9 N ð20þ The brush thickness, H, increases with the increasing strength of the electrostatic interactions as H µ l B 2/9 and scales with the chain s grafting density as H µ F g 1/3. This is precisely the scaling dependence observed in our simulations (see Figure 9). Using DOI: /la901839j 13165

9 Article eqs 19 and 20, we can obtain an expression for the aggregation number of the cylindrical aggregate: M cyl ðf g σ 2 Þ 1=3 ε 7=9 c ðuf 2 Þ -4=9 ð21þ The aggregation number of a cylindrical aggregate, M, scales with the value of the Bjerrum length as M µ l -4/9 B. The number of chains in the aggregate decreases with the increasing strength of the electrostatic interactions. This should not be surprising since electrostatic interactions control the aggregate thickness, which also decreases with the increasing value of the Bjerrum length. The scaling exponent is very close to the exponent -0.4 observed in our simulations (see Figure 4). With increasing the strength of the electrostatic interactions, the localized within brush layer counterions begin to control the chain s elongation. This is the so-called osmotic brush regime. The crossover to the osmotic brush regime is determined as a condition at which the counterion configurational entropy (the fourth term in the right hand side of the eq 18) dominates the brush stretching. This occurs when the brush grafting density exceeds F osm g σ 2 > ε2 c D2 σ 2 ðβfnþ 3 ε 8=3 c ð22þ N 3 u 2=3 f 13=3 In simplifying eq 22 we have assumed that all localized within brush layer counterions are osmotically active, β 1. The crossover to this regime shifts toward the lower brush grafting densities with increasing the chain s degree of polymerization. In the osmotic regime the aggregate height is determined by balancing the free energy of the strings connecting the aggregates to the chain s grafting points (the last term in the right hand side of the eq 18) with the free energy of the localized within brush layer counterions (the fourth term in the right hand side of the eq 18). This results in the following expression for the height of the brush layer: H osm cyl σβ 2 f 2 N 3 F g =ε c D 2 u 2=3 ε c -5=3 f 10=3 F g σ 3 N 3 ð23þ In this regime, the brush thickness has a strong N 3 dependence on the chain s degree of polymerization. This regime continues until the elastic free energy of the stretched chains becomes larger than the free energy of the strings. A crossover value of the brush grafting density where this happens is estimated as F osm - str g σ 2 ε 5=3 c N 2 f 17=6 u 2=3 ð24þ Forlargerbrushgraftingdensities,thebrushthicknessisdetermined by optimizing the chain s elastic energy with the counterion translational entropy. Thus, we recover the usual for the osmotic brush dependence of the brush thickness on the fraction of the charged monomers and the chain s degree of polymerization: osm - str Hcyl σðβf Þ 1=2 N ð25þ In this regime, the brush thickness is independent of the brush grafting density. Note that pure scaling regimes can only be observed for the relatively long chains. For our chain lengths we should expect a crossover-like behavior of the brush thickness on the brush grafting density. Indeed, in Figures 9 and 10 we can only clearly see one of the possible scaling regimes with H µ F g 1/3. Let us comment on a counterion condensation and its effect on the fraction of the osmotically active counterions. Counterions begin to condense on the core of the cylindrical aggregates when Figure 12. Dependence of the reduced brush height <H>/N on the brush grafting density F g for weakly charged, f=1/3, planar brushes with the value of the LJ interaction parameter ε LJ = 1.5 k B T, the value of the Bjerrum length l B = σ, and two different degrees of polymerization N=60 (filled circles) and N= 180 (filled squares). The lines are the results of the numerical optimization of the brush free energy eq 29 corresponding to the following set of the adjustable parameters β=0.65, γ=0.57 k B T/σ 2, Fσ 3 =0.37, ε c =25.6 k B T for N=60, and β=0.65, γ=0.45 k B T/σ 2, Fσ 3 =0.35,ε c =26.8 k B T for N=180. the linear charge density, ffd 2 fd 2 /σ 3,exceeds1/l B. 20,25-27 This occurs for the values of the Bjerrum length larger than l B g σ f ε 2 ð26þ c Above the counterion condensation threshold, the effective fraction of the charged monomers is reduced to its crossover value such that βl B fd 2 /σ 3 1. Solving for the parameter β we have β f uε 2 c 1=3 ð27þ In writing eq 27, we have assumed that counterions are excluded from the core of the cylindrical aggregates such that their bulk charge stays unchanged. This is in agreement with our simulation results that show exclusion of the counterions from the interior of the aggregates. Thus, above a counterion condensation threshold, a fraction β of the osmotically active counterion decreases which results in the decrease of the brush thickness with the increasing value of the Bjerrum length. osm - str H cyl σ f 2=3 u 1=6 ε 1=3 c N ð28þ So, the brush thickness first increases with the increasing value of the Bjerrum length (see eqs 20, 23), and then it begins to decrease (eq 28) when the value of the Bjerrum length exceeds the counterion condensation threshold. This is in qualitative agreement with our simulation results that show nonmonotonic dependence of the brush thickness on the value of the Bjerrum length (see Figure 3). While the scaling analysis is very useful in understanding what terms in the brush free energy are responsible for a particular brush behavior, the exact numerical optimization of the brush free energy could be a test on how well our approximate expression for the brush free energy describes the real situation. In Figures 12 and 13, DOI: /la901839j

10 Figure 13. Dependence of the aggregation number M on the brush grafting density F g for weakly charged, f = 1/3, planar brushes with the value of the LJ interaction parameter ε LJ =1.5k B T, the value of the Bjerrum length l B =0.125σ, and two different degrees of polymerization N = 60 (filled circles) and N = 180 (filled squares). The lines are results of the numerical optimization of the brush free energy eq 29 with the same set of the adjustable parameters as in Figure 12. we show the comparison of the simulation results with the results of the numerical optimization of the brush free energy: F cyl k B TF g S 1 Mk B T ðf surf þ F string þ U elect þ F osm þ F elast Þ ð29þ with respect to the brush height H and aggregation number M at a fixed value of the average monomer density F=MN /(πhd 2 ). For these plots, we considered the average monomer density inside the aggregate F, the fraction of the osmotically active counterions β, the aggregate surface energy γ, and the cohesive energy ε c as the adjustable parameters that minimize the difference between the height of the brush layer and the average aggregation numbers obtained in simulations and numerical minimization of the brush free energy eq 29: χðβ, γ, F, ε c Þ¼ Xnp ½ðH sim, i - HðF g, i ÞÞ2 þðm sim, i - MðF g, i ÞÞ2 Š i ¼1 ð30þ Summation in eq 30 is carried out over all the simulation points. As one can see, the agreement between both the simulation and numerical optimization results is very good. For both sets of simulation data with N = 60 and N = 180, the values of the adjustable parameters are very close. However, the values of the cohesive energy are relatively high ε c =25.6 k B T for N=60 and ε c = 26.8 k B T for N = 180. The reason for this could be that our expression for the string s free energy (see eq 12) underestimates the string deformation. 4. Conclusions We have studied the effect of the polymer solvent quality and the strength of the electrostatic interactions on the structure of the polyelectrolyte brush in poor solvent conditions for the polymer backbone. Our molecular dynamics simulations show that a polyelectrolyte brush can form vertically oriented cylindrical aggregates, maze-like aggregate structures, or dense polymeric layers covering a substrate (see Figure 2). These different brush morphologies appear Article as a result of the optimization of the electrostatic interactions between the charges and the short-range monomer-monomer interactions. The relative importance of the different factors controlling the brush conformations is manifested in a nonmonotonic dependence of the brush thickness on the value of the Bjerrum length (see Figure 3). First, the brush thickness increases with the increasing value of the Bjerrum length. The increase in the brush layer thickness is a result of the optimization of the electrostatic interactions between the charges and the surface energy contribution stemming from the short-range LJ interactions between the monomers. With increasing the value of the Bjerrum length, the aggregation number of each cylindrical aggregate decreases, while their number increases. At the value of the Bjerrum length l B >1σ, the condensed counterions start to play an important role in determining the brush morphology. The localized within brush layer counterions reduce electrostatic repulsion between the charged monomers resulting in a decrease of the brush height with increasing the value of the Bjerrum length. At large values of the Bjerrum length l B >2σ, the cylindrical aggregates disappear, and a brush forms a dense layer covering a substrate surface. At such high values of the Bjerrum length, the correlation-induced attractive interactions between charged monomers and the condensed within brush layer counterions determine the layer structure. It is interesting to point out that there are qualitative similarities between aggregation in 2D charged system studies by de la Cruz et al and our simulations. In both cases, the aggregation is driven by competition between the electrostatic and the short-range interactions. However, in our case there is an additional effect of the charge connectivity into polymer chains, resulting in an elastic free energy contribution to the brush free energy which is crucial for determining the stability of the particular type of brush morphology. We have also established that the brush layer thickness increases as the the brush grafting density increases. At low brush grafting densities, F g < σ -2, the brush layer consists of individual chains (so-called mushroom brush regime). In this interval of brush grafting densities, the brush height demonstrates a weak dependence on the brush grafting density (see Figures 9 and 10). At intermediate values of the brush grafting densities, σ -2 <F g <10-1 σ -2, our simulations show that first the brush thickness grows as H µ F g 1/3,and then it shows a slightly stronger dependence on the brush grafting density. Note, that for entire interval of the grafting densities, σ -2 < F g,a polyelectrolyte brush forms cylindrical aggregates. It is interesting to point out that one can also induce an aggregate formation (microphase separation) in weak polyelectrolyte brushes. 16,40 In such systems, the fraction of the ionized groups is controlled by the solution ph and the ionic strength. 16,40-42 By varying a solution ph or ionic strength, one can effectively change (37) Solis, F. J.; Stupp, S. I.; de la Cruz, M. O. Charge induced pattern formation on surfaces: Segregation in cylindrical micelles of cationic-anionic peptide-amphiphiles. J. Chem. Phys. 2005, 122, /1-9. (38) Loverde, S. M.; de la Cruz, M. O. Asymmetric charge patterning on surfaces and interfaces: Formation of hexagonal domains. J. Chem. Phys. 2007, 127, /1-6. (39) Loverde, S. M.; Velichko, Y. S.; de la Cruz, M. O. Competing interactions in two-dimensional Coulomb systems: Surface charge heterogeneities in coassembled cationic-anionic incompatable mixtures. J. Chem. Phys. 2006, 124, /1-7. (40) Witte, K. N.; Hur, J.; Sun, W.; Kim, S.; Won, Y. Y. Evidence of Lateral Nanoscale Heterogeneities in Weak Polyelectrolyte Brushes. Macromolecules 2008, 41, (23), (41) Nap, R.; Gong, P.; Szleifer, I. Weak polyelectrolytes tethered to surfaces: Effect of geometry, acid-base equilibrium and electrical permittivity. J. Polym. Sci., Part B: Polym. Phys. 2006, 44, (42) Gong, P.; Wu, T.; Genzer, J.; Szleifer, I. Behavior of surface-anchored poly(acrylic acid) brushes with grafting density gradients on solid substrates: 2. Theory. Macromolecules 2007, 40, DOI: /la901839j 13167