Polyampholyte Brushes Grafted on the Surface of a Spherical Cavity

May 26, 2015 - Responsive behavior of polyampholyte brushes in electric fields. Qianqian Cao , Lujuan Li , Chuncheng Zuo , Fengli Huang , Dongmei Hu. ...
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Polyampholyte Brushes Grafted on the Surface of a Spherical Cavity: Effect of the Charged Monomer Sequence, Grafting Density, and Chain Stiffness Qianqian Cao*,†,‡,§ and Hao You*,∥ †

College of Mechanical and Electrical Engineering, Jiaxing University, Jiaxing 314001, PR China Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany § College of Mechanical Science and Engineering, Jilin University, Changchun 130022, PR China ∥ Department of Chemistry and Physics, Georgia Regents University, 1120 15th Street, Augusta, Georgia 30912, United States ‡

ABSTRACT: Molecular dynamics simulations are used to study the conformational behaviors of the flexible and semiflexible polyampholytes coated onto the internal surface of a spherical cavity. Dependences of the brush structure and the local conformation of grafted chains on the sequence of charged monomers, the grafting density, and the chain stiffness are addressed. In the range of parameters studied, it was found that a significant transition of the brush structure occurs due to the variation of the charged monomer sequence. As the number of repeat charged monomers increases, both the flexible and semiflexible polyampholyte brushes change to the collapsed conformation. The spherical concave geometry tends to exclude the conformation of chains perpendicular to the grafting surface for the semiflexible case. In addition, we find that most counterions are depleted in the polyampholyte brush due to the strong electrostatic correlation between the oppositely charged monomers.

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INTRODUCTION Polyelectrolytes are water-soluble polymers containing ionizable groups that can be dissociated and release counterions into solution. They are ubiquitous in nature such as genetic material (DNA and RNA) and proteins and thus are of fundamental scientific interest and have a wide range of industrial applications.1,2 When polyelectrolyte chains are densely grafted onto interfaces with different geometries by covalent bonding or physical adsorption, strong interactions between adjacent chains occur and a polyelectrolyte brush (PEB) is formed. The PEBs are quite complicated systems in biology and materials science,3−5 in which the electrostatic interactions between the charged groups and the short-range excluded-volume interactions between the monomers result in rich conformational behaviors of the brush. Therefore, they play an important role in controlling material surface properties, such as colloidal stabilization,6 lubrication,7,8 drug delivery,9 and smart surfaces.10 A number of analytical and computational approaches have been applied to study the structural and dynamical properties of PEBs, for example, scaling laws,11−13 self-consistent field theories,14,15 molecular theories,16 and simulations.17−26 Several regimes for the behavior of the PEBs have been identified according to the structural parameters (such as the charge fraction of polyelectrolyte chains) and the environmental conditions (such as the grafting density, ionization degree, and salt concentration). The osmotic and the charged brush © XXXX American Chemical Society

regimes are two typical regimes, and theoretically they are wellstudied by self-consistent field theory and the scaling method. The osmotic regime is identified when the brush is sufficiently dense and strongly charged. Under these conditions, the counterions, which compensate for the immobilized charge of the polyelectrolytes, are trapped inside the brush, and the brush exhibits a stretched conformation due to the osmotic pressure of counterions inside the brush. On the contrary, the charged brush regime corresponds to a relatively sparse grafting of the polyelectrolytes and a low degree of ionization, where mobile counterions spread in the solution due to the weak electrostatic attraction between counterions and polyelectrolytes. As a result, the system forms a planarlike capacitor consisting of a cloud of mobile counterions and the PEBs as a charged interface. However, realistic brush systems generally cannot be classified into these two regimes. For example, the brush can be formed by intermediately tethering the chain ends to a substrate or by partial neutralization. If a polyelectrolyte chain bears both cationic and anionic groups, then such polyelectrolytes are composed of charged monomers of both signs and they are called polyampholytes. One typical example is the natural polypeptide chains with acidic or basic amino acids. Generally, acidic and basic amino Received: April 1, 2015 Revised: May 26, 2015

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DOI: 10.1021/acs.langmuir.5b01190 Langmuir XXXX, XXX, XXX−XXX

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effects, and the elastic bending energy leads to a variety of equilibrium properties of PABs. However, this problem has not yet been addressed systematically. To mention an example, for the viral capsid,45−47 the (semi)flexible peptide arms which are part of the protein subunits are densely distributed on the (spherical) capsid surface, forming the PABs. Each capsid peptide arm contains hundreds of amino acid residues. The assembly of the ssRNA viruses is mainly driven by the electrostatic interactions between genetic phosphate groups and basic amino groups located in the peptide arms. An investigation of the equilibrium structure of the PABs grafted onto the sphere can help us better understand the inner structure and assembly of viruses. It is also relevant to potential nanotechnology applications. Here, we answer the problem from molecular dynamics simulations of a simplified, coarsegrained model of the PAB. We find that when the monomer sequence carries more charged monomers in a block, the system tends to exclude the conformation of chains perpendicular to the grafting surface. In particular, the semiflexible diblock PAB adopts a collapsed (yet stretched) conformation, and there are no coexisting conformations. This is a unique feature due to the spherical concave geometry, as compared to a planar PAB. The article is organized as follows. In the next section, the simulation method and the model system are described. The results are presented and discussed subsequently. The effects of various parameters including the grafting density, the sequence of charged monomers, and the chain stiffness are addressed. Conclusions are in the final section.

acid units are distributed irregularly along the protein backbone. The aggregation behavior and the self-assembly process in the solution of peptides under isoelectric conditions are influenced by the sequence of charged amino acids along ionically complementary peptides.27 The electrostatic attraction between oppositely charged monomers causes the polyampholytes to contract, while the intrachain electrostatic repulsion between monomers with the same charge makes the polyelectrolyte swell. As a result, the oppositely charged groups can form a polyelectrolyte complex which leads to phase separation. In this case, a key factor which hinders the polyampholyte complexes from further aggregation is the repulsion between uncompensated charges. Compared to the PEB, many fewer studies were done on the structure of the polyampholyte brush (PAB). Experimentally, the responsive behavior and conformational properties of synthetic PABs have been studied.28−31 Recently, Srinivasan et al. created intrinsically disordered protein brushes which demonstrate stimuli-responsive behavior with respect to solution pH and ionic strength.32 Theoretically, most previous works focused on the block polyampholytes, which is an easy model to solve because of the independent concentration of oppositely charged blocks.33−35 Linse’s group investigated the diblock PABs using the Monte Carlo simulation and the lattice mean field theory.33,34,36 Recently, Qu et al. used the selfconsistent field theory to study the equilibrium structure and the stimuli-responsive behaviors of the diblock PABs.35 Baratlo and Fazli investigated the conformation and the thickness of the brush using the molecular dynamics simulations.37,38 They found that the conformations of grafted chains on the planar surface are significantly affected by the sequence of the charged monomers. The dependence of the equilibrium properties on the grafting density and the salt concentration are more obvious if the polyampholyte chains contain longer blocks of similarly charged monomers. For the case of flexible chains, the behavior of the polyampholytes with positive and negative charges randomly or regularly distributed along the chain may be similar to that of the diblock polyampholytes or the mixtures of oppositely charged polyelectrolytes because they can form aggregates under the electrostatic attraction between oppositely charged units. However, when the chains become stiffer, the chain or brush conformations depend noticeably on the sequence of charged groups. In this work, we investigate the equilibrium properties of the polyampholyte brushes grafted onto a spherical surface. Unlike typical spherical brushes with grafted chains extending from the spherical surface outward to the bulk of the solution,26,39,40 in our model the polyampholyte chains tethered onto the inner surface of the sphere extend toward the interior of the spherical cavity. This spherically confined brush model is motivated by several facts: (1) The curved geometry of the substrate surface on which the polymer chains are anchored has critical effects on the structure of polymer brushes, as can be seen in ref 41 and references therein. (2) The confinement of polymer chains effectively reduces the conformational entropy. There has been increasing interest in studies of the conformations of the polymer chains42−44 confined in spheres. When the radius of the grafting sphere is comparable to the size of the PABs, then the concave substrate geometry is of importance. Chain segments have less available space away from the sphere substrate, as compared to a planar brush. As a result, the substrate geometry affects the electrostatic interactions. The competition among the electrostatic interactions, the entropic

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MODEL AND SIMULATION METHOD In this work, a coarse-grained bead−spring model was used to study the grafted polyampholyte chains. In the simulation, each chain is assumed to be composed of Ngm = 30 beads (excluding the grafted monomers on the spherical surface) with a diameter σ. There are Ng polyampholyte chains uniformly coated on the inner surface of the spherical core with a core radius of R = 35σ. The corresponding grafting density is ρg = Ng/4πR2. To study the effect of the various sequences of the charged monomers in the chain, we denote the charge sequence as pxnx, where p and n correspond to positively and negatively charged segments, respectively, and x represents the number of repeat charged monomers in each segment with the same charge sign. Figure 1 shows the schematic structures of the grafted chains with different sequences of the charged monomers. For example, the p15n15 chain is composed of two oppositely charged blocks with equal numbers of monomers for each. It is often denoted as the diblock PAB. All monomer sequences under investigation correspond to the PABs with zero net charge except for the p2n2 case. The p2n2 brush carries 2Ng positive unit charges. In the next section, we will find that the influence of net charges is negligible due to a small fraction of the total number of charged monomers. For comparison, the PEB with completely negatively charged monomers is also investigated. In the model, monovalent counterions are considered: the cations and anions are dissociated from the negatively and positively charged polyelectrolytes, respectively. The numbers of cations and anions are denoted as Npi and Nni. The core− shell is assumed to be permeable to the counterions but impermeable to the polymer chains. The periodic boundary conditions are applied in all three directions of the cubic simulation box. The box length is set to L = 160σ such that it is B

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time unit. First, the system is equilibrated for 5 × 105 time steps, and then a production run of 2 × 106 time steps is performed to obtain the equilibrium properties of the brush.

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RESULTS AND DISCUSSION Monomer Density and End-Monomer Distribution. Figure 2a shows the monomer density for the flexible chains at

Figure 1. Schematic structures of the grafted chains with different charge sequences. The cyan and blue beads denote the positively and negatively charged monomers, respectively. The grafted monomers are not shown. The PEB is negatively charged. The p15n15 case corresponds to a diblock brush. Each chain is composed of 30 charged monomers.

large enough to prevent the short-range interactions of the chains and counterions with their periodic images. The short-range excluded volume interactions between particles are modeled by the pairwise Lennard-Jones (LJ) potential with the interaction strength ϵLJ = kBT, where kB is the Boltzmann constant and the system temperature is set to room temperature with T = 300K. The LJ potential is truncated and shifted at a cutoff distance of rc = 21/6σ. The neighboring monomers are linked by a finitely extendable nonlinear elastic (FENE) potential with a maximum bond length of R0 = 1.5σ and a spring constant of kb = 30ϵLJ/σ2.48 The chain stiffness is modeled via the harmonic angle potential. The potential parameter is chosen as kθ = 0 (for a flexible chain) and 100ϵLJ/ rad2 (for a semiflexible chain). For the semiflexible chain, we add a ghost bead in the radial direction pointing to the grafted monomer at a distance of σ outward from the core surface. The positions of the grafted and ghost beads are fixed. They do not interact with counterions and other movable monomers through the pair potential interactions, but there is an angle potential between them and other linked monomers. As a result, the semiflexible chains energetically prefer to be perpendicular to the core surface on which they are grafted. The Coulomb potential is used to model the electrostatic interactions between charged particles, which is calculated by means of the particle−particle/particle-mesh (PPPM) algorithm.49 The Bjerrum length λB = e2/(4πϵ0ϵrkBT) is set to σ, where ϵ0 and ϵr are the vacuum permittivity and the dielectric constant of the solvent. For water at room temperature, λB is approximately equal to 0.7 nm. The corresponding spherical cavity diameter is about 49 nm. LAMMPS50 is used to perform our simulations in an NVT ensemble. Constant temperature is implemented by coupling the system to a Langevin thermostat.51 Initially, the counterions are randomly dispersed within the box. LJ parameters ϵLJ and σ and bead mass m are set as the basic energy, length, and mass units. Molecular dynamics (MD) simulations are performed with the time step at Δt = 0.005τ where τ = (mσ2/ϵLJ)1/2 is the

Figure 2. Profiles of monomer density as a function of radial distance from the center of the cavity for (a) flexible and (b) semiflexible chains with various charged monomer sequences. The brush contains Ng = 150 grafted chains. The vertical dashed line represents the position of the spherical grafting surface.

various charged monomer sequences. For chains with a small number of repeat monomers such as p1n1 and p2n2, the density profiles of the flexible brush are insensitive to the change in the monomer sequence. With further increases in the number of repeat monomers, the brush undergoes a collapsed transition toward the grafting spherical surface. For comparison, when the brush is completely negatively charged, namely, the PEB, a large fraction of monomers is located closer to the sphere’s center, corresponding to a swelling brush conformation. Figure 2b shows the monomer density for the semiflexible chains. Similar to the flexible brush, increasing the number of repeat monomers leads to a collapsed brush conformation. However, unlike the flexible brush, an obvious density oscillation due to the harmonic angle potential is observed for the semiflexible chains. In addition, for the brush with a small number of repeat charged monomers, there is a higher density near the center compared to that of the flexible brush. This is due to the chain stiffness. As for the PEB chain, however, there is a uniform monomer density distribution corresponding to a stretched configuration. Figure 3 presents the distribution function P(r) of the end monomers at various charged monomer sequences for the flexible and semiflexible brushes, respectively. Obviously there is a significant dependence of P(r) on the sequence of charged monomers: as the number of repeat monomers increases, more C

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Figure 3. Distribution function P(r) of end monomers as a function of radial distance from the center of the cavity for (a) flexible and (b) semiflexible chains with various charged monomer sequences. The brush contains Ng = 150 grafted chains. The vertical dashed line represents the position of the spherical grafting surface.

Figure 5. Distribution function P(r) of end monomers as a function of radial distance from the center of the cavity for (a) flexible and (b) semiflexible PEBs for different numbers of grafted chains. The vertical dashed line represents the position of the spherical grafting surface.

repeat charged monomers further increases, a relatively apparent dependence of the conformational behavior on the charged monomer sequences is identified. We also find that P(r) of the flexible PABs has only one peak near the grafting surface whereas for the flexible PEBs, one peak appears near the surface and the other peak is located within the spherical cavity. It reveals that less flexible polyelectrolyte chains are in a collapsed state compared to flexible polyampholyte chains due to the stronger electrostatic repulsion in PEBs. For the semiflexible brush case (Figure 3b), the endmonomer distributions shift away from the grafting surface compared to the flexible brush case because of the stretching of the chains induced by the intrinsic chain rigidity. No chains fold back to the grafting surface in the p1n1 and p2n2 cases because of the electrostatic neutrality on small length scales. As the length of blocks with the same charged monomers becomes longer, more chains bend to the grafting surface. Particularly for the diblock case, there is a broad peak in the distribution P(r) near the surface with no end monomers near the sphere center, corresponding to a fully collapsed state. In contrast, two peaks in the distribution P(r) for the semiflexible PEB indicate that collapsed chains and straight chains coexist. The probability of the end monomers near the sphere center is less than that of p1n1 and p2n2 due to the stronger electrostatic repulsion in the PEB. Additionally, the peak near the spherical surface is much shaper than that of the flexible PEB. Interestingly, for the diblock case, the chain stiffness has an unnoticeable influence on the distribution profiles, as can be seen in Figures 2 and 3. This feature indicates that the electrostatic attraction between oppositely charged segments overwhelms the bending energy for diblock PABs. Figure 4 shows typical simulation snapshots at different charge sequences for the flexible and semiflexible brushes. All of the flexible PABs are in the collapsed state, and the flexible PEBs exhibit relatively stretched conformations. For the

Figure 4. Typical simulation snapshots of flexible and semiflexible brushes with different charged monomer sequences. The second and fourth rows show a small part of the brush. The number of grafted chains is Ng = 150. The red beads represent the grafted monomers. The cyan and blue segments are positively and negatively charged monomers, respectively.

end monomers appear in the region close to the sphere shell, corresponding to a collapsed brush conformation. It confirms the conclusion from Figure 2. For the flexible PABs, Figures 2a and 3a show that the p1n1 brush conformations are almost the same as the p2n2 case. It reveals that the effect of the charged monomer sequences on the brush conformations is negligible when the number of repeat charged monomers is small. In these situations, the electrostatic neutrality occurs on small length scales , and thus the charge sequences have negligible effects. As the number of D

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Figure 6. Distribution function P(r) of end monomers of flexible (a, c, and e) and semiflexible (b, d, and f) brushes as a function of radial distance from the center of the cavity for three different charged monomer sequences: (a and b) p1n1, (c and d) p5n5, and (e and f) p15n15. The vertical dashed line represents the position of the spherical grafting surface.

semiflexible brushes, the p1n1 chains are greatly stretched; some chains in the p5n5 and the PEB fold back toward the spherical surface, and all of the chains in the diblock brush are bent and there are no stretched chains coexisting. In fact, as seen from Figure 3b, the p1n1 brush has a higher density in the region away from the grafting surface than other PABs. Moreover, though the free ends of some semiflexible chains such as p5n5 fold back to the surface, these chains are not in a coiled state, which indicates that the stiffness still affects configurations. By modeling the adjacent charged segments as the electric dipole pairs, one can see that for the cases with a small number of repeat charged monomers, such as p1n1 and p2n2, there is a significant cancellation of the electrostatic interaction with other charged monomers. As a result, the chain appears electrostatically neutral on small length scales, thus the rigidity dominates the formation of the brush. When the length of blocks with the same charged monomers becomes longer (from p3n3 to p15n15), the electrostatic interaction of dipoles becomes stronger and thus begins to dominate the formation of the brush. Another interesting feature is that the semiflexible chains interlace with each other whereas the flexible chains tend

to contract by the electrostatic attraction between adjacent oppositely charged intrachain segments. As for the counterions (they are not shown in the snapshots), a considerable number of counterions accumulate within the PEB layer because of the electrostatic attraction between the monomers and counterions. However, for the PABs, most counterions disperse in the entire simulation box and are not adsorbed into the brush layer. The effect of the grafted-chain number on the end-monomer distribution function P(r) is shown in Figures 5 and 6, where the number of grafted chains is Ng = 30, 150 and 300, corresponding to grafting density ρg = 0.00195σ−2, 0.00974σ−2, and 0.0195σ−2, respectively. Figure 5 shows the distribution of end monomers for the PEBs. For the flexible cases, at low grafting Ng = 30, most chain ends are folded back onto the surface. With an increasing number of grafted chains, the endmonomer distribution shifts toward the center of the surface. The physical explanation is, with the increase in the grafting density, the excluded-volume effect is enhanced and tends to push the end monomer away from the sphere’s surface. However, it is not clear whether the chains adopt a coiled conformation. We will further study the conformational E

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excluded-volume effects the monomers tend to move toward the center when the grafting density increases. For p5n5, the peak slightly moves toward the sphere’s surface when the grafting density increases because of stronger electrostatic attraction. For the diblock (p15n15) brush, all chains are located in the vicinity of the spherical surface, and there is no discernible difference caused by the change in the grafting density. At a fixed grafting density, increasing the number of repeat charged monomers results in an offset toward the grafting surface, which confirms the observations in Figure 3. For the semiflexible brushes, the end-monomer distributions seem to be more sensitive to the grafting density than the flexible brush cases. As the number of grafted chains increases, the end-monomer distribution shifts toward the surface due to a stronger interchain attraction. At a fixed grafting density, similar to the flexible-brush case, increasing the number of repeat monomers results in the accumulation of end-monomers near the surface. Brush Thickness. To describe the conformational transition of the brush quantitatively, we calculated the average thickness H of the brush layer. The brush thickness is defined as H = [⟨(R − Rt) 2⟩]1/2, where Rt is the average distance of the terminal monomers located at rti from the sphere’s center Rc, namely Rt = [∑Ni=1g (rti − Rc)2/Ng]1/2. The brush thickness as a function of the number of grafted chains Ng is shown in Figure 7. It is obvious that the thickness increases with Ng for both flexible-brush and semiflexible-brush cases. However, the increase is not remarkable for the semiflexible brushes, except for the diblock brush (p15n15). The thickness of the semiflexible diblock brush grows fast when Ng increases from 30 to 150. Meanwhile, the end-monomer distribution of the diblock brush does not change significantly (Figure 6f) in this process, despite the fact that the brush actually undergoes a conformational transition from the collapsed state to the stretching state. From Figure 7b, one can see that the thickness of the semiflexible brushes (except for the diblock brush) exceeds 25σ, which corresponds to an extreme stretching conformation, considering that the chain length is 30σ. The diblock brush thickness is much smaller than other PABs due to the strong electric attractions. For the flexible-brush case, a much larger thickness of the PEB is observed compared to that of the PABs, which is different from the semiflexible-brush case. Figure 8 shows the effect of the charge sequence on the brush thickness. The increase in the number of repeat charged monomers induces a decrease in the brush thickness, which seems more remarkable from the p3n3. Compared to the PEB thickness, we find that at Ng = 30 the thickness of the brushes with the number of repeat charged monomers of less than 3 is larger than that of the PEB. For other cases, the PAB has a smaller thickness than the PEB. Radial Distribution Function and Shape Factor. Furthermore, we compute the radial distribution functions (RDFs) between the negatively charged monomers and their counterions, gnc(r), and the RDFs between negatively charged monomers and positively charged monomers, gnp(r). The results are presented in Figure 9. For every specific charge sequence of PABs, the values of gnc(r) are significantly smaller than those of gnp(r). This indicates that the strong electrostatic correlation between the positively and negatively charged monomers suppresses the binding of counterions to their corresponding charged monomers, irrespective of the chain stiffness. As a demonstration, the density of positively charged counterions for the semiflexible brush systems is shown in

Figure 7. Thickness of (a) flexible and (b) semiflexible brushes as a function of the number of grafted chains for various charged monomer sequences.

Figure 8. Thickness of flexible and semiflexible brushes with different numbers of grafted chains as a function of charged monomer sequences. The horizontal lines represent the corresponding PEB height.

characteristics in the following subsections. When the chains become stiffer, as can be seen in Figure 5b, there is more endmonomer stretch to the sphere’s center than in the flexible case, especially at a low grafting density. Interestingly, the peak near the grafting surface becomes much shaper for the semiflexible brushes, independent of the grafting density. In Figure 6, we show the end-monomer distributions of the PABs for different numbers of grafted chains and for different charged monomer sequences. For the flexible brushes case, the number of grafted chains has a weak effect on the distribution profiles. The distribution of p1n1 has a slight shift away from the surface as Ng increases. This is because the p1n1 brush behaves like a neutral brush, and for a neutral brush, due to F

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Figure 9. Radial distribution function (a and b) between negatively charged monomers and their corresponding counterions and (c and d) between oppositely charged monomers for (a and c) flexible and (b and d) semiflexible brushes with different charged monomer sequences. The number of grafted chains is fixed at Ng = 150.

Figure 10. Density profiles of positively charged counterions as a function of radial distance from the center of the cavity for semiflexible brushes with various charged monomer sequences. The number of grafted chains is Ng = 150. The vertical dashed line represents the position of the grafting sphere surface.

Figure 11. Shape factor S = Ree2/Rg2 for (bottom panel) flexible and (top panel) semiflexible brushes as a function of the number of grafted chains with various charged monomer sequences.

due to the compact layers formed by the collapsed flexible brushes. For the semiflexible brushes, as can be seen in Figure 9d, the electrostatic correlation for the p1n1 case is still the weakest and the diblock brush is still the strongest. However, the correlation function gnp(r) has several local maxima, and their values decay with the increase in the radial distance r for up to >8σ. Compared to the flexible brushes case, we know that this difference is caused by the ordered charge arrangement along the highly stretched semiflexible chains. Interestingly, the first peak of the semiflexible p1n1 brush appears at r ≈ 3σ, which is further away from the position of the first peak for other charge sequences, indicating that there is weaker electrostatic binding for the p1n1 case. It also implies that the p1n1 brush has some characteristics of its neutral counterpart.

Figure 10. In the PAB cases, almost all counterions are outside the cavity, and only a tiny fraction of counterions are dispersed in the cavity due to free diffusion. It verifies that for the PABs, the counterions distribution is not important in controlling the conformational behavior of the brush. The interchain or intrachain electrostatic interactions dominate the brush conformations. Contrary to the PABs, more counterions aggregate in the PEB. The counterion osmotic pressure has a significant effect on the brush structure. From Figure 9c, one can see that for the flexible brushes the electrostatic correlation between oppositely charged monomers for p1n1 is the weakest compared to other charge sequences; the correlation for the diblock brush is the strongest. The correlations mostly occur within the region of r < 2σ and are G

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spherical surface than on the planar surface. The concave geometry results in a higher charge density near the center and thus stronger repulsions between the end monomers, compared to those of the planar PEB.

Finally, we compute the shape factor which is used to describe the overall shape of a polymer chain.18,52 The shape factor is defined as S = Ree2/Rg2, where Ree is the end-to-end distance and Rg is the radius of gyration. For reference, the shape factors of a free Gaussian chain and a rigid rodlike chain are S = 6 and 12, respectively. As shown in Figure 11, the semiflexible brushes have shape factors of between 10 and 11, indicating that they adopt a highly stretched morphology. Even though most of the end monomers in the semiflexible diblock brush are folded back onto the surface of the spherical cavity (Figure 6f), the shape factor is still close to 10. It means that though the chain is bent, its free end is still far away from its fixed end, thus forming a stretched configuration. For the flexible chains, when the number of repeat charged monomers increases, the shape factors of the polyampholytes become smaller, indicating that they become more coiled. The shape factors of the p15n15 brushes are even smaller than that of the free Gaussian chain. In particular, at Ng = 30, the shape factor is close to 4 which corresponds to a highly compact packed structure. As for the flexible PEB chain, the shape factor of flexible polyelectrolyte chains is >8 and insensitive to the change in the grafting density. Therefore, we can conclude that the flexible PEB chains at Ng = 30 should be locally stretched conformations, although they are mostly folded back as shown in Figure 5a. Spherical Concave Geometry Effect on Conformational Structures. To investigate the substrate curvature effect, the chain length is set to be 30σ such that the brush size (Ree = 4−29σ) is comparable to the sphere radius (R = 35σ). Overall, when the number of repeat charged monomers increases from the p1n1 to the p15n15, more chains fold back to the grafting surface. The system tends to exclude the chain conformation that is straight and perpendicular to the grafting surface. In particular, the diblock PAB adopts a fully collapsed conformation. All chains fold back to the grafting surface, irrespective of the grafting density. As a comparison, a fraction of straight chains perpendicular to the grafting surface coexist with the collapsed chains in the semiflexible planar PABs at high grafting density.37,38 The difference originates from the geometrical effect. For spherical concave geometry, chain segments have less available spaces away from the sphere substrate, as compared to the planar geometry. The end monomers are confined within much less volume, resulting in a high charge density near the sphere’s center. For PABs with a small number of repeat charged monomers such as p1n1 and p2n2, there is a significant cancellation of the electrostatic interaction between dipoles, and electrostatic neutrality occurs on small length scales. The electrostatic repulsion is negligible. As the number of monomers in a single block increases, the electrostatic correlation is enhanced and tends to push those straight chains away from the sphere’s center. In particular for the spherical diblock case, the strong repulsion near the center forces all chains to fold back to the grafting surface, and thus there is no coexisting conformation, as compared to the planar diblock case.38 We also compare the end-monomer distribution P(r) of the PEB chains grafted on a spherical cavity surface with that of the PEB chains on a planar surface. Similarly, at a low grafting density, a peak also appears near the grafting surface for the planar PEB,18 which is considered to be reminiscent of the chain collapse of highly charged polyelectrolytes. However, given the same grafting density, the comparison shows that the grafted PEB chains are easier to collapse or fold onto the

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CONCLUSIONS We simulate the conformational behaviors of polyampholyte chains grafted onto a spherical cavity. The MD simulation results show that (1) the brush structure and the local conformation of the chains are weakly affected by the variance of the charged monomer sequences when the repeat monomer number in the sequence is small. The chain rigidity dominates the conformation of the brush for a small repeat monomer number. As the repeat monomer number increases, the conformational behavior depends more apparently on the variance of the charged monomer sequence, undergoing a phase transition from the swelling brush structure to the collapsed brush structure. Particularly for the diblock PAB, there is a weak effect of the chain stiffness on the brush structure. (2) Although the semiflexible chain is bent in a collapsed conformation, its free end is still far away from the fixed end and thus the chain adopts a stretched configuration. Additionally, the shape factors for the flexible diblock brushes are smaller compared to those of a free Gaussian chain, indicating that it is extremely contracted. (3) The semiflexible polyampholyte chains interlace with each other, and the flexible polyampholyte chains tend to contract themselves because of the electrostatic attraction between the adjacent oppositely charged segments in the intrachain. (4) Because of the spherical concave geometry, the interchain long-range repulsion tends to push grafted chains away from the cavity center. Particularly for the semiflexible diblock PAB, the conformation of chains perpendicular to the grafting surface is forbidden. (5) The strong electrostatic correlation between the positively and negatively charged monomers suppresses the binding of counterions to their corresponding charged monomers.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Q.C. gratefully acknowledges support from the Alexander von Humboldt Foundation. This work was partially supported by the National Natural Science Foundation of China under grant no. 51175223. We thank P. C. Stancil, I. O. Lebedyeva, and T. Albers for their useful comments.

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