Langmuir 2007, 23, 1465-1472
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Diblock Polyampholytes Grafted onto Spherical Particles: Effect of Stiffness, Charge Density, and Grafting Density Anna Akinchina† and Per Linse* Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund UniVersity, Box 124, SE-221 00 Lund, Sweden ReceiVed August 23, 2006. In Final Form: October 23, 2006 The structure of spherical brushes formed by symmetric diblock polyampholytes end-grafted onto small spherical particles in aqueous solution is examined within the framework of the so-called primitive model using Monte Carlo simulations. The properties of the two blocks are identical except for the sign of their charges. Three different chain flexibilities corresponding to flexible, semiflexible, and stiff blocks are considered at various polyampholyte linear charge densities and grafting densities. The link between the two blocks is flexible at all conditions, and the grafted segments are laterally mobile. Radial and lateral spatial distribution functions of different types and single-chain properties are analyzed. The brush structure strongly depends on the chain flexibility. With flexible chains, a disordered polyelectrolyte complex is formed at the surface of the particle, the complex becoming more compact at increasing linear charge density. With stiff blocks, the inner blocks are radially oriented. At low linear charged density, the outer blocks are orientationally disordered, whereas at increasing electrostatic interaction the two blocks of a polyampholyte are parallel and close to each other, leading to an ordered structure referred to as a polyampholyte star. As the grafting density is increased, the brush thickness responds differently for flexible and nonflexible chains, depending on a different balance between electrostatic interactions and excluded volume effects.
1. Introduction Polyampholytes are polymers carrying charges of both signs. The positive and negative charges can, e.g., be distributed randomly along the chain, or charges of one sign can be arranged in long sequences (blocks). If one type of charge is in large excess, properties of polyampholyte solutions are similar to those of polyelectrolyte solutions. If both charges are present in equal amount, the solution behavior depends noticeably on the charge sequence. The behavior of diblock polyampholytes in solution resembles those of mixtures of oppositely charged polyelectrolytes in the sense that they can precipitate or form soluble aggregates under certain conditions. From a theoretical point of view, block polyampholytes are easier to handle, since the concentration of oppositely charged blocks is not independent. The present understanding of, in particular, random polyampholytes in solution and their interaction with surfaces and polyelectrolytes has recently been reviewed.1,2 In addition, theoretical and computer studies of single diblock polyampholytes3-5 and diblock polyampholytes in solution6,7 have been performed. Uncompensated charges were found to be an important factor to prevent polyampholyte complexes from further aggregation.6,7 Polymers chemically attached to solid surfaces play an immense role in regulating the interaction between colloids and preventing * To whom correspondence should be addressed. E-mail: Per.Linse@ fkem1.lu.se. † Current address: Department of Food Science, University of Leeds, LS2 9JT Leeds, U.K. (1) Dobrynin, A. V.; Colby, R. H.; Rubinstein, M. J. Polym. Sci., Part B: Polym. Phys. 2004, 42, 3513. (2) Shusharina, N. P.; Rubinstein, M. Nanostructured Soft Matter: Experiment, Theory, Simulation and PerspectiVes; Springer-Verlag 2006; in press. (3) Imbert, J. B.; Victor, J. M.; Tsunekawa, N.; Hiwatari, Y. Phys. Lett. A 1999, 258, 92. (4) Baumketner, A.; Shimizu, H.; Isobe, M.; Hiwatari, Y. J. Phys.: Condens. Matter 2001, 13, 10279. (5) Wang, Z.; Rubinstein, M. Macromolecules 2006, 39, 5897. (6) Castelnovo, M.; Joanny, J. F. Macromolecules 2002, 35, 4531. (7) Shusharina, N. P.; Zhulina, E. B.; Dobrynin, A. V.; Rubinstein, M. Macromolecules 2005, 38, 8870.
macromolecules and particles from adsorbing on a surface.8 The use of charged polymers (polyions) instead of uncharged polymers introduces additional degrees of freedom to the system. Regarding grafted polyions, so far most experimental investigations concern surfaces with a single type of polyion. However, Stamm and co-workers have investigated and utilized surfaces with both polyanions and polycations simultaneously attached to them,9 and preliminary results on grafted diblock polyampholytes on planar surfaces have recently been reported by Genzer et al.10 On the theoretical size, (i) brushes at planar and curved surfaces formed by grafted homopolymers have extensively been investigated by various theoretical methods,11-15 (ii) those formed by block polymers of different block architectures have been studied by Monte Carlo simulations,16 and (iii) those formed by grafted diblock polyampholytes have been investigated by lattice meanfield modeling17-19 and simulation.19 In our previous study,19 spherical brushes composed of diblock polyampholytes grafted onto solid spherical particles in aqueous solution were investigated by using the primitive model solved with Monte Carlo simulations and by lattice mean-field theory. Such brushes may be obtained by self-association in an aqueous solution of triblock copolymers with two oppositely charged blocks and one uncharged hydrophobic end block. The charge of the end-grafted block was varied, whereas the charge of the outer block was fixed. In the limit of an uncharged end-grafted block, the chains were stretched and formed an extended (8) Hunter, R., J. Foundations of colloid science; Clarendon Press: Oxford, U.K., 1987. (9) Houbenov, N.; Minko, S.; Stamm, M. Macromolecules 2003, 36, 5897. (10) Bhat, R. R.; Tomlinson, M. R.; Wu, T.; Genzer, J. AdV. Polym. Sci. 2006, 198, 51. (11) Zhulina, E. B.; Borisov, O. V.; Birshtein, T. M. J. Phys. II 1992, 2, 63. (12) Wijmans, C. M.; Zhulina, E. B. Macromolecules 1993, 26, 7214. (13) Lindberg, E.; Elvingson, C. J. Chem. Phys. 2001, 114, 6343. (14) Klos, J.; Pakula, T. Macromolecules 2004, 37, 8145. (15) Almusallam, A. S.; Sholl, D. S. Nanotechnology 2005, 16, S409. (16) Chen, C. C.; Dormidontova, E. E. Langmuir 2005, 21, 5605. (17) Shusharina, N. P.; Linse, P. Eur. Phys. J. E 2001, 4, 399. (18) Shusharina, N. P.; Linse, P. Eur. Phys. J. E 2001, 6, 147. (19) Akinchina, A.; Shusharina, N. P.; Linse, P. Langmuir 2004, 20, 10351.
10.1021/la062481r CCC: $37.00 © 2007 American Chemical Society Published on Web 12/20/2006
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Akinchina and Linse The total potential energy of the system can be expressed as U ) Uhc + Uel + Ubond + Uangle + Ucell
(1)
where the hard-core repulsion Uhc and the Coulomb interaction Uel are given by Uhc + Uel )
Uij )
Figure 1. Schematic illustration of a system comprising five diblock polyampholytes grafted onto a spherical particle (large sphere) centered in a spherical cell. The outer A-blocks are composed of negatively charged segments (small light spheres) and the inner B-blocks of positively charged segments (small dark spheres). All radial distances are measured from the surface of the particle.
polyelectrolyte brush. In the other limit with the charges of the blocks exactly compensating each other, the chains were collapsed and formed a polyelectrolyte complex, surrounding the particles. At intermediate charge conditions, a polyelectrolyte brush and a polyelectrolyte complex coexisted, which constituted two substructures of the spherical brush. On the basis of our previous investigations, we here examine new aspects of diblock polyampholytes grafted onto spherical surfaces, where the polyampholytes are laterally mobile to represent the case of self-assembled particles. Our focus is on structural properties of such brushes at different (i) flexibilities of the polyampholytes with the link between the blocks being flexible, (ii) linear charge densities, and (iii) grafting densities of the polyampholytes. By introducing nonflexible blocks joined with a flexible link, new polymer brush structures may be expected. For simplicity, we consider throughout in this study symmetric diblock polyampholytes where the properties of the two blocks are identical except the sign of their charges. Thus, the polyelectrolyte brush has zero net charge.
∑U
(2)
ij
i Rcell
(7)
where Rcell is the radius of the cell. The cell size was chosen to be large enough (Rcell ) 1000 Å) to fit the stretched chains having a contour length Lchain ) 〈Rbb2〉1/2(NA + NB - 1) ≈ 580 Å. Only the simple ions are affected by Ucell. Throughout, the radius of the central particle is Rsph ) 18 Å, the segment radii RA ) RB ) 2 Å, and the simple ion radii R+ ) R) 2 Å. Simulations without salt and with monovalent salt corresponding to the counterions of the individual blocks have been performed. The brush properties were however found to be essentially identical at these two conditions. Therefore, only results for systems without salt are given. The brush structure has been examined at three different chain flexibilities by using the angle force constants kangle ) 0, 12.3, and 135 J/(mol deg2). The bare persistence lengths of the corresponding free and uncharged polymers become LP ) 8 Å (referred to as flexible), LP ) 100 Å (semiflexible), and LP ) 1000 Å (stiff), respectively, as evaluated by LP ) 〈Rbb2〉1/2/(1 + 〈cos R〉).20 (For a further discussion of different definitions of the persistence length and their relations, please see ref 21.) Moreover, three (20) Akinchina, A.; Linse, P. Macromolecules 2002, 35, 5183 (21) Ullner, M.; Woodward, C. E. Macromolecules 2002, 35, 1437.
Polyampholytes Grafted onto Spherical Particles Table 1. Parameters of the Systems Investigated cell radius temperature relative permittivity radius
General Parameters Rcell ) 1000 Å T ) 298 K r ) 78.5 Spherical Particle Rsph ) 18 Å
Grafted Diblock Polyampholytes degree of polymerization NA ) NB ) Nblock ) 50 segment radius RA ) RB ) 2 Å bare persistence length LP ) 8, 100, and 1000 Å fractional charge per segment τA ) τB ) τ ) 0, 0.1, 0.2, and 0.5 polyampholyte grafting density σ ) 1.2 × 10-3, 2.5 × 10-3, and 4.9 × 10-3 Å-2 simple ion radii simple ion valence
Simple Salt R + ) R- ) 2 Å Z+ ) -Z - ) 1
polyampholyte charge densities and corresponding uncharged systems obtained by using the fractional charges τA ) τB ) τ ) 0, 0.1, 0.2, and 0.5 are considered and combined with three grafting densities σ ) 1.2 × 10-3, 2.5 × 10-3, and 4.9 × 10-3 Å-2 obtained by using Nchain ) 5, 10, and 20 grafted polyampholytes. Hence, in total 36 different systems will be considered. The values of all the parameters including the temperature T and the relative permittivity r are compiled in Table 1. We note that τ ) 0.5 corresponds to a linear charge density of one unit charge per ≈12 Å. Some, less systematic, simulations with larger linear charge densities were also made to examine the formation of a bundle structure. In addition, results from complementary simulations of a free and uncharged polymer with 100 segments will be given. 2.2. Method. All Monte Carlo (MC) simulations were performed in the canonical ensemble (constant number of particles, volume, and temperature) according to the standard Metropolis algorithm.22 All interactions inside the cell were included in the potential energy evaluations; hence, no potential cutoff was applied. Initial configurations were generated by first placing the grafted segments in hard-sphere contact with the particle. Then successively and randomly the remaining segments of the chains were placed at a separation r0 from the previous one with checks to avoid hardsphere overlaps. To facilitate the assessment that equilibrium structures were achieved, in some cases of semiflexible and stiff chains back-folded polyampholytes arranged laterally separated in a star or/and laterally together in a single bundle were employed as initial configurations. Three types of trial displacements were applied: (i) translational movement of a single-chain segment or a simple ion, (ii) pivot rotation of a section of a chain that is not attached to the particle, and (iii) a rigid rotation of an entire polyampholyte with respect to the center of the particle. The last trail displacement is important for laterally moving the chains on the spherical surface. Single-segment trial displacements were attempted 100 times more often than the two other types. The displacement parameters of the three different trial moves were ∆segment ) 1-3 Å, ∆pivot ) 5-180°, and ∆rot ) 5-90°, respectively. Typically, the equilibration runs involved 106 MC passes (trial moves per particle), and 106-107 MC passes were used for the production runs. In the more demanding systems with the highest linear charge density and/or the highest grafting density, even longer simulations were performed. All the simulations were performed with the use of the integrated Monte Carlo/molecular dynamics/ Brownian dynamics simulation package MOLSIM.23
3. Results The structures formed by the grafted polyampholytes have been examined through snapshots, spatial distribution functions, (22) Allen, M. P.; Tildelsley, D. J. Computer simulation of liquids; Oxford: New York, 1987. (23) Linse, P. MOLSIM, version 3.4; Lund University: Lund, Sweden, 2003.
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brush thickness, and single-chain properties. In the following, we will first examine the effects of the chain stiffness and linear charge density at the intermediate grafting density σ ) 2.5 × 10-3 Å-2. Thereafter, we will report on the role of the grafting density. 3.1. Overview. The nine final configurations obtained from the MC simulations at σ ) 2.5 × 10-3 Å-2 are shown in Figure 2. Here and in the following figures, the results are arranged in a 3 × 3 matrix with increasing chain stiffness (LP ) 8, 100, and 1000 Å) from left to right and with increasing linear charge density (τ ) 0.1, 0.2, and 0.5) from top to bottom. The brush structures depend strongly on the chain stiffness. With flexible chains (left column), a relatively compact and disordered segment distribution is found, with the degree of segment-segment association and brush compactness depending on the linear charge density. Regarding the cases with semiflexible chains (central column), the blocks are more stretched and the brush becomes thicker and less dense. Also here the degree of segment association increases with the linear charge density. As for the stiff chains, the blocks are essentially fully stretched and the segment density in the brush is even lower. There is a considerable influence of the linear charge density; at τ ) 0.1 the degree of association of oppositely charged blocks is small, whereas at τ ) 0.5 a complete association appear and a starlike brush structure is formed. 3.2. Radial Spatial Distributions. The structure of the polyampholyte brush will now be examined in more detail by considering four different types of radial probability distributions. These are the probability of finding (i) a free end segment of the A-block, PA-end(r), (ii) a junction between the A- and B-blocks, PAB-junc(r), (iii) an A-segment, PA-seg(r), and (iv) a B-segment, PB-seg(r), at a distance r from the surface. The integrals of these probability distributions are unity. The distributions were obtained from the MC simulations by discrete sampling of the frequency at which an A-end, an AB-junction, an A-segment, and a B-segment, respectively, appeared at a distance r from the surface of the particle using a histogram width of 6 Å. Figure 3 displays PA-end(r) and PAB-junc(r), whereas PA-seg(r) and PB-seg(r) are given in Figure 4. The brush thicknesses, calculated as twice the first moment of the segment distribution probabilities, are collected in Table 2. 3.2.1. Flexible Chains. Again starting with the flexible chains (LP ) 8 Å), independent of the linear charge density, the distributions of the A-ends (solid curves) and AB-junctions (dashed curves) in Figure 3 are similar. The corresponding distributions of A-segments and B-segments in Figure 4 also show similar radial arrangement. All 12 distribution functions display a single peak at r ≈ 30-60 Å possessing a width of ≈100 Å. A closer inspection shows, first, that at τ ) 0.1 the distribution of A-ends is wider as compared to that of the ABjunctions, but the difference diminishes at increasing τ. Similarly, at τ ) 0.1 the layer of A-segments is displaced ca. 10-20 Å away from the particle as compared to the layer of B-segments, but the same radial distributions appear at τ ) 0.5. Second, at increasing τ the distributions also become narrower. The brush thicknesses given in Table 2 show that the thickness of the brush shrinks from H ) 100 Å at τ ) 0.1 to H ) 66 Å at τ ) 0.5, quantifying the compaction of the brush layer at increasing chain linear charge density. Finally, a comparison between PA-end(r) for τ ) 0 (dotted curve) and that for τ ) 0.1 (solid curve) in the top-left panel of Figure 3 shows that the electrostatic interaction already at τ ) 0.1 has a prominent effect on the brush compactness. Obviously, the A- and B-segments are radially well mixed, where the mixing and the compaction of the structure increase
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Figure 2. Snapshots of spherical particles with end-grafted diblock polyampholytes at bare persistence lengths LP ) 8, 100, and 1000 Å (from left to right) and fractional charges τ ) 0.1, 0.2, and 0.5 (from top to bottom) at grafting density σ ) 2.5 × 10-3 Å-2. Color code: particle (green), A-segments (light blue), and B-segments (red). The scale varies among the different cases.
at increasing chain charge density. This type of compact polyelectrolyte layer has previously been referred to as a polyelectrolyte complex.19 3.2.2. Semiflexible Chains. As for the semiflexible chains (LP ) 100 Å), Figure 3 shows that the distributions of A-ends and AB-junctions become more dissimilar at increasing τ, a behavior opposite that of the flexible chains. More specifically, at τ ) 0.1 the distribution of A-ends ranges from the surface to r ≈ 500 Å with the maximum at 100 Å, whereas the distribution of ABjunctions is restricted to 50-250 Å from the particle surface. Hence, the radial distributions of the segments are more extended and fluctuating as compared to those of the corresponding flexible chains. At increasing τ, the distribution of the location of the A-ends is shifted toward the particle, and at τ ) 0.5 all A-ends are within 100 Å from the surface of the particle. Oppositely, the radial positions of the AB-junctions are shifted outward, and at τ ) 0.5 a complete radial separation of A-ends and ABjunctions appears. The radial segment distributions in Figure 4 confirm the more extended brush structure as compared to the flexible chains. For τ ) 0.1 there are two regions within the brush: an inner one consisting of all the B-segments and some of the A-segments and an outer one containing only A-segments. With an increase
of the linear charge density, the A-blocks fold back, leading to basically equal distributions of A- and B-segments. The appearance of a maximum in the B-segment distribution at particle contact originates from the fact that one B-segment is grafted onto the particle surface. Moreover, the brush data given in Table 2 clearly demonstrate the much larger brush thickness for LP ) 100 Å, as compared to LP ) 8 Å, with the most expanded brush appearing at the lowest linear charge density. Hence, at increasing chain stiffness, the disordered polyelectrolyte complex is transformed to a more ordered, larger, and less dense structure. At the lowest linear charge density blocks of opposite charge are not yet strongly associated, whereas at the largest charge density completely paired blocks appear, still partly bent and with a hairpin bend at the AB-junction. 3.2.3. Stiff Chains. Figure 3 shows that the spatial segregation of A-ends and AB-junctions becomes even stronger for the stiff chains (LP ) 1000 Å). The location of the AB-junctions is restricted to a narrow range about r ) 270 Å at all τ. Since the contour length of the B-block is ≈290 Å, the inner B-blocks are oriented radially, as also was indicated in Figure 2 (right column). At τ ) 0.1, the probability of observing A-ends is nearly uniform between r ) 0 Å and r ) 550 Å, implying a wide distribution of the angle between the directions of the two blocks. This is in
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Langmuir, Vol. 23, No. 3, 2007 1469 Table 2. Brush Thickness Ha (Å)
Figure 3. Probability distributions of free A-ends, PA-end(r) (solid curves), and of AB-junctions, PAB-junc(r) (dashed curves), as a function of the distance r from the surface of the particle at bare persistence lengths LP ) 8, 100, and 1000 Å (from left to right) and linear charge densities τ ) 0.1, 0.2, and 0.5 (from top to bottom) at grafting density σ ) 2.5 × 10-3 Å-2. In the panels of the top row, the distributions PA-end(r) at τ ) 0 are also given (dotted curves).
Figure 4. Probability distributions of A-segments, PA-seg(r) (solid curves), and of B-segments, PB-seg(r) (dashed curves), as a function of the distance r from the surface of the particle at bare persistence lengths LP ) 8, 100, and 1000 Å (from left to right) and linear charge densities τ ) 0.1, 0.2, and 0.5 (from top to bottom) at grafting density σ ) 2.5 × 10-3 Å-2.
contrast to the case with uncharged polymers (τ ) 0), where PA-end(r) displays a prominent maximum at r ≈ 500 Å (dotted curve). At τ ) 0.2 nearly all of the A-blocks are folded back to the particle, and at τ ) 0.5 the back-folding is completed. In the latter case, Figure 2 (bottom right) shows that the A-blocks are back-folded on the B-block of the same chain. The nearly uniform distribution of B-segments extending to r ≈ 260 Å observed in Figure 4 is consistent with a radial orientation of the stiff B-blocks. At τ ) 0.1, the distribution of the A-segments displays a maximum at r ≈ 270 Å with nearly symmetrical wings. We propose that this symmetry appears as a balance of two main factors: (i) the larger volume at increasing r, which promotes a majority of A-segments to appear at r > 270 Å, and (ii) the electrostatic attraction between A- and B-blocks that favors the A-segments to be in the region of the B-segments, viz., r < 270 Å. As for the more flexible chains, at increasing linear charge density the distributions of the two types of segments become closer and eventually are merged at τ ) 0.5. At τ ) 0.1, where the intrachain attraction is weak and the A-blocks explore all orientations, the brush thickness becomes H ≈ 375 Å, whereas
t
σ (10-3 Å-2)
LP ) 8 Å
LP ) 100 Å
LP ) 1000 Å
0 0 0 0.1 0.1 0.1 0.2 0.2 0.2 0.5 0.5 0.5
1.2 2.5 4.9 1.2 2.5 4.9 1.2 2.5 4.9 1.2 2.5 4.9
112 120 132 96 100 109 79 82 94 64 66 73
330 333 339 264 248 236 201 193 195 222 208 187
524 524 532 409 375 341 296 280 275 268 267 268
a Evaluated according to H ) 2∑iri[(NAPA-seg(ri) + NBPB-seg(ri))/(NA + NB)], where the summation extends over the histograms. With this definition, we obtain H ≈ Lchain for fully stretched chains. Here, Lchain ) 〈Rbb2〉1/2(NA + NB - 1) ≈ 580 Å.
at τ ) 0.2 and 0.5 we have H ≈ 270 Å, in close agreement with the length of a stretched block. The structure with folded polyampholytes with stretched blocks as depicted in Figure 2 (bottom right) will be referred to as a polyampholyte star. 3.2.4. Grafting Density Dependence. The radial probability distributions were found to be dependent on the grafting density in a nontrivial way. With flexible chains at all τ values investigated, the distributions of AB-junctions and B-segments were shifted radially outward at increasing grafting density (data not shown). The largest shift of their maxima was ≈20 Å. Table 2 shows that for all τ values the brush thickness increases by ≈15% as σ is increased from 1.2 × 10-3 to 4.9 × 10-3 Å-2, thus demonstrating that the excluded volume influences the thickness of the polyelectrolyte complex layer formed by flexible diblock polyampholytes. An opposite grafting density dependence appears for semiflexible and stiff chains, in particular at τ ) 0.1. An increased grafting density shifted A-ends and AB-segments toward the particle (data not shown). As a consequence the brush thickness decreases, e.g., from H ) 409 Å to H ) 341 Å (by ∼17%) as σ is increased from 1.2 × 10-3 to 4.9 × 10-3 Å-2 for τ ) 0.1 and LP ) 1000 Å (Table 2). Hence, at increasing grafting density the cohesive effect originating from the electrostatic interaction among the polyampholytes becomes more important. For semiflexible and stiff chains the excluded volume effect is of less importance owing to the low bead densities in the brush. Similar nonmonotonic density dependencies also originating from repulsive excluded volume interactions and cohesive electrostatic interactions appear, e.g., in solutions of a simple salt. In such solutions, the osmotic pressure and the excess chemical potential of the ionic components first decrease and then increase with an increase of the ion concentration.24,25 3.3. Lateral Spatial Correlations. We will now consider the lateral arrangement and the degree of interpenetration of individual polyampholytes. These characteristics have been analyzed by using the center-of-mass (com) of the polyampholytes. Let rcom,i denote the position of the com of polyampholyte i with respect to the center of the particle. In the following, we will consider the com-com radial distribution function gsurf com,com(r) using the arc length separation r between rcom,i and rcom,j projected on the surface of the particle according to r ) Rsph cos-1(rˆ com,i‚rˆ com,j) (24) McQuarrie, D. A. Statistical Mechanics; Harper & Row: New York, 1973. (25) Edgecombe, S.; Schneider, S.; Linse, P. Macromolecules 2004, 37, 10089.
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surf Figure 5. Radial distribution function gcom,com (r) at bare persistence lengths LP ) 8, 100, and 1000 Å (from left to right) and linear charge densities τ ) 0.1, 0.2, and 0.5 (from top to bottom) at grafting density σ ) 2.5 × 10-3 Å-2. Here, r denotes the arc length separation between projected positions of the center-of-mass of a chain on the surface of the particle. The largest possible value of r is πRsph ) 56.6 Å.
with rˆ com,i ≡ rcom,i/|rcom,i|. As for 3-dimensional radial distribution functions, gsurf com,com(r) is normalized to unity for a uniform distribution. Figure 5 displays gsurf com,com(r) for the nine combinations of chain flexibility and linear charged density for the intermediate grafting density (σ ) 2.5 × 10-3 Å-2). For LP ) 8 Å and τ ) 0.1 the value of gsurf com,com(r) at r ) 0 is 0.2, and it increases monotonously and approaches unity at r ≈ 20 Å. Hence, the chains repel each other laterally with an effective soft repulsive potential. At increasing τ, the effective repulsion between neighboring chains is reduced, and at τ ) 0.5, the distribution displays a maximum at r ) 0, implying an interpenetration and weak attraction among the polyampholytes. Generally, at increasing stiffness, gsurf com,com(r) at short separation increases, which is in agreement with the radially more extended structure leading to a reduced lateral extension. Finally, for LP ) 1000 Å and τ ) 0.2 and 0.5 a pronounced maximum appears at r ≈ 0 originating from some interchain pairing of polyampholytes, which leads to basically the same lateral com positions of the chains involved. Hence, flexible and low charged chains tend to sterically repel each other, whereas the degree of interpenetration increases with increasing stiffness and linear charge density. 3.4. Properties of Individual Chains. Various probability distributions of properties of individual chains have been determined from the MC simulations. Below, we will focus on the asphericity parameter, which describes the shape of individual chains by a single number. The asphericity is related to the three principal moments of inertia, Li, i ) 1, 2, and 3, according to26
A≡
〈(L12 - L22)2 + (L22 - L32)2 + (L32 - L12)2〉 2〈(L12 + L22 + L32)2〉
(8)
where the moments are ordered in size. By the definition, the asphericity varies between zero and unity, and it becomes 0 for a sphere, 1/4 for a disk, and 1 for a rod. Figure 6 shows the normalized probability distributions of the asphericity P(A) for the nine different systems at σ ) 2.5 × 10-3 Å-2. In the top-central panel, the probability distribution for a (26) Zifferer, G.; Olaj, O. F. J. Chem. Phys. 1994, 100, 636.
Figure 6. Probability distributions of the asphericity P(A) at bare persistence lengths LP ) 8, 100, and 1000 Å (from left to right) and linear charge densities τ ) 0.1, 0.2, and 0.5 (from top to bottom) at grafting density σ ) 2.5 × 10-3 Å-2. In the top-central panel, P(A) for a free and uncharged polymer with 100 segments is also given (dotted curve).
100-segment-long uncharged and free chain with no angular potential and otherwise the same segment size and bond potential as for the diblock polyampholyte chains is also given (dotted curve). The asphericity distributions for the flexible chains have a similar shape, being fairly broad and possessing a maximum at A ) 0.20. At increasing linear charge density, the probability for large A is slightly decreased, indicating a smaller fraction of stretched chains, consistent with the previous conclusion of a more compact layer of polyions. Moreover, the distributions are shifted to smaller A as compared to that of the uncharged and free chain, showing that the electrostatic interaction contracts the polyampholytes. Regarding the semiflexible chains, at τ ) 0.1 and 0.2 the asphericity distributions are shifted to larger values of A, with maxima appearing at 0.25. A comparison with the distribution for the uncharged and free chain shows that the shape of the grafted chain possesses similarities to the former one, showing that the nonelectrostatic stretching and the contraction by the electrostatic interaction are similar in magnitude and to a large extent oppose each other. At τ ) 0.5, P(A) has qualitatively changed shape, the maximum appears at 0.9, and the probability for A < 0.5 is small. Thus, at τ ) 0.1 and 0.2 the chains are coil-like, whereas at τ ) 0.5 they are better described as extended objects with still some bending flexibility (cf. the middle column of Figure 2). Finally, an even larger variation of the asphericity distributions appears for the rigid chains. At τ ) 0.1, P(A) is zero for A < 0.2, has a prominent maximum at 0.25, and displays a plateau up to A near unity. This implies that the chains form mostly 2-dimensional objects, but there is the full spectrum from 2-dimensional objects (the two rigid blocks being perpendicularly oriented with respect to each other) to 1-dimensional objects (the two rigid blocks being parallel to each other either sequentially or folded). At the intermediate linear charge density (τ ) 0.2), the amplitude of P(A) at A ≈ 0.25 is reduced, and at A ≈ 1 it is increased, demonstrating a smaller probability for the perpendicular block arrangement and increased probability for the parallel arrangement, in agreement with the snapshots shown in Figure 2. Regarding the highest linear charge density, P(A) * 0 only for A > 0.85, consistent with essentially only rodlike structures.
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Figure 7. Schematic illustration of different polymer brush structures formed by diblock polyampholytes end-grafted onto a spherical particle in the bare persistence length (LP)-fractional charge (τ) space. Throughout, the link between the two blocks is flexible.
3.5. High Linear Charge Density. So far, we have considered grafted diblock polyampholytes with linear charged densities up to one unit charge per ≈12 Å (τ ) 0.5). Higher linear charge densities have also been examined. However, the larger electrostatic coupling led sometimes to structures that depended on the initial configuration. Therefore, we will only discuss the brush properties at τ > 0.5 qualitatively. The brush properties of the flexible chains (LP ) 8 Å) at 0.5 e τ e 1 are similar to those at τ ) 0.5; viz., a dense polyelectrolyte complex was formed. The radial extension of the complex continued to contract at increasing τ. Regarding semiflexible and stiff chains, at τ e 0.5 the three different types of initial configurations (random, star, and bundle) gave the same results, whereas at τ g 0.6 the results depended on the type of initial configuration used. Random initial configurations led to a disordered structure, whereas a star or bundle configuration ended up in a bundle formation. The bundle structure possessed a lower potential energy as compared to the disordered one. By employing the reasonable ground-state approximation, we hence have support that bundle formation appears at τ ≈ 0.6 or larger. Of course, the structure at large τ would have been different if the grafted chains were not mobile on the surface of the particle.
4. Discussion We will now briefly discuss (i) the transition from a disordered to an ordered state appearing at increasing stiffness at sufficiently large linear charge density and (ii) the orientation of the outer block for stiff chains at different electrostatic interactions. It was observed that at increasing linear charge density the structural reorganization of the brush became different for flexible chains as compared to semiflexible and stiff chains. Figures 2 and 3 showed that for flexible chains the polyampholyte brush remained disordered with a similar radial distribution of A-ends and AB-junctions, whereas for the stiffer chains and at not too low linear charge density a spatial separation of A-ends and AB-junctions appeared. This spatial separation is a signature of a disordered-ordered transition, at which the disordered brush (polyelectrolyte complex) is transformed into an ordered one (polyampholyte star), characterized by pair formation of oppositely charged blocks, radially oriented. The locations of the two types of structures in the LP-τ-parameter space are schematically illustrated in Figure 7. Figure 7 also displays a third region with radially ordered inner blocks and disordered outer blocks, appearing at high LP
Figure 8. (a) Reduced electrostatic potential energy for a polyampholyte composed of two rigid and straight blocks with a flexible junction as a function of the angle γ between the two blocks (see the illustration in the inset). Each block has 50 segments, and the separation between bonded segments is 6 Å except for that at the junction, which is 4 Å. A segment has a hard-sphere radius of 2 Å and carries fractional charge τ. The same temperature and dielectric permittivity as in the main study are employed. (b) Normalized probability distributions of diblock end-to-end separation Pee(r) from the MC simulations (solid curves) and predicted from the simple diblock model, see the text (dotted curves), at τ ) 0.1 and 0.2.
and low τ. In this region, the electrostatic attraction between the two blocks is not able to match the orientational entropy, the latter favoring a broad distribution of angles between the two blocks. The transition from disordered outer blocks to ordered outer blocks (polyampholyte star) has further been analyzed by using a simple diblock model. In this model, two contributions, one enthalpic and one entropic, are assumed to control the structure. The reduced electrostatic attraction between two rigid blocks constitutes the former one. Figure 8a displays Uelec/τ2kT as a function of the angle γ formed by the directions of the two blocks. The electrostatic potential energy for paired blocks (γ ) 0) becomes -4.8 kT at τ ) 0.1 and decreases to (becomes more negative) -120 kT at τ ) 0.5. The non-normalized probability of angle γ is given by P(γ) ) g(γ) exp[-Uelec(γ)/ kT], where the degeneration factor g(γ) is proportional to sin(γ). In Figure 8b, we compare the diblock end-to-end separation probability distribution Pee(r) for τ ) 0.1 and 0.2 obtained from simulations and the diblock model. For the latter case, Pee(r) is obtained from P(γ) by trivial geometrical consideration. At τ ) 0.1 even distributions of the end-to-end separation are observed, whereas at increasing linear charge density a preference for short end-to-end separations is found, implying folded diblock polyampholytes. Hence, we conclude that (i) the electrostatic interaction within a single polyampholyte chain and (ii) the entropic contribution from the angular degree of freedom in the 3-dimensional space together are sufficient to qualitatively predict the transition from randomly oriented outer blocks to the paired configuration dominating at strong electrostatic interaction (cf. top-right and bottom-right panels of Figure 4). For the present
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system, the transition between the disordered and ordered outer blocks appears at τ ≈ 0.2. Yet another structure is also possible for not too low persistence length. Figure 7 illustrates that the formation of a bundle involving folded polyampholytes aggregated together appears at higher linear charge density. For the present system the transition between a polyampholyte star and a bundle structure appears at τ ≈ 0.6. Finally, we envision that particles with end-grafted block polyampholytes could find applications in, e.g., biochemistry due to the tunable properties of the brush structure. By having titrated groups, the charge of the blocks can be controlled by the pH, and hence, different parts of the brush can be exposed to the solution. One idea is to have different affinity ligands27 at the diblock junctions and the ends with potential application in protein separation and drug delivery.
5. Summary Within the framework of the primitive model, spherical brushes composed of symmetric diblock polyampholytes end-grafted on a small spherical particle have been examined using Monte Carlo simulations. Three different chain flexibilities have been considered, corresponding to a bare persistence length of the blocks much less than the contour length of the block (flexible), less than the contour length (semiflexible), and larger than the contour length (stiff). Moreover, the electrostatic interaction has been varied (27) Longo, G.; Szleifer, I. Langmuir 2005, 21, 11342.
Akinchina and Linse
from weak to strong, and three different grafting densities have been considered. The link between the two blocks is kept flexible throughout. The structural properties of the spherical brush have been examined using various radial probability distribution functions characterizing the location of free ends and block junctions as well as segment distributions. The lateral spatial correlations have also been considered. In addition, single-chain properties have been analyzed, and asphericity data were used to describe the shape of the polyampholytes. The stiffness of the blocks had a decisive effect on the brush structure. With flexible blocks, the polyampholytes are always collapsed and form a disordered and compact polyelectrolyte complex layer at the particle surface, which becomes denser with increasing linear charge density and increasing grafting density. As for stiff blocks, the inner blocks are oriented perpendicular to the surface. At low linear charge density, the electrostatic interaction between oppositely charged blocks is too weak to substantially orient the outer blocks, whereas at the highest linear charge density oppositely charged blocks are tightly aligned with respect to each other through the formation of hairpin bends. We refer to this type of structure as a polyampholyte star. Finally, at even higher linear charge density bundle formation appears. Acknowledgment. We thank Jan Genzer for generous comments on our manuscript. Financial support from the Swedish Research Council (VR) is gratefully acknowledged. LA062481R
